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\section{Introduction}
\newcommand{\zk}{\ensuremath{\mathfrak{k}}}
Let $a$ and $q$ be two positive coprime integers, $0<a<q$.
By the Euclidean algorithm, a rational $a/q$ can be uniquely represented as a regular continued fraction
\begin{equation}\label{exe}
\frac{a}{q}=[0;b_1,\dots,b_s]=
\cfrac{1}{b_1 +\cfrac{1}{b_2 +\cfrac{1}{b_3+\cdots +\cfrac{1}{b_s}}}}
,\qquad b_s \ge 2.
\end{equation}
Assuming $q$ is known, we use $b_j(a)$, $j=1,\ldots,s=s(a)$ to denote the partial quotients of $a/q$; that is,
\[
\frac aq := [ 0; b_1(a),\ldots,b_{s}(a)].
\]
Zaremba's famous conjecture \cite{zaremba1972methode} posits that there is an absolute constant $\zk$ with the following property:
for any positive integer $q$ there exists $a$ coprime to $q$ such that in the continued fraction expansion (\ref{exe}) all partial quotients are bounded:
\[
b_j (a) \le \zk,\,\, 1\le j \le s = s(a).
\]
In fact, Zaremba conjectured that $\zk=5$.
For large prime $q$, even $ \zk=2$ should be enough, as conjectured by Hensley \cite{hensley_SL2}, \cite{hensley1996}.
This theme is rather popular especially at the last time, see, e.g.,
papers
\cite{bourgain2011zarembas,bourgain2014zarembas}, \cite{FK}, \cite{hensley1989distribution}--\cite{hensley1996},
\cite{KanIV}--\cite{Mosh_A+B},
\cite{Nied} and many others.
The history of the question can be found, e.g., in \cite{NG_S}.
Here we obtain the following "modular"\, version of Zaremba's conjecture.
The first theorem in this direction was
proved by Hensley
in \cite{hensley_SL2} and after that in \cite{MOW_MIX}, \cite{MOW_Schottky}.
\begin{theorem}
There is an absolute constant $\zk$ such that for any prime number $p$
there exist some positive integers $q = O(p^{30})$, $q\equiv 0 \pmod p$ and $a$, $a$ coprime with $q$ having the property that the ratio $a/q$ has partial quotients bounded by $\zk$.
\label{t:main_intr}
\end{theorem}
Also, we can say something nontrivial about finite continued fractions with $\zk=2$.
It differs our paper from \cite{bourgain2011zarembas}, \cite{bourgain2014zarembas}, \cite{KanIV}, \cite{MOW_MIX}, \cite{MOW_Schottky}.
\begin{theorem}
There is an absolute constant $C>0$ such that for any prime number $p$
there exist some positive integers $q = O(p^{C})$, $q\equiv 0 \pmod p$ and $a$, $a$ coprime with $q$ having the property that the ratio $a/q$ has partial quotients bounded by $2$.
\label{t:main_intr2}
\end{theorem}
Our
proof uses growth results in $\SL_2 (\F_p)$ and some well--known facts about the representation theory of $\SL_2 (\F_q)$.
We study a combinatorial question about intersection of powers of a certain set of matrices $A \subseteq \SL_2 (\F_q)$ with an arbitrary Borel subgroup and this seems like a new innovation.
In principle, results from \cite{hensley_SL2} can be written in a form similar to Theorem \ref{t:main_intr} in an effective way but the dependence of $q$ on $p$ in
\cite{hensley_SL2}
is rather poor.
Thus Theorem \ref{t:main_intr} can be considered as an explicit version (with very concrete constants)
of
Hensley's results as well as rather effective Theorem 2 from \cite{MOW_Schottky}.
Also, the methods of paper \cite{hensley_SL2} and papers \cite{MOW_MIX}, \cite{MOW_Schottky} are
very
different from ours.
We thank I.D. Kan for useful discussions and remarks.
\section{Definitions}
Let $\Gr$ be a group with the identity $1$.
Given two sets $A,B\subset \Gr$, define the \textit{product set} of $A$ and $B$ as
$$AB:=\{ab ~:~ a\in{A},\,b\in{B}\}\,.$$
In a similar way we define the higher product sets, e.g., $A^3$ is $AAA$.
Let $A^{-1} := \{a^{-1} ~:~ a\in A \}$.
The Ruzsa triangle inequality \cite{Ruz} says that
\[
|C| |AB| \le |AC||C^{-1}B|
\]
for any sets $A,B,C \subseteq \Gr$.
As usual, having two subsets $A,B$ of a group $\Gr$ denote by
\[
\E(A,B) = |\{ (a,a_1,b,b_1) \in A^2 \times B^2 ~:~ a^{-1} b = a^{-1}_1 b_1 \}|
\]
the {\it common energy} of $A$ and $B$.
Clearly, $\E(A,B) = \E(B,A)$ and by the Cauchy--Schwarz inequality
\[
\E(A,B) |A^{-1} B| \ge |A|^2 |B|^2 \,.
\]
We use representation function notations like $r_{AB} (x)$ or $r_{AB^{-1}} (x)$, which counts the number of ways $x \in \Gr$ can be expressed as a product $ab$ or $ab^{-1}$ with $a\in A$, $b\in B$, respectively.
For example, $|A| = r_{AA^{-1}}(1)$ and $\E (A,B) = r_{AA^{-1}BB^{-1}}(1) =\sum_x r^2_{A^{-1}B} (x)$.
In this paper we use the same letter to denote a set $A\subseteq \Gr$ and its characteristic function $A: \Gr \to \{0,1 \}$.
We write $\F^*_q$ for $\F_q \setminus \{0\}$.
The signs $\ll$ and $\gg$ are the usual Vinogradov symbols.
All logarithms are to base $2$.
\section{On the representation theory of $\SL_2 (\F_p)$ and basis properties of its subsets}
First of all, we recall some notions and simple facts from the representation theory, see, e.g., \cite{Naimark} or \cite{Serr_representations}.
For a finite group $\Gr$ let $\FF{\Gr}$ be the set of all irreducible unitary representations of $\Gr$.
It is well--known that size of $\FF{\Gr}$ coincides with the number of all conjugate classes of $\Gr$.
For $\rho \in \FF{\Gr}$ denote by $d_\rho$ the dimension of this representation.
We write $\langle \cdot, \cdot \rangle$ for the corresponding Hilbert--Schmidt scalar product
$\langle A, B \rangle = \langle A, B \rangle_{HS}:= \tr (AB^*)$, where $A,B$ are any two matrices of the same sizes.
Put $\| A\| = \sqrt{\langle A, A \rangle}$.
Clearly, $\langle \rho(g) A, \rho(g) B \rangle = \langle A, B \rangle$ and $\langle AX, Y\rangle = \langle X, A^* Y\rangle$.
Also, we have $\sum_{\rho \in \FF{\Gr}} d^2_\rho = |\Gr|$.
For any $f:\Gr \to \mathbb{C}$ and $\rho \in \FF{\Gr}$ define the matrix $\FF{f} (\rho)$, which is called the Fourier transform of $f$ at $\rho$ by the formula
\begin{equation}\label{f:Fourier_representations}
\FF{f} (\rho) = \sum_{g\in \Gr} f(g) \rho (g) \,.
\end{equation}
Then the inverse formula takes place
\begin{equation}\label{f:inverse_representations}
f(g) = \frac{1}{|\Gr|} \sum_{\rho \in \FF{\Gr}} d_\rho \langle \FF{f} (\rho), \rho (g^{-1}) \rangle \,,
\end{equation}
and the Parseval identity is
\begin{equation}\label{f:Parseval_representations}
\sum_{g\in \Gr} |f(g)|^2 = \frac{1}{|\Gr|} \sum_{\rho \in \FF{\Gr}} d_\rho \| \FF{f} (\rho) \|^2 \,.
\end{equation}
The main property of the Fourier transform is the convolution formula
\begin{equation}\label{f:convolution_representations}
\FF{f*g} (\rho) = \FF{f} (\rho) \FF{g} (\rho) \,,
\end{equation}
where the convolution of two functions $f,g : \Gr \to \mathbb{C}$ is defined as
\[
(f*g) (x) = \sum_{y\in \Gr} f(y) g(y^{-1}x) \,.
\]
Finally, it is easy to check that for any matrices $A,B$ one has $\| AB\| \le \| A\|_{o} \| B\|$ and $\| A\|_{o} \le \| A \|$, where the operator $l^2$--norm $\| A\|_{o}$ is just the absolute value of the maximal eigenvalue of $A$.
In particular, it shows that $\| \cdot \|$ is indeed a matrix norm.
Now consider the group $\SL_2 (\F_q)$ of matrices
\[
g=
\left( {\begin{array}{cc}
a & b \\
c & d \\
\end{array} } \right) = (ab|cd) \,, \quad \quad a,b,c,d\in \F_q \,, \quad \quad ad-bc=1 \,.
\]
Clearly, $|\SL_2 (\F_q)| = q^3-q$.
Denote by $\B$ the standard Borel subgroup of all upper--triangular matrices from $\SL_2 (\F_q)$, by $\U \subset \B$ denote the standard unipotent subgroup of $\SL_2 (\F_q)$ of matrices $(1u|01)$, $u \in \F_q$ and by $\D \subset \B$ denote the subgroup of diagonal matrices.
$\B$ and all its conjugates form all maximal proper subgroups of $\SL_2 (\F_p)$.
Also, let $I_n$ be the identity matrix and $Z_n$ be the zero matrix of size $n\times n$.
Detailed description of the representation theory of $\SL_2 (\F_q)$ can be found in \cite[Chapter II, Section 5]{Naimark}.
We formulate the main result from book \cite{Naimark} concerning this theme.
\begin{theorem}
Let $q$ be an odd power.
There are $q+3$ nontrivial representations of $\SL_2 (\F_q)$, namely,\\\
$\bullet~$ $\frac{q-3}{2}$ representations $T_\chi$ of dimension $q+1$ indexed via $\frac{q-3}{2}$ nontrival multiplicative characters $\chi$ on $\F^*_q$, $\chi^2 \neq 1$,\\
$\bullet~$ a representation $\tilde{T}_1$ of dimension $q$,\\
$\bullet~$ two representations $T^{+}_{\chi_1}$, $T^{-}_{\chi_1}$ of dimension $\frac{q+1}{2}$, $\chi^2_1 = 1$, \\
$\bullet~$ two representations $S^{+}_{\pi_1}$, $S^{-}_{\pi_1}$ of dimension $\frac{q-1}{2}$, \\
$\bullet~$ $\frac{q-1}{2}$ representations $S_\pi$ of dimension $q-1$ indexed via $\frac{q-1}{2}$ nontrival multiplicative characters $\pi$ on an arbitrary quadratic extension of $\F_q$, $\pi^2 \neq 1$.
\label{t:Naimark}
\end{theorem}
By $d_{\min}$, $d_{\max}$ denote the minimum/maximum over dimensions of all nontrivial representations of a group $\Gr$.
Thus the result above tells us that in the case $\Gr = \SL_2 (\F_q)$ these quantities differ just in two times roughly.
Below we assume that $q\ge 3$.
Theorem \ref{t:Naimark} has two consequences, although, a slightly weaker result than Lemma \ref{l:A^n_large} can be obtained via the classical Theorem of Frobenius \cite{Frobenius}, see, e.g., \cite{sh_as}. Originally, similar arguments were suggested in \cite{SX}.
\begin{lemma}
Let $n\ge 3$ be an integer, $A\subseteq \SL_2 (\F_q)$ be a set and $|A| \ge 2(q+1)^2 q^{2/n}$.
Then $A^n = \SL_2 (\F_q)$.
Generally, if for some sets $X_1, \dots, X_n \subseteq \SL_2 (\F_q)$ one has
\[
\prod_{j=1}^n |X_j| \ge (2q (q+1))^n (q-1)^{2} \,,
\]
then $X_1 \dots X_n = \SL_2 (\F_q)$.
\label{l:A^n_large}
\end{lemma}
\begin{proof}
Using formula \eqref{f:Parseval_representations} with $f=A$, we have for an arbitrary nontrivial representation $\rho$ that
\begin{equation}\label{f:Fourier_est}
\| A \|_{o} < \left(\frac{|A| |\SL_2 (\F_q)|}{d_{\min}} \right)^{1/2} = \left(\frac{|A| (q^3-q)}{d_{\min}} \right)^{1/2} \,.
\end{equation}
Hence for any $x\in \SL_2 (\F_q)$ we obtain via formulae \eqref{f:inverse_representations}, \eqref{f:Parseval_representations}
and estimate \eqref{f:Fourier_est} that
\[
A^n (x) > \frac{|A|^n}{|\SL_2 (\F_q)|} - \left(\frac{|A| (q^3-q)}{d_{\min}} \right)^{(n-2)/2} |A| \ge 0 \,,
\]
provided $|A|^n \ge 2^{n-2} (q+1)^n q^n (q-1)^2$.
The second part of the lemma can be obtained similarly.
This completes the proof.
$\hfill\Box$
\end{proof}
\begin{remark}
It is easy to see (or consult Lemma \ref{l:B_Wiener} below) that bound \eqref{f:Fourier_est} is sharp, e.g., take $A=\B$.
\end{remark}
\bigskip
For any function $f : \Gr \to \mathbb{C}$ consider the Wiener norm of $f$ defined as
\begin{equation}\label{def:Wiener}
\| f\|_W := \frac{1}{|\Gr|} \sum_{\rho \in \FF{\Gr}} d_\rho \| \FF{f} (\rho) \| \,.
\end{equation}
\begin{lemma}
We have $\| \B\|_W = 1$.
Moreover, $\| \FF{\B} (\tilde{T}_1) \| = \| \FF{\B} (\tilde{T}_1) \|_o = |\B|$ and the Fourier transform of $\B$ vanishes on all other nontrivial representations.
\label{l:B_Wiener}
\end{lemma}
\begin{proof}
We
introduce
even three proofs of upper and lower bounds of $\| \B\|_W$, although, the first and the third ones being shorter give slightly worse constants.
Also, they do not provide full description of non--vanishing representations of $\B$.
Since $\B$ is a subgroup, we see using \eqref{f:Parseval_representations} twice that
\[
|\B|^2 = |\{b_1 b_2 = b_3 ~:~ b_1,b_2,b_3 \in \B\}| = \frac{1}{|\SL_2 (\F_q)|} \sum_{\rho \in \FF{\Gr}} d_\rho \langle \FF{\B}^2 (\rho), \FF{\B} (\rho) \rangle
\le
\]
\[
\le
\frac{1}{|\SL_2 (\F_q)|} \sum_{\rho} d_\rho \langle \FF{\B} (\rho), \FF{\B} (\rho) \rangle \| \FF{\B} (\rho)\|_{o}
\le
\frac{|\B|}{|\SL_2 (\F_q)|} \sum_{\rho} d_\rho \langle \FF{\B} (\rho), \FF{\B} (\rho) \rangle = |\B|^2 \,,
\]
because, clearly, $\| \FF{\B} (\rho)\|_{o} \le |\B|$.
It means that for any representation $\rho$ either $\| \FF{\B} (\rho)\| = 0$ (and hence $\| \FF{\B} (\rho)\|_o = 0$) or $\| \FF{\B} (\rho)\|_{o} = |\B|$.
But another application of \eqref{f:Parseval_representations} gives us
\begin{equation}\label{tmp:01.10_1}
|\B| = \frac{1}{|\SL_2 (\F_q)|} \sum_{\rho} d_\rho \|\FF{\B} (\rho) \|^2
\end{equation}
and hence the number $m$ of nontrivial representations $\rho$ such that $\| \FF{\B} (\rho)\| \ge \| \FF{\B} (\rho)\|_{o} = |\B|$ is bounded in view of Theorem \ref{t:Naimark} as
\[
|\B| \ge \frac{|\B|^2}{|\SL_2 (\F_q)|} \left( 1 + \frac{m(q-1)}{2} \right) \,.
\]
In other words, $m\le 2q/(q-1)$.
Hence
\begin{equation}\label{f:Borel_1-}
\| \B\|_W \le \frac{|\B|}{|\SL_2 (\F_q)|} + \frac{m |\B|}{|\SL_2 (\F_q)|}\cdot d_{\max}
\le
\frac{|\B|}{|\SL_2 (\F_q)|} + \frac{2q (q+1) |\B|}{|\SL_2 (\F_q)|(q-1)} \le 4 \,.
\end{equation}
A similar argument gives us a lower bound for $\| \B\|_W$ of the same sort.
Let us give another proof which replaces $4$ to $1$ and uses the representation theory of $\SL_2 (\F_q)$ in a slightly more extensive way.
For $u_b \in \U$, $u_b=(1b|01)$, we have \cite[pages 121--123]{Naimark} that
in a certain orthogonal basis the following holds
$\tilde{T}_1 (u_b) = \mathrm{diag} (e(bj))$, $j=0,1,\dots,q-1$
and for $g_\la = (\la 0|0 \la^{-1}) \in \D$ the matrix $\tilde{T}_1 (g_\la)$ is the direct sum of $I_1$ and a permutation matrix of size $(q-1) \times (q-1)$.
Clearly, $\B = \D \U = \U \D$ and hence $\FF{\B} (\rho) = \FF{\D} (\rho) \FF{\U} (\rho)$ for any representation $\rho$.
But from above $\FF{\U} (\tilde{T}_1)$ is the direct sum $qI_1 \oplus Z_{q-1}$ and $\FF{\D} (\tilde{T}_1) = (q-1) I_1 \oplus 2\cdot J$, where
$J = (J_{ij})_{i,j=1}^{q-1}$ is a certain $(q-1) \times (q-1)$ matrix with all components equal one for $i/j$ belonging to the set of quadratic residues
(such precise description of $J$ is not really important for us).
Hence
\[
\FF{\B} (\tilde{T}_1) = \FF{\D} (\tilde{T}_1) \FF{\U} (\tilde{T}_1) = q(q-1) I_1 \oplus Z_{q-1} \,.
\]
Thus $\| \FF{\B} (\tilde{T}_1) \| = \| \FF{\B} (\tilde{T}_1) \|_o = |\B|$.
Applying formula \eqref{tmp:01.10_1}, we obtain
\begin{equation}\label{f:Borel_lower}
|\B| \ge \frac{|\B|^2}{|\SL_2 (\F_q)|} + \frac{q}{|\SL_2 (\F_q)|} \|\FF{\B} (\tilde{T}_1) \|^2
= \frac{|\B|^2}{|\SL_2 (\F_q)|} (1 + q ) = |\B| \,.
\end{equation}
It follows that for any other representations Fourier coefficients of $\B$ vanish.
Finally,
\begin{equation}\label{f:Borel_1}
\| \B\|_W = \frac{|\B|}{|\SL_2 (\F_q)|} + \frac{q |\B|}{|\SL_2 (\F_q)|} = 1
\end{equation}
as required.
For the last proof it is enough to look at inequality \eqref{f:Borel_lower} and apply Theorem \ref{t:Naimark}, which gives that $\FF{\B}(T_\chi)$ must vanish thanks to dimension of $T_\chi$.
Further
if we have two nontrivial non--vanishing representations $S_\pi$ or $T^{\pm}_{\chi_1}$, then it is again contradicts \eqref{f:Borel_lower} because
sum of their dimensions is too large.
Hence there is the only one nontrivial non--vanishing representation (and calculations from the second proof show that it is indeed $\tilde{T}_1$)
or one of the following pairs $(T^{\pm}_{\chi_1}, S^{\pm}_{\pi_1})$ or $(S^{+}_{\pi_1}, S^{-}_{\pi_1})$.
Thus a rough form of identity \eqref{f:Borel_1}, say, bound \eqref{f:Borel_1-} follows and, actually, we have not use any concrete basis in our first and the third
arguments.
This completes the proof of the lemma.
$\hfill\Box$
\end{proof}
\begin{remark}
One can show in the same way that an analogue of Lemma \ref{l:B_Wiener} takes place for any subgroup $\G$ of an arbitrary group $\Gr$, namely,
$\| \G\|_W \ll d_{\max}/d_{\min}$.
\end{remark}
Lemma \ref{l:B_Wiener} gives us an alternative way to show that $A^3 \cap \B \neq \emptyset$.
Indeed, just use estimate \eqref{f:Fourier_est} and write
\[
r_{A^3\B} (1) \ge \frac{|A|^3 |\B|}{|\SL_2 (\F_q)|} - \|\B \|_W \left(\frac{|A|(q^3-q)}{d_{\min}}\right)^{3/2}
=
\frac{|A|^3 |\B|}{|\SL_2 (\F_q)|} - \left(\frac{|A|(q^3-q)}{d_{\min}}\right)^{3/2} > 0 \,,
\]
provided $|A| \gg q^{8/3}$.
We improve this bound in the next section.
\section{On intersections of the product set with the Borel subgroup}
It was shown in the previous section (see Lemma \ref{l:A^n_large}) that for any $A \subseteq \SL_2 (\F_q)$ one has $A^3 = \SL_2 (\F_p)$, provided $|A|^3 \gg q^8$
and in the same way
the last result
holds for
three different sets, namely, given $X,Y,Z \subseteq \SL_2 (\F_q)$
with
$|X||Y||Z| \gg q^8$, we have $XYZ=\SL_2 (\F_q)$.
It is easy to see that in this generality the last result is sharp.
Indeed, let $X=S\B$, $Y=\B T$, where $S,T$ are two sets of sizes $\sqrt{q}/2$ which are chosen as $|X| \sim |S| |\B|$ and $|Y| \sim |T| |\B|$
(e.g., take $S,T$ from left/right cosets of $\B$ thanks to the Bruhat decomposition).
Then $XY=S\B T$, and hence $|XY| \le |S||T||B| \le |\SL_2 (\F_q)|/2$.
Thus we take $Z^{-1}$ equals the complement to $XY$ in $\SL_2 (\F_q)$ and we see that the product set $XYZ$ does not contain $1$ but $|X||Y||Z| \gg q^8$.
\bigskip
Nevertheless, in the "symmetric"\, case of the same set $A$ this $8/3$ bound can be improved, see Theorem \ref{t:8/3-c} below.
We need a simple lemma and the proof of this result, as well as the proof of Theorem \ref{t:8/3-c}
extensively play on non--commutative properties of $\SL_2 (\F_q)$.
\begin{lemma}
Let $g\notin \B$ be a fixed element from $\SL_2 (\F_q)$.
Then for any $x$ one has
\[
r_{\B g\B} (x) \le q-1 \,.
\]
\label{l:BgB}
\end{lemma}
\begin{proof}
Let $g=(ab|cd)$ and $x=(\a \beta |\gamma \d)$.
By our assumption $c\neq 0$.
We have
\begin{equation}\label{tmp:16.10_1}
\left( {\begin{array}{cc}
\la & u \\
0 & \la^{-1} \\
\end{array} } \right)
\left( {\begin{array}{cc}
a & b \\
c & d \\
\end{array} } \right)
\left( {\begin{array}{cc}
\mu & v \\
0 & \mu^{-1} \\
\end{array} } \right)
=
\left( {\begin{array}{cc}
(\la a + u c)\mu & * \\
\mu c/\la & vc/\la + d/(\la \mu) \\
\end{array} } \right)
=
\left( {\begin{array}{cc}
\a & \beta \\
\gamma & \d \\
\end{array} } \right) \,.
\end{equation}
In other words, $\mu = \la \gamma c^{-1} \neq 0$ (hence $\gamma \neq 0$ automatically) and from
\[
\a = (\la a + u c)\mu = \la \gamma c^{-1} (\la a + u c)
\]
we see that having $\la$ we determine $u$ uniquely (then, equation \eqref{tmp:16.10_1} gives us $\mu, v$ automatically).
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
Lemma \ref{l:BgB} quickly implies a result on the Bruhat decomposition of $\SL_2 (\F_q)$.
\begin{corollary}
Let $g\in \SL_2 (\F_q) \setminus \B$.
Then $\B g\B = \SL_2 (\F_q) \setminus \B$.
\end{corollary}
\begin{proof}
Clearly, $\B \cap \B g\B = \emptyset$ because $g\in \SL_2 (\F_q) \setminus \B$.
On the other hand, by the Cauchy--Schwartz inequality and Lemma \ref{l:BgB}, we have
\[
|\B g\B| \ge \frac{|\B|^4}{\E(\B,g\B)} \ge \frac{|\B|^4}{(q-1) |\B|^2} = q^3 - q^2 = |\SL_2 (\F_q) \setminus \B| \,.
\]
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
Using growth of products of $\B$ as in the last corollary, one can combinatorially improve the constant $8/3$ (to do this combine Lemma \ref{l:A^n_large} and bound \eqref{f:Borel_sum-product} below).
We suggest another method which uses the representation theory of $\SL_2 (\F_q)$ more extensively and which allows to improve this constant further.
\begin{theorem}
Let $A\subseteq \SL_2 (\F_q)$ be a set,
$|A| \ge 4 q^{18/7}$.
Then
$A^3 \cap \B \neq \emptyset$.
Generally, $A^n \cap \B \neq \emptyset$ provided $|A| \ge 4 q^{2+\frac{4}{3n-2}}$.
\label{t:8/3-c}
\end{theorem}
\begin{proof}
Let $g\notin \B$ and put $A^{\eps}_g = A^{\eps} \cap g\B$, where $\eps \in \{ 1,-1\}$.
Also, let $\D = \max_{\eps,\, g\notin \B} |A^\eps_g|$.
Since we can assume $A \cap \B = \emptyset$, it follows that
\begin{equation}\label{tmp:04.10_1-}
\E(A, \B) = \sum_{x} r^2_{A^{-1} \B} (x) = \sum_{x \notin \B} r^2_{A^{-1} \B} (x) \le \D |\B| |A|
\end{equation}
and similarly for $\E(A^{-1}, \B)$.
On the other hand, from \eqref{tmp:04.10_1-} and by the second part of Lemma \ref{l:B_Wiener}, we see that
\begin{equation}\label{tmp:12.10_1}
\D |\B| |A| \ge \E(A, \B) = \frac{1}{|\SL_2 (\F_q)|} \sum_{\rho} d_\rho \| \FF{A}^* (\rho) \FF{\B} (\rho) \|^2
=
\frac{q}{|\SL_2 (\F_q)|} \| \FF{A}^* (\tilde{T}_1) \FF{\B} (\tilde{T}_1) \|^2
\,,
\end{equation}
and, again, similarly for $\| \FF{A} (\tilde{T}_1) \FF{\B} (\tilde{T}_1) \|^2$.
Now consider the equation $b_1 a' a'' a b_2 = 1$ or, equivalently the equation $a'' a b_2 = (a')^{-1} b_1^{-1}$, where $a, a',a'' \in A$ and $b_1, b_2 \in \B$.
Clearly, if $A^3 \cap \B = \emptyset$, then this equation has no solutions.
Combining Lemma \ref{l:B_Wiener} with bound \eqref{tmp:12.10_1} and calculations as in the proof of Lemma \ref{l:A^n_large}, we see that this equation can be solved provided
\[
\frac{q}{|\SL_2 (\F_q)|} | \langle \FF{A}^2 (\tilde{T}_1) \FF{\B} (\tilde{T}_1), \FF{A}^* (\tilde{T}_1) \FF{\B}^* (\tilde{T}_1) \rangle |
\le \frac{q}{|\SL_2 (\F_q)|} \|\FF{A}^2 (\tilde{T}_1) \FF{\B} (\tilde{T}_1) \|
\cdot \| \FF{A}^* (\tilde{T}_1) \FF{\B} (\tilde{T}_1) \|
\le
\]
\[
\le \frac{q}{|\SL_2 (\F_q)|} \|\FF{A} (\tilde{T}_1) \FF{\B} (\tilde{T}_1) \| \|\FF{A}^* (\tilde{T}_1) \FF{\B} (\tilde{T}_1) \| \| \FF{A} \|_o
\le
\D |\B| |A| \| \FF{A} \|_o
<
\frac{|A|^3 |\B|^2}{|\SL_2 (\F_q)|} \,.
\]
In other words, in view of \eqref{f:Fourier_est}
it is enough
to have
\begin{equation}\label{tmp:04.10_2-}
|A|^4 \ge 2 (q+1)^2 \D^2 \cdot |A| q(q+1)
\end{equation}
or, equivalently,
\begin{equation}\label{tmp:04.10_2}
2 q (q+1)^3 \D^2 \le |A|^3 \,.
\end{equation}
Now let us obtain another bound which works well when $\D$ is large.
Choose $g\notin \B$ and $\eps \in \{1,-1\}$ such that $\D = |A^\eps_g|$.
Using Lemma \ref{l:BgB}, we
derive
\begin{equation}\label{f:B_Ag-}
\E(\B,A^\eps_g) = \sum_{x} r^2_{\B A^\eps_g} (x) \le \sum_{x} r_{\B A^\eps_g} (x) r_{\B g \B} (x) \le (q-1) |\B| |A^\eps_g| \,,
\end{equation}
and hence by the Cauchy--Schwarz inequality, we get
\begin{equation}\label{f:B_Ag}
|\B A^\eps_g| \ge \frac{|\B|^2 |A^\eps_g|^2}{\E(\B,A^\eps_g)} \ge \frac{|\B| |A^\eps_g|}{q-1} = q \D \,.
\end{equation}
Consider the equation $a_g (a' a'')^\eps =b$, where $b\in \B$, $a_g \in A^\eps_g$ and $a',a'' \in A$.
Clearly, if $A^3 \cap \B = \emptyset$, then this equation has no solutions.
To solve
$a_g (a' a'')^\eps =b$
it is enough to solve the equation $z (a' a'')^\eps = 1$, where now $z\in \B A^\eps_g$.
Applying the second part of Lemma \ref{l:A^n_large} combining with \eqref{f:B_Ag}, we obtain that it is enough to have
\[
8 q^3 (q+1)^3 (q-1)^{2} \le q\D |A|^2 \le |\B A^\eps_g| |A|^2
\]
or, in other words,
\begin{equation}\label{tmp:04.10_1}
8 q^2 (q+1)^3 (q-1)^{2} \le \D |A|^2 \,.
\end{equation}
Considering the second power of \eqref{tmp:04.10_1} and multiplying it with \eqref{tmp:04.10_2}, we obtain
\[
|A|^{7} \ge 2^{14} q^{18} \ge 2^7 q^5 (q+1)^{9} (q-1)^{4}
\]
as required.
In the general case inequality \eqref{tmp:04.10_2} can be rewritten as
\[
|A|^n \ge 2^{n-2} \D^2 (q+1)^n q^{n-2}
\]
and using the second part of Lemma \ref{l:A^n_large}, we obtain an analogue of \eqref{tmp:04.10_1}
\[
|A|^{n-1} \D \ge 2^n q^{n-1} (q+1)^n (q-1)^{2} \,.
\]
Combining the last two bounds, we derive the required result.
This completes the proof.
$\hfill\Box$
\end{proof}
\begin{remark}
It is easy to see that Theorem \ref{t:8/3-c}, as well as Lemma \ref{l:BgB} (and also Lemma \ref{l:B_Wiener}) take place for any Borel subgroup not just for the standard one.
\label{r:BgB_B*}
\end{remark}
\begin{remark}
It is easy to see that the arguments of the proof of Theorem \ref{t:8/3-c}
give the following combinatorial statement about left/right multiplication of an arbitrary set $A$ by $\B$
(just combine bounds \eqref{tmp:04.10_1-} and \eqref{f:B_Ag}), namely,
\begin{equation}\label{f:Borel_sum-product}
\max\{ |A\B|, |\B A| \} \gg \min\{q^{3/2} |A|^{1/2}, |A|^2 q^{-2} \} \,.
\end{equation}
\end{remark}
\bigskip
As we have seen by
Theorem \ref{t:8/3-c}
we know that
$A^n \cap \B \neq \emptyset$ for large $n$ but under the condition $|A| \gg q^{2+\eps}$ for a certain $\eps>0$.
For the purpose of the next section we need to break the described $q^2$--barrier and we do this for prime $q$, using growth in $\SL_2 (\F_p)$.
Let us recall quickly what is known about growth of generating sets in $\SL_2 (\F_p)$.
In paper \cite{H} Helfgott obtained his famous result in this direction
and we proved in \cite{RS_SL2} the following form of Helfgott's result.
\begin{theorem}
Let $A \subseteq \SL_2 (\F_p)$ be a set, $A=A^{-1}$ which generates the whole group.
Then $|AAA| \gg |A|^{1+1/20}$.
\label{t:Misha_Je}
\end{theorem}
Thus in the case of an arbitrary symmetric generating set and a prime number $p$
Theorem \ref{t:Misha_Je}, combining with Theorem \ref{t:8/3-c},
allow to obtain some bounds which guarantee that $A^n = \SL_2 (\F_p)$.
For example, if $A$ generates $\SL_2(\F_p)$, $A=A^{-1}$,
and $|A| \gg p^{2-\epsilon}$, $\epsilon < \frac{2}{21}$,
then $A^n \cap \B \neq \emptyset$ for $n\ge \frac{84-42\epsilon}{2-21\epsilon}$.
On the other hand,
the methods from \cite{H}, \cite{RS_SL2} allow to obtain the following result about generation of $\SL_2 (\F_p)$ via large and not necessary symmetric sets
(the condition of non--symmetricity of $A$ is rather crucial for us, see the next section).
\begin{theorem}
Let $A\subseteq \SL_2 (\F_p)$ be a generating set, $p\ge 5$ and $|A| \gg p^{2-\epsilon}$, $\epsilon < \frac{2}{25}$.
Then $A^n \cap \B \neq \emptyset$ for $n\ge \frac{100-50\epsilon}{2-25 \epsilon}$.
Also, $A^n = \SL_2 (\F_p)$, provided $n\ge \frac{144}{2-25 \epsilon}$.
\label{t:Misha_Je_large}
\end{theorem}
\begin{proof}
Put $K=|AAA|/|A|$.
We can assume that, say, $|A| \le p^{2+2/35}$ because otherwise one can apply Theorem \ref{t:8/3-c}.
We call an element $g\in \SL_2 (\F_p)$ to be regular if $\tr(g) \neq 0, \pm 2$ and let $\mathcal{C}_g$ be the correspondent conjugate class, namely,
\[
\mathcal{C}_g = \{s \in \SL_2 (\F_p) ~:~ \tr(s) = \tr(g) \} \,.
\]
Let $T$ be a maximal torus (in $\SL_2 (\F_p)$ it is just a maximal commutative subgroup) such that there is $g\in T\cap A^{-1}A$ and $g \neq 1$.
By \cite[Lemma 5]{RS_SL2} such torus $T_*$, containing a regular element $g$, exists,
otherwise $K\gg |A|^{2/3}$.
Firstly, suppose that for a certain $h\in A$ the torus $T'=hTh^{-1}$ has no such property, i.e., there are no
nontrivial
elements from $A^{-1}A \cap T'$.
Then for the element $g'=hgh^{-1} \in T'$ (in the case $T=T_*$ the element $g'$ is regular) the projection $a\to ag'a^{-1}$, $a\in A$ is one--to--one.
Hence $|A^2 A^{-1} A A^{-2} \cap \mathcal{C}_g| \ge |A|$.
By \cite[Lemma 11]{RS_SL2}, we have $|S \cap \mathcal{C}_g| \ll |S^{-1}S|^{2/3} + p$ for any set $S$ and regular $g$.
Using the Ruzsa triangle inequality, we obtain
\begin{equation}\label{tmp:15.10_I}
|(A^2 A^{-1} A A^{-2})^{-1}(A^2 A^{-1} A A^{-2})| \le |A|^{-1} |A^2 A^{-1} A A^{-3}| |A^3 A^{-1} A A^{-2}|
=
\end{equation}
\[
=
|A|^{-1} |A^3 A^{-1} A A^{-2}|^2 \le |A|^{-1} (|A|^{-1} |A^3 A^{-2}| |A^2 A^{-2}| )^2 \le |A|^{-1} (|A|^{-3} |A^4| |A^3|^3 )^2 \le K^{12} |A|
\]
and hence
\[
|A| \ll |(A^2 A^{-1} A A^{-2})^{-1}(A^2 A^{-1} A A^{-2})|^{2/3} + p \ll K^{8} |A|^{2/3} \,.
\]
It gives us $K\gg |A|^{1/24}$.
In the complementary second case (see \cite{RS_SL2}) thanks to the fact that $A$ is a generating set, we
suppose that for {\it any} $h\in \SL_2(\F_p)$ there is a
nontrivial
element from $A^{-1}A$ belonging to the torus $hTh^{-1}$.
Then $A^{-1}A$ is partitioned between these tori and hence again by \cite[Lemma 11]{RS_SL2}, as well as the Ruzsa triangle inequality, we obtain
\[
|(AA^{-1}AA^{-1})^{-1}(AA^{-1}AA^{-1})| \le |A|^{-1} |A^2 A^{-1} A A^{-1}|^2
\le
\]
\[
\le
|A|^{-1} (|A|^{-1} |A^2 A^{-2}| |A^2 A^{-1}|)^2
\le |A|^{-1} (|A|^{-3} |A^3|^4)^2 \le K^8 |A|
\]
and whence
\[
K^2 |A| \ge |A^{-1}A| \ge \sum_{h \in \SL_2 (\F_p)/N(T_*)} |A^{-1} A \cap h T_* h^{-1}|
\gg
\]
\[
\gg
p^2 \cdot \frac{|A|}{|(AA^{-1}AA^{-1})^{-1}(AA^{-1}AA^{-1})|^{2/3}}
\ge p^2 |A|^{1/3} K^{-16/3} \,,
\]
where $N(T)$ is the normalizer of any torus $T$, $|N(T)| \asymp |T| \asymp p$.
Hence thanks to our assumption $|A| \le p^{2+2/35}$, we have $K \gg p^{3/11} |A|^{-1/11} \gg |A|^{1/24}$.
In other words, we always obtain $|AAA| \gg p^{2+\frac{2-25\epsilon}{24}}$.
After that apply Theorem \ref{t:8/3-c} to find that $A^n \cap \B \neq \emptyset$ for $n\ge \frac{100-50\epsilon}{2-25 \epsilon}$.
If we use Lemma \ref{l:A^n_large} instead of Theorem \ref{t:8/3-c}, then we obtain $A^n = \SL_2 (\F_p)$, provided $n\ge \frac{144}{2-25 \epsilon}$.
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
Thus for sufficiently small $\epsilon>0$ one can take $n=51$ to get $A^n \cap \B \neq \emptyset$ (and $n=73$ to obtain $A^n = \SL_2 (\F_p)$).
In the next section we improve this bound for a special set $A$ but nevertheless the arguments of the proof of Theorem \ref{t:Misha_Je_large} will be used in the proof of Theorem \ref{t:main_intr2} from the Introduction.
\bigskip
We finish this section showing that generating sets $A$ of sizes close to $p^2$ (actually, the condition $|A| =\Omega(p^{3/2+\eps})$ is enough) with small tripling constant $K=|A^3|/|A|$ avoid all Borel subgroups.
\begin{lemma}
Let $A\subseteq \SL_2 (\F_p)$ be a generating set, $p\ge 5$ and $K=|A^3|/|A|$.
Then for any Borel subgroup $\B_*$ one has $|A\cap \B_*| \le 2p K^{5/3} |A|^{1/3}$.
\label{l:A_cap_B}
\end{lemma}
\begin{proof}
We obtain the result for the standard Borel subgroup $\B$ and after that apply the conjugation to prove our Lemma in full generality.
Let $\gamma \in \F_p^*$ be any number and $l_\gamma$ be the line
$$ l_\gamma = \{ (\gamma u| 0 \gamma^{-1}) ~:~ u\in \F_p \} \subset \SL_2 (\F_p) \,.$$
By \cite[Lemma 7]{RS_SL2}, we have $|A\cap l_\gamma| \le 2 |A^3 A^{-1} A|^{1/3}$.
Using the last bound, as well as the Ruzsa triangle inequality, we obtain
\[
|A\cap \B| \le \sum_{\gamma \in \F^*_p} |A\cap l_\gamma| \le 2 p |A^3 A^{-1} A|^{1/3}
\le
2p (|A^4||A^{-2} A|/|A|)^{1/3}
\le
2p K^{5/3} |A|^{1/3} \,.
\]
This completes the proof.
$\hfill\Box$
\end{proof}
\begin{remark}
Examining the proof of Lemma 7 from \cite{RS_SL2} one can equally write $|A\cap l_\gamma| \le 2 |A^3 A^{-2}|^{1/3}$ and hence by the calculations above
$|A\cap \B_*| \le 2p K^{4/3} |A|^{1/3}$.
Nevertheless, his better estimate has no influence to the final bound in Theorem \ref{t:main_intr}.
\end{remark}
\begin{remark}
Bounds for intersections of $A\subseteq \SL_2 (\F_q)$, $K=|A^3|/|A|$ with $g\B_*$, where $g\notin \B_*$ are much simpler and follow from Lemma \ref{l:BgB} (also, see Remark \ref{r:BgB_B*}).
Indeed, by this result putting $A_* = A\cap g\B_*$, we have
\[
K|A|\ge |AA| \ge |A_* A_*| \ge \frac{|A_*|^4}{\E(A^{-1}_*, A_*)} \ge \frac{|A_*|^4}{\E(A^{-1}_*,g\B_*)} \ge \frac{|A_*|^2}{q-1}
\]
without any assumptions on generating properties of $A$.
\label{r:sB}
\end{remark}
\section{On Zaremba's conjecture}
In this section we
apply methods of the proofs of Theorems \ref{t:8/3-c}, \ref{t:Misha_Je_large} to Zaremba conjecture but also we use the specific of this problem, i.e. the special form of the correspondent set of matrices from $\SL_2 (\F_p)$.
\bigskip
Denote by $F_M(Q)$ the set of all {\it rational} numbers $\frac{u}{v}, (u,v) = 1$ from $[0,1]$ with all partial quotients in (\ref{exe}) not exceeding $M$ and with $ v\le Q$:
\[
F_M(Q)=\left\{
\frac uv=[0;b_1,\ldots,b_s]\colon (u,v)=1, 0\leq u\leq v\leq Q,\, b_1,\ldots,b_s\leq M
\right\} \,.
\]
By $F_M$ denote the set of all {\it irrational}
numbers from $[0,1]$ with partial quotients less than or equal to $M$.
From \cite{hensley1992continued} we know that the Hausdorff dimension $w_M$ of the set $F_M$ satisfies
\begin{equation}
w_M = 1- \frac{6}{\pi^2}\frac{1}{M} -
\frac{72}{\pi^4}\frac{\log M}{M^2} + O\left(\frac{1}{M^2}\right),
\,\,\, M \to \infty \,,
\label{HHD}
\end{equation}
however here we need a simpler result from \cite{hensley1989distribution}, which states that
\begin{equation}\label{oop}
1-w_{M} \asymp \frac{1}{M}
\end{equation}
with absolute constants in the sign $\asymp$.
Explicit estimates for dimensions of $F_M$ for certain values of $M$ can be found in \cite{jenkinson2004density}, \cite{JP} and in other papers.
For example, see \cite{JP}
\[
w_2 = 0.5312805062772051416244686...
\]
In papers \cite{hensley1989distribution,hensley1990distribution} Hensley gives the bound
\begin{equation}
|F_M(Q)| \asymp_M Q^{2w_M} \,.
\label{QLOW}
\end{equation}
\bigskip
Now we are ready to prove Theorem \ref{t:main_intr} from the Introduction.
One has
\begin{equation}\label{f:continuants_aj}
\left( {\begin{array}{cc}
0 & 1 \\
1 & b_1 \\
\end{array} } \right)
\dots
\left( {\begin{array}{cc}
0 & 1 \\
1 & b_s \\
\end{array} } \right)
=
\left( {\begin{array}{cc}
p_{s-1} & p_s \\
q_{s-1} & q_s \\
\end{array} } \right) \,,
\end{equation}
where $p_s/q_s =[0;b_1,\dots, b_s]$ and $p_{s-1}/q_{s-1} =[0;b_1,\dots, b_{s-1}]$.
Clearly, $p_{s-1} q_s - p_s q_{s-1} = (-1)^{s}$.
Let $Q=p-1$ and consider the set $F_M(Q)$.
Any $u/v \in F_M(Q)$ corresponds to a matrix from \eqref{f:continuants_aj} such that $b_j \le M$.
The set $F_M(Q)$ splits into ratios with even $s$ and with odd $s$,
in other words
$F_M(Q) = F^{even}_M(Q) \bigsqcup F^{odd}_M(Q)$.
Let $A \subseteq \SL_2 (\F_p)$ be the set of matrices of the form above with even $s$.
It is easy to see from \eqref{QLOW}, multiplying if it is needed the set $F^{odd}_M (Q)$ by $(01|1b)^{-1}$, $1\le b \le M$ that
$|F^{even}_M(Q)| \gg_M |F_M (Q)| \gg_M Q^{2w_M}$.
It is easy to check that if for a certain $n$ one has $A^n \cap \B \neq \emptyset$, then $q_{s-1}$
equals zero modulo $p$ and hence there is $u/v \in F_M ((2p)^n)$ such that $v\equiv 0 \pmod p$.
In a similar way,
we can easily assume that for any $g = (ab|cd)\in A$ all entries $a,b,c,d$ are nonzero (and hence by the construction they are nonzero modulo $p$), see, e.g., \cite[page 46]{hensley_SL2} or the proof of Lemma \ref{l:M^3} below (the same paper \cite{hensley_SL2} contains the fact that $A$ is a generating subset of $\SL_2 (\F_p)$).
Analogously, we can suppose that all $g \in A$ are regular, that is, $\tr(g) \neq 0,\pm 2$.
Let $K = |AAA|/|A|$ and $\tilde{K} = |AA|/|A| = K^\a$, $0\le \a \le 1$.
We need to
estimate
from below cardinality of the set of all possible traces of $A$, that is, cardinality of the set of sums $q_{s} + p_{s-1}$
(this expression is called "cyclical continuant").
Fix $p_{s-1}$ and $q_s$.
Then $p_{s-1} q_s - 1 = p_s q_{s-1}$ and thus $p_s$ is a divisor of $p_{s-1} q_s - 1$.
In particular, the number of such $p_s$ is at most $p^\eps$ for any $\eps>0$.
But now knowing the pair $(p_s,q_s)$, we determine the correspondent matrix \eqref{f:continuants_aj} from $A$ uniquely.
Hence the number of different pairs $(p_{s-1}, q_s)$ is at least $\Omega_M (p^{-\eps}|F_M(Q)|)$
and thus the number of different traces of all matrices from $A$ is $\Omega_M (p^{-1-\eps} |A|)$.
Actually, one can improve the last bound to $\Omega_M (p^{-1} |A|)$.
\begin{lemma}
The number of all possible sums $q_{s} + p_{s-1}$ is at least $\Omega(|A|/(M^3 p))$.
\label{l:M^3}
\end{lemma}
\begin{proof}
As above fix
$q_s$ and $p_{s-1}$.
It is well--known (see, e.g., \cite{hensley_SL2}) that
$q_s = \langle b_1,\dots, b_s \rangle$,
$p_s = \langle b_2,\dots, b_s \rangle$,
$q_{s-1} = \langle b_1,\dots, b_{s-1} \rangle$,
$p_{s-1} = \langle b_2,\dots, b_{s-1} \rangle$, where by $\langle x_1,\dots, x_n \rangle$ we have denoted the corresponding continuant.
We know that
\begin{equation}\label{tmp:01.11_1}
-p_{s} q_{s-1} = - q_s p_{s-1} + 1 \,.
\end{equation}
Substituting the well--known formula $p_{s} = b_s p_{s-1} + p_{s-2}$ into \eqref{tmp:01.11_1}, we obtain
\begin{equation}\label{tmp:01.11_2}
-b_s p_{s-1} q_{s-1} \equiv - q_s p_{s-1} + 1 \pmod {p_{s-2}} \,.
\end{equation}
and thus for any fixed $b_s \neq 0 \pmod {p_{s-2}}$ the number $q_{s-1}$ is uniquely determined modulo $p_{s-2} = \langle b_2,\dots, b_{s-2} \rangle $.
But applying the recurrence formula for continuants again, we get
\[
q_{s-1} = b_{s-1} \langle b_1,\dots, b_{s-2} \rangle + \langle b_1,\dots, b_{s-3} \rangle \le (b_{s-1} + 1) \langle b_1,\dots, b_{s-2} \rangle =
\]
\[
= (b_{s-1} + 1) ( b_1 p_{s-2} + \langle b_3,\dots, b_{s-2} \rangle) \le (b_{s-1} + 1) (b_1+1) p_{s-2} \,.
\]
It follows that there are at most $(M+1)^2$ possibilities for $q_{s-1}$.
Now if $b_s \equiv 0 \pmod {p_{s-2}}$, then $M\ge b_s \ge p_{s-2} \ge \left(\frac{1+\sqrt{5}}{2}\right)^{s-2}$ and hence $s\ll \log M$.
It gives us, say, at most $M^s \ll M^{O(1)} \le |A|/2$
matrices from $A$.
This completes the proof of the lemma.
$\hfill\Box$
\end{proof}
\bigskip
Now recall \cite[Lemma 12]{RS_SL2}, which is a variant of the Helfgott map \cite{H} from \cite{Brendan_rich} (we have already used similar arguments in the proof of Theorem \ref{t:Misha_Je_large}).
For the sake of the completeness we give the proof of a "statistical"\,
version
of this result.
\begin{lemma}
Let $\Gr$ be any group and $A\subseteq \Gr$ be a finite set.
Then for an arbitrary $g\in \Gr$, there is $A_0 \subseteq A$, $|A_0| \ge |A|/2$ such that for any $a_0 \in A_0$ the following holds
\begin{equation}\label{f:CS_ineq}
|A|/2 \le |{\rm Conj} (g) \cap AgA^{-1}| \cdot |{\rm Centr}(g) \cap a_0^{-1} A| \,.
\end{equation}
Here ${\rm Conj} (g)$ is the conjugacy class and ${\rm Centr}(g)$ is the centrlizer of $g$ in $\Gr$.
\label{l:CS_ineq}
\end{lemma}
\begin{proof}
Let $\_phi : A \to {\rm Conj} (g) \cap AgA^{-1}$ be the Helfgott map $\_phi(a) := a g a^{-1}$.
One sees that $\_phi(a) = \_phi (b)$ iff
\[
b^{-1} a g = g b^{-1} a \,.
\]
In other words, $b^{-1} a \in {\rm Centr}(g) \cap A^{-1} A$.
Clearly, then
\[
|A| = \sum_{c\in {\rm Conj} (g) \cap AgA^{-1}} |\{ a\in A ~:~ \_phi (a) = c\}|
\le
\]
\begin{equation}\label{tmp:01.11_10}
\le
2 \sum_{c\in {\rm Conj} (g) \cap AgA^{-1} ~:~ |\{ a\in A ~:~ \_phi (a) = c\}| \ge |A|/(2|{\rm Conj} (g) \cap AgA^{-1}|)} |\{ a\in A ~:~ \_phi (a) = c\}| \,.
\end{equation}
For $c\in \_phi (A) \subseteq {\rm Conj} (g) \cap AgA^{-1}$ put $A (c) = \_phi^{-1} (c) \subseteq A$ and let
$$
A_0 = \bigsqcup_{c ~:~ |A (c)| \ge |A|/(2|{\rm Conj} (g) \cap AgA^{-1}|)} A (c) \,.
$$
In other words, estimate \eqref{tmp:01.11_10} gives us
\[
|A_0| = \sum_c |A (c)| \ge |A|/2 \,.
\]
But for any $b\in A_0$ one has $|{\rm Centr}(g) \cap b^{-1} A| \ge |A|/(2|{\rm Conj} (g) \cap AgA^{-1}|)$
as required.
This completes the proof of the lemma.
$\hfill\Box$
\end{proof}
\bigskip
Now summing inequality \eqref{f:CS_ineq} over all $g\in A$ with different traces, we obtain in view of the Ruzsa triangle inequality and Lemma \ref{l:M^3} that
\begin{equation}\label{tmp:15.10_1}
|A|^2 p^{-1} \ll_M |AAA^{-1}| \cdot \max_{g\in A} |{\rm Centr}(g) \cap a_0^{-1}(g) A|
\le
K \tilde{K} |A| \cdot \max_{g\in A} |{\rm Centr}(g) \cap a_0^{-1}(g) A| \,.
\end{equation}
Here for every $g\in A$ we have taken a concrete $a_0 (g) \in A_0 (g)$ but in view of Lemma \ref{l:CS_ineq} it is known that there are a lot of them and we will use this fact a little bit later.
Now by \cite[Lemma 4.7]{H}, we see
that
\[
|(a_0^{-1}(g) A) g_* (a_0^{-1}(g) A) g^{-1}_* (a_0^{-1}(g) A)^{-1}| \gg |{\rm Centr}(g) \cap a_0^{-1}(g) A|^3 \,,
\]
where $g_* = (ab|cd)$ is any element from $A$ such that $abcd \neq 0$ in the basis where $g$ has the diagonal form.
Thanks to Lemma \ref{l:A_cap_B} and Remark \ref{r:sB} we can choose $g_* = a_0 (g)$, otherwise $|A| \ll p^{3/2} K^{5/2}$.
In the last case if, say, $|A| \gg p^{2-1/35}$, then $K\gg p^{33/175}$ and hence $|A^3| \gg p^{2+4/25}$.
Using Theorem \ref{t:8/3-c}, we see that one can take $n=27$ and this is better than we want to prove.
Then with this choice of $g_*$, we have by the Ruzsa triangle inequality
\[
|A^2 g^{-1}_* A^{-1}| \le |A^2 A^{-2}| \le K^2 |A| \,,
\]
and hence $|{\rm Centr}(g) \cap a_0^{-1}(g) A| \ll K^{2/3} |A|^{1/3}$.
Substituting the last bound into \eqref{tmp:15.10_1}, we get
\begin{equation}\label{tmp:15.10_1-}
|A|^2 p^{-1} \ll_M K \tilde{K} |A| \cdot K^{2/3} |A|^{1/3}
\end{equation}
and hence
\begin{equation}\label{tmp:15.10_2}
K \gg_M (|A|^2 p^{-3})^{\frac{1}{5+3\a}} \gg p^{\frac{4w_M}{5+3\a} - \frac{3}{5+3\a}} \,.
\end{equation}
In other words, $|AAA| \gg_M p^{2+\frac{w_M (14+6\a) - 13- 6\a}{5+3\a}}$.
Take $M$ sufficiently large
such that $w_M (14+6\a) - 13- 6\a >0$.
Using Theorem \ref{t:8/3-c}, we see that for any
\begin{equation}\label{f:n_1}
n\ge \frac{w_M (28+12\a)- 6}{w_M (14+6\a) - 13- 6\a}
\end{equation}
one has $A^n \cap \B \neq \emptyset$.
On the other hand, from \eqref{tmp:15.10_2}, we get
\[
|AA| = |A|K^\a \gg p^{2+ \frac{w_M(10+10\a) -10 - 9 \a}{5+3\a}} \,.
\]
Suppose that $w_M(10+10\a) -10 - 9 \a > 0$.
It can be done if $\a>0$ and if we take sufficiently large $M$.
Applying Theorem \ref{t:8/3-c} one more time, we derive that for any
\begin{equation}\label{f:n_2}
n \ge \frac{2}{3} \cdot \frac{w_M(20+20\a) - 6 \a}{w_M(10+10\a) -10 - 9 \a}
\end{equation}
one has $A^n \cap \B \neq \emptyset$.
Comparing \eqref{f:n_1} and \eqref{f:n_2}, we choose $\a$ optimally when
\[
\a^2 (120 w^2_M - 12w_M - 72) + \a(400 w_M^2 -368 w_M + 6)
+
280 w_M^2 + 180 - 500 w_M = 0
\]
and it gives
\[
18 \a^2 + 19 \a - 20 = 0
\]
and whence
$\a = \frac{-19+\sqrt{1801}}{36} + o_M (1)$ as
$M\to +\infty$.
Hence from \eqref{f:n_1}, say, we obtain $n\ge \frac{47+\sqrt{1801}}{3} + o_M (1) > 29.81 + o_M (1)$.
Taking sufficiently large $M$,
we can choose $n=30$.
If $\a=0$, then for sufficiently large $M$ estimate \eqref{f:n_1} allows us to take $n = 23$.
This completes the proof.
$\hfill\Box$
\bigskip
Combining the arguments
above
with Theorems \ref{t:8/3-c}, \ref{t:Misha_Je_large}, we obtain Theorem \ref{t:main_intr2} from the Introduction.
Actually, if we apply the second part of Theorem \ref{t:Misha_Je_large}, then we generate the whole $\SL_2 (\F_p)$ (and this differs our method from \cite{MOW_Schottky}, say).
Because in the case $\zk =2$ we use results about growth in $\SL_2 (\F_p)$ for relatively small asymmetric set $A$ ($|A| \gg p^{2w_2} \gg p^{1.062}$)
our absolute constant $C$ is
large.
It is easy to see that the arguments of this section on trace of the set $A$ begin to work for $w_M > 3/4$ (see Lemma \ref{l:A_cap_B},
as well as
estimates \eqref{tmp:15.10_1}, \eqref{tmp:15.10_1-}) and in this case the constant $C$ can be decreased, although it remains rather large.
| 21,005 |
\section{Pion assisted $N\Delta$ and $\Delta\Delta$ dibaryons}
\label{sec:int}
Nonstrange $s$-wave dibaryon resonances ${\cal D}_{IS}$ with isospin $I$
and spin $S$ were predicted by Dyson and Xuong in 1964~\cite{DX64} as early
as SU(6) symmetry proved successful, placing the nucleon $N(939)$ and its
$P_{33}$ $\pi N$ resonance $\Delta(1232)$ in the same ${\bf 56}$ multiplet
which reduces to a ${\bf 20}$ SU(4) spin-isospin multiplet for nonstrange
baryons. For SU(3)-color singlet and spatially symmetric $L=0$ 6q
configuration, the spin-isospin 6q configuration ensuring a totally
antisymmetric color-spin-isospin-space 6q wavefunction is a ${\bf 50}$
dimensional SU(4) representation, denoted by its (3,3,0,0) Young tableau,
which is the lowest-dimension SU(4) multiplet in the $\bf{20\times 20}$
direct product~\cite{PPL15}. This ${\bf 50}$ SU(4) multiplet includes the
deuteron ${\cal D}_{01}$ and $NN$ virtual state ${\cal D}_{10}$, plus four
more nonstrange dibaryons, with masses listed in Table~\ref{tab:dyson} in
terms of SU(4) mass-formula constants $A$ and $B$.
\begin{table}[hbt]
\begin{center}
\caption{Predicted masses of non-strange $L=0$ dibaryons ${\cal D}_{IS}$
with isospin $I$ and spin $S$, using the Dyson-Xuong~\cite{DX64}
SU(6)$\to$SU(4) mass formula $M=A+B\,[I(I+1)+S(S+1)-2]$.}
\begin{tabular}{ccccccccccccc}
\hline
${\cal D}_{IS}$ & & ${\cal D}_{01}$ & & ${\cal D}_{10}$ & & ${\cal D}_{12}$
& & ${\cal D}_{21}$ & & ${\cal D}_{03}$ & & ${\cal D}_{30}$ \\
\hline
$BB'$ & & $NN$ & & $NN$ & & $N\Delta$ & & $N\Delta$ & & $\Delta\Delta$ & &
$\Delta\Delta$ \\
SU(3)$_{\rm f}$ & & $\overline{\bf 10}$ & & ${\bf 27}$ & & ${\bf 27}$ & &
${\bf 35}$ & & $\overline{\bf 10}$ & & ${\bf 28}$ \\
$M({\cal D}_{IS})$ & & $A$ & & $A$ & & $A+6B$ & & $A+6B$ & & $A+10B$ & &
$A+10B$ \\
\hline
\end{tabular}
\label{tab:dyson}
\end{center}
\end{table}
Identifying $A$ with the $NN$ threshold mass 1878~MeV, the value
$B\approx 47$~MeV was derived by assigning ${\cal D}_{12}$ to the
$pp\leftrightarrow \pi^+ d$ coupled-channel resonance behavior noted then
at 2160~MeV, near the $N\Delta$ threshold (2.171~MeV). This led in particular
to a predicted mass $M=2350$~MeV for the $\Delta\Delta$ dibaryon candidate
${\cal D}_{03}$ assigned at present to the recently established d$^\ast$(2380)
resonance~\cite{clement17}. Since the ${\bf 27}$ and $\overline{\bf 10}$
flavor-SU(3) multiplets accommodate $NN$ $s$-wave states that are close
to binding ($^1S_0$) or weakly bound ($^3S_1$), we focus here on the
${\cal D}_{12}$ and ${\cal D}_{03}$ dibaryon candidates assigned to these
flavor-SU(3) multiplets.
The idea behind the concept of pion assisted dibaryons~\cite{gal16} is that
since the $\pi N$ $p$-wave interaction in the $P_{33}$ channel is so strong
as to form the $\Delta$(1232) baryon resonance, acting on two nucleons it may
assist in forming $s$-wave $N\Delta$ dibaryon states, and subsequently also
in forming $s$-wave $\Delta\Delta$ dibaryon states. This goes beyond the
major role played by a $t$-channel exchange low-mass pion in binding or
almost binding $NN$ $s$-wave states.
As discussed below, describing $N\Delta$ systems in terms of
a stable nucleon ($N$) and a two-body $\pi N$ resonance ($\Delta$) leads to
a well defined $\pi NN$ three-body model in which $IJ=12$ and $21$ resonances
identified with the ${\cal D}_{12}$ and ${\cal D}_{21}$ dibaryons of
Table~\ref{tab:dyson} are generated. This relationship between $N\Delta$ and
$\pi NN$ may be generalized into relationship between a two-body $B\Delta$
system and a three-body $\pi NB$ system, where the baryon $B$ stands for $N,
\Delta, Y$ (hyperon) etc. In order to stay within a three-body formulation
one needs to assume that the baryon $B$ is stable. For $B=N$, this
formulation relates the $N\Delta$ system to the three-body $\pi NN$ system.
For $B=\Delta$, once properly formulated, it relates the
$\Delta\Delta$ system to the three-body $\pi N\Delta$ system, suggesting to
seek $\Delta\Delta$ dibaryon resonances by solving $\pi N\Delta$ Faddeev
equations, with a stable $\Delta$. The decay width of the $\Delta$ resonance
is considered then at the penultimate stage of the calculation. In terms
of two-body isobars we have then a coupled-channel problem $B\Delta
\leftrightarrow\pi D$, where $D$ stands generically for appropriate dibaryon
isobars: (i) ${\cal D}_{01}$ and ${\cal D}_{10}$, which are the $NN$ isobars
identified with the deuteron and virtual state respectively, for $B=N$, and
(ii) ${\cal D}_{12}$ and ${\cal D}_{21}$ for $B=\Delta$.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=0.8\textwidth]{BDel.eps}
\caption{Diagrammatic representation of the $B\Delta$ $T$-matrix integral
equation from $\pi NB$ Faddeev equations with separable pairwise
interactions where $B=N,\,\Delta$~\cite{GG13,GG14}.}
\label{fig:piNB}
\end{center}
\end{figure}
Within this model, and using separable pairwise interactions,
the coupled-channel $B\Delta -\pi D$ eigenvalue problem reduces to a single
integral equation for the $B\Delta$ $T$ matrix shown diagrammatically in
Fig.~\ref{fig:piNB}, where starting with a $B\Delta$ configuration the
$\Delta$-resonance isobar decays into $\pi N$, followed by $NB\to NB$
scattering through the $D$-isobar with a spectator pion, and ultimately
by means of the inverse decay $\pi N\to\Delta$ back into the $B\Delta$
configuration. The interaction between the $\pi$ meson and $B$ is neglected
for $B=\Delta$, for lack of known $\pi\Delta$ isobar resonances in the
relevant energy range.
The ${\cal D}_{12}$ dibaryon of Table~\ref{tab:dyson} shows up clearly in
the Argand diagram of the $NN$ $^1D_2$ partial wave which is coupled above
the $NN\pi$ threshold to the $I=1$ $s$-wave $N\Delta$ channel. Its $S$-matrix
pole position $W=M-{\rm i}\Gamma/2$ was given by 2148$-{\rm i}$63~MeV in $NN$
phase shift analyses~\cite{arndt87} and by 2144$-{\rm i}$55~MeV in dedicated
$pp \leftrightarrow np\pi^+$ coupled-channels analyses~\cite{hosh92}. It has
been observed, most likely, at $W$=(2.14$\pm$0.01)-i(0.09$\pm$0.01) GeV in
a recent $\gamma d\to \pi^0\pi^0 d$ ELPH experiment~\cite{ELPH19}. Values of
${\cal D}_{12}$ and ${\cal D}_{21}$ pole positions from our hadronic-model
three-body $\pi NN$ Faddeev calculations~\cite{GG13,GG14}, substituting
$N$ for $B$ in Fig.~\ref{fig:piNB}, are listed in Table~\ref{tab:BDel}.
The ${\cal D}_{12}$ mass and width values calculated in the Faddeev hadronic
model version using $r_{\Delta}\approx\,1.3$~fm are remarkably close to these
phenomenologically derived values. As for the ${\cal D}_{21}$ dibaryon, recent
$pp\to pp\pi^+\pi^-$ production data~\cite{wasa18} place it almost degenerate
with the ${\cal D}_{12}$. Our $\pi NN$ Faddeev calculations produce it about
10-20~MeV higher than the ${\cal D}_{12}$, see Table~\ref{tab:BDel}. The
widths of these near-threshold $N\Delta$ dibaryons are, naturally, close to
that of the $\Delta$ resonance. We note that only $^3S_1$ $NN$ enters the
calculation of the ${\cal D}_{12}$ resonance, while for the ${\cal D}_{21}$
resonance calculation only $^1S_0$ $NN$ enters, both with maximal strength.
Obviously, with the $^1S_0$ interaction the weaker of the two, one expects
indeed that the ${\cal D}_{21}$ resonance lies above the ${\cal D}_{12}$
resonance. Moreover, these two dibaryon resonances differ also in their
flavor-SU(3) classification, see Table~\ref{tab:dyson}, which is likely to
push up the ${\cal D}_{21}$ further away from the ${\cal D}_{12}$. Finally,
the $N\Delta$ $s$-wave states with $IJ=$ $11$ and $22$ are found not to
resonate in the $\pi NN$ Faddeev calculations~\cite{GG14}.
\begin{table}[hbt]
\begin{center}
\caption{${\cal D}_{IS}$ dibaryon $S$-matrix pole positions
$M-{\rm i}\frac{\Gamma}{2}$ (in MeV) obtained by solving the $N\Delta$
and $\Delta\Delta$ $T$-matrix integral equation Fig.~\ref{fig:piNB} are
listed for $\pi N$ $P_{33}$ form factors specified by radius parameter
$r_{\Delta}$~\cite{GG13,GG14}.}
\begin{tabular}{ccccc}
\hline
$r_{\Delta}$ & \multicolumn{2}{c}{$N\Delta$} & \multicolumn{2}{c}
{$\Delta\Delta$} \\
(fm) & ${\cal D}_{12}$ & ${\cal D}_{21}$ & ${\cal D}_{03}$ & ${\cal D}_{30}$
\\
\hline
1.3 & 2147$-{\rm i}$60 & 2165$-{\rm i}$64 & 2383$-{\rm i}$41 &
2411$-{\rm i}$41 \\
0.9 & 2159$-{\rm i}$70 & 2169$-{\rm i}$69 & 2343$-{\rm i}$24 &
2370$-{\rm i}$22 \\
\hline
\end{tabular}
\label{tab:BDel}
\end{center}
\end{table}
The ${\cal D}_{03}$ dibaryon of Table~\ref{tab:dyson} is best demonstrated
by the relatively narrow peak observed in $pn\to d\pi^0\pi^0$ by the
WASA-at-COSY Collaboration~\cite{wasa11} about 80~MeV above the $\pi^0\pi^0$
production threshold and 80~MeV below the $\Delta\Delta$ threshold, with
$\Gamma_{d^{\ast}}\approx 70$~MeV. Its $I=0$ isospin assignment follows from
the isospin balance in $pn \to d\pi^0\pi^0$, and the $J^P=3^+$ spin-parity
assignment follows from the measured deuteron angular distribution. The
d$^{\ast}$(2380) was also observed in $pn\to d\pi^+\pi^-$~\cite{wasa13},
with cross section consistent with that measured in $pn\to d\pi^0\pi^0$,
and studied in several $pn\to NN\pi\pi$ reactions~\cite{wasa15}. Recent
measurements of $pn$ scattering and analyzing power~\cite{wasa14} have
led to a $pn$ $^3D_3$ partial-wave Argand plot fully supporting the
${\cal D}_{03}$ dibaryon resonance interpretation.
Values of ${\cal D}_{03}$ and ${\cal D}_{30}$ pole positions $W=M-{\rm i}
\Gamma/2$ from our hadronic-model three-body $\pi N\Delta$ Faddeev
calculations~\cite{GG13,GG14} are also listed in Table~\ref{tab:BDel}.
The ${\cal D}_{03}$ mass and width values calculated in the Faddeev hadronic
model version using $r_{\Delta}\approx\,1.3$~fm are remarkably close to the
experimentally determined ones. The ${\cal D}_{30}$ dibaryon resonance is
found in our $\pi N\Delta$ Faddeev calculations to lie about 30~MeV above
the ${\cal D}_{03}$. These two states are degenerate in the limit of
equal $D={\cal D}_{12}$ and $D={\cal D}_{21}$ isobar propagators in
Fig.~\ref{fig:piNB}. Since ${\cal D}_{12}$ was found to lie lower than
${\cal D}_{21}$, we expect also ${\cal D}_{03}$ to lie lower than
${\cal D}_{30}$ as satisfied in our Faddeev calculations. Moreover, here
too the difference in their flavor-SU(3) classification will push the
${\cal D}_{30}$ further apart from the ${\cal D}_{03}$. The ${\cal D}_{30}$
has not been observed and only upper limits for its production in $pp\to pp
\pi^+\pi^+\pi^-\pi^-$ are available~\cite{wasa16}.
Finally, we briefly discuss the ${\cal D}_{03}$ mass and width
values from two recent quark-based resonating-group-method (RGM)
calculations~\cite{wang14,dong16} that add $\Delta_{\bf 8}\Delta_{\bf 8}$
hidden-color (CC) components to a $\Delta_{\bf 1}\Delta_{\bf 1}$ cluster.
The two listed calculations generate mass values
that are close to the mass of the d$^{\ast}$(2380). The calculated widths,
however, differ a lot from each other: one calculation generates a width of
150~MeV~\cite{wang14}, exceeding substantially the reported value $\Gamma_{
d^{\ast}(2380)}$=80$\pm$10~MeV~\cite{wasa14}, the other one generates a width
of 72~MeV~\cite{dong16}, thereby reproducing the d$^{\ast}$(2380) width.
While the introduction of CC components has moderate effect on the resulting
mass and width in the chiral version of the first calculation, lowering the
mass by 20~MeV and the width by 25~MeV, it leads to substantial reduction of
the width in the second (also chiral) calculation from 133~MeV to 72~MeV.
The reason is that the dominant CC $\Delta_{\bf 8}\Delta_{\bf 8}$ components,
with $68\%$ weight~\cite{dong16}, cannot decay through single-fermion
transitions $\Delta_{\bf 8}\to N_{\bf 1}\pi_{\bf 1}$ to asymptotically
free color-singlet hadrons. However, as argued in the next section, these
quark-based width calculations miss important kinematical ingredients that
make the width of a single compact $\Delta_{\bf 1}\Delta_{\bf 1}$ cluster
considerably smaller than $\Gamma_{d^{\ast}(2380)}$. The introduction of
substantial $\Delta_{\bf 8}\Delta_{\bf 8}$ components only aggravates the
disagreement.
\section{The width of d$^\ast$(2380), small or large?}
\label{sec:width}
The width derived for the ${\cal D}_{03}$ dibaryon
resonance d$^{\ast}$(2380) by WASA-at-COSY and SAID,
$\Gamma_{d^{\ast}(2380)}$=80$\pm$10~MeV~\cite{wasa14}, is dominated by
$\Gamma_{d^{\ast}\to NN\pi\pi}\approx 65$~MeV which is much smaller than
twice the width $\Gamma_{\Delta}\approx 115$~MeV~\cite{SP07,anisovich12} of
a single free-space $\Delta$, expected naively for a $\Delta\Delta$ quasibound
configuration. However, considering the reduced phase space, $M_{\Delta}=1232
\Rightarrow E_{\Delta}=1232-B_{\Delta\Delta}/2$~MeV in a bound-$\Delta$ decay,
where $B_{\Delta\Delta}=2\times 1232-2380=84$~MeV is the $\Delta\Delta$
binding energy, the free-space $\Delta$ width gets reduced to 81~MeV using
the in-medium single-$\Delta$ width $\Gamma_{\Delta\to N\pi}$ expression
obtained from the empirical $\Delta$-decay momentum dependence
\begin{equation}
\Gamma_{\Delta\to N\pi}(q_{\Delta\to N\pi})=\gamma\,
\frac{q^3_{\Delta\to N\pi}}{q_0^2+q^2_{\Delta\to N\pi}},
\label{eq:gamma}
\end{equation}
with $\gamma=0.74$ and $q_0=159$~MeV~\cite{BCS17}. Yet, this simple estimate
is incomplete since neither of the two $\Delta$s is at rest in a deeply
bound $\Delta\Delta$ state, as also noted by Niskanen~\cite{niskanen17}.
To take account of the $\Delta\Delta$ momentum distribution, we evaluate
the bound-$\Delta$ decay width ${\overline{\Gamma}}_{\Delta\to N\pi}$ by
averaging $\Gamma_{\Delta\to N\pi}(\sqrt{s_{\Delta}})$ over the $\Delta\Delta$
bound-state momentum-space distribution~\cite{gal17},
\begin{equation}
{\overline{\Gamma}}_{\Delta\to N\pi}\equiv\langle \Psi^{\ast}(p_{\Delta\Delta})
|\Gamma_{\Delta\to N\pi}(\sqrt{s_{\Delta}})|\Psi(p_{\Delta\Delta})\rangle
\approx \Gamma_{\Delta\to N\pi}(\sqrt{{\overline{s}}_{\Delta}}),
\label{eq:av}
\end{equation}
where $\Psi(p_{\Delta\Delta})$ is the $\Delta\Delta$ momentum-space
wavefunction and the dependence of $\Gamma_{\Delta\to N\pi}$ on $q_{\Delta
\to N\pi}$ for on-mass-shell nucleons and pions was replaced by dependence
on $\sqrt{s_{\Delta}}$. The averaged bound-$\Delta$ invariant energy squared
${\overline{s}}_{\Delta}$ is defined by ${\overline{s}}_{\Delta}=(1232-B_{
\Delta\Delta}/2)^2-P_{\Delta\Delta}^2$ in terms of a $\Delta\Delta$
bound-state r.m.s. momentum $P_{\Delta\Delta}\equiv{\langle p_{\Delta
\Delta}^2\rangle}^{1/2}$ inversely proportional to the r.m.s. radius
$R_{\Delta\Delta}$.
The d$^{\ast}$(2380) in the quark-based RGM calculations of Ref.~\cite{dong16}
appears quite squeezed compared to the diffuse deuteron. Its size, $R_{\Delta
\Delta}$=0.76~fm~\cite{huang15}, leads to unacceptably small upper limit of
about 47~MeV for $\Gamma_{d^{\ast}\to NN\pi\pi}$~\cite{gal17}. This drastic
effect of momentum dependence is missing in quark-based width calculations
dealing with pionic decay modes of $\Delta_{\bf 1}\Delta_{\bf 1}$ components,
e.g. Ref.~\cite{dong16}, and as presented by this Beijing group at MESON 2018
\cite{huang19} and at QNP 2018 \cite{dong19a}. Practitioners of quark-based
models ought therefore to ask ``what makes $\Gamma_{d^{\ast}(2380)}$ so much
larger than the width calculated for a compact $\Delta\Delta$ dibaryon?"
rather than ``what makes $\Gamma_{d^{\ast}(2380)}$ so much smaller than
twice a free-space $\Delta$ width?"
The preceding discussion of $\Gamma_{d^{\ast}(2380)}$ suggests that
quark-based model findings of a tightly bound $\Delta\Delta$ $s$-wave
configuration are in conflict with the observed width. Fortunately,
hadronic-model calculations~\cite{GG13,GG14} offer resolution of this
insufficiency by coupling to the tightly bound and compact $\Delta\Delta$
component of the d$^{\ast}$(2380) dibaryon's wavefunction a $\pi N\Delta$
resonating component dominated asymptotically by a $p$-wave pion attached
loosely to the near-threshold $N\Delta$ dibaryon ${\cal D}_{12}$ with size
about 1.5--2~fm. Formally, one can recouple spins and isospins in this
$\pi{\cal D}_{12}$ system, so as to assume an extended $\Delta\Delta$-like
object. This explains why a discussion of $\Gamma_{d^{\ast}\to NN\pi\pi}$
in terms of a $\Delta\Delta$ constituent model requires a size
$R_{\Delta\Delta}$ considerably larger than provided by quark-based RGM
calculations~\cite{dong16} to reconcile with the reported value of
$\Gamma_{d^{\ast}(2380)}$. We recall that the width calculated in our
diffuse-structure $\pi N\Delta$ model~\cite{GG13,GG14}, as listed in
Table~\ref{tab:BDel}, is in good agreement with the observed width of
the d$^{\ast}$(2380) dibaryon resonance.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig-plb5.eps}
\caption{$\sigma(\gamma d\to d\pi^0\pi^0)$ as a function of the total cm
energy $W$ from the ELPH experiment~\cite{ELPH17}. The red histogram shows
systematic errors, the dotted curve shows a nonresonant calculation by Fix
and Arenh\"{o}vel~\cite{fix05} and the solid curve is obtained by adding
a BW shape centered at $W_{\gamma d}=2370$~MeV with $\Gamma=68$~MeV.
A similar excitation spectrum has been reported by the MAMI A2
Collaboration~\cite{MAMI18}.}
\label{fig:fix}
\end{center}
\end{figure}
Support for the role of the $\pi{\cal D}_{12}$ configuration in the decay of
the d$^{\ast}$(2380) dibaryon resonance is provided by a recent ELPH $\gamma d
\to d \pi^0 \pi^0$ experiment~\cite{ELPH17}. The cross section data shown in
Fig.~\ref{fig:fix} agree with a relativistic Breit-Wigner (BW) resonance shape
centered at 2370~MeV and width of 68~MeV, but the statistical significance of
the fit is low, particularly since most of the data are from the energy region
above the d$^{\ast}$(2380). Invariant mass distributions from this experiment
at $W_{\gamma d}=2.39$~GeV, shown in Fig.~\ref{fig:ELPH}, are more instructive.
The $\pi\pi$ mass distribution shown in (a) suggests a two-bump structure,
fitted in solid red. The lower bump around 300~MeV is perhaps a manifestation
of the ABC effect~\cite{ABC60}, already observed in $pn\to d\pi^0\pi^0$ by
WASA-at-COSY~\cite{wasa11,BCS17} and interpreted in Ref.~\cite{gal17} as due
to a tightly bound $\Delta\Delta$ decay with reduced $\Delta \to N \pi$ phase
space. The upper bump in (a) is consistent then with the d$^{\ast}(2380)\to
\pi {\cal D}_{12}$ decay mode, in agreement with the $\pi d$ invariant-mass
distribution shown in (b) that peaks slightly below the ${\cal D}_{12}$(2150)
mass.
\begin{figure}[!b]
\begin{center}
\includegraphics[width=0.7\textwidth]{ELPH.eps}
\caption{Invariant mass distributions in ELPH experiment~\cite{ELPH17}
$\gamma d \to d \pi^0 \pi^0$ at $W_{\gamma d}=2.39$~GeV.}
\label{fig:ELPH}
\end{center}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.48\textwidth,height=5.5cm]{tot_xs_pn.eps}
\includegraphics[width=0.48\textwidth,height=5cm]{platonova.eps}
\caption{The $pn\to d\pi^0\pi^0$ peak (left) and its $M_{d\pi}$ invariant-mass
distribution (right) as observed by WASA-at-COSY \cite{wasa11}. The curves
in the right panel are from Ref.~\cite{PK16}: the dot-dashed line gives the
$\pi{\cal D}_{12}(2150)$ contribution to the two-body decay of the d$^{\ast}
$(2380) dibaryon, and the dashed line gives a $\sigma$-meson emission
contribution. The solid lines are calculated $M_{d\pi}$ distributions
for two input parametrizations of ${\cal D}_{12}(2150)$.}
\label{fig:PK}
\end{center}
\end{figure}
Theoretical support for the relevance of the ${\cal D}_{12}(2150)$ $N\Delta$
dibaryon to the physics of the d$^{\ast}$(2380) resonance is corroborated
in Fig.~\ref{fig:PK} by showing in the right panel a $d\pi$ invariant-mass
distribution peaking near the $N\Delta$ threshold as deduced from the
$pn\to d\pi^0\pi^0$ reaction in which the d$^{\ast}$(2380) was discovered
and which is shown for comparison in the left panel. However, the $M_{d\pi}$
peak is shifted to about 20 MeV below the mass of the ${\cal D}_{12}$(2150)
and its width is smaller by about 40 MeV than the ${\cal D}_{12}$(2150) width,
agreeing perhaps fortuitously with $\Gamma_{d^{\ast}(2380)}$. Both of these
features, the peak downward shift and the smaller width, can be explained
by the asymmetry between the two emitted $\pi^0$ mesons, only one of which
is due to the $\Delta\to N\pi$ decay within the ${\cal D}_{12}$(2150).{
\footnote{I'm indebted to Heinz Clement for confirming this explanation.}}
\begin{table}[hbt]
\begin{center}
\caption{d$^{\ast}$(2380) decay width branching ratios (BRs) calculated in
Ref.~\cite{gal17}, for a total decay width $\Gamma_{d^{\ast}(2380)}$=75~MeV,
are compared with BRs derived from experiment~\cite{BCS15,wasa17}.}
\begin{tabular}{cccccccccc}
\hline
\% & $d\pi^0\pi^0$ & $d\pi^+\pi^-$ & $pn\pi^0\pi^0$ &
$pn\pi^+\pi^-$ & $pp\pi^-\pi^0$ & $nn\pi^+\pi^0$ & $NN\pi$ & $NN$ & total \\
\hline
BR(th.) & 11.2 & 20.4 & 11.6 & 25.8 & 4.7 & 4.7 & 8.3 & 13.3 & 100 \\
BR(exp.) & 14$\pm$1 & 23$\pm$2 & 12$\pm$2 & 30$\pm$5 & 6$\pm$1 & 6$\pm$1 &
$\leq$9 &
12$\pm$3 & 103 \\
\hline
\end{tabular}
\label{tab:BR}
\end{center}
\end{table}
Recalling the $\Delta\Delta$ -- $\pi{\cal D}_{12}$ coupled channel nature of
the d$^{\ast}$(2380) in our hadronic model~\cite{GG13,GG14}, one may describe
satisfactorily the d$^{\ast}$(2380) total and partial decay widths in terms
of an incoherent mixture of these relatively short-ranged ($\Delta\Delta$)
and long-ranged ($\pi{\cal D}_{12}$) channels. This is demonstrated in
Table~\ref{tab:BR} where weights of $\frac{5}{7}$ and $\frac{2}{7}$ for
$\Delta\Delta$ and $\pi{\cal D}_{12}$, respectively, are assigned to an
assumed value of $\Gamma_{d^{\ast}\to NN\pi\pi}$=60~MeV~\cite{gal17}. This
choice yields a branching ratio for $\Gamma_{d^{\ast}\to NN\pi}$ which does
not exceed the upper limit of BR$\leq$9\% determined recently from {\it not}
observing the single-pion decay branch~\cite{wasa17}. A pure $\Delta\Delta$
description leads, as expected, to BR$\ll$1\%~\cite{dong17}.
\section{Hexaquark, diquark and you-name-it-quark models for d$^{\ast}$(2380)}
\label{sec:disc}
In this concluding section we comment on the applicabilty of several
quark-based models to the d$^{\ast}$(2380) dibaryon resonance. Interestingly,
the same main Beijing group arguing for a compact hexaquark structure of
the d$^{\ast}$(2380) has voiced recently~\cite{dong19b} reservations on
the ability of their underlying model to reach the observed level of the ELPH
measured cross section $\sigma(\gamma d\to d^{\ast}(2380)\to d\pi^0\pi^0)$
shown in Fig.~\ref{fig:fix}. The hexaquark cross section calculation
underestimates the BW contribution in the figure, $\approx$18~nb at
the nominal resonance energy, by about a factor of 20~\cite{dong19b}.
Another quark-based model suggestion was made recently by a faction of the
Beijing group~\cite{shi19}. These authors tried to fit the d$^{\ast}$(2380)
within a diquark ($\cal D$) model in terms of a bound system of three vector
diquarks. However, Gal and Karliner~\cite{gk19} noted that the $I=0,J^P=1^+$
deuteron-like and the $I=1,J^P=0^+$ virtual-like 3$\cal D$ states in the
particular diquark model considered are located about 200-250 MeV above the
physical deuteron, where no hint of irregularities in the corresponding $NN$
phase shifts analyses occur. In fact no resonance feature in the corresponding
partial-wave phase shifts up to at least $W=2.4$~GeV has ever been reliably
established~\cite{SAID}. Moreover, it was shown by these authors~\cite{gk19}
that if the d$^{\ast}$(2380) structure were dominated by a 3$\cal D$
component, its decay width would have been suppressed by at least an
isospin-color recoupling factor 1/9 with respect to the naive $\Delta\Delta$
hadronic estimate of 160~MeV width, bringing it cosiderably below the deduced
value of $\Gamma_{d^{\ast}(2380}\approx 70$~MeV.
We end with a brief discussion of possible 6q admixtures in the essentially
hadronic wavefunction of the d$^{\ast}$(2380) dibaryon resonance. For this
we refer to the recent 6q non-strange dibaryon variational calculation in
Ref.~\cite{PPL15} which depending on the assumed confinement potential
generates a $^3S_1$ 6q dibaryon about 550 to 700~MeV above the deuteron,
and a $^7S_3$ 6q dibaryon about 230 to 350~MeV above the d$^{\ast}$(2380).
Taking a typical 20~MeV potential matrix element from deuteron structure
calculations and 600~MeV for the energy separation between the deuteron and
the $^3S_1$ 6q dibaryon, one finds admixture amplitude of order 0.03 and
hence 6q admixture probability of order 0.001 which is compatible with that
discussed recently by Miller~\cite{miller14}. Using the same 20~MeV potential
matrix element for the $\Delta\Delta$ dibaryon candidate and 300~MeV for the
energy separation between the d$^{\ast}$(2380) and the $^7S_3$ 6q dibaryon,
one finds twice as large admixture amplitude and hence four times larger 6q
admixture probability in the d$^{\ast}$(2380), altogether smaller than 1\%.
These order-of-magnitude estimates demonstrate that long-range hadronic and
short-range quark degrees of freedom hardly mix also for $\Delta\Delta$
configurations, and that the d$^{\ast}$(2380) is extremely far from a pure 6q
configuration. This conclusion is at odds with the conjecture made recently
by Bashkanov, Brodsky and Clement~\cite{BBC13} that 6q CC components dominate
the wavefunctions of the $\Delta\Delta$ dibaryon candidates ${\cal D}_{03}$,
identified with the observed d$^{\ast}$(2380), and ${\cal D}_{30}$.
Unfortunately, most of the quark-based calculations discussed in the present
work combine quark-model input with hadronic-exchange model input in a loose
way~\cite{lu17} which discards their predictive power.
| 9,609 |
\section{INTRODUCTION}
Optical clocks, with natural linewidth at the millihertz level, have demonstrated great
improvements in stability and accuracy over the microwave frequency standards.
The research on optical atomic clocks have achieved remarkable progress in the past
several years especially in single-ion optical clocks.
All-optical atomic clock referenced to the 1.064 petahertz transition of a single trapped $^{199}$Hg$^{+}$
ion has been realized in 2001~\cite{Diddams}. Two Al$^{+}$ ion optical clocks, operated at
$^{1}$S$_{0}$ to $^{3}$P$_{0}$ clock transition with frequency near 1.121 petahertz and narrow natural linewidth of 8 mHz,
have been constructed with fractional frequency inaccuracy of the order of magnitude $10^{-18}$~\cite{Chou,Rosenband}.
Although these single-ion optical clocks have reached unprecedented stability and accuracy, there are still some
problems to be solved. The observed linewidth of the clock transition is limited by the linewidth of the
probe laser~\cite{Chou2,Jiang,Katori,Swallows}. Thus, narrow linewidth laser light source becomes the
key factor of the performance of single-ion optical clocks.
Since the proposal of active optical clock~\cite{Chen1,Chen2},
a number of neutral atoms with two-level, three-level and four-level at thermal,
laser cooling and trapping configurations have been investigated
recently~\cite{Chen1, Chen2, Zhuang1, Zhuang2, Zhuang3, Yu, Chen3, Meiser1, Meiser2, Xie, Zhuang4, Zhuang5}.
The potential quantum-limited linewidth of active optical clock is narrower
than mHz, and it is possible to reach this unprecedented linewidth since the thermal
noise of cavity mode can be reduced dramatically with the mechanism of
active optical clock~\cite{Chen1, Chen2, Zhuang1, Zhuang2, Zhuang3, Yu, Chen3, Meiser1, Meiser2, Xie, Zhuang4, Zhuang5}.
Therefore, a laser light source based on active ion optical clock
will be favorable to single-ion optical clocks.
One candidate is $^{171}$Yb$^{+}$ ion. With about $10^6$ $^{171}$Yb$^{+}$ ions in a Paul Trap~\cite{Casdorff},
the population inversion can be realized as shown in Fig.~1. The cooling laser and repumping laser at
$^{2}$S$_{1/2}(F=1)$ to $^{2}$P$_{1/2}(F=0)$ and $^{2}$S$_{1/2}(F=0)$ to $^{2}$P$_{1/2}(F=1)$
transitions are used as pumping laser. First, during the cycling $F=1$ to $F=0$ transition, there
is a branching probability of $6.6\times10^{-3}$ for ions decaying to $^{2}$D$_{3/2}(F=1)$
sublevel~\cite{Tamm1}. Second, when the ions that leaked to $^{2}$S$_{1/2}(F=0)$ are repumped to
$^{2}$P$_{1/2}(F=1)$, ions will decay to $^{2}$D$_{3/2}(F=1, 2)$ sublevels. Then the population
inversion is established between $^{2}$D$_{3/2}(F=1)$ and $^{2}$S$_{1/2}(F=0)$ sublevels. The ions at
$^{2}$D$_{3/2}(F=2)$ can be repumped with a 935 nm laser at transition from
$^{2}$D$_{3/2}(F=2)$ to $^{3}$D[3/2]$_{1/2}(F=1)$ and then decay to $^{2}$S$_{1/2}(F=1)$
by spontaneous transition very quickly. So this channel can be ignored when calculating the energy level
population. Small leakage channels to other
sublevels can be closed with additional repumping lasers if necessary.
\begin{figure}[!h]
\includegraphics[width= 6cm]{Fig1.eps}
\caption{(Color online) Scheme of $^{171}$Yb$^{+}$ ion laser
operating at 435.5 nm clock transition, pumped with 369.5 nm and
935 nm transitions.} \label{fig:Fig1}
\end{figure}
The calculations of dynamical process of
population inversion of $^{171}$Yb$^{+}$ ions in a Paul Trap
and lasing process with a Fabry-Perot type resonator are presented in detail.
We calculated the quantum-limited linewidth of active $^{171}$Yb$^{+}$ ion optical clock and got it should be narrower than 1 mHz.
Other proper candidate ions for active optical clock and ion laser are also discussed and compared with
$^{171}$Yb$^{+}$ in this paper.
\section{POPULATION INVERSION OF IONS IN A PAUL TRAP AND LASING OF YB IONS ACTIVE OPTICAL FREQUENCY STANDARD}
Considering the spontaneous decay rate of $^{2}$P$_{1/2}$ is $2\pi\times23$ MHz, it is reasonable to set
the pumping rate~\cite{Tamm1} from $^{2}$S$_{1/2}(F=1)$ to $^{2}$P$_{1/2}(F=0)$ $\Omega_{23}=10^{7} s^{-1}$.
So the decay rate to $^{2}$D$_{3/2}(F=1)$ sublevel is $6.6\times10^{4}s^{-1}$. Then we can set the repumping rate of
$^{2}$S$_{1/2}(F=0)$ to $^{2}$P$_{1/2}(F=1)$ transition $\Omega_{14}$ to be of the order of magnitude $10^{5}s^{-1}$
and the repumping rate of $^{2}$D$_{3/2}(F=2)$ to $^{3}$D[3/2]$_{1/2}(F=1)$ sublevel to be $10^{3}s^{-1}$.
The effective power is reduced to one third if only
the 0-0 clock transition is effective. In order to depopulate ions at $^{2}$D$_{3/2}(F=1)$
($m_{F}=1,-1$) Zeeman sublevels, each transition from these two Zeeman sublevels
$^{2}$D$_{3/2}(F=1)$ ($m_{F}=1,-1$) to $^{2}$S$_{1/2}(F=0)$ is also coupled with one mode of ion laser
cavity.
Based on the assumption that the ions have been trapped in a Paul Trap, and the theories of the
interaction between the ions and light, the density matrix equations for $^{171}$Yb$^{+}$
ions interacted with the cooling light and the repumping light can be written in RWA-approximation as follows:
\begin{equation}
\begin{aligned}
\nonumber\frac{d\rho_{11}}{dt}&=-\Omega_{14}\rho^{I}_{14}+\Gamma_{41}\rho_{44}+\Gamma_{51}\rho_{55}-Kn\rho_{11}+Kn\rho_{55}\\
\nonumber\frac{d\rho_{22}}{dt}&=-\Omega_{23}\rho^{I}_{23}+\Gamma_{32}\rho_{33}+\Gamma_{42}\rho_{44}+\Gamma_{52}\rho_{55}\\
\nonumber\frac{d\rho_{33}}{dt}&=\Omega_{23}\rho^{I}_{23}-(\Gamma_{32}+\Gamma_{35})\rho_{33}\\
\nonumber\frac{d\rho_{44}}{dt}&=\Omega_{14}\rho^{I}_{14}-(\Gamma_{41}+\Gamma_{42}+\Gamma_{45})\rho_{44}\\
\nonumber\frac{d\rho_{55}}{dt}&=\Gamma_{45}\rho_{44}+\Gamma_{35}\rho_{33}-(\Gamma_{51}+\Gamma_{52})\rho_{55}+Kn\rho_{11}-Kn\rho_{55}\\
\nonumber\frac{d\rho^{I}_{14}}{dt}&=\frac{1}{2}\Omega_{14}(\rho_{11}-\rho_{44})+\rho^{R}_{14}\Delta_{1}-\frac{1}{2}(\Gamma_{41}+\Gamma_{42}+\Gamma_{45})\rho^{I}_{14}\\
\nonumber\frac{d\rho^{R}_{14}}{dt}&=-\rho^{I}_{14}\Delta_{1}-\frac{1}{2}(\Gamma_{41}+\Gamma_{42}+\Gamma_{45})\rho^{R}_{14}\\
\nonumber\frac{d\rho^{I}_{23}}{dt}&=\frac{1}{2}\Omega_{23}(\rho_{22}-\rho_{33})+\rho^{R}_{23}\Delta_{2}-\frac{1}{2}(\Gamma_{32}+\Gamma_{35})\rho^{I}_{23}\\
\nonumber\frac{d\rho^{R}_{23}}{dt}&=-\rho^{I}_{23}\Delta_{2}-\frac{1}{2}(\Gamma_{32}+\Gamma_{35})\rho^{R}_{23}\\
\nonumber\frac{dn}{dt}&=Kn(\rho_{55}-\rho_{11})-\Gamma_{c}n
\end{aligned}
\end{equation}
The subscript numbers of the density matrix correspond to different
energy levels as shown in Fig.~1. The diagonal elements mean the
number of ions in corresponding states and off-diagonal elements
mean coherence between two states. The Rabi frequency is
$\Omega_{23}=d_{23}\varepsilon_{1}/\hbar$, where $d_{23}$ is the
electric dipole matrix between the $^{2}$S$_{1/2}(F=1)$ state and
$^{2}$P$_{1/2}(F=0)$ state, $\varepsilon_{1}$ is the electric
strength of the cooling light.
$\Omega_{14}=d_{14}\varepsilon_{2}/\hbar$, with the electric dipole
matrix $d_{14}$ between the $^{2}$S$_{1/2}(F=0)$ state and
$^{2}$P$_{1/2}(F=1)$ state and the electric strength
$\varepsilon_{2}$ of the repumping light.
$\Delta_{1}(=\omega_{23}-\omega_{1})$ and
$\Delta_{2}(=\omega_{14}-\omega_{2})$ are frequency detunings of
cooling and repumping light on the transition frequencies. They are
both set to be 0. $\Gamma_{32}$, $\Gamma_{35}$, $\Gamma_{41}$,
$\Gamma_{42}$, $\Gamma_{45}$, $\Gamma_{51}$ and $\Gamma_{52}$ are
decaying rates related to the lifetimes of the corresponding energy
levels described in Fig.~1. $\Gamma_{c}$ is the cavity loss rate and
$n$ is the photon number if there exists a cavity. $K\sim
g^{2}t_{int}$ is the laser emission coefficient~\cite{An} where $g$
is ion-cavity coupling constant and $t_{int}$ is the interaction
time. Without the cavity, $\Gamma_{c}$ and $K$ are both 0. The
effective decay rate of the channel, from 6p $^{2}$P$_{1/2}$ to 5d
$^{2}$D$_{3/2}(F=0)$ then repumped with 935 nm laser to
$^{3}$D[3/2]$_{1/2}(F=1)$ and decay to $^{2}$S$_{1/2}$ by
spontaneous transition, is ignored in calculations since it is much
smaller than $\Gamma_{32}$.
For the convenience of reading, here we summarize all of the relevant parameters mentioned above
in Table~1 as follows.
\begin{center}
\textbf{Table 1. Parameters related to\\active ion optical clock}
\end{center}
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|c|c|c|}
\hline
parameters & value & parameters & value\\
\hline
$\Gamma_{32}$ & $2\pi\times23$ MHz & $\Omega_{23}$ & $1\times10^{7}s^{-1}$\\
\hline
$\Gamma_{35}$ & $2\pi\times152$ kHz & $\Omega_{14}^{*}$ & $3.35\times10^{5}s^{-1}$\\
\hline
$\Gamma_{41}$ & $2\pi\times23$ MHz & $\Gamma_{c}^{**}$ & 500 kHz\\
\hline
$\Gamma_{42}$ & $2\pi\times23$ MHz & $K^{**}$ & 10$s^{-1}$\\
\hline
$\Gamma_{45}$ & $2\pi\times152$ kHz & $\Delta_{1}$ & 0\\
\hline
$\Gamma_{51}$ & $19$ Hz & $\Delta_{2}$ & 0\\
\hline
$\Gamma_{52}$ & $19$ Hz & & \\
\hline
\end{tabular}
\end{center}
\ \ \ \ \ \ $*$ Another value for discussion is $4.75\times10^{5}s^{-1}$.
\ \ $**$ Without the cavity, the value is 0.
\begin{figure}[!h]
\includegraphics[width=7.5cm]{Fig2.eps}
\caption{(Color online) The dynamical populations with the Rabi
frequency $\Omega_{23}=10^{7} s^{-1}$ and
$\Omega_{14}=3.35\times10^{5} s^{-1}$.} \label{fig:Fig2}
\end{figure}
Without the cavity, the numerical solution results for the above equations are shown in Fig.~2. Assuming there are $10^{6}$ ions
in the Paul Trap and a quarter of them are in the $^{2}$S$_{1/2}(F=0)$ state at the beginning, the population inversion between
$^{2}$D$_{3/2}(F=1)$ and $^{2}$S$_{1/2}(F=0)$ states is built up at the time scale of $10^{-3}$s. From Fig.~2,
it is obvious that under the action of cooling laser, the number of ions in the $^{2}$S$_{1/2}(F=1)$ state decrease rapidly. The lifetime
of $^{2}$P$_{1/2}(F=0)$ and $^{2}$P$_{1/2}(F=1)$ states are much shorter than that of $^{2}$D$_{3/2}(F=1)$ state,
so ions accumulate in $^{2}$D$_{3/2}(F=1)$ state and the population inversion is built up.
If we keep other conditions unchanged, the number of ions at the steady-state in $^{2}$D$_{3/2}(F=1)$ sublevel is 10 times as
much as that in $^{2}$S$_{1/2}(F=0)$ sublevel for $\Omega_{14}=3.35\times10^{5}s^{-1}$, and 20 times for
$\Omega_{14}=4.75\times10^{5}s^{-1}$.
An optical resonant bad cavity, whose linewidth of cavity mode is much wider than the linewidth of gain,
could be applied to the lasing transition between the inverted states
$^{2}$D$_{3/2}(F=1)$ and $^{2}$S$_{1/2}(F=0)$, as the population inversion occurs. The oscillating process
could start up once the optical gain exceeds the loss rate. The corresponding density matrix
equations together with the equations of emitted photons($n$) from the total ions($N$)
inside the cavity can be written as above. We use a Fabry-Perot resonator with mode volume about
$3.3\times10^{-7} m^{3}$ and the cavity loss rate $\Gamma_{c}=500$ kHz. The laser emission coefficient is $K=10 s^{-1}$.
Fig.~3 describes the solution of the photon number equation. From Fig.~3 we can conclude that
a stable laser field has been built up within laser cavity on condition that the population inversion is preserved.
The steady-state value of photon number $n$ is 160 and
the power of this $^{171}$Yb$^{+}$ ion laser lasing from $^{2}$D$_{3/2}(F=1)$ to
$^{2}$S$_{1/2}(F=0)$ is about 37pW.
\begin{figure}[!h]
\includegraphics[width=7.5cm]{Fig3.eps}
\caption{The average photon number inside the cavity for
$\Omega_{14}=3.35\times10^{5}s^{-1}$. There are $10^{6}$ ions in the
cavity. The steady-state value of photon number $n$ is 160.}
\label{fig:Fig3}
\end{figure}
The population inversion depends on the Rabi frequency $\Omega_{14}$ and $\Omega_{23}$.
Here we consider the effect of $\Omega_{14}$. The enhancement of $\Omega_{14}$ will increase the population inversion, thus
increasing the probability of lasing transition between the inverted states.
The steady-state value of photon number for $\Omega_{14}=4.75\times10^{5}s^{-1}$
is larger than that of $\Omega_{14}=3.35\times10^{5}s^{-1}$ in the case of the same total ions number(see Fig.~4).
Given $\Omega_{14}=4.75\times10^{5}s^{-1}$, the largest value of $\Omega_{14}$
we can get under general experimental condition, the steady-state value of photon number $n$ is 336.
The power of this $^{171}$Yb$^{+}$ ion laser is about 77 pW.
\begin{figure}[!h]
\includegraphics[width=7.5cm]{Fig4.eps}
\caption{(Color online) The steady-state value of photon number $n$
varies with the total ions number $N$ in the cavity. The red line is
for $\Omega_{14}=3.35\times10^{5}s^{-1}$, the blue line is for
$\Omega_{14}=4.75\times10^{5}s^{-1}$.} \label{fig:Fig4}
\end{figure}
As the number of ions in the cavity increases, the steady-state value of photon number is expected to rise accordingly.
The steady-state value of photon number $n$ varying with the total ions number $N$ in the cavity
is also shown in Fig.~4. Considering the actual experimental condition,
we set the largest number of total ions in the cavity to be $10^{6}$.
Compared with the traditional laser, the primary
three conditions are fulfilled for the active optical frequency standards based on
Paul Trap trapped ions just proposed in this paper. The very difference is that
the lifetime of the lasing upper energy level is so long that the natural linewidth
of the laser field is much narrower than the cavity mode linewidth, which is
500 kHz. Besides, the temperature of cold ions trapped in Paul Trap decreases the Doppler
broadening linewidth. Therefore, active ion optical clock operating at the condition of bad cavity
will be a very stable narrow-linewidth laser source since the cavity pulling effect will dramatically
reduce the cavity length noise due to Johnson thermal effect.
It is suitable for single-ion optical frequency standards due to its higher stability.
\section{LINEWIDTH OF ACTIVE YB ION LASER}
Although conventional single-ion optical clocks have reached unprecedented stability and accuracy, the observed linewidth
of the clock transition is limited by the linewidth of the
probe laser~\cite{Chou2,Jiang,Katori,Swallows}. Active ion optical clock, as which we presented in this paper, can offer a new
laser light source for single-ion optical clocks because of its narrow linewidth and high stability.
After laser cooling, the $\gamma=3.1$ Hz natural linewidth clock transition between $^{2}$S$_{1/2}(F=0)$ and
$^{2}$D$_{3/2}(F=1)$ states has been measured with a Fourier-limited linewidth of 30 Hz recently~\cite{Tamm2}. Based on the
mechanism of active optical clock~\cite{Chen1}, to reduce the effect caused by the thermal noise of cavity, the ion laser
cavity should be bad cavity, which means the linewidth of the cavity (or the cavity loss rate) should be much wider than
the linewidth of the gain (the broaden ion transition linewidth after considering the Doppler broadening, collision
broadening and light shift \emph{etc.}). Given the linewidth of ion gain to be 30 Hz as measured, one can set the cavity mode linewidth as
$\kappa=500$ kHz. Then the cavity related noise is reduced by 4 orders of magnitude to below 1 mHz with the best cavity design.
The ion-cavity coupling constant ~\cite{Chen1} can be set around $g=20$ Hz. The quantum-limited linewidth of active ion optical clock~\cite{Yu}
$\gamma_{ionlaser}=g^{2}/\kappa$ is narrower than 1 mHz.
There are still some effects that will shift and broaden the linewidth of active ion optical clock,
like light shift caused by the repumping laser. For $^{171}$Yb$^{+}$ active ion optical clock, the largest light
shift effect is caused by the repumping laser of $^{2}$S$_{1/2}(F=0)$ to $^{2}$P$_{1/2}(F=1)$ transition. By adjusting
this repuming laser at red detuning to $^{2}$P$_{1/2}(F=1)$, its light shift can be greatly reduced by suitable blue detuning to
$^{2}$P$_{1/2}(F=0)$ transition. The repumping laser can be stabilized to an uncertainty of $\Delta=100$ Hz.
Considering that the spontaneous decay rate of $^{2}$P$_{1/2}$ is $2\pi\times23$ MHz, far greater than 100 Hz,
we can write the light shift as $\Delta\nu=2\Omega^{2}_{14}\Delta/\pi\Gamma^{2}_{41}$.
Given the repumping rate $\Omega_{14}=4.75\times10^{5}s^{-1}$, the value
of light shift is 0.69 mHz.
\section{DISCUSSION AND CONCLUSION}
Other ions such as Ba$^{+}$, Ca$^{+}$ and Sr$^{+}$ are also potential candidates for active optical clock and narrow
linewidth ion laser with the same mechanism.
Compared with most clock transitions of traditional passive ion optical frequency standards, which are between
$^{2}$S$_{1/2}$ and $^{2}$D$_{5/2}$ states like Ba$^{+}$, Ca$^{+}$ and Sr$^{+}$,
we recommend the lasing from $^{2}$D$_{3/2}$ to $^{2}$S$_{1/2}$ states for active optical clock and
narrow linewidth ion laser. In this case, here taking the even isotopic $^{88}$Sr$^{+}$ as an example, it can
be a natural three-level laser configuration with the cooling laser used as repumping laser at the same time,
thus simpler than traditional passive ion optical clock. During the 422 nm cooling procedure, ions spontaneously
decay to $^{2}$D$_{3/2}$ state and accumulate at this state with 0.4s lifetime~\cite{Biemont}.
Then, coupled with a bad cavity via stimulated emission, the ions transit from $^{2}$D$_{3/2}$ to $^{2}$S$_{1/2}$ state in the
configuration of three-level ion laser. With a magnetic field along the cavity axis, the laser light will be circularly polarized.
In summary, we propose a scheme of active ion optical clock, i.e.
narrow linewidth bad cavity ion laser with
detailed pumping method, lasing states, output power, linewidth and light shift reduction.
We especially study the $^{171}$Yb$^{+}$ ions in a Paul
Trap and propose to utilize a Fabry-Perot resonator to realize lasing of active optical
frequency standards. The population inversion between
$^{2}$D$_{3/2}(F=1)$ and $^{2}$S$_{1/2}(F=0)$ states can be built up at the time scale of $10^{-3}$s.
The steady-state value of photon number $n$ in the Fabry-Perot cavity increases with $\Omega_{14}$ and the total ions number $N$.
If $\Omega_{14}=4.75\times10^{5}s^{-1}$ and $N=10^{6}$, the steady-state value of photon number is $n=336$
and the laser power is 77 pW. The quantum-limited linewidth of active $^{171}$Yb$^{+}$ ion optical clock
is narrower than 1 mHz. Other ions like Ba$^{+}$,
Ca$^{+}$ and Sr$^{+}$ are suitable for active optical clock. Active optical clock and narrow linewidth
ion laser may also be realized with these candidate ions.
\section{ACKNOWLEDGMENT}
This work is supported by the National Natural Science Foundation of China
(Grant No. 10874009).
| 6,877 |
\section{Code Listings}
\input{appendix.tex}
\end{document}
\section{Conclusion}
We presented a \name{GrGen} solution for both tasks, constant folding as well as instruction selection, covering all test cases.
The solution employing pure graph rewriting is pretty concise for the constant folding task but admittedly less so for the instruction selection task,
which requires a good deal of (very simple) graph relabeling rules, one rule per operation plus a further rule for the operations with immediate.
The conciseness could be improved by stepping an abstraction level up by implementing a generator emitting GrGen code, which is what we did in~\cite{Buchwald2010} and~\cite{SG:07} and what we recommend for the classes of applications which would lead to repetitive code.
\begin{figure}
\centering
\includegraphics[scale=0.38]{OptimizationSituation.png}
\caption{A situation from constant folding}
\label{fig:OptimizationSituation}
\end{figure}
The debugger included in the GrShell, which allows to execute the sequences step-by-step under user control, was a great help during development of the optimizations.
Especially its graph visualizations, which are depicted in \autoref{fig:OptimizationSituation} and in the case description \cite{compileroptimizationcase}.
Due to the wavefront algorithm only visiting relevant operations and the core speed of the generated code we were able to give a solution with excellent performance.
For the largest GReTL test case -- consisting of 27993 nodes and 55981 edges -- the execution speed is 6.3 seconds for constant folding and 111 milliseconds for instruction selection.
For the largest test shipped with this case -- consisting of 277529 nodes and 824154 edges -- we need 11 seconds for constant folding and 1.2 seconds for instruction selection.
This is more than one order of magnitude better than the next-closest competitor.
The measurements were made using Ubuntu 11.04 and Mono 2.6.7 on a Core2Duo E6550 with 2.33 GHz.
The numbers are even better for the SHARE machine(s) hosting the GrGen image \cite{share} at the time of writing.
Our solution does not apply rules in parallel.
However, we consider it as a benchmark for parallel solutions.
Unfortunately, nobody submitted a parallel solution, which was the main purpose of publishing the challenge: we were hoping for hints about the potential benefits of parallelization.
So the question which speed up can be gained by parallel rule application is still open.
\section{What is GrGen.NET?}
\name{GrGen.NET}\ is an application domain neutral graph rewrite system, which started as a helper tool for compiler construction.
It is well suited for solving this challenge due to its origin, thus our solution can be seen as a reference, even though the origins were left behind a while ago.
The feature highlights regarding practical relevance are:
\begin{description}
\item[Fully Featured Meta Model:] \name{GrGen.NET}\ uses attributed and typed multigraphs with multiple inheritance on node/edge types. Attributes may be typed with one of several basic types, user defined enums, or generic set, map, and array types.
\item[Expressive Rules, Fast Execution:] The expressive and easy to learn rule specification language allows straightforward formulation of even complex problems, with an optimized implementation yielding high execution speed at modest memory consumption.
\item[Programmed Rule Application:] \name{GrGen.NET}\ supports a high-level rule application control language, Graph Rewrite Sequences (GRS), offering sequential, logical, iterative and recursive control plus variables and storages for the communication of processing locations between rules.
\item[Graphical Debugging:] \name{GrShell}, \name{GrGen.NET}'s command line shell, offers interactive execution of rules, visualising together with yComp the current graph and the rewrite process. This way you can see what the graph looks like at a given step of a complex transformation and develop the next step accordingly. Or you can debug your rules and sequences.
\item[Extensive User Manual:] The \name{GrGen.NET} User Manual \cite{GrGenUserManual} guides you through the various features of \name{GrGen.NET}, including a step-by-step example for a quick start.
\end{description}
\section{Instruction Selection}
The instruction selection task transforms the intermediate representation (IR) into a target-dependent representation (TR).
It can be considered as some kind of model transformation.
The TR supports immediate instructions, i.e.\ a \texttt{Const} operand can be encoded within the instruction.
Our solution consists of one rule per target instruction which means we have at most two rules for each IR operation: one for the immediate variant and one for the variant without immediate.
Before we start matching all immediate operations, we apply the auxiliary rule shown in \autoref{fig:normalizeConst.grg}.
It matches commutative operations with a \texttt{Const} operand at \texttt{position} $0$, but not at \texttt{position} $1$, and then exchanges these operands to ensure that the constant operand is at \texttt{position} $1$.
Thus, the rule ensures deterministic behaviour in case of two constant operands.
\autoref{fig:iselAddI.grg} shows exemplarily the rule that creates an \texttt{AddI} operation.
It retypes the \texttt{Add} to an \texttt{TargetAddI}, deletes the \texttt{Dataflow} edge to the constant and stores the value of the \texttt{Const} node in the \texttt{value} attribute.
Since the other rules for immediate operations are very similar, we omit them here.
Instead we want to introduce the rule for creating \texttt{TargetLoad}s given in \autoref{fig:iselLoad.grg}.
Again, we retype the original operation to the corresponding target operation.
Since \texttt{Load} is a memory operation, we also need to set the \texttt{volatile} attribute.
The instruction selection rules are applied by the sequence shown in \autoref{fig:isel.grs}.
The \texttt{[rule]} operator matches and rewrites all occurrences of the given \texttt{rule}.
Hence, we first create all immediate operations, and then the non-immediate operations for the remaining nodes.
\section{Constant Folding}
The first task is to perform constant folding.
Constant folding transforms operations which have only \texttt{Const} operands into a \texttt{Const} itself, e.g.\ to transform $1 + 2$ into $3$.
This is a local optimization requiring only rules referencing local graph context.
\subsection{Driver and Data Flow}
Constant folding is carried out from the driver sequence given in \autoref{fig:driversequence.grsi},
which performs inside a main loop in each step
i) first constant folding along data flow with a wavefront algorithm,
and then ii) control flow folding together with clean-up tasks.
Graph rewrite sequences are the rule application control language of \name{GrGen}.
The most fundamental operation is a rule application denoted by the rule name, with parameters given in parenthesis, and optionally an assignment of output values to variables (given in parenthesis, too).
By using all bracketing \verb#[r]# we execute the rule \texttt{r} for all matches in the host graph.
The then-right operator \texttt{;>} executes the left sequence and then the right sequence, returning as result of execution the result of the execution of the right sequence.
The potential results of sequence execution are \emph{success} equaling \texttt{true} and \emph{failure} equaling \texttt{false}; a rule which matches (at least once) counts as success.
With the postfix star \texttt{*} we iterate the preceding sequence as long as it succeeds.
The result of a star iteration is always success, in contrast to the plus \texttt{+} postfix which requires the preceding sequence to match at least once in order to succeed; so a sequence of rules with plus postfixes linked by strict disjunction operators \verb#|# succeeds (\texttt{true}) if one of the rules matches.
Conjunction \verb#&# is available as well, so are the lazy versions \verb#&&# and \verb#||# of the operators not executing the right sequence in case the result of the left sequence already determines the outcome.
The prefix exclamation mark operator negates the result of sequence execution, in \autoref{fig:driversequence.grsi} we negate the
value from variable \texttt{isEmpty}.
The backslashes \verb#\# allow to concatenate multiple lines into one.
\begin{figure}[h]
\lstinputlisting[language=xgrs]{DriverSequence.grsi}
\caption{Driver sequence for constant folding.}
\label{fig:driversequence.grsi}
\end{figure}
\noindent In the driver sequence two empty storage sets are created initially for the wavefront algorithm: \texttt{now} which will contain the users of constant nodes which are visited in this step and \texttt{next} which will contain the users of constant nodes which will be visited in the next step.
Then the main loop is entered.
There the rule \texttt{collectConstUsers} is executed on all available matches, collecting all users of \texttt{Const} nodes available in the graph into the storage set \texttt{now} (the \texttt{dummy} variable was never assigned and the \texttt{collectConstUsers} rule was specified to search for a valid entity if the first parameter is undefined).
This is the initial stone thrown into the water of our program graph.
From it on a wavefront is iterated following the data flow edges until it comes to a halt because no new constants were created anymore.
A wavefront step is encapsulated in the subsequence \texttt{wavefront} given in \autoref{fig:wavefront.grsi};
it visits all operations of its first argument,
and adds the operations which are users of the constants it created newly with constant folding into the set given as its second argument.
After this subsequence was called on the \texttt{now} argument the \texttt{now} and \texttt{next} sets are swapped (the set referenced by \texttt{now} is cleared, \texttt{now} is assigned the set referenced by the \texttt{next}, and next is assigned the empty set previously pointed to by \texttt{now}).
The wavefront iteration is stopped when the \texttt{now} set of the next iteration step is empty.
\begin{figure}[ht]
\lstinputlisting[language=xgrs]{Wavefront.grsi}
\caption{Wavefront step for constant folding.}
\label{fig:wavefront.grsi}
\end{figure}
\noindent A wavefront step defined by the subsequence given in \autoref{fig:wavefront.grsi}
iterates with a \texttt{for} loop over the users of constants contained in the storage set parameter \texttt{constUsersNow},
binding them to an iteration variable \texttt{cu} of type \texttt{FirmNode}.
This constant is used as input to the rules \texttt{foldNot}, \texttt{foldBinary} and \texttt{foldPhi} really doing the folding in case all of the arguments to the operations they handle are constant.
They return the newly created constant by folding, the rule \texttt{collectConstUsers} then adds all users of this constant to the \texttt{constUsersNext} storage set.
Before we take a closer look at the rules, let us start with a short introduction into the syntax of the rule language:
Rules in GrGen consist of a pattern part specifying the graph pattern to match and a nested rewrite part specifying the changes to be made.
The pattern part is built up of node and edge declarations or references with an intuitive syntax:
Nodes are declared by \texttt{n:t}, where \texttt{n} is an optional node identifier, and \texttt{t} its type.
An edge \texttt{e} with source \texttt{x} and target \texttt{y} is declared by \texttt{x -e:t-> y}, whereas \texttt{-->} introduces an anonymous edge of type \texttt{Edge}.
Nodes and edges are referenced outside their declaration by \texttt{n} and \texttt{-e->}, respectively.
Attribute conditions can be given within \texttt{if}-clauses.
The rewrite part is specified by a \texttt{replace} or \texttt{modify} block nested within the rule.
With \texttt{replace}-mode, graph elements which are referenced within the replace-block are kept, graph elements declared in the replace-block are created, and graph elements declared in the pattern, not referenced in the replace-part are deleted.
With \texttt{modify}-mode, all graph elements are kept, unless they are specified to be deleted within a \texttt{delete()}-statement.
Attribute recalculations can be given within an \texttt{eval}-statement.
In addition to rewriting, \name{GrGen} supports relabeling, we prefer to call it retyping though.
Retyping is specified with the syntax \texttt{y:t<x>}: this defines \texttt{y} to be a retyped version of the original node \texttt{x}, retyped to the new type \texttt{t}; for edges the syntax is \texttt{-y:t<x>->}.
These and the language elements we introduce later on are described in more detail in our solution of the Hello World case \cite{helloworldsolutiongrgennet},
and especially in the extensive GrGen.NET user manual \cite{GrGenUserManual}.
\autoref{fig:foldBinary.grg} shows the constant folding rule for \texttt{Binary} operations.
It takes the node with the \texttt{Const} operand as parameter and returns the folded \texttt{Const}.
The rule matches both \texttt{Const} operands and has an \texttt{alternative} statement that contains one case for each binary operation.
Within the cases the value of the new \texttt{Const} is computed.
In almost all cases the computation is as simple as in the \texttt{foldAdd} case.
The only exception is the \texttt{foldCmp} case that also needs to consider the \texttt{relation} of the \texttt{Cmp} to compute the resulting value of the \texttt{Const}.
Finally, at the end of the rule execution, the \texttt{exec} statement relinks the users of the \texttt{Binary} to the newly created \texttt{Const} and deletes the \texttt{Binary} from the graph (by \texttt{exec}uting the given sequence).
The constant folding for unary \texttt{Not} nodes is straight forward and not further discussed;
$n$-ary \texttt{Phi} nodes are of interest again.
\autoref{fig:foldPhi.grg} shows the corresponding rule.
It first matches the \texttt{Phi} node and one \texttt{Const} operand and then iteratively edges to the same operand and to the \texttt{Phi} itself.
If this covers all edges we can replace the \texttt{Phi} with the \texttt{Const} operand.
\subsection{Control Flow and Cleanup}
Let us mentally return to the driver sequence in \autoref{fig:driversequence.grsi};
after the wavefront which folded alongside data flow has collapsed,
execution continues with folding condition nodes at the interface of data flow to control flow
and with folding control flow proper (jumps and blocks).
In addition there are some clean up task left to be executed.
If no control flow was folded leading to further data flow folding possibilities, the main loop is left.
The rule shown in \autoref{fig:foldCond.grg} is responsible for folding \texttt{Cond}itional jumps.
Depending on the value of the constant,
either the edge of type \texttt{True} gets deleted and the edge of type \texttt{False} retyped to an edge of type \texttt{Controlflow}, or the edge of type \texttt{False} gets deleted and the edge of type \texttt{True} retyped to an edge of type \texttt{Controlflow}.
Since there is only one jump target left, we also retype the conditional jump to a simple jump of type \texttt{Jmp}.
Due to the folding of condition jumps, there may be unreachable blocks which can be deleted.
We use two rules to remove unreachable code: the rule shown in \autoref{fig:removeUnreachableBlock.grg} deletes unreachable \texttt{Block}s, the rule shown in \autoref{fig:removeUnreachableNode.grg} deletes nodes without a \texttt{Block}.
Furthermore, we also need to adapt \texttt{Phi} nodes if they have an operand without a \texttt{Controlflow} counterpart in the \texttt{Block}.
\autoref{fig:removeUnreachablePhiOperand.grg} shows the corresponding rule that matches a \texttt{Phi} and deletes an operand that has no \texttt{Controlflow} counterpart.
After the deletion we fix the position of all edges using the \texttt{fixEdgePosition} rule.
The solution contains two rules \texttt{removeUnusedNode} and \texttt{mergeBlocks} that simplify the graph.
The first rule removes a node that is not used, i.e.\ a node which has no incoming edges.
This rule is similar to the \texttt{removeUnreachableBlock} rule.
If there is a \texttt{Block} \texttt{block1} with only one outgoing edge of type \texttt{Controlflow} and this edge leads to a node of type \texttt{Jmp} contained in \texttt{Block} \texttt{block2}, then we remove the \texttt{Jmp} and merge \texttt{block1} and \texttt{block2} with the second rule.
This rule is able to remove the chain of \enquote{empty} \texttt{Block}s that occurs in the running example of the case description.
The verifier mentioned in the case descpription was implemented with the tests in \texttt{Verifier.gri}, called from a subsequence \texttt{verify} defined in \texttt{Verifier.grsi}; the subsequence is used with the statement \texttt{validate xgrs verify} before and after the transformations checking the integrity of the graph.
\subsection{Folding More Constants}
\autoref{fig:foldAssociativeAndCommutative.grg} shows a rule that folds constants for the term $(x\ast c1) \ast c2$ where $\ast$ is a associative and commutative operation.
This rule enables us to solve the test cases provided by the GReTL solution.
In contrast to the GReTL rule, this rule does not change the structure of the graph.
| 4,467 |
\section{Introduction}
We live in the age of social computing. Social networks are everywhere, exponentially increasing in volume, and changing everything about our lives, the way we do business, and how we understand ourselves and the world around us. The challenges and opportunities residing in the social oriented ecosystem have overtaken the scientific, financial, and popular discourse.
\remove{
In recent years the social sciences have been undergoing a digital revolution, heralded by the emerging field of ``computational social science''. Lazer, Pentland et. al \cite{CSS-Lazer-Science-2009} describe the potential of computational social science to increase our knowledge of individuals, groups, and societies, with an unprecedented breadth, depth, and scale. Computational social science combines the leading techniques from network science \cite{CSS-BarabasiAlbert-Science-1999,CSS-Watts-Nature-1998,CSS-Newman-SIAM-2003} with new machine learning and pattern recognition tools specialized for the understanding of people's behavior and social interactions \cite{RealityMining}.
Marketing campaigns are essential facility in many areas of our lives, and specifically in the virtual medium. One of the main thrusts that propels the constant expansions and enhancement of social network based services is its immense impact on the ``real world'' in a variety of fields such as politics, traditional industry, currency and stock trading and more.
As a result, a constantly growing portion of commercial and government marketing budgets is being allocated to advertizement in social platforms the main goal of which is to spark viral phenomena that by spreading through the social networks would result in a global ``trend''.
Many large-scale networks are analyzed, and this field is becoming increasingly popular \cite{Eagle21052010,Leskovec:2007:DVM:1232722.1232727}, due to the possibility of increasing the impact of campaigns by using network related information in order to optimize the allocation of resources in the campaign. This relies on the understanding that a substantial impact of a campaign is achieved through the social influence of people on one another, rather than purely through the interaction of campaign managers with the people that are exposed to the campaign messages directly.
}
In this paper we study the evolution of trend spreading dynamics in social networks. Where there have been numerous works studying the topic of anomaly detection in networks (social, and others), literature still lacks a theoretic model capable of predicting \emph{how do network anomalies evolve}. When do they spread and develop into global trends, and when they are merely statistical \mbox{phenomena}, local fads that get quickly forgotten?
We give an analytically proven lower bound for the spreading probability, capable of detecting \mbox{``future trends'' -- spreading} behavior patterns that are likely to become prominent trends in the social \mbox{network}.
We demonstrate our model using social networks from two different domains. The first is the \emph{Friends and Family} experiment \cite{Aharony2011}, held in MIT for over a year, where the complete activity of 130 users was analyzed, including data concerning their calls, SMS, MMS, GPS location, accelerometer, web activity, social media activities, and more. The second dataset contains the complete financial transactions of the \emph{eToro community} members -- the world's largest ``social trading'' platform, allowing users to trade in currency, commodities and indices by selectively copying trading activities of prominent traders.
The rest of the paper is organized as follows~: Section \ref{sec.related_work} discusses related works. The information diffusion model is presented in Section \ref{sec.problem} and its applicability is demonstrated in Section \ref{sec.results}, and concluding remarks \mbox{are given in Section \ref{sec.conc}.}
\section{Related Work}
\label{sec.related_work}
\noindent \textbf{Diffusion Optimization}.
Analyzing the spreading of information has long been the central focus in the study of
social networks for the last decade~\cite{huberman2009social}~\cite{leskovec2009meme}. Researchers
have explored both the offline networks structure by asking and incentivizing
users to forward real mails and E-mails~\cite{dodds2003experimental},
and online networks by collecting and analyzing data from various sources such as \emph{Twitter} feeds~\cite{kwak2010twitter}.
\remove{
Researchers believe that this line of works can help understand the influence of
individuals in the modern WWW, which is fully equipped with all types of social media websites
and tools~\cite{cha2010measuring}, and it can eventually lead to accurate prediction and active optimization and
construction for successful and low-cost viral market campaigns such as the DARPA Challenge~\cite{pickard2011time}.
However,the information diffusion process on social networks
is overwhelmingly complicated: the outcome is clearly sensitive to many parameters and model
settings that are not entirely well understood and modeled correctly.
As a result, accurate prediction and optimization for promoting a certain trend still remains as an active topic.
}
The dramatic effect of the network topology on the dynamics of information diffusion in communities was demonstrated in works such as~\cite{choi2010role}~\cite{nicosia2011impact}. One of the main challenges associated with modeling of behavioral dynamics in social communities stems from the fact that it often involves stochastic generative processes.
While simulations on realizations from these models can help us
explore the properties of networks~\cite{herrero2004ising}, a theoretical analysis is much more
appealing and robust. In this work we present results are based on a pure theoretical analysis.
The identity and composition of an initial ``seed group'' in trends analysis has also been the topic of much research.
Kempe et al. applied theoretical analysis on the seeds selection problem~\cite{kempe2003maximizing} based on two simple adoption models:
\emph{Linear Threshold Model} and \emph{Independent Cascade Model}.
Recently, Zaman et al. developed a method to trace rumors back in the topological spreading path to identify sources
in a social network~\cite{shah2009rumors}, and suggest such method can be used to
locate influencers in a network. Some scholars express their doubts and concerns for
the influencer-driven viral marketing
approach, suggesting that ``everyone is an influencer''~\cite{bakshy2011everyone},
and companies ``should not rely on it''~\cite{watts2007viral}. They argue that the
content of the message is also important in determining its spreads, and likely the adoption
model we were using is not a good representation for the reality.
Our work, on the other hand, focuses on predicting emerging trends given a current snapshot of the network and
adoption status, rather than finding the most influential nodes. We provide a lower bound for the probability that an emerging trend would spread throughout the network, based on the the analysis of the diffusion process outreach, which is largely missing in current literature.
\noindent \textbf{Adoption Model}.
A fundamental building block in trends prediction that is not yet entirely clear to scholars
is the adoption model, modeling individuals' behavior based on the social signals they are exposed to. Centola has shown both theoretical
and empirically that a complex contagion model is indeed more precise for diffusion~\cite{centola2007complex,centola2010spread}.
Different adoption models can dramatically alter the model outcome~\cite{dodds2004universal}.
In fact, a recent work on studying mobile application diffusions using mobile
phones demonstrated that in real world the diffusion process
is a far more complicated phenomenon, and a more realistic model was proposed in ~\cite{funfaaai}.
Our results also incorporate this realistic diffusion model.
\remove{
\subsection{Social Influence}
The topic of social influence and how it can be inferred based on constrained information derived from online domains has been the topic of many works~\cite{onnela:2010facebookpnas}.
Much research concerning the prediction of users' behavior based on the dynamics in their community has been carried out in the past, using a variety of approaches such as sociological methods \cite{game-prediction-socio-HSU,friedman1-secondlife}, communities-oriented approaches \cite{game-prediction-communities-HSU}, game theory \cite{game-prediction-gametheory-CESABIANCHI} and various machine learning methods \cite{game-prediction-doppelganger-ORWANT}.
Still, to date there has been no attempts to generalize works regarding local influence models into a network-oriented prediction model.
}
\noindent \textbf{Trends Prediction and Our Proposed Model}.
In this work we study the following question~:
Given a snapshot of a social network with some behavior occurrences (i.e. an emerging trend), what is the probability that these occurrences (seeds) will result in a viral diffusion and a wide-spread trend (or alternatively, dissolve into oblivious). Though this is similar
to the initial seed selection problem~\cite{kempe2003maximizing}, we believe that the key factor to succeed in a viral marketing campaign optimization is a better analytical model for the diffusion process itself.
The main innovation of our model is the fact that it is based on a fully analytical framework
with a scale-free network model. Therefore, we manage to overcome the dependence on
simulations for diffusion processes that characterizes most of the works in this field~\cite{choi2010role,banerjee2004reaction}. We are able to do so by decomposing the diffusion process to the transitive random walk of ``exposure agents'' and the local adoption model based
on ~\cite{funfaaai}.
While there are some works that analyze scale-free network~\cite{meloni2009traffic} most of them come short to providing accurate results, due to the fact that they calculate the expected values of the global behaviors dynamics. Due to strong ``network effect'' however, many real world networks display much less coherent patterns, involving local fluctuations and high variance in observed parameters, rendering such methods highly inaccurate and sometimes impractical.
Our analysis on the other hand tackle this problem by modeling the diffusion process on scale-free networks in a way which takes into account such interferences, and can bound their overall effect on the network.
\section{Trend Prediction in Social Networks}
\label{sec.problem}
One of the main difficulty of trends-prediction stems from the fact that the first spreading phase of ``soon to be global trends'' demonstrate significant similarity to other types of anomalous network patterns. In other words, given several observed anomalies in a social network, it is very hard to predict which of them would result in a wide-spread trend and which will quickly dissolve into oblivious.
We model the community, or social network, as a graph $G$, that is comprised of $V$ (the community's members) and $E$ (social links among them). We use $n$ to denote the size of the network, namely $|V|$. In this network, we are interested in predicting the future behavior of some observed anomalous pattern $a$. Notice that $a$ can refer to a growing use of some new web service such as \emph{Groupon}, or alternatively a behavior such as associating oneself with the ``\emph{99\% movement}''.
Notice that ``exposures'' to trends are transitive. Namely, an ``exposing'' user generates ``exposure agents'' which can be transmitted on the network's social links to ``exposed users'', which can in turn transmit them onwards to their friends, and so on. We therefore model trend's exposure interactions as movements of random walking agents in a network. Every user that was exposed to a trend $a$ generates $\beta$ such agents, on average.
We assume that our network is (or can be approximated by) a scale free network $G(n,c,\gamma)$, namely, a network of $n$ users where the probability that user $v$ has $d$ neighbors follows a power law~:
\[
P(d) \sim c \cdot d^{-\gamma}
\]
We also define the following properties of the network~:
\begin{definition}
Let $V_{a}(t)$ denote the group of network members that at time $t$ advocate the behavior associated with the potential trend $a$.
\end{definition}
\begin{definition}
Let us denote by $\beta > 0$ the average ``\emph{diffusion factor}'' of a trend $a$. Namely, the average number of friends a user who have been exposed to the trend will be talking about the trend with (or exposing the trend in other ways).
\end{definition}
\begin{definition}
Let $P_{\Delta}$ be defined as the probability that two arbitrary members of the network vertices have degrees ratio of $\Delta$ or higher~:
\[
P_{\Delta} \triangleq Prob \left[ deg(u) > \Delta \cdot deg(v) \right]
\]
\end{definition}
\begin{definition}
\label{def.prob.wxpoaw.min}
\remove{For any $t$ and $\Delta_{t}$} We denote by \remove{$\sigma_{+}$ and } $\sigma_{-}$ the \remove{highest and } ``\emph{low temporal resistance}'' \mbox{of the network~:}
\remove{
\[
\sigma_{+} \triangleq \min \left\{ \begin{array}{c}
1 \leq d \leq 2 \\
1 \leq \Delta
\end{array}
\ \left| \
1 - e^{-\Delta \cdot d \cdot \frac{\beta^{\Delta_{t}} \cdot |V_{a}(t)}{n}} \cdot
\left(1 - c \cdot \frac{1-\frac{d-1}{d}^{\gamma-1}}{\gamma-1}\right) \cdot (1 - P_{\Delta})\right.\right\}
\]
}
\[
\forall t, \Delta_{t} \qquad , \qquad
\sigma_{-} \triangleq \max \left\{ 1 \leq \Delta
\ \left| \
1 - e^{-\Delta \cdot \frac{\beta^{\Delta_{t}} \cdot |V_{a}(t)|}{n}} \cdot
(1 - P_{\Delta})\right.\right\}
\]
\end{definition}
\begin{definition}
Let $P_{Local-Adopt}(a,v,t,\Delta_{t})$ denote the probability that at time $t + \Delta_{t}$ the user $v$ had adopted trend $a$ (for some values of $t$ and $\Delta_{t}$). This probability may be different for each user, and may depend on properties such as the network's topology, past interactions between members, etc.
\end{definition}
\begin{definition}
Let $P_{Local}$ denote that expected value of the local adoption probability throughout the network~:
\[
P_{Local} = \underset{u \in V}{\operatorname{E}} \left[ P_{Local-Adopt}(a,u,t,\Delta_{t})\right]
\]
\end{definition}
\begin{definition}
Let us denote by $P_{Trend} (\Delta_{t}, \frac{V_{a}(t)}{n}, \varepsilon)$ the probability that at time $t + \Delta_{t}$ the group of network members that advocate the trend $a$ has at least $\varepsilon \cdot n$ members (namely, that $|V_{a}(t + \Delta_{t})| \geq \varepsilon \cdot n$).
\end{definition}
We assume that the seed group of members that advocate a trend at time $t$ is randomly placed in the network. Under this assumption we can now present the main result of this work : the lower bound over the prevalence of an emerging trend. Note that we use $P_{Local-Adopt}$ as a modular function in order to allow future validation in other environments. The explicit result is \mbox{given in Theorem \ref{theorem.random.funfmodel}.}
\begin{theorem}
\label{theorem.random.generic}
For any value of $\Delta_{t}$, $|V_{a}(t)|$, $n$, $\varepsilon$, the probability that at time $t + \Delta_{t}$ at least $\epsilon$ portion of the network's users would advocate trend $a$ is lower bounded as follows~:
\remove{
\[
P_{Trend} \left(\Delta_{t}, \frac{|V_{a}(t)|}{n}, \varepsilon\right) \leq \left(E_{u \in V} \left[ P_{Local-Adopt}(a,u,t,\Delta_{t})\right]\right)^{\varepsilon \cdot n} \cdot
\left(1 - \Phi \left( \sqrt{n} \cdot \frac{\varepsilon - \tilde{P_{+}}}{\sqrt{\tilde{P_{+}} (1-\tilde{P_{+}}) }} \right)\right)
\]
}
\[
P_{Trend} \left(\Delta_{t}, \frac{|V_{a}(t)|}{n}, \varepsilon\right) \geq
P_{Local}^{\varepsilon \cdot n} \cdot
\left(1 - \Phi \left( \frac{\sqrt{n} \cdot (\varepsilon - \tilde{P_{-}})}{\sqrt{\tilde{P_{-}} (1-\tilde{P_{-}}) }} \right)\right)
\]
where~:
\remove{
\[
\tilde{P_{+}} =
e^{(\rho_{opt_{+}} - \Delta_{t} \cdot \sigma_{+} )} \cdot \left( \frac{\Delta_{t} \cdot \sigma_{+}}{\rho_{opt_{+}}} \right)^{\rho_{opt_{+}}}
\]
}
\[
\tilde{P_{-}} =
e^{-\left(\frac{\Delta_{t} \cdot \sigma_{-}} {2} -\rho_{opt_{-}} + \frac{\rho_{opt_{-}}^{2}}{2 \Delta_{t} \cdot \sigma_{-}}\right)}
\]
and where~:
\[
\rho_{opt_{-}} \triangleq \underset{\rho}{\operatorname{argmin}} \ \left(P_{Local}^{\varepsilon \cdot n} \cdot P_{Trend} \left(\Delta_{t}, \frac{|V_{a}(t)|}{n}, \varepsilon\right)\right)
\]
and provided that~:
\remove{
\[
\rho_{opt_{+}} > \Delta_{t} \cdot \sigma_{+}
\]
}
\[
\rho_{opt_{-}} < \Delta_{t} \cdot \sigma_{-}
\]
and as $\sigma_{-}$ depends on $P_{\Delta}$, using the following bound~:
\[
\forall v,u \in V \quad , \quad P_{\Delta} \leq \frac{c^{2} \cdot \Delta^{1-\gamma}}{2\gamma^{2}-3\gamma+1}
\]
\begin{proof}
See Appendix for the complete proof of the Theorem
\end{proof}
\end{theorem}
Recent studies examined the way influence is being conveyed through social links. In \cite{funfaaai} the probability of network users to install applications, after being exposed to the applications installed by the friends, was tested. This behavior was shown to be best modeled as follows, for some user $v$~:
\begin{equation}
\label{eq.funf1}
P_{Local-Adopt}(a,v,t,\Delta_{t}) = 1 - e^{-(s_{v} + p_{a}(v))}
\end{equation}
Exact definitions and methods of obtaining the values of $s_{v}$ and $w_{v,u}$ can be found in \cite{funfaaai}.
The intuition of these network properties is as follows~:
For every member $v \in V$, $s_{v} \geq 0$ captures the individual susceptibility of this member, regardless of the specific behavior (or trend) in question.
$p_{a}(v)$ denotes the \emph{network potential} for the user $v$ with respect to the trend $a$, and is defined as the sum of network agnostic ``\emph{social weights}'' of the user $v$ with the friends exposing him with the trend $a$.
Notice also that both properties are trend-agnostic. However, while $s_{v}$ is evaluated once for each user and is network agnostic, $p_{a}(v)$ contributes network specific information and can also be used by us to decide the identity of the network's members that we would target in our initial campaign.
Using Theorem \ref{theorem.random.generic} we can now construct a lower bound for the success probability of a campaign, regardless of the specific value of $\rho$~:
\begin{theorem}
\label{theorem.random.funfmodel}
For every $\Delta_{t}$, $|V_{a}(t)|$, $n$, $\varepsilon$, the probability that at time $t + \Delta_{t}$ at least $\epsilon$ portion of the network's users advocate the trend $a$ is~:
\remove{
\[
P_{Trend} \left(\Delta_{t}, \frac{|V_{a}(t)|}{n}, \varepsilon\right) \leq e^{-\varepsilon \cdot n \cdot \xi_{G} \cdot \xi_{N}^{\rho_{opt_{+}}}} \cdot
\left(1 - \Phi \left( \sqrt{n} \cdot \frac{\varepsilon - \tilde{P_{+}}}{\sqrt{\tilde{P_{+}} (1-\tilde{P_{+}}) }} \right)\right)
\]
}
\[
P_{Trend} \left(\Delta_{t}, \frac{|V_{a}(t)|}{n}, \varepsilon\right) \geq
e^{-\varepsilon \cdot n \cdot \xi_{G} \cdot \xi_{N}^{\rho_{opt_{-}}}} \cdot
\left(1 - \Phi \left( \sqrt{n} \cdot \frac{\varepsilon - \tilde{P_{-}}}{\sqrt{\tilde{P_{-}} (1-\tilde{P_{-}}) }} \right)\right)
\]
where~:
\remove{
\[
\tilde{P_{+}} =
e^{(\rho_{opt_{+}} - \Delta_{t} \cdot \sigma_{+} )} \cdot \left( \frac{\Delta_{t} \cdot \sigma_{+}}{\rho_{opt_{+}}} \right)^{\rho_{opt_{+}}}
\]
}
\[
\tilde{P_{-}} =
e^{-(\frac{\Delta_{t} \cdot \sigma_{-}} {2} -\rho_{opt_{-}} + \frac{\rho_{opt_{-}}^{2}}{2 \Delta_{t} \cdot \sigma_{-}})}
\]
and where~:
\remove{
\[
\rho_{opt_{+}} \triangleq \underset{\rho}{\operatorname{argmax}} \ \left(e^{-\varepsilon \cdot n \cdot \xi_{G} \cdot \xi_{N}^{\rho}} \cdot P_{Trend} \left(\Delta_{t}, \frac{|V_{a}(t)|}{n}, \varepsilon\right)\right)
\]
}
\[
\rho_{opt_{-}} \triangleq \underset{\rho}{\operatorname{argmin}} \ \left(e^{-\varepsilon \cdot n \cdot \xi_{G} \cdot \xi_{N}^{\rho}} \cdot P_{Trend} \left(\Delta_{t}, \frac{|V_{a}(t)|}{n}, \varepsilon\right)\right)
\]
and provided that~:
\remove{
\[
\rho_{opt_{+}} > \Delta_{t} \cdot \sigma_{+}
\]
}
\[
\rho_{opt_{-}} < \Delta_{t} \cdot \sigma_{-}
\]
and where $\xi_{G}$ denotes the network's \emph{adoption factor} and $\xi_{N}$ denotes the network's \emph{influence factor}~:
\[
\xi_{G} = e^{-\frac{1}{n}\sum_{v \in V}s_{v}}
\qquad , \qquad
\xi_{N} = e^{ - \frac{1}{n} \sum_{e(v,u) \in E} (\frac{w_{u,v}}{|\mathcal{N}_{v}|} + \frac{w_{v,u}}{|\mathcal{N}_{u}|})}
\]
\begin{proof}
See complete proof in the Appendix.
\end{proof}
\end{theorem}
\section{Experimental Results}
\label{sec.results}
We have validated our model using two comprehensive datasets, the \emph{Friends and Family} dataset that studied the casual and social aspects of a small community of students and their friends in Cambridge, and the \emph{eToro} dataset --- the entire financial transactions of over 1.5M users of a ``social trading;; community.
The datasets were analyzed using the model given in \cite{funfaaai}, based on which we have experimentally calculated the values of $\beta, \xi_{G}, \xi_{N}$ and $\sigma_{-}$.
Figures \ref{fig.etoro1} and \ref{fig.funf1} demonstrate the probabilistic lower bound for trend emergence, as a function of the overall penetration of the trend at the end of the time period, under the assumption that the emerging trend was observed in 5\% of the population. In other words, for any given ``magnitude'' of trends, what is the probability that network phenomena that are being advocated by 5\% of the network, would spread to this magnitude.
Notice that although a longer spreading time slightly improve the penetration probability, the ``maximal outreach'' of the trend (the maximal rate of global adoption, with sufficient probability) is dominated by the topology of the network, and the local adoption features.
\begin{figure}[htb]
\centering
\includegraphics[width=0.47 \textwidth,keepaspectratio]{etoro1linear.eps}
\includegraphics[width=0.47 \textwidth,keepaspectratio]{etoro1log.eps}
\caption{Trends spreading potential in the \emph{eToro} network, for various penetration rates. Initial seed group is defined as 5\% of the population. Each curve represents a different time period, from 2 weeks to 6 weeks.
}
\label{fig.etoro1}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.48 \textwidth,keepaspectratio]{funf1linear.eps}
\includegraphics[width=0.48 \textwidth,keepaspectratio]{funf1log.eps}
\caption{Trends spreading potential in the \emph{Friends and Family} network, for various penetration rates. Initial seed group is defined as 5\% of the population. Each curve represents a different time period, from 2 weeks to 5 weeks.
}
\label{fig.funf1}
\end{figure}
\section{Conclusions and Future Work}
\label{sec.conc}
In this work we have discussed the problem of trends prediction, that is --- observing anomelous network patterns and predicting which of them would become a prominent trend, spreading successfully throughout the network. We have analyzed this problem using information diffusion techniques, and have presented a lower bound for the probability of a pattern to become a global trend in the network, for any desired level of spreading.
In order to model the local interaction between members, we have used the results from \cite{funfaaai} that studied the local social influence dynamics between members of social networks.
\remove{
It is also interesting to mentioned that the influence model of \cite{funfaaai} is also a good approximated of the \emph{Gompertz Function} --- a model that is frequently used for predicting the dynamics of a great variety of processes, such as mobile phone uptake \cite{Gompertz-mobile}, population in a confined space \cite{Gompertz-population}, or growth of tumors \cite{Gompertz-tumors}.
}
Though our work provides a comprehensive theoretical framework to understand trends diffusion in social networks, there are still a few challenges ahead. For example, we wish to extend our model to other network models such as Erdos-Renyi random networks, as well as Small World networks. This is essential as more evidences are suggesting that some communities involve complex structures that cannot be easily approximated using s simplistic scale-free model~\cite{leskovec2005graphs}.
In addition, our results can be used in order to provide answers to other questions, such as what is the optimal group of members that should be used as a ``seed group'' in order to maximize the effects of marketing campaigns. Another example might be finding changes in the topology of the social network that would influence the information diffusion progress in a desired way (either to encourage or surpass certain emerging trends).
In order to achieve these goals we are planning a large-scale field test with a leading online social platform, that would give us access to collect more empirical supporting evidences, as well as conducting an active experiment in which we would try to predict trends in real time.
Finally, we are interested in comparing the prediction obtained from our model with the actual semantics of the trends, to better understand the connection between the trends semantics and the diffusion process they undergo.
\bibliographystyle{elsarticle-num}
| 7,761 |
\section*{Introduction}
Let $H$ be a Hopf algebra with bijective antipode over the base field $\fie$, and let $(R^{\vee},R)$
together with a bilinear form $\langle \;,\;\rangle : R^{\vee} \ot R \to \fie$ be a dual pair of
Hopf algebras in the braided category $\ydH$ of left Yetter-Drinfeld modules over $H$
(see Definition \ref{defin:pair}).
The smash products or bosonizations $R^{\vee} \# H$ and $R \# H$ are Hopf algebras in the usual sense.
We are interested in their braided monoidal categories of left Yetter-Drinfeld modules.
By our first main result, Theorem \ref{theor:third}, there is a braided monoidal isomorphism
\begin{align}\label{intro1}
(\Omega,\omega) : {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat} \to {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat},
\end{align}
where the index rat means Yetter-Drinfeld modules which are rational over $R$ and over $R^{\vee}$
(see Definition \ref{defin:pair}). In particular, $(\Omega,\omega)$ maps Hopf algebras to Hopf algebras.
For $X \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$, $\Omega(X)=X$ as a Yetter-Drinfeld module over $H$.
The origin of the
isomorphism \eqref{intro1} is the standard correspondence between comodule structures over a
coalgebra and module structures over the dual algebra. In Theorem \ref{theor:first} we first prove a monoidal isomorphism between right and left relative Yetter-Drinfeld modules, and hence a braided monoidal isomorphism between their Drinfeld centers. Then we show in Theorem \ref{theor:second} that this isomorphism preserves the subcategories of right and left Yetter-Drinfeld modules we want. Finally, in Theorem \ref{theor:third} we change the sides to left Yetter-Drinfeld modules on both sides. Without this strategy, it would be hard to guess and to prove the correct formulas.
Our motivation to find such an isomorphism of categories comes from the theory
of Nichols algebras which in turn are fundamental for the classification of
pointed Hopf algebras. If $M \in \ydH$, the Nichols algebra $\NA(M)$ is a
braided Hopf algebra in $\ydH$ which is the unique graded quotient of the
tensor algebra $T(M)$ such that $M$ coincides with the space of primitive
elements in $\NA(M)$.
A basic construction to produce new Nichols algebras is the reflection of
semisimple Yetter-Drinfeld modules $M_1 \oplus \cdots \oplus M_\theta$,
where $\theta \in \ndN $ and $M_1,\dots,M_\theta$ are finite-dimensional
and irreducible objects in $\ydH$. For $1 \leq i \leq \theta$, the $i$-th
reflection of $M =(M_1,\dots,M_\theta)$ is a certain $\theta$-tuple
$R_i(M) =(V_1,\dots,V_\theta)$ of finite-dimensional irreducible
Yetter-Drinfeld modules in $\ydH$. It is defined assuming a growth condition
of the adjoint action in the Nichols algebra $\NA(M)$ of
$M_1 \oplus \cdots \oplus M_\theta$. The Nichols algebras
$\NA(R_i(M))$ of $V_1 \oplus \cdots \oplus V_\theta$ and $\NA(M)$ have the
same dimension. The reflection operators allow to define the Weyl groupoid
of $M$, an important combinatorial invariant. In this paper we give a natural
explanation of the reflection operators in terms of the isomorphism
$(\Omega,\omega)$.
To describe our new approach to the reflection operators, fix $1 \leq i \leq \theta$, and let $K_i^M$ be the algebra of right coinvariant elements of $\NA(M)$ with respect to the canonical projection $\NA(M) \to \NA(M_i)$ coming from the direct sum decomposition of $M$. By the theory of bosonization of Radford-Majid, $K_i^M$ is a Hopf algebra in $ {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}$. To define $R_i(M)$ we have to assume that $K_i^M$ is rational as an $R$-module. Let $W= \ad \NA(M_i)(\oplus_{j \neq i} M_j) \subseteq \NA(M)$. Then $W$ is an object in $ {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$, and by Proposition \ref{prop:old} its Nichols algebra is isomorphic to $K_i^M$. This new information on $K_i^M$ is used to prove our second main result, Theorem \ref{theor:NtoN}, which says that
\begin{align}\label{intro2}
\Omega(K_i^M) \# \NA(M_i^*) \cong \NA(R_i(M)),
\end{align}
where the braided monidal functor $(\Omega,\omega)$ is defined with respect to the dual pair $(\NA(M_i^*),\NA(M_i))$. The left-hand side of \eqref{intro2} is the bosonization, hence a braided Hopf algebra in a natural way. In \cite[Thm.\,3.12(1)]{a-AHS10} a different algebra isomorphism
\begin{align}\label{intro3}
K_i^M \# \NA(M_i^*) \cong \NA(R_i(M)),
\end{align}
formally similar to \eqref{intro2}, was obtained. But there, the left-hand side is not a bosonization, and a priori it is only an algebra and not a braided Hopf algebra. This is the reason why the proof of \eqref{intro3} was quite involved. The Hopf algebra structure of $K_i^M \# \NA(M_i^*)$ induced from the isomorphism \eqref{intro3} was determined in \cite[Theorem 4.2]{p-HeckSchn09}.
If $R$ is an algebra and $M$ is a right $R$-module, we denote its module structure by $\mu^R_M = \mu_M : M \ot R \to M$. If $C$ is a coalgebra and $M$ is a right $C$-comodule, we denote by $\delta_M^C = \delta_M : M \to M \ot C$ the comodule structure map. The same notations $\mu^R_M$ and $\delta_M^C$ will be used for left modules and left comodules. In the following we assume that $H$ is a Hopf algebra over $\fie $ with comultiplication $\Delta = \Delta_H : H \to H \ot H, \; h \mapsto h\sw1 \ot h\sw2$, augmentation $\varepsilon = \varepsilon_H$, and
bijective antipode $\mathcal{S}$.
\section{Preliminaries on bosonization of Yetter-Drinfeld Hopf algebras}
\label{sec:bosonization}
We recall some well-known notions and results (see e.\,g.~\cite[Sect.~1.4]{a-AHS10}), and note some useful formulas from the theory of Yetter-Drinfeld Hopf algebras.
A left Yetter-Drinfeld module over $H$ is a left $H$-module and a left $H$-comodule with $H$-action and $H$-coaction denoted by
$H \ot V \to V,\; h \ot v \mapsto h \lact v$, and
$\delta=\delta_V : V \to H \ot V,\; v \mapsto \delta(v) = v\_{-1} \ot v\_0$,
such that
\begin{align}
\delta(h \lact v) = h\_1 v\_{-1} \mathcal{S}(h\_3) \ot h\_2 \lact v\_0\label{YD}
\end{align}
for all $v \in V,h \in H$.
The category of left Yetter-Drinfeld modules over $H$ with $H$-linear and $H$-colinear maps as morphisms is denoted by $\ydH$. It is a monoidal and braided category. If $V,W \in \ydH$, then the tensor product is the vector space $V \ot W$ with diagonal action and coaction given by
\begin{align}
h \lact (v \ot w) &= h\_1 \lact v \ot h\_2 \lact w,\\
\delta(v \ot w) &= v\_{-1} w\_{-1} \ot v\_{0} \ot w\_{0},\\
\intertext{and the braiding is defined by}
c_{V,W} : V &\ot W \to W \ot V,\; v \ot w \mapsto v\_{-1} \lact w \ot v\_0\label{braiding},
\intertext{with inverse}
c_{V,W}^{-1} : W &\ot V \to V \ot W,\; w \ot v \mapsto v\sw0 \ot \mathcal{S}^{-1}(v\sw{-1}) \lact w,
\end{align}
for all $h \in H, v \in V,w \in W$.
The category $\rydH$ is defined in a similar way, where the objects are the right Yetter-Drinfeld modules over $H$, that is, right $H$ modules and right $H$-comodules $V$ such that
\begin{align}
\delta(v \lact h) = v\_0 \lact h\_2 \ot \mathcal{S}(h\_1) v\_1 h\_3
\end{align}
for all $v \in V,h \in H$.
The monoidal structure is given by diagonal action and coaction, and the braiding is defined by
\begin{align}\label{braidingright}
c_{V,W} : V \ot W \to W \ot V,\; v \ot w \mapsto w\sw0 \ot v \lact w\sw1,
\end{align}
for all $V,W \in \rydH$.
We note that for any object $V \in \ydH$, there is a linear isomorphism
\begin{align}
&\theta_V : V \xrightarrow{\cong} V,\; v \mapsto \mathcal{S}(v\sw{-1}) \lact v\sw0,\\
\intertext{with inverse}
&V \xrightarrow{\cong} V, \; v \mapsto \mathcal{S}^{-2}(v\sw{-1}) \lact v\sw0.\label{ydiso}
\end{align}
The map $\theta_V$ is not a morphism in $\ydH$, but
\begin{align}
\theta _V(h\lact v) &= \mathcal{S}^2(h) \lact \theta_V(v),\label{theta1}\\
\delta (\theta_V(v)) &=\mathcal{S}^2(v\sw{-1}) \ot \theta_V(v\sw0)\label{theta2}
\end{align}
for all $v \in V, h \in H$, where $\delta(v) = v\sw{-1} \ot v\sw0$.
\begin{comment}
Thus $\theta = (\theta_V)_{\V \in \ydH}$ is a family of functorial isomorphisms from the identity functor of $\ydH$ to
the functor $I_{\mathcal{S}^2} : \ydH \to \ydH$ given by the Hopf algebra isomorphism $\mathcal{S}^2$. If $\varphi : H \to H$ is a Hopf algebra isomorphism, we define $I_{\varphi} : \ydH \to \ydH$ by $I_{\varphi}(f) = f$ for morphisms $f$. For all objects $V \in \ydH$, $I_{\varphi}(V) = V$ as a vector space with $H$-action and $H$-coaction defined by
\begin{align*}
hv &= \varphi(h) v,\\
\delta_{I_{\varphi}(V)}(v) &= \varphi^{-2}(v\sw{-1}) \ot \sw0
\end{align*}
for all $v \in I_{\varphi}(V)$.
\end{comment}
If $A,B$ are algebras in $\ydH$, then the algebra structure of the tensor product $A \ot B$ of the vector spaces $A,B$ is defined in terms of the braiding by
\begin{align}
(a \ot b)(a'\ot b') = a (b\sw{-1} \lact a') \ot b\sw0 b'
\end{align}
for all $a,a' \in A$ and $b,b'\in B$.
Let $R$ be a Hopf algebra in the braided monoidal category $\ydH $ with augmentation $\varepsilon_R : R \to \fie$, comultiplication $\copr _R:R\to R\ot R,\; r \mapsto
r\swo1 \ot r\swo2,$ and antipode $\mathcal{S}_R$. Thus $\varepsilon_R, \Delta_R, \mathcal{S}_R$ are morphisms in $\ydH$ satisfying the Hopf algebra axioms.
\begin{comment}
We note the explicit formulas for $H$-linearity and $H$-colinearity of $\varepsilon_R$ and $\Delta_R$:
\begin{align}
\varepsilon_R(h \lact r) &= \varepsilon(h) \varepsilon_R(r),\label{rulevarepsilon1}\\
r\sw{-1} \varepsilon_R(r\sw0) &= \varepsilon_R(r) 1,\label{rulevarepsilon2}\\
(h \lact r)\swo1 \ot (h \lact r)\swo2 &= h\sw1 \lact r\swo1 \ot h\sw2 \lact r\swo2,\label{ruleDelta1}\\
{r\swo1}\sw{-1} {r\swo2}\sw{-1} \ot {r\swo1}\sw0 \ot {r\swo2}\sw0 &= r\sw{-1} \ot {r\sw0}\swo1 \ot {r\sw0}\swo2,\label{ruleDelta2}
\end{align}
for all $h \in H, r \in R$.
\end{comment}
The map $\mathcal{S}_R$ anticommutes with multiplication and comultiplication in the following way.
\begin{align}
\label{ruleS1}
\mathcal{S}_R(rs)=&\mathcal{S}_R(r\_{-1}\lact s)\mathcal{S}_R(r\_0),\\
\copr _R(\mathcal{S}_R(r))=&\mathcal{S}_R(r\^1{}\_{-1}\lact r\^2)\ot \mathcal{S}_R(r\^1{}\_0)
\label{ruleS2}
\end{align}
for all $r,s\in R$.
Let $A = R\#H$ be the bosonization of $R$.
As an algebra, $A$ is the smash product given by the $H$-action on $R$ with multiplication
\begin{align}
\label{smashproduct}
(r\#h)(r'\#h')=r(h\_1\lact r')\#h\_2h'
\end{align}
for all $r,r'\in R$, $h,h'\in H$. We will identify $r \# 1$ with $r$ and $1 \# h$ with $h$. Thus we view $R \subseteq A$ and $H \subseteq A$ as subalgebras, and the multiplication map
$$ R \ot H \to A, \;r \ot h \mapsto rh = r \# h,$$
is bijective. Since $\lact$ denotes the $H$-action, we will always write $ab$ for the product of elements $a,b \in A$ (and not $a \lact b$). Note that
\begin{align}
hr &= (h\sw1 \lact r) h\sw2,\label{smash1}\\
rh &= h\sw2 (\mathcal{S}^{-1}(h\sw1) \lact r)\label{smash2}
\end{align}
for all $r \in R, h \in H$.
As a coalgebra, $A$ is the cosmash product given by the $H$-coaction of the coalgebra $R$. We will denote its comultiplication by
$$\Delta : A \to A \ot A,\; a \mapsto a\sw1 \ot a\sw2.$$
By definition,
\begin{align}
(rh)\sw1 \ot (rh)\sw2 = r\swo1 {r\swo2}\sw{-1} h\sw1 \ot {r\swo2}\sw0 h\sw2\label{cosmash}
\end{align}
for all $r \in R, h \in H$. Thus the projection maps
\begin{align}
&\pi : A \to H,\; r \# h \mapsto \varepsilon_R(r)h,\\
&\vartheta : A \to R, \; r \# h \mapsto r \varepsilon(h), \label{eq:defvartheta}
\end{align}
are coalgebra maps, and
$$A \to R \ot H, \;a \mapsto \vartheta(a\sw1) \ot \pi(a\sw2),$$
is bijective.
Then $A= R \# H$ is a Hopf algebra with antipode $\mathcal{S} = \mathcal{S}_A$, where the restriction of $\mathcal{S}$ to $H$ is the antipode of $H$, and
\begin{align}
\mathcal{S}(r)&=\mathcal{S}(r\_{-1})\mathcal{S}_R(r\_0),\label{bigS}\\
\intertext{hence}
\mathcal{S}^2(r) &= \mathcal{S}_R^2(\theta_R(r))\label{bigS2}
\end{align}
for all $r\in R$.
The map $\pi$ is a Hopf algebra projection, and the subalgebra $R \subseteq A$ is a left coideal subalgebra, that is, $\Delta(R) \subseteq A \ot R$, which is stable under $\mathcal{S}^2$.
The structure of the braided Hopf algebra $R$ can be expressed in terms of the Hopf algebra $R \# H$ and the projection $\pi$:
\begin{align}
R = A^{\co H} &= \{r \in A \mid r\sw1 \ot \pi(r\sw2) = r \ot 1 \},\\
h \lact r &= h\sw1 r \mathcal{S}(h\sw2),\label{action}\\
r\sw{-1} \ot r\sw0 &= \pi(r\sw1) \ot r\sw2,\label{coaction}\\
r\swo1 \ot r\swo2 &= r\sw1 \pi\mathcal{S}(r\sw2) \ot r\sw3,\label{comult}\\
\mathcal{S}_R(r)&=\pi(r\sw1)\mathcal{S}(r\sw2)\label{antipode}\\
\intertext{for all $h\in H$, $r\in R$. We list some formulas related to the projection $\vartheta$.}
\vartheta(a) &= a\sw1 \pi\mathcal{S}(a\sw2),\label{vartheta}\\
a &=\vartheta(a\sw1) \pi(a\sw2), \label{decomposition}\\
r\swo1 \ot r\swo2 &= \vartheta(r\sw1) \ot r\sw2,\label{comult1}\\
\vartheta(a)\swo1 \ot \vartheta(a)\swo2 &= \vartheta(a\sw1) \ot \vartheta(a\sw2),\label{comultvartheta}\\
\vartheta(a)\sw{-1} \ot \vartheta(a)\sw0 &= \pi(a\sw1 \mathcal{S}(a\sw3)) \ot \vartheta(a\sw2)\label{coactionvartheta}\\
\intertext{for all $r \in R, a \in A$.}
\intertext{By \eqref{action}, the inclusion $R \subseteq A$ is an $H$-linear algebra map, where the $H$-action on $A$ is the adjoint action. By \eqref{comultvartheta} and \eqref{coactionvartheta},
the map $\vartheta : A \to R$ is an $H$-colinear coalgebra map, where the $H$-coaction of $A$ is defined by}
A &\to H \ot A,\; a \mapsto \pi(a\sw1 S(a\sw3)) \ot a\sw2,
\intertext{that is, by the coadjoint $H$-coaction of $A$.}
\intertext{Finally we note the following useful formulas related to the behaviour of $\vartheta$ with respect to multiplication.}
\vartheta(ah) &= \varepsilon(h) \vartheta(a),\label{vartheta1}\\
\vartheta(ha) &= h \lact \vartheta(a),\label{vartheta2}
\end{align}
for all $h \in H,a \in A$.
\begin{lemma}\label{lem:vartheta3}
Let $R$ be a Hopf algebra in $\ydH$ and $A =R \# H$ its bosonization. Then
\begin{align*}
\vartheta\mathcal{S}\left(a \pi\mathcal{S}^{-1}(b\sw2) b\sw1\right) = \vartheta\mathcal{S}(b\sw2) \left(\pi\left(\mathcal{S}(b\sw1) b\sw3\right) \lact \vartheta\mathcal{S}(a)\right),
\end{align*}
for all $h \in H$ and $a,b \in A$.
\end{lemma}
\begin{proof}
\begin{align*}
\vartheta\mathcal{S}(b\sw2) \left(\pi\left(\mathcal{S}(b\sw1) b\sw3\right) \lact \vartheta\mathcal{S}(a)\right)&=\vartheta\mathcal{S}(b\sw3) \pi\left(\mathcal{S}(b\sw2) b\sw4\right) \vartheta\mathcal{S}(a)\pi\mathcal{S}\left(\mathcal{S}(b\sw1) b\sw5\right)\\
&=\mathcal{S}(b\sw2) \pi(b\sw3) \mathcal{S}(a\sw2) \pi\mathcal{S}^2(a\sw1)\pi\mathcal{S}\left(\mathcal{S}(b\sw1) b\sw4\right)\\
&=\vartheta\left(\mathcal{S}(b\sw1) \pi(b\sw2) \mathcal{S}(a)\right)\\
&=\vartheta\mathcal{S}\left(a \pi\mathcal{S}^{-1}(b\sw2) b\sw1\right),
\end{align*}
where the second equality follows from \eqref{decomposition} applied to $\mathcal{S}(b\sw2)$ and \eqref{vartheta} applied to $\mathcal{S}(a)$, and the third equality follows from \eqref{vartheta}.
\end{proof}
It follows from \eqref{bigS2} and \eqref{ydiso} that the antipode $\mathcal{S}_R$ of $R$ is bijective if and only if the antipode $\mathcal{S}$ of $R$ is bijective. In this case the following formulas hold for $\mathcal{S}_R^{-1}$ and $\mathcal{S}^{-1}$.
\begin{align}
\mathcal{S}_R^{-1}(r)&=\mathcal{S}^{-1}(r\_0)r\_{-1} = \vartheta \mathcal{S}^{-1}(r),\label{ruleS5}\\
\mathcal{S}^{-1}(rh) &= \mathcal{S}^{-1}(h) \mathcal{S}_R^{-1}(r\sw0) \mathcal{S}^{-1}(r\sw{-1})\label{bigSinvers}
\end{align}
for all $r,s \in R$.
\begin{comment}
The next lemma is not difficult to check.
\begin{lemma}\label{lem:Sinverse}
\begin{enumerate}
\item
Assume that the antipode $\mathcal{S}$ of $R \# H$ is bijective. Then $\mathcal{S}_R$ is bijective, and for all $r \in R$,
\begin{align}
\mathcal{S}_R^{-1}(r)&=\mathcal{S}^{-1}(r\_0)r\_{-1} = \vartheta \mathcal{S}^{-1}(r).\label{ruleS5}
\end{align}
\item Assume that $\mathcal{S}_R$ is bijective. Then the antipode of $R \# H$ is bijective, and for all $r \in R,h \in H$,
\begin{align}
\mathcal{S}^{-1}(rh) = \mathcal{S}^{-1}(h) \mathcal{S}_R^{-1}(r\sw0) \mathcal{S}^{-1}(r\sw{-1}).\label{bigSinvers}
\end{align}
\end{enumerate}
\end{lemma}
\end{comment}
\begin{comment}
\begin{align}
\mathcal{S}_R^{-1}(rs)&=\mathcal{S}_R^{-1}(s\_0)\mathcal{S}_R^{-1}(S^{-1}(s\_{-1})\lact r),\label{ruleS3}\\
\copr _R(\mathcal{S}_R^{-1}(r))&=\mathcal{S}_R^{-1}(r\^2{}\_0)\ot \mathcal{S}_R^{-1}(\mathcal{S}^{-1}(r\^2{}\_{-1})
\lact r\^1)
\label{ruleS4}
\end{align}
for all $r,s \in R$.
\end{comment}
\section{Dual pairs of braided Hopf algebras and rational modules}
The field $\fie$ will be considered as a topological space with the discrete topology.
We denote by $\mathcal{L}_{\fie}$ the category of {\em linearly topologized vector spaces} over $\fie$. Objects of $\mathcal{L}_{\fie}$ are topological vector spaces which have a basis of neighborhoods of $0$ consisting of vector subspaces. Morphisms in $\mathcal{L}_{\fie}$ are continuous $\fie$-linear maps.
Thus an object in $\mathcal{L}_{\fie}$ is a vector space and a topological space $V$, where the topology on $V$ is given by a set $\{V_i \subseteq V \mid i \in I\}$ of vector subspaces of $V$ such that for all $i,j \in I$ there is an index $k \in I$ with $V_k \subseteq V_i \cap V_j$. The set $\{V_i \subseteq V \mid i \in I\}$ is a basis of neighborhoods of $0$, and a subset $U \subseteq V$ is open if and only if for all $x \in U$ there is an index $i \in I$ such that $x + V_i \subseteq U$.
In particular, a vector subspace $U \subseteq V$ is open if and only if $V_i \subseteq U$ for some $i \in I$.
Let $V, W \in \mathcal{L}_{\fie}$, and let $\{V_i \subseteq V \mid i \in I\}$ and $\{W_j \subseteq W \mid j \in J\}$ be bases of neighborhoods of $0$. Then a linear map $f : V \to W$ is continuous if and only if for all $j \in J$ there is an index $i \in I$ with $f(V_i) \subseteq W_j$. We define the tensor product $V \ot W$ as an object in $\mathcal{L}_{\fie}$ with
$$ \{V_i \ot W + V \ot W_j \mid (i,j) \in I \times J\}$$
as a basis of neighborhoods of $0$.
Let $R,R^{\vee}$ be vector spaces, and let
$$\langle\;,\;\rangle : R^{\vee} \ot R \to \fie,\;\xi \ot x \mapsto \langle \xi,x \rangle,$$
be a $\fie$-bilinear form. If $X \subseteq R$ and $X' \subseteq R^{\vee}$ are subsets, we define
\begin{align*}
^{\perp}X &= \{ \xi \in R^{\vee} \mid \langle \xi,x\rangle = 0 \text { for all } x \in X\},\\
X'^{\perp} &= \{ x \in R \mid \langle \xi,x\rangle = 0 \text { for all } \xi \in X'\}.
\end{align*}
We endow $R^{\vee}$ with the {\em finite topology} ({or the \em weak topology}), which is the coarsest topology on $R^{\vee}$ such that the evaluation maps $\langle\;,x\rangle : R^{\vee} \to \fie,\; \xi \mapsto \langle\xi,x\rangle,$ for all $x \in R$ are continuous.
In the same way we view $R$ as a topological space with the finite topology with respect to the evaluation maps $\langle\xi,\;\rangle : R \to \fie,\; x \mapsto \langle \xi,x\rangle$, for all $\xi \in R^{\vee}$.
Let $\mathcal{E}$ be a cofinal subset of the set of all finite-dimensional subspaces of $R$ (that is, $\mathcal{E}$ is a set of finite-dimensional subspaces of $R$, and any finite-dimensional subspace $E \subseteq R$ is contained in some $E_1 \in \mathcal{E}$). Let $\mathcal{E}'$ be a cofinal subset of the set of all finite-dimensional subspaces of $R^{\vee}$. Then $R^{\vee}$ and $R$ are objects in $\mathcal{L}_{\fie}$, where
\begin{equation*}
\{^{\perp}E \mid E \in \mathcal{E}\} \text{ and }
\{E'^{\perp} \mid E' \in \mathcal{E}'\}
\end{equation*}
are bases of neighborhoods of $0$ of $R^{\vee}$ and $R$, respectively.
The pairing $\langle\;,\; \rangle$ is called {\em non-degenerate}
if $^{\perp}R =0 \text{ and } {R^{\vee}}^{\perp} =0$. Let $E \in \mathcal{E}$, and assume that $^{\perp}R = 0$. Then
$$E \to (R^{\vee}/^{\perp}E)^*, \; x \mapsto (\overline{\xi} \mapsto \langle \xi,x\rangle),$$
is injective. Since
$$R^{\vee}/^{\perp}E \to E^*,\; \overline{\xi} \mapsto (x \mapsto \langle \xi,x \rangle),$$
is injective by definition, it follows that
\begin{align}
R^{\vee}/^{\perp}E &\xrightarrow{\cong} E^*,\;\overline{\xi} \mapsto \langle \xi,\; \rangle,\label{nonde1}
\end{align}
is bijective.
By the same argument, for all $E' \in \mathcal{E}_{R^{\vee}}$
\begin{align}
R/E'^{\perp} &\xrightarrow{\cong} {E'}^*,\;\overline{x} \mapsto \langle \;,x\rangle,\label{nonde2}
\end{align}
is bijective, if ${R^{\vee}}^{\perp} =0$.
If $V,W$ are vector spaces, denote by
$$\Hom_{\mathrm{rat}}(R^{\vee} \ot V,W) \text{ (respectively }\Hom_{\mathrm{rat}}(V \ot R^{\vee},W))$$ the set of all linear maps $g : R^{\vee} \ot V \to W$ (respectively $g : V \ot R^{\vee} \to W$) such that for all $v \in V$ there is a finite-dimensional subspace $E \subseteq R$ with $g(^{\perp}E \ot v) = 0$ (respectively $g(v \ot {^{\perp}E}) = 0$).
\begin{lemma}\label{lem:rational}
Let $\langle \;,\; \rangle : R^{\vee} \ot R \to \fie$ be a non-degenerate $\fie$-bilinear form of vector spaces, and
let $V,W$ be vector spaces. Then the following hold.
\begin{enumerate}
\item The map
$$ \mathrm{D} : \Hom(V,R \ot W) \to \Hom_{\mathrm{rat}}(R^{\vee} \ot V,W),\;\; f \mapsto (\langle\;,\; \rangle \ot \id)(\id \ot f),$$
is bijective.
\item The map
$$ \mathrm{D}' : \Hom(V,R \ot W) \to \Hom_{\mathrm{rat}}(V \ot R^{\vee},W),\;\; f \mapsto (\id \ot \langle\;,\; \rangle)\tau(f \ot \id),$$
is bijective, where $\tau : R \ot W \ot R^{\vee} \to W \ot R^{\vee} \ot R$ is the twist map with $\tau(x \ot w \ot \xi) = w \ot \xi \ot x$ for all $x \in R,w \in W, \xi \in R^{\vee}$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) For completeness we recall the following well-known argument.
Let $f \in \Hom(V,R \ot W)$, and $g=\mathrm{D}(f)$. For all $v \in V$ there is a finite-dimensional subspace $E \subseteq R$ with $f(v) \in E \ot W$, hence $g(^{\perp}E \ot v) =0$. Thus $g \in \Hom_{\mathrm{rat}}(R^{\vee} \ot V,W)$.
Conversely, let $g \in \Hom_{\mathrm{rat}}(R^{\vee} \ot V,W)$. For any finite-dimensional subspace $U \subseteq V$ there is a finite-dimensional subspace $E \subseteq R$ with $g(^{\perp}E \ot U)=0$. Let $g_{U,E} \in \Hom(R^{\vee}/^{\perp}E \ot U, W)$ be the map induced by $g$, and $f_{U,E} \in \Hom(U,E \ot W)$ the inverse image of $g_{U,E}$ under the isomorphisms
\begin{align*}
\Hom(U,E \ot W) \xrightarrow{\cong} \Hom(E^* \ot U,W) \xrightarrow{\cong} \Hom(R^{\vee}/^{\perp}E \ot U,W),
\end{align*}
where the first map is the canonical isomorphism, and the second map is induced by the isomorphism in \eqref{nonde1}.
If $E'$ is a finite-dimensional subspace of $R$ containing $E$, then
$$f_{U,E}(v) = f_{U,E'}(v) \text{ for all } v \in U.$$
Hence $f_U \in \Hom(U,R \ot W)$, defined by $f_U(v) = f_{U,E}(v)$ for all $v \in U$, does not depend on the choice of $E$.
Since $f_{U'} \mid U = f_U$ for all finite-dimensional subspaces $U \subseteq U'$ of $V$, the inverse image $\mathrm{D}^{-1}(g)$ can be defined by the family $(f_U)$.
(2) follows from (1) since the twist map $V \ot R^{\vee} \to R^{\vee} \ot V$ defines an isomorphism $\Hom_{\mathrm{rat}}(R^{\vee} \ot V,W) \cong \Hom_{\mathrm{rat}}(V \ot R^{\vee},W)$.
\end{proof}
Let $R,R^{\vee}$ be Hopf algebras in the braided monoidal category $\ydH$, and let
$$\langle\;,\;\rangle : R^{\vee} \ot R \to \fie,\;\xi \ot x \mapsto \langle \xi,x \rangle,$$
be a $\fie$-bilinear form of vector spaces.
\begin{defin}\label{defin:pair}
Assume
that there are cofinal subsets $\mathcal{E}_R$ (respectively $\mathcal{E}_{R^{\vee}}$) of the sets of all finite-dimensional vector subspaces of $R$ (respectively of $R^{\vee}$) consisting of subobjects in $\ydH$.
Then the pair $(R,R^{\vee})$ together with the bilinear form $\langle \;,\; \rangle : R^{\vee} \ot R \to \fie$ is called a {\em dual pair of Hopf algebras} in $\ydH$ if
\begin{align}
\langle\;,\;\rangle& \text{ is non-degenerate},\label{pair1}\\
\langle h \lact \xi,x\rangle &= \langle \xi,\mathcal{S}(h) \lact x\rangle,\label{pair2}\\
\xi\sw{-1} \langle\xi\sw0,x\rangle &= \mathcal{S}^{-1}(x\sw{-1}) \langle \xi,x\sw0\rangle,\label{pair3}\\
\langle \xi,xy\rangle &= \langle \xi\swo1,y\rangle \langle \xi\swo2,x\rangle,\;\langle1,x \rangle = \varepsilon(x),\label{pair4}\\
\langle \xi \eta, x\rangle &= \langle \xi, x\swo2\rangle \langle \eta,x\swo1\rangle,\;\langle\xi,1\rangle = \varepsilon(\xi),\label{pair5}\\
\Delta_{R^{\vee}} : R^{\vee} &\to R^{\vee} \ot R^{\vee} \text{ is continuous},\label{pair7}\\
\Delta_{R} : R &\to R \ot R \text{ is continuous}\label{pair8}
\end{align}
for all $x,y \in R, \xi,\eta \in R^{\vee}$ and $h \in H$.
A left or right $R^{\vee}$-module (respectively $R$-module) $M$ is called {\em rational} if any element of $M$ is annihilated by $^{\perp}E$ (respectively $E'^{\perp}$) for some finite-dimensional vector subspace $E \subseteq R$ (respectively $E'\subseteq R^{\vee}$).
\end{defin}
\begin{lemma}\label{lem:pair}
Let $(R,R^{\vee})$ together with $\langle \;,\; \rangle : R^{\vee} \ot R \to \fie$ be a dual pair of Hopf algebras in $\ydH$. Then for all $x \in R, \xi \in R^{\vee}$ and for all $E \in \mathcal{E}_R, E' \in \mathcal{E}_{R^{\vee}}$,
\begin{align}
&\langle \mathcal{S}_{R^{\vee}}(\xi),x \rangle = \langle \xi,\mathcal{S}_R(x) \rangle,\label{pair6}\\
&^{\perp}E \subseteq R^{\vee} \text{ and } E'^{\perp} \subseteq R \text{ are subobjects in }\ydH. \label{pair9}
\end{align}
\end{lemma}
\begin{proof}
The vector space $\Hom(R^{\vee}, R^{*\op})$ is an algebra with convolution product. We define linear maps $\varphi_1,\varphi_2, \psi \in \Hom(R^{\vee}, R^{*\op})$ by
$$\varphi_1(\xi)(x) = \langle \xi, \mathcal{S}_R(x) \rangle,\; \varphi_2(\xi)(x) = \langle \mathcal{S}_{R^{\vee}}(\xi), x \rangle,\; \psi(\xi)(x) = \langle \xi,x \rangle,$$
for all $\xi \in R^{\vee}, x \in R$. Then by \eqref{pair4} and \eqref{pair5} the unit element in $\Hom(R^{\vee}, R^{*\op})$ is equal to $\varphi_1 * \psi$ and also to $\psi * \varphi_2$. Hence $\varphi_1 = \varphi_2$.
\eqref{pair9} follows from \eqref{pair2} and \eqref{pair3}.
\end{proof}
Note that the bilinear form $\langle\;,\;\rangle : R^{\vee} \ot R \to \fie$ is a morphism in $\ydH$ if and only if \eqref{pair2} and \eqref{pair3} are satisfied.
The continuity conditions \eqref{pair7} and \eqref{pair8} are equivalent to the following. For all $E \in \mathcal{E}_R$ and $E' \in \mathcal{E}_{R^{\vee}}$ there are $E_1 \in \mathcal{E}_R$ and $E'_1 \in \mathcal{E}_{R^{\vee}}$ such that
$$\Delta_{R^{\vee}}({}^{\perp}E_1) \subseteq {^{\perp}E} \ot R^{\vee} + R^{\vee} \ot {}^{\perp}E, \;
\Delta_{R}({E_1'^{\perp}}) \subseteq E'{^{\perp}} \ot R + R \ot E'{^{\perp}}.$$
By \eqref{nonde1} and \eqref{nonde2}, rational modules over $R$ or $R^{\vee}$ are locally finite. Recall that a module over an algebra is {\em locally finite} if each element of the module is contained in a finite-dimensional submodule.
\begin{examp}\label{exa:gradedpair}
Let $R^{\vee}= \oplus_{n \geq 0} R^{\vee}(n)$ and $R= \oplus_{n \geq 0} R(n)$ be $\mathbb{N}_0$-graded Hopf algebras in $\ydH$ with finite-dimensional components $R^{\vee}(n)$ and $R(n)$ for all $n \geq 0$, and let $\langle \;,\; \rangle : R^{\vee} \ot R \to \fie$ be a bilinear form of vector spaces such that
\begin{align}
&\langle R^{\vee}(m), R(n) \rangle = 0 \text{ for all } n \neq m \text{ in }\mathbb{N}_0.\label{gradedpair}
\end{align}
Assume \eqref{pair1} -- \eqref{pair5}.
For all integers $n \geq 0$ we define
$$\mathcal{F}_nR = \oplus_{i=0}^{n} R(i),\; \mathcal{F}_nR^{\vee} = \oplus_{i=0}^{n} R^{\vee}(i).$$
Then the subspaces $\mathcal{F}_n R \subseteq R, n \geq 0,$ and $\mathcal{F}_n R^{\vee} \subseteq R^{\vee}, n\geq 0,$ form cofinal subsets of the set of all finite-dimensional subspaces of $R$ and of $R^{\vee}$ consisting of subobjects in $\ydH$.
For all $n \geq 0$, let
$$\mathcal{F}^nR = \oplus_{i \geq n} R(i),\; \mathcal{F}^nR^{\vee} = \oplus_{i \geq n} R^{\vee}(i).$$
Then by \eqref{gradedpair} and \eqref{pair1}, for all $n \geq 0$,
\begin{align}\label{gradedperp}
{}^{\perp}(\mathcal{F}_{n-1}R) = \mathcal{F}^n R^{\vee}, \;({\mathcal{F}_{n-1}R^{\vee}})^{\perp} = \mathcal{F}^nR.
\end{align}
Since the coalgebras $R^{\vee}$ and $R$ are $\mathbb{N}_0$-graded, it follows that
$$\Delta_{R^{\vee}}(\mathcal{F}^{2n} R^{\vee}) \subseteq \mathcal{F}^{n} R^{\vee} \otimes R^{\vee} + R^{\vee} \otimes \mathcal{F}^{n} R^{\vee} ,\;\Delta_{R}(\mathcal{F}^{2n} R) \subseteq \mathcal{F}^n R \otimes R + R \otimes \mathcal{F}^n R$$
for all $n \geq 0$. Thus $\Delta_R$ and $\Delta_{R^{\vee}}$ are continuous.
Hence the pair $(R,R^{\vee})$ together with the bilinear form $\langle \;,\; \rangle $ is a dual pair of Hopf algebras in $\ydH$. Moreover, the remaining structure maps of $R^{\vee}$ and of $R$, that is multiplication, unit map, augmentation and antipode, are all continuous, since they are $\mathbb{N}_0$-graded. Here, the ground field is graded by $\fie(0) = \fie$, and $\fie(n) = 0$ for all $n \geq 1$.
Since $R(0)$ is a finite-dimensional Hopf algebra in $\ydH$, the antipode of $R(0)$ is bijective by \cite[Proposition 7.1]{inp-Takeuchi00}. Hence the Hopf subalgebra $\mathcal{F}_0 R\# H$ of $R \# H$ has bijective antipode by \eqref{bigS2} and \eqref{ydiso}. The filtration
$$\mathcal{F}_0R \# H \subseteq \mathcal{F}_1R \# H \subseteq \mathcal{F}_2R \# H \subseteq \cdots \subseteq R \# H$$
is a coalgebra filtration, and by the argument in \cite[Remark 2.1]{p-HeckSchn09}, the antipodes of $R \# H$ and of $R$ are bijective. The same proof shows that the antipodes of $R^{\vee} \# H$ and of $R^{\vee}$ are bijective.
\end{examp}
Let $(R,R^{\vee})$ together with $\langle \;,\; \rangle : R^{\vee} \ot R \to \fie$ be a dual pair of Hopf algebras in $\ydH$.
We denote by ${^R(\ydH)}$ the category of left $R$-comodules in the monoidal category $\ydH$, and by ${_{R^{\vee}}(\ydH)}_{\mathrm{rat}}$ the category of left $R^{\vee}$-modules in $\ydH$ which are rational as $R^{\vee}$-modules.
\begin{propo}\label{propo:rational}
\begin{enumerate}
\item For all $M \in {^R(\ydH)}$ let $D(M)=M$ as an object in $\ydH$ with $R^{\vee}$-module structure given by
\begin{align*}
\xi m = \langle \xi,m\sws{-1} \rangle m\sws0
\end{align*}
for all $\xi \in R^{\vee},m \in M$,where the left $R$-comodule structure of $M$ is denoted by $\delta_M(m)= m\sws{-1} \ot m\sws0$. Then $D(M) \in {_{R^{\vee}}(\ydH)}_{\mathrm{rat}}$.
\item The functor
\begin{align*}
D : {^R(\ydH)} \to {_{R^{\vee}}(\ydH)_{\mathrm{rat}}}
\end{align*}
mapping $M \in {^R(\ydH)}$ onto $D(M)$, and with $D(f) = f$ for all morphisms in ${^R(\ydH)}$, is an isomorphism of categories.
\end{enumerate}
\end{propo}
\begin{proof}
This follows from Lemma \ref{lem:rational} together with \eqref{pair2} -- \eqref{pair5}.
\end{proof}
\begin{lemma}\label{lem:rationaltensor}
The trivial left $R \# H$-module $\fie$ is rational as an $R$-module (by restriction). Let $V,W$ be left $R \# H$-modules, and $V \ot W$ the left $R \#H$-module given by diagonal action. If $V$ and $W$ are rational as left $R$-modules, then $V \ot W$ is a rational $R$-module.
\end{lemma}
\begin{proof}
The trivial $R$-module $\fie$ is rational since for all $x \in (\fie 1_{R^{\vee}})^{\perp}$,
\begin{align*}
x 1_{\fie} = \varepsilon(x) = \langle 1_{R^{\vee}},x \rangle = 0
\end{align*}
by \eqref{pair4}.
To prove that $V \ot W$ is rational as an $R$-module, let $v \in V,w \in W$. It is enough to show that $E^{\perp} (v \ot w) =0$ for some $E \in \mathcal{E}_{R^{\vee}}$. Since $V$ and $W$ are rational $R$-modules, there are $E_1, E_2 \in \mathcal{E}_{R^{\vee}}$ with $E_1^{\perp} v =0, E_2^{\perp} w =0$. Let $E_3 \in \mathcal{E}_{R^{\vee}}$ with $E_1 + E_2 \subseteq E_3$. Then $E_3^{\perp}v=0, E_3^{\perp}w=0$. By \eqref{pair8} there is a subspace $E \in \mathcal{E}_{R^{\vee}}$ such that
\begin{align}
&\Delta_R(E^{\perp}) \subseteq E_3^{\perp} \ot R + R \ot E_3^{\perp}.\label{continuity}
\end{align}
Let $r \in E^{\perp}$. Then by \eqref{cosmash},
\begin{align}
r(v \ot w) = r\swo1 {r\swo2}\sw{-1} v \ot {r\swo2}\sw0 w.\label{action1}
\end{align}
We rewrite the first tensorand on the right-hand side in \eqref{action1} according to the multiplication rule \eqref{smash2} for elements in $R \# H$. Then the equality $r(v \ot w)=0$ follows from \eqref{continuity}, \eqref{action1} and \eqref{pair9}.
\end{proof}
Lemma \ref{lem:rationaltensor} also holds for $R^{\vee}$ instead of $R$ using \eqref{pair5} and \eqref{pair7} instead of \eqref{pair4} and \eqref{pair8}.
\begin{lemma}\label{lem:inversepair}
Assume that the antipodes of $R$ and of $R^{\vee}$ are bijective. Define $\langle\;,\;\rangle' : R \ot R^{\vee} \to \fie$ by
\begin{align}
\langle x,\xi \rangle' = \langle \xi,\mathcal{S}^2(x) \rangle\label{defin:inversepair}
\end{align}
for all $x \in R,\xi \in R^{\vee}$, where $\mathcal{S}$ is the antipode of $R \# H$. Then $(R^{\vee},R)$ together with $\langle \;,\; \rangle' : R \ot R^{\vee} \to \fie$ is a dual pair of Hopf algebras in $\ydH$.
\end{lemma}
\begin{proof}
Using \eqref{bigS2}, \eqref{pair1} -- \eqref{pair5} for $\langle\;,\; \rangle'$ are easily checked.
We denote by $\perp$ (respectively $\perp'$) the complements with respect to $\langle\;,\; \rangle$ (respectively to $\langle \;,\; \rangle'$).
To prove \eqref{pair7} for $\langle\;,\; \rangle'$, we note that by \eqref{defin:inversepair} for all finite-dimensional subspaces $E \subseteq R$, $E^{\perp'} = {^{\perp}(\mathcal{S}^2(E))}$. By assumption and \eqref{bigS2}, $\mathcal{S}^2$ induces an isomorphism on $R$. Hence the weak topologies of $R^{\vee}$ defined with respect to $\langle\;,\; \rangle$ and to $\langle\;,\; \rangle'$ coincide, and \eqref{pair7} for $\langle\;,\; \rangle'$ follows.
To prove \eqref{pair8} for $\langle\;,\; \rangle'$, we again show that the weak topologies of $R$ defined with respect to $\langle\;,\; \rangle$ and to $\langle\;,\; \rangle'$ coincide. For all $x \in R, \xi \in R^{\vee}$,
\begin{align*}
\langle x, \xi \rangle' &= \langle \xi, \mathcal{S}^2(x) \rangle && \\
&= \langle \xi, \mathcal{S}_R^2(\mathcal{S}(x\sw{-1}) \lact x\sw0) \rangle && \text{(by \eqref{bigS2})}\\
&= \langle \mathcal{S}_{R^{\vee}}^2(\xi), \mathcal{S}(x\sw{-1}) \lact x\sw0 \rangle &&\text{(by \eqref{pair6})}.
\end{align*}
Hence for all $E_1 \in \mathcal{E}_{R^{\vee}}$,
\begin{align*}
{^{\perp'}E_1} &= \{ x \in R \mid \mathcal{S}(x\sw{-1}) \lact x\sw0 \in (\mathcal{S}_{R^{\vee}}^2(E_1))^{\perp}\}\\
&= (\mathcal{S}_{R^{\vee}}^2(E_1))^{\perp},
\end{align*}
where the second equality follows from \eqref{ydiso} and \eqref{pair9}.
This proves our claim, since $\{\mathcal{S}_{R^{\vee}}^2(E_1) \mid E_1 \in \mathcal{E}_{R^{\vee}} \}$
is a cofinal subset of $\mathcal{E}_{R^{\vee}}$ by the bijectivity of $\mathcal{S}_{R^{\vee}}$.
\end{proof}
\begin{comment}
(3)(a) Formally we write
\begin{align*}
f(v)= v\sws{-1} \ot v\sws0,\; g(\xi \ot v) = \langle \xi,v\sws{-1} \rangle v\sws0,
\end{align*}
for all $v \in V,\xi \in R^{\vee}$. Then $g$ is left $H$-linear if and only if
\begin{align*}
\langle \xi,v\sws{-1}\rangle hv\sws0 &= \langle h\sw1 \lact \xi,(h\sw2 v)\sws{-1} \rangle (h\sw2v)\sws0 &&\\
&= \langle \xi,\mathcal{S}(h\sw1) \lact (h\sw2v)\sws{-1} \rangle (h\sw2v)\sws0,&(\text{by } \eqref{pair2})
\end{align*}
for all $\xi \in R^{\vee}, h \in H,v \in V$, or equivalently,
\begin{align*}
\mathcal{S}(h\sw1) \lact (h\sw2v)\sws{-1} \ot (h\sw2v)\sws0 = v\sws{-1} \ot hv\sws0
\end{align*}
for all $h \in H,v \in V$, that is, $f$ is $H$-linear.
(b) is proved similarly using \eqref{pair3}.
\end{comment}
\section{Review of monoidal categories and their centers}\label{sec:categories}
Our reference for monoidal categories is \cite{b-Kassel1}, where the term tensor categories is used. Let $\cC$ and $\mathcal{D}$ be strict monoidal categories, and $F : \cC \to \mathcal{D}$ a functor. We assume that $F(I)$ is the unit object in $\mathcal{D}$. Let
$$\varphi = (\varphi_{X,Y} : F(X) \ot F(Y) \to F(X \ot Y))_{X,Y \in \cC}$$
be a family of natural isomorphisms. Then $(F,\varphi)$ is a {\em monoidal functor} if for all $U,V,W \in \cC$
\begin{align}
\varphi_{I,U} = \id_{F(U)} = \varphi_{U,I},
\end{align}
and the diagram
\begin{align}\label{monoidal}
\begin{CD}
F(U) \ot F(V) \ot F(W) @>{\id \ot \varphi_{V,W}}>> F(U) \ot F(V \ot W)\\
@V{\varphi_{U,V} \ot \id}VV @V{\varphi_{U,V \ot W}}VV\\
F(U \ot V) \ot F(W) @>{\varphi_{U \ot V,W}}>> F(U \ot V \ot W)
\end{CD}
\end{align}
commutes. A monoidal functor $(F,\varphi)$ is called {\em strict} if $\varphi = \id$. If $\cC$ and $\mathcal{D}$ are strict braided monoidal categories, then a monoidal functor $(F,\varphi)$ is {\em braided} if
for all $X,Y \in \cC$ the diagram
\begin{align}\label{braidedmonoidal}
\begin{CD}
F(X) \ot F(Y) @>{\varphi_{X,Y}}>> F(X \ot Y)\\
@V{c_{F(X),F(Y)}}VV @V{F(c_{X,Y})}VV\\
F(Y) \ot F(X) @>{\varphi_{Y,X}}>> F(Y \ot X)
\end{CD}
\end{align}
commutes. A {\em monoidal equivalence} (respectively {\em isomorphism}) is a monoidal functor $(F,\varphi)$ such that $F$ is an equivalence (respectively an isomorphism) of categories. Recall that a functor $F : \cC \to \mathcal{D}$ is called an isomorphism if there is a functor $G : \mathcal{D} \to \cC$ with $FG = \id_{\mathcal{D}}$ and $GF = \id_{\cC}$.
A {\em braided monoidal equivalence} (respectively {\em isomorphism}) is a monoidal equivalence (respectively isomorphism) $(F,\varphi)$ such that $(F,\varphi)$ is a braided monoidal functor.
If $(F,\varphi) : \cC \to \mathcal{D}$ and $(G ,\psi): \mathcal{D} \to \mathcal{E}$ are monoidal (respectively braided monoidal) functors, then the composition
\begin{align}\label{monoidalcomposition}
(GF,\lambda) : \cC \to \mathcal{E}, \;\lambda_{X,Y} = G(\varphi_{X,Y})\psi_{F(X),F(Y)}, \text{ for all }X,Y \in \cC,
\end{align}
is a monoidal (respectively braided monoidal) functor.
Let $(F,\varphi) : \cC \to \mathcal{D}$ be a monoidal isomorphism of categories with inverse functor $G : \mathcal{D} \to \cC$. Then $(G,\psi)$ is a monoidal functor with
\begin{align}
\psi_{U,V}= G(\varphi_{G(U),G(V)})^{-1}: G(U) \ot G(V) \to G(U \ot V)\label{inversemonoidal}
\end{align}
for all $U,V \in \mathcal{D}$.
For later use we note the following lemma.
\begin{lemma}\label{lem:diagram}
Let $\cC,\mathcal{D}$ and $\mathcal{E}$ be strict monoidal and braided categories, and $F : \cC \to \mathcal{D}$ a functor. Let $(G,\psi) : \mathcal{D} \to \mathcal{E}$ and $(GF, \lambda) : \cC \to \mathcal{E}$ be braided monoidal functors. Assume that the functor $G$ is fully faithful. Then there is exactly one family $\varphi = (\varphi_{X,Y})_{X,Y \in \cC}$ such that $(F,\varphi)$ is a braided monoidal functor and $$(GF,\lambda)=(\cC \xrightarrow{(F,\varphi)} \mathcal{D} \xrightarrow{(G,\psi)} \mathcal{E}).$$
\end{lemma}
\begin{proof}
Since $G$ is fully faithful, for all $X,Y \in \cC$ there is exactly one morphism $\varphi_{X,Y} : F(X) \ot F(Y) \to F(X \ot Y)$ with $\lambda_{X,Y} = G(\varphi_{X,Y})\psi_{F(X),F(Y)}$. Then one checks that $(F,\varphi)$ is a braided monoidal functor.
\end{proof}
We recall the notion of the (left) {\em center} $\mathcal{Z}(\cC)$ of a strict monoidal category $\cC$ with tensor product $\ot$ and unit object $I$ (see \cite[XIII.4]{b-Kassel1}, where the right center is discussed). Objects of $\mathcal{Z}(\cC)$ are pairs $(M,\gamma)$, where $M \in \cC$, and
$$\gamma = (\gamma_X : M \ot X \to X \ot M)_{X \in \cC}$$
is a family of natural isomorphisms such that
\begin{align}
\gamma_{X \ot Y} &= (\id_X \ot \gamma_Y)(\gamma_X \ot \id_Y)\label{objectcenter}
\end{align}
for all $X,Y \in \cC$.
Note that by \eqref{objectcenter}
\begin{align}
\gamma_I = \id_M\label{objectcenterI}
\end{align}
for all $(M,\gamma) \in \mathcal{Z}(\cC)$.
A morphism $f : (M,\gamma) \to (N,\lambda)$ between objects $(M,\gamma)$ and $(N,\lambda)$ in $\mathcal{Z}(\cC)$ is a morphism $f : M \to N$ in $\cC$ such that
\begin{align}\label{morphismcenter}
(\id_X \ot f) \gamma_X = \lambda_X (f \ot \id_X)
\end{align}
for all $X \in \cC$. Composition of morphisms is given by the composition of morphisms in $\cC$. The category $\mathcal{Z}(\cC)$ is braided monoidal. For objects $(M,\gamma),(N,\lambda)$ in $\mathcal{Z}(\cC)$ the tensor product is defined by
\begin{align}
(M,\gamma) \ot (N,\lambda) &= (M \ot N, \sigma),\label{center1}\\
\sigma_X &= (\gamma_X \ot \id_N)(\id_M \ot \lambda_X)\label{center2}
\end{align}
for all $X \in \cC$. The pair $(I, \id)$, where $\id_X = \id_{I \ot X}$ for all $X \in \cC$, is the unit in $\mathcal{Z}(\cC)$. The braiding is defined by
\begin{align}\label{braidingcenter}
\gamma_N : (M,\gamma) \ot (N,\lambda) \to (N,\lambda) \ot (M,\gamma).
\end{align}
We note that a monoidal isomorphism $(F,\varphi) : \cC \to \mathcal{D}$ defines in the natural way a braided monoidal isomorphism between the centers of $\cC$ and $\mathcal{D}$. For all objects $(M,\gamma) \in \cC$ let
\begin{align}
F^{\mathcal{Z}}(M,\gamma) = (F(M), \ti{\gamma}),
\end{align}
and for all $X \in \cC$, the isomorphism $\ti{\gamma}_{F(X)}$ is defined by the commutative diagram
\begin{align}
\begin{CD}
F(M) \ot F(X) @>{\ti{\gamma}_{F(X)}}>> F(X) \ot F(M)\\
@V{\varphi_{M,X}}VV @V{\varphi_{X,M}}VV\\
F(M \ot X) @>{F(\gamma_X)}>> F(X \ot M).
\end{CD}\label{defin:tilde}
\end{align}
For morphisms $f$ in $\mathcal{Z}(\cC)$ we define $F^{\mathcal{Z}}(f) = F(f)$. For $(M,\gamma),(N,\lambda) \in \mathcal{Z}(\cC)$ let
\begin{align}
{\varphi}^{\mathcal{Z}}_{(M,\gamma),(N,\lambda)} = \varphi_{M,N}.\label{monoidalcenter2}
\end{align}
Then the next lemma follows by carefully writing down the definitions.
\begin{lemma}\label{lem:center}
Let $(F,\varphi) : \cC \to \mathcal{D}$ be a monoidal isomorphism. Then
\begin{align*}
(F^{\mathcal{Z}},{\varphi}^{\mathcal{Z}}) : \mathcal{Z}(\cC) \to \mathcal{Z}(\mathcal{D})
\end{align*}
is a braided monoidal isomorphism.
\end{lemma}
Finally we note that we may view the categories of vector spaces and of modules or comodules over a Hopf algebra as strict monoidal categories since the associativity and unit constraints are given by functorial maps.
\section{Relative Yetter-Drinfeld modules}\label{sec:relative}
In this section we assume that $B,C$ are Hopf algebras with bijective antipode, $\rho: B \to C$ is a Hopf algebra homomorphism, and $\mathcal{R} \subseteq {_B\cM}$ is a full subcategory of the category of left $B$-modules closed under tensor products and containing the trivial left $B$-module $\fie$.
\begin{defin}\label{defin:ydBC}
We denote by $ {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}$ the following monoidal category (depending on the map $\rho$). Objects of $ {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}$ are left $B$-modules and left $C$-comodules $M$ with comodule structure $\delta : M \to C \ot M, m \mapsto m\sw{-1} \ot m\sw0,$ such that $M \in \mathcal{R}$ as a module over $B$ and
\begin{align}\label{relativeYD}
\delta(bm) = \rho(b\sw1) m\sw{-1} \rho\mathcal{S}(b\sw3) \ot b\sw2 m\sw0
\end{align}
for all $m \in M$ and $b \in B$. Morphisms are left $B$-linear and left $C$-colinear maps.
The tensor product $M \ot N$ of $M,N \in {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}$ is the tensor product of the vector spaces $M,N$ with diagonal action of $B$ and diagonal coaction of $C$.
We define $ {}^{\phantom{.}C}_B\mathcal{YD} = {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}$, when $\mathcal{R} = {_B\cM}$ is the category of all $B$-modules. The full subcategory of $ {}^{\phantom{.}{B}}_{B}\mathcal{YD}$ consisting of all objects $M \in {}^{\phantom{.}{B}}_{B}\mathcal{YD}$ with $M \in \mathcal{R}$ as a $B$-module is denoted by $ {}^{\phantom{.}{B}}_{B}\mathcal{YD}_{\mathcal{R}}$.
\end{defin}
The Hopf algebra map $\rho : B \to C$ defines a functor
\begin{align}
{^{\rho}(\;)} : {}^{\phantom{.}{B}}_{B}\mathcal{YD} \to {}^{\phantom{.}C}_B\mathcal{YD},
\end{align}
mapping an object $M \in {}^{\phantom{.}{B}}_{B}\mathcal{YD}$ onto ${^{\rho}M}$, where ${^{\rho}M} = M$ as a $B$-module, and where ${^{\rho}M}$ is a $C$-comodule by $M \xrightarrow{\delta_M} B \ot M \xrightarrow{\rho \ot \id_M} C \ot M$.
Let
\begin{align}
\Phi : {}^{\phantom{.}{B}}_{B}\mathcal{YD}_{\mathcal{R}} \to \mathcal{Z}( {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}})
\end{align}
be the functor defined on objects $M \in {}^{\phantom{.}{B}}_{B}\mathcal{YD}_{\mathcal{R}}$ by
\begin{align}
\Phi(M) = (^{\rho}M,c_M),\;
c_{M,X} : M \ot X \to X \ot M,\; m \ot x \mapsto m\sw{-1} x \ot m \sw0,\label{defin:Phi}
\end{align}
for all $X \in {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}$, where $M \to B \ot M,\;m \mapsto m\sw{-1} \ot m\sw0,$ denotes the $B$-comodule structure of $M$. We let $\Phi(f) = f$ for morphisms $f$ in $ {}^{\phantom{.}{B}}_{B}\mathcal{YD}_{\mathcal{R}}$. It is easy to see that $\Phi$ is a well-defined functor.
We need the existence of enough objects in $ {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}$.
\begin{defin}
The category $ {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}$ is called {\em $B$-faithful} if the following conditions are satisfied.
\begin{align}
\text{For any }0 \neq b \in B, \;&bX \neq 0 \text{ for some }X \in {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}.\label{faithful1}\\
\text{For any }0 \neq t \in B \ot B, \;&t(X \ot Y) \neq 0 \text{ for some }X,Y \in {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}.\label{faithful2}
\end{align}
\end{defin}
\begin{examps}\label{exa:regularrep}
(1) Let $B$ be the left $B$-module with the regular representation, and the left $C$-comodule with the coadjoint coaction
\begin{align}
B \to C \ot B, \;b \mapsto \rho(b\sw1 \mathcal{S}(b\sw3)) \ot b\sw2.\label{coadjoint}
\end{align}
Then $B$ is an object in $ {}^{\phantom{.}C}_B\mathcal{YD}$. Since $bB\neq 0$, $t (B \ot B) \neq 0$ for all
$ 0 \neq b \in B$, $0 \neq t \in B \ot B$, the category $ {}^{\phantom{.}C}_B\mathcal{YD}$ is $B$-faithful.
(2) Let
\begin{align*}
R = \bigoplus_{n \in \mathbb{N}_0} R(n)
\end{align*}
be an $\mathbb{N}_0$-graded Hopf algebra in $\ydH$, and $A = R \# H$ the bosonization. We define $ {}^{\phantom{.}H}_A\mathcal{YD}$ with respect to the Hopf algebra map $\pi : A \to H$. As in (1), $A$ with the regular representation and the coadjoint coaction with respect to $\pi$ defined in \eqref{coadjoint}, is an object in $ {}^{\phantom{.}H}_A\mathcal{YD}$. The $H$-coaction $\delta_A : A \to H \ot A$ can be computed explicitly as
\begin{align*}
\delta_A(rh) = r\sw{-1} h\sw1 \mathcal{S}(h\sw3) \ot r\sw0 h\sw2
\end{align*}
for all $r \in R,h \in H$.
Hence it follows that for all $n \geq 0$,
\begin{align*}
\mathcal{F}^nA = \bigoplus_{i \geq n} R(i) \ot H \subseteq A
\end{align*}
is an ideal and a left $H$-subcomodule of $A \in {}^{\phantom{.}H}_A\mathcal{YD}$. Note that
\begin{align}
\bigcap_{n \geq 0} \mathcal{F}^nA = 0,\;\bigcap_{n \geq 0} (\mathcal{F}^nA \ot A + A \ot \mathcal{F}^nA) = 0.
\end{align}
Hence for any $0 \neq a \in A, 0 \neq t \in A \ot A$ there is an integer $n \geq 0$ with
\begin{align*}
a(A/\mathcal{F}^nA) \neq 0, \; t (A/\mathcal{F}^nA \ot A/\mathcal{F}^nA) \neq 0.
\end{align*}
Thus $ {}^{\phantom{.}H}_A\mathcal{YD}_{\mathcal{R}}$ is $A$-faithful for all full subcategories $\mathcal{R}$ of ${_A\cM}$ such that $A/\mathcal{F}^nA \in {}^{\phantom{.}H}_A\mathcal{YD}_{\mathcal{R}}$ for all $n \geq 0$.
Note that for all $n \geq0$, $A/\mathcal{F}^nA$ as an $R$-module is annihilated by $\oplus _{i \geq n} R(i)$.
\end{examps}
\begin{propo}\label{propo:modulecenter}
Assume that $ {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}$ is $B$-faithful.
\begin{enumerate}
\item The functor $\Phi : {}^{\phantom{.}{B}}_{B}\mathcal{YD}_{\mathcal{R}} \to \mathcal{Z}( {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}})$ is fully faithful, strict monoidal and braided.\label{modulecenter1}
\item Let $(M,\gamma) \in \mathcal{Z}( {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}})$ with comodule structure $\delta_M : M \to C \ot M$. Assume that there is a $\fie$-linear map $\widetilde{\delta}_M : M \to B \ot M$, denoted by $\widetilde{\delta}_M(m) = m\swe{-1} \ot m\swe0$ for all $m \in M$, with
\begin{align}
\gamma_X (m \ot x) &= m\swe{-1} x \ot m\swe0\label{module1}\\
\delta_M &= (\rho \ot \id_M)\widetilde{\delta}_M,\label{module2}
\end{align}
for all $X \in {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}, x \in X$ and $m \in M$. Then the map $\widetilde{\delta}_M$ is uniquely determined. Let $\ti{M} =M$ as a $B$-module. Then $\ti{M} \in {}^{\phantom{.}{B}}_{B}\mathcal{YD}_{\mathcal{R}}$ with $B$-comodule structure $\widetilde{\delta}_M$, and $\Phi(\ti{M}) = (M,\gamma)$.\label{modulecenter2}
\end{enumerate}
\end{propo}
\begin{proof}
(1) It is clear from the definitions that $\Phi$ is strict monoidal and braided, see \eqref{braidingright}, \eqref{braidingcenter} and \eqref{defin:Phi}. To prove that $\Phi$ is fully faithful, let $M,N \in {}^{\phantom{.}{B}}_{B}\mathcal{YD}$, and $f : \Phi(M) \to \Phi(N)$ a morphism in $\mathcal{Z}( {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}})$. In particular, $f : M \to N$ is a left $B$-linear and left $C$-colinear homomorphism. We have to show that $f$ is left $B$-colinear.
Let $X \in {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}$, $m \in M$ and $x \in X$. Then
\begin{align}\label{equationfaithful}
f(m)\sw{-1}x \ot f(m)\sw0 = m\sw{-1}x \ot f(m\sw0),
\end{align}
since $f$ is a morphism in $\mathcal{Z}( {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}})$. It follows from \eqref{equationfaithful} and \eqref{faithful1} that
\begin{align*}
f(m)\sw{-1} \ot f(m)\sw0 = m\sw{-1} \ot f(m\sw0)
\end{align*}
in $B \ot M$ for all $m \in M$, that is, $f$ is $B$-colinear.
(2) The map $\widetilde{\delta}_M$ is uniquely determined by \eqref{faithful1} and \eqref{module1}. We have to show that $\ti{M}$ is a $B$-comodule with structure map $\widetilde{\delta}_M$, and that $\ti{M} \in {}^{\phantom{.}{B}}_{B}\mathcal{YD}_{\mathcal{R}}$ with comodule structure $\widetilde{\delta}_M$ and the given $B$-module structure.
Let $X,Y \in {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}, x \in X,y \in Y$ and $m \in M$. By \eqref{objectcenter},
\begin{align*}
\Delta(m\swe{-1})(x \ot y) \ot m\swe0 = m\swe{-1}x \ot m \swe0{\swe{-1}}y \ot {m\swe0}{\swe0}.
\end{align*}
Hence $\widetilde{\delta}_M$ is coassociative by \eqref{faithful2}. Let $\fie \in {}^{\phantom{.}C}_B\mathcal{YD}{_\mathcal{R}}$ be the trivial object. Then by \eqref{objectcenterI},
\begin{align*}
1 \ot m = \gamma_{\fie}(m \ot 1) = m\swe{-1} 1 \ot m \swe0 = 1 \ot \varepsilon(m\swe{-1}) m\sw0
\end{align*}
for all $m \in M$. Hence the comultiplication $\widetilde{\delta}_M$ is counitary.
For all $X \in {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}}$, the map $\gamma_X$ is $B$-linear. Hence
\begin{align*}
(b\sw1m)\swe{-1} b\sw2x \ot (b\sw1m)\swe0 = b\sw1 m\swe{-1}x \ot b\sw2m \swe0
\end{align*}
for all $b \in B, m \in M$ and $x \in X$. Hence $\ti{M} \in {}^{\phantom{.}{B}}_{B}\mathcal{YD}_{\mathcal{R}}$ by \eqref{faithful1}.
Finally $\Phi(\ti{M}) = (M,\gamma)$ by \eqref{module1} and \eqref{module2}.
\end{proof}
\begin{remar}
In general, $\Phi : {}^{\phantom{.}{B}}_{B}\mathcal{YD}_{\mathcal{R}} \to \mathcal{Z}( {}^{\phantom{.}C}_B\mathcal{YD}_{\mathcal{R}})$ is not an equivalence.
However, in the case when $C = \fie$ and $\rho = \varepsilon$, hence $ {}^{\phantom{.}C}_B\mathcal{YD} = {_B\cM}$,
it is well-known (compare \cite{b-Kassel1} XIII.5) that $\Phi : {}^{\phantom{.}{B}}_{B}\mathcal{YD} \to \mathcal{Z}({_B\cM})$
is an equivalence. Indeed, let $(M,\gamma) \in \mathcal{Z}({_B\cM})$. Define
$m\swe{-1} \ot m\swe0 = \gamma_B(m \ot 1)$ for all $m \in M$,
where the $B$-module structure of $B \in {_B\cM}$ is given by multiplication.
Then for any $X \in {_B\cM}$ and $x \in X$ there is a $B$-linear map
$f : B \to X$ with $f(1) =x$, and $\gamma_X(m \ot x) = m\swe{-1} x \ot m\swe0$
by the naturality of $\gamma$. This proves \eqref{module1}.
Similarly, \eqref{module2} follows by considering the trivial $B$-module $\fie$ and the $B$-linear map $\varepsilon$.
Moreover, ${_B\cM}$ is $B$-faithful by Example \ref{exa:regularrep} (1).
Thus in this case the assumption in Proposition \ref{propo:modulecenter} \eqref{modulecenter2} is always satisfied.
\end{remar}
\begin{defin}\label{defin:rydBC}
We denote by $\mathcal{YD}^{\phantom{.}C}_B$ the monoidal category whose objects are right $B$-modules and right $C$-comodules $M$ with comodule structure denoted by $\delta: M \to M \ot C,\; m \mapsto m\sw{0} \ot m\sw{-1}$, such that
\begin{align}
\delta(mb) = m\sw0 b\sw2 \ot \mathcal{S}(\rho(b\sw1)) m\sw{1} \rho(b\sw3)\label{rydBC}
\end{align}
for all $m \in M$ and $b \in B$. Morphisms are right $B$-linear and right $C$-colinear maps.
The tensor product $M \ot N$ of $M,N \in \mathcal{YD}^{\phantom{.}C}_B$ is the tensor product of the vector spaces $M,N$ with diagonal action of $B$ and diagonal coaction of $C$. The monoidal category $\mathcal{YD}^{\phantom{.}C}_C$ is braided by \eqref{braidingright}.
\end{defin}
We define a functor
\begin{align}
\Psi : \mathcal{YD}^{\phantom{.}C}_C \to \mathcal{Z}(\mathcal{YD}^{\phantom{.}C}_B)
\end{align}
on objects $M \in \mathcal{YD}^{\phantom{.}C}_C$ by
\begin{align}
\Psi(M) = (M_{\rho},c_M),\;
c_{M,X} : M \ot X \to X \ot M,\; m \ot x \mapsto x\sw0 \ot m x\sw1,\label{defin:Psi}
\end{align}
for all $X \in \mathcal{YD}^{\phantom{.}C}_B$, where $M_{\rho}$ is $M$ as a $B$-module via $\rho$. We let $\Psi(f) = f$ for morphisms $f$ in $\mathcal{YD}^{\phantom{.}C}_C$.
\begin{examp}\label{exa:regularcorep}
Let $C$ be the regular corepresentation with right $C$-comodule structure given by the comultiplication $\Delta_C$ of $C$. We define a right $B$-module structure on $C$ by the adjoint action, that is
\begin{align}
c \vartriangleleft b = \rho\mathcal{S}(b\sw1)c \rho(b\sw2)
\end{align}
for all $c \in C,b \in B$. Then $C$ is an object in $\mathcal{YD}^{\phantom{.}C}_B$.
\end{examp}
\begin{propo}\label{propo:comodulecenter}
\begin{enumerate}
\item The functor $\Psi : \mathcal{YD}^{\phantom{.}C}_C \to \mathcal{Z}(\mathcal{YD}^{\phantom{.}C}_B)$
is fully faithful, strict monoidal and braided. \label{comodulecenter1}
\item Let $(M,\gamma) \in \mathcal{Z}(\mathcal{YD}^{\phantom{.}C}_B)$ with module structure $\mu_M : M \ot B \to M$. Assume that there is a $\fie$-linear map $\widetilde{\mu}_M : M \ot C \to M$ such that
\begin{align}
\gamma_X (m \ot x) &= x\sw0 \ot \widetilde{\mu}_M(m \ot x\sw1),\label{comodule1}\\
\mu_M &=\widetilde{\mu_M} (\id \ot \rho)\label{comodule2}
\end{align}
for all $X \in \mathcal{YD}^{\phantom{.}C}_B, x \in X$ and $m \in M$. Then the map $\widetilde{\mu}_M$ is uniquely determined.
Let $\ti{M} = M$ as a $C$-comodule.
Then $\ti{M} \in \mathcal{YD}^{\phantom{.}C}_C$ with $C$-module structure $\widetilde{\mu}_M$, and
$\Psi(\ti{M}) = (M,\gamma)$.
\label{comodulecenter2}
\end{enumerate}
\end{propo}
\begin{proof}
(1) Again it is clear that $\Psi$ is strict monoidal and braided. To see that $\Psi$ is fully faithful, let $M,N \in \mathcal{YD}^{\phantom{.}C}_C$ and $f : \Psi(M) \to \Psi(N)$ a morphism in $\mathcal{Z}(\mathcal{YD}^{\phantom{.}C}_B)$. We have to show that $f$ is right $C$-linear. Let $X =C \in \mathcal{YD}^{\phantom{.}C}_B$ in Example \ref{exa:regularcorep}. Since $f$ is a morphism in $\mathcal{Z}(\mathcal{YD}^{\phantom{.}C}_B)$,
$$x\sw1 \ot f(mx\sw2) = x\sw1 \ot f(m)x\sw2$$
for all $x \in C, m \in M$. By applying $\varepsilon \ot \id$ to this equation it follows that $f$ is right $C$-linear.
(2) Let $C \in \mathcal{YD}^{\phantom{.}C}_B$ as in Example \ref{exa:regularcorep}. Then $(\varepsilon \ot \id)\gamma_C = \widetilde{\mu}_M$. Hence $\widetilde{\mu}_M$ is uniquely determined. Let $X=Y = C \in \mathcal{YD}^{\phantom{.}C}_B$. By \eqref{objectcenter}
$$x\sw1 \ot y\sw1 \ot \widetilde{\mu}_M(m \ot x\sw2y\sw2) = x\sw1 \ot y\sw1 \ot \widetilde{\mu}_M(\widetilde{\mu}_M(m \ot x\sw2) \ot y\sw2)$$
for all $x,y \in C, m \in M$. By applying $\varepsilon \ot \varepsilon \ot \id$ it follows that $\widetilde{\mu}_M$ is associative. By \eqref{objectcenterI}, $\widetilde{\mu}_M$ is unitary.
We will write $mc = \widetilde{\mu}_M(m \ot c)$ for all $m \in M,c \in C$.
Since $\gamma_C$ is right $C$-colinear,
$$x\sw1 \ot (m x\sw3)\sw0 \ot x\sw2 (m x\sw3)\sw1 = x\sw1 \ot m\sw0 x\sw2 \ot m\sw1 x\sw3$$
for all $x \in C, m \in M$. By applying $\varepsilon \ot \id$ it follows that $\ti{M} \in \mathcal{YD}^{\phantom{.}C}_C$.
Finally $\Psi(\ti{M}) = (M,\gamma)$ by \eqref{comodule1} and \eqref{comodule2}.
\end{proof}
We fix an odd integer $l$, and assume that the antipodes of $B$ and $C$ are bijective.
Let $M \in \mathcal{YD}^{\phantom{.}C}_B$ with $C$-comodule structure $\delta_M : M \to M \ot C, \;m \mapsto m\sw{0} \ot m\sw1$. We define an object $S_l(M) \in {}^{\phantom{.}C}_B\mathcal{YD}$ by $S_l(M) = M$ as a vector space with left
$B$-action and left $C$-coaction given by
\begin{align}
bm &= m \mathcal{S}^{-l}(b),\label{leftright1}\\
\delta_{S_l(M)}(m) &= \mathcal{S}^l(m\sw1) \ot m\sw0\label{leftright2}
\end{align}
for all $b \in B,m\in M$. For morphisms $f$ in $\mathcal{YD}^{\phantom{.}C}_B$ we set $S_l(f) = f$.
Let $M \in {}^{\phantom{.}C}_B\mathcal{YD}$ with comodule structure $\delta_M : M \to C \ot M, \; m \mapsto m\sw{-1} \ot m\sw0$. We define $S_l^{-1}(M) = M$ as a vector space with right $B$-action and right $C$-coaction given by
\begin{align}
mb &= \mathcal{S}^l(b)m,\\
\delta_{S_l^{-1}(M)} (m) &= m \sw0 \ot \mathcal{S}^{-l}(m\sw{-1})
\end{align}
for all $b \in B,m \in M$. For morphisms $f$ in $ {}^{\phantom{.}C}_B\mathcal{YD}$ we set $S_l^{-1}(f) = f$.
\begin{lemma}\label{lem:leftright}
Let $l$ be an odd integer, and assume that the antipodes of $B$ and $C$ are bijective.
\begin{enumerate}
\item The functor $S_l : \mathcal{YD}^{\phantom{.}C}_B \to {}^{\phantom{.}C}_B\mathcal{YD}$ mapping an object $M \in \mathcal{YD}^{\phantom{.}C}_B$ onto $S_l(M)$, and a morphism $f$ onto $f$, is an isomorphism of categories with inverse $S_l^{-1}$.
\item Let $B=C=H$, and $\rho =\id_H$. Then $(S_l,\varphi) : \rydH \to \ydH$ is a braided monoidal isomorphism, where $\varphi$ is defined by
\begin{align*}
\varphi_{M,N} : S_l(M) \ot S_l(N) &\to S_l(M \ot N),\\
m \ot n &\mapsto m \mathcal{S}^{-1}(n\sw1) \ot n\sw0
= \mathcal{S}^{-1}(n\sw{-1})m \ot n\sw0,
\end{align*}
for all $M,N \in \rydH$.
The inverse braided monoidal isomorphism is $(S_l^{-1},\psi) : \ydH \to \rydH$, where $\psi$ is defined by
\begin{align*}
\psi_{M,N} : S_l^{-1}(M) \ot S_l^{-1}(N) &\to S_l^{-1}(M \ot N),\\
m \ot n &\mapsto n\sw{-1} m \ot n\sw0
= m n\sw1 \ot n\sw0,
\end{align*}
for all $M,N \in \ydH$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Let $M \in \mathcal{YD}^{\phantom{.}C}_B$. Then $S_l(M) \in {}^{\phantom{.}C}_B\mathcal{YD}$ since for all $m \in M,b \in B$,
\begin{align*}
\delta_{S_l(M)}(bm) = \delta_{S_l(M)}(m \mathcal{S}^{-l}(b)) &= \mathcal{S}^l\left(\rho\mathcal{S}\cS^{-l}(b\sw3) m\sw1 \rho\mathcal{S}^{-l}(b\sw1)\right) \ot m\sw0\mathcal{S}^{-l}(b\sw2)\\
&=\rho(b\sw1) \mathcal{S}^l(m\sw1) \mathcal{S}\rho(b\sw3) \ot b\sw2 m\sw0.
\end{align*}
Thus $S_l$ is a well-defined functor. Similarly it follows that $S_l^{-1}$ is a well-defined functor.
(2) is shown in \cite[Proposition 2.2.1, 1.]{a-AndrGr99} for $l= -1$.
\end{proof}
\begin{remar}
In general, it is not clear whether the functor $S_l$ in Lemma \ref{lem:leftright} is monoidal.
This is one of the reasons why in the proof of our braided monoidal isomorphism of left Yetter-Drinfeld modules given in Theorem~\ref{theor:third}
we have to change sides starting in Theorem \ref{theor:first} with a monoidal isomorphism between relative right and left Yetter-Drinfeld modules.
\end{remar}
\section{The first isomorphism}\label{sec:first}
\begin{defin}\label{defin:relative}
Let $R$ be a Hopf algebra in $\ydH$.
We denote by $ {}^{\phantom{}R \#H}_{\phantom{aa.}H}\mathcal{YD}$ and $\mathcal{YD}^{\phantom{.}R \# H}_H$ the categories $ {}^{\phantom{}R \#H}_{\phantom{aa.}H}\mathcal{YD}$ and $\mathcal{YD}^{\phantom{.}R \# H}_H$ in Definition \ref{defin:ydBC} and \ref{defin:rydBC} with respect to the inclusion $H \subseteq R \#H$ as the Hopf algebra map $\rho$.
We denote by ${}_{R \#H}^{\phantom{aaa.}H}\mathcal{YD}$ the category ${}_{R \#H}^{\phantom{aaa.}H}\mathcal{YD}$ in Definition \ref{defin:ydBC} where $\rho$ is the Hopf algebra projection $\pi : R\#H \to H$ of $R \#H$.
Assume that $(R, R^{\vee})$ together with $\langle\;,\; \rangle$ is a dual pair of Hopf algebras in $\ydH$ with bijective antipodes. Then the antipodes of $R \# H$ and of $R^{\vee} \# H$ are bijective by \eqref{bigS2} and \eqref{ydiso}. We denote by ${{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}$ (respectively $ {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$) the full subcategory of objects of ${{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}}$ (respectively of $ {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}$) which are rational as $R^{\vee}$-modules by restriction. The full subcategories of $ {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}$ (respectively of $\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}$) consisting of objects which are rational over $R$ will be denoted by $ {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$ (respectively ${}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$).
\end{defin}
\begin{comment}
\begin{remar}\label{smashmod}
Let $R$ be Hopf algebra in $\ydH$.
(1) Let $M$ be a left $H$-module with module structure $\mu^H : H \ot M \to M$, and a left $R$-module with module structure $\mu^R : R \ot M \to M$.
\end{remar}
\begin{lemma}
Let $R$ be a Hopf algebra in $\ydH$, and $M$ a vector space with $H$-module structure $\mu^H : H \ot M \to M$ and $H$-comodule structure $\delta^H : M \to H \ot M$, denoted by $\delta^H(m) = m\sw{-1} \ot m\sw0$ for all $m \in M$.
\begin{enumerate}\label{lem:mod}
\item Let $\mu^R : R \ot M \to M$ be a left module structure on $M$. Then the following are equivalent:
\begin{enumerate}
\item $M \in {}_{R \#H}^{\phantom{aaa.}H}\mathcal{YD}$, where $R \#H \ot M \xrightarrow{\id_R \ot \mu^H} R \ot M \xrightarrow{\mu^R} M$ is the left $R \#H$-module structure on $M$.\label{lem:mod1}
\item $M \in {_R(\ydH)}$.\label{lem:mod2}
\end{enumerate}
\item Let $\delta^R : M \to R \ot M$ be a left comodule structure on $M$. Then the following are equivalent:
\begin{enumerate}\label{lem:comod}
\item $M \in {}^{\phantom{}R \#H}_{\phantom{aa.}H}\mathcal{YD}$, where $M \xrightarrow{\delta^R} R \ot M \xrightarrow{\id_R \ot \delta^H} R \# H \ot M$ is the $R \#H$-comodule structure of $M$.\label{lem:comod1}
\item $M \ {^R(\ydH)}$.\label{lem:comod2}
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}
(1) It follows from the definition of the smash product that $M$ is a left $R \#H$-module if and only if $\mu^R_M$ is $H$-linear.
The set of all elements $a \in R \#H$ satisfying the following Yetter-Drinfeld condition
\begin{align}
\delta^H(am) = \pi(a\sw1) m\sw{-1} \pi\mathcal{S}(a\sw3) \ot a\sw2 m \sw0\label{piyd}
\end{align}
for all $m \in M$ and $a \in R\#H$, is a subalgebra of $R \#H$. Hence \eqref{piyd} holds for all $a \in R\#H$ and $m \in M$ if and only if \eqref{piyd} holds for all $m \in M$ and $a \in R \cup H$. Note that \eqref{piyd} for all $m \in M$ and $a \in H$ is the Yetter-Drinfeld condition of $\ydH$, and \eqref{piyd} for all $m \in M$ and $a \in R$ says that $\mu^R$ is $H$-colinear, since for all $a \in R$, $a\sw1 \ot a\sw2 \ot a\sw3 \in R\#H \ot R\#H \ot R$, hence
\begin{align*}
a\sw1 \ot a\sw2 \ot \pi\mathcal{S}(a\sw3) = a\sw1 \ot a\sw2 \ot 1.
\end{align*}
This proves the equivalence in (1).
(2) As in (1), it follows from the definition of the cosmash product, that $M$ is a left $R\#H$-comodule if and only if $\delta^R$ is $H$-colinear.
\end{proof}
\end{comment}
\begin{lemma}\label{lem:Rmod}
Let $R$ be a Hopf algebra in $\ydH$, and let $_R(\ydH)$ be the category of left $R$-modules in the monoidal category $\ydH$.
\begin{enumerate}
\item Let $M \in {}_{R \#H}^{\phantom{aaa.}H}\mathcal{YD}$. Define $V_1(M) = M$ as a vector space and as a left $H$- and a left $R$-module by restriction of the $R\#H$-module structure. Then $V_1(M) \in \ydH$ with the given $H$-comodule structure, and the multiplication map $ R \ot M \to M$ is a morphism in $\ydH$.
\item The functor
\begin{align*}
V_1 : {}_{R \#H}^{\phantom{aaa.}H}\mathcal{YD} \to {_R(\ydH)}
\end{align*}
mapping objects $M \in \ydH$ to $V_1(M)$ and morphisms $f$ to $f$, is an isomorphism of categories. The inverse functor $V_1^{-1}$ maps an object $M \in {_R(\ydH)}$ onto the vector space $M$ with given $H$-comodule structure and $R\#H$-module structure $R \#H \ot M \xrightarrow{\id_R \ot \mu^H_M} R \ot M \xrightarrow{\mu^R_M} M$.
\end{enumerate}
\end{lemma}
\begin{proof}
It follows from the definition of the smash product that $M$ is a left $R \#H$-module if and only if $\mu^R_M$ is $H$-linear.
The set of all elements $a \in R \#H$ satisfying the following Yetter-Drinfeld condition
\begin{align}
\delta^H(am) = \pi(a\sw1) m\sw{-1} \pi\mathcal{S}(a\sw3) \ot a\sw2 m \sw0\label{piyd}
\end{align}
for all $m \in M$ and $a \in R\#H$, is a subalgebra of $R \#H$. Hence \eqref{piyd} holds for all $a \in R\#H$ and $m \in M$ if and only if \eqref{piyd} holds for all $m \in M$ and $a \in R\cup H$. Note that \eqref{piyd} for all $m \in M$ and $a \in H$ is the Yetter-Drinfeld condition of $\ydH$, and \eqref{piyd} for all $m \in M$ and $a \in R$ says that $\mu^R_M$ is $H$-colinear, since for all $a \in R$, $a\sw1 \ot a\sw2 \ot a\sw3 \in R\#H \ot R\#H \ot R$, hence
\begin{align*}
a\sw1 \ot a\sw2 \ot \pi\mathcal{S}(a\sw3) = a\sw1 \ot a\sw2 \ot 1.
\end{align*}
This proves the Lemma.
\end{proof}
\begin{lemma}\label{lem:Rcomod}
Let $R$ be a Hopf algebra in $\ydH$, and let $^R(\ydH)$ be the category of left $R$-comodules in the monoidal category $\ydH$.
\begin{enumerate}
\item Let $M \in {}^{\phantom{}R \#H}_{\phantom{aa.}H}\mathcal{YD}$ with comodule structure $\delta_M : M \to R \#H \ot M$. Define $V_2(M) = M$ as a vector space with left $H$-comodule structure $\delta_M^H$ and left $R$-comodule structure $\delta_M^R$ given by
\begin{align*}
\delta_M^H = (\pi \ot \id_M)\delta_M,\;\delta_M^R = (\vartheta \ot \id_M)\delta_M.
\end{align*}
Then $V_2(M) \in \ydH$ with $H$-comodule structure $\delta_M^H$ and the given $H$-module structure, and $\delta_M^R : M \to R \ot M$ is a morphism in $\ydH$.
\item The functor
\begin{align*}
V_2 : {}^{\phantom{}R \#H}_{\phantom{aa.}H}\mathcal{YD} \to {^R(\ydH)}
\end{align*}
mapping objects $M \in \ydH$ to $V_2(M)$ and morphisms $f$ to $f$, is an isomorphism of categories. The inverse functor $V_2^{-1}$ maps an object $M \in {^R(\ydH)}$ onto the vector space $M$ with given $H$-module structure and $R \#H$-comodule structure $M \xrightarrow{\delta^R_M} R \ot M \xrightarrow{\id_R \ot \delta^H_M} R \# H \ot M$.
\end{enumerate}
\end{lemma}
\begin{proof}
This is shown similarly to the proof of Lemma \ref{lem:Rmod}.
\end{proof}
For later use we note a formula for the right $R \#H$-comodule structure of a left $R \#H$-comodule defined via $\mathcal{S}^{-1}$.
\begin{lemma}\label{lem:reformulation}
Let $R$ be a Hopf algebra in $\ydH$ with bijective antipode, $M$ a left $H$-comodule with $H$-coaction
$\delta^H : M \to H \ot M$, $m \mapsto m\sw{-1} \ot m\sw0$,
and
\begin{align*}
\delta^R : M &\to R \ot M,\; m \mapsto m\sws{-1} \ot m\sws0
\end{align*}
a linear map. Define $\delta : M \to R \#H \ot M, \; m \mapsto m\swe{-1} \ot m\swe0$,
by $\delta = (\id \ot \delta^H)\delta^R$. Then
\begin{align}
\vartheta\mathcal{S}^{-1}(m\swe{-1}) \ot m\swe0 = \mathcal{S}_R^{-1}\left(\mathcal{S}^{-1}({m\sws0}\sw{-1}) \lact m\sws{-1}\right) \ot {m\sws0}\sw0
\end{align}
for all $m \in M$.
\end{lemma}
\begin{proof}
Let $m \in M$. Then $m\swe{-1} \ot m\swe0 = m\sws{-1} {m\sws0}\sw{-1} \ot {m\sws0}\sw0$, and
\begin{align*}
\mathcal{S}^{-1}({m\sws0}\sw{-1}) \lact m\sws{-1} \ot {m\sws0}\sw0&= \mathcal{S}^{-1}({m\sws0}\sw{-1}) m\sws{-1} {m\sws0}\sw{-2} \ot {m\sws0}\sw0\\
&= \mathcal{S}^{-1}({m\swe0}\sw{-1}) m\swe{-1} \ot {m\swe0}\sw0.
\end{align*}
Hence
\begin{align*}
&\mathcal{S}_R^{-1}\left(\mathcal{S}^{-1}({m\sws0}\sw{-1}) \lact m\sws{-1}\right) \ot {m\sws0}\sw0 &&\\
&\phantom{aa.}=\mathcal{S}_R^{-1}\left(\mathcal{S}^{-1}({m\swe0}\sw{-1}) m\swe{-1}\right) \ot {m\swe0}\sw0&&\\
&\phantom{aa.}= \vartheta \mathcal{S}^{-1}\left(\mathcal{S}^{-1}({m\swe0}\sw{-1}) m\swe{-1}\right) \ot {m\swe0}\sw0 &\text{ (by } \eqref{ruleS5})&\\
&\phantom{aa.}= \vartheta \mathcal{S}^{-1}(m\swe{-1}) \ot {m\swe0}. &\text{ (by } \eqref{vartheta1})&
\end{align*}
\end{proof}
\begin{theor}\label{theor:first}
Let $(R, R^{\vee})$ be a dual pair of Hopf algebras in $\ydH$ with bijective antipodes and with bilinear form $\langle\;,\; \rangle$.
A monoidal isomorphism
$$(F,\varphi) : \mathcal{YD}^{\phantom{.}R \# H}_H \to {{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}$$
is defined as follows.
For any object $M \in \mathcal{YD}^{\phantom{.}R \# H}_H$ with right $R \# H$-comodule structure denoted by
$$\delta_M: M \to M \ot R \# H,\; m \mapsto m\swe0 \ot m\swe1,$$
let $F(M)=M$ as a vector space and $F(M) \in {{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}}$
with left $H$-action, $H$-coaction $\delta_{F(M)}^H$ and $R^{\vee}$-action, respectively, given by
\begin{align}
hm &= m \mathcal{S}^{-1}(h),\label{first1}\\
\delta_{F(M)}^H(m) &= \pi\mathcal{S}(m\swe1) \ot m\swe0,\label{first2}\\
\xi m &= \langle \xi,\vartheta\mathcal{S}(m\swe1)\rangle m\swe0\label{first3}
\end{align}
for all $h \in H,m \in M, \xi \in R^{\vee}$. For any morphism $f$ in $\mathcal{YD}^{\phantom{.}R \# H}_H$ let $F(f)=f$.
The natural transformation $\varphi$ is defined by
\begin{align}
\varphi_{M,N} : F(M) \ot F(N) \to F(M \ot N),\; m \ot n \mapsto m \pi\mathcal{S}^{-1}(n\swe1) \ot n\swe0,
\end{align}
for all $M,N \in \mathcal{YD}^{\phantom{.}R \# H}_H$.
\end{theor}
\begin{proof}
The functor $F$ is the composition of the isomorphisms
\begin{align*}
\mathcal{YD}^{\phantom{.}R \# H}_H \xrightarrow{S} {}^{\phantom{}R \#H}_{\phantom{aa.}H}\mathcal{YD} \xrightarrow{V_2} {^R(\ydH)} \xrightarrow{D} {_{R^{\vee}}(\ydH)_{\mathrm{rat}}} \xrightarrow{V_1^{-1}} {{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}},
\end{align*}
where $S=S_1$ is the isomorphism of Lemma \ref{lem:leftright}, $V_2$ is the isomorphism of Lemma \ref{lem:Rcomod}, $D$ is the isomorphism of Proposition \ref{propo:rational}, and where the last isomorphism is the restriction of $V_1^{-1}$ for $R^{\vee}$ of Lemma \ref{lem:Rmod} to rational objects.
Let $M,N \in \mathcal{YD}^{\phantom{.}R \# H}_H$. The map
\begin{align*}
\varphi = \varphi_{M,N} : F(M) \ot F(N) \to F(M \ot N)
\end{align*}
is a linear isomorphism with $\varphi^{-1}(m \ot n)= m\pi(n\swe1) \ot n \swe0$ for all $m \in M,n\in N$. It follows from the Yetter-Drinfeld condition \eqref{rydBC} that $\varphi$ is an $H$-linear and $H$-colinear map, since for all $m \in M,n\in N$ and $h \in H$,
\begin{align*}
\varphi(h(m \ot n)) &= \varphi(m\mathcal{S}^{-1}(h\sw1) \ot n \mathcal{S}^{-1}(h\sw2))\\
&= m \mathcal{S}^{-1}(h\sw1) \pi\mathcal{S}^{-1}(h\sw4 n\swe1 \mathcal{S}^{-1}(h\sw2)) \ot n\swe0 \mathcal{S}^{-1}(h\sw3)\\
&=m\mathcal{S}^{-1}(h\sw1) \mathcal{S}^{-2}(h\sw2) \pi\mathcal{S}^{-1}(n\swe1) \mathcal{S}^{-1}(h\sw4) \ot n\swe0 \mathcal{S}^{-1}(h\sw3)\\
&= h \varphi(m \ot n),
\end{align*}
\begin{align*}
\delta _{F(M\ot N)}^H\varphi(m \ot n) &= \pi\mathcal{S}(\pi(n\swe4) m\swe1 \pi\mathcal{S}^{-1}(n\swe2) n\swe1) \ot m\swe0 \pi\mathcal{S}^{-1}(n\swe3) \ot n\swe0\\
&= \pi\mathcal{S}(n\swe2 m\swe1) \ot m\swe0 \pi\mathcal{S}^{-1}(n\swe1) \ot n\swe0\\
&=(\id_H \ot \varphi)\delta_{F(M) \ot F(N)}^H(m \ot n).
\end{align*}
To prove that $\varphi$
is a left $R^{\vee}$-linear map, let $\xi \in R^{\vee}, m \in M$ and $n \in N$.
We first show that
\begin{align}\label{NR}
\xi\sw{-2} \ot \xi\sw{-1} \langle \xi\sw0,\vartheta\mathcal{S}(a)\rangle = \pi( \mathcal{S}(a\sw2) a\sw4) \ot \pi(\mathcal{S}(a\sw1) a\sw5) \langle\xi,\vartheta\mathcal{S}(a\sw3)\rangle
\end{align}
for all $a \in R \# H$.
By \eqref{coactionvartheta},
\begin{align*}
(\vartheta\mathcal{S}(a))\sw{-2} \ot (\vartheta\mathcal{S}(a))\sw{-1} &\ot (\vartheta\mathcal{S}(a))\sw{0}\\
&= \Delta(\pi(S(a\sw3) \mathcal{S}^2(a\sw1)) \ot \vartheta\mathcal{S}(a\sw2)\\
&=\pi(\mathcal{S}(a\sw5) \mathcal{S}^2(a\sw1)) \ot \pi(\mathcal{S}(a\sw4) \mathcal{S}^2(a\sw2)) \ot \vartheta\mathcal{S}(a\sw3).
\end{align*}
Hence \eqref{NR} follows from \eqref{pair3}.
Then
\begin{align*}
\varphi(\xi(m \ot n)) &= \varphi(\xi\sw1m \ot \xi\sw2 n)&&\\
&= \varphi(\xi\swo1 \xi\swo2\sw{-1}m \ot \xi\swo2\sw0 n)&&\\
&= \varphi\left(\xi\swo1 \left(m\mathcal{S}^{-1}(\xi\swo2\sw{-1})\right) \ot \xi\swo2\sw0 n\right)&&\\
&= \varphi\Big(\Big\langle \xi\swo1,\vartheta\mathcal{S}\left(\xi\swo2\sw{-1} m\swe1 \mathcal{S}^{-1}(\xi\swo2\sw{-3})\right) \Big\rangle m\swe0 \mathcal{S}^{-1}(\xi\swo2\sw{-2}) &&\\
&\phantom{aa}\ot \langle \xi\swo2\sw0,\vartheta\mathcal{S}(n\swe1)\rangle n\swe0\Big)& &\\
&= \varphi(m\swe0 \mathcal{S}^{-1}(\xi\swo2\sw{-1}) \ot n \swe0) &(\text{by } \eqref{vartheta1})&\\
&\phantom{aa}\times \Big\langle \xi\swo1,\vartheta\left(\xi\swo2\sw{-2} \mathcal{S}(m\swe1)\right)\Big\rangle \langle\xi\swo2\sw{0},\vartheta\mathcal{S}(n\swe1) \rangle.&&\\
\end{align*}
Hence by \eqref{NR} we obtain
\begin{align*}
\varphi(\xi(m \ot n)) &= \varphi(m\swe0 \mathcal{S}^{-1}(\pi(\mathcal{S}(n\swe1) n\swe5)) \ot n \swe0) & &\\
&\phantom{aa}\times \langle \xi\swo1,\vartheta\left(\pi( \mathcal{S}(n\swe2) n\swe4) \mathcal{S}(m\swe1)\right)\rangle \langle\xi\swo2,\vartheta\mathcal{S}(n\swe3) \rangle &&\\
&=m\swe0 \pi(\mathcal{S}^{-1}(n\swe6)n\swe2) \pi\mathcal{S}^{-1}(n\swe1) \ot n\swe0 &(\text{by }\eqref{pair4})&\\
&\phantom{aa}\times \langle \xi, \vartheta\mathcal{S}(n\swe4) \vartheta\left(\pi(\mathcal{S}(n\swe3) n\swe5) \mathcal{S}(m\swe1)\right)\rangle&&\\
&=m\swe0 \pi\mathcal{S}^{-1}(n\swe4) \ot n\swe0 &&\\
&\phantom{aa}\times \Big\langle\xi,\vartheta\mathcal{S}(n\swe2) \vartheta\left(\pi(\mathcal{S}(n\swe1) n\swe3) \mathcal{S}(m\swe1)\right) \Big\rangle&&\\
&=m\swe0 \pi\mathcal{S}^{-1}(n\swe3) \ot n\swe0 \langle\xi,\vartheta\mathcal{S}(m\swe1 \pi \mathcal{S}^{-1}(n\swe2) n\swe1) \rangle, &
\end{align*}
where the last equality follows from Lemma \ref{lem:vartheta3} and from \eqref{vartheta2}.
On the other hand
\begin{align*}
\xi \varphi(m \ot n) &= \xi (m \pi\mathcal{S}^{-1}(n\swe1) \ot n\swe0)&&\\
&= \langle \xi,\vartheta\mathcal{S}(\pi(n\swe4) m\swe1 \pi\mathcal{S}^{-1}(n\swe2) n\swe1) \rangle m\swe0\pi\mathcal{S}^{-1}(n\swe3) \ot n\swe0 &&\\
&= m\swe0 \pi\mathcal{S}^{-1}(n\swe3) \ot n\swe0 \langle\xi, \vartheta\mathcal{S}(m\swe1 \pi\mathcal{S}^{-1}(n\swe2) n\swe1) \rangle. &(\text{by } \eqref{vartheta1})&
\end{align*}
Hence $\varphi(\xi(m \ot n)) = \xi \varphi(m \ot n)$.
It is easy to check that the diagrams \eqref{monoidal} commute for $(F,\varphi)$. Hence $(F,\varphi)$ is a monoidal functor.
\end{proof}
\section{The second isomorphism}\label{sec:second}
In this section we assume that $(R, R^{\vee})$ is a dual pair of Hopf algebras in $\ydH$ with bijective antipodes and bilinear form $\langle\;,\; \rangle$. The monoidal isomorphism
$(F,\varphi) : \mathcal{YD}^{\phantom{.}R \# H}_H \to {{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}$
of Theorem \ref{theor:first} induces by Lemma \ref{lem:center} a braided monoidal isomorphism between the centers
$$(F^{\mathcal{Z}},{\varphi}^{\mathcal{Z}}) : \mathcal{Z}(\mathcal{YD}^{\phantom{.}R \# H}_H) \to \mathcal{Z}({{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}).$$
Assume that ${{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}$ is $R^{\vee} \#H$-faithful. By Propositions \ref{propo:comodulecenter} and \ref{propo:modulecenter}, the functors
\begin{align*}
\Psi : {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}} &\to \mathcal{Z}(\mathcal{YD}^{\phantom{.}R \# H}_H),\\
\Phi : {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat} &\to \mathcal{Z}({{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}})
\end{align*}
are fully faithful, strict monoidal and braided. The functor $\Psi$ is defined
with respect to the Hopf algebra inclusion $\iota : H \to R \#H$.
We denote the image of $M \in {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$ in $\mathcal{YD}^{\phantom{.}R \# H}_H$ defined
by restriction by $M_{\mathrm{res}}$. The functor $\Phi$ is defined
with respect to the Hopf algebra projection $\pi : R^\vee \#H \to H$,
and we denote the image of $M \in {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$ in ${{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}$
by ${}^{\pi}M$.
Our goal is to show in Theorem \ref{theor:second} that $(F,\varphi)$ induces a braided monoidal isomorphism
\begin{align*}
{}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}} \to {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}.
\end{align*}
Let $G : {{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}} \to \mathcal{YD}^{\phantom{.}R \# H}_H$ be the inverse functor of the isomorphism $F$ of Theorem \ref{theor:first}. Then
$(G,\psi) : {{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}} \to \mathcal{YD}^{\phantom{.}R \# H}_H$ is a monoidal isomorphism, where $\psi$ is defined by \eqref{inversemonoidal}.
We first construct functors
$$\widetilde{F} : {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}} \to {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}, \;\widetilde{G} : {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat} \to {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$$ such that the diagrams
\begin{align}\label{diagramsecondF}
\begin{CD}
{}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}} @>{\widetilde{F}}>> {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}\\
@V{\Psi}VV @V{\Phi}VV\\
\mathcal{Z}(\mathcal{YD}^{\phantom{.}R \# H}_H) @>{F^{\mathcal{Z}}}>> \mathcal{Z}({{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}})
\end{CD}
\intertext{and}
\begin{CD}
{}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat} @>{\widetilde{G}}>> {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}\\
@V{\Phi}VV @V{\Psi}VV\\
\mathcal{Z}({{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}) @>{G^{\mathcal{Z}}}>> \mathcal{Z}(\mathcal{YD}^{\phantom{.}R \# H}_H)
\end{CD}\label{diagramsecondG}
\end{align}
commute.
The existence of $\widetilde{F}$ will follow from the next two lemmas.
\begin{lemma}\label{lem:second1}
Let $(F^{\mathcal{Z}},{\varphi}^{\mathcal{Z}}) : \mathcal{Z}(\mathcal{YD}^{\phantom{.}R \# H}_H) \to \mathcal{Z}({{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}})$ be the monoidal isomorphism induced by the isomorphism $(F,\varphi)$ of Theorem \ref{theor:first}. Let $M \in {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$, and $\Psi(M)= (M_{\mathrm{res}},\gamma)$, where $\gamma = c_M$ is defined in \eqref{defin:Psi}. Then
$$F^{\mathcal{Z}} \Psi(M)= (F(M_{\mathrm{res}}), \ti{\gamma}),$$
and $\widetilde{\gamma}_{F(X)} : F(M_{\mathrm{res}}) \ot F(X) \to F(X) \ot F(M_{\mathrm{res}})$ is given by
\begin{align}
\ti{\gamma}_{F(X)}(m \ot x) = x\swe0 \pi\left(\mathcal{S}(x\swe1) x\swe4 m\swe1\right) \ot m\swe0 \pi\mathcal{S}^{-1}(x\swe3) x\swe2\label{computeZ1}
\end{align}
for all $X \in \mathcal{YD}^{\phantom{.}R \# H}_H, x \in X$ and $m \in M$.
\end{lemma}
\begin{proof}
Let $X \in \mathcal{YD}^{\phantom{.}R \# H}_H$ with comodule structure
$$X \to X \ot R \#H,\; x \mapsto x\swe0 \ot x\swe1.$$
Recall that $\ti{\gamma}_{F(X)} = {\varphi_{X,M_{\mathrm{res}}}}^{-1} F(c_{M,X}) \varphi_{M_{\mathrm{res}},X}$ by \eqref{defin:tilde}. It follows from the definition of $\varphi_{X,M_{\mathrm{res}}}$ in Theorem \ref{theor:first} that
\begin{align}
{\varphi_{X,M_{\mathrm{res}}}}^{-1}(x \ot m) = x \pi(m\swe1) \ot m\swe0 \label{computeZ2}
\end{align}
for all $x \in X, m\in M$. Hence
\begin{align*}
\tilde{\gamma}_{F(X)}(m \ot x)&= {\varphi_{X,M_{\mathrm{res}}}}^{-1} F(c_{M,X}) \varphi_{M_{\mathrm{res}},X}(m \ot x)\\
&={\varphi_{X,M_{\mathrm{res}}}}^{-1} F(c_{M,X})(m \pi \mathcal{S}^{-1}(x\swe1) \ot x\swe0)\\
&={\varphi_{X,M_{\mathrm{res}}}}^{-1}\left(x\swe0 \ot m \pi\mathcal{S}^{-1}(x\swe2) x\swe1\right)\\
&= x\swe0 \pi\left(\mathcal{S}\left(\left(\pi\mathcal{S}^{-1}(x\swe2) x\swe1\right)\swe1\right) m\swe1 \left(\pi\mathcal{S}^{-1}(x\swe2) x\swe1\right)\swe3\right)\\
&\phantom{aa}\ot m\swe0 \left(\pi\mathcal{S}^{-1}(x\swe2) x\swe1\right)\swe2\\
&= x\swe0 \pi\left(\mathcal{S}(\mathcal{S}^{-1}(x\swe6) x\swe1) m\swe1 \mathcal{S}^{-1}(x\swe4) x\swe3\right) \ot m\swe0 \pi\mathcal{S}^{-1}(x\swe5) x\swe2\\
&=x\swe0 \pi\left(\mathcal{S}(x\swe1) x\swe4 m\swe1\right) \ot m\swe0 \pi\mathcal{S}^{-1}(x\swe3) x\swe2.
\end{align*}
\end{proof}
In the next lemma we define a map $\delta_{\widetilde{F}(M)}$ which will be the coaction of $R \# H$ on $\widetilde{F}(M)$ in Theorem \ref{theor:second}.
\begin{lemma}\label{lem:second2}
Let $M \in {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$. We denote the left $H$-comodule structure of $F(M_{\mathrm{res}})$ by $M \to H \ot M,\; m \mapsto m\sw{-1} \ot m\sw0$. Define a linear map
\begin{align*}
\delta_M^{R^{\vee}} : M \to R^{\vee} \ot M,\; m \mapsto m\swos{-1} \ot m\swos0,
\end{align*}
by the equation
\begin{align}
mr = \langle r, \mathcal{S}_{R^{\vee}}^{-1}\left(\mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact m\swos{-1}\right)\rangle' {m\swos0}\sw0\label{deltaRvee}
\end{align}
for all $r \in R,m \in M$. Let
\begin{align}
\delta_{\widetilde{F}(M)} : M \to R^{\vee} \#H \ot M, \;m \mapsto m\swoe{-1} \ot m\swoe0 = m\swos{-1} {m\swos0}\sw{-1} \ot {m\swos0}\sw0.\label{delta}
\end{align}
Then the following hold.
\begin{enumerate}
\item For all $m \in M,a \in R \#H$,
\begin{align*}
\langle \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact m\swos{-1}, \vartheta\mathcal{S}(a)\rangle {m\swos0}\sw0 = m \pi\mathcal{S}^{-1}(a\sw2) a\sw1.
\end{align*}
\item Let $X \in \mathcal{YD}^{\phantom{.}R \# H}_H$, and let $\ti{\gamma}_{F(X)} : F(M_{\mathrm{res}}) \ot F(X) \to F(X) \ot F(M_{\mathrm{res}})$ be the isomorphism in ${{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}}$ defined in Lemma \ref{lem:second1}. Then for all $x \in X$ and $m \in M$, $m\swoe{-1}x \ot m\swoe0 = \ti{\gamma}_{F(X)}(m \ot x)$.
\item For all $m \in M$, $\pi(m\swoe{-1}) \ot m\swoe0 = m\sw{-1} \ot m\sw0$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) The map $\delta_M^{R^{\vee}}$ is well-defined since $M$ is a rational right $R$-module, $\langle\;,\;\rangle$ is non-degenerate, and the maps $\mathcal{S}_{R^{\vee}}$ and
\begin{align*}
R^{\vee} \ot M \to R^{\vee} \ot M,\; \xi \ot m \mapsto \mathcal{S}^{-1}({m}\sw{-1}) \lact \xi \ot {m}\sw0,
\end{align*}
are bijective.
Note that if (1) holds for $a \in R \#H$ then it holds for $ha$ for all $h \in H$. Thus it is enough to assume in (1) that $a \in \mathcal{S}^{-1}(R)$. For all $r \in R$ and $a= \mathcal{S}^{-1}(r)$,
\begin{align*}
\pi\mathcal{S}^{-1}(a\sw2) a\sw1 = \pi\mathcal{S}^{-2}(r\sw1) \mathcal{S}^{-1}(r\sw2) = \mathcal{S}_R(\mathcal{S}^{-2}(r))
\end{align*}
by \eqref{antipode}. Therefore (1) is equivalent to
\begin{align*}
\langle \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact m\swos{-1}, r\rangle {m\swos0}\sw0 = m \mathcal{S}_R(\mathcal{S}^{-2}(r))
\end{align*}
for all $r \in R, m\in M$. This last equation holds by our definition of $\delta_M^{R^{\vee}}$ since
\begin{align*}
m \mathcal{S}_R(\mathcal{S}^{-2}(r)) &= \langle \mathcal{S}_R(\mathcal{S}^{-2}(r)), \mathcal{S}_{R^{\vee}}^{-1}\left(\mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact m\swos{-1}\right)\rangle' {m\swos0}\sw0&\text{(by } \eqref{deltaRvee})&\\
&=\langle \mathcal{S}^{-2}(r), \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact m\swos{-1}\rangle' {m\swos0}\sw0 & \text{(by } \eqref{pair6})&\\
&=\langle \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact m\swos{-1}, r\rangle {m\swos0}\sw0.&&
\end{align*}
Here, we used that by Lemma \ref{lem:inversepair}, $(R^{\vee},R)$ together with $\langle \;,\; \rangle' : R \ot R^{\vee} \to \fie$ is a dual pair of Hopf algebras in $\ydH$.
(2) Let $X \in \mathcal{YD}^{\phantom{.}R \# H}_H$. By Lemma \ref{lem:second1} we have to show that
\begin{align}
m\swoe{-1} x \ot m\swoe0 =x\swe0 \pi(\mathcal{S}(x\swe1) x\swe4 m\swe1) \ot m\swe0 \pi\mathcal{S}^{-1}(x\swe3) x\swe2\label{second21}
\end{align}
for all $x \in X,m\in M$.
By \eqref{delta} and \eqref{deltaRvee}, the left-hand side of \eqref{second21} can be written as
\begin{align*}
m\swoe{-1}x \ot m\swoe0 &= m\swos{-1} ({m\swos0}\sw{-1}x) \ot {m\swos0}\sw0\\
&=m\swos{-1} (x\mathcal{S}^{-1}({m\swos0}\sw{-1})) \ot {m\swos0}\sw0\\
&= \langle m\swos{-1}, \vartheta\mathcal{S}\left({m\swos0}\sw{-1} x\swe1 \mathcal{S}^{-1}({m\swos0}\sw{-3})\right) \rangle x\swe0 \mathcal{S}^{-1}({m\swos0}\sw{-2}) \ot {m\swos0}\sw0\\
&= \langle m\swos{-1}, \vartheta\mathcal{S}\left( x\swe1 \mathcal{S}^{-1}({m\swos0}\sw{-2})\right) \rangle x\swe0 \mathcal{S}^{-1}({m\swos0}\sw{-1}) \ot {m\swos0}\sw0,
\end{align*}
where the last equality follows from \eqref{vartheta1}. Thus \eqref{second21} is equivalent to the equation
\begin{align}
&\phantom{aa.}\langle m\swos{-1}, \vartheta\mathcal{S}\left( x\swe1 \mathcal{S}^{-1}({m\swos0}\sw{-2})\right) \rangle x\swe0 \mathcal{S}^{-1}({m\swos0}\sw{-1}) \ot {m\swos0}\sw0 \label{second22}\\
&=x\swe0 \pi(\mathcal{S}(x\swe1) x\swe4 m\swe1) \ot m\swe0 \pi\mathcal{S}^{-1}(x\swe3) x\swe2\notag
\end{align}
for all $x \in X,m \in M$.
To simplify \eqref{second22} we apply the isomorphism
\begin{align}
X \ot M \to X \ot M,\; x \ot m \mapsto x \mathcal{S}^{-2}(m\sw{-1}) \ot m\sw0.\label{iso}
\end{align}
Under the isomorphism \eqref{iso} the left-hand side of \eqref{second22} becomes
\begin{align*}
&\phantom{aa.}\langle m\swos{-1}, \vartheta\mathcal{S}\left( x\swe1 \mathcal{S}^{-1}({m\swos0}\sw{-1})\right) \rangle x\swe0 \ot {m\swos0}\sw0&&\\
&=\langle m\swos{-1}, \vartheta\left({m\swos0}\sw{-1}\mathcal{S}( x\swe1)\right)\rangle x\swe0 \ot {m\swos0}\sw0&&\\
&=\langle m\swos{-1}, {m\swos0}\sw{-1} \lact \vartheta \mathcal{S}( x\swe1)\rangle x\swe0 \ot {m\swos0}\sw0 & \text{(by } \eqref{vartheta2})&\\
&=\langle \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact m\swos{-1}, \vartheta\mathcal{S}(x\swe1)\rangle x\swe0 \ot {m\swos0}\sw0,& \text{(by } \eqref{pair2})&
\end{align*}
and the right-hand side equals
\begin{align*}
&\phantom{aa.}x\swe0 \pi\left(\mathcal{S}(x\swe1) x\swe4 m\swe1\right) \mathcal{S}^{-2}\mathcal{S}\pi\left((m\swe0 \pi\mathcal{S}^{-1}(x\swe3) x\swe2)\swe1\right) \ot (m\swe0 \pi\mathcal{S}^{-1}(x\swe3) x\swe2)\swe0\\
&=x\swe0 \pi\left(\mathcal{S}(x\swe1) x\swe8 m\swe2\right) \mathcal{S}^{-1} \pi\left(\mathcal{S}\left(\pi\mathcal{S}^{-1}(x\swe7) x\swe2\right) m\swe1 \pi\mathcal{S}^{-1}(x\swe5) x\swe4\right)\\
&\phantom{aa.}\ot m\swe0 \pi \mathcal{S}^{-1}(x\swe6) x\swe3\\
&= x\swe0 \pi\left(\mathcal{S}(x\swe1) x\swe8 m\swe2\right) \mathcal{S}^{-1}\pi\left(\mathcal{S}(x\swe2)x\swe7 m\swe1 \mathcal{S}^{-1}(x\swe5)x\swe4 \right)\\
&\phantom{aa.}\ot m\swe0 \pi \mathcal{S}^{-1}(x\swe6) x\swe3\\
&=x\swe0 \ot m \pi\mathcal{S}^{-1}(x\swe2) x\swe1.
\end{align*}
Thus the claim follows from (1).
(3) Let $m \in M$. By \eqref{deltaRvee} and \eqref{pair4}, $m = m1 = \varepsilon(m\swos{-1}) m\swos0$. Hence
$$\pi(m\swoe{-1}) \ot m\swoe0 = \varepsilon(m\swos{-1}) {m\swos0}\sw{-1} \ot {m\swos0}\sw0 = m\sw{-1} \ot m\sw0.$$
\end{proof}
The existence of $\widetilde{G}$ will follow from the next two lemmas.
Let $M \in { {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}}$. We denote the left $R^{\vee} \# H$-comodule structure of $M$ by
$$M \to R^{\vee} \# H \ot M, \; m \mapsto m\swoe{-1} \ot m\swoe0 = m\swos{-1}{m\swos0}\sw{-1} \ot {m\swos0}\sw0,$$
where $M \to R^{\vee} \ot M,\; m \mapsto m\swos{-1} \ot m\swos0,$ is the $R^{\vee}$-comodule structure of $M$.
For all $X \in {{{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}}$ the right $R \# H$-comodule structure of $G(X)$ is denoted by
$$X \to X \ot R \# H,\; x \mapsto x\swe0 \ot x\swe1.$$
Note that $G(X) = X$ as a vector space. The right $H$-module structure of $G(X)$ is defined by
\begin{align}
xh = \mathcal{S}(h)x\label{ruleX}
\end{align}
for all $x \in X, h \in H$. Since $FG({}^{\pi}M) ={}^{\pi}M$, it follows that
\begin{align}
\pi\mathcal{S}(m\swe1) \ot m\swe0 = \pi(m\swoe{-1}) \ot m\swoe0\label{ruleM}
\end{align}
for all $m \in M$.
\begin{lemma}\label{lem:second3}
Let $(G^{\mathcal{Z}},\psi^{\mathcal{Z}}) : \mathcal{Z}({{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}) \to \mathcal{Z}(\mathcal{YD}^{\phantom{.}R \# H}_H)$ be the monoidal isomorphism induced by the monoidal isomorphism $(G,\psi)$. Let $M \in {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$, and $\Phi(M) = (^{\pi}M, \gamma)$, where $\gamma = c_M$ is defined in \eqref{defin:Phi}.
Then
$$G^{\mathcal{Z}} \Phi(M) = (G(^{\pi}M), \widetilde{\gamma}),$$
and $\widetilde{\gamma}_{G(X)} : G({}^{\pi}M) \ot G(X) \to G(X) \ot G({}^{\pi}M)$ is given by
\begin{align}
\widetilde{\gamma}_{G(X)}(m \ot x) = \left(\mathcal{S}^{-1}({m\swos{0}}\sw{-1} \pi\mathcal{S}^2(x\swe1)) \lact m\swos{-1}\right) x\swe0 \ot \pi \mathcal{S}(x\swe2) {m\swos0}\sw0\label{resulttilde}
\end{align}
for all $X \in {{{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}}, x \in X$ and $m \in M$.
\end{lemma}
\begin{proof}
Let $X \in {{{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}}$. By \eqref{defin:Phi}, $\gamma_X : {}^{\pi}M \ot X \to X \ot {^{\pi}M}$ is defined by
$$\gamma_X(m \ot x) = m\swoe{-1} x \ot m\swoe0$$
for all $x \in X,m \in M$.
By \eqref{inversemonoidal} and \eqref{defin:tilde}, the isomorphism $\widetilde{\gamma}_{G(X)}$ is defined by the equation
\begin{align}
\widetilde{\gamma}_{G(X)}G(\varphi_{G({}^{\pi}M),G(X)}) = G(\varphi_{G(X),G({}^{\pi}M)})G(\gamma_X).\label{equationtilde}
\end{align}
We apply both sides of \eqref{equationtilde} to an element $m \ot x, m \in M,x \in X$. Then
\begin{align*}
\widetilde{\gamma}_{G(X)}G(\varphi_{G({}^{\pi}M),G(X)})(m \ot x) &= \widetilde{\gamma}_{G(X)}(m \pi\mathcal{S}^{-1}(x\swe1) \ot x\swe0),&&\\
\intertext{and}
G(\varphi_{G(X),G({}^{\pi}M)})G(\gamma_X)(m \ot x) &= (m\swoe{-1} x) \pi\mathcal{S}^{-1}({m\swoe0}\swe1) \ot {m\swoe0}\swe0&&\\
&=\pi({m\swoe0}\swe1) (m\swoe{-1} x) \ot {m\swoe0}\swe0 &&(\text{by } \eqref{ruleX})\\
&=\pi\mathcal{S}^{-1}(m\swoe{-1}) m\swoe{-2}x \ot m\swoe0 && (\text{by } \eqref{ruleM})\\
&=\left(\mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact m\swos{-1}\right)x \ot {m\swos0}\sw0,
\end{align*}
where in the proof of the last equality the following formula in $R^{\vee} \# H$ is used for $a = m\swoe{-1} = m\swos{-1} {m\swos0}\sw{-1}$. Let $\xi \in R^{\vee},h \in H$ and $a=\xi h \in R^{\vee} \# H$. Then
\begin{align*}
\pi\mathcal{S}^{-1}(a\sw2)a\sw1 &= \pi\mathcal{S}^{-1}({\xi\swo2}\sw0 h\sw2) \xi\swo1 {\xi\swo2}\sw{-1} h\sw1\\
&= \mathcal{S}^{-1}(h\sw2) \xi h\sw1\\
&= \mathcal{S}^{-1}(h) \lact \xi.
\end{align*}
We have shown that
\begin{align}
\widetilde{\gamma}_{G(X)}(m \pi\mathcal{S}^{-1}(x\swe1) \ot x\swe0)= \left(\mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact m\swos{-1}\right)x \ot {m\swos0}\sw0.\label{formulatilde}
\end{align}
Since $m \ot x = m \pi(x\swe2) \pi\mathcal{S}^{-1}(x\swe1) \ot x\swe0 = (\pi\mathcal{S}(x\swe2)m) \pi\mathcal{S}^{-1}(x\swe1) \ot x\swe0$, we obtain from \eqref{formulatilde} and the Yetter-Drinfeld condition for $M$
\begin{align*}
\widetilde{\gamma}_{G(X)}&(m \ot x) \\
&=\left(\mathcal{S}^{-1}\left({(\pi\mathcal{S}(x\swe1)m)\swos0}\sw{-1}\right) \lact \left(\pi\mathcal{S}(x\swe1)m\right)\swos{-1}\right)x\swe0
\ot {(\pi\mathcal{S}(x\swe1)m)\swos0}\sw0\\
&=\left(\mathcal{S}^{-1}\left((\pi\mathcal{S}(x\swe1)m\swos0)\sw{-1}\right) \lact \pi\mathcal{S}(x\swe2) m\swos{-1}\right) x\swe0 \ot \left(\pi\mathcal{S}(x\swe1) m\swos0\right)\sw0\\
&=\left(\mathcal{S}^{-1}({m\swos{0}}\sw{-1} \pi\mathcal{S}^2(x\swe1)) \lact m\swos{-1}\right) x\swe0 \ot \pi \mathcal{S}(x\swe2) {m\swos0}\sw0.
\end{align*}
\end{proof}
\begin{lemma}\label{lem:second4}
Let $M \in {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$, and $G^{\mathcal{Z}} \Phi(M) = (G(^{\pi}M), \widetilde{\gamma})$ as in Lemma \ref{lem:second3}. Let $\langle\;,\; \rangle' : R \ot R^{\vee} \to \fie$ be the form defined in \eqref{defin:inversepair}. Define a linear map
$\mu_{\widetilde{G}(M)} : M \ot R \# H \to M$
by
\begin{align}
\mu_{\widetilde{G}(M)}(m \ot a)= \langle m\swos{-1},{m\swos0}\sw{-1} \lact \vartheta(\pi\mathcal{S}^2(a\sw2) \mathcal{S}(a\sw1)) \rangle \pi\mathcal{S}(a\sw3) {m\swos0}\sw0 \label{tildeaction}
\end{align}
for all $m \in M,a \in R \# H$. Then the following hold.
\begin{enumerate}
\item For all $X \in {{{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}}, x \in X$ and $m \in M$,
$$\widetilde{\gamma}_{G(X)} (m \ot x) = x\swe0 \ot \mu_{\widetilde{G}(M)}(m \ot x\swe1).$$
\item For all $m \in M$ and $h \in H$, $\mu_{\widetilde{G}(M)}(m \ot h) = mh$.
\item For all $m \in M$ and $r \in R$, $\mu_{\widetilde{G}(M)}(m \ot r)=
\langle r, \vartheta\mathcal{S}^{-1}(m\swoe{-1}) \rangle' m\swoe0$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Let $X \in {{{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}}, x \in X$ and $m \in M$. Then by \eqref{resulttilde},
\begin{align*}
\widetilde{\gamma}_{G(X)}(m \ot x) &= \left(\mathcal{S}^{-1}\left({m\swos{0}}\sw{-1} \pi\mathcal{S}^2(x\swe1)\right) \lact m\swos{-1}\right) x\swe0 \ot \pi \mathcal{S}(x\swe2) {m\swos0}\sw0 \\
&= \langle \mathcal{S}^{-1}\left({m\swos0}\sw{-1} \pi\mathcal{S}^2(x\swe2)\right) \lact m\swos{-1}, \vartheta \mathcal{S}(x\swe1) \rangle x\swe0 \ot \pi \mathcal{S}(x\swe3) {m\swos0}\sw0\\
&=x\swe0 \ot \langle m\swos{-1}, {m\swos0}\sw{-1} \lact \vartheta\left(\pi\mathcal{S}^2(x\swe2) \mathcal{S}(x\swe1)\right) \rangle \pi\mathcal{S}(x\swe3) {m\swos0}\sw0\\
&= x\swe0 \ot \mu_{\widetilde{G}(M)}(m \ot x\swe1),
\end{align*}
where we used \eqref{first3} and the equality $X = FG(X)$ together with \eqref{vartheta2} and \eqref{pair2}.
(2) Let $m \in M$ and $h \in H$. Then
\begin{align*}
\mu_{\widetilde{G}(M)}(m \ot h) &= \langle m\swos{-1},{m\swos0}\sw{-1} \lact \vartheta\left(\pi\mathcal{S}^2(h\sw2) \mathcal{S}(h\sw1)\right) \rangle \pi\mathcal{S}(h\sw3) {m\swos0}\sw0&&\\
&= \langle m\swos{-1},{m\swos0}\sw{-1} \lact 1 \rangle \pi\mathcal{S}(h) {m\swos0}\sw0&&\\
&= \langle m\swos{-1},1 \rangle \pi\mathcal{S}(h) m\swos0&& \text{(by } \eqref{pair5})\\
&= \pi\mathcal{S}(h) m&&\\
&= mh.&&
\end{align*}
(3) Let $m \in M$ and $r \in R$. Then $r\sw1 \ot \pi(r\sw2) = r \ot 1$. Hence
\begin{align*}
\mu_{\widetilde{G}(M)}(m \ot r)&= \langle m\swos{-1}, {m\swos0}\sw{-1} \lact \vartheta(\mathcal{S}(r)) \rangle {m\swos0}\sw0\\
&=\langle m\swos{-1}, {m\swos0}\sw{-1} \lact \vartheta(\mathcal{S}(r\sw{-1}) \mathcal{S}_R(r\sw0)) \rangle {m\swos0}\sw0&&\text{(by } \eqref{bigS})\\
&=\langle m\swos{-1}, \left({m\swos0}\sw{-1}\mathcal{S}(r\sw{-1})\right) \lact \mathcal{S}_R(r\sw0) \rangle {m\swos0}\sw0 &&\text{(by } \eqref{vartheta2})\\
&= \langle \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact m\swos{-1}, \mathcal{S}(r\sw{-1}) \lact \mathcal{S}_R(r\sw0) \rangle {m\swos0}\sw0&&\text{(by } \eqref{pair2})\\
&= \langle \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact \mathcal{S}_{R^{\vee}}^{-1}(m\swos{-1}), \mathcal{S}(r\sw{-1}) \lact \mathcal{S}_R^2(r\sw0) \rangle {m\swos0}\sw0&&\text{(by } \eqref{pair6})\\
&=\langle \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact \mathcal{S}_{R^{\vee}}^{-1}(m\swos{-1}), \mathcal{S}^2(r) \rangle {m\swos0}\sw0&&\text{(by } \eqref{bigS2})\\
&=\langle \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact \vartheta\mathcal{S}^{-1}(m\swos{-1}), \mathcal{S}^2(r) \rangle {m\swos0}\sw0&&\text{(by } \eqref{ruleS5})\\
&=\langle \vartheta\mathcal{S}^{-1}(m\swos{-1}{m\swos0}\sw{-1}), \mathcal{S}^2(r) \rangle {m\swos0}\sw0&&\text{(by } \eqref{vartheta2})\\
&=\langle r, \vartheta\mathcal{S}^{-1}(m\swoe{-1}) \rangle' m\swoe0.
\end{align*}
\end{proof}
\begin{theor}\label{theor:second}
Let $(R, R^{\vee})$ be a dual pair of Hopf algebras in $\ydH$ with bijective antipodes and bilinear form $\langle\;,\; \rangle : R^{\vee} \ot R \to \fie$. Let $\langle\;,\; \rangle' : R \ot R^{\vee} \to \fie$ be the form defined in \eqref{defin:inversepair}. Assume that ${{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}$ is $R^{\vee} \#H$-faithful.
Then the functor
$$(\widetilde{F},\widetilde{\varphi}) : {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}} \to {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$$
as defined below is a braided monoidal isomorphism.
For any object $M \in {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$ with right $R \# H$-comodule structure denoted by
$$\delta_M: M \to M \ot R \# H,\; m \mapsto m\swe0 \ot m\swe1,$$
let $\widetilde{F}(M)=M$ as a vector space and $\widetilde{F}(M) \in {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$
with left $H$-action, $H$-coaction $\delta_{\widetilde{F}(M)}^H$, $R^{\vee}$-action, and $R^{\vee} \#H$-coaction
$$\delta_{\widetilde{F}(M)} : M \to R^{\vee} \#H \ot M,\; m \mapsto m\swoe{-1} \ot m\swoe0,$$
respectively, given by
\begin{align}
hm &= m \mathcal{S}^{-1}(h),\label{tilde1}\\
\delta_{\widetilde{F}(M)}^H(m) &= \pi\mathcal{S}(m\swe1) \ot m\swe0,\label{tilde2}\\
\xi m &= \langle \xi,\vartheta\mathcal{S}(m\swe1)\rangle m\swe0,\label{tilde3}\\
mr&= \langle r, \vartheta\mathcal{S}^{-1}(m\swoe{-1})\rangle' m\swoe0\label{tilde4}
\end{align}
for all $h \in H,m \in M, \xi \in R^{\vee}$ and $r \in R$. For any morphism $f$ in ${}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$ let $\widetilde{F}(f) = f$.
The natural transformation $\widetilde{\varphi}$ is defined by
\begin{align}
\widetilde{\varphi}_{M,N} : \widetilde{F}(M) \ot \widetilde{F}(N) &\to \widetilde{F}(M \ot N),\\
m \ot n &\mapsto m \pi\mathcal{S}^{-1}(n\swe1) \ot n\swe0 = \pi \mathcal{S}^{-1}(n\swoe{-1})m \ot n\swoe0,
\end{align}
for all $M,N \in {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$.
\end{theor}
\begin{proof}
Let $M \in {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$. As in Lemma \ref{lem:second1} we write $\Psi(M)= (M_{\mathrm{res}},\gamma)$. Then
$$F^{\mathcal{Z}} \Psi(M)= (F(M_{\mathrm{res}}), \ti{\gamma}).$$
By Lemma \ref{lem:reformulation}, the definitions of $\delta_{\widetilde{F}(M)}$ in Lemma \ref{lem:second2} and in \eqref{tilde4} coincide. Thus, by Lemma \ref{lem:second2} (2), for all $X \in \mathcal{YD}^{\phantom{.}R \# H}_H$, the isomorphism
$$\ti{\gamma}_{F(X)} : F(M_{\mathrm{res}}) \ot F(X) \to F(X) \ot F(M_{\mathrm{res}})$$
has the form
$$\ti{\gamma}_{F(X)}(m \ot x) = m\swoe{-1}x \ot m\swoe0$$
for all $m \in M,x \in X$, where $\delta_{\widetilde{F}(M)}(m) = m\swoe{-1} \ot m\swoe0$ is defined in Lemma \ref{lem:second2}. By Lemma \ref{lem:second2} (3), the left $H$-comodule structure of $F(M_{\mathrm{res}})$ is $(\pi \ot \id) \delta_{\widetilde{F}(M)}$.
The left $H$-action, $H$-coaction and $R^{\vee} \#H$-action of $\widetilde{F}(M)$ are those of $F(M_{\mathrm{res}})$, see Theorem \ref{theor:first}.
We now conclude from Proposition \ref{propo:modulecenter} that $\widetilde{F}(M)$ with $R^{\vee} \#H$-comodule structure $\delta_{\widetilde{F}(M)}$ is an object in ${}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$, and $\Phi(\widetilde{F}(M)) = F^{\mathcal{Z}}\Psi(M)$.
Thus we have defined a functor $\widetilde{F} : {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}} \to {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$ such that the diagram \eqref{diagramsecondF} commutes. By Lemma \ref{lem:diagram} there is a uniquely determined family $\widetilde{\varphi}$ such that $(\ti{F},\ti{\varphi})$ is a braided monoidal functor with
$$(F^{\mathcal{Z}},\varphi^{\mathcal{Z}})(\Psi,\id) = (\Phi,\id)(\ti{F},\ti{\varphi}).$$
Let $M,N \in {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$. Then $\Phi(\ti{\varphi}_{M,N}) = \varphi^{\mathcal{Z}}_{\Psi(M),\Psi(N)}$ by \eqref{monoidalcomposition}, that is, for all $m \in M,n \in N$,
$$\ti{\varphi}_{M,N}(m \ot n)=\varphi_{M_{\mathrm{res}},N_{\mathrm{res}}}(m \ot n) = m\pi\mathcal{S}^{-1}(n\swe1) \ot n\swe0$$
by Theorem~\ref{theor:first}.
\begin{comment}
Let $M,N \in {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$. Since $(F^{\mathcal{Z}},{\varphi}^{\mathcal{Z}})$ is a monoidal functor,
\begin{align*}
{\varphi}^{\mathcal{Z}}_{\Psi(M),\Psi(N)} : F^{\mathcal{Z}}\Psi(M) \ot F^{\mathcal{Z}}\Psi(N) \to F^{\mathcal{Z}}\Psi(M\ot N)
\end{align*}
is a morphism in $\mathcal{Z}({{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}})$. By \eqref{monoidalcenter2},
\begin{align*}
{\varphi}^{\mathcal{Z}}_{\Psi(M),\Psi(N)} = \varphi_{M_{\mathrm{res}},N_{\mathrm{res}}} : F(M_{\mathrm{res}}) \ot F(N_{\mathrm{res}}) \to F(M_{\mathrm{res}} \ot N_{\mathrm{res}}).
\end{align*}
Hence it follows from the commutative diagram \eqref{diagramsecondF} that the $R^{\vee} \#H$-linear and $H$-colinear map
$$\varphi_{M,N} = \widetilde{\varphi}_{M,N} : \widetilde{F}(M) \ot \widetilde{F}(N) \to \widetilde{F}(M \ot N)$$
is a morphism
\begin{align*}
\Phi(\ti{F}(M) \ot \ti{F}(N)) \to \Phi\ti{F}(M \ot N)
\end{align*}
in $\mathcal{Z}({{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}})$. Thus $\ti{\varphi}_{M,N}$ is left $R^{\vee} \#H$-colinear by Proposition \ref{propo:modulecenter} (1).
\end{comment}
To define the inverse functor of $\widetilde{F}$ let $M \in {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$.
Let $\widetilde{G}(M) = M$ as a vector space with right $R \# H$-comodule structure
and $H$-module structure given by ${}^{\pi}M$, and with right $R \# H$-module structure
$\mu_{\widetilde{G}}$ defined in \eqref{tildeaction}.
Then $\widetilde{G}(M) \in \mathcal{YD}^{\phantom{.}R \# H}_{R \# H}$ by Proposition \ref{propo:comodulecenter} and
Lemma \ref{lem:second4} (1), (2). It follows from Lemma \ref{lem:second4} (3)
that $\widetilde{G}(M)$ is rational as an $R$-module. We let $\widetilde{G}(f) =f$
for morphisms in $ {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$.
Thus we have defined a functor $\widetilde{G} : {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat} \to {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}$,
and it is clear form the explicit definitions of $\widetilde{F}$ and $\widetilde{G}$
that $\widetilde{F}\widetilde{G} = \id$ and $\widetilde{G}\widetilde{F} = \id$.
\end{proof}
\section{The third isomorphism}\label{sec:third}
Finally we compose the isomorphism in Theorem \ref{theor:second} with the isomorphism in Lemma \ref{lem:leftright}.
\begin{comment}
\begin{theor}\label{theor:third}
Let $(R, R^{\vee})$ be a dual pair of Hopf algebras in $\ydH$ with bijective antipodes and with bilinear form $\langle\;,\; \rangle : R^{\vee} \ot R \to \fie$. Let $\langle\;,\; \rangle' : R \ot R^{\vee} \to \fie$ be the form defined in \eqref{defin:inversepair}. Assume that ${{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}$ is $R^{\vee} \#H$-faithful.
Then the functor
$$(\Omega,\omega) : {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat} \to {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$$
as defined below is a braided monoidal isomorphism.
Let $M \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$ with left $R \# H$-comodule structure denoted by
$$\delta_M: M \to R \# H \ot M,\; m \mapsto m\swe{-1} \ot m\swe0.$$
Then $M \in \ydH$ with $H$-action defined by restriction, and $H$-coaction $(\pi \ot \id)\delta_M$. Let $\Omega(M)=M$ as an object in $\ydH$, and $\Omega(M) \in {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$
with $R^{\vee}$-action and $R^{\vee} \#H$-coaction denoted by
$$\delta_{\Omega(M)} : M \to R^{\vee} \#H \ot M,\; m \mapsto m\swoe{-1} \ot m\swoe0,$$
respectively, given by
\begin{align}
\xi m &= \langle \xi,\vartheta(m\swe{-1})\rangle m\swe0,\label{actionO}\\
\mathcal{S}(r)m&= \langle r, \vartheta\mathcal{S}^{-1}(m\swoe{-1})\rangle' m\swoe0\label{coactionO}
\end{align}
for all $m \in M, \xi \in R^{\vee}$ and $r \in R$. For any morphism $f$ in $ {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$ let $\Omega(f)=f$.
The natural transformation $\omega$ is defined by
\begin{align}
\omega_{M,N} : \Omega(M) \ot \Omega(N) \to \Omega(M \ot N),\; m \ot n \mapsto n\swe{-1} \pi \mathcal{S}^{-1}(n\swe{-2}) m \ot n \swe0,\label{fu}
\end{align}
for all $M,N \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$.
\end{theor}
\end{comment}
We recall from Lemma \ref{lem:Rmod} and Lemma \ref{lem:Rcomod} the description of left modules and left comodules over $R \# H$, where $R$ is a Hopf algebra in $\ydH$. In particular, the restriction of an object $M \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}$ with $R \# H$-comodule structure $\delta_M$ is an object in $\ydH$, where the $H$-action is defined by restriction and the $H$-coaction is $(\pi \ot \id)\delta_M$.
\begin{theor}\label{theor:third}
Let $(R, R^{\vee})$ be a dual pair of Hopf algebras in $\ydH$ with bijective antipodes and with bilinear form $\langle\;,\; \rangle : R^{\vee} \ot R \to \fie$. Assume that ${{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}$ is $R^{\vee} \#H$-faithful.
Then the functor
$$(\Omega,\omega) : {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat} \to {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$$
as defined below is a braided monoidal isomorphism.
Let $M \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$ with left $R$-comodule structure denoted by
$$\delta_M^R : M \to R \ot M,\; m \mapsto m\sws{-1} \ot m\sws0.$$
Let $\Omega(M)=M$ as an object in $\ydH$ by restriction, and $\Omega(M) \in {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$
with $R^{\vee}$-action and $R^{\vee}$-coaction $\delta_{\Omega(M)}^{R^{\vee}}$,
respectively, given by
\begin{align}
\xi m &= \langle \xi,m\sws{-1}\rangle m\sws0,\label{actions}\\
\delta_{\Omega(M)}^{R^{\vee}} (m) &= c^2_{R^{\vee},M}(m\swoss{-1} \ot m\swoss0),\label{coactionss}\\
\intertext{where}
rm &= \langle m\swoss{-1}, \theta_R(r) \rangle m\swoss0\label{coactionss1}
\end{align}
for all $m \in M, \xi \in R^{\vee}$ and $r \in R$. For any morphism $f$ in $ {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$ let $\Omega(f)=f$.
The natural transformation $\omega$ is defined by
\begin{align}
\omega_{M,N} : \Omega(M) \ot \Omega(N) \to \Omega(M \ot N),\; m \ot n \mapsto \mathcal{S}^{-1} \mathcal{S}_R(n\sws{-1}) m \ot n \sws0,\label{fu}
\end{align}
for all $M,N \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$.
\end{theor}
\begin{proof}
Let $(S_1^{-1}, \psi) : {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD} \to \mathcal{YD}^{\phantom{.}R \# H}_{R \# H}$ be the braided monoidal isomorphism
defined in Lemma \ref{lem:leftright} (2).
Let $M \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}$, and assume that $M$ is rational as a left $R$-module.
By definition, $S_1^{-1}(M) =M$ as a vector space, and $mr = \mathcal{S}(r)m$ for all $m \in M,r \in R$,
where $\mathcal{S}$ is the antipode of $R \# H$.
Let $m \in M$. Since $M$ is a rational left $R$-module, $E'^{\perp}m =0$ for some $E'\in \mathcal{E}_{R^{\vee}}$.
Choose a subspace $E''\in \mathcal{E}_{R^{\vee}}$ with $\mathcal{S}_{R^{\vee}}(E') \subseteq E''$.
Then $\mathcal{S}(r)m=\mathcal{S}(r\sw{-1}) \mathcal{S}_R(r\sw0)m=0$ for all $r \in E''^{\perp}$ by \eqref{bigS} and \eqref{pair6}.
Hence $S_1^{-1}(M)$ is rational as a right $R$-module.
Thus $(S_1^{-1}, \psi)$ induces a functor on the rational objects. We denote the induced functor again by
$$(S_1^{-1}, \psi) : {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat} \to {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}.$$
Let
$$(\widetilde{F},\widetilde{\varphi}) : {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}} \to {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$$
be the braided monoidal isomorphism of Theorem \ref{theor:second}. Then the composition
\begin{align}\label{def:fu}
(\Omega,\omega) =(\widetilde{F},\widetilde{\varphi})(S_1^{-1}, \psi)
\end{align}
is a braided monoidal isomorphism.
Let $M \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$. The $R^{\vee} \# H$-coaction denoted by
$$\delta_{\Omega(M)} : M \to R^{\vee} \# H \ot M,\; m \mapsto m\swoe{-1} \ot m\swoe0,$$
is given by
\begin{align}
\mathcal{S}(r)m= \langle r, \vartheta\mathcal{S}^{-1}(m\swoe{-1})\rangle' m\swoe0\label{coactions}
\end{align}
for all $m \in M$ and $r \in R$.
Let
$$\delta_{\Omega(M)}^{R^{\vee}} = (\vartheta \ot \id)\delta_{\Omega(M)} : M \to R^{\vee} \ot M,\;m \mapsto m\swos{-1} \ot m\swos0,$$
be the $R^{\vee}$-coaction of $\Omega(M)$.
To prove \eqref{coactionss}, let $m \in M, r \in R$. Then by \eqref{coactions} and \eqref{bigS2},
\begin{align*}
\mathcal{S}(r)m &= \langle \vartheta \mathcal{S}^{-1}(m\swoe{-1}), \mathcal{S}_R^2(\theta_R(r)) \rangle m\swoe0,
\end{align*}
hence
\begin{align*}
\mathcal{S}_R(r) m &= \langle \vartheta \mathcal{S}^{-1}(m\swoe{-1}), \mathcal{S}^2_R(\theta_R(r\sw0) \rangle r\sw{-1} m\swoe0 &&\text{(by }\eqref{bigS})\\
&=\langle \mathcal{S}^2_{R^{\vee}} \vartheta \mathcal{S}^{-1}(m\swos{-1} {m\swos0}\sw{-1}),\theta_R(r\sw0) \rangle r\sw{-1} {m\swos0}\sw0 &&\text{(by }\eqref{pair6})\\
&=\langle \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact \mathcal{S}^2_{R^{\vee}} \vartheta \mathcal{S}^{-1}(m\swos{-1}) ,\theta_R(r\sw0) \rangle r\sw{-1} {m\swos0}\sw0 &&\text{(by }\eqref{vartheta2})\\
&=\langle \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact \mathcal{S}_{R^{\vee}} (m\swos{-1}) ,\theta_R(r\sw0) \rangle r\sw{-1} {m\swos0}\sw0 &&\text{(by }\eqref{ruleS5})\\
&=\langle \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact \mathcal{S}_{R^{\vee}} (m\swos{-1}) ,\theta_R(r)\sw0 \rangle \mathcal{S}^{-2}(\theta_R(r)\sw{-1}){m\swos0}\sw0 &&\text{(by }\eqref{theta2}).
\end{align*}
Since $\theta_R \mathcal{S}_R^{-1} = \mathcal{S}_R^{-1} \theta_R$, we obtain by \eqref{pair6}
\begin{align}\label{formulathird}
rm &= \langle \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact m\swos{-1}, \theta_R(r)\sw0 \rangle \mathcal{S}^{-2}(\theta_R(r)\sw{-1}) {m\swos0}\sw{0}.
\end{align}
Note that $c^{-1}_{R^{\vee},M}(m\swos{-1} \ot m\swos0) = {m\swos0}\sw0 \ot \mathcal{S}^{-1}({m\swos0}\sw{-1}) \lact m\swos{-1}$. Hence by \eqref{formulathird} and \eqref{pair3},
\begin{align*}
rm &= \langle m\swoss{-1}, \theta_R(r) \rangle m\swoss0,
\end{align*}
where $m\swoss{-1} \ot m\swoss0 = c_{M,R^{\vee}}^{-1} c_{R^{\vee},M}^{-1}(m\swos{-1} \ot m\swos0)$.
Finally, by \eqref{def:fu} and \eqref{monoidalcomposition} the natural transformation $\omega$ is given by
\begin{align}\label{fuold}
\omega_{M,N} : \Omega(M) \ot \Omega(N) \to \Omega(M \ot N),\; m \ot n \mapsto n\swe{-1} \pi \mathcal{S}^{-1}(n\swe{-2}) m \ot n \swe0,
\end{align}
for all $M,N \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$, where
$$N \to R \# H \ot N,\; n \mapsto n\swe{-1} \ot n\swe0 = n\sws{-1} {n\sws0}\sw{-1} \ot {n\sws0}\sw0,$$
denotes the $ R \# H$- coaction of $N$. Let $r \in R, h \in H$ and $a = rh \in R \# H$. Then
\begin{align*}
a\sw2 \pi\mathcal{S}^{-1}(a\sw1) &= \varepsilon(h) r\sw2 \pi\mathcal{S}^{-1}(r\sw1)&&\\
&= \varepsilon(h) {r\swo2}\sw0 \pi\mathcal{S}^{-1}(r\swo1 {r\swo2}\sw{-1})&&\\
&= \varepsilon(h) r\sw0 \mathcal{S}^{-1}(r\sw{-1})&&\\
&= \varepsilon(h) \mathcal{S}^{-1}\mathcal{S}_R(r). &&\text{by \eqref{antipode}}
\end{align*}
Hence \eqref{fu} follows from \eqref{fuold}.
\end{proof}
\begin{comment}
\begin{proof}
Let $(S_1^{-1}, \psi) : {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD} \to \mathcal{YD}^{\phantom{.}R \# H}_{R \# H}$ be the braided monoidal isomorphism defined in Lemma \ref{lem:leftright} (2). Let $M \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}$, and assume that $M$ is rational as a left $R$-module.
By definition, $S_1^{-1}(M) =M$ as a vector space, and $mr = \mathcal{S}(r)m$ for all $m \in M,r \in R$, where $\mathcal{S}$ is the antipode of $R \# H$. Let $m \in M$. Since $M$ is a rational left $R$-module, $E'^{\perp}m =0$ for some $E'\in \mathcal{E}_{R^{\vee}}$. Choose a subspace $E'' \in \mathcal{E}_{R^{\vee}}$ with $\mathcal{S}_{R^{\vee}}(E') \subseteq E''$. Then $\mathcal{S}(r)m=\mathcal{S}(r\sw{-1}) \mathcal{S}_R(r\sw0)m=0$ for all $r \in E''$ by \eqref{bigS} and \eqref{pair6}.
Hence $S_1^{-1}(M)$ is rational as a right $R$-module.
Thus $(S_1^{-1}, \psi)$ induced a functor on the rational objects. We denote the induced functor again by
$$(S_1^{-1}, \psi) : {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat} \to {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}}.$$
Let
$$(\widetilde{F},\widetilde{\varphi}) : {}_{\rat}{\mathcal{YD}^{\phantom{.}R \# H}_{R \# H}} \to {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$$
be the braided monoidal isomorphism of Theorem \ref{theor:second}. Then the composition $(\Omega,\omega) =(\widetilde{F},\widetilde{\varphi})(S_1^{-1}, \psi)$ is a braided monoidal isomorphism, where $\Omega=\widetilde{F}S_1^{-1}$, and $\omega$ is determined by \eqref{monoidalcomposition}.
\end{proof}
\end{comment}
We specialize the last theorem to the case of $\mathbb{N}_0$-graded dual pairs of braided Hopf algebras in $\ydH$.
Let $R = \oplus_{n\geq 0} R(n)$ be an $\mathbb{N}_0$-graded Hopf algebra in $\ydH$. We view the bosonization $R \# H$ as an $\mathbb{N}_0$-graded Hopf algebra with $\deg R(n)=n$ for all $n\geq 0$, and $\deg H =0$.
For any Yetter-Drinfeld module $W \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}$ we define two ascending filtrations of Yetter-Drinfeld modules in $\ydH$ by
\begin{align}
\Fd n W&= \{ w \in W \mid \delta^R_W(w) \in \oplus_{i=0}^n R(i) \ot W \},\\
\Fm n W&= \{ w \in W \mid R(i) w = 0 \text{ for all } i > n \}
\end{align}
for all $n \geq 0$. Then
$\cup_{n \geq 0} \Fd n W = W$. But in general,
$\cup_{n \geq 0} \Fm n W \neq W$.
Given an abelian monoid $\Gamma $ and a $\Gamma $-graded Hopf algebra $A$
with bijective antipode, we say that $M \in {}^{\phantom{.}{A}}_{A}\mathcal{YD} $ is $\Gamma $-\textit{graded}
if $M=\oplus_{\gamma \in \Gamma }M(\gamma )$
is a vector space grading and if the module and comodule maps of $M$ are $\Gamma $-graded
of degree $0$.
\begin{corol}\label{cor:third}
Let $R^{\vee}= \oplus_{n \geq 0} R^{\vee}(n)$ and $R= \oplus_{n \geq 0} R(n)$ be $\mathbb{N}_0$-graded Hopf algebras in $\ydH$ with finite-dimensional components $R^{\vee}(n)$ and $R(n)$ for all $n \geq 0$, and let $\langle \;,\; \rangle : R^{\vee} \ot R \to \fie$ be a bilinear form of vector spaces satisfying \eqref{pair1} -- \eqref{pair5} and \eqref{gradedpair}. Then the functor
$$(\Omega,\omega) : {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat} \to {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$$
as defined in Theorem \ref{theor:third} is a braided monoidal isomorphism.
Moreover, the following hold.
\begin{enumerate}
\item A left $R$- (respectively $R^{\vee}$)-module $M$ is rational if and only if for any $m \in M$ there is a natural number $n_0$ such that $R(n)m =0$ (respectively $R^{\vee}(n)m =0$) for all $n \geq n_0$.
\item Let $M \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$ be $\mathbb{Z}$-graded. Then $\Omega(M)$ is a $\mathbb{Z}$-graded object in $ {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$ with $\Omega(M)(n) = M(-n)$ for all $m \in \mathbb{Z}$.
\item For any $M \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$ and $n \geq 0$,
\begin{align*}
\Fm n \Omega(W) = \Fd n W,\;
\Fd n \Omega(W) = \Fm n W.
\end{align*}
\end{enumerate}
\end{corol}
\begin{proof}
By Example \ref{exa:gradedpair}, the antipodes of $R$ and of $R^{\vee}$ are bijective,
and $(R, R^{\vee})$ together with $\langle\;,\; \rangle$ is a dual pair of Hopf algebras in $\ydH$.
By Example \ref{exa:regularrep} (2), the category ${{}^{\phantom{aaaaa.}H}_{\phantom{a}R^{\vee} \# H}\mathcal{YD}_{\rat}}$ is $R^{\vee} \#H$-faithful.
Thus $(\Omega,\omega)$ is a braided monoidal isomorphism by Theorem \ref{theor:third}.
(1) is clear from Example \ref{exa:gradedpair}, and (2) and (3) can be checked using \eqref{actions} and \eqref{coactionss}.
\end{proof}
\begin{propo}\label{propo:Fdm}
Let $R = \oplus_{n\geq 0} R(n)$ be an $\mathbb{N}_0$-graded Hopf algebra in $\ydH$ with finite-dimensional components $R(n)$ for all $n\geq 0$. Let $W$ be an irreducible object in the category of $\mathbb{Z}$-graded left Yetter-Drinfeld modules over $R \# H$. Assume that $W$ is locally finite as an $R$-module, or equivalently finite-dimensional.
Let $n_0 \leq n_1$ in $\ndZ$, and $W = \oplus_{i=n_0}^{n_1} W(i)$ be the decomposition into homogeneous components such that $W( n_0) \neq 0, W(n_1) \neq 0$.
Then
\begin{align}\label{Fdm}
\Fd n W = \mathop{\oplus} _{i=n_0}^{n_0 + n} W(i),\quad \Fm n W = \mathop{\oplus} _{i=n_1 -n}^{n_1} W(i)
\end{align}
for all $n \geq 0$.
Moreover, $W(n_0)$ and $W(n_1)$ are irreducible Yetter-Drinfeld modules over $R(0)\#H$,
where the action and coaction arise from the action and coaction of $R\#H$ on $W$
by restriction and projection, respectively.
\end{propo}
\begin{proof}
The inclusions $\supseteq$ in \eqref{Fdm} follow from the definitions
since $W$ is a $\ndZ$-graded Yetter-Drinfeld module. On the other hand, assume that
$\Fd n W \neq \mathop{\oplus} _{i=n_0}^{n_0 + n} W(i)$ for some $n \geq 0$. Then there exist $l > n_0 + n$ and $w \in W(l) \cap \Fd n (W)$ with $w \neq 0$, since $W$ is a $\ndZ$-graded Yetter-Drinfeld module. Then the Yetter-Drinfeld submodule of $W$ generated by $w$ is contained in $\oplus_{n > n_0} W(n)$. This is a contradiction to $W(n_0) \neq 0$ and the irreducibility of $W$. The proof of the second equation in \eqref{Fdm} is similar.
By degree reasons, $W(n_0)$ is a Yetter-Drinfeld module over $R(0)\#H$
in the way explained in the claim. It is irreducible, since $W$ is irreducible
and hence it is the $R\#H$-module generated by any nonzero Yetter-Drinfeld submodule
over $R(0)\#H$ of $W(n_0)$.
Similarly, $W(n_1)$ is an irreducible Yetter-Drinfeld module over $R(0)\#H$,
since $W$ is the $R\#H$-comodule generated by any nonzero Yetter-Drinfeld submodule
over $R(0)\#H$ of $W(n_1)$.
\end{proof}
Let $R$ be a braided Hopf algebra in $\ydH$, and let $K$ be a Hopf algebra in
$ {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}$. Then
\begin{align*}
K \# R := (K \# (R \# H))^{\co H}
\end{align*}
denotes the braided Hopf algebra in $\ydH$ of $H$-coinvariant elements with
respect to the canonical projection $K \# (R \# H) \to R \# H \to H$.
\begin{corol} \label{corol:Hopfproj}
In the situation of Theorem \ref{theor:third} assume that $R $ is a Hopf subalgebra of a Hopf algebra $B$ in $\ydH$ with a Hopf algebra projection onto $R$, and let $K:=B^{\co R}$.
\begin{enumerate}
\item $K= (B \# H)^{\co R \# H}$ is a Hopf algebra in $ {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD} $, and the multiplication map $K \# R \to B$ is an isomorphism of Hopf algebras in $\ydH$. \item Assume that $K$ is rational as an $R$-module. Then $\Omega(K) \# R^{\vee}$ is a Hopf algebra in $\ydH$ with a Hopf algebra projection onto $R^{\vee}$.
\end{enumerate}
\end{corol}
\begin{proof}
(1) is shown in \cite[Lemma 3.1]{a-AHS10}. By Theorem \ref{theor:third}, $\Omega(K)$ is a Hopf algebra in $ {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD} $. This proves (2).
\end{proof}
\section{An application to Nichols algebras}\label{sec:Nichols}
In the last section we want to apply the construction in Corollary~\ref{corol:Hopfproj}
to Nichols algebras. We show in Theorem~\ref{theor:NtoN} that if $B$
is a Nichols algebra of a semisimple Yetter-Drinfeld module,
then the Hopf algebra $\Omega(K) \# R^{\vee}$ constructed in Corollary~\ref{corol:Hopfproj} is again a Nichols algebra.
The advantage of the construction is that the new Nichols algebra is usually not
twist equivalent to the original one.
We start with some
general observations.
\begin{remar}\label{rem:primitive}
Let $R = \oplus_{n \in \mathbb{N}_0} R(n)$ be an $\mathbb{N}_0$-graded bialgebra in $\ydH$.
(1) The space
$$P(R) = \{ x \in R \mid \Delta_R(x) = 1 \ot x + x \ot 1 \}$$
of primitive elements of $R$ is an $\mathbb{N}_0$-graded subobject of $R$ in $\ydH$, since it is the kernel of the graded, $H$-linear and $H$-colinear map
$$R \to R \ot R,\; x \mapsto \Delta_R(x) - 1 \ot x - x \ot 1.$$
(2) Assume that $R(0) = \fie$. Then $R(1) \subseteq P(R)$. Moreover, $R$ is an $\mathbb{N}_0$-graded braided Hopf algebra in $\ydH$.
\end{remar}
Let $M \in \ydH$. A {\em pre-Nichols algebra} \cite{a-Masuo08} of $M$ is an $\mathbb{N}_0$-graded braided bialgebra $R = \oplus_{n \geq \mathbb{N}_0} R(n)$ in $\ydH$ such that
\begin{enumerate}
\item [(N1)]$R(0) = \fie$,
\item [(N2)]$R(1) = M$,
\item [(N3)]$R$ is generated as an algebra by $M$.
\end{enumerate}
The Nichols algebra of $M$ is a pre-Nichols algebra $R$ of $M$ such that
\begin{enumerate}
\item [(N4)]$P(R) \cap R(n) = 0$ for all $n \geq 2$.
\end{enumerate}
It is denoted by $\NA(M)$. Up to isomorphism, $\NA(M)$ is uniquely determined by $M$. By Remark \ref{rem:primitive}, our definition of $\NA(M)$ coincides with \cite[Def.~2.1]{inp-AndrSchn02}.
The Nichols algebra $\NA(M)$ has the following {\em universal property}:
For any pre-Nichols algebra $R$ of $M$ there is exactly one map
$$\rho : R \to \NA(M),\; \rho \mid M = \id,$$
of $\mathbb{N}_0$-graded braided bialgebras in $\ydH$. Thus $\NA(M)$ is the smallest pre-Nichols algebra of $M$.
In the situation of Theorem \ref{theor:third}, the functor
$$(\Omega,\omega) : {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat} \to {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$$
is a braided monoidal isomorphism. Hence for any $\mathbb{N}_0$-graded braided bialgebra $B$ in $ {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$ with multiplication $\mu_B$ and comultiplication $\Delta_B$, the image $\Omega(B)$ is an $\mathbb{N}_0$-graded braided bialgebra in $ {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$ with multiplication
$$\Omega(B) \ot \Omega(B) \xrightarrow{\omega_{B,B}} \Omega(B \ot B) \xrightarrow{\Omega(\mu_B)} \Omega(B)$$
and comultiplication
$$\Omega(B) \xrightarrow{\Omega(\Delta_{B})} \Omega(B \ot B) \xrightarrow{\omega_{B,B}^{-1}}\Omega(B) \ot \Omega(B).$$
The unit elements and the augmentations in $B$ and $\Omega(B)$ coincide.
\begin{corol}\label{cor:Nichols}
Under the assumptions of Theorem \ref{theor:third}, let $M \in {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$. Then
$$\Omega(\NA(M)) \cong \NA(\Omega(M))$$
as $\mathbb{N}_0$-graded braided Hopf algebras in $ {}^{\phantom{.}R^{\vee} \# H}_{R^{\vee} \# H}\mathcal{YD}_{\rat}$.
\end{corol}
\begin{proof}
By (N3) and \eqref{pair8}, $\NA(M)$ is rational as an $R$-module, since $M$ is rational. By Theorem \ref{theor:third}, $(\Omega,\omega)$ is a braided monoidal isomorphism. Hence $\NA(M)$ is an $\mathbb{N}_0$-graded braided bialgebra in $ {}^{\phantom{.}R \# H}_{R \# H}\mathcal{YD}_{\rat}$. Since $\Omega$ is the identity on morphisms, (N1) -- (N4) hold for $\Omega(\NA(M))$. This proves the Corollary.
\end{proof}
\begin{comment}
For a braided Hopf algebra $R$ in $\ydH $ and a left $R\#H$-comodule $W$
define
\begin{align}
{}^{\co R}W=\{w\in W\,|\,(\vartheta \otimes \id )\delta (w)=1\otimes w\},
\label{eq:coRW}
\end{align}
where $\vartheta :R\#H \to R$ is the coalgebra projection defined in
\eqref{eq:defvartheta}.
We note that ${}^{\co R}W = \{ w \in W \mid \delta(w) \in H \ot W\}$. Thus ${}^{\co R}W$ is the largest Yetter-Drinfeld module over $H$ which is contained in $W$.
Similarly, for a right $R\#H$-comodule $W$
define
\begin{align}
W^{\co R}=\{w\in W\,|\,(\id \otimes \vartheta )\delta (w)=w\otimes 1\}.
\label{eq:WcoR}
\end{align}
One should not confuse this notation with $W^{\co H}$ defined in
Equation~\eqref{action}.
\end{comment}
Let $B$ be a coalgebra. An $\ndN _0$-filtration $\mathcal{F} = (\mathcal{F}_n B)_{n \in \mathbb{N}_0}$ of $B$
is a family of subspaces $\mathcal{F}_nB, n \geq 0,$ of $B$ such that
\begin{itemize}
\item []$\mathcal{F} _nB$ is a subspace of $\mathcal{F} _mB$ for all $m,n\in \ndN _0$ with $n\le m$,
\item []$B=\bigcup _{n\in \ndN _0}\mathcal{F} _nB $, and
\item []$\Delta _B(x)\in \sum _{i=0}^n\mathcal{F} _iB\otimes \mathcal{F} _{n-i}B$ for all $x\in \mathcal{F} _nB$,
$n\in \ndN _0$.
\end{itemize}
\begin{lemma}\label{lem:F0}
Let $B$ be a coalgebra having an $\ndN _0$-filtration $\mathcal{F} $.
Let $U\in {}^B\cM $ be a non-zero object.
Then there exists $u\in U\setminus \{0\}$ such that
$\delta (u)\in \mathcal{F}_0B \otimes U$.
\label{le:YDBNH}
\end{lemma}
\begin{proof}
The coradical $B_0$ of $B$ is contained in $\mathcal{F}_0B$ by \cite[Lemma 5.3.4]{b-Montg93}. Hence $\delta^{-1}(\mathcal{F}_0B \ot U) \neq0$, since for any irreducible subcomodule $U'\subseteq U$ there is a simple subcoalgebra $C'$ with $\delta(U') \subseteq C'\ot U'$.
We give an alternative and more explicit proof. Let $x\in U\setminus \{0\}$. Then there exists $n\in \ndN _0$ with $\delta (x)\in \mathcal{F} _nB\otimes U$.
If $n=0$, we are done. Assume now that $n\ge 1$ and let $\pi _0:B\to B/\mathcal{F} _0B$ be the canonical
linear map.
Since $\mathcal{F} $ is a coalgebra filtration, there is a maximal $m\in \ndN _0$ such that
\[ \pi _0(x_{(-m)})\otimes \cdots \otimes \pi _0(x_{(-1)})\otimes x_{(0)}\not=0, \]
where $\delta (x)=x\_{-1}\otimes x\_0$.
Let $f_1,\dots,f_m\in B^*$ with $f_i|_{B_0}=0$ for all $i\in \{1,\dots,m\}$
such that
\[ y:=f_1(x_{(-m)})\cdots f_m(x_{(-1)})x_{(0)}\not=0. \]
Then $\delta (y)=f_1(x_{(-m-1)})\cdots f_m(x_{(-2)})x_{(-1)}\otimes x_{(0)}
\in \mathcal{F} _0B \otimes U$ by the maximality of $m$.
\end{proof}
\begin{lemma}\label{lem:ZgradK}
Let $\Gamma $ be an abelian group with neutral element $0$, and $A$ a $\Gamma $-graded Hopf algebra.
\begin{enumerate}
\item Let $K$ be a Nichols algebra in $ {}^{\phantom{.}{A}}_{A}\mathcal{YD} $, and $K(1)=\oplus _{\gamma \in \Gamma }K(1)_\gamma $
a $\Gamma $-graded object in $ {}^{\phantom{.}{A}}_{A}\mathcal{YD} $.
Then there is a unique $\Gamma$-grading on $K$ extending the grading on $K(1)$. Moreover, $K(n)$ is $\Gamma$-graded in $ {}^{\phantom{.}{A}}_{A}\mathcal{YD}$ for all $n \geq 0$.
\item Let $K$ be a $\Gamma $-graded braided Hopf algebra in $ {}^{\phantom{.}{A}}_{A}\mathcal{YD} $. Then the bosonization $K \# A$ is a $\Gamma$-graded Hopf algebra with $\deg K(\gamma) \# A(\lambda) = \gamma + \lambda$ for all $\gamma, \lambda \in \Gamma$.
\item Let $H \subseteq A$ be a Hopf subalgebra of degree {0}, and $\pi : A \to H$ a Hopf algebra map with $\pi \mid H = \id$. Define $R= A^{\co H}$. Then $R$ is a $\Gamma$-graded braided Hopf algebra in $\ydH$ with $R(\gamma) = R \cap A(\gamma)$ for all $\gamma \in \Gamma$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) The module and comodule maps of $K(1)$ are $\Gamma $-graded and hence the infinitesimal
braiding $c\in \Aut (K(1)\otimes K(1))$, being determined by the module and comodule maps,
is $\Gamma $-graded.
Now the claim of the lemma follows from the fact that
$K(n)$ for $n\in \ndN $ as well as
the structure maps of $K$ as a braided Hopf algebra
are determined by $c$ and $K(1)$.
(2) and (3) are easily checked.
\end{proof}
We now study the projection of $H$-Yetter-Drinfeld Hopf algebras in Corollary \ref{corol:Hopfproj} in the case of Nichols algebras. Recall that for any $M,N\in \ydH $ there is a canonical surjection
$$\pi_{\NA(N)} : \NA(M \oplus N) \to \NA(N),\; \pi_{\NA(N)} \mid N = \id, \pi_{\NA(N)} \mid M = 0,$$
of braided Hopf algebras in $\ydH$.
It defines a canonical projection
$$\pi _{\NA (N) \# H} = \pi_{\NA(N)} \# \id :\NA (M\oplus N) \# H \to \NA (N) \# H$$
of Hopf algebras.
Let $K =(\NA(M \oplus N) \# H)^{\co \NA(N) \# H}$ be the space of right $\NA(N) \# H$-coinvariant elements with respect to the projection $\pi_{\NA(N) \# H}$. Thus $K$ is a braided Hopf algebra in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$ with $\NA(N) \# H$-action
$$\ad : \NA(N) \# H \ot K \to K,\; a \ot x \mapsto (\ad a)x= a\sw1 x \mathcal{S}(a\sw2),$$
and $\NA(N) \# H$-coaction
$$\delta_K : K \to \NA(N) \# H \ot K, x \mapsto \pi_{\NA(N) \# H}(x\sw1) \ot x\sw2.$$
Then by ~\cite[Lemma 3.1]{a-AHS10}, $K = \NA(M \oplus N)^{\co \NA(N)}$,
the space of right $\NA(N)$-coinvariant elements with respect to $\pi_{\NA (N)}$.
The bosonization $\NA(N) \# H$ is a $\mathbb{Z}$-graded Hopf algebra with $\deg N =1$ and $\deg H = 0$. We always view the bosonizations of Nichols algebras in $\ydH$ as graded Hopf algebras in this way.
\begin{lemma}\label{lem:old}
Let $M,N\in \ydH$ and
$K=(\NA (M \oplus N) \# H)^{\co \NA (N) \# H}$.
\begin{enumerate}
\item The standard $\mathbb{N}_0$-grading of $\NA(M \oplus N)$ induces an $\mathbb{N}_0$-grading on
$$W = (\ad \NA(N))(M) = \oplus_{n \in \mathbb{N}_0} (\ad N)^n(M)$$ with $\deg (\ad N)^n(M)=n +1$. Then $W$ is a $\mathbb{Z}$-graded object in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$, where $W \subseteq K$ is a subobject in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$.
\item Assume that $M = \oplus_{i \in I} M_i$ is a direct sum of irreducible objects in $\ydH$. Let $W_i = (\ad\NA(N))(M_i)$ for all $i \in I$. Then $W = \oplus_{i \in I} W_i$ is a decomposition into irreducible subobjects $W_i$ in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$. For all $i \in I$, $W_i = \oplus_{n \geq 0}(\ad N)^n(M_i)$ is a $\ndZ$-graded object in the category of left Yetter-Drinfeld modules over $\NA(N) \# H$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Let $a \in N$ and $x \in \NA(M \oplus N)$ a homogeneous element. Then $\Delta_{\NA(M \oplus N) \# H}(a) = a \ot 1 + a\sw{-1} \ot a\sw0$, since $a$ is primitive in $\NA(N)$. Hence
$$(\ad a)(x) = ax - (a\sw{-1} \lact x) a\sw0$$
is of degree $\deg x + 1$ in $\NA(M \oplus N)$. This implies the decomposition of $W$. Moreover, $W \subseteq K$, since $M \subseteq K$.
Since $W= (\ad \NA(N) \# H)(M)$, it is clear that $W$ is stable under the adjoint action of $\NA(N) \# H$, and that
$$\ad : \NA(N) \# H \ot W \to W$$
is $\mathbb{Z}$-graded. To see that $W \subseteq K$ is a $\NA(N) \# H$-subcomodule, and that the comodule structure
$$W \to \NA(N) \# H \ot W$$
is $\mathbb{Z}$-graded, we compute $\delta_K$ on elements of $W$. For all $a \in \NA(N) \# H$ and $x \in M$,
\begin{align*}
\delta_K(\ad a)(x) &= (\pi_{\NA(N) \# H} \ot \id)\Delta_{\NA(M \oplus N) \# H}(\ad a)(x)\\
&=\pi_{\NA(N) \# H}\left(a\sw1 x\sw1 \mathcal{S}(a\sw4)\right) \ot a\sw2 x\sw2 \mathcal{S}(a\sw3)\\
&=\pi_{\NA(N) \# H}\left(a\sw1 x \mathcal{S}(a\sw4)\right) \ot a\sw2 \mathcal{S}(a\sw3)\\
&\phantom{aa}+ \pi_{\NA(N) \# H}\left(a\sw1 x\sw{-1} \mathcal{S}(a\sw4)\right) \ot a\sw2 x\sw0 \mathcal{S}(a\sw3)\\
&= a\sw1 x\sw{-1} \mathcal{S}(a\sw3) \ot (\ad a\sw2)(x\sw0).
\end{align*}
Thus the $\NA(N) \#H$-costructure of $W$ is well-defined and $\mathbb{Z}$-graded.
(2) is shown in ~\cite[Prop. 3.4, Prop. 3.5]{a-AHS10}.
\end{proof}
\begin{propo}\label{prop:old}
Let $M,N\in \ydH$ and
$K=(\NA (M \oplus N) \# H)^{\co \NA (N) \# H}$. Then there is a unique isomorphism
$$K\cong \NA \big( (\ad \NA (N))(M) \big)$$
of braided Hopf algebras in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$ which is the identity on $(\ad \NA (N))(M)$.
\end{propo}
\begin{proof}
Since $M \oplus N$ is a $\mathbb{Z}$-graded object in $\ydH$ with $\deg M = 1$ and $\deg N =0$, the Nichols algebra $\NA(M \oplus N)$ is a $\mathbb{Z}$-graded braided Hopf algebra in $\ydH$ by Lemma \ref{lem:ZgradK} (1). Hence the bosonization $\NA(M \oplus N) \# H$ is a $\mathbb{Z}$-graded Hopf algebra with $\deg M =1, \deg N =0, \deg H =0$. By Lemma \ref{lem:ZgradK} (3), $K$ is a $\mathbb{Z}$-graded Hopf algebra in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$. By ~\cite[Prop. 3.6]{a-AHS10}, $K$ is generated as an algebra by $K(1)= (\ad \NA(N))(M)$. Hence $K(n) = K(1)^n$ for all $n \geq 1$, and $K(0) = \fie$.
It remains to prove that all homogeneous primitive elements of $K$ are of
degree one. Let $n\in \ndN _{\ge 2}$ and let $U\subseteq K(n)$ be a subspace
of primitive elements. We have to show that $U=\{0\}$.
By Remark~\ref{rem:primitive} (1) we may assume that $U\in {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD} $.
Since $\NA (N)\#H$ has a coalgebra filtration $\mathcal{F} $ with $\mathcal{F} _0=H$ and $\mathcal{F} _1=NH+H$,
Lemma~\ref{le:YDBNH} implies that there exists a nonzero primitive element $u\in U$ with
$\delta (u)\in H\otimes U$. Then $u$ is primitive in $\NA (M \oplus N)$. Indeed,
$$ \Delta _{K\#(\NA (N)\#H)}=u\otimes 1+1u_{[-1]}\otimes u_{[0]}
=u\otimes 1+u_{(-1)}\otimes u_{(0)},
$$
and hence $\Delta _{K\#\NA (N)}(u)=(\vartheta \otimes \id )\Delta _{K\#(\NA (N)\#H)}(u)
=u\otimes 1+1\otimes u$.
Since $K(n) = \left(\ad \NA(N)(M)\right)^n$,
$u$ is an element of degree at least $n$ in the usual grading of $\NA(M \oplus N)$.
This contradicts the assumption that
$\NA (M \oplus N)$ is a Nichols algebra.
\end{proof}
Next we prove the converse of the above proposition under additional restrictions, see
Proposition~\ref{propo:KBNichols}.
Let $C$ be a coalgebra, $D \subseteq C$ a subcoalgebra, and $W$ a left $C$-comodule with comodule structure $\delta : W \to C \ot W$.
We denote the largest $D$-subcomodule of $W$ by
\begin{align*}
W(D) = \{ w \in W \mid \delta(w) \in D \ot W\}.
\end{align*}
\begin{comment}
Given an abelian group $\Gamma $ and a $\Gamma $-graded Hopf algebra $A$
with bijective antipode, we say that a $\Gamma$-graded object in $W\in {}^{\phantom{.}{A}}_{A}\mathcal{YD} $ is $\Gamma $-\textit{graded-semisimple}
if $W$ is the direct sum of $\ndZ $-graded and (not necessarily homogeneous)
irreducible subobjects in $ {}^{\phantom{.}{A}}_{A}\mathcal{YD} $.
\end{comment}
\begin{lemma}
Let $N\in \ydH $ and $W\in {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD} $.
Assume that $\oplus _{i\in I}W_i$ is a decomposition of $W$ into irreducible objects
in the category of $\ndZ$-graded left Yetter-Drinfeld modules over $\NA(N) \# H$.
Let $M=W(H)$, and $M_i=M\cap W_i$ for all $i \in I$.
\begin{enumerate}
\item $M=\oplus _{i\in I}M_i$ is a decomposition into irreducible
objects in $\ydH $.
\item For all $i\in I$, $M_i$ is the $\ndZ $-homogeneous component
of $W_i$ of minimal degree, and $W_i=B(N)\cdot M_i = \oplus_{n \geq 0} N^n \lact M_i$.
\end{enumerate}
\label{lem:ydBNH}
\end{lemma}
\begin{proof}
Let $W= \oplus_{n \in \mathbb{Z}} W(n)$ be the $\mathbb{Z}$-grading of $W$ in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD} $. Then
$M$ is a $\mathbb{Z}$-graded object in $\ydH$ with homogeneous components $M(n) = M \cap W(n)$ for all $n \in \mathbb{Z}$.
It is clear that $M = \oplus_{i \in I} M_i$, where $M_i = M \cap W_i = W_i(H)$ for all $i$.
Let $i \in I$. By Lemma \ref{lem:F0}, $M_i \neq 0$.
Let $0 \neq M'_i$ be a homogeneous subobject of $M_i$ in $\ydH $, and let $n$ be its degree.
Then the $\NA (N)\#H$-module $W'_i:=\NA (N)\cdot M'_i$ is a $\ndZ $-graded
subobject of $W_i$ in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD} $,
the homogeneous components of $W'_i$ have degrees $\ge n$,
and the degree $n$ component of $W'_i$ coincides with $M'_i$ since $\NA (N)(0)=\fie $
and $\deg N = 1$.
Thus the irreducibility of $W_i$ implies that $M_i=M'_i$ is irreducible and it is the
homogeneous component of $W_i$ of minimal degree.
Finally, for all $i \in I$ and $n \in \mathbb{N}_0$,
$$\deg (N^n \lact M_i) = n + \deg M_i,$$
since the multiplication map $\NA(N) \# H \ot W_i \to W_i$ is graded. It follows that $W_i = \oplus_{n \geq 0}N^n \lact M_i$ for all $i$.
\end{proof}
\begin{propo}
Let $N\in \ydH$ and $W\in {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$. Assume that $W$ is a semisimple object in the category of $\ndZ$-graded left Yetter-Drinfeld modules over $\NA(N) \# H$. Let $K=\NA (W)$ be the Nichols algebra
of $W$ in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD} $, and define $M = W(H)$.
Then there is a unique isomorphism
$$K\#\NA (N)\cong \NA (M \oplus N)$$
of braided Hopf algebras in $\ydH $ which is the identity on $M \oplus N$.
\label{propo:KBNichols}
\end{propo}
\begin{proof}
Let $\oplus _{i\in I}W_i$ be a decomposition of $W$ into irreducible objects in the category of $\ndZ$-graded left Yetter-Drinfeld modules over $\NA(N) \# H$. For all $i \in I$, let $M_i = W_i \cap M$. By Lemma~\ref{lem:ydBNH} (2), we can define a new $\mathbb{Z}$-grading on $W$ by
$$\deg (N^n\cdot M_i)=n+1$$
for all $n\in \ndN _0, i \in I$. Then $W$ is a $\mathbb{Z}$-graded object in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$. Because of Lemma~\ref{lem:ZgradK} (1),
and since $W=K(1)$, we know that $K$ is a $\ndZ $-graded braided Hopf algebra with this new
$\ndZ $-grading on $K(1)$. Thus by Lemma~\ref{lem:ZgradK} (2) and (3), $K\#(\NA (N) \# H)$ is a $\ndZ $-graded Hopf algebra, and
$$R:=K\#\NA (N)= \left(K\#(\NA (N) \# H)\right)^{\co H}$$
is a $\ndZ $-graded braided Hopf algebra
in $\ydH $
with $\fie 1$ as degree $0$ part and with
$M \oplus N$ as degree $1$ part.
Let $m \in M$ and $b \in \NA(N)$. Then
\begin{align}\label{ruleR}
b \lact m = b\swo1 ({b\swo2}\sw{-1} \lact m) \mathcal{S}_{\NA(N)}({b\swo2}\sw0)
\end{align}
in the algebra $R=K\#\NA (N)$.
Since $K$ is generated as an algebra by $K(1)$, and since
$K(1)=\NA (N)\cdot M$, we conclude from \eqref{ruleR} that
$R$ is generated as an algebra by $R(1)=M \oplus N$. Thus $R$ is a pre-Nichols algebra of $M \oplus N$.
By the universal property of the Nichols algebra $\NA(M \oplus N)$, there is a surjective homomorphism
$$\rho : R \to \NA(M \oplus N),\; \rho \mid M \oplus N = \id,$$
of $\mathbb{N}_0$-graded Hopf algebras in $\ydH$. Then
$$ \rho \# \id : R \# H \to \NA(M \oplus N) \# H$$
is a surjective map of Hopf algebras. Let $K'= (\NA(M \oplus N) \# H)^{\co \NA(N) \# H}$. Since the multiplication maps
$$R \# H \to K \# (\NA(N) \# H),\; K' \# (\NA(N) \# H) \to \NA(M \oplus N) \# H$$
are bijective maps of Hopf algebras, the map $\rho \# \id$ defines a surjective map of Hopf algebras
$$\rho': K \# (\NA(N) \# H) \to K'\# (\NA(N) \# H),\; \rho'\mid (M \oplus N) = \id.$$
The action of $\NA(N) \# H$ on $K$ is the adjoint action in $K \# (\NA(N) \# H)$.
Since the algebras $K$ and $K'$ are generated by $(\ad \NA(N))(M)$ on both sides, $\rho'$ induces a map
$$\rho_1 : K \to K', \; \rho_1 \mid M = \id,$$
of $\mathbb{N}_0$-graded braided Hopf algebras in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$, and a map
$$\rho_2 : \NA(N) \lact M \to (\ad\NA(N))(M), \; \rho_2 \mid M = \id,$$
in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$. Since $(\ad\NA(N))(M_i)$ is irreducible in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$ for all $i \in I$, it follows that $\rho_2$ is bijective. Hence $\rho_1$ is bijective by the universal property of the Nichols algebra $K = \NA(W)$. Thus $\rho = \rho_1 \# \id_{\NA(N)}$ is bijective.
\end{proof}
We now apply Corollary \ref{cor:third} to Nichols algebras.
Let $N \in \ydH$ be finite-dimensional. Then the dual vector space $N^* = \Hom(N,\fie)$ is an object in $\ydH$ with
\begin{align*}
\langle h \lact \xi,x\rangle &= \langle \xi,\mathcal{S}(h) \lact x\rangle,\\
\xi\sw{-1} \langle\xi\sw0,x\rangle &= \mathcal{S}^{-1}(x\sw{-1}) \langle \xi,x\sw0\rangle
\end{align*}
for all $\xi \in N^*, x \in N, h \in H$, where $\langle \;,\; \rangle : N^* \ot N \to \fie$ is the evaluation map. The Nichols algebras of the finite-dimensional Yetter-Drinfeld modules $N^*$ and $N$ have finite-dimensional $\mathbb{N}_0$-homogeneous components, and there is a canonical pairing
$\langle \;,\; \rangle : \NA(N^*) \ot \NA(N) \to \fie$
extending the evaluation map such that the conditions \eqref{pair1} -- \eqref{pair5} and \eqref{gradedpair} hold, see for example \cite[Prop.\, 1.10]{a-AHS10}. Let
$$(\Omega_N,\omega_N) : {}^{\phantom{.}\NA(N) \# H}_{\NA(N) \# H}\mathcal{YD}_{\rat} \to {}^{\phantom{.}\NA(N^*) \# H}_{\NA(N^*) \# H}\mathcal{YD}_{\rat}$$
be the functor of Corollary \ref{cor:third} with respect to the canonical dual pairing.
\begin{theor}
\label{theor:NtoN}
Let $n \geq 1$, and let $M_1,\dots, M_n,N$ be finite-dimensional objects in $\ydH$. Assume that for all $1 \leq i \leq n$, $M_i$ is irreducible in $\ydH$, and
that $(\ad \NA(N))(M_i)$ is a finite-dimensional subspace of $\NA (\oplus_{i = 1}^n M_i \oplus N)$. For all $i$
let $V_i = \Fm 0(\ad \NA (N))(M_i)$, and let
$K=\NA (\oplus_{i = 1}^n M_i \oplus N)^{\co \NA(N)}$.
\begin{enumerate}
\item The modules $V_1,\dots, V_n$ are irreducible in $\ydH$, $\Omega_N(K)$ is a braided Hopf algebra in $ {}^{\phantom{.}\NA(N^*) \# H}_{\NA(N^*) \# H}\mathcal{YD}_{\rat}$, and there is a unique isomorphism
$$\Omega_N(K)\#\NA (N^*)\cong \NA (\oplus_{i=1}^n V_i\oplus N^*)$$
of braided Hopf algebras in $\ydH $ which is the identity on $\oplus_{i=1}^n V_i\oplus N^*$.
\item For all $1 \leq i \leq n$, let $m_i = \max \{m \in \mathbb{N}_0 \mid (\ad N)^m(M_i) \neq 0 \}$, and $W_i = (\ad \NA(N))(M_i)$. Then
\begin{align*}
W_i &= \mathop\oplus_{n=0}^{m_i} (\ad N)^n(M_i),& V_i &= (\ad N)^{m_i}(M_i),\\
\Omega_N(W_i) &\cong \mathop\oplus_{n=0}^{m_i} (\ad N^*)^n(V_i),& M_i &\cong (\ad N^*)^{m_i}(V_i)
\end{align*}
for all $i$.
\end{enumerate}
\end{theor}
\begin{proof}
(1) Let $W = (\ad \NA(N))(M)$. By Lemma \ref{lem:old} (2), $W_1,\dots,W_n$ are
irreducible objects in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$, $W = \oplus_{i=1}^n W_i$, and for all
$1 \leq i \leq n$, $M_i$ is the $\mathbb{Z}$-homogeneous component of $W_i$ of minimal degree.
By Proposition~\ref{propo:Fdm}, the Yetter-Drinfeld modules $V_1,\dots,V_n\in \ydH $ are
irreducible.
By Proposition \ref{prop:old}, $K$ is isomorphic to the Nichols algebra of $W$ in
$ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$. Since $(\ad \NA(N))(M)$ is a finite-dimensional and $\mathbb{Z}$-graded object
in $ {}^{\phantom{.}\NA (N)\#H}_{\NA (N)\#H}\mathcal{YD}$, it is a rational $\NA(N)$-module.
Therefore $\Omega_N(\NA(W)) \cong \NA(\Omega_N(W))$ by Corollary \ref{cor:Nichols}.
Hence there is a unique isomorphism $\Omega_N(K) \cong \NA(\Omega_N(W))$ of braided Hopf algebras in $ {}^{\phantom{.}\NA(N^*) \# H}_{\NA(N^*) \# H}\mathcal{YD}_{\rat}$ which is the identity on $\Omega_N(W)$. Recall that
$$\Omega_N(W)(H) = \Fd0 \Omega_N(W) = \Fm0 W = \oplus_{i=1}^n V_i$$
by Corollary \ref{cor:third} (3). Then by Proposition \ref{propo:KBNichols} there is a unique isomorphism
$$\Omega_N(K) \# \NA(N^*) \cong \NA (\oplus_{i=1}^n V_i\oplus N^*)$$
of braided Hopf algebras in $\ydH$ which is the identity on $\oplus_{i=1}^n V_i\oplus N^*$ which proves (1). For the last conclusion we have to check the assumptions of Proposition \ref{propo:KBNichols}, that is, $\Omega_N(W) \in {}^{\phantom{.}\NA(N^*) \# H}_{\NA(N^*) \# H}\mathcal{YD}_{\rat}$ is a semisimple $\ndZ$-graded Yetter-Drinfeld module. By Corollary \ref{cor:third} (2), $\Omega_N(W)$ is $\ndZ$-graded, and it is semisimple since $W$ is semisimple by Lemma \ref{lem:old} and $\Omega_N$ is an isomorphism by Corollary \ref{cor:third}.
(2) Let $i\in \{1,\dots,n\}$. The first equation follows from the definition of $W_i$
and the second from Proposition~\ref{propo:Fdm} for $R=\NA (N)$ with $\deg N=1$ and
$\deg (\ad N)^n(M_i)=1+n$ for all $n\ge 0$.
By Corollary~\ref{cor:third}, $\Omega _N(W_i)=W_i$ is $\ndZ $-graded with homogeneous components
$(\ad N)^n(M_i)$ of degree $-n-1$. Moreover, $V_i=\Fd 0\Omega _N(W_i)$ by the proof of (1),
and hence $\Omega _N(W_i)=\oplus _{n=0}^{m_i}(N^*)^nV_i$ since $\Omega _N(W_i)$
is irreducible. In particular, $M_i=(N^*)^{m_i}V_i$. These equations imply
the remaining claims of (2).
\end{proof}
\begin{remar}
Theorem \ref{theor:NtoN} still holds if we replace the canonical pairing in the definition of $(\Omega_N,\omega_N)$ by any dual pairing
$\langle \;,\; \rangle : \NA(N^*) \ot \NA(N) \to \fie$
satisfying \eqref{pair1} -- \eqref{pair5} and \eqref{gradedpair}.
\end{remar}
The definition of
the Weyl groupoid of a Nichols algebra of a semisimple Yetter-Drinfeld module over $H$
is based on
\cite[Thm.\,3.12]{a-AHS10},
see also \cite[Sect.\,3.5]{a-AHS10} and \cite[Thm.\,6.10, Sect.\,5]{a-HeckSchn10}.
To see that Theorem~\ref{theor:NtoN} can be considered as an alternative approach to the
definition of the Weyl groupoid, we introduce some notations.
\begin{comment}
\begin{corol}
Let $\theta \in \ndN $, $i\in \{1,\dots,\theta \}$
and let $M_1,\dots,M_\theta \in \ydH $ be finite-dimensional irreducible objects.
Assume that the subspace $(\ad \NA (M_i))(M_j)$ of
the Nichols algebra $\NA (\oplus _{n=1}^\theta M_n)$
is finite-dimensional for all $j\not=i$.
Let $V_i=M_i^*$
and $V_j=\Fm 0 (\ad \NA (M_i))(M_j)$ for all $j\not=i$.
Then the following hold.
\begin{enumerate}
\item
For all $j$, $V_j$ is a finite-dimensional irreducible object in $\ydH $.
\item
For all $j\not=i$, $(\ad \NA (V_i))(V_j)$ is a finite-dimensional subspace
of $\NA (\oplus _{n=1}^\theta V_n)$
and $\Fm 0 (\ad \NA (V_i))(V_j)\cong M_i$ in $\ydH $.
\item
For all $j\not=i$ and $n\ge 0$,
$(\ad V_i)^n(V_j)=0$ if and only $(\ad M_i)^n(M_j)=0$.
\end{enumerate}
\end{corol}
\end{comment}
Let $\theta \geq 1$ be a natural number. Let $\mathcal{F}_{\theta}$ denote the class of all families
$M = (M_1,\dots,M_{\theta})$, where $M_1,\dots,M_{\theta} \in \ydH$
are finite-dimensional irreducible Yetter-Drinfeld modules. If $M \in \mathcal{F}_{\theta}$, we define
$$\NA(M)=\NA(M_1 \oplus \cdots \oplus M_{\theta}).$$
For families $M,M'\in \mathcal{F}_\theta $, we write $M \cong M'$, if $M_j \cong M_j'$ in $\ydH $ for all $j$.
For $1 \leq i \leq \theta$ and $M \in \mathcal{F}_{\theta}$, we say that the
{\em $i$-th reflection $R_i(M)$ is defined} if for
all $j \neq i$ there is a natural number $m_{ij}^M \geq 0$ such that
$(\ad M_i)^{m_{ij}^M}(M_j)$ is a non-zero finite-dimensional subspace
of $\NA(M)$, and $(\ad M_i)^{m_{ij}^M+1}(M_j)=0$. Assume that $R_i(M)$ is defined.
Then we set $R_i(M) = (V_1,\dots,V_{\theta})$, where
\begin{align*}
V_j &= \begin{cases}
V_i^*, &\text{ if } j=i,\\
(\ad M_i)^{m^M_{ij}}(M_j), &\text{ if } j \neq i,
\end{cases}
\end{align*}
For all $j \neq i$, let $a_{ij}^M = -m_{ij}^M$.
By \cite[Lemma\,3.22]{a-AHS10}, $(a_{ij}^M)$ with $a_{i i}^M=2$ for all $i$
is a generalized Cartan matrix.
The next Corollary follows from a restatement of Theorem \ref{theor:NtoN}. Thus we obtain a new proof of \cite[Thm.\,3.12(2)]{a-AHS10} which allows to define the Weyl groupoid of $M \in \mathcal{F}_{\theta}$.
\begin{corol} \cite[Thm.\,3.12(2)]{a-AHS10}
Let $M \in \mathcal{F}_{\theta}$, and $1 \leq i \leq \theta$. Assume that $R_i(M)$ is defined. Then
$R_i(M) \in \mathcal{F}_{\theta}$, $R_i^2(M)$ is defined, $R_i^2(M) \cong M$, and $a_{ij}^M = a_{ij}^{R_i(M)}$ for all $1 \leq j \leq \theta$.
\end{corol}
In the situation of the last Corollary, let $K_i^M = \NA(M)^{\co \NA(M_i)}$ with respect to the projection $\NA(M) \to \NA(M_i)$. Then
$$K_i^M \# \NA(M_i) \cong \NA(M)$$
by bosonization, and
$$\Omega(K_i^M) \# \NA(M_i^*) \cong \NA(R_i(M))$$
by Theorem \ref{theor:NtoN}.
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| 68,663 |
\section{Introduction and statement of the problem \label{sec1}}
Granular gases are dilute systems made of inelastic particles that can be maintained in a fluidized state by the application of external drivings to
compensate for the dissipation of kinetic energy due to collisions. These systems are always out of equilibrium and exhibit a wealth of
intriguing complex phenomena \cite{C90,JNB96a,JNB96b,K99,OK00,K00,D00,PL01,D02,HM02,G03,BP04}.
They are important from an applied point of view but also at the level of fundamental physics. As Kadanoff stated in his review paper \cite{K99}, ``one might even say that the study of
granular materials gives one a chance to reinvent statistical mechanics in a new context.''
One of the most controversial issues in granular fluids refers to the validity of a hydrodynamic description \cite{K99,TG98}. In conventional fluids, the densities of
the conserved quantities (mass, momentum, and energy) satisfy formally exact balance (or continuity) equations involving the divergence of the associated fluxes.
In the case of granular fluids, however, energy is dissipated on collisions and this gives rise to a sink term in the energy balance
equation. As a consequence, except perhaps in quasielastic
situations, the role of the energy density (or, equivalently, of the granular temperature) as a hydrodynamic
variable is not evident. Both for conventional and granular fluids, the mass, momentum, and energy balance equations do not form a closed set due to the appearance of the momentum and energy fluxes (plus the energy sink in the granular case). On the other hand, by assuming ``hydrodynamic'' conditions, the balance equations are closed by the addition of approximate constitutive equations relating the momentum and energy fluxes (again plus the energy sink in the granular case) to the mass, momentum, and energy fields.
The simplest constitutive equations consist of replacing the fluxes by their local equilibrium forms, thus neglecting the influence of the hydrodynamic gradients. This gives rise to the Euler hydrodynamic equations, which fail to account for irreversible effects, even in the case of conventional fluids. This is corrected by the Navier--Stokes (NS) constitutive equations, where the fluxes
are assumed to be linear in the hydrodynamic gradients. On the other hand, if the gradients are not weak enough (i.e., if the Knudsen number is not small enough), the NS equations are insufficient and, thus, nonlinear (i.e., non-Newtonian) constitutive equations are needed in a hydrodynamic description \cite{CC70,GS03}.
\begin{figure}
\includegraphics[width=.9 \columnwidth]{fig1.eps}
\caption{(Color online) Schematic description of the two-stage evolution of the velocity distribution function in a conventional gas.}
\label{fig1}
\end{figure}
In conventional fluids, applicability of a hydrodynamic description (Euler, NS, or non-Newtonian) requires two basic conditions, one spatial and another one temporal. On the one hand, one must focus on the \emph{bulk} region of the system, i.e., outside the \emph{boundary layers}, whose width is on the order of the mean free path. On the other hand, one must let the system \emph{age} beyond the \emph{initial layer}, whose duration is on the order of the mean free time. Let us consider the latter condition in more detail.
In a conventional gas, the typical evolution scenario starting from an arbitrary initial state represented by an arbitrary initial velocity distribution $f_0(\mathbf{r},\mathbf{v})$ proceeds along two successive stages \cite{DvB77}. First,
during the so-called \emph{kinetic} stage, the velocity distribution $f(\mathbf{r},\mathbf{v},t)$, which depends functionally on $f_0$, experiences a fast relaxation (lasting a few collision times) toward a ``normal'' form $f[\mathbf{v}|n,\mathbf{u},T]$, where all the spatial and temporal dependence occurs through a
\emph{functional} dependence on the hydrodynamic fields (number density $n$, flow velocity $\mathbf{u}$, and temperature $T$). Next, during the \emph{hydrodynamic} stage, a slower evolution of the hydrodynamic fields takes place until either equilibrium or an externally imposed nonequilibrium
steady state is eventually reached. While the first stage is very sensitive to the initial preparation of the system, the details of the initial
state are practically ``forgotten'' in the hydrodynamic regime. Figure \ref{fig1} depicts a schematic summary of this two-stage evolution in a conventional gas.
The absence of energy conservation in granular fluids sheds some reasonable doubts on the applicability of the above scenario beyond the quasielastic regime. While the usefulness of a \emph{non-Newtonian} hydrodynamic description in \emph{steady states} has been validated by computer simulations \cite{BRM97,TTMGSD01,VSG10,VGS11}, it is not obvious that a hydrodynamic treatment holds as well during the \emph{transient} regime toward the steady state.
\begin{figure}
\includegraphics[width=.7 \columnwidth]{fig2.eps}
\caption{(Color online) Sketch of the USF.}
\label{fig2}
\end{figure}
\begin{figure}
\includegraphics[width=.9 \columnwidth]{fig3.eps}
\caption{(Color online) Sketch of the ULF for (a) $a(t)>0$ and (b) $a(t)<0$.}
\label{fig3}
\end{figure}
In order to address the problem described in the preceding paragraph, it seems convenient to focus on certain prototypical classes of flows. Let us assume a ($d$-dimensional) granular gas with uniform density $n(t)$, uniform temperature $T(t)$, and a flow velocity along a given axis (say $x$) with a linear spatial variation with respect to a certain Cartesian coordinate $\h$, i.e.,
\beq
\nabla_j u_i(\mathbf{r},t)=a(t)\delta_{ix}\delta_{j\h}.
\label{1}
\eeq
Here $a(t)$ is a uniform rate of strain. Two distinct possibilities arise: either $\h\neq x$ (say $\h=y$) or $\h=x$. The first case defines an
incompressible flow ($\nabla\cdot \mathbf{u}=0$) commonly known as simple or uniform shear flow (USF)
\cite{TM80,LSJC84,CG86,WB86,JR88,C89,C90,S92,HS92,SK94,LB94,GT96,SGN96,C97,BRM97,CR98,MGSB99,F00,K00a
K00b,K01,C00,C01,C01b,AH01,CH02,BGS02,G02,MG02,MG03,GM03,G03a,GS03,AL03a,AL03b,G03,SGD04,L04,AS05,MGAS06,SG07,AS07,S08,S08a}, the associated rate of strain $a$ being the \emph{shear rate}. The second case is an example of compressible flow ($\nabla\cdot \mathbf{u}=a\neq 0$) that will be referred to as uniform longitudinal flow (ULF) \cite{GK96,KDN97,UP98,KDN98,UG99,S00a,S08a,S09}; the corresponding rate of strain in this case will be called \emph{longitudinal rate}. These two states are particular cases of a more general class of
homo-energetic affine flows characterized by $\partial^2
u_i/\partial x_j\partial x_k=0$ \cite{TM80}. The USF and ULF flows are sketched in Figs.\ \ref{fig2} and \ref{fig3}, respectively.
Assuming that the velocity distribution function $f(\mathbf{r},\mathbf{v},t)$ depends on the spatial variable $\h$ only, the Boltzmann equation reads
\beq
\frac{\partial f}{\partial t}+v_\h\frac{\partial f}{\partial \h}=J[f,f],
\label{2}
\eeq
where $J[f,f]$ is the (inelastic) Boltzmann collision operator, whose explicit form can be found, for instance, in Refs.\ \cite{GS95,DBS97,NE01}.
{Multiplying both sides of Eq.\ \eqref{2} by $\{1,\mathbf{v},v^2\}$, and integrating over velocity, we get the balance equations for mass, momentum, and energy densities},
\begin{equation}
{D_tn=-n\frac{\partial u_\h}{\partial \h} ,}
\label{nbal}
\end{equation}
\begin{equation}
{D_t{u_i}=-\frac{1}{mn}\frac{\partial P_{\h i}}{\partial\h} ,}
\label{Pbal}
\end{equation}
\begin{equation}
{D_tT+\zeta T=-\frac{2}{dn}\left(P_{\h i}\frac{\partial u_i}{\partial \h}+\frac{\partial q_\h}{\partial \h}\right) .}
\label{Tbal}
\end{equation}
{In Eqs.\ \eqref{nbal}--\eqref{Tbal}, $D_t\equiv \partial_t+{u}_\h\partial_\h$ is the material derivative, and the number density $n$, flow velocity $\mathbf{u}$, temperature $T$, pressure tensor $P_{ij}$, heat flux vector $\mathbf{q}$, and cooling rate $\zeta$ are defined by}
\beq
n(\h,t)=\int \dd\mathbf{v}\, f(\h,\mathbf{v},t),
\label{n}
\eeq
\beq
n({\h},t)\mathbf{u}({\h},t)=\int \dd{\mathbf{v}}\,{\mathbf{v}}{f}({\h},{\mathbf{v}},{t}),
\label{u}
\eeq
\beq
n({\h},t)T({\h},t)=p({\h},t)=\frac{1}{d}\text{tr}\mathsf{P}({\h},t),
\label{T}
\eeq
\beq
P_{ij}({\h},t)={m}\int \dd{\mathbf{v}}\,[{v_i}-{u_i}({\h},t)][{v_j}-{u_j}({\h},t)]{f}({\h},{\mathbf{v}},{t}),
\label{PPij}
\eeq
\beq
{\boldsymbol{q}({\h},t)=\frac{m}{2}\int \dd\mathbf{v}\,[\mathbf{v}-\mathbf{u}({\h},t)]^2[\mathbf{v}-\mathbf{u}({\h},t)]f({\h},\mathbf{v},t),}
\label{q}
\eeq
\beq
{n({\h},t)T({\h},t)\zeta({\h},t)=-\frac{m}{d}\int \dd\mathbf{v}\,v^2 J[f,f].}
\label{zeta}
\eeq
{The first equality of Eq.\ \eqref{T} defines the hydrostatic pressure $p$, which is just given by the ideal-gas law in the Boltzmann limit.}
As said above, the density $n(t)$ and temperature $T(t)$, and so the hydrostatic pressure $p(t)$, are uniform in the (fully developed) USF and ULF. On physical grounds, it can also be assumed that the whole pressure tensor $P_{ij}(t)$ is uniform as well. Moreover, in the absence of thermal and density gradients, the heat flux can be expected to vanish. Taking all of this into account, {as well as Eq.\ \eqref{1}}, the balance equations [Eqs.\ {\eqref{nbal}--\eqref{Tbal}}] become
\beq
\dot{n}(t)=-n(t)a(t)\delta_{x\h},
\label{3}
\eeq
\beq
\dot{a}(t)=-a^2(t)\delta_{x\h},
\label{4}
\eeq
\beq
\dot{T}(t)=-\frac{2a(t)}{dn(t)}P_{x\h}(t)-\zeta(t)T(t).
\label{5}
\eeq
In the case of the USF ($\h=y$), Eqs.\ \eqref{3} and \eqref{4} imply that both the density and the shear rate are constant quantities. As for the temperature, it evolves in time subject to two competing effects: viscous heating [represented by the term $-(2/dn)aP_{xy}>0$] and inelastic cooling [represented by $\zeta T$]. Both effects eventually cancel each other in the steady state.
In the case of the ULF ($\h=x$), the solution to Eqs.\ \eqref{3} and \eqref{4} is
\beq
\frac{n(t)}{a(t)}=\frac{n_0}{a_0},\quad a(t)=\frac{a_0}{1+a_0 t},
\label{6}
\eeq
where $n_0$ is the initial density and $a_0$ is the initial longitudinal rate.
In contrast
to the USF, the sign of $a(t)$ (or, equivalently, the sign of $a_0$) plays a relevant role
and defines two separate situations (see Fig.\ \ref{fig3}). The
case $a>0$ corresponds to a progressively slower {\em
expansion\/} of the gas from the plane $x=0$ into all of space. On
the other hand, the case $a<0$ corresponds to a progressively
faster {\em compression\/} of the gas toward the plane $x=0$. The
latter takes place over a {\em finite\/} time period $t=|a_0|^{-1}$.
However, since the collision frequency rapidly increases with time,
the finite period $t=|a_0|^{-1}$ comprises an {\em infinite\/}
number of collisions per particle \cite{S09}.
Given that $P_{xx}>0$, the energy balance equation [see Eq.\ \eqref{5}] implies that the temperature monotonically decreases with time in the ULF with $a>0$. On the other hand, if $a<0$, we have again a competition between viscous heating and inelastic cooling, so the temperature either increases or decreases (depending on the initial state) until a steady state is eventually reached.
The main characteristic features of the USF and the ULF are summarized in Table \ref{table1}.
\begin{table}
\caption{Main characteristic features of the USF and the ULF.\label{table1}}
\begin{ruledtabular}
\begin{tabular}{cccc}
&USF&ULF ($a>0$)&ULF ($a<0$)\\
\hline
Inelastic cooling&Yes&Yes&Yes\\
Viscous heating&Yes&No&Yes\\
$T(t)\downarrow$ \& $|a^*(t)|\uparrow$&Yes, if&Yes&Yes, if\\
&$\zeta>\frac{2|aP_{xy}|}{dp}$&&$\zeta>\frac{2|a|P_{xx}}{dp}$\\
$T(t)\uparrow$ \& $|a^*(t)|\downarrow$&Yes, if&No&Yes, if\\
&$\zeta<\frac{2|aP_{xy}|}{dp}$&&$\zeta<\frac{2|a|P_{xx}}{dp}$\\
Steady state&Yes&No&Yes\\
\end{tabular}
\end{ruledtabular}
\end{table}
Regardless of whether the rate of strain $a$ is constant (USF) or changes with time (ULF), the relevant parameter is the ratio
\beq
a^*(t)=\frac{a(t)}{\nu(t)}
\label{a*}
\eeq
between $a(t)$ and a characteristic collision frequency $\nu(t)\propto n(t)[T(t)]^{1/2}$. Note that the absolute value of $a^*(t)$ represents the Knudsen number of the problem, i.e., the ratio between the mean free path and the characteristic length associated with the velocity gradient \cite{S08,S08a}. Since $a(t)/n(t)=\text{const}$ both in the USF and the ULF, we have $a^*(t)\propto [T(t)]^{-1/2}$. Consequently, the qualitative behavior of $|a^*(t)|$ is the opposite to that of $T(t)$, as indicated in Table \ref{table1}.
The scenario depicted in Fig.\ \ref{fig1}, if applicable to a granular gas in the USF or in the ULF, means that, after the kinetic stage, the velocity distribution function $f[\mathbf{r},\mathbf{v},t|f_0]$ should adopt a hydrodynamic (or normal) form
\beq
f[\mathbf{r},\mathbf{v},t|f_0]\to n(t)\left[\frac{m}{2T(t)}\right]^{d/2}f^*\left(\mathbf{C}(\mathbf{r},t), a^*(t)\right),
\label{7}
\eeq
where
\beq
\mathbf{C}(\mathbf{r},t)=\frac{\mathbf{v}-\mathbf{u}(\mathbf{r},t)}{\sqrt{2T(t)/m}}
\label{8}
\eeq
is the peculiar velocity scaled with the thermal speed.
For a given value of the
coefficient of restitution,
the scaled velocity distribution
function $f^*(\mathbf{C},a^*)$ must be independent of the details of the initial state $f_0$ and depend on the applied shear or longitudinal rate $a(t)$ through the reduced scaled
quantity $a^*$ only. In other words, if a hydrodynamic description
is possible, the form \eqref{7} must ``attract'' the manifold of solutions $f[\mathbf{r},\mathbf{v},t|f_0]$ to the
Boltzmann equation \eqref{2} for sufficiently long times, even before the steady state (if it exists) is reached.
Equation \eqref{7} has its counterpart at the level of the velocity
moments. In particular, the pressure tensor $P_{ij}[t|f_0]$ would
become
\beq
P_{ij}[t|f_0]\to n(t)T(t)P_{ij}^*(a^*(t))
\label{10}
\eeq
with well-defined hydrodynamic functions $P_{ij}^*(a^*)$.
A few years ago we reported a preliminary study \cite{AS07} where the validity of the unsteady hydrodynamic forms \eqref{7} and \eqref{10} for the USF was confirmed by means of the direct simulation Monte Carlo (DSMC) method to solve the Boltzmann equation and by a simple rheological model. The aim of the present paper is twofold. On the one hand, we want to revisit the USF case by presenting a more extensive and efficient set of simulations, by providing a detailed derivation of the rheological model (which was just written down without derivation in Ref.\ \cite{AS07}), and by including in the analysis the second viscometric function (which was omitted in Ref.\ \cite{AS07}). On the other hand, we perform a similar analysis (both computational and theoretical) in the case of the ULF. This second study is relevant because, despite the apparent similarity between the USF and the ULF, the latter differs from the former in that it is compressible, the two signs of $a$ physically differ, and a steady state is possible only for negative $a$.
The remainder of the paper is organized as follows. The formal kinetic theory description for both types of flow is presented in Sec.\ \ref{sec2} within a unified framework. Next, Sec.\ \ref{sec3} offers a more specific treatment based on a simple model kinetic equation. In particular, a fully analytical rheological model is derived. Section \ref{sec4} describes the simulation method employed to solve the Boltzmann equation and the classes of initial conditions considered. The most relevant part of the paper is contained in Sec.\ \ref{sec5}, where the results obtained from the simulations are presented and discussed. It is found that the scenario depicted by Fig.\ \ref{fig1} and Eqs.\ \eqref{7} and \eqref{10} is strongly supported by the simulations. Moreover, the simple analytical rheological model is seen to agree quite well with the simulation results. The paper is closed in Sec.\ \ref{sec6} with a summary and conclusions.
\section{Boltzmann equation for USF and ULF\label{sec2}}
Let us consider a granular gas modeled as a system of smooth inelastic hard spheres (of mass $m$, diameter $\sigma$, and constant coefficient of normal restitution $\al$), subject to the USF or to the ULF sketched in Figs.\ \ref{fig2} and \ref{fig3}, respectively.
In the dilute regime, the velocity distribution function $f(\h,\mathbf{v},t)$ obeys the Boltzmann equation \eqref{2}. As is well known, the adequate boundary conditions for the USF are Lees--Edwards's boundary conditions \cite{LE72}, which are not but periodic boundary conditions in the comoving Lagrangian frame \cite{DSBR86}. The appropriate boundary conditions for the ULF are much less obvious. In order to construct them, it is convenient to perform a series of mathematical changes of variables. Also, we will proceed by encompassing the ULF and the USF in a common framework.
\subsection{Changes of variables}
We start by defining scaled time and spatial variables $\widetilde{t}$ and $\widetilde{\h}$ as
\beq
\widetilde{t}=\lb(t),\quad \widetilde{\h}=\dot{\lb}(t)\h,
\label{11}
\eeq
where
\beq
\lb(t)=\begin{cases}
t,&\text{USF } (\h=y),\\
a_0^{-1}\ln(1+a_0 t),&\text{ULF } (\h=x),
\end{cases}
\label{12}
\eeq
\beq
\dot{\lb}(t)=\begin{cases}
1,&\text{USF } (\h=y),\\
(1+a_0 t)^{-1},&\text{ULF } (\h=x).
\end{cases}
\label{13}
\eeq
Also, a new velocity variable $\widetilde{\mathbf{v}}$ is defined as
\beq
\widetilde{\mathbf{v}}=\mathbf{v}-a(t)\h \mathbf{e}_x,
\label{14}
\eeq
where $\mathbf{e}_x$ is the unit vector in the $x$ direction and we have taken into account that $a(t)=a_0\dot{\lb}(t)$ both in the USF and in the ULF.
The velocity distribution function corresponding to the variables $\widetilde{\h}$, $\widetilde{\mathbf{v}}$, and $\widetilde{t}$ is
\beq
\widetilde{f}(\widetilde{\h},\widetilde{\mathbf{v}},\widetilde{t})=\frac{1}{\dot{\lb}(t)}f(\h,\mathbf{v},t).
\label{15n}
\eeq
Consequently,
\beq
\frac{1}{\dot{\lb}^2}\frac{\partial f}{\partial t}=\frac{\partial \widetilde{f}}{\partial \widetilde{t}}-a_0\delta_{x\h}\left[\widetilde{f}+\dot{\lb}{\h}\left(
\frac{\partial \widetilde{f}}{\partial \widetilde{\h}}-a_0\frac{\partial \widetilde{f}}{\partial \widetilde{v}_x}\right)\right],
\label{17}
\eeq
where we have taken into account that
\beq
\ddot{\lb}(t)=-a_0[\dot{\lb}(t)]^2\delta_{x\h}.
\label{16}
\eeq
Similarly,
\beq
\frac{1}{\dot{\lb}^2}\frac{\partial f}{\partial \h}=\frac{\partial \widetilde{f}}{\partial \widetilde{\h}}-a_0\frac{\partial \widetilde{f}}{\partial \widetilde{v}_x},
\label{18}
\eeq
\beq
\frac{1}{\dot{\lb}^2} J[f,f]=J[\widetilde{f},\widetilde{f}].
\label{19a}
\eeq
Inserting Eqs.\ \eqref{17}, \eqref{18}, and \eqref{19a} into Eq.\ \eqref{2}, and taking into account Eq.\ \eqref{14}, one finally gets
\beq
\frac{\partial \widetilde{f}}{\partial \widetilde{t}}+\widetilde{v}_\h\frac{\partial \widetilde{f}}{\partial \widetilde{\h}}-a_0\frac{\partial}{\partial \widetilde{v}_x}\left(\widetilde{v}_\h \widetilde{f}\right)=J[\widetilde{f},\widetilde{f}].
\label{19}
\eeq
It is important to remark that no assumption has been made. Therefore,
Eqs.\ \eqref{2} and \eqref{19} are mathematically equivalent, so any solution to Eq.\ \eqref{2} can be mapped onto a solution to Eq.\ \eqref{19} and vice versa. While Eq.\ \eqref{2} describes a gas in the absence of external forces, Eq.\ \eqref{19} describes a gas under the influence of a \emph{nonconservative} external force $\widetilde{\mathbf {F}}=-ma_0 \widetilde{v}_\h\mathbf{e}_x$.
Note that in the case of the ULF with $a_0<0$ the finite time interval $0<t<|a_0|^{-1}$ translates into the infinite scaled time interval $0<\widetilde{t}<\infty$.
The density $\widetilde{n}(\widetilde{\h},\ot)$, flow velocity $\widetilde{\mathbf{u}}(\widetilde{\h},\ot)$, temperature $\widetilde{T}(\widetilde{\h},\ot)$, and pressure tensor $\widetilde{P}_{ij}(\widetilde{\h},\ot)$ associated with the scaled distribution \eqref{15n} are defined analogously to Eqs.\ \eqref{n}--\eqref{PPij}. The quantities with and without tilde are related by
\beq
\widetilde{n}(\widetilde{\h},\ot)=\frac{n(\h,t)}{\dot{\lb}(t)} , \quad \widetilde{P}_{ij}(\widetilde{\h},\ot)=\frac{P_{ij}(\h,t)}{\dot{\lb}(t)},
\label{20n}
\eeq
\beq
\widetilde{\mathbf{u}}(\widetilde{\h},\ot)=\mathbf{u}(\h,t)-a(t) \h \mathbf{e}_x,\quad \widetilde{T}(\widetilde{\h},\ot)=T(\h,t).
\label{21n}
\eeq
At a microscopic level, we now define the USF ($\h=y$) and the ULF ($\h=x$) as spatially \emph{uniform} solutions to Eq.\ \eqref{19}, i.e.,
\beq
\widetilde{f}(\widetilde{\h},\widetilde{\mathbf{v}},\widetilde{t})=\widetilde{f}(\widetilde{\mathbf{v}},\widetilde{t}).
\label{22n}
\eeq
Thus, conservation of mass and momentum implies that $\widetilde{n}=\text{const}$ and $\widetilde{\mathbf{u}}=\text{const}$.
Without loss of generality we can take $\widetilde{\mathbf{u}}=0$.
It seems quite natural that periodic boundary conditions (at $\widetilde{\h}=\pm\widetilde{L}/2$) are the appropriate ones to complement the (scaled) Boltzmann equation \eqref{19} in order to ensure the consistency with uniform solutions \eqref{22n}, i.e.,
\beq
\widetilde{f}(-\widetilde{L}/2,\widetilde{\mathbf{v}},\widetilde{t})=\widetilde{f}(\widetilde{L}/2,\widetilde{\mathbf{v}},\widetilde{t}).
\label{23}
\eeq
Assuming uniform solutions of Eq.\ \eqref{19} and going back to the original variables, Eqs.\ \eqref{20n} and \eqref{21n} yield $n(t)=\dot{\lb}(t)\widetilde{n}$,
$\mathbf{u}(\h,t)=a(t)\h \mathbf{e}_x$, $T(t)=\widetilde{T}(\widetilde{t})$, and $P_{ij}(t)=\dot{\lb}(t)\widetilde{P}_{ij}(\widetilde{t})$.
Therefore, uniform solutions to Eq.\ \eqref{19} map onto USF ($\h=y$) or ULF ($\h=x$) solutions to Eq.\ \eqref{2}.
The periodic boundary conditions \eqref{23} translate into
\beq
f\left(-\frac{\widetilde{L}}{2\dot{\lb}(t)},\mathbf{v},t\right)=
f\left(\frac{\widetilde{L}}{2\dot{\lb}(t)},\mathbf{v}+a_0\widetilde{L}\mathbf{e}_x,t\right).
\label{26}
\eeq
In the case of the USF, these are the well-known Lees--Edwards's boundary conditions \cite{LE72}.
While the forms \eqref{2} and \eqref{19} of the Boltzmann equation are fully equivalent, as are the respective boundary conditions \eqref{26} and \eqref{23}, it is obvious that Eqs.\ \eqref{19} and \eqref{23} are much simpler to implement in computer simulations than Eqs.\ \eqref{2} and \eqref{26}. This is especially important if one restricts oneself to uniform solutions of the form \eqref{22n}. In that case, Eq.\ \eqref{19} becomes
\beq
\frac{\partial \widetilde{f}}{\partial \widetilde{t}}-a_0\frac{\partial}{\partial \widetilde{v}_x}\left(\widetilde{v}_\h \widetilde{f}\right)=J[\widetilde{f},\widetilde{f}].
\label{31}
\eeq
The corresponding energy balance equation is
\beq
\dot{\widetilde{T}}(\widetilde{t})=-\frac{2a_0}{d\widetilde{n}}\widetilde{P}_{x\h}(\widetilde{t})-
\widetilde{\zeta}(\widetilde{t})\widetilde{T}(\widetilde{t}),
\label{32}
\eeq
where
\beq
\widetilde{\zeta}=-\frac{m}{d\widetilde{n}\widetilde{T}}\int\dd\widetilde{\mathbf{v}}\, \widetilde{v}^2 J[\widetilde{f},\widetilde{f}].
\label{33}
\eeq
Note that
\beq
\zeta(t)=\dot{\lb}(t)\widetilde{\zeta}(\widetilde{t})
\label{25b}
\eeq
and, thus, Eqs.\ \eqref{5} and \eqref{32} are equivalent.
Although $T=\oT$, in Eq.\ \eqref{32} we keep the notation $\oT$ to emphasize that here the temperature is seen as a function of the scaled time $\ot$.
In cooling situations, i.e., in the USF if $\zeta>2|aP_{xy}|/dp$, in the ULF with $a_0<0$ if $\zeta>2|a|P_{xx}/dp$, or in the ULF with $a_0>0$, the temperature can reach values much smaller than the initial one, {which} can cause technical difficulties (low signal-to-noise ratio) in the simulations. This is especially important in the ULF with $a_0>0$ since no steady state exists and the temperature keeps decreasing without any lower bound. To manage this problem, it is convenient to introduce a velocity rescaling (or thermostat). {}From a mathematical point of view, let us perform the additional change of variables
\beq
\widehat{t}=\lc(\widetilde{t}),\quad \wv=\frac{\ov}{\dot{\lc}(\ot)},
\label{34}
\eeq
\beq
\wf(\wv,\wt)=\left[\dot{\lc}(\ot)\right]^d \of(\ov,\ot),
\label{35}
\eeq
where so far $\lc(\widetilde{t})$ is an arbitrary (positive definite) protocol function. The following identities are straightforward,
\beq
\left[\dot{\lc}(\ot)\right]^{d-1}\frac{\partial\of}{\partial\ot}=\frac{\partial\wf}{\partial\wt}
-\frac{\ddot{\lc}(\ot)}{\left[\dot{\lc}(\ot)\right]^2}\frac{\partial}{\partial\wv}\left(\wv\wf\right),
\label{36}
\eeq
\beq
\left[\dot{\lc}(\ot)\right]^{d-1}J[\of,\of]=J[\wf,\wf].
\label{37}
\eeq
Therefore, Eq.\ \eqref{31} becomes
\beq
\frac{\partial \wf}{\partial \wt}-\wa(\wt)\frac{\partial}{\partial \widehat{v}_x}\left(\widehat{v}_\h \wf\right)-\mu(\wt)\frac{\partial}{\partial\wv}\left(\wv\wf\right)=J[\wf,\wf],
\label{38}
\eeq
where
\beq
\wa(\wt)\equiv \frac{a_0}{\dot{\lc}(\ot)},\quad \mu(\wt)\equiv \frac{\ddot{\lc}(\ot)}{\left[\dot{\lc}(\ot)\right]^2}.
\label{mu}
\eeq
The Boltzmann equation [Eq.\ \eqref{38}] represents the action of a {nonconservative} external force $\widehat{\mathbf {F}}(\wv,\wt)=-m\wa(\wt)\widehat{v}_\h\mathbf{e}_x-m\mu(\wt)\wv$.
The relationship between the granular temperatures defined from $\wf$ and $\of$, respectively, is $\widehat{T}(\wt)=\widetilde{T}(\ot)/\left[\dot{\lc}(\ot)\right]^2$. Thus, the \emph{thermostat} choice $\dot{\lc}(\ot)\propto \left[\widetilde{T}(\ot)\right]^{1/2}$ keeps the rescaled temperature $\widehat{T}(\wt)$ constant. While, at a theoretical level, Eq.\ \eqref{31} is simpler and more transparent than Eq.\ \eqref{38}, the latter is more useful from a computational point of view in cooling situations.
\subsection{Rheological functions}
In order to characterize the non-Newtonian properties of the unsteady USF and ULF, it is convenient to introduce generalized transport coefficients.
As is well known, the NS shear viscosity $\eta_\text{NS}$ is defined through the linear constitutive equation
\beq
P_{ij}=p\delta_{ij}-\eta_\text{NS} \left(\nabla_i u_j+\nabla_j u_i-\frac{2}{d}\nabla\cdot \mathbf{u}\delta_{ij}\right),
\label{etaNS}
\eeq
{where we have taken into account that the NS bulk viscosity is zero in the low-density limit \cite{CC70,BDKS98}. This is especially relevant in compressible flows like the ULF.
Making use of} Eq.\ \eqref{1}, we define (dimensionless) non-Newtonian viscosities $\eta^*(a^*)$ for the USF and the ULF by the relation
\beq
P_{ij}^*(a^*)=\delta_{ij}-\eta^*(a^*)a^*\left(\delta_{ix}\delta_{j\h}+\delta_{jx}\delta_{i\h}-\frac{2}{d}\delta_{x\h}\delta_{ij}\right).
\label{Pij}
\eeq
More specifically, setting $ij=x\h$, Eq.\ \eqref{Pij} yields
\beq
\eta^*(a^*)=\frac{\delta_{x\h}-P_{x\h}^*(a^*)}{a^*\left(1+\frac{d-2}{d}\delta_{x\h}\right)}.
\label{eta}
\eeq
The rheological function $\eta^*(a^*)$ differs in the USF from that in the ULF. In the latter flow it is related to the \emph{normal} stress $P_{xx}^*$. In that case, by symmetry, one has $P_{xx}^*+(d-1)P_{yy}^*=d$, so
\beq
P_{xx}^*-P_{yy}^*=-2\eta^*(a^*)a^*.
\label{Pxx-Pyy}
\eeq
Since $0<P_{xx}^*<d$, the viscosity $\eta^*(a^*)$ in the ULF must be positive definite but upper bounded:
\beq
\eta^*(a^*)<\begin{cases}
\frac{d}{2(d-1)a^*},&a^*>0,\\
\frac{d}{2|a^*|},&a^*<0.
\end{cases}
\eeq
In contrast, in the USF the viscosity function is related to the \emph{shear} stress $P_{xy}^*$,
\beq
P_{xy}^*=-\eta^*(a^*)a^*.
\label{etaUSF}
\eeq
In this state the normal stress differences are characterized by the viscometric functions
\beq
\Psi_1^*(a^*)=\frac{P_{yy}^*(a^*)-P_{xx}^*(a^*)}{{a^*}^2},
\label{43}
\eeq
\beq
\Psi_2^*(a^*)=\frac{P_{zz}^*(a^*)-P_{yy}^*(a^*)}{{a^*}^2}.
\label{43bis}
\eeq
\section{Kinetic model and nonlinear hydrodynamics\label{sec3}}
\subsection{Kinetic model}
In order to progress on the theoretical understanding of the USF and the ULF, it is convenient to adopt an extension of the Bhatnagar--Gross--Krook (BGK) kinetic model \cite{C88}, in which the (inelastic) Boltzmann collision operator $J[f,f]$ is replaced by a simpler form \cite{BDS99,SA05}:
\beq
J[f,f]\to-\beta(\alpha)\nu(f-f_\text{hcs})+\frac{\zeta}{2}\frac{\partial}{\partial \mathbf{v}}\cdot[(\mathbf{v}-\mathbf{u})f].
\label{n1}
\eeq
Here $f_\text{hcs}$ is the local version of the homogeneous cooling state distribution {\cite{BP04}} and
\beq
\nu=
\frac{8\pi^{(d-1)/2}}{(d+2)\Gamma(d/2)}n\sigma^{d-1}\sqrt{\frac{T}{m}}
\label{0.1}
\eeq
is a convenient choice for the effective collision frequency.
The factor $\beta(\alpha)$ can be freely chosen to optimize agreement with the original Boltzmann equation. Although it is not necessary to fix it in the remainder of this section, we will take
\beq
\beta(\alpha)=\frac{1}{2}(1+\alpha)
\eeq
at the end \cite{SA05}. The cooling rate is defined by Eq.\ \eqref{zeta} but here we will take the expression obtained from the Maxwellian approximation, namely
\beq
\zeta=\frac{d+2}{4d}(1-\alpha^2)\nu.
\label{0.1a}
\eeq
{This is sufficiently accurate from a practical point of view \cite{AS05}, especially at the level of the simple kinetic model \eqref{n1}.}
Using the replacement \eqref{n1}, Eq.\ \eqref{31} becomes
\beq
\frac{\partial \widetilde{f}}{\partial \widetilde{t}}-a_0\frac{\partial}{\partial \widetilde{v}_x}\left(\widetilde{v}_\h \widetilde{f}\right)=-\beta\onu(\of-\of_\text{hcs})+\frac{\ozeta}{2}\frac{\partial}{\partial \ov}\cdot(\ov\of),
\label{3.1}
\eeq
where $\onu$ and $\ozeta$ are given by Eqs.\ \eqref{0.1} and \eqref{0.1a}, respectively, except for the change $n\to \widetilde{n}$ (recall that $T=\widetilde{T}$).
Taking second-order velocity moments on both sides of Eq.\ \eqref{3.1} one gets
\beq
\dot{\oP}_{ij}=-a_0\left(\oP_{j\h}\delta_{ix}+\oP_{i\h}\delta_{jx}\right)-{\ozeta}\,{\oP_{ij}}-\beta\onu\left(\oP_{ij}-\on \oT\delta_{ij}\right).
\label{3.2}
\eeq
{}From the trace of both sides of Eq.\ \eqref{3.2} we recover the exact energy balance equation \eqref{32}. The advantage of the BGK-like model kinetic equation \eqref{3.1} is that it allows one to complement Eq.\ \eqref{32} with a closed set of equations for the elements of the pressure tensor \cite{BRM97,S09}. It is interesting to note that Eq.\ \eqref{3.2}, with $\beta=(1+\al)[d+1+(d-1)\al]/4d$, can also be derived from the original Boltzmann equation in the Grad approximation \cite{SGD04,VSG10,VGS11}.
As discussed in Sec.\ \ref{sec1} [cf.\ Eq.\ \eqref{10}], the relevant quantity is the \emph{reduced} pressure tensor defined as
\beq
P_{ij}^*(\ot)=\frac{P_{ij}(t)}{n(t)T(t)}=\frac{\oP_{ij}(\ot)}{\on\oT(\ot)}
\label{3.6}
\eeq
Combining Eqs.\ \eqref{32} and \eqref{3.2} we obtain
\beq
\frac{\dot{P}_{ij}^*}{\onu}=-a^*\left(P_{j\h}^*\delta_{ix}+P_{i\h}^*\delta_{jx}\right)+\frac{2a^*}{d}
P_{ij}^*P_{x\h}^*-\beta\left(P_{ij}^*-\delta_{ij}\right),
\label{3.2bis}
\eeq
where, according to Eq.\ \eqref{a*},
\beq
a^*(\ot)=\frac{a(t)}{\nu(t)}=\frac{a_0}{\onu(\ot)}
\label{3.5}
\eeq
is the reduced shear ($\h=y$) or longitudinal ($\h=x$) rate.
Taking $ij=x\h$ and $ij=\h\h$, Eq.\ \eqref{3.2bis} yields
\beq
\frac{\dot{P}_{x\h}^*}{\onu}=-a^*\left(P_{\h\h}^*+P_{x\h}^*\delta_{x\h}\right)+\frac{2a^*}{d}\left(P_{x\h}^*\right)^2-
\beta\left(P_{x\h}^*-\delta_{x\h}\right),
\label{3.3}
\eeq
\beq
\frac{\dot{P}_{\h\h}^*}{\onu}=-2a^*P_{\h\h}^*\delta_{x\h}+\frac{2a^*}{d}P_{\h\h}^*P_{x\h}^*-\beta\left(P_{\h\h}^*-1\right).
\label{3.4}
\eeq
Note that Eq.\ \eqref{3.4} is identical to Eq.\ \eqref{3.3} in the ULF ($\h=x$).
Equations \eqref{3.3} and \eqref{3.4} must be complemented with the evolution equation for $a^*$. We recall that $\onu\propto \oT^q$ and $a^*\propto \oT^{-q}$ with $q=\frac{1}{2}$. Thus, using Eq.\ \eqref{32} one simply obtains
\beq
\frac{\dot{a}^*}{\onu}=qa^*\left(\frac{2a^*}{d}P_{x\h}^*+\zeta^*\right),
\label{3.7}
\eeq
where, according to Eq.\ \eqref{0.1a},
\beq
\zeta^*=\frac{d+2}{4d}(1-\alpha^2).
\label{3.8}
\eeq
Here we will temporarily view $q$ as a free parameter, so the solutions to Eqs.\ \eqref{3.3}--\eqref{3.7} depend parametrically on $q$. The exponent $q$ is directly related to the wide class of dissipative gases introduced by Ernst \emph{et al.} \cite{ETB06a,ETB06b,TBE07}.
Equations \eqref{3.3}--\eqref{3.7} constitute a closed set of nonlinear equations for $\{P_{x\h}^*(\ot),P_{\h\h}^*(\ot),a^*(\ot)\}$ that can be numerically solved subject to a given initial condition
\beq
\of_0(\ov)\Rightarrow\{P_{x\h,0}^*,P_{\h\h,0}^*,a^*_0\}.
\label{ini}
\eeq
\subsection{Steady state}
Setting $\dot{P}_{x\h}^*=0$, $\dot{P}_{\h\h}^*=0$, and $\dot{a}^*=0$ in Eqs.\ \eqref{3.3}--\eqref{3.7}, they become a set of three (USF) or two (ULF) independent algebraic equations whose solution provides the steady-state values. In the case of the USF ($\h=y$) the solution is
\beq
|a^*_s|=\sqrt{\frac{d\zeta^*}{2\beta}}(\beta+\zeta^*),
\label{3.9}
\eeq
\beq
P_{xy,s}^*=-\sqrt{\frac{d\beta\zeta^*}{2}}\frac{\text{sgn}(a^*_s)}{\beta+\zeta^*},\quad
P_{yy,s}^*=\frac{\beta}{\beta+\zeta^*}.
\label{3.10}
\eeq
In contrast, the solution for the ULF ($\h=x$) is
\beq
a^*_s=-\frac{d\zeta^*}{2}\frac{\beta+\zeta^*}{\beta+d\zeta^*},
\label{3.11a}
\eeq
\beq
P_{xx,s}^*=\frac{\beta+d\zeta^*}{\beta+\zeta^*}.
\label{3.11b}
\eeq
Note that the steady-state values are independent of $q$. A linear stability analysis in the case of the USF \cite{SGD04} shows that the steady state, Eqs.\ \eqref{3.9} and \eqref{3.10}, is indeed a stable solution of Eqs.\ \eqref{3.3}--\eqref{3.7}. The proof can be easily extended to the ULF.
\subsection{Unsteady hydrodynamic solution}
In the USF ($\h=y$), Eq.\ \eqref{3.2bis} implies that $(\dot{P}_{zz}^*-\dot{P}_{yy}^*)/\widetilde{\nu}=(2a^*P_{xy}^*/d-\beta)({P}_{zz}^*-{P}_{yy}^*)$. Since, on physical grounds, $a^*P_{xy}^*<0$, we conclude that ${P}_{zz}^*-{P}_{yy}^*=0$ in the hydrodynamic regime. Therefore, according to the kinetic model description, the second viscometric function identically vanishes, i.e.,
\beq
\Psi_2^*(a^*)=0.
\label{Psi2_0}
\eeq
Next, by symmetry,
$P_{xx}^*+P_{yy}^*+(d-2)P_{zz}^*=d$. This mathematical identity, combined with ${P}_{zz}^*={P}_{yy}^*$, allows one to rewrite Eq.\ \eqref{43} as
\beq
\Psi_1^*=-d\frac{1-{P}_{yy}^*}{{a^*}^2}.
\label{Psi1_0}
\eeq
As sketched in Fig.\ \ref{fig1} and described by Eqs.\ \eqref{7} and \eqref{10}, the hydrodynamic solution requires the whole time dependence of $P_{ij}^*$ to be captured through a dependence on $a^*$ common to every initial state.
As a first step to obtain such a hydrodynamic solution, let us eliminate time in favor of $a^*$ in Eqs.\ \eqref{3.3} and \eqref{3.4} with the help of Eq.\ \eqref{3.7}, i.e.,
\beqa
q\left(\frac{2a^*}{d}P_{x\h}^*+\zeta^*\right)\frac{\partial{P}_{x\h}^*}{\partial a^*}&=&-P_{\h\h}^*-P_{x\h}^*\delta_{x\h}+\frac{2}{d}\left(P_{x\h}^*\right)^2\nn
&&-\frac{\beta}{a^*}\left(P_{x\h}^*-\delta_{x\h}\right),
\label{3.12}
\eeqa
\beqa
q\left(\frac{2a^*}{d}P_{x\h}^*+\zeta^*\right)\frac{\partial{P}_{\h\h}^*}{\partial a^*}&=&-2P_{\h\h}^*\delta_{x\h}+\frac{2}{d}P_{\h\h}^*P_{x\h}^*\nn
&&-\frac{\beta}{a^*}\left(P_{\h\h}^*-1\right).
\label{3.13}
\eeqa
This set of two nonlinear coupled differential equations must be solved in general with the initial
conditions stemming from Eq.\ \eqref{ini}, namely
\beq
a^*=a^*_0\Rightarrow \{P_{x\h}^*=P_{x\h,0}^*, P_{\h\h}^*=P_{\h\h,0}^*\}.
\label{3.14}
\eeq
Equations \eqref{3.12} and \eqref{3.13} must be solved in agreement with the physical direction of time. This means that the solutions uncover the region $|a^*_0|<|a^*|<|a_s^*|$ in conditions of cooling and the region $|a^*_0|>|a^*|>|a_s^*|$ in conditions of heating (see Table \ref{table1}).
In the case of the ULF with $a^*_0>0$, one has $a^*_0<a^*<\infty$ due to the absence of steady state.
Equations \eqref{3.12} and \eqref{3.13} describe both the kinetic and hydrodynamic
regimes. In order to isolate the hydrodynamic solution, one must
apply appropriate boundary conditions \cite{SGD04,S09}.
An alternative route to get the hydrodynamic solution consists of expanding $P_{x\h}^*$ and $P_{\h\h}^*$ in powers of $a^*$ (Chapman--Enskog expansion),
\beq
P_{x\h}^*(a^*)=\delta_{x\h}+\sum_{j=1}^\infty c_{x\h}^{(j)}{a^*}^j,\quad P_{\h\h}^*(a^*)=1+\sum_{j=1}^\infty c_{\h\h}^{(j)}{a^*}^j.
\label{3.15}
\eeq
Inserting the expansions in both sides of Eqs.\ \eqref{3.12} and \eqref{3.13} one can get the Chapman--Enskog coefficients $c_{x\h}^{(j)}$ and $c_{\h\h}^{(j)}$ in a recursive way. The first- and second-order coefficients are
\beq
c_{x\h}^{(1)}=-\frac{d+(d-2)\delta_{x\h}}{d(\beta+q\zeta^*)},\quad c_{\h\h}^{(1)}=c_{x\h}^{(1)}\delta_{x\h},
\label{3.16}
\eeq
\beq
c_{x\h}^{(2)}=c_{\h\h}^{(2)}\delta_{x\h},
\label{3.17}
\eeq
\beq
c_{\h\h}^{(2)}=2\frac{2(d-1)(d-2+q)\delta_{x\h}+d(\delta_{x\h}-1)}{d^2(\beta+q\zeta^*)(\beta+2q\zeta^*)}.
\label{3.18}
\eeq
Equation \eqref{3.16} gives NS coefficients, while Eqs.\ \eqref{3.17} and \eqref{3.18} correspond to Burnett coefficients.
{}From Eqs.\ \eqref{3.15} and \eqref{3.16}, Eq.\ \eqref{eta} yields
\beq
\lim_{a^*\to 0}\eta^*(a^*)=\frac{1}{\beta+q\zeta^*}.
\label{eta0NS}
\eeq
Thus, the NS viscosity coincides in the USF and in the ULF, as expected.
Regarding the USF first viscometric function, Eqs.\ \eqref{Psi1_0}, \eqref{3.15}, and \eqref{3.18} gives the Burnett coefficient
\beq
\lim_{a^*\to 0}\Psi_1^*(a^*)=-\frac{2}{(\beta+q\zeta^*)(\beta+2q\zeta^*)}.
\label{Burnett}
\eeq
In general, all the even (odd) coefficients of $P_{xy}^*$ ($P_{yy}^*$) vanish in the USF. In the ULF, however, all the coefficients of $P_{xx}^*$ are non-zero.
It is interesting to remark that, in contrast to the elastic case ($\zeta^*=0$) \cite{SBD86,S00a}, the Chapman--Enskog expansions \eqref{3.15} are \emph{convergent} \cite{S08,S08a,S09} if $\zeta^*>0$. On the other hand, the radius of convergence is finite and coincides with the stationary value $|a_s^*|$.
The series \eqref{3.15} clearly correspond to the hydrodynamic solution since they give $P_{x\h}^*$ and $P_{\h\h}^*$ as unambiguous functions of $a^*$, regardless of the details of the initial conditions \eqref{3.14}. However, the series have two shortcomings. First, since they diverge for $|a^*|>|a_s^*|$, they do not provide $P_{x\h}^*(a^*)$ and $P_{\h\h}^*(a^*)$ in a direct way for that region. Second, even if $|a^*|<|a_s^*|$, closed expressions for $P_{x\h}^*(a^*)$ and $P_{\h\h}^*(a^*)$ are not possible.
In order to get closed and explicit (albeit approximate) solutions, we formally take $q$ as a small parameter and
perturb around $q=0$ \cite{KDN97,S00a},
\beq
P_{ij}^*(a^*)=P_{ij}^{*(0)}(a^*)+q P_{ij}^{*(1)}(a^*)+\cdots.
\label{3.19}
\eeq
Setting $q=0$ in Eqs.\ \eqref{3.12} and \eqref{3.13} one gets
\beq
P_{x\h}^{*(0)}=d\left[\delta_{x\h}-\frac{\beta\gamma_\h(a^*)}{a^*}\right],
\label{20}
\eeq
\beq
P_{\h\h}^{*(0)}=\frac{1}{1+2\gamma_\h(a^*)},
\label{21}
\eeq
where $\gamma_\h(a^*)$ is the physical solution of a cubic (USF, $\h=y$) or a quadratic (ULF, $\h=x$) equation:
\beq
\gamma_y(1+2\gamma_y)^2=\frac{{a^*}^2}{\beta^2d},
\label{17USF}
\eeq
\beq
d\left(1-\frac{\beta\gamma_x}{a^*}\right)(1+2\gamma_x)=1.
\label{17ULF}
\eeq
The respective solutions are
\beq
\gamma_y(a^*)=
\frac{2}{3}\sinh^2\left[\frac{1}{6}\cosh^{-1}\left(1+27\frac{{a^*}^2}{\beta^2 d}\right)\right],
\label{15}
\eeq
\beq
\gamma_x(a^*)=\frac{a^*}{2\beta}-\frac{1}{4}+\frac{1}{2}\sqrt{\left(\frac{a^*}{\beta}+\frac{1}{2}\right)^2-\frac{2a^*}{\beta d}}.
\label{15ULF}
\eeq
For small $|a^*|$ one has
\beq
\gamma_y(a^*)=\frac{{a^*}^2}{\beta^2d}+\mathcal{O}({a^*}^4),
\eeq
\beq
\gamma_x(a^*)=\frac{d-1}{d}\frac{{a^*}}{\beta}\left(1+\frac{2}{d}\frac{{a^*}}{\beta}\right)+\mathcal{O}({a^*}^3),
\eeq
so
\beq
P_{x\h}^{*(0)}(a^*)=\delta_{x\h}-\frac{d+(d-2)\delta_{x\h}}{\beta d}a^*+\mathcal{O}({a^*}^2),
\label{NS0}
\eeq
in agreement with Eq.\ \eqref{3.16}. Furthermore, it is easy to check that in the steady state [cf.\ Eqs.\ \eqref{3.9} and \eqref{3.11a}] one gets
\beq
\gamma_y(a^*_s)=\frac{1}{2}\frac{\zeta^*}{\beta},
\label{22USF}
\eeq
\beq
\gamma_x(a^*_s)=-\frac{d-1}{2}\frac{\zeta^*}{\beta+d\zeta^*},
\label{22ULF}
\eeq
so Eqs.\ \eqref{20} and \eqref{21} yield Eqs.\ \eqref{3.10} and \eqref{3.11b}, as expected.
Once $P_{ij}^{*(0)}$ are known, Eqs.\ \eqref{3.12} and \eqref{3.13} provide $P_{ij}^{*(1)}$. In the USF case ($\h=y$) the results are
\beq
P_{xy}^{*(1)}=-P_{xy}^{*(0)}h_y(a^*),
\label{31bis}
\eeq
\beq
P_{yy}^{*(1)}=6 P_{yy}^{*(0)}\gamma_y(a^*)H_y(a^*),
\label{32bis}
\eeq
where
\beq
h_y(a^*)\equiv \frac{\left[\zeta^*/\beta-2\gamma_y(a^*)\right]\left[1-6\gamma_y(a^*)\right]}{\left[1+6\gamma_y(a^*)\right]^2},
\label{hUSF}
\eeq
\beq
H_y(a^*)\equiv
\frac{\zeta^*/\beta-2\gamma_y(a^*)}{\left[1+6\gamma_y(a^*)\right]^2},
\label{HUSF}
\eeq
and use has been made of Eq. \eqref{17USF} and the relation
\beq
\frac{\partial \gamma_y(a^*)}{\partial
a^*}=\frac{2a^*}{\beta^2d}\frac{1}{[1+2\gamma_y(a^*)][1+6\gamma_y(a^*)]}.
\label{33b}
\eeq
Analogously, taking into account Eq.\ \eqref{17ULF}, the final expression in the ULF case ($\h=x$) is
\beq
P_{xx}^{*(1)}=
2 P_{xx}^{*(0)}\gamma_x(a^*)h_x(a^*),
\label{32ULF}
\eeq
where
\beq
h_x(a^*)\equiv\frac{2a^*/\beta-2\gamma_x(a^*)+\zeta^*/\beta}{\left[1-2a^*/\beta+4\gamma_x(a^*)\right]^2}.
\label{hULF}
\eeq
Note that in the steady state $P_{ij}^{*(1)}=0$ both in the USF and the ULF. This is consistent with the fact that the steady-state values are independent of $q$. For small $|a^*|$,
\beq
P_{x\h}^{*(1)}(a^*)=\frac{d+(d-2)\delta_{x\h}}{\beta^2 d}\zeta^*a^*+\mathcal{O}({a^*}^2),
\label{NS1}
\eeq
{which} again agrees with Eq.\ \eqref{3.16}.
In principle, it is possible to proceed further and get the terms of
orders $q^2$, $q^3$, \ldots, in Eq.\ \eqref{3.19} \cite{S00a}. However, for our purposes it is
sufficient to retain the linear terms only.
\subsection{Rheological model}
The definitions of the rheological functions \eqref{eta}, \eqref{43}, and \eqref{43bis} are independent of any specific model employed to obtain $P_{ij}^*(a^*)$. Here we make use of the kinetic model \eqref{n1} and the expansion \eqref{3.19} truncated to first order in $q$. Next, by means of a Pad\'e approximant we construct (approximate) explicit expressions for $\eta^*(a^*)$. Let us start with the ULF ($\h=x$), in which case Eqs.\ \eqref{20} and \eqref{32ULF} yield
\beq
\eta^*(a^*)\approx\frac{d}{d-1}\frac{\gamma_x(a^*)}{a^*[1+2\gamma_x(a^*)]}\frac{1}{1+q h_x(a^*)}.
\label{etaULF}
\eeq
Analogously, in the USF case ($\h=y$), Eqs.\ \eqref{20}, \eqref{17USF}, and \eqref{31bis} give
\beq
\eta^*(a^*)\approx\frac{1}{\beta
\left[1+2\gamma_y(a^*)\right]^{2}}\frac{1}{1+qh_y(a^*)}.
\label{etaUSFPade}
\eeq
Let us analyze now the USF first viscometric function.
Using Eqs.\ \eqref{21}, \eqref{17USF}, and \eqref{32bis}, Eq.\ \eqref{Psi1_0} gives
\beq
\Psi_1^*(a^*)=-\frac{2}{\beta^{2}
\left[1+2\gamma_y(a^*)\right]^{3}}\left[1-3qH_y(a^*)\right]+\mathcal{O}(q^2).
\eeq
Again, it is convenient to construct a Pad\'e approximant of $\Psi_1^*(a^*)$. Here we take
\beq
\Psi_1^*(a^*)\approx-\frac{2}{\beta^{2}\left[1+2\gamma_y(a^*)\right]^{3}}\frac{1}{[1+q H_y(a^*)][1+2q H_y(a^*)]}.
\label{Psi1}
\eeq
In principle, one should have written $1+3qH_y$ instead of $(1+qH_y)(1+2qH_y)$ in Eq.\ \eqref{Psi1}, but the form chosen has the advantage of being consistent with the Burnett coefficient \eqref{Burnett} for any $q$.
In summary, our simplified rheological model consists of Eq.\ \eqref{etaULF}, complemented with Eqs.\ \eqref{15ULF} and \eqref{hULF}, for the ULF and Eqs.\ \eqref{etaUSFPade} and \eqref{Psi1}, complemented with Eqs.\ \eqref{15}, \eqref{hUSF}, and \eqref{HUSF}, for the USF. Since we are interested in hard spheres, we must take $q=\frac{1}{2}$ in those equations.
This approximation has a number of important properties. First, as said in connection with the Chapman--Enskog expansion \eqref{3.15}, Eqs.\ \eqref{etaULF}, \eqref{etaUSFPade}, and \eqref{Psi1} qualify as a (non-Newtonian) \emph{hydrodynamic} description. Second, in contrast to the full expansions \eqref{3.15} and \eqref{3.19}, they provide the relevant elements $P_{x\h}^*$ of the pressure tensor as \emph{explicit} functions of both the shear or longitudinal rate $a^*$ and the coefficient of normal restitution $\alpha$ (through $\beta$ and $\zeta^*$). Third, as seen from Eq.\ \eqref{eta0NS}, Eqs.\ \eqref{etaULF} and \eqref{etaUSFPade} agree with the \emph{exact} NS coefficients predicted by the kinetic model \eqref{n1} for arbitrary values of the parameter $q$; this agreement extends to the Burnett-order coefficient \eqref{Burnett}. Next, the correct \emph{steady-state} values [Eqs.\ \eqref{3.9}--\eqref{3.11b}] are included in the Pad\'e approximants \eqref{etaULF}, \eqref{etaUSFPade}, and \eqref{Psi1}. Finally, it can be checked that the correct \emph{asymptotic} forms in the limit $|a^*|\to\infty$ \cite{SGD04,S09} are preserved.
Moreover, even in the physical case of hard spheres
($q=\frac{1}{2}$), Eqs.\ \eqref{etaULF}, \eqref{etaUSFPade}, and \eqref{Psi1} represent an excellent
analytical approximation to the numerical solution of the set of Eqs.\
\eqref{3.12} and \eqref{3.13} with appropriate initial conditions \cite{SGD04,S09}. The results for $\alpha=0.5$ and $\alpha=0.9$ are presented in Figs.\ \ref{fig4} and \ref{fig5} in the cases of the USF and the ULF, respectively.
We observe a generally good agreement between the numerical and the simplified results. This is especially true in the USF, where the limitations of the rheological model are only apparent for the most inelastic system ($\alpha=0.5$) and for shear rates smaller than the steady-state value, the agreement being better for the viscosity than for the first viscometric function. In the ULF case the differences are more important, both for $\alpha=0.5$ and $\alpha=0.9$, although they are restricted to longitudinal rates near the maximum and also to values more negative than that of the maximum.
\begin{figure}
\includegraphics[width=.9 \columnwidth]{BGK_USF.eps}
\caption{(Color online) Shear-rate dependence of (a) the viscosity function and (b) the first viscometric function for $\alpha=0.5$ and $\alpha=0.9$ in the USF with $d=3$. The solid lines have been obtained from numerical solutions of Eqs.\ \protect\eqref{3.12} and \protect\eqref{3.13}, while the dashed lines correspond to the simplified rheological model \protect\eqref{etaUSFPade} and \protect\eqref{Psi1}. The circles represent the steady-state values [cf.\ Eqs.\ \protect\eqref{3.9} and \protect\eqref{3.10}]. Note that the simplified solution deviates practically from the numerical solution only for the most inelastic system ($\alpha=0.5$) and in the region $a^*<a_s^*$.}
\label{fig4}
\end{figure}
\begin{figure}
\includegraphics[width=.9 \columnwidth]{BGK_ULF.eps}
\caption{(Color online) Longitudinal-rate dependence of the viscosity for $\alpha=0.5$ and $\alpha=0.9$ in the ULF with $d=3$. The solid lines have been obtained from numerical solutions of Eq.\ \protect\eqref{3.12}, while the dashed lines correspond to the simplified rheological model \protect\eqref{etaULF}. The circles represent the steady-state values [cf.\ Eqs.\ \protect\eqref{3.11a} and \protect\eqref{3.11b}].}
\label{fig5}
\end{figure}
\section{Simulation details\label{sec4}}
We have performed DSMC simulations of the three-dimensional ($d=3$) USF and ULF. The details are similar to those described elsewhere \cite{AS05}, so here we provide only some distinctive features.
Since the original Boltzmann equation \eqref{2} is fully equivalent to the scaled form \eqref{19} [with the change of variables \eqref{11}--\eqref{15n}], we have solved the latter and have applied the periodic boundary conditions \eqref{23}. We call this the ``inhomogeneous'' problem since the scaled distribution function $\of(\widetilde{\h},\ov,\ot)$ is, in principle, allowed to depend on the scaled spatial variable $\widetilde{\h}$. On the other hand, if one restricts oneself to \emph{uniform} solutions \eqref{22n}, then Eq.\ \eqref{19} reduces to Eq.\ \eqref{31}. The solution to Eq.\ \eqref{31} by the DSMC method will be referred to as the ``homogeneous'' problem. Most of the results that will be presented in Sec.\ \ref{sec5} correspond to the homogeneous problem.
A wide sample of initial conditions $\of_0(\widetilde{\h},\widetilde{\mathbf{v}})$ has been considered, as described below.
Note that, since $\dot{\theta}(0)=1$ [cf.\ Eq. \eqref{13}], $\widetilde{\h}=\h$ and $\widetilde{\mathbf{v}}=\mathbf{v}-a_0\h \mathbf{e}_x$ at $t=0$. Consequently, $\widetilde{n}_0(\widetilde{\h})=n_0(\h)$ and $\widetilde{\mathbf{u}}_0(\widetilde{\h})=\mathbf{u}_0(\h)-a_0\h \mathbf{e}_x$. However, for consistency, we will keep the tildes in the expressions of the initial state. An exception will be the initial temperature because $\widetilde{T}=T$ for all times.
The inhomogeneous problem for the USF was already analyzed in Ref.\ \cite{AS05} and, thus, only the inhomogeneous problem for the ULF is considered in this paper. The chosen initial condition is
\beq
\of_0(\widetilde{x},\widetilde{\mathbf{v}})=\frac{\on_0(\widetilde{x})}{4\pi}\frac{m}{3T_0}
\delta\left(|\widetilde{\mathbf{v}}-\widetilde{\mathbf{u}}_0(\widetilde{x})|-\sqrt{3T_0/m}\right),
\label{4.2}
\eeq
where $\delta(x)$ is Dirac's distribution and the initial density and velocity fields are
\beq
\on_0(\widetilde{x})=\langle \on\rangle\left(1+\frac{1}{2}\sin\frac{2\pi \widetilde{x}}{\widetilde{L}}\right),
\label{4.3}
\eeq
\beq
\widetilde{\mathbf{u}}_0(\widetilde{x})=a_0\widetilde{L}\left(\cos\frac{\pi
\widetilde{x}}{\widetilde{L}}-\frac{2}{\pi}\right)\mathbf{e}_x,
\label{4.4}
\eeq
respectively, while the initial temperature $T_0$ is uniform.
In Eq.\ \eqref{4.3} $\langle \on\rangle$ is the density spatially averaged between $\widetilde{x}=-\widetilde{L}/2$ and $\widetilde{x}=\widetilde{L}/2$. This quantity is independent of time.
In our simulations of the inhomogeneous problem we have taken $\alpha=0.5$ for the coefficient of restitution, $a_0=-4/\tau_0$ for the initial longitudinal rate, and $\widetilde{L}=2.5\lambda$ for the (scaled) box length. Here,
\beq
\lambda=\frac{1}{\sqrt{2}\pi\langle \on\rangle \sigma^2},\quad \tau_0=\frac{\lambda}{\sqrt{2T_0/m}}
\label{4.1}
\eeq
are a characteristic mean free path and an initial characteristic collision time, respectively.
\begin{table}
\caption{Values of the initial pressure tensor for the four initial conditions of the class \protect\eqref{4.6}.\label{table2}}
\begin{ruledtabular}
\begin{tabular}{cccccc}
Label&$\phi$&$\widetilde{P}_{xx,0}$&$\widetilde{P}_{yy,0}$&$\widetilde{P}_{zz,0}$&$\widetilde{P}_{xy,0}$\\
\hline
B0&$0$&$2\on T_0$&$0$&$\on T_0$&$0$\\
B1&$\pi/4$&$\on T_0$&$\on T_0$&$\on T_0$&$-\on T_0$\\
B2&$\pi/2$&$0$&$2\on T_0$&$\on T_0$&$0$\\
B3&$3\pi/4$&$\on T_0$&$\on T_0$&$\on T_0$&$\on T_0$\\
\end{tabular}
\end{ruledtabular}
\end{table}
As for the homogeneous problem (both for USF and UFL) two classes of initial conditions have been chosen. First, we have taken the local equilibrium state (initial condition A), namely
\beq
\of_0(\widetilde{\mathbf{v}})=\on\left(\frac{m}{2\pi T_0}\right)^{3/2}e^{-m\widetilde{v}^2/2T_0}.
\label{4.5}
\eeq
The other class of initial conditions is of the anisotropic form
\begin{eqnarray}
\of_0(\widetilde{\mathbf{v}})&=&\frac{\on}{2}\left(\frac{m}{2\pi T_0}\right)^{1/2}
e^{-{m \widetilde{v}_{z}^{2}}/{2T_0}}\nn
&&\times\left[\delta\left(\widetilde{v}_{x}-V_{0}\cos
\phi\right)\delta\left(\widetilde{v}_{y}+V_{0}
\sin \phi\right)\right. \nonumber \\
&&\left.+\delta\left(\widetilde{v}_{x}+V_{0}\cos
\phi\right)\delta\left(\widetilde{v}_{y}-V_{0}\sin \phi\right)\right],
\label{4.6}
\end{eqnarray}
where $V_0\equiv\sqrt{2T_0/m}$
is the initial thermal speed and $\phi\in [0,\pi]$ is an angle
characterizing each specific condition. The pressure tensor
corresponding to Eq.\ \eqref{4.6} is given by
$\widetilde{P}_{xx,0}=2\on T_0\cos^2\phi$, $\widetilde{P}_{yy,0}=2\on T_0\sin^2\phi$,
$\widetilde{P}_{zz,0}=\on T_0$, and $\widetilde{P}_{xy,0}=-\on T_0\sin 2\phi$. The four values of
$\phi$ considered are $\phi=k \pi/4$ with $k=0$, $1$, $2$, and $3$;
we will denote the respective initial conditions of type \eqref{4.6}
as B0, B1, B2, and B3. The values of the elements of the pressure tensor for these four initial conditions are displayed in Table \ref{table2}.
In the fully developed USF it is expected that $\widetilde{P}_{xy}<0$ (if $a>0$) and $\widetilde{P}_{xx}>\widetilde{P}_{yy}$. As we see from Table \ref{table2}, the four initial conditions are against those inequalities, especially in the case of condition B3. As for the fully developed ULF, the physical expectations are $\widetilde{P}_{xy}=0$, $\widetilde{P}_{yy}=\widetilde{P}_{zz}$, and $\widetilde{P}_{xx}<\widetilde{P}_{yy}$ if $a_0>0$ and $\widetilde{P}_{xx}>\widetilde{P}_{yy}$ if $a_0<0$ [cf.\ Eq.\ \eqref{Pxx-Pyy}]. Again, none of the four initial conditions is consistent with those physical expectations, especially in the case of condition B0 if $a_0>0$ and B2 if $a_0<0$.
The ``artificial'' character of the initial conditions \eqref{4.6} represents a stringent test of the scenario depicted in Fig.\ \ref{fig1}.
As summarized in Table \ref{table1}, when $\ozeta >2|a_0\oP_{xy}|/3\on\oT$ in the USF or when $\ozeta >-2a_0\oP_{xx}/3\on\oT$ in the ULF, the temperature
decreases with time (cooling states) either without lower bound (ULF with $a_0>0$) or until reaching the steady state (ULF with $a_0<0$ and USF). In those cooling states the temperature can decrease so much (relative to the initial value) that this might create technical problems (low signal-to-noise ratio), as mentioned at the end of Sec.\ \ref{sec2}. This can be corrected by the application of a \emph{thermostatting} mechanism, as represented by the change of variables \eqref{34} and \eqref{35} with $\dot{\lc}(\ot)\propto \left[\widetilde{T}(\ot)\right]^{1/2}$. The DSMC implementation of Eqs.\ \eqref{38} and \eqref{mu} is quite simple. Let us denote by $\{\wv_i(\wt);i=1,\ldots,N\}$ the (rescaled) velocities of the $N$ simulated particles at time $\wt$.
The corresponding rescaled temperature and shear (or longitudinal) rate are $\wT(\wt)$ and $\wa(\wt)$, respectively. During the time step $\delta\wt$
the velocities change due to the action of the (deterministic) nonconservative external force $-m\wa(\wt) \widehat{v}_\h \mathbf{e}_x$ and
also due to the (stochastic) binary collisions. Let us denote by $\{\wv_i'(\wt+\delta\wt);i=1,\ldots,N\}$ and by $\wT'(\wt+\delta\wt)$ the velocities and temperature after this stage. Thus, the action of the thermostat force $-m\mu(\wt)\wv$ is equivalent to the velocity rescaling
$\wv_i'(\wt+\delta\wt)\to \wv_i(\wt+\delta\wt)=\wv_i'(\wt+\delta\wt)\sqrt{\wT(\wt)/\wT'(\wt+\delta\wt)}$, so $\wT'(\wt+\delta\wt)\to \wT(\wt+\delta\wt)=\wT(\wt)$. Similarly, the rescaled shear or longitudinal rate is updated as $\wa(\wt+\delta\wt)=\wa(\wt)\sqrt{\wT'(\wt+\delta\wt)/\wT(\wt)}$, so $\wa(\wt+\delta\wt)/\sqrt{\wT'(\wt+\delta\wt)}=\wa(\wt)/\sqrt{\wT(\wt)}$.
For each one of the five initial conditions for the homogeneous problem we have considered three coefficients of restitution: $\alpha=0.5$, $0.7$, and $0.9$. In the case of the USF, the values taken for the shear rate have been $a=0.01/\tau_0$, $a=0.1/\tau_0$, $a=4/\tau_0$, and $a=10/\tau_0$. The
two first values ($a=0.01/\tau_0$ and
$a=0.1/\tau_0$) are small enough to correspond to cooling cases,
even for the least inelastic system ($\alpha=0.9$), while the other
two values ($a=4/\tau_0$ and $a=10/\tau_0$) are large enough to
correspond to heating cases, even for the most inelastic system
($\alpha=0.5$). In the case of the ULF we have chosen $a_0=0.01/\tau_0$, $a_0=-0.01/\tau_0$, and $a_0=-10/\tau_0$. The first and second values correspond to cooling states (without and with a steady state, respectively), while the third value corresponds to heating states.
Therefore, the total number of independent systems simulated in the homogeneous problem is 60 for the USF and 45 for the ULF.
The technical parameters of the simulations have been the following ones: $N=10^6$ simulated particles, an adaptive time step $\delta t=10^{-3}\tau_0\sqrt{T_0/\langle T\rangle}$, and a layer thickness (inhomogeneous problem) $\delta \widetilde{x}=0.05\lambda$.
Moreover, in order to improve the statistics, the results have been averaged over 100 independent realizations.
\section{Results\label{sec5}}
\subsection{USF. Homogeneous problem}
We have simulated the Boltzmann equation describing the homogeneous problem of the USF, i.e., Eq.\ \eqref{31} with $\h=y$, by means of the DSMC method. As said in Sec.\ \ref{sec4}, three coefficients of restitution ($\alpha=0.5$, $0.7$, and $0.9$) and four shear rates ($a=0.01/\tau_0$, $0.1/\tau_0$, $4/\tau_0$, and $10/\tau_0$) have been considered. For each combination of $\alpha$ and $a$, five different initial conditions (A and B0--B3) have been chosen.
In the course of the simulations we focus on the temporal evolution of the elements of the reduced pressure tensor $P_{ij}^*$, Eq.\ \eqref{3.6}, and of the reduced shear rate $a^*$, Eq.\ \eqref{3.5}. {}From these quantities one can evaluate the viscosity $\eta^*$, Eq.\ \eqref{etaUSF}, the first viscometric function $\Psi_1^*$, Eq.\ \eqref{43}, and the second viscometric function $\Psi_2^*$, Eq.\ \eqref{43bis}. The effective collision frequency $\nu$ in Eq.\ \eqref{3.5} is defined by
\beq
\nu= {\frac{p}{\eta_{\text{NS}}^{\text{el}}}=}\frac{1}{1.016}\frac{16\sqrt{\pi}}{5}n\sigma^2\sqrt{\frac{T}{m}}.
\label{nuB}
\eeq
{The factor $1.016$ comes from an elaborate Sonine approximation employed to determine the NS shear viscosity $\eta_{\text{NS}}^{\text{el}}$ of a gas of elastic hard spheres \cite{CC70}. For simplicity, and to be consistent with the approximate character of the kinetic model \eqref{n1}, this factor is not included in Eq.\ \eqref{0.1}.}
Note that the initial reduced shear rate is $a_0^*=a\tau_0/0.8885$.
Time is monitored through the accumulated number of collisions per particle, i.e., the total number of collisions in the system since the initial state, divided by the total number of particles. The reduced quantities and the number of collisions per particle are not affected by the changes of variables discussed in Sec.\ \ref{sec2}.
\begin{figure}
\includegraphics[width=.9 \columnwidth]{USF_Evol_cooling.eps}
\caption{(Color online) (a) Reduced shear rate $a^*$ and (b) reduced viscosity $\eta^*$ versus the number of collisions per particle for the USF with $\alpha=0.5$ in the cooling states
$a=0.01 /\tau_0$ [blue (dark gray) lines] and $a=0.1/\tau_0$ [orange (light gray) lines]. The legend refers to the five initial conditions considered. The dotted horizontal line in panel (a) denotes the value $a_h^*=0.4$ above which the hydrodynamic regime is clearly established (see text).
\label{fig6}}
\end{figure}
\begin{figure}
\includegraphics[width=.9 \columnwidth]{USF_Evol_heating.eps}
\caption{(Color online) (a) Reduced shear rate $a^*$ and (b) reduced viscosity $\eta^*$ versus the number of collisions per particle for the USF with $\alpha=0.5$ in the heating states
$a=4 /\tau_0$ [orange (light gray) lines] and $a=10/\tau_0$ [blue (dark gray) lines]. The legend refers to the five initial conditions considered. The dotted horizontal line in panel (a) denotes the value $a_h^*=1.25$ below which the hydrodynamic regime is clearly established (see text).
\label{fig7}}
\end{figure}
As a representative case, we first present results for the most inelastic system ($\alpha=0.5$).
Figures \ref{fig6} and \ref{fig7} show the evolution of $a^*$ and $\eta^*$ for the cooling states ($a=0.01/\tau_0$ and $0.1/\tau_0$) and the heating states ($a=4/\tau_0$ and $10/\tau_0$), respectively.
We clearly observe that after about 30 collisions per particle (cooling states) or 20 collisions per particle (heating states) both $a^*$ and $\eta^*$ have reached their stationary values.
Figure \ref{fig6} shows that, for each value of $a$, the full temporal evolution of
$a^*\propto T^{-1/2}$ is practically independent of the
initial condition, especially in the case $a=0.01/\tau_0$. This is due to the fact that for these low values of $a\tau_0$ the
viscous heating term $-2aP_{xy}/dn$ in Eq.\ \eqref{5} can be neglected versus the inelastic cooling term $\zeta T$ for short times,
so the temperature initially evolves as in the homogeneous
cooling state (decaying practically exponentially with the number of
collisions), hardly affected by the details of the initial state. On the other hand, the first stage in the evolution of the reduced viscosity $\eta^*$ is widely dependent on the type of initial condition, as expected from the values of $\widetilde{P}_{xy,0}$ shown in Table \ref{table2}. In the heating cases, Fig.\ \ref{fig7} shows that the evolution followed by both $a^*$ and $\eta^*$ is distinct for each initial condition, except when the steady state is practically reached.
\begin{figure}
\includegraphics[width=.9 \columnwidth]{USF_Param_cooling.eps}
\caption{(Color online) (a) Reduced viscosity $\eta^*$, (b) first viscometric function $-\Psi_1^*$, and (c) second viscometric function $\Psi_2^*$ versus the reduced shear rate $a^*$ for the USF with $\alpha=0.5$ in the cooling states
$a=0.01 /\tau_0$ [blue (dark gray) lines] and $a=0.1/\tau_0$ [orange (light gray) lines]. The legend refers to the five initial conditions considered. The circles represent the steady-state points $(a_s^*,\eta_s^*)$, $(a_s^*,-\Psi_{1,s}^*)$, and $(a_s^*,\Psi_{2,s}^*)$, respectively. The dotted vertical line denotes the value $a_h^*=0.4$ above which the curves collapse to a common one.
\label{fig8}}
\end{figure}
\begin{figure}
\includegraphics[width=.9 \columnwidth]{USF_Param_heating.eps}
\caption{(Color online) (a) Reduced viscosity $\eta^*$, (b) first viscometric function $-\Psi_1^*$, and (c) second viscometric function $\Psi_2^*$ versus the reduced shear rate $a^*$ for the USF with $\alpha=0.5$ in the heating states
$a=10 /\tau_0$ [blue (dark gray) lines] and $a=4/\tau_0$ [orange (light gray) lines]. The legend refers to the five initial conditions considered. The circles represent the steady-state points $(a_s^*,\eta_s^*)$, $(a_s^*,-\Psi_{1,s}^*)$, and $(a_s^*,\Psi_{2,s}^*)$, respectively. The dotted vertical line denotes the value $a_h^*=1.25$ below which the curves collapse to a common one.
\label{fig9}}
\end{figure}
In any case, the interesting point is whether an unsteady hydrodynamic regime is established \emph{prior} to the steady state.
If so, a parametric plot of $\eta^*$ versus $a^*$ must approach a well-defined function $\eta^*(a^*)$, regardless of the initial
condition. Figures \ref{fig8} and \ref{fig9} present such a parametric plot, also for the viscometric functions, for the cooling and heating states, respectively.
We observe that, for each class of states (either cooling or heating) , the 10 curves are \emph{attracted} to a common smooth ``universal'' curve, once the kinetic stage (characterized by strong variations, especially in the case of the second viscometric function for the cooling states) is over. One can safely say that the hydrodynamic regime extends to the range $0.4\lesssim a^*\leq a^*_s$ for the considered cooling states and to the range $1.25\gtrsim a^*\geq a_s^*$ for the considered heating states. We will denote the above threshold values of $a^*$ by $a_h^*$. It is expected that $a_h^*$ depends on the initial value $a_0^*$ (apart from a weaker dependence on the details of the initial distribution); in fact, Figs.\ \ref{fig8} and \ref{fig9} show that the hydrodynamic regime is reached at a value $a_h^*<0.4$ by the states with $a=0.01/\tau_0$ and at a value $a_h^*>1.25$ by the states with $a=10/\tau_0$. Here, however, we adopt a rather conservative criterion and take a common value $a_h^*=0.4$ for $a=0.01/\tau_0$ and $0.1\tau_0$ and a common value $a_h^*=1.25$ for $a=4/\tau_0$ and $10\tau_0$. It is also quite apparent from Figs.\ \ref{fig8} and \ref{fig9} that the collapse to a common curve takes place earlier for $\eta^*$ than for $\Psi_1^*$, $\Psi_2^*$ being the quantity with the largest ``aging'' period.
\begin{table}
\caption{This table shows, for each value of the coefficient of restitution $\alpha$ and each value of $a\tau_0$, the duration of the aging period toward the unsteady hydrodynamic regime and the total duration of the transient period toward the steady state, both measured by the number of collisions per particle. The aging period is defined as the number of collisions per particle needed to reach a common threshold value $a_h^*$ above (below) which the hydrodynamic regime is established for the cooling (heating) states. The table also includes the stationary value $a_s^*$ of the reduced shear rate.\label{table3}}
\begin{ruledtabular}
\begin{tabular}{cccccc}
$\alpha$&$a_s^*$&$a\tau_0$&$a_h^*$&Aging period&Transient period\\
\hline
$0.5$&$0.92$&$0.01$&$0.4$&$15$&$30$\\
&&$0.1$&$0.4$&$5$&$20$\\
&&$4$&$1.25$&$5$&$20$\\
&&$10$&$1.25$&$5$&$20$\\
\hline
$0.7$&$0.68$&$0.01$&$0.35$&$20$&$50$\\
&&$0.1$&$0.35$&$8$&$30$\\
&&$4$&$1$&$5$&$30$\\
&&$10$&$1$&$5$&$30$\\
\hline
$0.9$&$0.37$&$0.01$&$0.2$&$50$&$100$\\
&&$0.1$&$0.2$&$10$&$60$\\
&&$4$&$0.8$&$5$&$50$\\
&&$10$&$0.8$&$5$&$50$\\
\end{tabular}
\end{ruledtabular}
\end{table}
{}From Fig.\ \ref{fig6} it can be seen that the value $a_h^*=0.4$ is reached after about $5$ collisions per particle in the states with $a=0.1/\tau_0$ and after about 15 collisions per particle in the states with $a=0.01/\tau_0$. Similarly, Fig.\ \ref{fig7} shows that the value $a_h^*=1.25$ is reached after about $5$ collisions per particle in the states with $a=4/\tau_0$ and $a=10/\tau_0$. Given that, as said before, the values $a_h^*=0.4$ and $a_h^*=1.25$ are conservative estimates, we find that, as expected, the duration of the kinetic stage is shorter than the duration of the subsequent hydrodynamic stage, before the steady state is eventually attained.
\begin{figure}
\includegraphics[width=.9 \columnwidth]{USF_hydro.eps}
\caption{(Color online) (a) Reduced viscosity $\eta^*$, (b) first viscometric function $-\Psi_1^*$, and (c) second viscometric function $\Psi_2^*$ versus the reduced shear rate $a^*$ for the USF with, from top to bottom, $\alpha=0.5$, $\alpha=0.7$, and $\alpha=0.9$. In order to focus on the hydrodynamic regime, the curves have been truncated at the values of $a_h^*$ given by Table \protect\ref{table3}. The circles represent the steady-state points. The thin solid lines in panels (a) and (b) represent the predictions of our simplified rheological model, Eqs.\ \protect\eqref{etaUSFPade} and \protect\eqref{Psi1}. Note that the model is unable to predict a non-zero second viscometric function.
\label{fig10}}
\end{figure}
We have observed behaviors similar to those of Figs.\ \ref{fig6}--\ref{fig9}
for the other two coefficients of restitution ($\alpha=0.7$ and $\alpha=0.9$, not shown). Table \ref{table3} gives the values of $a_h^*$ and the duration of the aging and transient periods for the 12 classes of states analyzed. It turns out that the number of
collisions per particle the system needs to lose memory of its initial state is
practically independent of $\alpha$ for the heating states. However, the total duration of the
transient period increases with $\alpha$ \cite{AS07}. In fact, there is no true steady state in the elastic limit $\alpha\to 1$. Therefore, the less inelastic the system, the smaller the fraction of the transient period (as measured by the number of collisions per particle) spent by the heating states in the kinetic regime. In the
cooling cases, both the aging and the transient periods increase with $\alpha$.
Figure \ref{fig10} displays the viscosity $\eta^*(a^*)$ and the viscometric functions $-\Psi_1^*(a^*)$ and $\Psi_2^*(a^*)$ for $\alpha=0.5$, $0.7$, and $0.9$, both for the cooling and the heating states. Here we have focused on the ranges of $a^*$ where the hydrodynamic regime can safely be assumed to hold, namely $0.4\leq a^*\leq 1.25$ for $\alpha=0.5$, $0.35\leq a^*\leq 1$ for $\alpha=0.7$, and $0.2\leq a^*\leq 0.8$ for $\alpha=0.9$. The curves describing the predictions of the simplified rheological model for $\eta^*$, Eq.\ \eqref{etaUSFPade}, and for $\Psi_1^*$, Eq.\ \eqref{Psi1}, are also included. We can see that the curves corresponding to the cooling states ($a^*<a^*_s$) and those corresponding to the heating states ($a^*> a^*_s$) smoothly match at the steady-state point. In the case of the nonlinear viscosity function $\eta^*$, the 10 curves building each branch (cooling or heating) for each value of $\alpha$ exhibit a very high degree of overlapping. Due to fluctuations associated with the normal stress differences, the common hydrodynamic curves for the viscometric functions are much more coarse grained, especially in the case of $\Psi_2^*$, whose magnitude is at least 10 times smaller than that of $\Psi_1^*$. The impact of fluctuations is higher in the cooling branches ($a^*<a_s^*$) than in the heating branches ($a^*>a_s^*$). In fact, the definition of the viscometric functions [see Eqs.\ \eqref{43} and \eqref{43bis}] shows that the signal-to-noise ratio is expected to deteriorate as the reduced shear rate $a^*$ decreases. Since $\eta^*$, $-\Psi_1^*$, and $\Psi_2^*$ decrease with decreasing inelasticity, the role played by fluctuations increase as $\alpha$ increases.
It is also interesting to remark that, despite its simplicity and analytical character, the rheological model described by Eqs.\ \eqref{etaUSFPade} and \eqref{Psi1} describes very well the nonlinear dependence of $\eta^*(a^*)$ and $\Psi_1^*(a)^*$. On the other hand, the simple kinetic model \eqref{n1} does not capture any difference between the normal stresses $P_{yy}$ and $P_{zz}$ in the hydrodynamic regime and, thus, it predicts a vanishing second viscometric function [cf.\ Eq.\ \eqref{Psi2_0}].
Figure \ref{fig10} strongly supports Eq.\ \eqref{10} in the USF, {i.e.},
the existence of well-defined hydrodynamic rheological functions
$P_{ij}^*(a^*)$ [or, equivalently, $\eta^*(a^*)$ and $\Psi_{1,2}^*(a^*)$] acting as ``attractors'' in the evolution of the
pressure tensor $P_{ij}(t|f_0)$, regardless of the initial
preparation $f_0$. The stronger statement \eqref{7} (see Fig.\ \ref{fig1}) is also supported by the simulation results \cite{AS07}.
\subsection{ULF. Inhomogeneous problem}
\begin{figure}
\includegraphics[width=.9 \columnwidth]{ULF_profiles.eps}
\caption{(Color online) Profiles of (a) density, (b) flow velocity, and (c) temperature in the ULF, starting from the initial condition of Eqs.\ \protect\eqref{4.2}--\protect\eqref{4.4}. The curves correspond to (1) $\ot=0.046 \tau_0$ ($\approx 0.03~\text{coll}/\text{part}$), (2) $\ot=0.158 \tau_0$ ($\approx 0.29~\text{coll}/\text{part}$), (3) $\ot=0.306 \tau_0$ ($\approx 0.72~\text{coll}/\text{part}$), and (4) $\ot=0.400 \tau_0$ ($\approx 1.07~\text{coll}/\text{part}$). The dashed lines represent the initial profiles.
\label{fig11}}
\end{figure}
Now we turn to the ULF sketched by Fig.\ \ref{fig3}. By performing the changes of variables \eqref{11}, \eqref{14}, and \eqref{15n} (with $\h=x$), we have numerically solved the Boltzmann equation \eqref{19} by means of the DSMC method. Although Eq.\ \eqref{19} admits homogeneous solutions in the fully developed ULF [see Eq.\ \eqref{22n}], it is worth checking that Eq.\ \eqref{19}, complemented by the periodic boundary conditions \eqref{23}, indeed leads an inhomogeneous initial state toward a (time-dependent) homogeneous state. A similar test was carried out in the case of the USF in Ref.\ \cite{AS05}.
As described in Sec.\ \ref{sec4}, we have considered the highly inhomogeneous initial state given by Eqs.\ \eqref{4.2}--\eqref{4.4} with $a_0=-4/\tau_0$ and $\widetilde{L}=2.5\lambda$, and solved Eq.\ \eqref{19} for a coefficient of restitution $\alpha=0.5$. The instantaneous density, flow velocity, and temperature profiles are plotted in Fig.\ \ref{fig11} at four representative times.
In order to decouple the relaxation to a homogeneous state from the increase of the global temperature (here viscous heating prevails over inelastic cooling), panel (c) of Fig.\ \ref{fig11} displays the ratio $\widetilde{T}/\langle \widetilde{T}\rangle$, with $\langle \widetilde{T}\rangle=\langle \widetilde{p}\rangle/\langle \widetilde{n}\rangle$, where $\langle \widetilde{n}\rangle$ and $\langle \widetilde{p}\rangle$ are the density and hydrostatic pressure, respectively, averaged across the system.
We observe that at $\ot=0.046 \tau_0$ the density, velocity, and temperature profiles are still reminiscent of the initial fields, except in the region $-0.4\lesssim \widetilde{x}/\widetilde{L}\lesssim -0.3$, where the density and the temperature present a maximum and the flow velocity exhibits a local minimum. At $\ot=0.158 \tau_0$ the inhomogeneities are still quite strong, with a widely depopulated region $0.2\lesssim \widetilde{x}/\widetilde{L}\leq 0.5$ of particles practically moving with the local mean velocity (which implies an almost zero temperature). By $\ot=0.306 \tau_0$ the profiles have smoothed out significantly. Finally, the system becomes practically homogeneous at $\ot=0.400 \tau_0$. Thus, the relaxation to the homogeneous state lasts about $1$ collision per particle only. The system keeps evolving to the steady state, {which} requires about $20$ collisions per particle. In fact, $\langle \widetilde{T}\rangle\simeq 58 T_0$ after $1$ collision per particle, while $\langle\widetilde{T}\rangle\simeq 211 T_0$ in the steady state. Recall that $\widetilde{\mathbf{u}}(\widetilde{x},\ot)=0$ translates into $\mathbf{u}(x,t)=a(t)x\mathbf{e}_x$ [see Eq.\ \eqref{21n}].
It is important to remark that the relaxation to the ULF geometry observed in Fig.\ \ref{fig11} does not discard the possibility of spatial instabilities for sufficiently large values of the system size $\widetilde{L}$, analogously to what happens in the USF case \cite{S92,SK94,GT96,K00a,K00b,K01,G06}.
\subsection{ULF. Homogeneous problem}
\begin{figure}
\includegraphics[width=.9 \columnwidth]{ULF_Evol.eps}
\caption{(Color online) (a) Absolute value of the reduced longitudinal rate $|a^*|$ and (b) reduced viscosity $\eta^*$ versus the number of collisions per particle for the ULF with $\alpha=0.5$ in the cooling states
$a_0=-0.01 /\tau_0$ [blue (dark gray) lines] and $a_0=0.01/\tau_0$ (black lines) and in the heating states $a_0=-10/\tau_0$ [orange (light gray) lines]. The legend refers to the five initial conditions considered. The dotted horizontal lines in panel (a) denote the values $a_h^*=\pm 0.08$ and $a_h^*=-0.4$ (see text).
\label{fig12}}
\end{figure}
\begin{figure}
\includegraphics[width=.9 \columnwidth]{ULF_Param.eps}
\caption{(Color online) Reduced viscosity $\eta^*$ versus the absolute value of the reduced longitudinal rate $|a^*|$ for the ULF with $\alpha=0.5$ in the cooling states
$a_0=-0.01 /\tau_0$ [blue (dark gray) lines] and $a_0=0.01/\tau_0$ (black lines) and in the heating states $a_0=-10/\tau_0$ [orange (dark gray) lines]. The legend refers to the five initial conditions considered. The circle represents the steady-state point $(|a_s^*|,\eta_s^*)$. The dotted vertical lines denote the values $a_h^*=\pm 0.08$ and $a_h^*=-0.4$ (see text).
\label{fig13}}
\end{figure}
Now we restrict to the homogeneous ULF problem. The homogeneous Boltzmann equation \eqref{31} wit $\h=x$ has been solved via the DSMC method outlined in Sec.\ \ref{sec4} for $\alpha=0.5$, $0.7$, and $0.9$, starting from the initial conditions \eqref{4.5} and \eqref{4.6} with $a_0=-10/\tau_0$, $-0.01/\tau_0$, and $0.01/\tau_0$. Note that the B1 initial state becomes B3, and vice versa, under the change $v_y\to -v_y$, and so both are formally equivalent in the ULF geometry. As said before, a steady state is only possible if $a_0<0$. Moreover, the choices $a_0=\pm 0.01/\tau_0$ correspond to cooling states, while the choice $a_0=-10/\tau_0$ corresponds to heating states. In the course of the simulations the reduced longitudinal rate $a^*$, Eq.\ \eqref{3.5}, and the reduced nonlinear viscosity $\eta^*$, Eq.\ \eqref{Pxx-Pyy}, are evaluated.
As in the USF case, let us adopt $\alpha=0.5$ to illustrate the behaviors observed. Figure \ref{fig12} displays the time evolution of $|a^*|$ and $\eta^*$ for the 15 simulated states (5 for each value of $a_0$). The states with negative $a_0$ become stationary after about $20$ collisions per particle, a value comparable to what we observed in the USF case for $\alpha=0.5$ (see Table \ref{table3}). As for the states with $a_0=0.01/\tau_0$, $a^*$ monotonically increases and $\eta^*$ monotonically decreases (except for a possible transient stage) without bound. It is worth noticing that the first stage of evolution (up to about $7$ collisions per particle) of $|a^*|$ and $\eta^*$ for the initial condition B0 (B2) with $a_0=-0.01/\tau_0$ is very similar to those for the initial condition B2 (B0) with $a_0=0.01/\tau_0$.
Eliminating time between $a^*$ and $\eta^*$ one obtains the parametric plot shown in Fig.\ \ref{fig13}. In the cooling states $a_0=\pm 0.01/\tau_0$ we observe that the hydrodynamic regime is reached at approximately $|a_h^*|=0.08$. {}From Fig.\ \ref{fig12} we see that this corresponds to about seven to eight collisions per particle. The heating states $a_0=-10/\tau_0$ deserve some extra comments. In those cases the time evolution is so rapid that, strictly speaking, the collapse of the five curves takes place only for $a_s^*\geq a^*\geq a_h^*=-0.4$, {which} corresponds to an aging period of about three to four collisions per particle (see Fig.\ \ref{fig12}). On the other hand, it can be clearly seen from Fig.\ \ref{fig13} that the three curves corresponding to the initial states B0, B1, and B3 have collapsed much earlier and are not distinguishable on the scale of the figure. It seems that the isotropic local equilibrium initial state A and the highly artificial anisotropic initial state B2 require a longer kinetic stage than in the cases of the initial states B0, B1, and B3.
{While the relatively slower convergence of the initial condition B2 can be expected because of its associated ``incorrect'' negative viscosity, it seems paradoxical that the local equilibrium initial condition A also relaxes more slowly than the initial conditions B1 and B3, the three of them having a zero initial viscosity. This might be due to the isotropic character of the local equilibrium distribution, in contrast to the high anisotropy of conditions B1 and B3.}
In what follows we will discard the initial conditions A and B2 for $a_0=-10/\tau_0$ and assume that the states starting from the initial conditions B0, B1, and B3 have already reached the hydrodynamic stage for say $a^*\geq -2$. A stricter limitation to $a^*>-0.4$ would miss the interesting maximum of $\eta^*$ at $a^*<a_s^*$ predicted by the BGK-like kinetic model (see Fig.\ \ref{fig5}). In any case, as observed in Fig.\ \ref{fig13}, the behavior of the curves with $a_0=-10/\tau_0$ corresponding to the initial conditions B2 and, especially, A is very close to the one corresponding to the initial conditions B0, B1, and B3.
\begin{figure}
\includegraphics[width=.9 \columnwidth]{ULF_hydro.eps}
\caption{(Color online) Reduced viscosity $\eta^*$ versus the reduced longitudinal rate $a^*$ for the ULF with, from top to bottom, $\alpha=0.5$, $\alpha=0.7$, and $\alpha=0.9$. The circles represent the steady-state points. The thin dashed lines represent the predictions of our simplified rheological model, Eq.\ \protect\eqref{etaULF}.
\label{fig14}}
\end{figure}
The analysis for the cases $\alpha=0.7$ and $0.9$ is similar to the one for $\alpha=0.5$ and, thus, it is omitted here. As in the USF (see Table \ref{table3}), the main effect of increasing $\alpha$ is to slow down the dynamics: the steady state (if $a_0<0$) is reached after a larger number of collisions and the hydrodynamic stage requires a longer period.
The $a^*$-dependence of $\eta^*$ for the three values of $\alpha$ is shown in Fig.\ \ref{fig14}, where we have focused on the intervals $-2\leq a^*\leq a_s^*$ for $a_0=-10/\tau_0$ (initial conditions B0, B1, and B3), $-0.08\geq a^*\geq a_s^*$ for $a_0=-0.01/\tau_0$ (initial conditions A and B0--B3), and $ a^*\geq 0.08$ for $a_0=0.01/\tau_0$ (initial conditions A and B0--B3). Analogously to the case of Fig.\ \ref{fig10}, it can be seen that the heating and cooling branches with negative $a_0$ smoothly match at the steady state. Moreover, the cooling branch with positive $a_0$ is a natural continuation of the cooling branch with negative $a_0$, even though the zero longitudinal rate $a^*=0$ represents a repeller in the time evolution of both branches. Figure \ref{fig14} also includes the predictions of the rheological model \eqref{etaULF}. The agreement with the simulation results is quite satisfactory, although the model tends to underestimate the maxima. This discrepancy is largely corrected by the true numerical solution of Eq.\ \eqref{3.12} (see Fig.\ \ref{fig5}). However, as done in Fig.\ \ref{fig10}, we prefer to keep the rheological model due to its explicit and analytical character.
\section{Conclusions\label{sec6}}
In this paper we have investigated whether the scenario of \emph{aging to
hydrodynamics} depicted in Fig.\ \ref{fig1} for conventional gases applies to granular gases as well, even at
high dissipation. Here the term \emph{hydrodynamics} means that the velocity distribution function, and, hence, the irreversible fluxes, is a functional of the hydrodynamic fields (density, flow velocity, and granular temperature) and, thus, it is not limited to the NS regime. To address the problem, we have restricted ourselves to unsteady states in two classes of flows, the USF and the ULF (see Figs.\ \ref{fig2} and \ref{fig3}). While the USF is an incompressible flow ($\nabla\cdot\mathbf{u}=0$) and the ULF is a compressible one ($\nabla\cdot\mathbf{u}\neq 0$), they share some physical features (uniform density, temperature, and rate of strain tensor) that allow for a unified theoretical framework. Both flows admit heating states (where viscous heating prevails over inelastic cooling) and cooling states (where inelastic cooling overcomes viscous heating), until a steady state is eventually reached. Moreover, in the ULF with positive longitudinal rates only ``super-cooling'' states (where inelastic cooling adds to ``viscous cooling'') are possible and, thus, no steady state exists.
Two complementary routes have been adopted. First, the BGK-like model kinetic equation \eqref{n1} has been used in lieu of the true inelastic Boltzmann equation, which allows one to derive a closed set of nonlinear first-order differential equations [cf.\ Eq.\ \eqref{3.2bis}] for the temporal evolution of the elements of the pressure tensor. A numerical solution with appropriate initial conditions and elimination of time between the reduced pressure tensor $P_{ij}^*$ and the reduced rate of strain $a^*$ provides the \emph{non-Newtonian} functions $P_{ij}^*(a^*)$, from which one can construct the viscosity function $\eta^*(a^*)$ [cf.\ Eq.\ \eqref{eta}] and (only in the USF case) the viscometric functions $\Psi_1^*(a^*)$ [cf.\ Eq.\ \eqref{43}] and $\Psi_2^*(a^*)$ [cf.\ Eq.\ \eqref{43bis}]. The numerical task can be avoided at the cost of introducing approximations. The one we have worked out consists of expanding the solution in powers of a parameter $q$ measuring the hardness of the interaction ($q=0$ for inelastic Maxwell particles and $q=\frac{1}{2}$ for inelastic hard spheres), truncating the expansion to first order, and then constructing Pad\'e approximants. This yields explicit expressions for the rate of strain dependence of the rheological functions. In the USF case the viscosity and the first viscometric functions are given by Eqs.\ \eqref{etaUSFPade} and \eqref{Psi1}, respectively, complemented by Eqs.\ \eqref{15}, \eqref{hUSF}, and \eqref{HUSF}; the second viscometric function vanishes in the BGK-like kinetic model \eqref{n1}. As for the ULF, only one rheological function (viscosity) exists and it is given by Eq.\ \eqref{etaULF}, complemented by Eqs.\ \eqref{15ULF} and \eqref{hULF}. While one could improve the approximation by including terms in the $q$ expansion beyond the first-order one \cite{S00a}, the approximation considered in this paper represents a balanced compromise between simplicity and accuracy (see Figs.\ \ref{fig4} and \ref{fig5}). In fact, the results predicted by the kinetic model \eqref{n1} and the simplified rheological model described by Eqs.\ \eqref{etaULF}, \eqref{etaUSFPade}, and \eqref{Psi1} share the non-Newtonian solution for $q=0$ as well as the steady-state values and the values at $a^*=0$ for arbitrary $q$.
The second, and most important, route has been the numerical solution of the true Boltzmann equation by the stochastic DSMC method for three values of the coefficient of restitution and a wide sample of initial conditions. The most relevant results are summarized in Figs.\ \ref{fig10} (USF) and \ref{fig14} (ULF). Those figures show an excellent degree of overlapping of the rheological functions obtained by starting from the different initial conditions, although the USF viscometric functions may exhibit large fluctuations. The overlapping takes place after a kinetic stage lasting about $5$ collisions per particle for the heating states considered and between $5$ and $50$ collisions per particle for the cooling states considered (see Table \ref{table3}). In the latter states the whole dynamics is slowed down with respect to the heating states, so the increase of the duration of the kinetic stage is correlated with a similar increase of the duration of the subsequent hydrodynamic stage. We have also observed that the characteristic time periods increase as the inelasticity decreases. Figures \ref{fig10} and \ref{fig14} also show that the rheological model, despite its simplicity, captures reasonably well, even at a quantitative level, the main features of the DSMC results. An exception is the USF second viscometric function which, although about 10 times smaller than the first viscometric function, is unambiguously nonzero.
In summary, we believe that our results provide strong extra support to the validity of a hydrodynamic description of granular gases outside the quasielastic limit and the NS regime.
{Of course, it is important to bear in mind that the USF and ULF are special classes of flows where no density or thermal gradients exist, except during the early kinetic stage (see, for instance, Fig.\ \ref{fig11}). Thus, the results presented here do not guarantee \emph{a priori} the applicability of a non-Newtonian hydrodynamic approach for a general situation in the presence of density and thermal gradients. On the other hand, recent investigations for Couette--Fourier flows \cite{BKR09,VSG10,VGS11,VGS11b,TTMGSD01} nicely complement the study presented in this paper in favor of a (non-Newtonian) hydrodynamic treatment of granular gases.}
\acknowledgments
The authors acknowledge support from the Ministerio de Ciencia e Innovaci\'on (Spain) through Grant No.\ FIS2010-16587 and from the Junta de Extremadura (Spain) through Grant No.\ GR10158, partially financed by FEDER (Fondo Europeo de Desarrollo Regional) funds.
\ed
| 33,068 |
\section{Introduction}
The transport of a polymer across a nanopore is vital to many biological processes, such as DNA and RNA translocation through nuclear pores, protein transport across membrane channels and virus injection~\cite{albertsbook}. Due to various potential technological applications such as rapid DNA sequencing, gene therapy and controlled drug delivery~\cite{meller2003}, polymer translocation has received considerable experimental~\cite{meller2003, kasi1996, storm2005} and theoretical interest~\cite{chuang2001,kantor2004, sung1996, muthu1999, dubbeldam2007,vocks2008,sakaue2007,sakaue2008,sakaue2010,saito2011, rowghanian2011,luo2008,luo2009,milchev2011,huopaniemi2006,bhatta2009,bhatta2010,metzler2010,huopaniemi2007,lehtola2008,lehtola2009,gauthier2008a,gauthier2008b,dubbeldam2011}. Of particular technological interest is the case of driven translocation, where the process is facilitated by an external driving force. The key theoretical issue here is to find a unifying physical description that yields the correct dynamical behavior, e.g., the dependence of the translocation time $\tau$ on the chain length $N_0$. Experiments and numerical simulations have indicated that $\tau \propto N_0^\alpha$. However, numerous different values of $\alpha$ have been observed, suggesting explicit dependence on the various physical parameters (cf. Ref.~\cite{milchev2011} for a recent review). Several theories of driven polymer translocation have emerged~\cite{vocks2008,dubbeldam2007,sakaue2007,sakaue2008,sakaue2010,saito2011,rowghanian2011,kantor2004,sung1996,muthu1999,dubbeldam2011}, some claiming agreement with the experimental or numerical results within a certain subset of the physical parameter space. However, to date no single theory has been able to capture the wide range of observed values of $\alpha$, nor quantitatively explain the reason for their dependence on the system's parameters. Therefore, the need for a unifying theory of driven translocation remains.
In Refs.~\cite{sung1996,muthu1999}, polymer translocation was described as a one-dimensional barrier crossing problem of the translocation coordinate $s$ (the length of the subchain on the {\it trans} side). Here, the chain starts from the {\it cis} side with one end inside the pore ($s=0$) and is considered as translocated once $s=aN_0$, with $a$ the segment length. The free-energy due to chain entropy and the chemical potential difference $\Delta\mu$ is
$\mathcal{F}(s)=(1-\gamma')k_BT\ln\left[\frac{s}{a}\left(N_0-\frac{s}{a}\right)\right] +\frac{s}{a}\Delta\mu.$
Here $\gamma'$ is the surface exponent ($\gamma'=0.5,~\approx 0.69,~\approx 0.95$ for an ideal chain, self-avoiding chain in 2D and 3D, respectively), and $k_BT$ is the thermal energy. From $\mathcal{F}(s)$, the Brownian dynamics equation for $s$ in the overdamped limit follows as
$\Gamma\frac{ds}{dt}=(1-\gamma')k_BT\left[ \frac{1}{aN_0-s} -\frac{1}{s} \right] - \frac{\Delta\mu}{a} + \zeta(t).$
Here $\Gamma$ is the (constant) effective friction, and $\zeta(t)$ is Gaussian white noise satisfying $\langle \zeta(t) \rangle=0$ and $\langle \zeta(t)\zeta(t') \rangle = 2\Gamma k_BT\delta(t-t')$. For moderate to large $\Delta\mu$, this model describes translocation at constant mean velocity $\langle ds/dt \rangle = -\Delta\mu / a\Gamma$. However, it is known that the translocation process initially slows down and finally speeds up towards the end~\cite{storm2005, gauthier2008a, gauthier2008b, huopaniemi2006, lehtola2008, lehtola2009}. Qualitatively, this observation has been explained by a simple force-balance argument, where the friction $\Gamma$ depends on the number of moving monomers on the {\it cis} side subchain~\cite{storm2005, lehtola2008, lehtola2009,gauthier2008a, gauthier2008b}. In Refs.~\cite{storm2005,gauthier2008a, gauthier2008b}, it is assumed that the whole subchain on the {\it cis} side is set into motion immediately after the force at the pore is applied. However, this assumption is only valid in the limit of extremely weak driving force, where the {\it cis} side subchain is always at equilibrium.
In most cases, the driving force is substantially larger, implying that even the subchains are out of equilibrium~\cite{lehtola2008, lehtola2009, bhatta2009, bhatta2010}. It has been proposed that in this regime, the out-of-equilibrium dynamics can be described by tension propagation (TP) along the chain backbone, which leads to nontrivial time-dependence of the drag force and gives the non-monotonic translocation velocity~\cite{sakaue2007,sakaue2008,sakaue2010,saito2011,rowghanian2011}. However, this idea has not been quantitatively verified, since most of the studies have considered the asymptotic limit $N_0\rightarrow\infty$, which is out of reach of experiments and numerical simulations.
Therefore, it is imperative to study the TP mechanism for finite $N_0$, which is the regime that is experimentally relevant and where numerical simulation data are available.
To this end, in this work we adopt the TP formalism in context of the Brownian dynamics (BD) equation of motion for $s$ mentioned above, in which we introduce a time-dependent friction coefficient $\Gamma=\Gamma(t)$ that is determined by the TP equations. We introduce a TP formalism for finite chain lengths by incorporating the pore-polymer interactions to the TP equations. We solve the resulting Brownian dynamics -- tension propagation (BDTP) model at finite chain length $N_0$, and validate it through extensive comparisons with molecular dynamics (MD) simulations. We verify that the tension propagation mechanism dominates the dynamics of driven translocation. In addition, we show that the model quantitatively reproduces the numerical values of $\alpha$ in various regimes without any free parameters, explaining the diversity in $\alpha$ as a finite chain length effect. Finally, we address the recent theoretical disagreement between the constant-velocity TP theory of Refs.~\cite{sakaue2007,sakaue2008,sakaue2010,saito2011} and the constant-flux TP theory of Ref.~\cite{rowghanian2011} and show that at the asymptotic limit, $N_0\rightarrow\infty$, $\alpha$ approaches $\alpha=1+\nu$.
\section{Model}
\subsection{General formulation}
We introduce dimensionless units for length, force, time, velocity and friction as $\tilde{s}=s/a$, $\tilde{f}\equiv fa/k_BT$, $\tilde{t}\equiv tk_BT/\eta a^2$, $\tilde{v}\equiv v \eta a / k_BT$ and $\tilde{\Gamma}=\Gamma/\eta$, where $\eta$ is the solvent friction per monomer. In these units, the BD equation reads
\begin{equation}
\tilde{\Gamma}(\tilde{t})\frac{d\tilde{s}}{d\tilde{t}}=(1-\gamma')\left[ \frac{1}{N_0-\tilde{s}} -\frac{1}{\tilde{s}} \right] + \tilde{f} + \tilde{\zeta}(t) \equiv \tilde{f}_\mathrm{tot}, \label{eq:motion_dimless}
\end{equation}
where, for simplicity, we have assumed that the pore length $l_p=a$. Generalization of Eq.~(\ref{eq:motion_dimless}) to different pore lengths is straightforward (see, e.g., Ref~\cite{gauthier2008a}). Eq.~(\ref{eq:motion_dimless}) is, of course, approximative rather than rigorously exact. It contains two approximations, which we will show to be valid by quantitative comparison with MD simulations. First, we postulate that the friction $\tilde{\Gamma}(\tilde{t})$ is determined by TP on the {\it cis} side subchain. While there is no conclusive {\it a priori} reason to neglect the non-equilibrium effects of the {\it trans} side subchain, we will show that those effects are negligibly small in the experimentally and computationally relevant regimes. Second, we note that Eq.~(\ref{eq:motion_dimless}) includes the entropic force, whose form is strictly valid only for small driving forces $\tilde{f}$, when the translocation time $\tilde{\tau}$ is comparable to the Rouse relaxation time~\cite{rouse1953,chuang2001,dehaan2010}. However, for larger $\tilde{f}$, the average contribution of the entropic force to the total force $\tilde{f}_\mathrm{tot}$ is very small (see results below for discussion). Therefore, even for large forces, the model will be shown to give excellent agreement with MD simulations.
The effective friction $\tilde{\Gamma}(\tilde{t})$ actually consists of two contributions. The first one is the drag force of the {\it cis} side subchain that is solved with the TP formalism. The other one is the frictional interaction between the pore and the polymer. Formally, we can write $\tilde{\Gamma}$ as the sum of the {\it cis} side subchain and pore frictions, $\tilde{\Gamma}(\tilde{t})=\tilde{\eta}_\mathrm{cis}(\tilde{t})+\tilde{\eta}_p$. While for $N_0\rightarrow\infty$ the first term dominates, for finite $N_0$ the pore friction can significantly affect the translocation dynamics. We will come back to this issue later, but let us first look at how the time-dependent part of the friction can be determined from the TP formalism. In the special case of extremely large driving force, one can find $\tilde{\eta}_\mathrm{cis}$ directly from the TP equations. More generally, however, it is easier to derive the velocity of the monomers at the pore entrance, $\tilde{v}_0$. In such a case, the effective friction is in a natural way defined as
\begin{equation}
\tilde{\Gamma}(\tilde{t})= \frac{\tilde{f}_\mathrm{tot}}{\tilde{\sigma}_0(\tilde{t})\tilde{v}_0(\tilde{t})}, \label{eq:effective_friction}
\end{equation}
where $\tilde{\sigma}_0$ is the line density of monomers near the pore and $\tilde{\sigma}_0\tilde{v}_0\equiv d\tilde{s}/d\tilde{t}$ is the flux of monomers through the pore entrance. In either case, determining $\tilde{\Gamma}(\tilde{t})$ essentially reduces to calculating the number of moving monomers, whose combined drag force then constitutes the time-dependent part of the friction. As the driving force is applied, the chain begins to move in stages, with the segments closest to the pore being set into motion first. A close analogue is a coil of rope pulled from one end, which first uncoils before starting to move as a whole. To keep track of the moving part of the chain, one defines a {\it tension front}, which divides the chain into the moving part that is under tension, and the nonmoving part outside the front (see Fig.~\ref{fig:configuration}). The front is located at $\tilde{x}=-\tilde{R}(\tilde{t})$, and propagates in time as parts of the chain further away from the pore are set in motion. The last monomer within the tension front is labeled as $N(\tilde{t})$. Using the TP formalism, one can derive an equation of motion for the tension front, using either $\tilde{R}$ or $N$ as the dynamical variable. The details of this calculation can be found in the Appendix.
\begin{SCfigure}
\includegraphics[width=0.48\columnwidth]{cinfs_hif_1b.eps}
\caption{(Color online) A snapshot of a translocating polymer in a stem-flower configuration. A tension front at $\tilde{x}=-\tilde{R}$ (black arc) divides the chain into moving and nonmoving parts, with the last moving monomer labeled as $N$. The number of translocated monomers is $\tilde{s}$. }
\label{fig:configuration}
\end{SCfigure}
\subsection{Different regimes}
Depending on the magnitude of the driving force, the equation of motion for the tension front attains a slightly different form. In the simplest case, when the driving force is very large compared to temperature and chain length, $\tilde{f} \gtrsim N_0^\nu$, the moving part of the chain is almost completely straight. In this {\it strong stretching} (SS) regime, the equation of motion is~\cite{derivation_ss}
\begin{equation}
\frac{dN}{d\tilde{t}}=\frac{\tilde{f}_\mathrm{tot}}{\tilde{\Gamma}(\tilde{t}) \left(1-\nu A_\nu N^{\nu-1} \right)}.\label{eq:tp_ss}
\end{equation}
Here, $\tilde{\Gamma}(\tilde{t})=N(\tilde{t})-\tilde{s}(\tilde{t})+\tilde{\eta}_p$, with $N(\tilde{t})-\tilde{s}(\tilde{t})$ being the number of moving monomers on the {\it cis} side. The Flory exponent $\nu$ and the prefactor $A_\nu$ are related to the end-to-end distance of the polymer, $\tilde{R}_\mathrm{ee}=A_\nu N_0^\nu$. In the SS approximation, Eq.~(\ref{eq:motion_dimless}) is solved simultaneously with Eq.~(\ref{eq:tp_ss}), using $\tilde{\Gamma}$ from Eq.~(\ref{eq:tp_ss}) as a input in Eq.~(\ref{eq:motion_dimless}), and vice versa for $\tilde{s}$.
For slightly smaller driving forces, $1 \lesssim \tilde{f} \lesssim N_0^\nu$, the force is not sufficient to completely straighten the chain. Due to thermal fluctuations, a flower-shaped tail develops (see Fig.~\ref{fig:configuration}). In this {\it stem-flower} (SF) regime, the line density and and velocity of the monomers are not constant in space. Therefore, one also has to solve the density $\tilde{\sigma}_R$ and the velocity $\tilde{v}_R$ near the tension front. As a result, one gets a system of equations,
\begin{align}
&\frac{d\tilde{R}}{d\tilde{t}}=\tilde{v}_{R} \left[ \frac{1}{\nu} A_\nu^{-1/\nu} \tilde{\sigma}_{R}^{-1} \tilde{R}^{1/\nu -1} \right]^{-1}, \label{eq:motionR}\\
&\tilde{\sigma}_R^{1/(\nu-1)}= \frac{\tilde{v}_0\tilde{R}}{\nu b \tanh(b)}\ln\left[\cosh\left(b\frac{\tilde{\sigma}_R^{\nu/(1-\nu)}}{\tilde{R}}\right)\right], \label{eq:sigmaR}\\
&\tilde{v}_R=\tilde{v}_0\frac{\tanh \left(b\tilde{\sigma}_R^{\nu/(1-\nu)} /\tilde{R} \right) }{\tanh(b)}, \label{eq:vR}\\
&\tilde{v}_0\tilde{R}\frac{\ln [\cosh (b)]}{b\tanh(b)}=\left[ \tilde{f}_\mathrm{tot}-\tilde{\eta}_p\tilde{v}_0 \right] + \nu -1, \label{eq:v0_sf}
\end{align}
that can be solved numerically for $\tilde{v}_0$. Here, $b$ is a (fixed) dimensionless parameter related to the spatial dependence of the velocity, and ensures global conservation of mass (see Appendix A). In the SF regime, $\tilde{\sigma}_0=1$, since the stem close to the pore is in a single-file configuration. The effective friction is given by Eq.~(\ref{eq:effective_friction}).
Finally, in the regime where the force insufficient to straighten even a small part of the chain, $\tilde{f} \lesssim N_0^{-\nu}$, the chain adopts a trumpet-like shape. In this {\it trumpet} (TR) regime, the dynamics can be described by Eqs.~(\ref{eq:motionR})--(\ref{eq:vR}), with the velocity $\tilde{v}_0$ and density $\tilde{\sigma}_0$ given by
\begin{align}
&\tilde{v}_0\tilde{R}\frac{\ln [\cosh (b)]}{b\tanh(b)}=\nu \left[ \tilde{f}_\mathrm{tot}-\tilde{\eta}_p\tilde{v}_0 \right]^{1/\nu}, \label{eq:v0_tr}\\
&\tilde{\sigma}_0= \left[ \tilde{f}_\mathrm{tot}-\tilde{\eta}_p\tilde{v}_0 \right]^{1-1/\nu}.
\end{align}
\subsection{Pore friction}
The time evolution of the tension front ($\tilde{R}$ or $N$) gives the contribution of the {\it cis} side subchain to the friction $\tilde{\Gamma}$. To complete the BDTP model, we still need to determine the pore friction $\eta_p$. In general, $\eta_p$ is a complicated function of the pore geometry, but here we restrict our study to the geometries used in our benchmark MD simulations. In order to fix $\eta_p$, we examine the waiting time per monomer $w(\tilde{s})$, defined as the time that the individual monomer spends inside the pore. With $\tilde{f}$ sufficiently large, $\tilde{w}\propto \tilde{\Gamma}/\tilde{f}$. For small $\tilde{s}$, the friction $\tilde{\Gamma}$ is mostly determined by $\tilde{\eta}_p$. Therefore, by comparing the $w(\tilde{s})$ of the BDTP model with MD simulations of Refs.~\cite{huopaniemi2006,luo2009,kaifu_private} for the first few monomers,we have measured $\eta_p^\mathrm{3D}\approx 5$ and $\eta_p^\mathrm{2D}\approx 4$ for the respective pore geometries. It should be noted that $\eta_p$ is fitted only once, as opposed to being done separately for each combination of $\tilde{f},\eta$, etc. Thus, $\eta_p$ is {\it not} a freely adjustable parameter.
\section{Results and discussion}
First, we validate the BDTP model through quantitative comparisons with MD simulations. In Fig.~\ref{fig:wt_N128}, we compare the waiting time $w(\tilde{s})$, which is the most important and sensitive measure of the translocation dynamics. As is shown, the match between BDTP and MD is almost exact. We stress that this agreement tells that the translocation dynamics is reproduced correctly at the most fundamental level and that such an agreement is a vital requirement for any correct theoretical model. The comparison also reveals an extremely lucid picture of the translocation process: first, as tension propagates along the chain, the effective friction increases and translocation slows down. In the second stage, the number of dragged monomers is reduced as the tail retracts, and translocation speeds up.
\begin{figure}
\includegraphics[width=0.75\columnwidth]{wt_N128.eps}
\caption{(Color online) Comparison of waiting times $w$ in both 2D and 3D for MD and the BDTP model. The agreement of the BDTP model with MD simulations is excellent, and reveals the two stages of translocation: the tension propagation stage of increasing $w(\tilde{s})$ and the tail retraction stage characterized by decreasing $w(\tilde{s})$. The parameters used were the same for both MD and BDTP ($N_0=128$, $f=5$, $k_BT=1.2$, $\eta=0.7$). The 3D MD results are from~\cite{kaifu_private}.}
\label{fig:wt_N128}
\end{figure}
\begin{figure}
\includegraphics[width=0.75\columnwidth]{alpha_asym.eps}
\caption{(Color online) The effective exponent $\alpha(N_0)\equiv\frac{d\ln \tau}{d\ln N_0}$ in 2D (circles) and 3D (squares) as a function of $N_0$ from the BDTP model, showing the extremely slow approach to the asymptotic limit $\alpha=1+\nu$. Most of experimental and simulation studies in the literature involve chain lengths of $N_0 \lesssim 10^3$ (shaded region), being clearly in the finite chain length regime. The inset shows the raw data $\tau(N_0)$. Model parameters are the same as in Fig.~\ref{fig:wt_N128}.
}
\label{fig:alpha_asym}
\end{figure}
\begin{table}
\caption{Values of $\alpha$ ($\tau\sim N_0^\alpha$) from the BDTP model as compared to the corresponding values from MD simulations.}
\label{tb:alphas}
\begin{tabular} { l l l }
$\alpha$ (BDTP) &$\alpha$ (MD) & Dimension and parameter values \\
\hline
\hline
& & 2D, $T=1.2$, Ref.~\cite{huopaniemi2006} \\
$1.51 \pm 0.02$ & $1.50 \pm 0.01$ & $f=5.0$, $\gamma=0.7$, $20 \leq N_0 \leq 70$ \\
$1.71 \pm 0.02$ & $1.69 \pm 0.04$ & $f=5.0$, $\gamma=0.7$, $500 \leq N_0 \leq 800$ \\
$1.52 \pm 0.02$ & $1.50 \pm 0.02$ & $f=2.4$, $\gamma=0.7$, $20 \leq N_0 \leq 70$ \\
$1.71 \pm 0.02$ & $1.65 \pm 0.04$ & $f=2.4$, $\gamma=0.7$, $500 \leq N_0 \leq 800$ \\
$1.66 \pm 0.02$ & $1.64 \pm 0.01$ & $f=5.0$, $\gamma=3.0$, $20 \leq N_0 \leq 70$ \\
$1.71 \pm 0.02$ & $1.67 \pm 0.03$ & $f=5.0$, $\gamma=3.0$, $500 \leq N_0 \leq 800$ \\
\hline
& & 3D, $T=1.2$, Ref.~\cite{luo2009} \\
$1.59 \pm 0.02$ & $1.58 \pm 0.03$ & $f=0.5$, $\gamma=0.7$, $16 \leq N_0 \leq 128$ \\
$1.35 \pm 0.02$ & $1.37 \pm 0.05$ & $f=5.0$, $\gamma=0.7$, $16 \leq N_0 \leq 256$ \\
$1.34 \pm 0.02$ & $1.37 \pm 0.02$ & $f=10.0$, $\gamma=0.7$, $16 \leq N_0 \leq 256$ \\
\hline
& & 3D, $T=1.2$, Ref.~\cite{luo2008} \\
$1.41 \pm 0.01$ & $1.42 \pm 0.01$ & $f=5.0$, $\gamma=0.7$, $40 \leq N_0 \leq 800$ \\
$1.39 \pm 0.02$ & $1.41 \pm 0.01$ & $f=5.0$, $\gamma=0.7$, $64 \leq N_0 \leq 256$ \\
\hline
& & 3D, $T=1.0$, Ref.~\cite{lehtola2008} \\
$1.46 \pm 0.02$ & $1.47 \pm 0.05$ & $f=3.0$, $\gamma=11.7$, $70 \leq N_0 \leq 200$ \\
$1.49 \pm 0.02$ & $1.50 \pm 0.01$ & $f=30.0$, $\gamma=11.7$, $200 \leq N_0 \leq 800$ \\
\hline
\end{tabular}
\end{table}
Next, we compare the exponents $\alpha$ obtained from the BDTP model with the corresponding numerical values from MD simulations. The parameter range has been chosen to span the TR, SF and SS regimes, and to cover both short and long chain regimes in both 2D and 3D. The results are shown in Table I. The diversity of $\alpha$ in these regimes is evident, yet in all of them, the BDTP model is accurate to three significant numbers within the margin of error. This clearly shows, that while the values of $\alpha$ depend on several parameters, they all share a common physical basis: non-equilibrium tension propagation on the {\it cis} side subchain. Then why is the exponent $\alpha$ not universal? The answer lies in the chain length regimes studied both in experiments and simulations. Typically, $N_0\lesssim 10^3$. However, in this regime $\alpha$ is not independent of the chain length! As shown in Fig.~\ref{fig:alpha_asym}, $\alpha$ retains a fairly strong dependence on $N_0$ up to $N_0 \approx 10^4$ in 2D and $N_0 \approx 10^5$ in 3D. Therefore, {\it the observed scatter in $\alpha$ is a finite chain length effect}, a fact that has been mostly ignored in the literature.
Two additional remarks about the results of Table I are in order. First, the effect of the entropic term in Eq.~(\ref{eq:motion_dimless}) on $\alpha$ is extremely small. To show this, we solved Eq.~(\ref{eq:motion_dimless}) also without the entropic term. The results match exactly with those given in Table I, except for the low force case $f=0.5$, $T=1.2$ in 3D, where, without the entropic term, $\alpha=1.56$ instead of 1.59. Second, regarding $f$ and $N_0$, the BDTP model gives two general trends for $\alpha$: i) for a fixed $f$, $\alpha$ increases with $N_0$ (as shown in Fig.~\ref{fig:alpha_asym}) and ii) for a fixed $N_0$, $\alpha$ decreases with $f$, as shown in Table I for 2D and 3D (Refs.~\cite{huopaniemi2006,luo2009}). For $f/k_BT \ll 1$, this trend is consistent with the value of $\alpha$ in absence of $f$~\cite{luo2008}. For $f/k_BT \gg 1$, $\alpha$ is almost independent of $f$. Therefore, the increase of $\alpha$ in the last two lines of Table I (Ref.~\cite{lehtola2008}) is in fact due to increase in $N_0$, not in $f$.
Finally, we have estimated the asymptotic value of $\alpha$ in the SF regime by solving the BDTP model up to $N_0=10^{10}$ (Fig.~\ref{fig:alpha_asym}). In 2D, the numerical estimate is $\alpha^\mathrm{2D}_\infty \approx 1.750\pm 0.001$, for $N_0\gtrsim 10^9$, and, in 3D, $\alpha^\mathrm{3D}_\infty \approx 1.590\pm 0.002$, for $N_0\gtrsim 10^9$. In both cases, we recover the value $1+\nu$ as predicted in Ref.~\cite{rowghanian2011} with the constant-flux TP theory, and recently also using a different approach~\cite{dubbeldam2011}. However, the value is different from Sakaue's original prediction of $\frac{1+\nu+2\nu^2}{1+\nu}$~\cite{sakaue2007,sakaue2010,saito2011}. The reason for the different predictions is that in both Refs.~\cite{rowghanian2011, dubbeldam2011}, and in our model, the number of monomers is globally conserved, whereas in Refs.~\cite{sakaue2007,sakaue2010,saito2011} the conservation is guaranteed only locally in the neighborhood of $\tilde{x}=-\tilde{R}$. Therefore, asymptotically, $\alpha_\infty = 1+\nu$ in both 2D and 3D, also in agreement with the prediction of Ref.~\cite{kantor2004}.
\section{Conclusions}
To summarize, we have introduced a new theoretical model of driven polymer translocation that has only two degrees of freedom and no free parameters. The model gives near-exact agreement with high-accuracy molecular dynamics simulations in a wide range of parameters. Our study shows that the dynamics of driven translocation is dictated by non-equilibrium tension propagation on the {\it cis} side subchain. The model also reveals that the majority of experiments and simulations in the literature are performed in the regime, where finite chain length effects have significant impact on the translocation dynamics. Although mostly overlooked in the literature, this is an important observation, since the finite chain length effects persists for chain lengths of at least several tens of thousands of monomers. Therefore, in most studies of polymer translocation, finite chain length effects cannot be neglected. This fact is also vital for the theoretical study of driven polymer translocation.
\acknowledgments
This work has been supported in part by the Academy of Finland through its COMP Center of Excellence and Transpoly Consortium grant. TI acknowledges the financial support of FICS and TES. The authors also wish to thank CSC, the Finnish IT center for science, for allocation of computer resources.
| 7,650 |
\section{Introduction}
Recently the D0 collaboration has measured a deviation
from the standard model (SM) prediction in the CP violating like-sign dimuon
charge asymmetry in semileptonic $b$ hadron decay
with the 9 fb$^{-1}$ integrated luminosity of $p \bar{p}$ data
at Tevatron \cite{d0new}:
\begin{equation} \label{asym_exp}
A_{sl}^b = -0.00787 \pm 0.00172~({\rm stat.}) \pm 0.00093~({\rm syst.}) .
\end{equation}
The like-sign dimuon events
comes from direct semileptonic decays
of one of $b$ hadrons and a semileptonic decay of the other $b$ hadron
following the $B^0-\bar{B}^0$ oscillation in $b \bar{b}$ pair production at Tevatron,
defined by
\begin{equation}
A_{sl}^b \equiv \frac{ \Gamma(b \bar{b} \to \mu^+ \mu^+ X)
- \Gamma(b \bar{b} \to \mu^- \mu^- X)}
{ \Gamma(b \bar{b} \to \mu^+ \mu^+ X)
+ \Gamma(b \bar{b} \to \mu^- \mu^- X)}.
\end{equation}
At Tevatron experiment,
both decays of $B_d$ and $B_s$ mesons contribute to the asymmetry.
If we define the charge asymmetry of semileptonic decays of neutral $B^0_q$ mesons as
\begin{equation}
a_{sl}^q \equiv \frac{\Gamma(\bar{B}^0_q(t) \to \mu^+ X)
- \Gamma(B^0_q(t) \to \mu^- X)}
{\Gamma(\bar{B}^0_q(t) \to \mu^+ X)
+ \Gamma(B^0_q(t) \to \mu^- X)},
\end{equation}
the like-sign dimuon charge asymmetry can be expressed
in terms of $a_{sl}^q$ as \cite{grossman}
\begin{equation} \label{Aslb}
A_{sl}^b = \frac{1}{f_d Z_d + f_s Z_s}
\left( f_d Z_d a_{sl}^d + f_s Z_s a_{sl}^s \right),
\end{equation}
assuming that $\Gamma(B_d^0 \to \mu^+ X)=\Gamma(B_s^0 \to \mu^+ X)$ to a very good approximation,
where $f_q$ are the production fractions of $B_q$ mesons, and
$Z_q = 1/(1-y_q^2)-1/(1+x_q^2)$
with $y_q = \Delta \Gamma_q/(2 \Gamma_q)$, $x_q=\Delta M_q/\Gamma_q$.
These parameters are measured to be
$f_d = 0.402 \pm 0.013$,
$f_s = 0.112 \pm 0.013$,
$x_d = 0.771 \pm 0.007$,
$x_s = 26.3 \pm 0.4$,
and
$y_d = 0$,
$y_s = 0.052 \pm 0.016$
\cite{HFAG}.
With these values, Eq. (\ref{Aslb}) is rewritten by
\begin{equation}
A_{sl}^b = (0.572 \pm 0.030) a_{sl}^d + (0.428 \pm 0.030) a_{sl}^s.
\end{equation}
The non-zero dimuon asymmetry implies
a difference between the $B^0 \leftrightarrow \bar{B}^0$ transitions
and the CP violation in the $B$ system.
In the SM, the source of the CP violation
in the neutral $B^0_q$ system is the phase of
the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements involved
in the box diagram.
Using the SM values for the semileptonic charge asymmetries
$a^d_{sl}$ and $a^s_{sl}$ of $B^0_d$ and $B^0_s$ mesons,
repectively \cite{Lenz07},
the prediction of the dimuon asymmetry in the SM is given by
\begin{equation}
A_{sl}^b = (-2.7^{+0.5}_{-0.6}) \times 10^{-4},
\label{Asl_SM}
\end{equation}
which shows that the D0 measurement of Eq. (1) deviates
about 3.9 $\sigma$ from the SM prediction.
If the deviation is confirmed with other experiments,
it indicates the existence of the new physics beyond the SM.
Recently several works are devoted to explaning
the D0 dimuon asymmetry in the SM and beyond \cite{dimuon}.
As an alternative model solution to the mismatch
between the measurement and the SM prediction of the dimuon charge asymmetry,
we consider the left-right model (LRM)
based on the SU(2)$_L \times$ SU(2)$_R \times$ U(1) gauge symmetry
which is one of the attractive extensions of the SM \cite{LRM}.
The current measurement of the dimuon charge asymmetry
can be explained in the LRM due to
the sizable right-handed current contributions to $B^0-\bar{B}^0$ mixing
\cite{Nam02}.
The manifest left-right symmetry provides an natural answer
to the origin of the parity violation.
Involving the triplet Higgs field $\Delta_{L,R}$
to break the additinal SU(2)$_R$ symmetry,
the lepton number violating Yukawa terms are introduced
and the see-saw mechanism for light neutrino masses is exploited in the LRM.
This model arises as an intermediate theory
in the SO(10) grand unified theory (GUT).
In the LRM, the right-handed fermions transform as doublets
under SU(2)$_R$ and singlets under SU(2)$_L$,
and the left-handed fermions behave reversely.
Thus a bidoublet Higgs field is required for the Yukawa couplings
and also responsible for the electoweak symmetry breaking (EWSB).
The scale of the masses of the new gauge bosons in the LRM
is constrained by direct searches and indirect analysis
\cite{czagon,cheung,chay,erler},
and we will discuss the constraints on the model in further detail.
This paper is organized as follows.
In section 2, we briefly review the charged sector in the general LRM.
We explicitly show the right-handed current contributions
in the neutral $B$ meson system in section 3,
and present the numerical analysis of $B^0-\bar{B}^0$ mixing
and the dimuon charge asymmetry of $B$ mesons in the general LRM in section 4.
Finally we conclude in section 5.
\section{The left-right model}
We briefly review the main features of the LRM, which are necessary for our analysis.
The gauge group of the left-right symmetric model is
SU(2)$_L \times$SU(2)$_R \times$U(1).
There exist a bidoublet Higgs field $\phi (2, \bar{2}, 0)$
and two triplet Higgs fields,
$\Delta_L (3,1,2)$ and $\Delta_R (1,3,2)$
in the minimal LR model
represented by
\begin{equation}
\phi =
\left(
\begin{array}{cc}
\phi_1^0 & \phi_1^+ \\
\phi_2^- & \phi_2^0 \\
\end{array}
\right),
~~~~~~~~~~~~
\Delta_{L,R} =
\frac{1}{\sqrt{2}}
\left(
\begin{array}{cc}
\delta^+_{L,R} & \sqrt{2} \delta^{++}_{L,R} \\
\sqrt{2} \delta^0_{L,R} & -\delta^{+}_{L,R} \\
\end{array}
\right) ,
\end{equation}
of which kinetic terms are given by
\begin{equation}
{\cal L} = {\bf Tr} \left[ ( D_\mu \Delta_{L,R} )^\dagger
( D^\mu \Delta_{L,R} )\right]
+ {\bf Tr} \left[ (D_\mu \phi)^\dagger
(D^\mu \phi) \right],
\end{equation}
where the covariant derivatives are defined by
\begin{eqnarray}
D_\mu \phi &=& \partial_\mu \phi - i \frac{g_L}{2} W^a_{L \mu} \tau^a \phi
+ i\frac{g_R}{2} \phi W^a_{R \mu} \tau^a ,
\nonumber\\
D_\mu \Delta_{L,R} &=& \partial_\mu \Delta_{L,R}
- i\frac{g_{L,R}}{2} \left[W^a_{L,R \mu} \tau^a , \Delta_{L,R} \right]
- i g^\prime B_\mu \Delta_{L,R}.
\end{eqnarray}
The gauge symmetries are spontaneously broken by
the vacuum expectation values (VEV)
\begin{equation}
\langle \phi \rangle = \frac{1}{\sqrt{2}}\left(
\begin{array}{cc}
k_1&0 \\
0&k_2 \\
\end{array}
\right),
~~~~~~~~~~~~
\langle \Delta_{L,R} \rangle = \frac{1}{\sqrt{2}}\left(
\begin{array}{cc}
0&0 \\
v_{L,R}&0 \\
\end{array}
\right),
\end{equation}
where $k_{1,2}$ are complex in general
and $v_{L,R}$ are real, which lead to the charged gauge boson masses
\begin{equation}
{\cal M}^2_{W^\pm} = \frac{1}{4} \left(
\begin{array}{cc}
g_L^2(k_+^2 + 2v_L^2) & -2 g_L g_R k_1^\ast k_2 \\[3pt]
-2g_L g_R k_1 k_2^\ast & g_R^2 (k_+^2 +2v_R^2) \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
M_{W_L}^2 & M_{W_{LR}}^2 e^{i\alpha} \\[3pt]
M_{W_{LR}}^2 e^{-i\alpha} & M_{W_R}^2 \\
\end{array}
\right) ,
\end{equation}
where $k_+^2 = |k_1|^2 + |k_2|^2$ and $\alpha$ is the phase of $k_1^\ast k_2$.
Since the SU(2)$_{\rm R}$ breaking scale $v_R$
should be higher than the electroweak scale,
$k_{1,2} \ll v_R$, $W_R$ is heavier than $W_L$.
Note that $v_L$ is irrelevant for the symmetry breaking
and just introduced in order to manifest the left-right symmetry.
If the neutrino mass is purely determined by the see-saw relation
$m_\nu \sim v_{L}+k^2_{+}/v_{R}$,
$v_R$ is typically very large $\sim 10^{11}$ GeV.
It indicates that the heavy gauge bosons are too heavy
to be produced at the accelerator experiments
and the direct search of the SU(2)$_R$ structure is hardly achieved.
Therefore we assume that $v_R$ is only moderately large,
$v_R \sim {\cal O} ({\rm TeV})$,
for the heavy gauge bosons to be founnd at the LHC,
and the Yukawa couplings are suppressed
in order that the neutrino masses are at the eV scale.
We let $v_L$ be very small or close to 0 without loss of generality.
This is achieved when the quartic couplings of
$(\phi \phi \Delta_L \Delta_R)$-type terms in the Higgs potential
are set to be zero \cite{gunion2,kiers}
and warranted
by the approximate horizontal U(1) symmetry \cite{khasanov}
as well as the see-saw picture for light neutrino masses.
We adopt this limit here and note that the Higgs boson masses are
not affected by taking this limit \cite{kiers}.
The general Higgs potential in the LRM has been studied in Refs.
\cite{gunion1,gunion2,kiers}.
After the mass matrix is diagonalized by a unitary transformation,
the mass eigenstates are written as
\begin{equation}
\left(
\begin{array}{c}
W^\pm \\
W'^\pm \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
\cos{\xi} & e^{-i\alpha}\sin{\xi} \\
-\sin{\xi} & e^{-i\alpha}\cos{\xi} \\
\end{array}
\right)
\left(
\begin{array}{c}
W_L^\pm \\
W_R^\pm \\
\end{array}
\right),
\end{equation}
with the mixing angle
\begin{equation}
\tan{2\xi} = -\frac{2 M_{W_{LR}}^2}{M_{W_R}^2 - M_{W_L}^2} .
\end{equation}
For $v_R \gg |k_{1,2}|$, the mass eigenvalues and the mixing angle reduce to
\begin{equation}
M^2_W \approx \frac{1}{4}g^2_L (|k_1|^2+|k_2|^2), \quad
M^2_{W^\prime} \approx \frac{1}{2}g^2_Rv_R^2, \quad
\xi \approx \frac{g_L|k_1^*k_2|}{g_Rv_R^2}.
\end{equation}
Here, the Schwarz inequality requires that
$\zeta_g \equiv (g_R/g_L)^2\zeta \geq \xi_g \equiv (g_R/g_L)\xi$
where $\zeta \equiv M_W^2/M_{W'}^2 $.
From the global analysis of muon decay measurements
\cite{MacDonald08},
the lower bound on $\zeta_g$ can be obtained without imposing discrete symmetry as follows:
\begin{equation}
\zeta_g < 0.031 \qquad \textrm{or} \qquad M_{W^\prime} > (g_R/g_L) \times 460\
\textrm{GeV} .
\label{MWRbound}
\end{equation}
The new gauge boson mass $M_{W'}$ is severly constrained from $K_L - K_S$ mixing
if the model has manifest ($V^R = V^L$) left-right symmetry ($g_R = g_L$):
$M_{W'} > 2.5$ TeV \cite{Zhang07},
where $V^L(V^R)$ is the left(right)-handed quark mixing matrice.
But, in general, the form of $V^R$ is not necessarily restricted
to manifest or pseudomanifest ($V^R = V^{L*}K$) symmetric type,
where $K$ is a diagonal phase matrix \cite{LRM}.
Instead, if we take the following form of $V^R$,
the limit on $M_{W'}$ may be significantly relaxed to approximately 300 GeV,
and the $W'$ boson contributions to $B_{d(s)}\bar{B}_{d(s)}$ mixings can be large
\cite{Langacker89}:
\begin{equation}
V^R_I = \left( \begin{array}{ccc} e^{i\omega} & \sim 0 & \sim 0 \\
\sim 0 & c_R e^{i\alpha_1} & s_R e^{i\alpha_2} \\
\sim 0 & -s_R e^{i\alpha_3} & c_R e^{i\alpha_4} \end{array} \right) ,\quad
V^R_{II} = \left( \begin{array}{ccc} \sim 0 & e^{i\omega} & \sim 0 \\
c_R e^{i\alpha_1} & \sim 0 & s_R e^{i\alpha_2} \\
-s_R e^{i\alpha_3} & \sim 0 & c_R e^{i\alpha_4} \end{array} \right) ,
\label{UR}
\end{equation}
where $c_R\ (s_R)\equiv \cos\theta_R\ (\sin\theta_R)$ $(0^\circ \leq \theta_R \leq 90^\circ )$.
Here the matrix elements indicated $\sim 0$ may be $\lesssim 10^{-2}$ and the unitarity
requires $\alpha_1+\alpha_4=\alpha_2+\alpha_3$.
From the $b\rightarrow c$ semileptonic decays of the $B$ mesons,
we can get an approximate bound $\xi_g\sin\theta_R \lesssim 0.013$
by assuming $|V^L_{cb}|\approx 0.04$ \cite{Voloshin97}.
\section{$B^0-\bar{B^0}$ mixing}
The neutral $B_q$ meson system $(q=d,s)$
is described by the Schr\"odinger equation
\begin{equation}
i \frac{d}{dt} \left(
\begin{array}{c}
B_q(t) \\
\bar{B}_q(t) \\
\end{array}
\right)
= \left( M - \frac{i}{2} \Gamma \right)
\left(
\begin{array}{c}
B_q(t) \\
\bar{B}_q(t) \\
\end{array}
\right),
\end{equation}
where $M$ is the mass matrix and $\Gamma$ the decay matrix.
The $\Delta B = 2$ transition amplitudes
\begin{equation}
\langle B_q^0 | {\cal H}_{\rm eff}^{\Delta B = 2} | \bar{B}_q^0 \rangle
= M_{12}^q,
\label{M12}
\end{equation}
yields the mass difference between the heavy and the light states of $B$ meson,
\begin{equation}
\Delta M_q \equiv M_H^{q} - M_L^{q} = 2 | M_{12}^q |,
\end{equation}
where $M_H^{q}$ and $M_L^{q}$
are the mass eigenvalues for the heavy and the light eigenstates, respectively.
The decay width difference is defined by
\begin{equation}
\Delta \Gamma_q \equiv \Gamma_L^q - \Gamma_H^q
= 2 |\Gamma_{12}^q| \cos \phi^q,
\end{equation}
where the decay widths $\Gamma_L$ and $\Gamma_H$ are corresponding to
the physical eigenstates $B_L$ and $B_H$, respectively,
and the CP phase is
$\phi^q \equiv {\rm arg}\left( - M_{12}^q/\Gamma_{12}^q \right)$.
The charge asymmetry in Eq. (3) is expressed as
\begin{equation} \label{aslq}
a_{sl}^q = \frac{|\Gamma_{12}^q|}{|M_{12}^q|} \sin \phi^q
= \frac{\Delta \Gamma_q}{\Delta M_q} \tan \phi^q,
\end{equation}
of which the SM predictions are given by
\cite{Lenz07}
\begin{eqnarray} \label{aslSM}
a_{sl}^{d} = (-4.8^{+1.0}_{-1.2}) \times 10^{-4},
\quad
a_{sl}^{s} = (2.1 \pm 0.6) \times 10^{-5}, \cr
\phi^d = (-9.1^{+2.6}_{-3.8})\times 10^{-2},
\quad
\phi^s = (4.2\pm 1.4)\times 10^{-3} .
\end{eqnarray}
In the SM, $\Delta \Gamma_d/\Gamma_d$ is less than $1 \%$,
while $\Delta \Gamma_s/\Gamma_s \sim 10 \%$ is rather large.
The decay matrix elements $\Gamma_{12}^q$ is obtained from
the tree level decays $b \to c \bar{c} q$
where the dominent right-handed current contribution
is suppressed by the heavy right-handed gauge boson mass $M_{W_R}$ \cite{Ecker86}.
Therefore, we ignore the contributions of our model to $\Gamma_{12}^q$ in this work.
We first consider the right-handed current contributions in the $B_d^0-\bar{B_d^0}$ system.
The $\Delta B = 2$ transition amplitudes in Eq. (\ref{M12}) is given by the following
effective Hamiltonian in the LRM \cite{Nam02}:
\begin{equation}
H^{B\bar{B}}_{eff} = H^{SM}_{eff} + H^{RR}_{eff} + H^{LR}_{eff},
\end{equation}
where
\begin{eqnarray} H^{SM}_{eff} &=&
\frac{G_F^2M_W^2}{4\pi^2}(\lambda_t^{LL})^2S(x^2_t)
(\bar{d_L}\gamma_\mu b_L)^2 , \label{HSMeff}\\
H^{LR}_{eff} &=& \frac{G_F^2M_W^2}{2\pi^2} \biggl\{
\left[\lambda_c^{LR} \lambda_t^{RL}x_cx_t\zeta_g A_1(x_t^2,\zeta)
+ \lambda_t^{LR} \lambda_t^{RL}x_t^2\zeta_g A_2(x_t^2,\zeta)\right]
(\bar{d_L}b_R)(\bar{d_R}b_L) \cr
&&\hspace{0.2in} +\ \lambda_t^{LL} \lambda_t^{RL}x_b\xi_g^-
\left[x_t^3A_3(x_t^2)(\bar{d_L}\gamma_\mu b_L)(\bar{d_R}\gamma_\mu b_R)
+ x_tA_4(x_t^2)(\bar{d_L}b_R)(\bar{d_R}b_L)\right]\biggr\} ,
\label{HLReff}
\end{eqnarray}
and
\begin{equation}
\lambda_i^{AB} \equiv V_{id}^{A*}V_{ib}^{B} , \quad x_i \equiv \frac{m_i}{M_W} \ (i = u,c,t),
\quad \xi_g^{\pm} \equiv e^{\pm\alpha}\xi_g ,
\end{equation}
with
\begin{eqnarray} \label{loopfn}
S(x) &=& \frac{x(4 - 11x + x^2)}{4(1-x)^2} - \frac{3x^3\ln x}{2(1-x)^3} , \cr
A_1(x,\zeta) &=& \frac{(4-x)\ln x}{(1-x)(1-x\zeta)}
+ \frac{(1-4\zeta)\ln \zeta}{(1-\zeta)(1-x\zeta)} , \cr
A_2(x,\zeta) &=& \frac{4-x}{(1-x)(1-x\zeta)} + \frac{(4-2x+x^2(1-3\zeta))\ln x}
{(1-x)^2(1-x\zeta)^2} + \frac{(1-4\zeta)\ln\zeta}{(1-\zeta)(1-x\zeta)^2} , \\
A_3(x) &=& \frac{7-x}{4(1-x)^2} + \frac{(2+x)\ln x}{2(1-x)^3} , \cr
A_4(x) &=& \frac{2x}{1-x} + \frac{x(1+x)\ln x}{(1-x)^2} . \nonumber
\end{eqnarray}
Note that $S(x)$ is the usual Inami-Lim function,
$A_1(x,\zeta)$ is obtained by taking the limit
$x_c^2=0$, and $H_{eff}^{RR}$ is suppressed because it is proportional to $\zeta^2$.
Also in the case of $V^R_I$, one can see from Eq. (\ref{UR}) that there is no significant contribution of
$H^{LR}_{eff}$ to $B_d^0-\bar{B_d^0}$ mixing,
so we only consider the $V^R_{II}$ type mixing matrix for $B_d^0-\bar{B_d^0}$ mixing.
The dispersive part of the $B_d^0-\bar{B_d^0}$ mixing matrix element can then be written as
\begin{equation} \label{massmixing}
M_{12}^d = M_{12}^{SM} + M_{12}^{LR} = M_{12}^{SM}\left( 1 + r^d_{LR} \right) ,
\end{equation}
where
\begin{equation}
r^d_{LR} \equiv \frac{M_{12}^{LR}}{M_{12}^{SM}}
= \frac{\langle \bar{B_d^0}|H_{eff}^{LR}|B_d^0 \rangle}
{\langle \bar{B_d^0}|H_{eff}^{SM}|B_d^0 \rangle} .
\end{equation}
For specific phenomenological estimates one needs the hadronic matrix elements of the operators
in Eqs. (\ref{HSMeff},\ref{HLReff}) in order to evaluate the mixing matrix element.
We use the following parametrization:
\begin{eqnarray}
\langle \bar{B_d^0}|(\bar{d_L}\gamma_\mu b_L)^2|B_d^0 \rangle &=& \frac{1}{3}B_1f_B^2m_B , \cr
\langle \bar{B_d^0}|(\bar{d_L}\gamma_\mu b_L)(\bar{d_R}\gamma_\mu b_R)|B_d^0 \rangle
&=& -\frac{5}{12}B_2f_B^2m_B , \\
\langle \bar{B_d^0}|(\bar{d_L}b_R)(\bar{d_R}b_L)|B_d^0 \rangle &=& \frac{7}{24}B_3f_B^2m_B \nonumber,
\end{eqnarray}
where
\begin{equation}
\langle 0|\bar{d_\beta}\gamma^\mu\gamma_5 b_\alpha|B_d^0 \rangle
= - \langle \bar{B_d^0}|\bar{d_\beta}\gamma^\mu\gamma_5 b_\alpha|0 \rangle
= - \frac{if_Bp_B^\mu}{\sqrt{2m_B}}\frac{\delta_{\alpha\beta}}{3} ,
\end{equation}
and where $f_B$ is the $B$ meson decay constant and $B_i\ (i=1,2,3)$ are the bag factors.
In the vacuum-insertion method \cite{Gaillard}, $B_i=1$ in the limit $m_b\simeq m_B$.
We will use $f_BB_i^{1/2}=(216\pm 15)$ MeV for our numerical estimates \cite{HPQCD09}.
Using the standard values of the quark masses and
$|V^L_{cd}|\approx 0.225$, one can express $r^d_{LR}$
in terms of the mixing angle and phases in the case of $V_{II}^R$ in Eq. (\ref{UR}) as
\begin{eqnarray} \label{rLRd}
r^d_{LR} &\approx& 17.5 \biggl(
\frac{1 - \zeta_g - (4.08 - 16.3\zeta_g)\ln(1/\zeta_g)) }{ 1 - 5.58\zeta_g }\biggr)
\zeta_g s_R^2e^{-i(2\beta - \alpha_2 + \alpha_3)} \cr
&-& 756 \biggl( \frac{ 1 - 5.03\zeta_g - (0.490 - 1.96\zeta_g )\ln(1/\zeta_g) }
{1 - 10.2\zeta_g + 30.1\zeta_g^2}\biggr) \zeta_g s_Rc_Re^{-i(\beta + \alpha_3 - \alpha_4)} \
- \ 7.94\xi_g s_R e^{-i(\beta + \alpha_3)} ,
\end{eqnarray}
where the mixing phase $\alpha$ was absorbed in $\alpha_i$ by redefining $\alpha_i + \alpha \rightarrow \alpha_i$,
and we used the approximation $A_i(x,\zeta) \simeq A_i(x,\zeta_g) (i=1,2)$
because $\zeta$ dependence on $A_i$ in Eq. (\ref{loopfn}) is rather weak for $M_{W'} > 100$ GeV
unless $g_R/g_L$ is drastically different from unity.
On the other hand, the right-handed current contributions to $B_s^0-\bar{B_s^0}$ mixing is sizable
only in the case of $V_{I}^R$ as one can see from Eq. (\ref{UR}).
Similarly to $r^d_{LR}$, we obtain $r^s_{LR}$ in the case of $V_I^R$ as
\begin{eqnarray} \label{rLRs}
r^s_{LR} &\approx& -3.47 \biggl(
\frac{1 - \zeta_g - (4.08 - 16.3\zeta_g)\ln(1/\zeta_g)) }{ 1 - 5.58\zeta_g }\biggr)
\zeta_g s_R^2e^{-i(-\alpha_2 + \alpha_3)} \cr
&+& 162 \biggl( \frac{ 1 - 5.03\zeta_g - (0.490 - 1.96\zeta_g )\ln(1/\zeta_g) }
{1 - 10.2\zeta_g + 30.1\zeta_g^2}\biggr) \zeta_g s_Rc_Re^{-i(\alpha_3 - \alpha_4)} \
+ \ 1.70\xi_g s_R e^{-i\alpha_3} .
\end{eqnarray}
The charge asymmtry $a^q_{sl}$ in Eq. (\ref{aslq}) can then be written in terms of $r^q_{LR}$ in the LRM as
\begin{equation}
a^q_{LR}=a^q_{SM}\frac{\cos{\phi^q_{LR}}}{|1+r^q_{LR}|}
\left(1 + \frac{\tan{\phi^q_{LR}}}{\tan{\phi^q_{SM}}} \right), \quad
\phi^q_{LR} \equiv \textsl{arg}(1+r^q_{LR}),
\end{equation}
where we omitted the subscript `$sl$'
and the SM values of $a^q_{sl}$ and $\phi^q$ are given in Eq. (\ref{aslSM}).
We use the above results for our numerical investigation of the right-handed current contributions
to the like-sign dimuon charge asymmetry in semi-leptonic $B$ decays in the next section.
\section{Results}
For our numerical analysis,
we use the following $2\sigma$ bounds obtained from the deviation of
the present experimental data from the SM predictions on $B$ meson mixing \cite{Lenz11}:
\begin{equation} \label{mixing_exp}
0.62 < |1+r^d_{LR}| < 1.15, \quad 0.79 < |1+r^s_{LR}| < 1.23 .
\end{equation}
Note from Eqs. (\ref{rLRd},\ref{rLRs}) that we have six independent new parameters
($\zeta_g, \xi_g, \theta_R, \alpha_{2,3,4}$), and it is beyond the scope of this paper
to perform a complete analysis by varying all six parameters.
For simple illustration of the possible effect of the new interaction on $B$ meson mixing, instead,
we set $\xi_g = \zeta_g/2$ and $\alpha_{2,4}=0$
because $\xi_g$ contributions to $B$ meson mixing is expected to be much smaller than $\zeta_g$'s
and $\alpha_3$ is important as the overall phase of $r^q_{LR}$.
\begin{figure}[!hbt]
\centering%
\includegraphics[width=8.3cm]{BsMmix1.png}
\caption{Allowed regions for $\alpha_3$ and $\theta_R$ at 2 $\sigma$ level for $M_{W'} = 800$ GeV
in the case of $V_I^R$.
The red and blue regions are allowed by the current measurements
of the like-sign dimuon charge asymmetry and $B_s\bar{B_s}$ mixing, respectively.}
\label{BsMmix1}
\end{figure}
\begin{figure}[!hbt]
\centering%
\includegraphics[width=8cm]{BsMmix2.png}
\caption{Allowed regions for $\theta_R$ and $\zeta_g$ at 2 $\sigma$ level for $\alpha_3=90^\circ$
in the case of $V_I^R$.
The red and blue regions are allowed by the current measurements
of the like-sign dimuon charge asymmetry and $B_s\bar{B_s}$ mixing, respectively.}
\label{BsMmix2}
\end{figure}
In the case of $V_I^R$, as discussed earlier,
the right-handed current contributions to $B_s-\bar{B_s}$ mixing could be sizable
while those to $B_d-\bar{B_d}$ mixing is negligible.
With the present experimental bounds of the dimuon charge asymmetry
and $B_s-\bar{B_s}$ mixing given in Eqs. (\ref{asym_exp},\ref{mixing_exp}),
we first plot the allowed region of $\alpha_3$ and $\theta_R$
for $M_{W'} = 800$ GeV at 2 $\sigma$ level in Fig. \ref{BsMmix1}.
One can see from the overlapped allowed region in the figure
that large values of $\theta_R$ are preferred.
This is the clear indication that manifest or pseudomanifest LRM is disfavored in this case.
In Fig. \ref{BsMmix2},
we plot the allowed region of $\theta_R$ and $\zeta_g$ for $\alpha_3=90^\circ$
at 2 $\sigma$ level. One can obtain the lower bound of $\zeta_g \gtrsim 0.004$ from the figure
which corresponds to the upper bound of $W'$ mass $M_{W'} \lesssim (g_R/g_L)\times 1.3$ TeV.
For different values of $\alpha_3$, this mass bound can change, but not very much.
In other words,
if it happens that the mass of $W'$ is much larger than the obtained upper bound,
the right-handed contributions are not big enough
to explain the present measurement of the dimuon charge asymmetry.
\begin{figure}[!hbt]
\centering%
\includegraphics[width=8.3cm]{BdMmix1.png}
\caption{Allowed regions for $\alpha_3$ and $\theta_R$ at 2 $\sigma$ level for $M_{W'} = 800$ GeV
in the case of $V_{II}^R$.
The red and blue regions are allowed by the current measurements
of the like-sign dimuon charge asymmetry and $B_d\bar{B_d}$ mixing, respectively.}
\label{BdMmix1}
\end{figure}
\begin{figure}[!hbt]
\centering%
\includegraphics[width=8cm]{BdMmix2.png}
\caption{Allowed regions for $\theta_R$ and $\zeta_g$ at 2 $\sigma$ level for $\alpha_3=90^\circ$
in the case of $V_{II}^R$.
The red and blue regions are allowed by the current measurements
of the like-sign dimuon charge asymmetry and $B_d\bar{B_d}$ mixing, respectively.}
\label{BdMmix2}
\end{figure}
In the case of $V_{II}^R$, on the other hand,
the right-handed current contributions to $B_d-\bar{B_d}$ mixing could be sizable
while those to $B_s-\bar{B_s}$ mixing is negligible.
Similarly to the $V_I^R$ case, we plot the allowed region of $\alpha_3$ and $\theta_R$
for $M_{W'} = 800$ GeV at 2 $\sigma$ level in Fig. \ref{BdMmix1}.
The figure shows that small or large values of $\theta_R$ are allowed unlike the $V_I^R$ case.
In order for direct comparison with the $V_I^R$ case,
we plot again the allowed region of $\theta_R$ and $\zeta_g$ for $\alpha_3=90^\circ$
at 2 $\sigma$ level in Fig. \ref{BdMmix2}. The figure shows that $V_{II}^R$ senario allows more wide range of
allowed area of new parameter space and the lower bound of $\zeta_g$ is approximately $\zeta_g \gtrsim 0.0004$.
We obtain the corresponding upper bound of $W'$ mass $M_{W'} \lesssim (g_R/g_L)\times 4$ TeV.
We found that this mass bound could be somewhat lower for different values of $\alpha_3$.
It should also be noted that we have similar results for different $\alpha_{2,4}$ in both senarios.
\section{Concluding Remarks}
In this paper,
we studied the right-handed current contributions to
the CP violating like-sign dimuon charge asymmetry
in semi-letonic $B$ decays in general left-right models.
Without imposing manifest or pseudomanifest left-right symmetry,
we consider two types of mass mixing matrix $V^R$
with which $W'$ contributions are big enough to explain the current mismatch of
the present measurents of the dimuon charge asymmetry and the SM prediction.
We evaluated the sizes of $W'$ contributions to $B_d-\bar{B_d}$ and $B_s-\bar{B_s}$ mixings
which govern the dimuon charge asymmetry,
and obtained the allowed regions of NP parameter spaces.
With the given parameter sets, we have the following mass bounds of $W'$:
$M_{W'} \lesssim (g_R/g_L)\times 1.3$ TeV for Type I ($V_I^R$) or
$M_{W'} \lesssim (g_R/g_L)\times 4$ TeV for Type II ($V_{II}^R$),
which represent the amount of NP effects enough to explain the present measurent of the dimuon charge asymmetry.
If we consider the early LHC bound on $W'$ \cite{LHCW},
Type I model including manifest or pseudomanifest LRM is disfavored if $g_R = g_L$.
This analysis can affect other $B$ meson mixing related observables such as $\sin{2\beta}$
and mixing induced CP violation in B decays.
A detailed discussion on such mixing induced CP asymmetries in general LRM can be found in Ref. \cite{Nam03},
and a combined study including other decays with new experimental results will be discussed in the follow-up paper.
\acknowledgments
KYL is supported in part by WCU program through the KOSEF funded by the MEST (R31-2008-000-10057-0)
and the Basic Science Research Program through the National Research Foundation of Korea (NRF)
funded by the Korean Ministry of Education, Science and Technology (2010-0010916).
\def\PRD #1 #2 #3 {Phys. Rev. D {\bf#1},\ #2 (#3)}
\def\PRL #1 #2 #3 {Phys. Rev. Lett. {\bf#1},\ #2 (#3)}
\def\PLB #1 #2 #3 {Phys. Lett. B {\bf#1},\ #2 (#3)}
\def\NPB #1 #2 #3 {Nucl. Phys. {\bf B#1},\ #2 (#3)}
\def\ZPC #1 #2 #3 {Z. Phys. C {\bf#1},\ #2 (#3)}
\def\EPJ #1 #2 #3 {Euro. Phys. J. C {\bf#1},\ #2 (#3)}
\def\JHEP #1 #2 #3 {JHEP {\bf#1},\ #2 (#3)}
\def\IJMP #1 #2 #3 {Int. J. Mod. Phys. A {\bf#1},\ #2 (#3)}
\def\MPL #1 #2 #3 {Mod. Phys. Lett. A {\bf#1},\ #2 (#3)}
\def\PTP #1 #2 #3 {Prog. Theor. Phys. {\bf#1},\ #2 (#3)}
\def\PR #1 #2 #3 {Phys. Rep. {\bf#1},\ #2 (#3)}
\def\RMP #1 #2 #3 {Rev. Mod. Phys. {\bf#1},\ #2 (#3)}
\def\PRold #1 #2 #3 {Phys. Rev. {\bf#1},\ #2 (#3)}
\def\IBID #1 #2 #3 {{\it ibid.} {\bf#1},\ #2 (#3)}
| 11,492 |
\chapter*{Acknowledgments}
This project would not have been possible without the support of many people. Many thanks to my advisor, Anthony J. Leggett, who carefully and patiently read countless preliminary ideas which lead to the present one. Without his extraordinary physical intuition and formulation of the problem, this outcome would not have been possible. Thanks to my lovely wife P\i nar Zorlutuna. Having endured this process to its end with all the conversation that comes with it, she is now equally knowledgable about the field as I, if not more. Also, thanks to my mother and father for producing me, and persistently encouraging me to ``finish writing up''.
I acknowledge helpful discussions over many years with numerous colleagues, in particular Pragya Shukla, Karin Dahmen, Alexander Burin, Zvi Ovadyahu and Philip Stamp. Thanks to University of British Columbia, University of Waterloo, and Harvard University for generously providing accommodation during long visits.
Finally, thanks to all the coffee shops in Champaign, specifically Green Street Coffee Shop and Bar-Guilliani for generously nourishing me with espresso shots every day for free, for many years.
\\\\
This work was supported by the National Science Foundation under grant numbers NSF-DMR-03-50842 and NSF-DMR09-06921.
\tableofcontents
\chapter{List of Symbols and Abbreviations}
\begin{symbollist}[0.7in]
\item[$l$] Phonon Mean-Free-Path
\item[$\lambda$] Wavelength
\item[$\chi_{ij:kl}$] Stress-Stress Linear Response Function
\item[$Q^{-1}$] Attenuation Coefficient ($\equiv l^{-1}\lambda/(2\pi^2)=(\pi\mbox{Re}\chi)^{-1})\mbox{Im}\chi$)
\item[$Q_0^{-1}$] Attenuation Coefficient of a microscopic block
\item[$\langle Q^{-1}\rangle$] Frequency and initial state averaged attenuation coefficient
\item[$r_0$] Size of a ``microscopic'' block
\item[$v_i$] Volume of Block i
\item[$v_{ij}$] Volume of Sphere with Radius Equal to Distance between Blocks i and j
\item[$T$] Temperature
\item[$T_0$] Reference Temperature
\item[$T_c$] Critical Temperature
\item[$\beta$] Inverse temperature
\item[$U_0$] Critical Energy / High-Energy Cutoff
\item[$U$] Renormalized $U_0$
\item[$N_b$] Number of Energy Levels Below Cutoff
\item[$\omega$] Probe Frequency
\item[$\omega_D$] Debye Frequency
\item[$\vec{k}$] Wavenumber
\item[$D(\omega)$] Phonon Density of States
\item[$f(\omega)$] Non-Phonon (block) Density of States
\item[$c_{\nu}(T)$] Phonon Specific Heat
\item[$C_{\nu}(T)$] Non-Phonon (block) Specific Heat
\item[$l,t$] Longitudinal, Transverse
\item[$c_l,c_t$] Longitudinal and Transverse Speed of Sound
\item[$K(T)$] Thermal Conductivity
\item[$\alpha$] Specific Heat Exponent
\item[$\beta$] Thermal Conductivity Exponent
\item[$\hat{H}$] Full Hamiltonian
\item[$\hat{H_{(ph)}}$] Phonon Hamiltonian
\item[$\hat{V}$] Phonon-Induced Stress-Stress Coupling
\item[$\hat{H_0^{s}}$] Noninteracting Block Hamiltonian of block $s$
\item[$\tau$] Pulse Duration
\item[$r_i$] Position of Block $i$
\item[$\hat{n}$] Unit Vector Along two Blocks
\item[$u$] Atomic Displacement
\item[$x_i$] Position Coordinate ($i=1,2,3$)
\item[$\rho$] Mass Density
\item[$e_{ij}$] Strain Tensor
\item[$\hat{T}_{ij}$] Stress Tensor Operator
\item[$L$] Block Linear Dimension
\item[$p_m$] Probability of Occupation of the $m^{th}$ state
\item[$Z$] Partition Function
\item[$z$] Renormalization step
\item[$\Lambda_{ijkl}$] Amorphous block Stress-Stress Coupling Coefficient
\item[p.v.] Principal Value
\item[TTLS] Tunneling Two Level System
\item[$\Omega$, $V$, $d$, $M_0$] TTLS Ground Frequency, Barrier Height, Well Separation, Mass
\item[$\Delta$, $\Delta_0$, $\gamma$] TTLS Well Asymmetry, Transition Matrix Element, Phonon Coupling Constant
\item[$I$, $I_0$] Sound Intensity, Critical Sound Intensity
\item[$P(\Delta,\Delta_0,\gamma)$] Probability distribution for TTLS parameters
\end{symbollist}
\mainmatter
\chapter{Introduction}
The generic term ``disordered solid'' (or ``amorphous solid'') is typically used to describe a large variety of condensed matter including metallic and dielectric glasses, disordered crystals, polymers, quasi-crystals and proteins. Starting from the pioneering work of Zeller and Pohl \cite{pohl71} it has become clear that the thermal, acoustic and electromagnetic response of disordered solids are (1) remarkably different than crystals, and (2) are remarkably similar among themselves. To put the problem posed by these two statements in proper context, it will be necessary to give a little background on the low temperature properties of both disordered and non-disordered condensed matter.
\section{A Comparison of Crystalline and Disordered Solids}
According to Debye's theory of crystalline insulators, at sufficiently low temperatures the amplitude $\vec{r}$ of atomic motion remains small enough to consider only the first order term in the inter-atomic force $F_i=2\mu_{ij} r_j$. Thus, the low temperature acoustic and thermal properties of all crystalline insulators are determined by a Hamiltonian which can be diagonalized into phonon modes, quantized with uniform increments of $E=\omega$.
In this framework, the standard text-book procedure \cite{kittel} to obtain crystalline specific heat is to simply integrate over these modes to find the average energy density,
\begin{eqnarray}\label{debye}
\langle E\rangle=\sum_p\int_0^{\omega_D} \omega D(\omega)n(\omega,\beta)d\omega,
\end{eqnarray}
where the summation is over phonon polarizations, and $n$ is the number of phonons at a given temperature $T=1/\beta$,
\begin{eqnarray}
n=\frac{1}{e^{\beta\omega}-1}
\end{eqnarray}
and $D$ is the phonon density of states per volume $v$, at wave number $\vec{k}$
\begin{eqnarray}
D(\omega)=\frac{|k|^2}{2\pi^2}\frac{d|k|}{d\omega}.
\end{eqnarray}
The upper cut-off $\omega_D$ of the integral in eqn(\ref{debye}) is the energy of a phonon with wavelength $\lambda$ equal to inter-atomic distance $a_0$. Such high end cutoffs are commonly used in condensed matter theories, including the present study (cf. Chapter 2), since they conveniently exclude smaller length scales for which the physical assumptions no longer hold. This of course, limits the validity of the theory to energy and temperatures below $\sim hc/a_0$.
In the continuum limit, $\lambda\gg a_0$, the linear dispersion relation
\begin{eqnarray}
\omega_{l,t}=kc_{l,t}
\end{eqnarray}
holds, where $c_l,t$ is the transverse $t$ or longitudinal $l$ speed of sound in the limit $\omega_{l,t}\to0$. At low temperatures $T\ll\omega_D$, eqn(\ref{debye}) can be easily integrated to yield the famous $T^3$ temperature dependence for the specific heat.
\begin{align}\label{famous}
c_v=\frac{\partial\langle E\rangle}{\partial T}=\frac{12\pi^4}{5}N\left(\frac{T}{\omega_D}\right)^3.
\end{align}
Let us now turn to the thermal conductivity, $K$. Since phonons are the main heat carriers of heat in a dielectric crystal, $K(T)$ can be calculated analogously to the kinetic theory of gases. If $\bar{l}$ and $\bar{c}$ is the frequency averaged mean free path and velocity of a phonon,
\begin{eqnarray}\label{K}
K\approx\frac{1}{3}c_v\bar{c}\bar{l}
\end{eqnarray}
Of course, for a plane wave propagating in a perfect infinite crystal whose atoms interact harmonically, $l$ and thus $K$ is infinite. However in practice, the phonons are restricted by the finite geometric scale $l=L$ of the sample. Thus, if there are no impurities and the temperature is low enough to neglect anharmonic interaction between atoms, from eqn(\ref{famous}) we roughly get $K\sim T^3$.
At first sight, the Debye theory seems equally applicable to disordered solids, since these materials are known to exhibit well-defined phonon modes, as seen from Brillouin scattering experiments below 33 GHz \cite{vacher76} and phonon interference experiments below 500 GHz \cite{rothenfusser83}. Thus, at energy scales for which the thermal or acoustic wavelength exceeds $a_0$ there is no a priori reason for the thermo-acoustic response of a solid, disordered or not, to deviate from that of the elastic continuum described above.
However, experimentally, disordered solids behave nothing like their crystalline counterparts (for two comprehensive reviews, see \cite{phillips81,esquinazi-rev}). The specific heat, measurements reveal\cite{pohl71} an super-linear temperature dependence of specific heat below $T_{0}$,
\begin{eqnarray}
C_v\propto T^{1+\alpha}
\end{eqnarray}
with $\alpha\sim0.1-0.3$. Remarkably, at $25mK$ this term dominates the Debye phonon contribution by a factor of about $1000$. Above $T_c$
the specific heat raises faster before reaching the Debye limit, which manifests as a bump in the $C_v/T^3$ vs $T$ curve (a historical convention used to compare the glassy specific heat to a ``crystalline reference''). $T_0$ is a more-or-less material dependent temperature, and typically varies between $1-30 K$.
According to specific heat measurements, it is clear that something other than phonons is being thermally excited in disordered solids. In the established model of amorphous solids, these are assumed to be ``tunneling two level systems''. In the present work, we acknowledge the presence of these degrees of freedoms but do not make any specific assumptions regarding their nature. A more complete description of both models will follow below.
Turning now to the thermal conductivity, measurements reveal a sub quadratic temperature dependence \cite{pohl71,anderson86}
\begin{eqnarray}
K\propto T^{2-\beta}.
\end{eqnarray}
with $\beta\sim0.05-0.2$. Again, around a temperature that empirically coincides with the ``bump'' in specific heat, $T_0$, this gradually changes to a temperature independent plateau. it is important to note in the present context, that even though $C_v$ and $K$ are traditionally fit to power laws, as we will see below, logarithmic corrections $C_v\sim T/\ln T$ and $K\sim T^2\ln T$, lead to indistinguishable functional forms below $T_0$)
The universality of disordered solids go as beyond these ``exponents'' in temperature dependences: In a disordered solid, heat is transported by phonons \cite{zaitlin75}, allowing the use of eqn(\ref{K}) to obtain the ultrasonic attenuation coefficient defined in terms of the phonon mean free path,
\begin{eqnarray}
Q^{-1}\equiv \lambda/(2\pi^2l).
\end{eqnarray}
Below $T<T_0$, with $T_0$ again roughly coinciding with that defined above, the mean free path displays an amazing degree of universality\cite{anderson86,mb,pohl2002}: In this regime sound waves travel about 150 times their own wavelength, regardless of the chemistry or composition of the amorphous matrix. Above $T_0$, $\lambda/l$ rapidly increases to a (non-universal) constant of the order one.
Two other universalities are the ratios \cite{mb} of transverse and longitudinal sound velocity and phonon coupling,
\begin{eqnarray}
\frac{c_t}{c_l}\approx0.6\\
\frac{\chi_t}{\chi_l}\approx0.4
\end{eqnarray}
In the present work we will provide some theoretical justification to these ratios, and use their experimental values \cite{mb} as inputs to obtain the value of $Q^{-1}$, and related quantities.
In addition to these quantitative universalities, disordered solids display a series of very unusual acoustic non-linearities that are not present in crystals. For example, above a critical sound intensity $I_c$, the acoustic absorption is ``saturated''\cite{hunklinger72}; i.e. above $I_c$, the solid becomes transparent to sound:
\[\chi\sim\frac{1}{\sqrt{1+I/I_c}}.\]
A second example is the spectacular echo experiments. These are the acoustic analogs of the magnetic response of a spin particle in a nuclear magnetic resonance setup, where the amorphous sample is cooled down below $T<\omega$. Then two consequent pulses of equal frequency $\omega$ separated by time $\tau_s\gg\tau_p$ are applied for a duration of $\tau_p$ and $2\tau_p$ respectively. The two elastic pulses surprisingly induces a ``spontaneous'' third one, precisely $\tau_s$ after the second pulse.
It is remarkable that, although less pronounced, the universalities hold for metallic glasses as well. However, we limit our considerations in this thesis only to amorphous insulators, since the electron-electron and electron-phonon interactions add an extra layer of complexity to the problem. Further, we will focus only on the universality of \emph{linear} acoustic properties of disordered solids in the resonance regime, and the thermal conductivity below $T_c$. We will not discuss the nonlinear properties, or the acoustic response in the relaxation regime.
\section{A Brief Review of the Tunneling Two Level System Model}
The theoretical interpretation of the low temperature data on amorphous materials has for 40 years been dominated by the phenomenological ``tunneling two state system'' (TTLS) model \cite{ahv,p}, which has two main assumptions; the first is that, within all disordered solids there exists entities that tunnel between two metastable states $|L\rangle$ and $|R\rangle$ (conventionally called ``left'' and ``right'', although the tunneling coordinate is not specified; cf. below). The tunneling entity is described by a reduced double well Hamiltonian,
\[H_{TTLS}=\left(\begin{array}{cc}
\Delta & -\Delta_0\\
-\Delta_0 & -\Delta
\end{array}\right)+e\left(\begin{array}{cc}
\gamma & 0\\
0 & -\gamma
\end{array}\right)\]
Here $\Delta$ is the double-well asymmetry, $\Delta_0$ is the tunneling matrix element expressed by the standard WKB formula
\begin{eqnarray}
\Delta_0\sim\Omega e^{-\xi}\nonumber\\
\xi=d\sqrt{2M_{0}V}
\end{eqnarray}
$M_{0}$ is the mass of the tunneling entity, $\Omega$ is the ground frequency of each (harmonic) well, $d$ is the separation between wells, and $V$ is the double well barier height. The TTLS couples to strain $e$ (and of course, the thermal phonons) through the coupling constant $\gamma$.
The second fundamental assumption of the TTLS theory is that the two level parameters of an ensemble of TTLS are distributed according to the probability density
\[P(\Delta,\Delta_0,\gamma)=\frac{\bar{P}\delta(\gamma-\gamma_0)}{\Delta_0}\]
The $1/\Delta_0$ factor can be obtained by assuming that the parameter $\xi$ in the exponent is uniformly distributed. $\gamma$ is assumed to be the same for all tunneling entities.
At the cost of introducing a fairly large number of fitting parameters, the TTLS theory gives an attractive explanation to the nonlinear acoustic effects\cite{phillips81} and a specific heat linear with temperature $c_v\sim T$ and quadratic in thermal conductivity $K\sim T^2$ \cite{ahv,p} reasonably close to experiment below $T_0$ (see however, Fig\ref{fig:Figure2} and corresponding discussion in Chapter 5). Moreover it gives the temperature and frequency dependences of the mean free path and sound velocity\cite{jackle72,golding76} (although the latter requires an additional fit function with no physical basis).
The dominant response of a TTLS to a sound wave in the regime $T<\omega$ occurs through ``resonant absorption''. A single phonon interacts with a TTLS with matching energy separation at thermal equilibrium. The mean free path is found from Fermi's golden rule for transition rates $1/\tau$. Using $l=c\tau$,
\begin{eqnarray}\label{mfp}
\lambda l^{-1}(T,\omega)\equiv2\pi^2Q^{-1}=2\pi^2\frac{\gamma^2\bar{P}}{\rho c^2}\mbox{tanh}\left(\frac{\omega}{T}\right)
\end{eqnarray}
this equation can be plugged in the Kramers-Kronig principal value integral
\begin{eqnarray}
c(T)-c(0)\equiv\Delta c=\frac{1}{\pi}\mbox{p.v.}\int_0^\infty\frac{c^2l^{-1}(\omega')}{\omega^2-\omega'^2}d\omega'
\end{eqnarray}
to yield a logarithmic velocity shift for $kT\sim\hbar\omega$ \cite{phyac76}
\begin{eqnarray}
\frac{\Delta c}{c}=-\frac{\gamma^2\bar{P}}{\rho c^2}\log\left(\frac{\omega}{T}\right)
\end{eqnarray}
Since $T\ll\omega$ is experimentally difficult to attain, this equation is usually tested in the regime $T>\omega$ with respect to an (arbitrary) reference temperature $T_0$,
\begin{eqnarray}\label{tlsvel}
\frac{c(T)-c(T_0)}{c(T)}=\frac{\gamma^2\bar{P}}{\rho c^2}\log\left(\frac{T}{T_0}\right)
\end{eqnarray}
Therefore, (with one exception \cite{golding76}) most of our direct knowledge regarding $Q^{-1}$ (and thus its universality) in the resonance regime comes indirectly, through measuring the velocity shift at high temperatures and then obtaining $Q^{-1}$ from the Kramers-Kronig integral that goes the other way around.
For higher temperatures $T>\omega$, the dominant mechanism for sound absorption is quite different than that described above, through a ``relaxation'' process. This is when a low frequency sound wave modulates the TTLS energy $E$, causing the population ratio $n$ of excited to unexcited TTLS to differ from its equilibrium value
\[\bar{n}\propto e^{-\beta E}.\]
It is not difficult to see how acoustic energy is converted to heat this way: The pulse must do work to widen the levels of an excited TTLS, which then is ``lost'' when the TTLS emits a thermal phonon to decrease the overpopulated $n$. In the limit $\omega\tau_{min}\ll 1$ (which corresponds to higher temperatures), the theory predicts half the value of the absorption in the resonance regime, again independent of temperature and frequency
\begin{eqnarray}
Q^{-1}=\frac{\bar{P}\gamma^2}{2\rho c^2}
\end{eqnarray}
where $\tau_{min}$ is the minimum relaxation time among the TTLS with splitting $E=kT$.
\section{Problems with the Standard TTLS Model}
The first difficulty of the TTLS model is the uncertainty regarding its microscopic nature. Although the configuration coordinate is typically referred to as ``left'' and ``right'', neither the entity undergoing the tunneling motion, nor the nature of its motion is known. The TTLS may correspond to a single atom translating between two relatively low density regions in the solid, or two atoms may be sliding or rotating by each other. A recently discovered isotopic effect\cite{nagel04} proposes that the TTLS is as large as ten or twenty atoms.
Various mechanisms that could give rise to tunneling two levels were suggested for individual amorphous solids, such as the mixed phase metallic NbZr alloy domains\cite{lou76} or the hydroxyl ion impurities in vitreous silica \cite{phillips81a}. The tunneling impurities are also known to be present in disordered crystals, such as KBr-KCN solutions \cite{deyoreo86} and are extensively studied. However the following questions remain: First of all, even if two level excitations are present in (at least some) disordered media, obviously the are not the \emph{only} kind of excitation that is possible. Why then should the thermo-acoustic response of disordered media be entirely governed by \emph{two level} excitations? And furthermore, even if these excitations are present and dominant among all, it is not clear why their distribution of parameters $P(\Delta,\Delta_0,\gamma)$ should be so similar in every amorphous material.
Ideally, a theory for disordered condensed matter should start from assumptions that holds equally true for a very wide class of materials. Even if the TTLS hypothesis is one that is effectively correct, it seems important to base its assumptions on a firm physical basis.
Secondly, although the standard TTLS model can explain qualitative features such as phonon echoes and saturated absorption as well as approximate temperature dependences of quantities such as the specific heat and ultrasonic absorption \cite{phillips81,esquinazi-rev}, in its simplest form (i.e. without ad-hoc additional fit parameters and fit functions) it fails to explain velocity shift and absorption data at low temperatures\cite{classen00,fefferman08,golding76}, in addition to the the ``bump'' and ``plateau'' that appears in the specific heat $C_v(T)$ and thermal conductivity $K(T)$ data at temperatures above $T_c$\cite{pohl71,anderson86}.
Thirdly, in the temperature regime where the TTLS model is supposed to work ($T<1K$), it fails to explain the quantitative and qualitative universalities \cite{leggett91,leggett88} the most striking one of which is the ultrasonic attenuation $Q^{-1}$ \cite{pohl2002}; it is not at all obvious\cite{anderson86,leggett88,leggett91} that the coefficient of eqn(\ref{mfp})
\begin{eqnarray}
\alpha=\frac{\gamma^2\bar{P}}{\rho c^2}
\end{eqnarray}
should be material independent. In fact, the quantities in the numerator and denominator are independent parameters of the TTLS theory, and if we restrict ourselves to dielectric glasses and polymers, while $\mu_l=\rho c_l^2$ varies from material to material by nearly three orders of magnitude, $Q^{-1}$ remains around $(3\pm2)\times10^{-4}$. It seems extremely unlikely that $\gamma^2$ and $\bar{P}$ happen to correlate with $\mu$ to this extent by pure coincidence.
Finally, the model, in its original form, neglects the fact that, as a result of interaction with the the phonon field, the stress of each TTLS's must be coupled\cite{joffrin75}. Although the original paper was formulated in the language of the TTLS, its outcome is neither sensitive to the number of levels of the interacting excitations, nor to the probability distribution function of their parameters. The only requirement is that the stress is coupled to the phonon field linearly (c.f. Appendix-A).
Recently this interaction was incorporated in the TTLS paradigm\cite{burin96}, and was shown that it leads to the experimentally observed small universal value of $Q^{-1}$. However, it seems legitimate to ask whether the original features of TTLS, such as those that give rise to saturation of absorption and echoes will survive this significant modification. Furthermore, it may be that the original assumptions of the TTLS model may be \emph{unnecessary} after this modification. It is possible that the presence of elastic interactions gives rise to the anomalous glassy behavior, regardless of what is interacting.
In \cite{leggett88,leggett91} it was conjectured that if one starts from a \emph{generic} model in which at short length scales there is a contribution to the stress tensor from some anharmonic degrees of freedom, and take into account their phonon mediated mutual interaction, one will obtain the significant features of glasses below 1K. The goal of this project is to quantitatively justify this conjecture, by calculating the frequency and temperature dependence of $Q^{-1}$ and the quantities related to it in the low temperature ``resonance'' regime $T\ll\omega$. The relaxation regime, non-linear effects or the intermediate temperature ($T>T_c$) behavior will not be considered in this work.
The layout of the thesis is as follows: In chapter-2 the precise details of the model is defined, and the central object of the study, namely, the dimensionless stress-stress correlation, whose thermally-averaged imaginary part is the measured ultrasonic absorption $Q^{-1}$ is introduced. In section 3 a real-space renormalization calculation of the \emph{average} of $Q_m^{-1}(\omega)$ over the frequency $\omega$ and the starting state $m$ (for details of the notation see below) is carried out. Here it is shown that this quantity vanishes logarithmically with the volume of the system and for experimentally realistic volumes, and has a small value $\sim0.015$. In section 4, on the basis of a heuristic calculation up to second order in the phonon-induced interaction, it is argued that the functional form of $Q^{-1}(\omega)$ at $T=0$ should be ($\mbox{ln}\omega)^{-1}$, and that when we combine this result with that of section 3, the numerical value of $Q^{-1}$ for experimentally relevant frequencies should be universal up to logarithmic accuracy and numerically close to the observed value $3\times10^{-4}$. In section 5 we attempt to assess the significance of our calculations.
\chapter{The Model}
Imagine the disordered solid as being composed of many (statistically identical) cubes of size $L$. The precise value of $L$ need not be specified, so long as it is much smaller than the experimental sample size, yet much larger than the (average) inter-atomic distance $a$. For this system we can define the strain tensor as usual
\begin{eqnarray}\label{2.1.1}
e_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_j}+\frac{\partial u_j}{x_i}\right)
\end{eqnarray}
where $\vec{u}(\vec{r})$ denotes the displacement relative to some arbitrary reference frame of the matter at point $\vec{r}=(x_1,x_2,x_3)$.
The Hamiltonian can be defined as a Taylor expansion up-to terms first order in strain $e_{ij}$
\begin{eqnarray}\label{2.1.2}
\hat{H}=\hat{H}_0+\sum_{ij}e_{ij}\hat{T}_{ij}+\mathcal{O}(e^2)
\end{eqnarray}
where the stress tensor operator $\hat{T}_{ij}$ is defined by
\begin{eqnarray}\label{2.1.3}
\hat{T}_{ij}=\partial \hat{H}/\partial e_{ij}
\end{eqnarray}
Note that, in general, in a representation in which $\hat{H}_0$ is diagonal, $\hat{T}_{ij}$ will have both diagonal and off-diagonal elements.
We can define the static elasticity modulus $\chi^{(0)}$, a fourth order tensor, as
\begin{eqnarray}\label{2.1.4}
\chi^{(0)}_{ij:kl}\equiv \frac{1}{L^3}\left.\frac{\partial\langle\hat{T}_{ij}\rangle}{\partial e_{ij}}\right|_{e(T)}\equiv \frac{1}{L^3}\left.\left\langle\frac{\partial^2\hat{H}}{\partial e_{ij}\partial e_{kl}}\right\rangle\right|_{e(T)}
\end{eqnarray}
where the derivative must be taken at the thermal equilibrium configuration. For $L\gg a$ an amorphous solid is rotationally invariant. Due to this symmetry, any component of $\chi_{ij:kl}^{(0)}$ can be written in terms of two independent constants \cite{landau-elastic}, for which we pick the transverse $\chi_t$ and longitudinal $\chi_l$ response;
\begin{eqnarray}\label{2.1.5}
\chi^{(0)}_{ij:kl}=(\chi_l-2\chi_t)\delta_{ij}\delta_{kl}+\chi_t(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})
\end{eqnarray}
In the approximation of an elastic continuum, these are related to the velocities $c_l$ and $c_t$ of the corresponding longitudinal and transverse sound waves (of wavelength $\lambda$ such that $a\ll\lambda\ll L$) by
\begin{eqnarray}\label{2.1.6}
\chi_{l,t}=\rho c_{l,t}^2
\end{eqnarray}
where $\rho$ is the mass density of the material. As expected, the continuum approximation leads to $Q^{-1}=0$. Clearly, we must take into account a Hamiltonian more general than (\ref{2.1.2}) to describe the acoustic properties of disordered solids.
\section{The Stress-Stress Correlation Function}
To go beyond the continuum approximation, we add an arbitrary ``non-phonon'' term $\hat{H}'(e_{ij})$ to eqn(\ref{2.1.2}),
\begin{eqnarray}\label{2.2.3}
\hat{H}(e_{ij})\equiv \hat{H}_{el}(e_{ij})+\hat{H}'(e_{ij})
\end{eqnarray}
Here, the purely elastic contribution $H_{el}$, is given by
\begin{eqnarray}\label{2.2.1}
\hat{H}_{el}(e_{ij})=\mbox{const.}+\int \frac{1}{2}d^3r\sum_{ijkl}\chi_{ij:kl}^{(0)}e_{ij}(\vec{r})e_{kl}(\vec{r})+\frac{1}{2}\sum_i\rho\dot{\vec{u}}_i^2(\vec{r})
\end{eqnarray}
Of course, the second term is meaningful if the velocity field $\dot{\vec{u}}$ is slowly varying over distances $a$. The ``non-phonon'' term $H'(e_{ij})$ is completely general; we neither assume that it is small compared to the purely elastic contribution $H_{el}$, nor do we identify it with any particular excitation or defect. As above, we define the ``elastic'' contribution to the stress tensor $\hat{T}_{ij}$ by
\begin{eqnarray}\label{2.2.2}
\hat{T}_{ij}^{(el)}\equiv\sum_{ijkl}\chi_{ij:kl}^{(0)}e_{kl},
\end{eqnarray}
and as we have done in (\ref{2.1.2}) and (\ref{2.1.3}) we can define the ``non-phonon'' contribution to the stress tensor by
\begin{eqnarray}\label{2.2.4}
\hat{H}'&=&\hat{H}'_0+\sum_{ij}e_{ij}\hat{T}_{ij}'+\mathcal{O}(e^2)\\
\hat{T}_{ij}'&=&\partial \hat{H}'/\partial e_{ij}
\end{eqnarray}
Note that the strain $e_{ij}$ may be due to thermal phonons, as well as experimental probing strains. Since all the contribution to $Q^{-1}$ comes from the non-phonon contributions, for the sake of simplifying our notation we will omit the primes in $\hat{H}_0'$ and $\hat{T}_{ij}'$ from now on, and used the unprimed symbols $\hat{H}_0$ and $\hat{T}_{ij}$ to denote non-phonon contributions.
We will now define the non-phonon linear response function at scale $L$. A sinusoidal strain field with infinitesimal (real) amplitude $e_{ij}$,
\begin{eqnarray}\label{2.2.6}
e_{ij}(\vec{r},t)=e_{ij}[e^{i(\vec{k}.\vec{r}-\omega t)}+e^{-i(\vec{k}.\vec{r}-\omega t)}]
\end{eqnarray}
will give rise to a stress response, $\langle T_{ij}\rangle$ (which in general is complex):
\begin{eqnarray}\label{2.2.7}
\langle T_{ij}\rangle(\vec{r},t)=\langle T_{ij}\rangle e^{i(\vec{k}.\vec{r}-\omega t)}+\langle T_{ij}\rangle^* e^{-i(\vec{k}.\vec{r}-\omega t)},
\end{eqnarray}
The complex linear response function $\chi_{ij:kl}(\vec{q},\omega)$ is defined in the standard way
\begin{eqnarray}\label{2.2.8}
\chi_{ij,kl}(q,\omega)=\frac{1}{V}\frac{\partial \langle T_{ij}\rangle(\vec{q},\omega)}{\partial e_{kl}}
\end{eqnarray}
In practice, we will be interested in the $\lambda\gg a$ limit. We may therefore work in the linear dispersion limit
\begin{eqnarray}
\chi(\omega,\vec{q})\approx\chi(\omega,\omega/c_{l,t})\equiv\chi(\omega)
\end{eqnarray}
It is not immediately obvious that the non-phonon response function $\chi_{ij:kl}(\vec{q},\omega)$ will have the isotropic form analogous to (\ref{2.1.5}), especially at the latter stages of the renormalization where we will be considering amorphous cubes with sizes comparable to the wavelength; however, since it is clear that any complications associated with this consideration are sensitive at our arbitrary choice of building-block shape, we will assume that a more rigorous (q-space) calculation will get rid of them, and thus assume that $\chi_{ijkl}(\omega)$ will have the same isotropic form as (\ref{2.1.5}), thereby defining ``longitudinal'' and ``transverse'' response functions $\chi_{l,t}(\omega)$ for cubes of size $L\gg a$.
We may now calculate the absorption time $\tau^{-1}=cl^{-1}$, in terms of the imaginary part of the linear response function. From Fermi's Golden rule,
\begin{eqnarray}\label{2.2.9}
Q_\alpha^{-1}(\omega)\equiv\frac{\lambda}{2\pi^2l} =\frac{1}{\pi\rho c_\alpha^2}\mbox{Im}\chi_\alpha(\omega)
\end{eqnarray}
where the quantity $\mbox{Im}\chi_\alpha(\omega)$ is given explicitly, in the representation in which $\hat{H}_0$ is diagonal, by the formula
\begin{eqnarray}
\mbox{Im}\chi_{ij:kl}(\omega)=\sum_mp_m\chi^{(m)}_{ij:kl}(\omega)\label{2.2.10}\\
\chi_{ij:kl}^{(m)}(\omega)=\frac{\pi}{L^3}\sum_n\langle m|T_{ij}|n\rangle\langle n|T_{kl}|m\rangle\delta(E_{n}-E_{m}-\omega)\label{2.2.11}
\end{eqnarray}
where $|m\rangle$ and $|n\rangle$ denote exact many-body eigenstates of $\hat{H}_{0}$, with energies $E_m$, $E_n$, and $p_m$ is the probability that at thermal equilibrium the system initially occupies state $m$,
\begin{eqnarray}
p_m=\frac{1}{Z_\beta}e^{-\beta E_m}
\end{eqnarray}
where $Z$ is the partition function.
Our main objective is to calculate (\ref{2.2.10}), which depends on many-body energy levels and stress tensor matrix elements. Note that the formula is quite general; substituting TTLS assumptions and parameters in it directly gives eqn(\ref{mfp}).
As an interesting side note, we point out that the TTLS mean free path is independent of frequency at zero temperature, which causes the Kramers-Kronig integral
\begin{align}\label{kkdiscuss}
\frac{\Delta c}{c}=\frac{2}{\pi}\int_0^\infty\frac{\mbox{Im}\chi(\omega')}{\omega'}d\omega'
\end{align}
to diverge logarithmically at zero frequency. This suggests that the actual zero temperature form of $Q_{\alpha}(\omega)$ is a weakly decreasing function of decreasing $\omega$. We will discuss this matter further in Chapter-4, and propose that the actual frequency dependence is $\sim\log^{-1} (U/\omega)$, which is the closest form to a constant that prevents (\ref{kkdiscuss}) from diverging\footnote{strictly speaking, we require $Q^{-1}(\omega)=\lim_{\epsilon\to0^+}1/\log^{1+\epsilon}U/\omega$.}
\section{Virtual Phonon Exchange between Blocks}
Let us define a ``block'' to be all the non-phonon degrees of freedom in a region enclosed by a cube of volume $L^3$ and consider a large collection of \emph{bare} uncorrelated blocks as described by the Hamiltonian (\ref{2.2.4}). Since the strain $e_{ij}$ includes the phonon field, the exchange of phonons must give rise to an effective coupling between pairs of block stress tensors. The phonon degrees of freedom are harmonic; therefore the stress-stress coupling should have the generic form
\begin{eqnarray}\label{2.3.1}
H_{int}^{(12)}=\int_{V_1}d\vec{r}\int_{V_2}d\vec{r}'\sum_{ijkl}\Lambda_{ijkl}(\vec{r}-\vec{r}')T_{ij}(\vec{r})T_{kl}(\vec{r'}).
\end{eqnarray}
The function $\Lambda_{ijkl}(\vec{r}-\vec{r}')$ is calculated in the paper of Joffrin and Levelut\cite{joffrin75}. For ``large'' $r-r'$. it has the form\cite{esquinazi-rev}
\begin{eqnarray}\label{interaction}
\Lambda_{ijkl}(\vec{r}-\vec{r}')=\frac{1}{\rho c^2_t}\frac{1}{2\pi|\vec{r}-\vec{r}'|^3}\tilde{\Lambda}_{ijkl}(\vec{n})\label{2.3.2}\nonumber\\
\tilde{\Lambda}_{ijkl}=-(\delta_{jl}-n_jn_l)\delta_{ik}+\left(1-\frac{c_t^2}{c_l^2}\right)[-\delta_{ij}\delta_{kl}-\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}\nonumber\\
+3(\delta_{ij}n_kn_l+\delta_{ik}n_jn_l+\delta_{il}n_kn_j+\delta_{jk}n_in_l+\delta_{jl}n_in_k+\delta_{kl}n_in_j)-15n_in_kn_kn_l]\label{2.3.3}
\end{eqnarray}
where $\vec{n}$ is the unit vector along $\vec{r}-\vec{r}'$. The interaction ceases to have the $1/r^3$ form for length scales smaller than $r_0$, where
\begin{align}\label{cutoff}
\langle V(r_0)\rangle\sim\frac{hc}{r_0}\equiv U_0.
\end{align}
Beneath this length-scale (beyond this energy scale) the interaction becomes oscillatory, and this can be taken into account by introducing a cutoff energy level $U_0$ to any integral or sum over energies. In the present theory, the cutoff energy associated with the ``microscopic'' blocks is one of the input parameters, and will be discussed further in the next section.
Due to condition (\ref{cutoff}), the virtual phonon wavelengths considered in any stage of this calculation will be larger than the block size. Thus, we will assume that $\vec{r}-\vec{r'}$ can be replaced with the distance between the center points of two blocks $R_1-R_2$, and that the stress \[T_{ij}\approx\int_{v_s}\hat{T}_{ij}(\vec{r})d\vec{r}\] is \emph{uniform} throughout a single block. Then, the interacting many-body Hamiltonian for a collection of blocks can be written as
\begin{eqnarray}\label{2.3.4}
\hat{H}_N=\sum_{s=1}^N\hat{H}_0^{(s)}+\sum_{\substack{s,s'=1\\s< s'}}^N\sum_{ijkl}\Lambda_{ijkl}(\vec{R}_s-\vec{R}_s')T_{ij}^{(s)}T_{kl}^{(s')}.
\end{eqnarray}
Eqn(\ref{2.3.4}) represents the Hamiltonian $H_0$ of the ``super block'' (of side $\sim N^{1/3}L$) composed by the $N$ blocks of side $L$; in principle, we can redefine the energy levels and stress tensor matrix elements for a super block and iterate the procedure untill we reach experimentally realistic length scales. At a given stage of renormalization, every pair $T^{(s)}T^{(s')}$ comes together with a $\Lambda$ which is proportional to $L^{-3}$ (note the same factor in the definition of $\chi$). Thus it is not difficult to see that the procedure is scale invariant, and the renormalization group equation of $Q^{-1}$ might converge to a fixed point with increasing volume.
Given infinite computational resources, one could start from arbitrary microscopic forms of $\hat{H}_0$ and $\hat{T}_0$, turn-on the interactions (\ref{2.3.1}) between blocks, diagonalize the N-body Hamiltonian $H_N$, and $T_{ij}=\partial H_N/\partial e_{ij}^{(N)}$ and iterate the procedure to see whether or not the final forms $H_N$ and $T_{ij}$ at the macroscopic scale are independent of the starting $\hat{H}_0$ and $\hat{T}_0$.
Unfortunately, the number of levels and matrix elements grow exponentially with increasing volume. Therefore it is not feasible to solve the problem stated in this form. We will instead narrow down our input parameters and focus on few output observables that can be deduced analytically.
\section{Input Parameters}
We believe that it is one of the strengths of the present work that our results do not rely on adjustable parameters, or the existence of \emph{other} microscopic (unmeasurable) universal ratios to explain the observable one \cite{yu89, wolynes01,turlakov04,parshin94,stamp09} (though cf.\cite{burin96}). The only two inputs on which our outcome depends sensitively are the ratios $c_l/c_t$ and $\chi_l/\chi_t$ (cf. below for details of the notation) both of which are observed \emph{experimentally} to vary little between different amorphous systems (cf. also Appendix). Our third input $r_0$, which is the size of a ``microscopic amorphous block'' (defined below) only enters into our equations logarithmically.
\subsection{Microscopic Building-Block Size}
As discussed in the introduction, the temperature dependence of thermal conductivity and specific heat changes at a critical temperature $T_c\sim 1K-30K$. At $10K$ the dominant phonons have wavelengths of the order $50\AA$, and in \cite{leggett91} it is argued that this is just the scale at which we get a crossover from ``Ising'' to ``Heisenberg'' behavior. Essentially, it is this length scale at which the approximations used in obtaining the simple $R^{-3}$ form of $\Lambda$, eqn(\ref{2.3.1}) breaks down. Therefore, we take the microscopic starting size to be about $r_0\sim 50\AA$, which is still much greater than $a$. We identify the ultraviolet cutoff $U_0$ with $T_c$, and expect the universality to break down as the block size becomes comparable to atomic size.
$U_0$ is the only quantity in the present work that is not \emph{directly} measurable. However we should emphasize that (a) strictly speaking $U_0$ is not a free parameter, since on a priori grounds we can assign an approximate value to it (b) it can be experimentally obtained \emph{indirectly}, from thermal conductivity and specific heat data through $T_c=hc/r_0$. Most importantly, (c) our results, if they depend at all, depend on $U_0$ only logarithmically.
\subsection{Meissner-Berret Ratios}
The second and third input parameters we will be using are the ratio of longitudinal to transverse speed of sound $c_t/c_l$ and phonon coupling constants $\chi_t/\chi_l$, which are known to be universal among materials up to a factor of 1.2 (cf. ref\cite{mb}, Fig.1 and 3). Even though we use as inputs the experimentally obtained values, it is not difficult to produce first-principle theoretical justifications for either (cf. below).
Suppose that the inter-atomic interactions (which reduce to (\ref{2.2.1}) for small strains) is due to some arbitrary inter-atomic (or inter-block) interaction $\phi(r)$. Namely,
\begin{eqnarray}
\hat{H}_{el}(e_{ij})=\mbox{const.}+\sum_{ab}^N\phi(r_{ab})+\frac{1}{2}\sum_i\rho\dot{\vec{u}}_i^2(\vec{r})
\end{eqnarray}
According to the virial theorem, the potential energy expectation value of a harmonic degree of freedom is equal to that of the kinetic energy, therefore for the purposes of obtaining the ratio $c_l/c_t$ without loss of generality we will drop the latter, and the const. term.
\begin{eqnarray}
\langle H\rangle=\sum_{ab}\langle\phi_{ab}\rangle.
\end{eqnarray}
Further, suppose that the relative displacement $u_{ab}$ of two blocks $a$ and $b$ are proportional to the distance $r_{ab}$ between them. By definition of $e$, it follows that.
\begin{eqnarray}\label{trivial}
u_{ab,x}=e_{xy}r_{ab,y}\\
u_{ab,x}=e_{xx}r_{ab,x}
\end{eqnarray}
While this is clearly true for length scales for which $r\gg a$, in an amorphous structure large deviations from this may occur locally for $r$ of the same order as $a$. However these should even out in the thermodynamic limit $N\to\infty$.
The speed of sound is related to the real part of the zero frequency response function according to (\ref{2.1.6}).
\begin{eqnarray}\label{mb-main}
\chi_{0t,l}\approx\left.\frac{\partial^2}{\partial e_{ij}^2}\sum_{a<b}^N\langle\phi(r_{ab})\rangle\right|_{e_{ij}=0}=\sum_{a<b}^N\left[\frac{\partial|r_{ab}|}{\partial e_{ij}^2}\frac{\partial \phi(r_{ab})}{\partial r_{ab}}+\frac{\partial^2\phi(r_{ab})}{\partial r_{ab}^2}\left(\frac{\partial r_{ab}}{\partial e_{ij}}\right)^2\right]_{e_{ij}=0}
\end{eqnarray}
where $i\neq j$ and $i=j$ give $\chi_t$ and $\chi_l$ respectively. The first term of the right hand side must be zero due to stability requirements. Thus all we need to do is to substitute (\ref{trivial}) into the second term. For a purely transverse strain
\begin{eqnarray}
r_{ab}=\sqrt{(r_{ab,x}+e_{xy}r_{ab,y})^2+r_{ab,y}^2+r_z^2},
\end{eqnarray}
which can be differentiated twice and substituted in (\ref{mb-main}). Letting $\sum_{ab}\to L^{-3}\int r^2drd\Omega$,
\begin{eqnarray}\label{chit}
\chi_{0t}=\frac{1}{L^3}\int \frac{\phi''(r)}{r^2}r^2dr\int r_x^2r_y^2 d\Omega.
\end{eqnarray}
where $\Omega$ is the solid angle. Similarly, for a purely longitudinal strain
\begin{eqnarray}
r_{ab}=\sqrt{(r_{ab,x}+e_{xx}r_{ab,x})^2+r_{ab,y}^2+r_z^2}.
\end{eqnarray}
Doing the same as above, we get
\begin{eqnarray}\label{chit}
\chi_{0l}=\frac{1}{L^3}\int \frac{\phi''(r)}{r^2}r^2dr\int r_x^4 d\Omega.
\end{eqnarray}
Then, the ratio for the speeds of sounds only depend on the angular integrals,
\begin{eqnarray}
\frac{c_t}{c_l}=\sqrt{\frac{\chi_{0t}}{\chi_{0l}}}=\frac{1}{\sqrt{3}}
\end{eqnarray}
which is $6\%$ larger than the experimental (average) value.
The second ratio $\mbox{Im}\chi_t/\mbox{Im}\chi_l$ is not as trivial to obtain, since these ratios come from the non-phonon degrees of freedom. However an argument similar to the above can be made if we consider a second order perturbation expansion of the ground state in the interaction $V$, and differentiate it twice with respect to strain $e$. Then the (short range) interaction $\phi(r)$ must be replaced with the square of the (long range) elastic coupling (\ref{2.3.1}), and the above argument goes through.
\subsection{Many-Body Density of States}
On general grounds, we can assume that the normalized density of states of the interacting and noninteracting system, can be written as a power series,
\begin{eqnarray}
f(\omega)=\sum_kc_k\omega^k\label{dosassumption1}\\
f_0(\omega)=\sum_kc_{0k}\omega^k\label{dosassumption2}
\end{eqnarray}
with dominating powers much larger than unity. Note that the density of states $F(E)$ of a composite system is given by the convolution of the density of states $f_i(E)$ of the constituent objects;
\begin{eqnarray}
F(E)=(f1*f2*f3*\ldots)(E)
\end{eqnarray}
where
\begin{eqnarray}
(f1*f2)(E)\equiv\int_{-\infty}^{\infty}f(E-\omega')f(\omega')d\omega'
\end{eqnarray}
From which it follows the dominating power of the density of states must be proportional to the number of particles if the system is extensive in energy. Thus, (\ref{dosassumption1}) and (\ref{dosassumption2}) holds quite generally.
Finally, while the actual many-body density of states we deduce from the specific heat data (cf. Appendix-B),
\begin{align}\label{2.4.11}
f(E)= \mbox{const.}e^{(NE/\epsilon_0)^{1/2}}
\end{align}
is consistent with the form assumed above, as we will see, none of our results will be sensitive to the precise choice of $c_n$.
\subsection{Initial State Dependence}
Finally, we will specify the initial state $|m\rangle$ dependence of the response function, which will be the simplest possible choice consistent with our general assumptions, namely the ``random form'',
\begin{eqnarray}\label{isi}
\chi_\alpha^{(m)}(\omega)=\mbox{const}.\theta(E_m+\omega)=(\rho c_\alpha^2)Q_0^{-1}\theta(E_m+\omega)
\end{eqnarray}
Important: This form will not be used until we consider the frequency dependence of $Q^{-1}$ in Chapter 4 (cf. eqn. \ref{freqdependence}). Any conclusion we reach in Chapter 3 is not sensitive to this assumption!
It should be carefully noted that the TTLS form of $\chi_\alpha^{(m)}$ is \emph{not} a special case of eqn(\ref{isi}); this may be seen by noting that the form of $Q^{-1}$ given by the latter is approximately,
\begin{eqnarray}\label{mfpdcv}
Q^{-1}(\omega,T)=Q_0^{-1}(1-e^{-\omega/T}).
\end{eqnarray}
which is different from eqn (\ref{mfp}), though not qualitatively so. An important point that should be emphasized is, if the non-phononic hamiltonian consisted entirely of harmonic oscillators $Q^{-1}$ would be independent of temperature, which is qualitatively very different than eqn(\ref{mfp}). Intuitively, the ansatz (\ref{mfpdcv}) describes a model intermediate between a harmonic-oscillator and the TTLS one, but in some sense close to the latter. In principle, with the knowledge of the many body density of states (and thus partition function), a more accurate functional form for the initial state dependence can be found. It is our hope that a more realistic m-dependence does not qualitatively alter the calculations that will follow in the next sections.
\chapter{The Universality of the Average Attenuation}
We introduce this chapter by discussing a simpler system (cf. \cite{fisch80,leggett91}) in which a collection of spin-like objects couple to strain through a coupling coefficient $\gamma$. As a result of this interaction we will get an effective ``spin-spin'' interaction which is roughly of the form $g/r^3$, where \[g=\frac{\eta \gamma^2}{\rho c^2}\]
with $\eta$ a dimensionless number of order 1. If the single-spin excitation spectrum $\bar{P}$ is assumed to be independent of energy, it follows on dimensional grounds \cite{fisch80} that \[\bar{P}\propto\frac{1}{g}.\]
In this model, the dimensionless attenuation coefficient is simply \[Q^{-1}=\frac{\pi\gamma^2\bar{P}}{2\rho c^2}.\]
Furthermore if we include all phonon modes (longitudinal and transverse) this number if reduced by a factor of 3. Our purpose in this chapter is to generalize this argument \cite{fisch80} to a more generic model. We will (a) not necessarily assume ``single particle'' excitations, and (b) take into account not only different phonon modes, but all tensor components, and show that (a) and (b) alone lead to a surprisingly small value of frequency and initial state averaged attenuation, $\langle Q^{-1}\rangle$.
\section{Coupling Two Generic Blocks}
The central quantity we will be interested in this sectionis the frequency and initial state averaged ultrasound attenuation coefficient, defined as
\[\langle Q_0^{-1}\rangle=\frac{1}{U_0N_b}\sum_{n}\int_0^{U_0} Q_{n}^{-1}(\omega-E_n)d\omega\]
where $N_b$ is the number of levels of the block, and $U_0$ is an energy level of the order $U_0=hc_\alpha/L$, but the precise value is not essential for our purposes below.
Our strategy will be to evaluate the quantity $M=\mbox{Tr}(V^2)$ in the $H_0$ and $H_0+V$ eigenbasis. Let us start by considering two blocks only, labeled by $1$ and $2$. We will follow Einstein's summation convention for tensor indices.
\begin{eqnarray*}
\hat{V}^2&=&\left(\Lambda_{ijkl}\hat{T}_{1,ij}\hat{T}_{2,kl}\right)^2\\
&=&\Lambda_{ijkl}\Lambda_{i'j'k'l'}T_{1,ij}T_{1,i'j'}T_{2,kl}T_{2,k'l'}
\end{eqnarray*}
If $H_0$ has eigenvectors $\{|n_0\rangle\}$ $M$ can be evaluated as
\begin{align}\label{tr2}
M=\sum_{mn}\Lambda_{ijkl}\Lambda_{i'j'k'l'}\langle m_0|T_{1,ij}T_{1,i'j'}|n_0\rangle\langle n_0|T_{2,kl}T_{2,k'l'}|m_0\rangle
\end{align}
Remember that $\Lambda$ depends on the relative positions of block 1 and 2.
Any expression of the form
\begin{align}
I= \sum_nF(E_{n}-E_{m})\langle n_0 |T_{ij}|m_0 \rangle\langle n_0 |T_{kl}|m_0 \rangle
\end{align}
can be written in terms of an integral of $\chi_0$ by inserting unity inside the sum,
\begin{align}\label{unity}
I&=\int\sum_nF(E_{n}-E_{m})\delta(E_{n}-E_{m}-\omega)\langle n_0 |T_{ij}|m_0 \rangle\langle n_0 |T_{kl}|m_0 \rangle d \omega\\
&=\int F(\omega)\chi_{0m,ijkl}(\omega)d\omega
\end{align}
Doing this twice in eqn(\ref{tr2}) the the trace can be written in terms of $\chi_0(\omega)$
\begin{eqnarray}\label{trace}
M=v_1v_2\sum_{n_1n_2}\int\int\Lambda_{ijkl}\Lambda_{i'j'k'l'}\chi_{0,n_1,iji'j'}(\omega')\chi_{0,n_2,klk'l'}(\omega'')d\omega'd\omega''
\end{eqnarray}
Here $v_1$ and $v_2$ are the volumes of block $1$ and $2$ respectively and both the sums and integrals are over the whole spectrum. A brief reminder of notation: The first subscript 0 means $\chi$ is the response of ``noninteracting'' blocks. the second ones $n_1$ and $n_2$ denote the eigenstate block 1 and 2 initially occupy. The primed and unprimed $i,j,k,l$ are tensor components.
Since an amorphous solid is isotropic all $3^8$ components of $\chi_{iji'j'}\chi_{klk'l'}$ can be expressed in terms of two independent constants. We chose the longitudinal $\chi_l$ and transverse $\chi_t$ elastic coefficients.
\[\chi_{ijkl}=(\chi_l-2\chi_t)\delta_{ij}\delta_{kl}+\chi_t(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\]
This holds true for any initial state (to avoid clutter we drop the initial state subscript till eqn(\ref{finaltr})). Let us define $x$,
\[x+2\equiv\frac{\chi_l}{\chi_t}\]
so that
\[\chi_{ijkl}=(x\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\chi_t.\]
which simplifies the massive sum of eqn(\ref{trace}) into,
\begin{align}
\!\Lambda_{ijkl}\Lambda_{i'j'k'l'}\chi_{iji'j'}\chi_{klk'l'}=
\!\Lambda_{ijkl}\Lambda_{i'j'k'l'}(x\delta_{ij}\delta_{i'j'}+\delta_{ii'}\delta_{jj'}+\delta_{ij'}\delta_{ji'})\\
\times(x\delta_{kl}\delta_{k'l'}+\delta_{kk'}\delta_{ll'}+\delta_{kl'}\delta_{lk'})\chi_t^2.
\end{align}
The symmetries of $\Lambda_{ijkl}$ simplifies matters a bit further,
\[\Lambda_{ijkl}=\Lambda_{kjil}=\Lambda_{ilkj}=\Lambda_{klij}.\]
Substituting the experimental values \cite{mb} of $\mu_l/\mu_t$ and $\chi_l/\chi_t$ into $\tilde{\Lambda}_{ijkl}$, the entire sum can be evaluated in terms of $x$,
\[\frac{1}{\mu_t^2}\tilde{\Lambda}_{ijkl}\tilde{\Lambda}_{i'j'k'l'}\chi_{0,iji'j'}\chi_{0,klk'l'}\approx122Q^{-2}_{0t}\equiv KQ^{-2}_0\]
a number independent of the relative orientation of the blocks. Thus, eqn(\ref{trace}) becomes
\begin{eqnarray}\label{finaltr}
M=K\frac{v_1v_1}{v_{12}^2}\sum_{n_1n_2}\int\!\!\int\!\! Q_{0,n_1}^{-1}(\omega')Q_{0,n_2}^{-1}(\omega'')d\omega'd\omega''
\end{eqnarray}
where $v_{12}$ is the volume of the sphere with radius equal to the distance between the two blocks. Notice that $M$ is precisely proportional to the square of $\langle Q^{-1}\rangle$. Assuming that the blocks are statistically identical, we can write
\begin{align}\label{t00}
M_0=K\frac{v_av_b}{v_{ab}^2}U_0^2N_b^2\langle Q_{0}^{-1}\rangle^2
\end{align}
Now that we have evaluated $M$ in the eigenbasis of $H_0$ which allowed us to express it in terms of $Q_0$, $N_b$ and $U_0$, we could also repeat the same steps, this time in the eigenbasis of $H=H_0+V$, and express the trace in terms of the above quantities $Q$, $N_b$ and $U$, now modified due to the presence of $V$.
\begin{eqnarray}\label{t0}
M=K\frac{v_1v_1}{v_{12}^2}U^2N_b^2\langle Q^{-1}\rangle^2
\end{eqnarray}
Note that $\langle Q^{-1}\rangle$ is \emph{not} necessarily the averaged absorption of the superblock 1+2 (or even related to it by a simple numerical factor), because the definition of the latter involves the squared matrix elements of the \emph{total} stress tensor of the superblock, $(\hat{T}_{ij}^{(1)}+\hat{T}_{ij}^{(2)})$ and thus contains terms like $\langle n|\hat{T}_{ij}^{(1)}|m\rangle\langle m|\hat{T}_{ij}^{(2)}|n\rangle$ (where $|m\rangle, |n\rangle$ now denote eigenstates of $\hat{H}$); while such terms were originally (in the absence of $\hat{V}$) uncorrelated, it is not obvious that they remain uncorrelated after $\hat{V}$ is taken into account. We shall, however, argue that on average those terms are likely to be small compared to terms of the form $|\langle m|T_{ij}^{(s)}|n\rangle|^2$, because $\hat{V}$ involves \emph{all} tensor components of $\hat{T}^{(1)}$ and $\hat{T}^{(2)}$ while the correlation only involves the \emph{same} component of $\hat{T}^{(1)}$ and $\hat{T}^{(2)}$. If this argument is accepted, we can identify the $\langle Q^{-1}\rangle$ in (\ref{t0}) with the physical inverse absorption of the superblock.
$\mbox{Tr}\hat{V}$ is a scalar, and of course its value is independent of which basis we evaluate it. Equating (\ref{t0}) to (\ref{t00})
\begin{eqnarray}\label{invariance}
U_0^2\langle Q_0^{-1}\rangle^2=U^2\langle Q^{-1}\rangle^2.
\end{eqnarray}
To relate $\langle Q_0^{-1}\rangle$ to the interacting one $\langle Q^{-1}\rangle$, we must find $(U)/(U_0)$. This will be done using the following argument: Let us square our interacting Hamiltonian of eqn(\ref{2.3.4}) and consider its trace over the same manifold as that defined by the integration limits in (\ref{tr2}).
\begin{align}\label{uncorrelatedT}
\mbox{Tr}(H^2)=\mbox{Tr}(H_0^2)+\mbox{Tr}(H_0V)+\mbox{Tr}(VH_0)+\mbox{Tr}(V^2)
\end{align}
Since pairs of stress tensors are uncorrelated, we can neglect the second and third terms when evaluating the expression in the noninteracting basis. Thus,
\begin{eqnarray}\label{tr1}
\mbox{Tr}(H^2)-\mbox{Tr}(H_0^2)=\mbox{Tr}V^2
\end{eqnarray}
The \textit{lhs} is the change in the variance of energy levels. Since the \textit{rhs} is positive, we can see that the physical effect of the interaction is to spread the energy levels. This equation can simply be written in terms of the density of states and eqn(\ref{t0}),
\begin{eqnarray}\label{dos1}
\int_0^U\omega^2 f(\omega)d\omega-\int_0^{U_0}\omega^2f_0(\omega)d\omega=K\frac{v_av_b}{v_{ab}^2}U_0^2N_b^2\langle Q_{0a}^{-1}\rangle\langle Q_{0b}^{-1}\rangle.
\end{eqnarray}
In general, a macroscopic quantum mechanical system has a density of states roughly given by
$f(\omega)\propto\omega^n$, where $n$ is a large exponent proportional to the number of constituent particles. However we can afford to be more general and assume
\begin{align}\label{assumef}
f_0(\omega)&=\sum_nc_{0n}\omega^{n_0} \\
f(\omega)&=\sum_nc_{n}\omega^n.
\end{align}
where the dominating terms have power much greater than unity (cf. eqn(\ref{dosintegral})). Note that the experimentally measured temperature dependence of specific heat $C_\nu\sim T$ is consistent with this form for the many-body density of states. In Appendix-B this consistency is shown, and (although not necessary for the present argument) the precise form of $c_n$, as well as an approximate closed form for $f$ is derived. For each set of coefficients $c_n$ and $c_{0n}$ the normalization conditions require that the two body system has $N_b^2$ levels whether they interact or not
\begin{eqnarray}\label{N}
\sum_{n=0}^{p}c_{0n}\frac{U_0^{n_0+1}}{n_0+1}=\sum_{n=0}^{p}c_n\frac{U^{n+1}}{n+1}=N_b^2\nonumber\\
\end{eqnarray}
Since both integrals in eqn(\ref{dos1}) have the form
\[I=\int_0^U\omega^2f(\omega)d\omega=\sum_{n=0}c_n\frac{U^{n+3}}{n+3},\]
if the terms for which $n+3\approx n+1$ dominate the density of states, we can use eqn(\ref{N}) to write
\begin{eqnarray}\label{dosintegral}
I=U^2\sum_{n=0}c_n\frac{U^{n+1}}{n+3}\approx U^2N_b^2.
\end{eqnarray}
Thus, eqn(\ref{dos1}) becomes
\begin{eqnarray}\label{connection}
N_b^2(U^2-U^2_0)=\langle Q_{0a}^{-1}\rangle\langle Q_{0b}^{-1}\rangle N_b^2U_0^2K\frac{v_av_b}{v_{ab}^2}.
\end{eqnarray}
Eqn(\ref{connection}) and eqn(\ref{invariance}) are sufficient to solve for $\langle Q^{-1}\rangle\equiv \langle Q_{a+b}^{-1}\rangle$.
\begin{eqnarray}\label{final}
\langle Q^{-1}\rangle=\left[\frac{1}{\langle Q_{0a}^{-1}\rangle\langle Q_{0b}^{-1}\rangle}+K\frac{v_av_b}{v_{ab}^2}\right]^{-1/2}.
\end{eqnarray}
This equation connects the attenuation coefficient of two non-interacting blocks to that of two interacting blocks.
\section{Coupling N Generic Blocks: Renormalization Group}
We shall use eqn(\ref{final}) to continue adding blocks till we reach experimental length scales. Let start by putting two blocks of side $r_0$ next to each other, so that we have a super-block with dimensions $2r_0\times r_0\times r_0$ (Such as A + H in Fig\ref{fig:cube}). For this super-block ($sb_1$).
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{cube-eps-converted-to.pdf}
\caption{\small \sl A superblock of unit length consisting of 8 blocks.\label{fig:cube}}
\end{center}
\end{figure}
\[\left(\frac{v_av_b}{v_{ab}^2}\right)_{sb_1}=\frac{1}{16\pi^2/9}\]
and
\[\langle Q_{sb_1}^{-1}\rangle=\left[\frac{1}{\langle Q_{0}^{-1}\rangle^2}+\frac{9K}{16\pi^2}\right]^{-1/2}.\]
Next we combine two $sb_1$'s to obtain a larger super-block ($sb_2$) with dimensions $2r_0\times 2r_0\times r_0$ (Such as AH + BC in Fig\ref{fig:cube}). This time,
\[\left(\frac{v_av_b}{v_{ab}^2}\right)_{sb_2}=\frac{4}{16\pi^2/9}\]
and
\[\langle Q_{sb_2}\rangle=\left[\frac{1}{\langle Q_{0}^{-1}\rangle^2}+\frac{9K}{16\pi^2}+4\frac{9K}{16\pi^2}\right]^{-1/2}.\]
Finally, we extend one more time to obtain a super-block with the same shape as the original ($sb_3$), but with dimensions $2r_0\times2r_0\times2r_0$ (Such as AHBC + FEGD in Fig\ref{fig:cube}).
\[\left(\frac{v_av_b}{v_{ab}^2}\right)_{sb_3}=\frac{16}{16\pi^2/9}\]
and denoting $Q_{sb_3}^{-1}=Q_{2L}^{-1}$ and $Q_{0}^{-1}=Q_{L}^{-1}$, we relate the attenuation of a cube with sides $2L$ to that of a cube with sides $L$
\begin{eqnarray}\label{selfsimilar}
\langle Q_{2L}^{-1}\rangle&=&\left[\frac{1}{\langle Q_{L}^{-1}\rangle^2}+\frac{9K}{16\pi^2}+4\frac{9K}{16\pi^2}+16\frac{9K}{16\pi^2}\right]^{-1/2}\nonumber\\
&=&\left[\frac{1}{\langle Q_{L}^{-1}\rangle^2}+K_0\right]^{-1/2}
\end{eqnarray}
where $K_0\approx150$. Eqn(\ref{selfsimilar}) is our central result. It has the very attractive feature that the value of $\langle Q^{-1}\rangle$ is very weakly dependent on $v_0=r_0^3$, and $Q_0^{-1}$, since $K\gg1/\langle Q_0^{-1}\rangle$. We now consider the effect of iterating the step which led to (\ref{selfsimilar}), by combining eight cubes of size $2L$ to make one of size $4L$; for convenience we keep the original definition of the low-energy manifold ($E_m<U_0$), though other choices are also possible. Since the only point at which $U_0$ actually enters the result is implicitly in the definition of the ``average'' in $\bar{Q}^{-1}$ the scale invariant nature of the problem implies that all considerations are exactly the same as at the first stage, and we simply recover (\ref{selfsimilar}) with the replacement of $L$ by $2L$ and $2L$ by $4L$. Continuing the iteration up to a spatial scale $R$, we find
\begin{align}\label{3.2.12}
\langle Q^{-1}(R)\rangle=\left[\langle Q_0^{-1}\rangle^2+K_0\log_2(R/r_0)\right]^{-1/2}
\end{align}
where $r_0$ is the linear dimension of the starting block.
Eqn(\ref{3.2.12}) predicts a remarkable counter-intuitive low-temperature effect that would be interesting to test experimentally: As $v\to\infty$, the attenuation vanishes logarithmically (see Fig\ref{fig:vanish}). While we know of no reason why this behavior must be unphysical, in practice we would guess that for any finite ultrasound wavelength $\lambda$, $R$ would be replaced by a quantity of the order $\lambda$.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=3.6in]{elasticabsorption-eps-converted-to.pdf}
\end{center}
\caption{
{The (numerical) renormalization of $\langle Q\rangle$}. Each increment in $z$ corresponds to an 8-fold increase in volume. $z=27$ corresponds to $r\sim1$m}\label{fig:vanish}
\label{fig:Figure2}
\end{figure}
It is well known that the temperature dependence of the thermal conductivity of amorphous materials changes at a temperature of the order of $1K$; in fact, most such materials show a pronounced ``plateau'' extending very roughly between 1 and 30K. At 10K the dominant phonons have wavelengths of the order of $50$\AA, and in \cite{leggett91} it is argued that this is just the scale at which we get a crossover from ``Ising'' to ``Heisenberg'' behavior (formally, at smaller scales the approximations used in obtaining the simple $R^{-3}$ form of $\Lambda$, eqn(\ref{2.3.2}) breaks down). Thus, we take the ``starting'' block size, $r_0$, to be $\sim50$\AA (which is still comfortably greater than $a$). Notice that the result (\ref{3.2.12}) depends only logarithmically on $r_0$, and thus the value of $Q^{-1}$ in the experimentally accessible lengthscales will not be particularly sensitive to this choice. Thus, for experimentally realistic values of $R$ we find
\begin{align}\label{result}
\langle Q^{-1} \rangle\sim0.015.
\end{align}
This value is surprisingly small, and more importantly very weakly dependent on the inputs $Q_0$ and $r_0$; however $\langle Q\rangle$ is larger than the experimental values measured in the MHz-GHz range. This is likely due to the contribution to the average value, of the rapid increase in $Q^{-1}$ at higher frequencies, as manifest in the thermal conductivity data around 10K. To obtain the experimentally observed absorption in the MHz-GHz range, we will consider the frequency dependence of $Q^{-1}(\omega)$ in the following chapter.
\chapter{The Frequency Dependence of the Attenuation Coefficient}
\section{Level Shifts}
The reader should be warned that the argument presented in this section is rather heuristic and unorthodox. Suppose that upon turning on the interactions the change in $\chi$ is solely due to the change in the two body density of states $f_0(\omega)$. In other words, we will be ignoring the changes in the matrix elements of $\hat{T}$. The two body density of states $f(\omega)$ after turning on the interactions $V=\Lambda T_1T_2$ can be approximately written as
\[f(\omega)=\int_0^{U_0}f_0(x)\delta(x-\omega-\Delta(\omega))dx\]
where
\begin{eqnarray}\label{pt}
\Delta(E_n)=\sum_k\frac{V^2_{kn}}{E_n-E_k}
\end{eqnarray}
is the second order correction to the energy level $E_n$. From these two equations,
\begin{align}\label{levelcross}
\frac{Q^{-1}(\omega)}{Q^{-1}_0(\omega)}\approx\frac{f(\omega)}{f_0(\omega)}\approx\frac{1}{|1+d\Delta(\omega)/d\omega|}
\end{align}
We could write $\Delta$ in terms of $Q$, as described in eqn(\ref{unity})
\begin{align}\label{pertintegral}
\Delta(\omega)=-K_0\int_{-E_{n_10_1}}^\infty\int_{-E_{n_20_2}}^{\infty}\frac{Q_{n_1}(\omega')Q_{n_2}(\omega'')\theta(U_0-|\omega'+\omega''|)}{\omega'+\omega''-\omega}d\omega'd\omega'',
\end{align}
where $E_{n_10_1}+E_{n_20_2}=\omega$. The unit step function $\theta$ imposes the ultraviolet cutoff such that $|E_n-E_k|<U$. In order to take the integral we must know the initial state $|n_1\rangle$ and $|n_2\rangle$ dependence of the absorption. For this, we use the ``random'' form discussed in subsection (2.3.4),
\begin{align}
Q_n^{-1}(\omega)=\theta(\omega+E_{n})Q^{-1}_0
\end{align}
where the attenuation coefficient of an input block $Q^{-1}_0$ is a constant. This would happen for example, if the level distribution and matrix elements were both uniformly distributed for a starting block. Then the integral in eqn.(\ref{pertintegral}) can be obtained analytically
\begin{eqnarray}\label{delta}
\Delta(\omega)=K_0[-\omega-U-\omega \log(\omega/U_0)]
\end{eqnarray}
Thus, upon turning on the interactions the density of states transforms as
\begin{align}\label{deriv}
f(\omega)=\frac{f_0(\omega)}{|1+K_0Q^{-2}_0\ln(\omega/U_0)|}
\end{align}
This of course, can be generalized to a ``single shot'' calculation where all blocks within volume $R^3$ contribute. The result is to simply replace the factor $K_0$ by $K_0\log_2(R/L)$:
\begin{align}\label{4.1.5}
Q^{-1}(\omega)&=\frac{Q_0^{-1}}{|K_0Q_0^{-2}\log_2(R/L)\ln(U_0/\omega)-1|}\\
&\approx\mbox{const.}(\ln(U_0/\omega)) \mbox{\,\,\,\,\,\,\,\,if\,\,\,\,} \omega\ll\omega_0
\end{align}
\section{Heuristic Considerations}
While eqn \ref{4.1.5} seems at least qualitatively consistent with the experimental data (see chapter 5), it is not even approximately universal since the constant is inversely proportional to $Q_0^{-1}$, and even given a cutoff at $\omega\sim U_0$ does not satisfy (\ref{3.2.12}).
There are two obvious reasons form (\ref{4.1.5}) cannot be taken to be literally. First of all, the denominator of (\ref{deriv}) is the absolute value of a negative number. Physically this means that the perturbation is so strong that the levels are crossing. Of course, we know that this cannot happen due to the ``no-level-crossing'' theorem. We will nevertheless assume that when the perturbation is calculated up to infinite order, the overall effect of the higher order terms are small in the level structure, and that the density of states is, at least in the limit $\omega\to0$ qualitatively similar to that given by the first order correction.
The second difficulty is the high-frequency divergence of $\chi$, which indicates that the levels are coming too close. While this too can obviously not be literally true (due to level repulsion) there may be some physical truth in this divergence too, since we know from the thermal conductivity data, that around $T_c$ the phonon mean free path rapidly increases by few orders of magnitude, and the divergence we are seeing in the first order correction might correspond to this rapid rise. We suppose that higher order terms in the perturbation expansion will prevent $Q^{-1}$ from diverging, and therefore place a cut off to the high end by a large value $\tilde{Q}_0^{-1}$ (which need not be equal to the microscopic $Q_0$ defined above, but presumably of the same order of magnitude). Thus, we will consider the following qualitative functional form for the attenuation coefficient
\begin{align}\label{freqdependence}
Q^{-1}(\omega)=\frac{1}{\tilde{Q}_0+A\log(U/\omega)}.
\end{align}
It is possible to support the ansatz (\ref{freqdependence}) further: We know from mean free path measurements, that $Q^{-1}$ is not ``noticeably dependent'' on $\omega$. However if we accept this statement literally, the zero frequency velocity shift as obtained by the Kramers-Kronig formula
\begin{align}\label{kramerskronig}
\frac{\Delta c}{c}=\Delta\int_0^\infty \frac{Q^{-1}(\omega')d\omega'}{\omega'}
\end{align}
diverges logarithmically. To obtain a finite speed of sound, $Q^{-1}$ must vanish faster with decreasing $\omega$. In fact, it can be shown that the closest form to a constant with non-divergent speed of sound is $Q^{-1}\sim \lim_{epsilon\to0^+}1/\log^{1+\epsilon} (U/\omega)$.
We shall therefore accept eqn(\ref{freqdependence}) to be the frequency dependence of $Q^{-1}(\omega)$, without necessarily assuming ``its value'' $A$.
\section{The Value of $Q^{-1}(1MHz)$}
Let us now ask what the value of $A$ must be, in order for eqn (\ref{freqdependence}) be consistent with result (\ref{result}). The frequency average can be evaluated analytically,
\[\langle Q\rangle=\frac{1}{U_0}\int_0^{U_0}\frac{d\omega}{1/Q_0+A\log (U_0/\omega)}=\frac{-e^{1/(AQ_0)}}{A}\mbox{Ei}(-1/(AQ_0)).\]
Here, Ei$(x)$ is a special function, defined as
\[\mbox{Ei}(x)=\int_{-\infty}^x\frac{e^t}{t}dt\]
which has the asymptotic form
\[\lim_{x\to0^+}\frac{e^{2\mbox{Ei}(-x)}}{x^2}=e^{2\gamma}\]
where $e^{2\gamma}\approx3.172\ldots$ is the Euler-Mascheroni constant. Thus,
\begin{eqnarray}\label{robust}
\langle Q\rangle=\frac{-e^{1/(AQ_0)}}{2A}\log \left(\frac{e^{2\gamma}}{A^2Q_0^2}\right),
\end{eqnarray}
from which one can obtain the value $A\sim350$ that produces $\langle Q\rangle=0.015$. Now that we know $A$, we may use eqn(\ref{freqdependence}) to find the value of $Q$ in the universal regime. The experimental probing frequencies $\omega$ are typically of the order of MHz, therefore we find
\[Q(\omega=1\mbox{MHz})=2.7\times10^{-4}.\]
which is precisely the ``typical'' experimental value. The dimensionless inverse mean free path $\lambda/l$ is then
\[\lambda/l\approx1/200\]
We emphasize that our central result, namely eqn({\ref{selfsimilar}}), concerns the value of $Q$, which is averaged over $\omega$ and $m$ symmetrically. Therefore if our assumptions on $\omega$ and $m$ were interchanged (or if an assumption pair suitably ``in between'' was used), one would still get similar numbers.
Let us relax some of our assumptions and see how sensitive the value of $Q$ is. For example, if dominating power in the density of states was not a large number, but an arbitrary one, it is not difficult to see that this introduces an extra factor between $1/3$ and $1$ in eqn(\ref{dosintegral}), which in turn alters the value of $\langle Q\rangle$ by a factor of about 1.7, very much within experimental variability per material. Note also that eqn(\ref{robust}) is very robust to fluctuations in $Q_0$. Namely,
\[\left.\frac{\partial\langle Q\rangle}{dQ_0}\right|_{Q_0\approx1}\ll1.\]
Since the equation (\ref{freqdependence}) was obtained by heuristically, it is also interesting to see how sensitive the value of $Q^{-1}$ is to the precise details of the functional form. For example, consider more general forms that are ``near-constant'', and repeat the above procedure to find $A(s)$.
\begin{align}\label{generalform}
Q^{-1}(\omega)=\frac{1}{[Q_0^{-s}+A(s)\log(U/\omega)]^s}
\end{align}
The dependence of the value of $Q$ to the exponent $s$ and microscopic value $Q_0$ (cf. eqn(\ref{generalform})) is displayed in table-1, which suggests that the universality is more general than that required by the precise form (\ref{freqdependence}).
\begin{table}[h]
\begin{center}
\begin{tabular}{c|c|c|c|c}
& $s=0.5$ & $s=0.7$ & $s=0.9$\\ \hline
$Q_0=0.1$ & 0.0024 & 0.0011 & 0.0006\\ \hline
$Q_0=1$ & 0.0024 & 0.0010 & 0.0004\\ \hline
$Q_0=10$ & 0.0024 & 0.0010 & 0.0003\\ \hline
\end{tabular}
\caption{The dependence of the average value of $Q^{-1}(1MHz)$ to the details of its functional form}
\end{center}
\end{table}
\chapter{Predictions}
In the previous two chapters we deduced the ultrasonic absorption at zero temperature from generic assumptions and arguments. Unfortunately, to the best of our knowledge no experiment has directly measured $Q^{-1}$ in the regime $T\ll\omega$. Therefore we will compare our results with indirect measurements from which $Q^{-1}$ is deduced. Two such quantities are the thermal conductivity $K$ and the ultrasound velocity shift $\Delta c/c$. As we have done in chapter 4, we will assume that the renormalized $\chi_m(\omega)$ is
roughly independent of initial state $|m\rangle$ (apart from $\theta(\omega+E_m)$) and hope that a more realistic $m$ dependence will not qualitatively alter the results.
\section{Temperature Dependence of Ultrasonic Velocity Shift}
In the TTLS model, the absorption coefficient is calculated by averaging over that of many TTLS. As discussed in the introduction, in the ``resonance regime'' this is given by
\begin{align}\label{tlsmfp}
Q^{-1}= Q^{-1}_{hf}\mbox{tanh}\left(\frac{\omega}{2T}\right)
\end{align}
where the coefficient $Q_{hf}^{-1}$ is predicted to be independent of frequency. The resonance behavior of a generic amorphous block is rather similar to that of a TTLS; since $T\ll\omega$ the unexcited distribution of occupied levels are well separated from the excited distribution. Let us therefore designate $p_1$ and $p_2$ as the probability that the amorphous block occupies a level in the former and latter groups respectively. In thermal equilibrium we have, as usual
\begin{align}
p_1+p_2=1\\
\frac{p_2}{p_1}\approx e^{\omega/T}.
\end{align}
The transitions are due to absorption of a phonon and stiumlated emission,
\[\frac{dp_1}{dt}=-W_{12}p_1+W_{21}p_2\]
where the transition rates $W_{12}$ and $W_{21}$ are given by Fermi's Golden Rule. From these equations it is trivial to calculate the phonon lifetime,
\[\tau_{ph}^{-1}=(1-e^{-\omega/T})\sum_{m}\pi\omega Q_{m}^{-1}\frac{e^{-E_m/T}}{Z(T)}\]
Thus, taking into account the assumption regarding initial state inpdependence $Q_m^{-1}$ can be calculated from the phonon mean free path
\[Q^{-1}(\omega,T)=Q^{-1}_{hf}(1-e^{-\omega/T})\]
where now,
\[Q^{-1}_{hf}\propto\frac{1}{\ln (U_0/\omega)}.\]
Note that for $\omega\gg T$, $Q^{-1}$ is only logarithmically dependent on frequency, and independent of temperature, consistent with experiment \cite{golding76}, and qualitatively similar to the TTLS prediction (\ref{tlsmfp}).
We can find the velocity shift from the Kramers Kronig relation;
\begin{align}\label{velshift}
\frac{\Delta c}{c}=-\Delta\frac{\chi_0}{\rho c^2}=-\Delta\int_0^{\infty}\frac{d\omega}{\omega}Q^{-1}(\omega,T)
\end{align}
If we insert (\ref{tlsmfp}) into (\ref{velshift}) we obtain for $T\gg\omega$
\begin{align}\label{tlsvelshift}
\frac{\Delta c}{c}=Q_{hf}^{-1}\ln\left(\frac{T}{T_0}\right)
\end{align}
Note that while this form works quite satisfactorily for low temperatures, it significantly departs from experimental data for higher temperatures \cite{golding76b}. The standard way this difficulty is resolved is by introducing an additional fit function to the TTLS density of states $P(\omega)=\bar{P}$, so that
\[P(\omega)=\bar{P}(1+a\omega^2)\]
where $a$ is a free parameter. The present model does a better job without such additional fit functions. If we treat $Q^{-1}_{hf}$ as a constant the integral (\ref{velshift}) can be taken analytically.
\begin{figure}[!ht]
\includegraphics[width=3.1in]{mfp-eps-converted-to.pdf}
\includegraphics[width=3.1in]{velshift-eps-converted-to.pdf}
\caption{
{\bf Normalized inverse mean free path (left) and velocity shift (right).} The present theory (solid) is compared against TTLS without the $\omega^2$ term (dashed) and experiment \cite{golding76,golding76b} (dots). While the functional forms predicted by both models are qualitatively similar at low temperatures, the TTLS model must use an additional fitting function $n(\omega)=n_0(1+a\omega^2)$ for the density of states to resolve the discrepancy in fitting $\Delta c/c$ data (right).}
\label{fig:Figure1}
\end{figure}
\begin{align}
\Delta c/c= \Delta (Q_{hf}^{-1}/2)[-e^{\omega/T}\mbox{Ei}(-\omega/T)-e^{-\omega/T}\mbox{Ei}(\omega/T)]
\end{align}
which is precisely the same result as (\ref{tlsvelshift}) in the $\omega\ll T$ limit. In the presence of $[\ln(U_0/\omega)]^{-1}$, the integration must be done numerically, and is shown in Fig\ref{fig:Figure1} and compared with the TTLS result (with unmodified density of states) as well as experimental data \cite{golding76,golding76b}.
\begin{figure}[!ht]
\includegraphics[width=3.1in]{K-eps-converted-to.pdf}
\caption{
{\bf Temperature Dependence of Thermal Conductivity}. The present theory (solid) $K\sim T^2\ln U_0/T$ is compared against TTLS prediction $K\sim T^2$ and experiment \cite{stephens73} (dots) below the ``plateau''.}
\label{fig:Figure2}
\end{figure}
\section{Temperature Dependence of Thermal Conductivity}
As already mentioned in the introduction, despite the anharmonic degrees of freedom, phonons are the main heat carriers in disordered solids. Thus, using the kinetic formula (\ref{K}),
\begin{align}
K(T)\approx c_\nu^{ph}(T)\bar{c}(T)\bar{l}_{ph}\\
=\mbox{const.} T^2Q(\omega,T)
\end{align}
where $c_\nu^{ph}(T)=\mbox{const.} T^2$ is the phononic specific heat and $\bar{c}(T)$ and $\bar{l}_{ph}$ are the speed of sound and phonon mean free path averaged over frequency. Since at temperature $T$, the frequency distribution function is sharply peaked at $\omega\sim4K$ we can evaluate the frequency averages (``dominant phonon approximation''),
\[K(T)=\mbox{const.}T^2\ln(U_0/T),\]
which is similar to TTLS prediction of $K=\mbox{const.}T^2$, but fits the experimental data better (see Fig\ref{fig:Figure2}).
\newpage
\newpage\pagebreak
\include{1-introduction}
\include{2-related}
\include{3-model}
\include{4-predictions}
\chapter{Conclusions}
Our main goal in this work has been motivated by providing an explanation to the universal acoustic response of disordered solids at low temperatures, which for us entails a description that does not use ad-hoc fit functions and parameters or postulate other (unobservable) universal quantities or entities.
We believe that we have been successful in doing so, in that we have shown that (a) simply starting from \emph{arbitrary} uncorrelated blocks and coupling them elastically yields a $\langle Q^{-1}\rangle$ factor that is universal, in the sense that its precise value depends sensitively only to measurable quantities $c_t/c_l$ and $\chi_t/\chi_l$, both of which fluctuate only by a factor of 1.2 across different materials. (b) That given the ansatz derived in chapter 4 for the frequency dependence of $Q(\omega)$, the experimentally probed $Q(1MHz-1GHz)$ is close to the measured value. (c) Other quantities related to $Q^{-1}$, namely the temperature dependences of the mean free path, thermal conductivity and velocity shift, are consistent with our generic model.
\begin{figure}[!ht]
\includegraphics[width=3.1in]{universality-eps-converted-to.pdf}
\caption{
{\bf $\langle Q^{-1}\rangle$ vs $\langle Q_0^{-1}\rangle$ }. Notice that the former can never be larger than $\mathcal{O}(10^{-2})$}
\label{fig:Figure4}
\end{figure}
Finally, we note that the essential ingredient of the present model is the existence of non-phonon degrees of freedom that couples the the strain field linearly. Thus, our arguments should apply equally well to disordered crystals (cf. \cite{stamp09}), with a slightly different angular dependence of the coupling coefficient $\Lambda_{ijkl}$. That being said, our conclusion does not imply that \emph{all} disordered solids must have the same $Q^{-1}$: If the attenuation coefficient of the ``microscopic blocks'' are significantly smaller than the canonical value, then so will be the macroscopic one.
However, a $Q^{-1}$ appreciably larger than the canonical value would tell against our hypothesis (see Fig\ref{fig:Figure4}).
\newpage
| 25,399 |
\section{Introduction}
The solar chromosphere exhibits three different classes of small scale intensity brightenings: flare-, plage-, and compact-brightenings. Although each is characterized by an enhanced temporal H$\alpha\ $ brightness relative to a background quiet Sun, they each have distinct physical processes governing their spatial and temporal evolution. Typically brightenings have been identified and characterized manually from a single data source~\citep[e.g.,][]{Kurt2000,Ruzdjak1989,Veronig2002}. However in order to form a better understanding of the underlying dynamics, data from multiple sources must be utilized and numerous similar features must be statistically analyzed.
This work focuses on flare brightenings and associated compact brightenings called sequential chromospheric brightenings (SCBs). SCBs were first observed in 2005 and appear as a series of spatially separated points that brighten in sequence \citep{Bala2005}. SCBs are observed as multiple trains of brightenings in association with a large-scale eruption in the chromosphere or corona and are interpreted as progressive propagating disturbances. The loci of brightenings emerge predominantly along the axis of the flare ribbons. SCBs are correlated with the dynamics which cause solar flares, coronal restructuring of magnetic fields, halo CMEs, EIT waves, and chromospheric sympathetic flaring \citep{Bala2005}. \citet{Pevtsov2007} demonstrate that SCBs have properties consistent with aspects of chromospheric evaporation.
This article presents a new description of the dynamical properties of SCBs resulting from applying a new automated method~\citep{Kirk2011} of identifying and tracking SCBs and associated flare ribbons. This tracking technique differs from previous flare tracking algorithms in that it identifies and tracks spatial and temporal subsections of the flare and all related brightenings from pre-flare stage, through the impulsive brightening stage, and into their decay. Such an automated measurement allows for tracking dynamical changes in intensity, position, and derived Doppler velocities of each individual subsection. The tracking algorithm is also adapted to follow the temporal evolution of ephemeral SCBs that appear with the flare. In Section~\ref{S-Data} we describe the data used to train the algorithm and the image processing involved in the detection routine. In Section~\ref{S-Flare} we present the results of tracking the evolution of flare kernels through an erupting flare. In Section~\ref{S-SCB} we present the application of the tracking algorithm to ephemeral SCBs. We present a physical interpretation of the SCBs and provide a heuristic a model of the origin of SCBs in Section~\ref{S-Interp}. Finally, in Section~\ref{S-Discussion} we discuss implications of these results and provide future direction for this work.
\section{Data and data processing}
\label{S-Data}
In this study we use chromospheric H$\alpha\ $ (6562.8 \AA) images from the USAF's {\it Improved Solar Observing Optical Network} (ISOON;~\citealp{Neidig1998}) prototype telescope to study flare ribbons and SCBs. ISOON is an automated telescope producing 2048$\times$2048 pixel full-disk images at a one-minute cadence. Each image has a 1.1 arc-second spatial sampling, is normalized to the quiet Sun, and corrected for atmospheric refraction (Figures~\ref{0513_boxes}(a), ~\ref{0506_boxes}(a), and \ref{1109_boxes}(a)). Nearly coincident to the spectral line center images, ISOON also records H$\alpha\ $ $\pm0.4$ \AA\ off-band Doppler images, from which Doppler signals are derived.
For this study, we chose to apply the brightening detection algorithms to three flares where~\citet{Bala2006} had previously identified SCB events (Table~\ref{T-events}). Each of the events selected has a two-ribbon configuration and an associated halo CME. Of these, both the 6 May 2005 and 13 May 2005 events were both located near disk center while the 9 November 2004 event was near the western limb. Images were extracted from the archive from $\pm 3.5$ hours of the eruption start time, yielding a data cube with $\approx 400$ images for each event. An H$\alpha\ $ relative intensity curve, an H$\alpha\ $ maximum intensity curve, and {\it Geostationary Operational Environmental Satellites} (GOES) hard and soft x-ray fluxes are plotted to characterize the flare (Figures~\ref{0513_boxes}(d), ~\ref{0506_boxes}(d), and \ref{1109_boxes}(d)) as described in~\citet{Kirk2011}.
\begin{table}
\caption{
The events selected for study. Each event was visually identified as having a two-ribbon configuration and SCBs associated with the flare.
}
\label{T-events}
\begin{tabular}{lcccc}
\hline
Date & Start (UT) & Duration (h) & Flare Class & CME \\
\hline
9 Nov. 2004& 16:59 & 0.5 & M8.9 & Halo \\
6 May 2005& 16:03 & 2.1 & C8.5& Halo \\
13 May 2005& 16:13 & 1.3 & M8.0& Halo \\
\hline
\end{tabular}
\end{table}
Each ISOON image is reconditioned to remove solar limb-darkening. The images are then de-projected into conformal coordinates using a Guyou projection~\citep[an oblique aspect of the Peirce projection,][]{Peirce1879}, which removes the projection effects of imaging the solar sphere. Each image is then cropped to the region of interest (ROI) (Figures~\ref{0513_boxes}(b), ~\ref{0506_boxes}(b), and \ref{1109_boxes}(b)). The set of images are aligned using a cross-correlation algorithm eliminating the rotation effects of the Sun. Frames that contain bad pixels or excessive cloud interference are removed.
The red and blue wings of ISOON Doppler images are each preprocessed using the same technique as applied to the line center images described above (Figures~\ref{0513_boxes}(c), ~\ref{0506_boxes}(c), and \ref{1109_boxes}(c)). In order to produce velocity measurements, a Doppler cancelation technique is employed in which red images are subtracted from blue images \citep[for a modern example see:][]{Connes1985}. Typically, red and blue images are separated by about 4 seconds and no more than 5 seconds. A mean zero redshift in the subtracted Doppler image in the quiet-Sun serves as a reference. In dynamical situations such as solar flares, H$\alpha\ $ profiles are often asymmetric especially in emission making this Doppler technique invalid. To avoid asymmetric profiles, only values derived outside of the flaring region are considered such that the areas of interest at $\pm 0.4$ \AA\ are still in absorption even if raised in intensity
\citep{Bala2004}. The values from the subtracted images are translated into units of km~s$^{-1}$\ by comparison to a measured response in spectral line shifts, which are calibrated against an intensity difference for the ISOON telescope. In this context, we interpret the Doppler shift as a line of sight velocity. Thus, Doppler velocity [$v_D$] is defined as:
\begin{equation}
v_D=k\left(\mathcal{I}_{\rm blue} - \mathcal{I}_{\rm red}\right),
\end{equation}
where $\mathcal{I}$ is the measured intensity and $k$ is a linear fitted factor that assumes the intensity changes due to shifts in the symmetric spectral line, which can be attributed to a Doppler measure as a first-order approximation. The linear factor [$k$] has been independently determined for the ISOON telescope by measuring the full H$\alpha\ $ spectral line Doppler shift across the entire solar disk~\citep{Bala2011}.
\subsection{Detection and tracking}
\label{S-detection}
An animated time series of sequential images covering an erupting flare reveals several physical characteristics of evolving ribbons: the ribbons separate, brighten, and change their morphology. Adjacent to the eruption, SCBs can be observed brightening and dimming in the vicinity of the ribbons. \citet{Kirk2011} describe in detail techniques and methods used to extract quantities of interest such as location, velocity, and intensity of flare ribbons and SCBs. The thresholding, image enhancement, and feature identification are tuned to the ISOON data. The detection and tracking algorithms are specialized for each feature of interest and requires physical knowledge (e.g. size, peak intensity, and longevity) of that feature being detected to isolate the substructure.
Briefly, the detection algorithm first identifies candidate bright kernels in a set of images. In this context, we define a kernel to be a locus of pixels that are associated with each other through increased intensity as compared with the immediately surrounding pixels. Each kernel has a local maximum, must be separated from another kernel by at least one pixel, and does not have any predetermined size or shape. Next, the algorithm links time-resolved kernels between frames into trajectories. A filter is applied to eliminate inconsistent or otherwise peculiar detections. Finally, characteristics of bright kernels are extracted by overlaying the trajectories over complementary datasets. To aid in this detection, tracking software developed by Crocker, Grier and Weeks was used as a foundation and modified to fit the needs of this project\footnote[1]{Crocker's software is available online at \url{www.physics.emory.edu/$\sim$weeks/idl/}.} \citep{Crocker1996}.
In order to characterize a kernel, we calculate its integrated intensity, radius of gyration, and eccentricity. The eccentricity of the kernel (as defined by its semi-major [a] and semi-minor [b] axis) is calculated using,
\begin{equation}
\label{Eq-ecc}
e=\sqrt{1-\frac{b^2}{a^2}}.
\end{equation}
The integrated intensity of a given kernel is defined by,
\begin{equation}
\label{Eq-tot_inten}
m= \sum_{i^2+j^2 \le \omega^2}\mathcal{A}(x+i, y+j)
\end{equation}
where $\mathcal{A}(x+i,y+j)$ is the intensity of the pixel located at $(x+i,y+j)$, the kernel has a brightness weighted centroid with coordinates $(x,y)$, and $\omega$ is the radius of the mask \citep{Crocker1996}. The radius of the mask is chosen to be eight pixels in this case. A radius of gyration, $R_g$, is related to the moment of inertia [$I$] by using:
\begin{equation}
\label{Eq-inertia}
I = \sum_{k} m_k r_k^2 = m R_g^2,
\end{equation}
\begin{equation}
r=(a+b)/2,
\end{equation}
where $m_k$ is the mass of particle $k$, $m$ is the total mass of the system, and $r$ is the distance to the rotation axis. In this context, we interpret the radius of gyration to be
\begin{equation}
\label{Eq-radius}
R_g^2 = \sum_{k} \frac{\mathcal{A}_k r_k^2}{m}
\end{equation}
where $m$ is the integrated intensity as defined in Equation~\ref{Eq-tot_inten} \citep{Crocker1996}. Figure~\ref{Kernel_diagram} diagrams a hypothetical kernel. The results of this algorithm identify and characterize several kernels in the flaring region (Figure~\ref{Kernel_marks}). The total number of flare kernels detected is typically between 100 -- 200, while the number of SCB kernels detected is typically three to four times that number.
\section{Flare ribbon properties}
\label{S-Flare}
The properties of kernels identified in flare ribbons can be examined in two ways: (i) each kernel can be considered as an independent aggregation of compact brightenings or (ii) kernels can be considered as dependent on each other as fragments of a dynamic system. Each category brings about contrasting properties. If one considers the kernels as independent elements, this provides a way to examine changes to subsections of the flare and is discussed in Section~\ref{S-Individual}. Associating kernels with their contextual surroundings allows a way to examine the total evolution of the flare without concern of how individual kernels behave. This type of examination is addressed in Section~\ref{S-total}.
\subsection{Qualities of individual kernels}
\label{S-Individual}
A sample of six kernels from the 13 May 2005 event, letters U - X in Figure~\ref{Kernel_map}, is representative of the majority of flare kernels tracked. The kernels were selected from three different regions of this flare. The normalized H$\alpha\ $ intensity of each of these kernels is shown in Figure~\ref{Kernel_curves}. In contrast with the integrated flare light curve, an individual flare kernel often has a shorter lifetime. This is because an individual kernel may first appear in the impulsive phase of the flare (as demonstrated in kernels W - Y) while other kernels may disappear as the flare begins to decay (as in kernels U and Z). Most likely these kernels merged with one another, at which point their unique identity was lost. Kernels W and X are the only two that show the characteristic exponential dimming found in the reference curves in Figure~\ref{0513_boxes}~(d). All of these kernels have peak intensities within a few minutes of the peak of the total flare intensity and have sustained brightening above background levels for up to an hour.
The integrated speed of displaced kernels provides context to the evolution of intensities. We define the integrated speed of displacement to be the sum over all time steps of the measured velocity of an individual kernel and thus small velocity perturbations are minimized by the sum. The peak integrated speed measured for each kernel peaks at $\sim 2.3$ km s$^{-1}$ and has a mean of $\sim 0.2$ km s$^{-1}$. There is significant motion along the flare ribbons as well as outflow away from flare center. The motions are complex but generally diverge. The total spatial (angular) displacement of each flare kernel peaks at $\sim 20.7$ Mm and has a mean of $\sim 5.0 $ Mm. Flare kernels that exhibited the greatest speeds did not necessarily have the greatest displacements. As the flare develops, most of the motion in flare ribbons indicated is synchronous with the separation of the ribbons. A few tracked kernels indicate motion along flare ribbons. These flows along flare ribbons are consistently observed in all three flares.
\subsection{Derived flare quantities}
\label{S-total}
Examining trajectories of flare kernels offers insight into the motions of the flare ribbons as they evolve through the eruption. Figure~\ref{Flare_vel} shows a time series of images centered about the peak-time of the flare. The velocities of the detected flare kernels are superposed as vectors. Notice the initial outflow of the two ribbons near the flare peak at 16:49 UT. But, as the flare continues to evolve, there is more motion along the flare ribbons. This is probably a result of the over-arching loops readjusting after the reconnection event. Beneath each image is a histogram of the distribution of velocities at each time step. The mean apparent lateral velocity of each kernel remains below 0.5 km~s$^{-1}$\ throughout the flare even though the total velocity changes significantly. Qualitatively, the bulk of the apparent motion appears after the flare peak intensity at 16:49 UT.
Grouping the kernels' velocities into appropriate bins provides another way to analyze the dynamics of the flare (Figure~\ref{Flare_vel_bins}). The kernels with apparent lateral speeds above 0.4 km~s$^{-1}$\ are highly coincident with the peak intensity of the flare. The number of these fast moving kernels peaks within a couple minutes of flare-peak, and then quickly decays back to quiet levels. The velocity bins between 0.1 and 0.4 km~s$^{-1}$\ show a different substructure. These velocity bins peak $\approx 30$ minutes after the peak of the flare intensity. They show a much slower decay rate, staying above the pre-flare velocity measured throughout the decay phase of the H$\alpha\ $ flare curve.
Figure~\ref{Integrated_speed} (left column) shows the integrated unsigned kernel velocity (speed) for each image. In general, the evolution of the integrated kernel speed has a shape similar to the intensity curve. For the 6 May 2005 event, the speed curve is more similar to the GOES x-ray intensity curve than the H$\alpha\ $ curve. For the November 2004 event, the limb geometry of the flare, as well as the noise in the original dataset, results in a noisy curve. Despite these difficulties, a clear increase in total speed is apparent near the peak of the flare. A continuum integrated speed level of 1--2 km~s$^{-1}$\ is consistent in each event, indicating the quality of data and of the tracking software. The peak integrated speed is about 15 km~s$^{-1}$\ for all three flares.
Examining the change in the speed of each kernel between successive images yields a derived acceleration for each tracked kernel. The integrated unsigned acceleration is plotted in Figure~\ref{Integrated_speed} (middle column). Both the 13 May 2005 and 6 May 2005 events show peak acceleration after the peak of the intensity curve. Hence the majority of the acceleration in the apparent motion of the flare comes in the decay phase of H$\alpha\ $ emission, in concert with the formation of post-peak flare loops, as seen in movies of TRACE images of other flares. The peak acceleration appears uncorrelated with the strength of the flare since the peak of the M class 13 May event is just over 300 km s$^{-2}$, while the peak of the C class flare on 6 May is nearly 400 km s$^{-2}$. The acceleration curve for the 9 November flare has too much noise associated with it to decisively determine the peak value.
The right column of Figure~\ref{Integrated_speed} shows a derived kinetic energy associated with the measured motion of the flare kernels. This was accomplished by assuming a chromospheric density [$\sigma$] of $10^{-5}$ kg m$^{-3}$ and a depth [h] of 1000 km. The mass contained in each volume is then calculated by
\begin{equation}
m=\pi h \sigma R_g^2 \sqrt{1-e^2}
\end{equation}
where $R_g$ is defined in Equation~\ref{Eq-radius} and $e$ is defined in Equation~\ref{Eq-ecc}. The kinetic energy under a kernel is defined in the standard way to be
\begin{equation}
\label{eq-energy}
E_k=\frac{1}{2} m v^2
\end{equation}
where $v$ (speed) is a measured quantity for each kernel. The derived kinetic energy curves show a $\approx$ 30 fold increase in kinetic energy during the flare with a decay time similar to the decay rate of the H$\alpha\ $ intensity. The measure of a kernel's kinetic energy is imprecise because the measured motion is apparent motion of the underlying plasma. In the model of two ribbon solar flares put forth by~\citet{Priest2002}, the apparent velocity is a better indicator of rates of magnetic reconnection rather than of plasma motion. Despite this caveat, the derived kernel kinetic energy is a useful measure because it combines the size of the kernels with their apparent velocities to characterize the flare's evolution, thus representing the two in a quantitative way.
The non-thermal 25--50 keV emission between 16:42--16:43 UT from the {\it Reuven Ramaty High Energy Solar Spectroscopic Imager} (RHESSI,~\citealp{Lin2002}) of the 13 May flare is contoured in Figure~\ref{RHESSI_over} over an H$\alpha\ $ image taken at 16:42 UT. The high-energy emission is centered over the flare ribbons but is discontinuous across the ribbon. There are three localized points from which the majority of the x-ray emission comes. The parts of the flare ribbons exhibiting the most displacement show generally lower x-ray intensity than the more stationary segments of the flare ribbons.
\section{SCB properties}
\label{S-SCB}
The measured properties of SCBs can be studied using two scenarios: each SCB kernel is considered as independent and isolated compact brightenings; each compact brightening is considered to be fragments of a dynamic system that has larger structure. This is similar to the approach used for flares. Again, considering both scenarios has its benefits. Identifying kernels as independent elements reveals structural differences in compact brightenings and is discussed in Section~\ref{S-IndividualSCB}. Associating kernels with their contextual surroundings provides some insight into the physical structure causing SCBs. An aggregate assessment of SCBs is addressed in Section~\ref{S-totalSCB}. To mitigate the effects of false or marginal detections, only SCBs with intensities two standard deviations above the mean background intensity are considered for characterizing the nature of the ephemeral brightening.
\subsection{Qualities of individual SCBs}
\label{S-IndividualSCB}
SCBs, although related to erupting flare ribbons, are distinctly different from the flare kernels discussed in Section~\ref{S-Flare}. Six SCBs are chosen from the 13 May 2005 event as an example of these ephemeral phenomena. The locations of these six events are letters A - F in Figure~\ref{Kernel_map}. The H$\alpha\ $ normalized intensity of each of these kernels is shown in Figure~\ref{Kernel_curves}. The SCB intensity curve is significantly different from the flare kernel curves shown on the left side of the figure. SCB curves are impulsive; they have a sharp peak and then return to background intensity in the span of about 12 minutes. Nearly all of the SCBs shown here peak in intensity before the peak of the flare intensity curve, shown as a vertical dashed line in Figure~\ref{Kernel_curves}. SCBs B and E both appear to have more internal structure than the other SCBs and last noticeably longer. This is most likely caused by several unresolved SCB events occurring in succession.
Nearly all SCBs have H$\alpha\ $ line center intensities 40 -- 60\% above the mean intensity of the quiet chromosphere (Figure~\ref{SCB_types}). From this increased initial intensity, SCBs brighten to intensities 75 -- 130\% above the quiet Sun.
Examining Doppler velocity measurements from the SCB locations reveals three distinct types of SCBs (Figure~\ref{SCB_types}). A type Ia SCB has an impulsive intensity profile and an impulsive negative Doppler profile that occurs simultaneously or a few minutes after the peak brightening. In this study, a negative velocity is associated with motion away from the observer and into the Sun. A type Ib SCB has a similar intensity and Doppler profile as a type Ia but the timing of the impulsive negative velocity occurs several minutes before the peak intensity of the SCB. The type Ib SCB shown in Figure~\ref{SCB_types} has a negative velocity that peaks 10 minutes before the peak intensity and returns to a stationary state before the H$\alpha\ $ line center intensity has decayed. A type II SCB has a positive Doppler shift perturbation that often lasts longer than the emission in the H$\alpha\ $ intensity profile. The timing of both are nearly coincidental. A type III SCB demonstrates variable dynamics. It has a broad H$\alpha\ $ intensity line center with significant substructure. A type III SCB begins with a negative Doppler profile much like a type I. Before the negative velocity perturbation can decay back to continuum levels, there is a dramatic positive velocity shift within three minutes with an associated line center brightening. In all types of SCBs the typical magnitude of Doppler velocity perturbation is between 2 and 5 km~s$^{-1}$\ in either direction perpendicular to the solar surface.
Of all off-flare compact brightenings detected using the automated techniques, 59\% of SCBs have no discernible Doppler velocity in the 6 May event and 21\% in the 13 May event. Out of the SCBs that do have an associated Doppler velocity, the 6 May event has 35\% of type I, 54\% of type II, and 11\% of type III. The 13 May event has 52\% of type I, 41\% of type II, and 17\% of type III. When totaling all of the Doppler velocity associated SCBs between both events, 41\% are of type I, 46\% of type II, and 13\% of type III. Both line center brightenings and Doppler velocities are filtered such that positive detections have at least a two standard deviation peak above the background noise determined in the candidate detection \citep{Kirk2011}.
\subsection{SCBs in aggregate}
\label{S-totalSCB}
Examining SCBs as a total population, the intensity brightenings have a median duration of 3.1 minutes and a mean duration is 5.7 minutes (Figure~\ref{SCB_duration}). The duration is characterized by the full-width half-maximum of the SCB intensity curve (examples of these curves are shown in Figure~\ref{Kernel_curves}). A histogram of the distribution of the number of SCB events as a function of duration shows an exponential decline in the number of SCB events between 2 and 30 minutes (Figure~\ref{SCB_duration}). The duration of SCBs is uncorrelated with both distance from flare center and the peak intensity of the SCB.
Examining individual SCBs' distance from the flare center as a function of time provides a way to extract the propagation trends of SCBs around the flare (Figure~\ref{SCB_stats}). The bulk SCBs appear between 1.2 and 2.5 Mm away from flare center in a 1 hour time window. At the extremes, SCBs are observed at distances up to $\approx 5$ Mm and several hours after the flare intensity maximum. On the top two panels of Figure~\ref{SCB_stats}, the shade of the mark corresponds to the intensity of the SCB measured where the lighter the mark, the brighter the SCB. The center of the flare is determined by first retaining pixels in the normalized dataset above 1.35. Then all of the images are co-added and the center of mass of that co-added image is set to be the ``flare center." The SCBs tend to clump together in the time-distance plot in both the 6 May and 13 May events. Generally, the brighter SCBs are physically closer to the flare and temporally occur closer to the flare peak. This intensity correlation is weak and qualitatively related to distance rather than time of brightening. Statistics from the November 9, 2004 event are dominated by noise.
Identifying SCBs with negative Doppler shifts within 3 minutes of the H$\alpha\ $ peak intensity limits SCBs to type I only. With this method of filtering, some trends become more apparent in a time-distance plot of SCBs (Figure~\ref{SCB_stats}, bottom panels). In this view of SCBs, they tend to cluster together in time as well as distance, with two larger groups dominating the plots.
To fit a slope to these populations, a forward-fitting technique is employed similar to a linear discriminate analysis. This method requires the user to identify the number of groups to be fitted and the location of each group. The fitting routine then searches all linear combinations of features for the next ``best point" to minimize the chi-square to a regression fit of the candidate group. Repeating this method over all the points in the set produces an ordered set of points that when fit, have an increasing chi-square value. A threshold is taken where the derivative of the chi-square curve increases to beyond one standard deviation, which has the effect of identifying where the chi-square begins to increase dramatically. Running this routine several times minimizes the effect of the user and provides an estaminet of the error associated with the fitted line. This method has two caveats. First, this fitting method relies on the detections having Poissonian noise. This is not necessarily correct since the detection process of compact brightenings introduces a selection bias. Second, the fitting method makes the assumption that no acceleration occurs in the propagation of SCBs. This is a reasonable approximation but from studies of Moreton waves in the chromosphere \citep[e.g.][]{Bala2010}, a constant velocity propagation is unlikely.
Applying the forward-fitting technique these two data sets yields two propagation speeds: a fast and a slow group. The 6 May event has propagation speeds of $71 \pm 2$ km~s$^{-1}$\ and $49 \pm 2$ km~s$^{-1}$\ . The 13 May event has propagation speeds of $106 \pm 7$ km~s$^{-1}$\ and $33 \pm 4$ km s$^{-1}$. Beyond these detections, the 13 May event has a third ambiguous detection with a propagation speed of $-7 \pm 5$ km s$^{-1}$. The implications of these statistics are discussed in Section~\ref{S-Discussion}.
The x-ray intensity measured with RHESSI temporally covers a part of the flaring event on 13 May 2005. RHESSI data coverage begins at 16:37 UT during the impulsive phase of the flare. Comparing the integrated x-ray intensity curve between 25 -- 50 keV to the aggregate of SCB intensities integrated over each minute yields some similarities (Figure~\ref{SCB_RHESSI}). The x-ray intensity peaks about a minute after the integrated SCBs reach their maximum intensity. The decay of both the x-ray intensity and SCB integrated curve occur approximately on the same time-scale of $\approx 50$ minutes.
\section{Physical interpretation of results}
\label{S-Interp}
From the results of the tracking method, we can gain insight into the physical processes associated with SCBs and flares. First, the flare ribbons studied are both spatially and temporally clumped into discrete kernels. The relatively smooth ribbon motions observed in H$\alpha\ $ and the nearly ideal exponential decay characterized in x-ray flux measurements belie the substructure that makes up a two-ribbon flare. This is discussed in Section~\ref{S-Model_flare}. Second, the brightening in the flare ribbons is caused by a distinctly different physical process than the one causing SCBs. In this sense, SCBs are different than micro-flares. Our interpretation of the physical origin of SCBs is addressed in Section~\ref{S-Model_SCB}.
\subsection{Two ribbon chromospheric flares}
\label{S-Model_flare}
Flare kernels are observed to appear and disappear as the underlying flare ribbons evolve, as discussed in Section~\ref{S-Individual}. Examining individual kernel structure suggests there is substructure within a flare ribbon whose elements impulsively brighten and dim within the brightness that encompasses the intensity curve. These results support the premise that flares are made up of several magnetic field lines reconnecting~\citep[e.g.,][]{Priest2002}. There is no evidence to claim an individual flare kernel is directly tracking a loop footpoint. Within a tracked flare kernel, multiple coronal reconnection events are superimposed to produce the observed asymmetries a single flare kernel's light curve.
\citet{Maurya2010} tracked subsections of the 28 October 2003 X17 flare as it evolved. They reported peak speeds ranging from $\sim 10 - 60$ km s$^{-1}$ (depending on the spatially tracked part of the flare) over the observed span of 13 minutes. \citet{Maurya2010} also reported that the total apparent distances the ribbons traveled were $\sim 10^4$ km. The 28 October flare is several orders of magnitude greater in GOES x-ray intensity than the flares considered for this study. In the present study, despite this difference, peak speeds of flare kernels observed in this work are measured at $\sim 2.3$ km s$^{-1}$ and the mean velocity of all flare kernels is $\sim 0.2$ km s$^{-1}$. The maximum distance flare kernels traversed was $\sim 2 \times 10^4$ km and the average distance traveled was $\sim 5 \times 10^3$ km; both of these values are similar to the 28 October 2003 event. Although the velocities of the flare ribbons studied here are at least an order of magnitude less, the two to three orders of magnitude difference in the GOES peak intensity between the 28 October flare and the flares studied here implies that the apparent velocities and distance traveled by flare ribbons do not scale linearly with the strength of the eruption.
The bulk of the apparent motion and acceleration in the flare ribbons are observed after the peak of the flare intensity. Integrating this into the dynamical model of a two-ribbon flare~\citep{Demoulin1988} implies that the peak intensity of the flare occurs in the low lying arcade and loses intensity as the x-point reconnection progresses vertically to higher levels~\citep{Baker2009}. The measured divergence of the ribbons dominate the motion but there is significant apparent flow tangential to the flare ribbons. This predicates that there is lateral propagation to the x-point as well as vertical propagation.
\subsection{A physical model of SCBs}
\label{S-Model_SCB}
Figure~\ref{SCB_loops} is a proposed phenomenological model of the overlying physical topology. SCBs are hypothesized to be caused by electron beam heating confined by magnetic loop lines over-arching flare ribbons. These over-arching loops are analogous to the higher lying, unsheared tethers in the breakout model of CMEs \citep{Antiochos1999}. As the flare erupts, magnetic reconnection begins in a coronal x-point. The CME escapes into interplanetary space, the remaining loop arcade produces a two ribbon flare, and the tethers reconnect to a new equilibrium position. The tether reconnection accelerates trapped plasma which impacts the denser chromosphere causing observed brightening. This description implies the driver of SCBs is the eruption of an CME. \citet{Bala2006} found a strong correlation between CMEs and the presence of SCBs.
A simple loop configuration without localized diffusion or anomalous resistance implies that the length of the loop directly proportional to the travel time of the electron beam. This means that the different propagation groups observed in Figure~\ref{SCB_stats} result from different physical orientations of over-arching loops. The time it takes to observe a chromospheric brightening can therefore be described as:
\begin{equation}
\label{SCB_formula}
\tau_{\rm SCB}=\gamma \frac{L}{v_e},
\end{equation}
where $\tau_{\rm SCB}$ is the time it takes for accelerated plasma to travel from the coronal x-point, along the magnetic loop lines, impact the chromosphere, and produce a brightening, $L$ is the length of the tether before reconnection, $v_e$ is the electron velocity along the loop line, and $\gamma$ is a function of changing diffusion and resistance between chromosphere and corona. In physical scenarios, $v_e$ is likely a combination of the Alfv\'en speed and the electron thermal velocity.
The slopes plotted in Figure~\ref{SCB_stats} show two different propagation speeds for SCBs: $33 - 49$ km s$^{-1}$ and $71 - 106$ km s$^{-1}$. The sound speed in the chromosphere is approximately $c_s\approx10$ km s$^{-1}$ \citep{Nagashima2009}, while the Alfv\'en speed in the upper chromosphere is approximated to be between $v_A \approx 10 - 100$ km $s^{-1}$ \citep{Aschwanden2005}. Both of the SCB propagation speeds are significantly above the approximated sound speed, however they fall reasonably well into the range of the Alfv\'en speed. Assuming the propagation speed of non-thermal plasma along the loop line is consistent between flare loops, the different propagation speeds would then imply different populations of loops being heated as the flare erupts.
Examining single SCBs, the coincident Doppler recoil with the H$\alpha\ $ intensity presents a contradiction. If an SCB is an example of compact chromospheric evaporation \citep[e.g.][]{Dennis1989}, then the only Doppler motion should be outward, the opposite of what is observed. Figure~\ref{SCB_cartoon} presents a possible solution to this. Since the scattering length of electrons is significantly smaller than that of protons, the electron beam impacts the mid chromosphere and deposit energy into the surrounding plasma while the protons penetrate deeper. This deposited heat cannot dissipate effectively through conduction or radiation and thus expands upward into the flux tube. To achieve an expansion with rates of a few km s$^{-1}$ as observed, the chromospheric heating rate must be below $E_H \le 10^{10}$ erg cm$^{-2}$ s$^{-1}$ \citep{Fisher1984}. As a reaction to this expansion, a reaction wave propagates toward the solar surface in the opposite direction of the ejected plasma. Since the Doppler measurements are made in the wings of the H$\alpha\ $ line, the location of the Doppler measurement is physically closer to the photosphere than the H$\alpha\ $ line center. Thus the observer sees the heating of the H$\alpha\ $ line center and coincidentally observes the recoil in the lower chromosphere.
In type II SCBs an up-flow is observed. This is an example of a classic chromospheric evaporation where the heated plasma in the bulk of the chromosphere is heated and ablated back up the flux tube. In contrast, the type III class of SCB shown in Figure~\ref{SCB_types} present an interesting anomaly to both the other observed SCBs and the model proposed in Figure~\ref{SCB_cartoon}. Since there is an initial down-flow, the beginning state of type III SCBs are similar to type Is. As the recoil is propagated, a continual bombardment of excited plasma (both protons and electrons) impacts the lower chromosphere, causing ablation and changing the direction of flow.
\section{Summary and Conclusions}
\label{S-Discussion}
Two conclusions can be made about the flares studied in this work: first, the asymmetrical motions of the flare ribbons imply the peak flare energy occurs in the low lying arcade, and second, flare related SCBs appear at distances on the order of $10^5$ km and have properties of sites of compact chromospheric evaporation. \citet{Maurya2010} estimated reconnection rates from a measurement of the photospheric magnetic field and the apparent velocity of chromospheric flare ribbons. Therefore, it would be possible to estimate the reconnection rates using this technique with the addition of photospheric magnetograms. Associating vector magnetograms with this technique and a careful consideration of the observed Doppler motions underneath the ribbons would provide a full 3D method for estimating the Lorentz force for subsections of a flare ribbon.
SCBs originate during the impulsive rise phase of the flare and often precede the H$\alpha\ $ flare peak. SCBs are found to appear with similar qualities as compact chromospheric ablation confirming the results of \citet{Pevtsov2007}. The heuristic model proposed in Figure~\ref{SCB_cartoon} requires unipolar magnetic field underneath SCBs. The integration of high resolution magnetograms and a field extrapolation into the chromosphere would confirm this chromospheric evaporation model. One consequence of chromospheric evaporation is that the brightening should also be visible in EUV and x-ray observations due to non-thermal protons interacting with the lower chromosphere or photosphere.
SCBs are a special case of chromospheric compact brightening that occur in conjunction with flares. The distinct nature of SCBs arises from their impulsive brightenings, unique Doppler velocity profiles, and origin in the impulsive phase of flare eruption. These facts combined demonstrate that SCBs have a non-localized area of influence and are indicative of the conditions of the entire flaring region. They can possibly be understood by a mechanism in which a destabilized overlying magnetic arcade accelerates electrons along magnetic tubes that impact a denser chromosphere to result in an SCB. This distinguishes SCBs from the flare with which they are associated. In a future work, we plan to address physical mechanisms to explain the energetic differences between SCBs and flares to explain the coupled phenomena.
\acknowledgments
The authors thank: (1) USAF/AFRL Grant FA9453-11-1-0259, (2) NSO/AURA for the use of their Sunspot, NM facilities, (3) AFRL/RVBXS, and (4) Crocker and Weeks for making their algorithm available online.
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\section{Introduction}
Let $G$ be a simply connected, semisimple complex linear algebraic group, split over $\mathbb R$.
The Peterson variety $\mathcal Y$ may be viewed as the compactification of the stabilizer $X:=G^\vee_F$ of
a standard principal nilpotent $F$ in $(\mathfrak g^\vee)^*$ (with respect to the coadjoint
representation of $G^\vee$), which one obtains by embedding $X$ into the Langlands dual
flag variety $G^\vee/B_-^\vee$ and taking the closure there.
Ginzburg \cite{Gin:GS} and Peterson \cite{Pet:QCoh} independently showed that the coordinate ring $\mathcal O(X)$ of the variety $X$ was isomorphic to the homology $H_*(\operatorname{Gr}_G)$ of the affine Grassmannian $\operatorname{Gr}_G$ of $G$, and Peterson discovered
moreover that the compactification $\mathcal Y$ encodes the quantum
cohomology rings of all of the flag varieties $G/P$. Peterson's remarkable work in particular
exhibited explicit homomorphisms between localizations of $qH^*(G/P,\mathbb C)$ and $H_*(\operatorname{Gr}_G,\mathbb C)$ taking quantum Schubert classes $\sigma_w^P$ to affine homology Schubert classes $\xi_x$. These homomorphisms were verified in \cite{LaSh:QH}.
The first aim of this paper is to compare different notions of positivity for the real points of $X$: (i) the {\it affine Schubert positive} part $X^{\mathrm{af}}_{>0}$ where affine Schubert classes $\xi_x$ take positive values via Ginzburg and Peterson's isomorphism $H_*(\operatorname{Gr}_G) \simeq \mathcal O(X)$; (ii) the {\it totally positive} part $X_{>0} :=X \cap U^\vee_{-,>0}$ in the sense of Lusztig \cite{Lus:TotPos94}; and (iii) the {\it Mirkovic-Vilonen positive} part $X^{\mathrm{MV}}_{>0}$ where the classes of the Mirkovic-Vilonen cycles from the geometric Satake correspondence \cite{MV:GS} take positive values.
Our first main theorem (Theorem \ref{thm:three}) states that these three notions of positivity coincide. For $G$ of type $A$ the coincidence $X^{\mathrm{af}}_{>0} = X_{>0}$ was already established in \cite{Rie:QCohPFl}, where instead of $X^{\mathrm{af}}_{>0}$, the notion of {\it quantum Schubert positivity} was used. In general quantum Schubert positivity is possibly weaker than affine Schubert positivity. It follows from \cite{Rie:QCohPFl} that the notions coincide in type A, and we verify that they coincide in type C in Appendix \ref{s:proof1}.
Our second main theorem (Theorem \ref{t:main}) is a parametrization of the totally positive $X_{>0}$ and totally nonnegative $X_{\geq 0}$ parts of $X$. We show that they are homeomorphic to $\mathbb R_{>0}^n$ and $\mathbb R_{\geq 0}^n$ respectively. This was conjectured by the second author in \cite{Rie:QCohPFl} where it was established in type $A$. In type $A_n$ we have that $X=G^\vee_F$ is the $n$-dimensional subgroup of lower-triangular unipotent Toeplitz matrices, and thus the parametrization $X_{\geq 0} \simeq \mathbb R_{\geq 0}^n$ is a ``finite-dimensional'' analogue of the Edrei-Thoma theorem \cite{Edr:ToeplMat} parametrizing {\it infinite} totally nonnegative Toeplitz matrices, appearing in the classification of the characters of the infinite symmetric group. The results of this article give an arbitrary type generalization.
The strategy of our proof is as follows: to show that $X^{\mathrm{af}}_{>0} \subseteq X^{\mathrm{MV}}_{>0}$ we use a result of Kumar and Nori \cite{KuNo:pos} stating that effective classes in $H_*(\operatorname{Gr}_G)$ are Schubert-positive. We then use the geometric Satake correspondence \cite{Gin:GS,MV:GS,Lus:GS} to describe $X^{\mathrm{MV}}_{>0}$ via matrix coefficients, and a result of Berenstein-Zelevinsky \cite{BeZe:Chamber} to connect to the totally positive part $X_{>0}$.
Finally, to connect $X_{>0}$ back to $X^{\mathrm{af}}_{>0}$, we parametrize the latter directly by combining the positivity of the 3-point Gromov-Witten invariants of $qH^*(G/B)$ with the Perron-Frobenius theorem. This argument follows the strategy of \cite{Rie:QCohPFl}.
There is a general phenomenon \cite{Lus:TotPos94,BeZe:Chamber} that totally positive parts have ``nice parametrizations''. This phenomenon is closely related to the relation between total positivity and the canonical bases \cite{Lus:Can}, and also the cluster algebra structures on related stratifications \cite{Fom}. Indeed our work suggests that the coordinate ring $\mathcal O(X)$ has the affine homology Schubert basis $\{\xi_w\}$ as a ``dual canonical basis'', and that the Hopf-dual universal enveloping algebra $U(\mathfrak g^\vee_F)$ has the cohomology affine Schubert basis $\{\xi^w\}$ as a ``canonical basis''. Certainly the affine Schubert bases have the positivity properties expected of canonical bases.
In \cite{Rie:TotPosGBCKS} the type A parameterization result for the totally positive part $X_{>0}$ of the
Toeplitz matrices $X$ is proved in a completely different way, using a mirror symmetric construction of $X$.
This approach does not however prove the interesting positivity properties of the bases we study in this paper.
The mirror symmetric approach was partly generalized to other types in \cite{Rie:QToda}, where the
existence of a totally positive point in $X$ for any choice of positive quantum parameters is proved (but not its uniqueness).
\vskip.2cm {\it Acknowledgements.} The authors would
like to thank Dale Peterson for his beautiful results which underly
this work. The second author also thanks Dima Panov for some helpful
conversations. The authors thank Victor Ginzburg for a question which
led to the inclusion of Section~6.2.
\section{Preliminaries and notation}\label{s:Prelims}
Let $G$ be a simple linear algebraic group over
$\mathbb C$ split over $\mathbb R$. Usually $G$ will be simply connected.
Denote by $\operatorname{Ad}:G\to GL(\mathfrak
g)$ the adjoint representation of $G$ on its Lie algebra $\mathfrak
g$. We fix opposite Borel subgroups $B^+$ and $B^-$ defined over
$\mathbb R$ and intersecting in a split torus $T$. Their Lie algebras
are denoted by $\mathfrak b^+$ and $\mathfrak b^-$ respectively. We
will also consider their unipotent radicals $U^+$ and $U^-$ with
their Lie algebras $\mathfrak u^+$ and $\mathfrak u^-$.
Let $X^*(T)$ be the character group
of $T$ and $X_*(T)$ the group of cocharacters together with the
usual perfect pairing $\ip{\, ,\,}: X^*(T)\times X_*(T)\to\mathbb Z$.
We may identify $X^*(T)$ with a lattice inside $\mathfrak h^*$,
and $X_*(T)$ with the dual lattice inside $\mathfrak h$. These
span the real forms $\mathfrak h^*_\mathbb R$ and $\mathfrak h_\mathbb R$,
respectively.
Let $\Delta_+\subset X^*(T)$ be the set of positive roots corresponding
to $\mathfrak b^+$, and $\Delta_-$ the set of negative roots.
There is a unique highest root in $\Delta_+$
which is denoted by $\theta$.
Let $I=\{1,\dotsc, n\}$ be an indexing set for the set $\Pi:=\{\alpha_i\ |\ i\in I\}$
of positive simple
roots .
The $\alpha_i$-root space $\mathfrak g_{\alpha_i}\subset\mathfrak g$ is
spanned by Chevalley generator $e_i$ and $\mathfrak g_{-\alpha_i}$
is spanned by $f_i$. The split real form of $\mathfrak g$, denoted $\mathfrak g_\mathbb R$
is generated by the Chevalley generators $e_i,f_i$.
Let $Q:=\left < \alpha_1,\dotsc,\alpha_{n}\right >_\mathbb Z$ be the root lattice.
We also have the
fundamental weights $\omega_1,\dotsc, \omega_{n}$, and the weight lattice
$L:=\left < \omega_1,\dotsc,\omega_{n}\right >_\mathbb Z$ associated to $G$. If $G$ is simply connected, we have the relations
$$
Q\subset X^*(T)=L\subset \mathfrak h^*.
$$
Let $Q^\vee$ denote the lattice spanned by the simple coroots,
$\alpha_1^\vee,\dotsc, \alpha_n^\vee$, and $L^\vee$ the lattice
spanned by the fundamental coweights $\omega^\vee_1,\dotsc,\omega^\vee_{n}$.
Then $Q^\vee$ is the dual lattice to $L$ and $L^\vee$ the dual lattice to $Q$,
giving
$$
Q^\vee=X_*(T)\subset L^\vee\subset \mathfrak h,
$$
in the case where $G$ is simply connected. We set $\rho=\sum_{i\in I}\omega_i$ and
write $\operatorname{ht}(\lambda^\vee)=\left<\rho,\lambda^\vee\right>$ for the
height of $\lambda^\vee\in Q^\vee$.
For any Chevalley generator $e_i, f_i$ of $\mathfrak g$ we may define
a `simple root subgroup' by
$$
x_i(t)=\exp(t e_i),\qquad y_i(t)=\exp(t f_i),\qquad\qquad \text{for $t\in\mathbb C$.}
$$
Let $W=N_G(T)/T$ be the Weyl group of $G$. It is generated by simple
reflections $s_1,\dotsc, s_{n}$. The length function $\ell: W\to \mathbb N$
gives the length of a reduced expression of $w\in W$ in the simple
reflections. The unique longest element is denoted $w_0$, and for a
root $\alpha$, we let $r_\alpha$ denote the corresponding reflection. For any
simple reflection $s_i$ we choose a representative $\dot s_i$ in $G$
defined by
$$
\dot s_i:=x_i(-1) y_i(1)x_i(-1).
$$
If $w=s_{i_1}\dotsc s_{i_m}$ is a reduced expression, then $\dot
w:=\dot s_{i_1}\dotsc \dot s_{i_m}$ is a well-defined
representative for $w$, independent of the reduced expression
chosen. $W$ is a poset under the Bruhat order $\le$.
We denote the Langlands dual group of $G$ by $G^\vee$, or $G^\vee_\mathbb C$
to emphasize that we mean the algebraic group over $\mathbb C$.
The notations for $G^\vee$ are the same as those for $G$ but with added
${}^\vee$ and any other superscripts moved down, for example $B^\vee_+$ for the
analogue of $B^+$.
\subsection{Parabolic subgroups}\label{s:parabolics}
Let $P$ denote a parabolic subgroup of $G$ containing $B^+$, and
let $\mathfrak p$ be the Lie algebra of $P$. Let $I_P$ be the subset
of $I$ associated to $P$ consisting of all the $i\in I$ with
$\dot s_i\in P$ and consider its complement $I^P:=I\setminus I_P$.
Associated to $P$ we have the parabolic subgroup
$W_P=\left<s_i\ |\ i\in I_P\right>$ of $W$.
We let $W^P\subset W$ denote the set of minimal coset representatives
for $W/W_P$. An element $w$ lies
in $W^P$ precisely if for all reduced expressions
$w=s_{i_1}\cdots s_{i_m}$ the last index $i_m$ always lies in
$I^P$. We write $w^P$ or $w_0^P$ for the longest element in $W^P$,
while the longest element in $W_P$ is denoted $w_P$. For example
$w^B_0=w_0$ and $w_B=1$. Finally $P$ gives rise to a decomposition
\begin{equation*}
\Delta_+= \Delta_{P,+}\sqcup \Delta_{+}^P.
\end{equation*}
Here $\Delta_{P,+}=\{\alpha\in\Delta_+ \ | \ \langle\alpha,
\omega_i^\vee\rangle=0
\text{ all $i\in I^P$} \} $, so that
$$
\mathfrak p=\mathfrak b^+\oplus\bigoplus_{\alpha\in\Delta_{P,+}}\mathfrak g_{-\alpha},
$$
and $\Delta_+^P$ is the complement of $\Delta_{P,+}$ in $\Delta_+$.
For
example $\Delta_{B,+}=\emptyset$ and $\Delta^B_+=\Delta_+$.
\section{Total Positivity}\label{s:totpos}
\subsection{Total positivity}
A matrix $A$ in $GL_n(\mathbb R)$ is called {\it totally positive} (or {\it
totally nonnegative}) if all the minors of $A$ are positive
(respectively nonnegative). In other words $A$ acts by positive or
nonnegative matrices in all of the fundamental representations
$\bigwedge^k\mathbb R^n$ (with respect to their standard bases). In the
1990's Lusztig \cite{Lus:TotPos94} extended this theory dating back
to the 1930's to all reductive algebraic groups. This work followed
his construction of canonical bases and utilized their deep
positivity properties in types ADE.
Let $G$ be a simple algebraic group, split over the
reals. For the rest of this paper the definitions here will be applied to $G^\vee$ rather than $G$.
The totally nonnegative part $U^+_{\geq 0}$ of $U_+$ is the
semigroup generated by $\{x_i(t) \mid i \in I \; \text{and} \; t \in
\mathbb R_{\geq 0}\}$. Similarly the totally nonnegative part $U^-_{\geq
0}$ of $U_-$ is the semigroup generated by $\{y_i(t) \mid i \in I \;
\text{and} \; t \in \mathbb R_{\geq 0}\}$. The totally positive parts are
given by $U^+_{>0} = U^+_{\geq 0} \cap B^-\dot w_0 B^-$ and
$U^-_{>0} = U^-_{\geq 0} \cap B^+\dot w_0 B^+$.
\subsection{Matrix coefficients}
Suppose $\lambda \in X^*(T)$ is dominant. Then we have a highest weight
irreducible representation $V_\lambda$ for $G$. The Lie algebra
$\mathfrak g$ also acts on
$V_\lambda$ as does its universal enveloping algebra $U(\mathfrak g)$.
We fix a highest weight vector $v_\lambda^+$ in $V_\lambda$.
The vector space $V_\lambda$ has a real form given by
$V_{\lambda,\mathbb R}=U(\mathfrak g_{\mathbb R})\cdot v^+_\lambda$.
Let $(\ )^T: U(\mathfrak g)\to U(\mathfrak g)$ be the
unique involutive anti-automorphism satisfying $e_i^T=f_i$.
We let $\ip{.,.}: V_\lambda\times V_\lambda \to \mathbb C$ denote the unique symmetric, non-degenerate
bilinear form (Shapovalov form)\cite[II, 2.3]{Kum:Book} satisfying
\begin{eqnarray} \label{E:adjoint}
\ip{u \cdot v,v'} &= & \ip{v,u^T \cdot v'} \qquad \text{for all $u\in U(\mathfrak g), v,v'\in V_\lambda$,}
\end{eqnarray}
normalized so that $\ip{v^+_\lambda,v^+_\lambda} =1$. The Shapovalov form is
real positive definite on $V_{\lambda,\mathbb R}$, see \cite[Theorem 2.3.13]{Kum:Book}.
We will be
studying total positivity in the Langlands dual group $G^\vee$ of a
simply-connected group $G$. Thus $G^\vee$ will be adjoint. Let
$G^*$ be the simply-connected cover of $G^\vee$. Then the unipotent
subgroups of $G^*$ and $G^\vee$ can be identified, and so can their
totally positive (resp.~negative) parts. The purpose of this
observation is to allow the evaluation of matrix coefficients of
fundamental representations on the unipotent subgroup of $G^\vee$.
(The adjoint group $G^\vee$ itself may not act on these
representations.)
Thus for a fundamental weight $\omega_i$ (not necessarily a
character of $G$!) and a vector $v \in V_{\omega_i}$ we have a
matrix coefficient
$$
y\longmapsto \ip{v, y \cdot v^+_{\omega_i}}
$$
on $U^-$.
The following result follows from a theorem (\cite[Theorem 1.5]{BeZe:Chamber}) of Berenstein and Zelevinsky (note that every chamber weight is a $w_0$-chamber weight in the terminology of \cite{BeZe:Chamber}).
\begin{prop}\label{P:BZ}
Let $y \in U^-$. Then $y$ is totally positive if and only if for
any $i \in I$ we have
$$
\ip{\dot w\cdot v^+_{\omega_i}, y\cdot v^+_{\omega_i} }> 0
$$
for each $w \in W$, where $v^+_{\omega_i}$ denotes a highest
weight vector in the irreducible highest weight representation
$V_{\omega_i}$.
\end{prop}
We will need the following generalization of the above Proposition.
\begin{prop}\label{P:genBZ}
Let $y \in U^-$. Suppose for any irreducible
representation $V_\lambda$ of $G$
with highest weight vector $v^+_\lambda$,
and any weight vector $v$ which lies in a
one-dimensional weight space of $V_{\lambda}$
such that $\ip{v,x \cdot v^+_{\lambda}}>0$ for all totally positive $x \in U^{-}_{>0}$ we have
\begin{equation}\label{e:allowable}
\ip{v, y\cdot v^+_{\lambda} }> 0.
\end{equation}
Then $y$ is totally positive.
\end{prop}
The proof of Proposition~\ref{P:genBZ} is delayed until Section \ref{s:genBZ}. If $G$ is simply connected then this Proposition \ref{P:genBZ} follows from
Proposition~\ref{P:BZ}. The difference arises if $G$ is not simply connected, in which case the fundamental weights may not be
characters of the maximal torus of $G$.
\begin{remark}\label{rem:can}
Suppose $G$ is simply-laced. Then the matrix coefficients of $x \in U^{-}_{>0}$ in the canonical basis of any irreducible representation $V_\lambda$ are positive. It follows that for any $v \neq 0$ lying in a one-dimensional weight space of $V_\lambda$, either $v$ or $-v$ has the property that $\ip{v,x \cdot v^+_{\lambda}}>0$ for all $x \in U^-_{>0}$.
\end{remark}
\section{The affine Grassmannian and geometric Satake}
In this section, $G$ is a simple simply-connected linear algebraic group over
$\mathbb C$. Let $\mathcal O = \mathbb C[[t]]$ denote the ring of formal power series and $\mathcal K
=\mathbb C((t))$ the field of formal Laurent series. Let $\operatorname{Gr}_G =
G(\mathcal K)/G(\mathcal O)$ denote the affine Grassmannian of $G$.
\subsection{Affine Weyl group}\label{s:affWeylGroup}
Let $W_\mathrm{af} = W \ltimes X_*(T)$ be the affine Weyl group of $G$. For
a cocharacter $\lambda \in X_*(T)$ we write $t_\lambda \in W_\mathrm{af}$
for the translation element of the affine Weyl group. We then have
the commutation formula $wt_\lambda w^{-1} = t_{w \cdot \lambda}$.
The affine Weyl group is also a Coxeter group, generated by simple
reflections $s_0,s_1,\ldots,s_n$, where $s_0=r_\theta t_{-\theta^\vee}$.
It is a graded poset with its usual length function $\ell:W_{\mathrm{af}}\to \mathbb Z_{\ge 0}$,
and Bruhat order $\ge$.
Let $W_\mathrm{af}^-$ denote the minimal length coset representatives of
$W_\mathrm{af}/W$. Thus we have canonical bijections
\begin{equation}\label{e:Wafminus}
X_*(T) \longleftrightarrow W_\mathrm{af}/W \longleftrightarrow W_\mathrm{af}^-.
\end{equation}
The intersection $X_*(T) \cap W_\mathrm{af}^-$ is given by the anti-dominant
translations, that is $t_\lambda$ where $\ip{\alpha_i,\lambda} \leq
0$ for each $i \in I$.
Note that an element $\lambda$ of $X_*(T)$ viewed as a map from $\mathbb C^*$ to $T$
can also be reinterpreted as an element of $T(\mathcal K)$. We denote this element
by $t^\lambda$. The two should not be confused since the isomorphism
$W_{\mathrm{af}}\to N_{G(\mathcal K)}(T)/T$ sends $t_{\lambda}$ to $t^{-\lambda}$.
\subsection{Geometric Satake and Mirkovic-Vilonen cycles}
The affine Grassmannian is an ind-scheme \cite{Kum:Book,Gin:GS,MV:GS}. The $G(\mathcal O)$-orbits
$\operatorname{Gr}_\lambda$ on $\operatorname{Gr}_G$ are parametrized by the dominant cocharacters
$\lambda \in X^+_*(T)$. Namely,
$$
\operatorname{Gr}_\lambda:=G(\mathcal O)t^{\lambda}G(\mathcal O)/G(\mathcal O).
$$
The geometric Satake correspondence \cite{Gin:GS, Lus:GS, MV:GS}
(with real coefficients) states that the tensor category
$\mathrm{Perv}(\operatorname{Gr}_G)$ of $G(\mathcal O)$-equivariant perverse sheaves on $\operatorname{Gr}_G$
with $\mathbb C$-coefficients is equivalent to the tensor category
$\mathrm{Rep}(G^\vee_\mathbb C)$ of finite-dimensional representations of the Langlands dual group $G^\vee_\mathbb C$. (For our
purposes the tensor structure will be unimportant.) The simple
objects of $\mathrm{Perv}(\operatorname{Gr}_G)$ are the intersection cohomology complexes
$IC_\lambda$ of the $G(\mathcal O)$-orbit closures $\overline{\operatorname{Gr}}_\lambda$.
They correspond under the geometric Satake correspondence to the
highest weight representations $V_\lambda$ of $G^\vee$.
Furthermore, we have a canonical isomorphism
\begin{equation}\label{E:IC}
IH^*(\overline{\operatorname{Gr}}_\lambda) = H^*(\operatorname{Gr}_G, IC_\lambda) \simeq
V_\lambda.
\end{equation}
Mirkovic and Vilonen found explicit cycles in $\operatorname{Gr}_G$ whose
intersection homology classes give rise to a weight-basis of
$V_\lambda$ under the isomorphism $\eqref{E:IC}$. We denote by
$\mathrm{MV}_{\lambda,v}$ the MV-cycle with corresponding vector $v \in V_\lambda$.
For $w \in W$, the weight-space
$V_\lambda(w\lambda)$ is one-dimensional. We denote by
$\mathrm{MV}_{\lambda,w{\lambda}}$ the corresponding MV-cycle. Thus
$[\mathrm{MV}_{\lambda,w\lambda}]_{IH} \in IH^*(\overline{\operatorname{Gr}}_\lambda) \simeq
V_\lambda$ has weight $w\lambda$. All the statements of this section hold with $\mathbb R$-coefficients: we take perverse sheaves with $\mathbb R$-coefficients, and consider the representations of a split real form $G^\vee_\mathbb R$ of the Langlands dual group.
\subsection{Schubert varieties in $\operatorname{Gr}_G$}
Let $\mathcal I \subset G(\mathcal O)$ denote the Iwahori
subgroup of elements $g(t)$ which evaluate to $g\in B^+$ at $t=0$.
The $\mathcal I$-orbits $\Omega_\mu$ on $\operatorname{Gr}_G$, called {\it
Schubert cells}, are labeled by all (not necessarily dominant)
cocharacters $\mu \in X_*(T)$. Explicitly,
$$\Omega_\mu=\mathcal I\, t^{\mu} G(\mathcal O)/G(\mathcal O).
$$
Alternatively, we may label
Schubert cells by cosets $xW \in W_\mathrm{af}/W$ or minimal coset
representatives $x \in W_\mathrm{af}^-$, using the bijection \eqref{e:Wafminus}.
Choosing a representative $\dot x$ of $x$ we have
$$
\Omega_x=\mathcal I\, \dot x G(\mathcal O)/G(\mathcal O).
$$
The Schubert cell $\Omega_\mu= \Omega_x$ is isomorphic
to $\mathbb C^{\ell(x)}$ whenever $x \in
W_\mathrm{af}^-$. We note that $\Omega_\mu=\Omega_{t_{-\mu}}$ if $\mu$ is dominant, compare
Section~\ref{s:affWeylGroup}. The Schubert varieties $X_x = \overline{\Omega_x}$, alternatively denoted $X_\mu=\overline{\Omega_\mu}$, are
themselves unions of Schubert cells: $X_x = \sqcup_{v \leq x}
\Omega_v$. The $G(\mathcal O)$ orbits are also unions of Schubert
cells:
$$
\operatorname{Gr}_{\lambda}=\bigsqcup_{w\in W} \Omega_{w\cdot\lambda}.
$$
In particular the largest one of these, $\Omega_{\lambda}\cong \mathbb C^{\ell(t_{-\lambda})}$,
is open dense in $\operatorname{Gr}_\lambda$ (where we assumed $\lambda$ dominant), and so
\begin{equation}\label{e:GrLambdaIsSchub}
\overline{\operatorname{Gr}}_{\lambda}=\overline{\Omega_{\lambda}}=X_{ \lambda}.
\end{equation}
Thus every $G(\mathcal O)$-orbit closure
is a Schubert variety, but not conversely.
Moreover $\overline{\operatorname{Gr}}_{\lambda}$ has dimension $\ell(t_{-\lambda})$, which
equals $2\operatorname{ht}(\lambda)$.
We note that the $MV$-cycle $MV_{\lambda,v}$ is an irreducible
subvariety of $\overline{\operatorname{Gr}_\lambda}$ of dimension $\operatorname{ht}(\lambda)+\operatorname{ht}(\nu)$
if $v$ lies in the $\nu$-weight space of $V_\lambda$, see \cite[Theorem~3.2]{MV:GS}. In particular
$MV_{\lambda,\lambda}=\overline{\operatorname{Gr}_\lambda}$ and $MV_{\lambda,w_0\lambda}$
is just a point.
\subsection{The (co)homology of $\operatorname{Gr}_G$}
The space $\operatorname{Gr}_G$ is homotopic to the based loop group
$\Omega K$ of polynomial maps of $S^1$ into the compact form $K
\subset G$ \cite{Qui:unp,PS:loopgroups}. Thus the homology
$H_*(\operatorname{Gr}_G; \mathbb C)$ and cohomology $H^*(\operatorname{Gr}_G; \mathbb C)$ are commutative and
co-commutative graded dual Hopf algebras over $\mathbb C$.
Ginzburg
\cite{Gin:GS} (see also \cite{BFM:K-hom}) and Dale Peterson \cite{Pet:QCoh} described $H_*(\operatorname{Gr}_G, \mathbb C)$
as the coordinate ring of the stabilizer subgroup
of a principal nilpotent in $\mathfrak (g^\vee)^*$. Namely, in our conventions, let
$F\in (\mathfrak g^\vee)^*$ be the principal nilpotent
element defined by
$$F=\sum_{i\in I} (e_i^\vee)^*,
$$
where $(e_i^\vee)^*(\zeta )=0$
if $\zeta \in\mathfrak g_\alpha^\vee$ for $\alpha\ne\alpha_i$,
and $(e_i^\vee)^*(e_i^\vee)=1$.
Let $X= (G^\vee)_F$ denote the stabilizer of $F$ inside
$G^\vee$, under the coadjoint action. It is an abelian
subgroup of $U^\vee_-$ of dimension equal to the rank of $G$.
Then the result from \cite{Gin:GS,Pet:QCoh} says that $H_*(\operatorname{Gr}_G)$ is Hopf-isomorphic to the ring of
regular functions on $X$.
Moreover, the cohomology, $H^*(\operatorname{Gr}_G,\mathbb C)$ is Hopf-isomorphic to the
universal enveloping algebra $U(\mathfrak g^\vee_F)$ of the centralizer of $F$,
as graded dual
We note that Ginzburg \cite{Gin:GS} works over $\mathbb C$ while Peterson
\cite{Pet:QCoh} works over $\mathbb Z$, but the details of Peterson's work
are so far unpublished.
Our choice of principal nilpotent $F$ is
compatible via Peterson's isomorphism \eqref{e:PetIsoLoopHom}, see \cite{LaSh:QH},
with the conventions in
\cite{Kos:QCoh,Kos:QCoh2,Rie:MSgen}, and is
related to the choice in \cite{Gin:GS,Pet:QCoh} by switching the roles of $B^+$
and $B^-$.
In terms of the above presentation
of $H_*(\operatorname{Gr}_G)$, the fundamental class of an MV-cycle can be described
as follows. Let $\ip{.,.}:H^*(\operatorname{Gr}_G) \times H_*(\operatorname{Gr}_G) \to \mathbb C$ be the
pairing obtained from cap product composed with pushing forward to a
point.
\begin{prop}\label{p:MVmatrix}
Suppose $\mathrm{MV}_{\lambda,v}$ is the MV-cycle with corresponding
weight vector $v \in V_\lambda$ under \eqref{E:IC}. Let $u \in
U(\mathfrak g^\vee_F) \simeq H^*(\operatorname{Gr}_G)$. Then the fundamental class
$[\mathrm{MV}_{\lambda,v}] \in H_*(\operatorname{Gr}_G)$ satisfies
$$
\ip{u,[\mathrm{MV}_{\lambda,v}]} = \ip{ u\cdot v,v^-_\lambda},
$$
where $v^-_\lambda$ is the lowest weight vector of $V_\lambda$ (in the MV-basis).
\end{prop}
\begin{proof}
The argument is essentially the same as \cite[Proposition
1.9]{Gin:GS}; the main difference is that in our conventions $u$ is lower unipotent,
rather than upper unipotent, however accordingly $\overline{\operatorname{Gr}_\lambda}$ is
in our conventions the MV-cycle representing the
highest weight vector, whereas it is the lowest weight
vector in \cite{Gin:GS}. So the difference is that everywhere
the roles of $B^+$ and $B^-$ are interchanged.
By \cite[Theorem 1.7.6]{Gin:GS}, the action of
$u\in U(\mathfrak g^\vee_F)$ on $V_\lambda$ is compatible with the action
of the corresponding element in $H^*(\operatorname{Gr}_G)$ on $IH^*(\overline{\operatorname{Gr}}_\lambda)$.
Under \eqref{E:IC}, the vector $v$ is sent to $[\mathrm{MV}_{\lambda,v}]_{IH}$
which maps to the fundamental class $[\mathrm{MV}_{\lambda,v}]$ under the natural
map from the intersection cohomology $IH^*(\overline{\operatorname{Gr}}_\lambda)$
to the homology $H_*(\overline{\operatorname{Gr}}_\lambda)$.
Also, under the fundamental class map the
action of $H^*(\operatorname{Gr}_G)$
on $IH^*(\overline{\operatorname{Gr}}_\lambda)$ is sent to
the cap product of $H^*(\operatorname{Gr}_G)$ on $H_*(\overline{\operatorname{Gr}}_\lambda)$.
Finally, pushing forward to a point is the same as pairing with
$v^-_\lambda$ (in our conventions). So we get the identity
$$
\ip{u,[\mathrm{MV}_{\lambda,v}]}= \pi_*(u\cap [\mathrm{MV}_{\lambda,v}]) = \ip{v^-_\lambda, u\cdot v}.
$$
where $\pi:X\to\{\, pt\}$.
\end{proof}
\subsection{Schubert basis}
We have
$$
H_*(\operatorname{Gr}_G) = \bigoplus_{x \in W_\mathrm{af}^-} \mathbb C \cdot \xi_x, \ \ \ \ H^*(\operatorname{Gr}_G)
= \bigoplus_{x \in W_\mathrm{af}^-} \mathbb C \cdot \xi^x,
$$
where the $\xi_w$ are the fundamental classes $[X_w]$ of the
Schubert varieties, and $\{\xi^w\}$ is the cohomology basis (dual
under the cap product). Suppose $\lambda$ is dominant, then we also have
\begin{equation*}
H_*(\overline{\operatorname{Gr}}_\lambda)=\bigoplus_{\small\begin{matrix} x \in W_\mathrm{af}^- \\ x\le t_{-\lambda} \end{matrix}} \mathbb C \cdot \xi_x,
\end{equation*}
because of \eqref{e:GrLambdaIsSchub} and the decomposition of $X_{\lambda}$ into Schubert cells.
By Ginzburg/Peterson's isomorphism, we will often
think of a Schubert basis element $\xi_w$ as a function on $X$.
The Schubert basis of $H_*(\operatorname{Gr}_G)$ has the following factorization property:
\begin{prop}[{\cite{Pet:QCoh,LaSh:QH}}]
Suppose $wt_\nu, t_\mu \in W_\mathrm{af}^-$. Then
$\xi_{wt_\nu}\xi_{t_\mu} = \xi_{wt_{\nu+\mu}}$.
\end{prop}
We remark that if $wt_\nu \in W_\mathrm{af}^-$, then necessarily $\nu$ is anti-dominant.
\section{The quantum cohomology ring of $G/P$}
\subsection{The usual cohomology of $\mathbf{G/P}$ and its Schubert
basis~}\label{s:ClassCoh} For our purposes it will suffice to
take homology or cohomology with complex coefficients, so
$H^*(G/P)$ will stand for $H^*(G/P,\mathbb C)$. By the well-known result of C.~Ehresmann,
the singular homology of $G/P$ has a
basis indexed by the elements $w\in W^P$ made up of the
fundamental classes of the Schubert varieties,
\begin{equation*}
X^P_w:=\overline{(B^+wP/P)}\subseteq G/P.
\end{equation*}
Here the bar stands for (Zariski) closure. Let $\sigma^P_{w}\in
H^*(G/P)$ be the Poincar\'e dual class to $[X^P_w]$. Note
that $X^P_w$ has complex codimension $\ell(w)$ in $G/P$ and
hence $\sigma^P_w$ lies in $H^{2\ell(w)}(G/P)$. The set
$\{\sigma^P_{w}\ |\ w\in W^P\}$ forms a basis of $H^*(G/P)$ called
the Schubert basis. The top degree cohomology of $G/P$ is spanned
by $\sigma^P_{w_0^P}$ and we have
the Poincar\'e duality pairing
\begin{equation*}
H^*(G/P)\times H^*(G/P)\longrightarrow \mathbb C ,\qquad(\sigma,\mu)\mapsto \left<\sigma\cup \mu\right>
\end{equation*}
which may be interpreted as taking $(\sigma,\mu)$ to the coefficient
of $\sigma^P_{w^P_0}$ in the basis expansion of the product $\sigma\cup
\mu$. For $w\in W^P$ let $PD(w)\in W^P$ be the minimal length
coset representative in $w_0wW_P$. Then this pairing is
characterized by
\begin{equation*}
\left <\sigma^P_w\cup \sigma^P_v\right>=\delta_{w, PD(v)}.
\end{equation*}
\subsection{The quantum cohomology ring $\mathbf{qH^*(G/P)}$}
The (small) quantum cohomology ring $qH^*(G/P)$ is a deformation of
the usual cohomology ring by $\mathbb C[q^P_1,\dotsc, q^P_k]$, where $k=\dim H^2(G/P)$,
with structure constants defined by $3$-point genus $0$ Gromov-Witten
invariants. For more background on quantum cohomology, see \cite{FuWo:SchubProds}.
We have
$$
qH^*(G/P) = \oplus_{w \in W^P} \mathbb C[q^P_1,\dotsc, q^P_k] \cdot \sigma^P_w
$$
where $\sigma^P_w$ now (and in the rest of the paper) denotes the quantum Schubert class. The quantum
cup product is defined by
\begin{equation*}
\sigma^P_v\cdot \sigma^P_w=\sum_{\begin{smallmatrix}u\in W^P\\ \mathbf
d\in \mathbb N^k
\end{smallmatrix} }\left<\sigma_u^P,\sigma_v^P,\sigma_w^P\right>_{\mathbf
d}\ \mathbf q^\mathbf d\sigma^P_{PD(u)},
\end{equation*}
where $\mathbf
q^\mathbf d$ is multi-index notation for $\prod_{i=1}^k
q_i^{d_i}$, and the $\left<\sigma_u^P,\sigma_v^P,\sigma_w^P\right>_{\mathbf d}$
are genus $0$, $3$-point Gromov-Witten invariants. These enumerate rational curves in $G/P$, with a fixed degree determined by $\mathbf d$, which pass through generaic translates of three Schubert varieties. In particular, $\left<\sigma_u^P,\sigma_v^P,\sigma_w^P\right>_{\mathbf d}$ is a nonnegative integer.
The quantum cohomology ring $qH^*(G/P)$ has an analogue of the
Poincar\'e duality pairing which may be defined as the symmetric $\mathbb C[q^P_1,\dotsc,
q^P_k]$-bilinear pairing
\begin{equation*} qH^*(G/P)\times qH^*(G/P)\longrightarrow
\mathbb C[q^P_1,\dotsc, q^P_k], \qquad (\sigma,\mu)\mapsto \left<\sigma\cdot
\mu\right>_{\mathbf q}
\end{equation*}
where $\left<\sigma\cdot \mu\right>_{\mathbf q}$ denotes the coefficient of
$\sigma^P_{w_0^P}$ in the Schubert basis expansion of the product
$\sigma \cdot \mu$.
In terms of the Schubert basis the quantum Poincar\'e duality
pairing on $qH^*(G/P)$ is given by
\begin{equation}\label{E:FW}
\left <\sigma^P_w\cdot\sigma^P_v\right>_{\mathbf q}=\delta_{w, PD(v)},
\end{equation}
where $v,w\in W^P$, and $PD:W^P\to W^P$ is the involution defined
in Section \ref{s:ClassCoh}. Equation \eqref{E:FW} can for example be deduced from Fulton and Woodward's results on the minimal coefficient of $q$ in a quantum product.\footnote{We thank L.~Mihalcea for pointing out that it also follows from Proposition 3.2 of ``Finiteness of cominuscule quantum $K$-theory'' by Buch, Chaput, Mihalcea, and Perrin.}
\section {Peterson's theory}\label{PetersonTheory}
In this section we summarize Peterson's results concerning his
geometric realizations of $qH^*(G/P)$ and their relationship with
$H_*(\operatorname{Gr}_G)$.
\subsection{Definition of the Peterson variety.}\label{s:Peterson}
Each $\operatorname{Spec}(qH^*(G/P))$ turns out to be most naturally viewed as
lying inside the Langlands dual flag variety $G^\vee/B^\vee$, where
it appears as a stratum (non-reduced intersection with a Bruhat
cell) of one $n$-dimensional projective variety called the {\it
Peterson variety}. This remarkable fact was discovered and shown by
Dale Peterson \cite{Pet:QCoh}.
The condition
\begin{equation*}
(\operatorname{Ad}(g^{-1})\cdot F)(X)=0\ \text{ for all $X\in[\mathfrak u^\vee_-,\mathfrak u^\vee_-]$,}
\end{equation*}
defines a closed subvariety of $G^\vee$ invariant under right
multiplication by $B_-^\vee$. Thus they define a closed subvariety of
$G^\vee/B_-^\vee$. This subvariety $\mathcal Y$ is the {\it Peterson variety}
for $G$. Explicitly we have
\begin{equation*}
\mathcal Y=\left\{gB_-^\vee\in G^\vee/B_-^\vee\ \left |\ \operatorname{Ad}(g^{-1})\cdot F \in [\mathfrak u^\vee_-,\mathfrak u^\vee_-]^\perp \right.\right\}.
\end{equation*}
For any parabolic subgroup $W_P\subset W$ with longest element
$w_P$ define $\mathcal Y_P$ as non-reduced intersection,
\begin{equation*}
\mathcal Y_P:= \mathcal Y\times_{G^\vee/B_-^\vee} \left(B_+^\vee w_P B_-^\vee/B_-^\vee\right ).
\end{equation*}
\begin{rem}
For $P=B$ we have a map
\begin{equation}
\label{e:IsoTodaLeaf}
\mathcal Y_B\to\mathcal A_G\ :\quad u B^\vee_- \mapsto u^{-1} \cdot F,
\end{equation}
where $\mathcal A_G\subset (\mathfrak g^\vee)^*$ is
the degenerate leaf of the Toda lattice.
This
map is an isomorphism
as follows from classical work of Kostant \cite{Kos:Toda}. Kostant also showed that
$\mathcal Y_B$ is irreducible \cite{Kos:QCoh}.
The isomorphism between $qH^*(G/B)$ and the functions on the degenerate leaf of the Toda lattice was
established by B.~Kim \cite{Kim} building on \cite{GiKi:FlTod}.
\end{rem}
\subsection{Irreducibility of $\mathcal Y$.}\label{s:Peterson}
It is not immediately obvious from the above definition that the Peterson variety $\mathcal Y$
is irreducible. In other words apart from the the closure of $\mathcal Y_B$
it could a priori contain some other irreducible components coming from intersections with other Bruhat cells.
We include a sketch of proof (put together from \cite{Pet:QCoh}) that this doesn't happen,
and that therefore $\mathcal Y$ is irreducible, $n$-dimensional and equal to the closure of $\mathcal Y_B$.
Namely we have the following proposition.
\begin{prop}[Dale Peterson]
If $w=w_P$, the longest element in $W_P$ for some parabolic subgroup $P$, then $\mathcal Y\cap B^\vee_+ w B^\vee_-/B^\vee_-$ is nonempty and of
dimension $|I^P|$. Otherwise $w^{-1}\cdot (-\Pi^\vee)\not\subset \Delta^\vee_-\cup\Pi^\vee$
and $\mathcal Y\cap B^\vee_+ w B^\vee_-/B^\vee_-=\emptyset$.
\end{prop}
\bigskip
\begin{proof}[Sketch of proof]
Clearly $w^{-1}\cdot F$ needs to lie in $b_{\Pi}:= [\mathfrak u^\vee_-,\mathfrak u^\vee_-]^\perp$ for
$\mathcal Y\cap B^\vee_+ w B^\vee_-/B^\vee_-$ to be non-empty. So $w^{-1}\cdot (-\Pi^\vee)\subset \Delta^\vee_-\cup\Pi^\vee$.
This is the case if and only if $w=w_P$ for some parabolic $P$, by a lemma from \cite{Pet:QCoh} reproduced in \cite[Lemma~2.2]{Rie:QCohGr}.
Consider the map
\begin{eqnarray*}
\psi: U^\vee_+ &\to & (\mathfrak u^\vee_-)^*, \\
u &\mapsto & (u^{-1}\cdot F)|_{\mathfrak u^\vee_-}.
\end{eqnarray*}
The coordinate rings of $U_+^\vee $ and $ (\mathfrak u_-)^*$ are
polynomial rings. On $U_+^\vee$ consider the $\mathbb C^*$-action coming from conjugation
by the one-parameter subgroup of $T^\vee$ corresponding to $\rho\in X_*(T^\vee)$.
On $(\mathfrak u^\vee_-)^* $
let $\mathbb C^*$ act by $z\cdot X_{\alpha}=z^{<\alpha,\rho>+1}X_{\alpha}$
for $X_{\alpha}$ in the $\alpha$-weight space of $(\mathfrak u^\vee_-)^*$
and $\alpha\in\Delta^\vee_+$.
Then $\psi^*$ is a homomorphism of
(positively) graded rings, namely it is straightforward to check that $\psi$ is $\mathbb C^*$-equivariant.
Also $\psi$ has the property that $\psi^{-1}(0)=\{0\}$ in terms of $\mathbb C$-valued points, or indeed over any algebraically closed field.
Peterson proves this in \cite{Pet:QCoh} by considering the $B^\vee_-$-Bruhat decomposition intersected
with $U^\vee_+$. Namely, the only way $\psi(u)$ can be $0$ for $u\in U^\vee_+\cap B^\vee_-\dot w B^\vee_-$
is if $w=e$, wherefore $u$ must be the identity element in $U^\vee_+$.
It follows from these two properties that $\psi$ is finite.
For example by page 660
in Griffiths-Harris and using the $\mathbb C^*$-action to go from the statement
locally around zero, to a global statement, or by another proposition in Peterson's lectures \cite{Pet:QCoh}.
Let
$$
U_P:=\psi^{-1}((w_P\cdot b_{\Pi})|_{\mathfrak u^\vee_-} ) =\{u\in U^\vee_+\ |\ (u^{-1}\cdot F)|_{\mathfrak u^\vee_-}\in
(w_P\cdot b_{\Pi})|_{\mathfrak u^\vee_-}\}.
$$
Since $u^{-1}\cdot F\in F+\mathfrak h + (\mathfrak u^\vee_-)^*$
for $u\in U^\vee_+$ and $F+\mathfrak h\subset w_P\cdot b_{\Pi}$, we can drop the restriction to ${\mathfrak u^\vee_-}$
on both sides of the condition above, and we have a projection map
$$
U_P=\{u\in U^\vee_+\ |\ u^{-1}\cdot F\in
w_P\cdot b_{\Pi}\}=\{u\in U^\vee_+\ |\ w_P^{-1} u^{-1}\cdot F\in b_{\Pi}\}\to \mathcal Y\cap B^\vee_+ w_P B^\vee_-/B^\vee_-
$$
taking $u\in U_P$ to $u w_P B^\vee_-$, which is a fiber bundle with fiber $\cong \mathbb C^{\ell(w_P)}$.
Since $\psi$ is finite the dimension of $U_P$ is equal to the dimension of the subspace
$(w_P\cdot b_{\Pi})|_{\mathfrak u^\vee_-}$ inside ${(\mathfrak u^\vee_-)^*} $. This dimension is just $|I^P|+\ell(w_P)$, by
looking at the weight space decomposition.
So $\mathcal Y\cap B^\vee_+w_P B^\vee_-/B^\vee_-$ has dimension $| I^P|$.
\end{proof}
\subsection{Geometric realization of $qH^*(G/P)$}
Recall the stabilizer $X$ of the principal nilpotent $F$, which
is an $n$-dimensional abelian subgroup of $U_-^\vee$.
Using an idea of Kostant's
\cite[page 304]{Kos:qToda}, the Peterson variety may also be understood as a compactification
of $X$. Namely,
$$
\mathcal Y=\overline{X\dot w_0 B^\vee_-/B^\vee_-}\subset G^\vee/B^\vee_-.
$$
For the parabolic $P$ let
$$
\mathcal Y_P^*:=\mathcal Y_P\times_{G^\vee/B_-^\vee} X\dot w_0 B^\vee_-/B^\vee_-=(X\times_{G^\vee} B^\vee_+\dot w_P\dot w_0 B^\vee_+)\dot w_0/B^\vee_- \cong X\times_{G^\vee}B^\vee_+\dot w_P\dot w_0 B_+^\vee,
$$
or equivalently,
$$
\mathcal Y_P^*=\mathcal Y_P\times_{G^\vee/B^\vee_-}B^\vee_-\dot w_0 B^\vee_-/B^\vee_-.
$$
We define
$$
X_P:=X\times_{G^\vee}B^\vee_+\dot w_P\dot w_0 B_+^\vee,
$$
so that the above is an isomorphism $\mathcal Y_P^*\cong X_P$.
\begin{thm}[Dale Peterson]\label{t:Pet}\
\begin{enumerate}
\item The $\mathcal Y_P$ give rise to a decomposition
\begin{equation*}
\mathcal Y(\mathbb C)=\bigsqcup_{P} \mathcal Y_P(\mathbb C).
\end{equation*}
\item
For $P=B$ we have
\begin{equation}\label{e:PetIso1}
qH^*(G/B)\overset\sim\to \mathcal O(\mathcal Y_B),
\end{equation}
via the isomorphism \eqref{e:IsoTodaLeaf} of $\mathcal Y_B$ with the degenerate leaf
of the Toda lattice of $G^\vee$.
\item If $w\in W^P$, then the the function
$S^w\in \mathcal O(\mathcal Y_B)$ associated
to the Schubert class $\sigma_w$ defines a
regular function $S_P^w$ on $\mathcal O(\mathcal Y_P)$. There is
an (uniquely determined) isomorphism
$$
qH^*(G/P)\overset\sim \to \mathcal O(\mathcal Y_P)
$$
which takes $\sigma^P_w$ to $S_P^w$.
\item
The isomorphisms above restrict, to give
isomorphisms
$$
qH^*(G/P)[q_1^{-1},\dotsc, q_k^{-1}\ ]\overset\sim\to \mathcal O(\mathcal Y_P^*).
$$
\end{enumerate}
\end{thm}
In particular, Theorem \ref{t:Pet}(1) gives
\begin{equation*}
X(\mathbb C)=\bigsqcup_{P} X_P(\mathbb C).
\end{equation*}
\subsection{Quantum cohomology and homology of the affine Grassmannian}
\begin{thm}[Dale Peterson] \label{t:PetIsoLoop}\
\begin{enumerate}
\item
The composition of isomorphisms
\begin{equation}\label{e:PetIsoLoopHom}
H_*(\operatorname{Gr}_G)[\xi_{(t_\lambda)}^{-1}] \cong \mathcal O(X_B)\cong \mathcal O(\mathcal Y_B^*)\cong qH^*(G/B)[q_1^{-1},\dotsc, q_n^{-1}\ ]
\end{equation}
is given by
\begin{equation}\label{e:PetIsoSchub}
\xi_{wt_\lambda} \xi_{t_\mu}^{-1} \longmapsto
q_{\lambda-\mu}\sigma^B_w
\end{equation}
where $q_{\nu} = q_1^{a_1}q_2^{a_2} \cdots q_n^{a_n}$ if $\nu = a_1
\alpha_1^\vee + \cdots + a_n \alpha_n^\vee$.
\item
More generally, for an arbitrary parabolic $P$ the composition
\begin{equation}\label{e:PetIsoLoopHomP}
(H_*(\operatorname{Gr}_G)/J_P)[(\xi_{(\pi_P(t_\lambda))}^{-1}] \cong \mathcal O(X_P)\cong \mathcal O(\mathcal Y_P^*)\cong qH^*(G/P)[q_1^{-1},\dotsc, q_k^{-1}\ ]
\end{equation}
is given by
\begin{equation}\label{e:PetIsoSchubP}
\xi_{w\pi_P(t_\lambda)} \xi_{\pi_P(t_\mu)}^{-1} \longmapsto
q_{\eta_P(\lambda-\mu)}\sigma^P_w
\end{equation}
where $J_P \subset H_*(\operatorname{Gr}_G)$ is an ideal, $\pi_P$ maps $W_\mathrm{af}$ to a subset $(W^P)_\mathrm{af}$, and $\eta_P$ is the natural projection $Q^\vee \mapsto Q^\vee/Q^\vee_P$ where $Q^\vee_P$ is the root lattice of $W_P$.
\end{enumerate}
\end{thm}
Lam and Shimozono \cite{LaSh:QH} verified that the maps
\eqref{e:PetIsoSchub} (resp. \eqref{e:PetIsoSchubP}) are isomorphisms from $H_*(\operatorname{Gr}_G)[\xi_{(t_\lambda)}^{-1}]$ to
$qH^*(G/B)[q_1^{-1},\dotsc, q_n^{-1}\ ]$ (respectively from, in the parabolic case, $(H_*(\operatorname{Gr}_G)/J_P)[(\xi_{(\pi_P(t_\lambda))}^{-1})]$ to $qH^*(G/P)[q_1^{-1},\dotsc, q_k^{-1}\ ]$).
We do not review the definitions of $J_P$ and $\pi_P$ here, but refer the reader to \cite{LaSh:QH}.
\begin{rem}
In \cite{LaSh:QH} it is not shown that the isomorphism \eqref{e:PetIsoSchub} is the one induced by the geometry of $X$. We sketch how this can be achieved.
First, Kostant \cite[Section 5]{Kos:QCoh}
expresses the quantum parameters as certain ratios of `chamber minors' on $X$. Ginzburg's \cite[Proposition 1.9]{Gin:GS} also expresses the translation affine Schubert classes $\xi_{t_{-\lambda}}$ as matrix coefficients,
since $\xi_{t_{-\lambda}}=[X_\lambda]=[\overline{Gr_\lambda}]=[MV_{\lambda,\lambda}]$. This allows one to compare $\xi_{t_{-\lambda}}$ with $q_{\lambda}$ as functions on $X_B$, and see that they agree. Namely both are equal to $x\mapsto \left<x\cdot v^+_{\lambda},v^-_{\lambda}\right>$.
Let $\lambda=m\omega_i^\vee$ be a positive multiple of $\omega_i^\vee$ contained in $Q^\vee$. We now compare the functions $\xi_{s_it_{-\lambda}}$ and $q_{-\lambda}\sigma_{s_i}$ on $X_B$. For the function $\sigma_{s_i}$, Kostant gives a formula in \cite[(119)]{Kos:QCoh} as a ratio of matrix coefficients on $X$. For the function $\xi_{s_it_{-\lambda}}$, one notes that since $\lambda$ is a multiple of $\omega_i^\vee$, then
$t_{-\lambda} \in W_\mathrm{af}^-$ covers only $s_i t_{-\lambda}$ in the
Bruhat order of $W_\mathrm{af}^-$. It follows that
$H_{2\ell(t_{-\lambda})-2}(\overline{\operatorname{Gr}}_{\lambda})$ is one-dimensional,
spanned by $\xi_{s_it_{-\lambda}}$. Similarly the weight space
$V_{\lambda}(\lambda-\alpha_i^\vee)$ is one-dimensional. If we let $[\mathrm{MV}_{ \lambda,v}]$ be the unique MV-cycle
with $v$ of weight $\nu= \lambda-\alpha_i^\vee$, then this gives a cycle in
homology of degree
$2(\operatorname{ht}(\lambda) + \operatorname{ht}(\nu))=4\operatorname{ht}(\lambda) - 2=2\ell(t_{-\lambda})-2$. So we have that the homology
class $[\mathrm{MV}_{ \lambda,v}]$
is a positive integer multiple of the Schubert class $\xi_{s_i t_{-\lambda}}$.
Proposition \ref{p:MVmatrix} allows one to write
$[\mathrm{MV}_{ \lambda,v}]$ as a matrix coefficient on $X$ and compare it with Kostant's formula for $q_{\lambda}\sigma_{s_i}$.
Finally we see in this way that $q_{\lambda}\sigma_{s_i}$ is a positive integral multiple of $ \xi_{s_i t_{-\lambda}}$ as
function on $X_B$.
Now we can compose this isomorphism $qH^*(G/B)[q_1^{-1},\dotsc,q_n^{-1}]\cong\mathcal O(X_B)\cong H_*(Gr_G)[\xi_{t_\mu}^{-1}]$ which we have
just seen takes $\sigma_{s_i}$ to a positive integral multiple of $ \xi_{s_i t_{-\lambda}}\xi_{t_{-\lambda}}^{-1}$,
with the isomorphism $ H_*(Gr_G)[\xi_{t_\mu}^{-1}]\to qH^*(G/B)[q_1^{-1},\dotsc,q_n^{-1}]$ defined by \eqref{e:PetIsoSchub} going the other way.
This way we get a map $qH^*(G/B)[q_1^{-1},\dotsc,q_n^{-1}]\to qH^*(G/B)[q_1^{-1},\dotsc,q_n^{-1}]$ which takes every $q_i$ to $q_i$ but
$\sigma_{s_i}$ to a positive integral multiple of $\sigma_{s_i}$. However the images of the $\sigma_{s_i}$ still need to
obey the unique quadratic relation in the quantum cohomology ring, which identifies a quadratic form in the $\sigma_{s_i}$'s
with a linear form in the $q_i$'s. The $\sigma_{s_i}$'s cannot be rescaled by positive integer multiples and this relation
still hold, unless all of the integer factors are $1$, which means that $\sigma_{s_i}$ must go to $\sigma_{s_i}$.
Since $qH^*(G/B)[q_1^{-1},\dotsc,q_n^{-1}]$ is generated by
the $\sigma_{s_i}$ as ring over $\mathbb C[q_1^{\pm 1},\dotsc q_n^{\pm 1}]$ the map $qH^*(G/B)[q_1^{-1},\dotsc,q_n^{-1}]\to qH^*(G/B)[q_1^{-1},\dotsc,q_n^{-1}]$
considered above is the identity. Therefore \eqref{e:PetIsoSchub} is the inverse to the map
$qH^*(G/B)[q_1^{-1},\dotsc,q_n]\cong\mathcal O(X_B)\cong H_*(Gr_G)[\xi_{t_\mu}^{-1}]$, and we are done.
\end{rem}
\section{Main results}
In the rest of the paper we will be working with the $\mathbb R$-structures on all our main objects. Since $X, \mathcal Y, H_*(\operatorname{Gr}_G), qH^*(G/P)$ are in fact all defined over $\mathbb Z$ there is no problem with this. All our notations for positivity and nonnegativity refer to $\mathbb R$-points. We now define
\begin{enumerate}
\item[(AP)] The subset
$$
X^{\mathrm{af}}_{>0} = \{x \in X \mid \xi_w(x) > 0 \,\text{for all}\, w \in
W_\mathrm{af}^-\}
$$
of {\it affine Schubert positive} elements.
\item[(MVP)] The subset
$$
X^{\mathrm{MV}}_{>0} = \{x \in X \mid [\mathrm{MV}_{\lambda,v}](x) > 0 \,\text{for
all $\mathrm{MV}$-cycles}\}
$$
of {\it MV-positive} elements.
\item[(TP)] The {\it totally positive subset} defined by Lusztig's theory, $$X_{>0}:=X\cap U^\vee_{-,>0}.$$
\item[(QP)]
The subset
$$
X^{\mathrm{qSchubert}}_{>0}:=\{x\in X_B\ | \sigma^w_B(x)>0\text{ all $w\in W$}\}
$$
of {\it quantum Schubert positive} elements defined in terms of the
quantum Schubert basis and Peterson's isomorphism
$qH^*(G/B)[q_1^{-1},\dotsc, q_n^{-1}]\cong \mathcal O(X_B)$.
\end{enumerate}
Define the totally nonnegative part $X_{\geq 0} = X \cap U^\vee_{-,\geq 0}$ of $X$, and the affine Schubert nonnegative part $X^\mathrm{af}_{\geq 0}$ of $X$ as the set of points $x \in X$ such that $\xi_w(x) \geq 0$ for every affine Schubert class $\xi_w$.
We can now state our first main theorem.
\begin{thm}\label{thm:three}
The first three notions of positivity in $X$ agree: we
have $X_{>0} = X^{\mathrm{af}}_{>0} = X^\mathrm{MV}_{>0}$, and for the fourth we have $X_{>0} \subset X^{\mathrm{qSchubert}}_{>0}$. Furthermore, we have $X_{\geq 0} = \overline{X_{> 0}} = \overline{X^{\mathrm{af}}_{> 0}} = X^{\mathrm{af}}_{\geq 0}$.
\end{thm}
Therefore the first three notions of positivity are equivalent, and
the fourth is at worst weaker. We note that by \eqref{e:PetIsoLoopHom} affine Schubert positivity is
equivalent to quantum Schubert positivity with additional positivity
of the quantum parameters. Therefore $X^{\mathrm{af}}_{> 0} \subset X^{\mathrm{qSchubert}}_{> 0}$ is immediate.
\begin{conjecture}\label{c:QP}
We have $X_{> 0} = X^{\mathrm{qSchubert}}_{>0}$.
\end{conjecture}
Conjecture \ref{c:QP} was shown in \cite{Rie:QCohPFl} for type $A$. We will verify it in Appendix \ref{s:proof1} for type $C$.
By the isomorphism
$$
\mathbb C
[\mathcal Y^*_{P}]\cong qH^*(G/P)[(q^P_1)^{-1},\dotsc,(q^P_k)^{-1}]
$$
from Theorem~\ref{t:Pet} combined with $X_{P}\cong \mathcal Y^*_P$ we have a morphism
$$
\pi^P=(q_1^P,\dotsc, q_k^P): X_P \to (\mathbb C^*)^k.
$$
Let $X_{P,>0}:=X_P\cap X_{\ge 0}$. In particular
$X_{B,>0}=X_{>0}$.
\begin{thm}\label{t:main}\
\begin{enumerate}
\item
$\pi^P$ restricts to a bijection
\begin{equation*}
\pi^P_{>0}:X_{P,>0}\to\mathbb R^k_{>0}.
\end{equation*}
\item
$X_{P,>0}$ lies in the smooth locus of $X_{P}$, and the map
$\pi^P$
is etale on $X_{P,>0}$.
\item
The maps $\pi^P_{>0}$ glue to give a homeomorphism
\begin{equation*}
\Delta_{\ge 0}: X_{\ge 0}\longrightarrow \mathbb R_{\ge 0}^{n}.
\end{equation*}
\end{enumerate}
\end{thm}
\section{One direction of Theorem \ref{thm:three}}
The main goal of this section is to show that
$$
X^{\mathrm{af}}_{>0} \subseteq X^\mathrm{MV}_{>0} \subseteq X_{>0}.
$$
\begin{lem}\label{l:AFtoMV} If $x\in X$ is affine Schubert positive, then it is MV-positive.
\end{lem}
\begin{proof}
The main result of Kumar and Nori \cite{KuNo:pos}, applied to
$\operatorname{Gr}_G$, shows that every effective cycle in $\operatorname{Gr}_G$ is homologous to
a positive sum of Schubert cycles. It follows that the fundamental class $[\mathrm{MV}_{\lambda,v}]$ of an MV-cycle is a positive linear combination of the Schubert classes $\xi_w$.
\end{proof}
\begin{lem}\label{l:MVtoTP} Suppose $ X^{\mathrm{MV}}_{>0}\cap X_{>0}\ne\emptyset$. Then $ X^{\mathrm{MV}}_{>0}\subseteq X_{>0}$.
\end{lem}
\begin{proof}
Suppose $V_{\lambda}$ is a representation of $G^\vee$ with highest weight $\lambda$ and $ \mu$ is a weight of $V_{\lambda}$ with
one-dimensional weight space.
Let $[\mathrm{MV}_{\lambda,\mu}]$ be the $MV$-cycle representing a weight vector $v$ with weight $\mu$ in $V_\lambda$ under the isomorphism \eqref{E:IC}. Then for $x \in X$, we have by Proposition \ref{p:MVmatrix}, $[MV_{\lambda,\mu}](x) = \ip{x,[MV_{\lambda,\mu}]} = \ip{x \cdot v, v_\lambda^-}= \ip{ v, x^T \cdot v_\lambda^-}$.
Note that we are really thinking of $V_\lambda$ as a lowest weight representation by fixing the lowest weight vector $v_\lambda^-$ (of weight $w_0\lambda$).
Now suppose that that there is a vector $v'$ in the weight space $V_\lambda(\mu)$ satisfying $\ip{v', y \cdot v_{\lambda}^-} > 0$ for all $y \in U^+_{>0}$.
Since the weight space $V_\lambda(\mu)$ is one-dimensional it follows that the $MV$-basis element $v = c_{\lambda,\mu} \, v'$ for
a scalar $c_{\lambda,\mu}$. Choose $x_0\in X^{\mathrm{MV}}_{>0}\cap X_{>0}$. Such an $x_0$ exists by our assumption. Then we see that
the scalar is positive,
$$
c_{\lambda,\mu}=\frac{\left<x_0\cdot v,
v_{\lambda}^-\right>}{\left<x_0\cdot v',
v_{\lambda}^-\right>}=\frac{[MV_{\lambda,\mu}](x_0)}{\left<v',x_0^T\cdot
v_{\lambda}^-\right>}>0.
$$
Now suppose $x\in X^{\mathrm{MV}}_{>0} $ is an arbitrary element.
Then
$$
\left<v',x^T\cdot v_{\lambda}^-\right> = \frac
1{c_{\lambda,\mu} }\ip{v,x^T\cdot v_{\lambda}^-} = \frac
1{c_{\lambda,\mu} }[\mathrm{MV}_{\lambda,\mu}](x)
>0.
$$
By Proposition \ref{P:genBZ} (applied with `positive Borel' $B^+$ taken to be $B^\vee_-$) , this implies that $x^T$ is totally positive.
Clearly then $x$ is totally positive.
\end{proof}
\begin{lem}\label{l:fcheck}
The principal nilpotent $f^\vee=\sum f^\vee_i$ goes to a
positive multiple of the affine Schubert class $\xi^{s_0}\in H^2(\operatorname{Gr}_G)$
under the isomorphism $\mathcal U((\mathfrak g^{\vee})^{F})\cong H^*(\operatorname{Gr}_G) $.
\end{lem}
\begin{proof}
Since $H^2(\operatorname{Gr}_G)$ is $1$-dimensional we know that $f^\vee=c\xi^{s_0}$
under the identification $\mathcal U((\mathfrak g^{\vee})^{F})\cong H^*(\operatorname{Gr}_G) $.
We want to show that $c$ is positive.
Consider the exponential $\exp(f^\vee)=\exp(c\xi^{s_0})$ as an element
of the completion $\widehat{ H^*(\operatorname{Gr}_G)}$, and
choose $\lambda$ such that $s_it_\lambda\in W^-_{\mathrm{af}}$.
If we evaluate the (localized) homology class
$\xi_{s_it_\lambda}\xi_{t_\lambda}^{-1}$ on this element we obtain
\begin{align*}
\xi_{s_it_\lambda}\xi_{t_\lambda}^{-1}(\exp(f^\vee))&=\xi_{s_it_\lambda}\xi_{t_\lambda}^{-1}(\exp(c\xi^{s_0}))
\\
&=\xi_{s_it_\lambda}\xi_{t_\lambda}^{-1}(1+c\xi^{s_0} +
\frac{c^2}{2!}(a_2) + \frac{c^3}{3!}(a_3) + \cdots)
\\
&=
\frac{c^{k-1}}{(k-1)!}\frac{k!}{c^{k}}\xi_{s_it_\lambda}(a_{k-1})\xi_{t_\lambda}(a_{k})^{-1}
\\
&
=\frac{k}{c}\ \xi_{s_it_\lambda}(a_{k-1})\xi_{t_\lambda}(a_{k})^{-1},
\end{align*}
where $\ell(t_\lambda) = k = \ell(s_it_\lambda) + 1$ and the $a_i
\in H^{2i}(\operatorname{Gr}_G)$ are positive linear combinations of Schubert
classes, since all cup product Schubert structure constants of
$H^*(\operatorname{Gr}_G)$ are positive \cite{KuNo:pos}. Therefore $c$ is positive
if and only if
$\xi_{s_it_\lambda}\xi_{t_\lambda}^{-1}(\exp(f^\vee))$ is positive.
We now compute $\xi_{s_it_\lambda}\xi_{t_\lambda}^{-1}(\exp(f^\vee))$ in
a different way. Under
Peterson's isomorphism \eqref{e:PetIsoLoopHom}
\begin{equation}\label{e:ExamplePetIso}
\xi_{s_it_\lambda}\xi_{t_\lambda}^{-1}\mapsto \sigma_{s_i}.
\end{equation}
Consider the element of $X_{>0}$, the totally positive part of $X_B$, given by $\exp(f^\vee)$.
Using \eqref{e:ExamplePetIso} and identifying both $H_*(\operatorname{Gr}_G)$ and $qH^*(G/B)[q_1^{-1},\dotsc, q_n^{-1} ]$ with
$\mathbb C[X_B]$ as in \eqref{e:PetIsoLoopHom}, we can evaluate
\begin{equation}\label{e:qCohEval}
\xi_{s_it_\lambda}\xi_{t_\lambda}^{-1}(\exp(f^\vee))=\sigma_{s_i}
(\exp(f^\vee)).
\end{equation}
The right hand side here is a quotient of two `chamber minors' of $\exp(f^\vee)$ by
Kostant's formula \cite[Proposition 33]{Kos:QCoh}. Therefore the total positivity of
$\exp(f^\vee)$ implies
\begin{equation}\label{e:xipos}
\xi_{s_it_\lambda}\xi_{t_\lambda}^{-1}(\exp(f^\vee))> 0.
\end{equation}
\end{proof}
\begin{lem}\label{l:SchubPosx} The element $x=\exp(f^\vee)$ lies in $X^{\mathrm{af}}_{>0}$.
\end{lem}
\begin{proof}
A Schubert class $\xi_w$ in $H_*(\operatorname{Gr}_G)$ can be evaluated against
$\exp(f^\vee)$ by viewing $f^\vee$ as element of $H^2(\operatorname{Gr}_G)$,
expanding $\exp(f^\vee)$ as a power series and pairing $\xi_w$ with
each summand. But Lemma~\ref{l:fcheck} implies that $\exp(f^\vee)$
expands as a positive linear combination of Schubert
classes. This implies that $\xi_w(\exp(f^\vee))>0$ for all $w\in
W_{\mathrm{af}}^-$.
\end{proof}
\begin{lem} We have $X_{>0}^{\mathrm{MV}}\subseteq X_{>0}$ and
$X_{\ge 0}^{\mathrm{MV}}\subseteq X_{\ge 0}$.
\end{lem}
\begin{proof}
In Lemma~\ref{l:SchubPosx} we found a totally positive point $x \in X_{>0}$,
namely $x=\exp(f^\vee)$, which is also affine Schubert positive.
Since affine Schubert positive implies MV-positive, by Lemma~\ref{l:AFtoMV}, this means that $x\in X^{\mathrm{MV}}_{>0}\cap X_{>0}$.
Now Lemma~\ref{l:MVtoTP} implies that $ X_{>0}^{\mathrm{MV}}\subseteq
X_{>0}$. The second inclusion is an immediate consequence.
\end{proof}
\begin{cor}\label{c:AFtoTN} We have
$X^{\mathrm{af}}_{>0} \subseteq X^\mathrm{MV}_{>0} \subseteq X_{>0}$ and $X^{\mathrm{af}}_{\geq 0} \subseteq X^\mathrm{MV}_{\geq 0} \subseteq X_{\geq 0}$.
\end{cor}
\section{Parametrizing the affine Schubert-positive part of $X_P$}
Let $X_{P,>0}^{\mathrm{af}} = X^{\mathrm{af}}_{\geq 0} \cap X_P$ denote the points $x \in X_P$ such that $\xi_w(x) \geq 0$ for every affine Schubert class $\xi_w$.
First we note that $\pi^P$ takes
values in $\mathbb R_{>0}^k$ on $X_{P,>0}^{\mathrm{af}}$. Indeed, by definition $q^P_i(x) \neq 0$ for $x \in X_P$, and expressing
the quantum parameters $q^P_i$ in terms of affine Schubert classes
$\xi_{t_\lambda}$ using
Theorem~\ref{t:PetIsoLoop}(2), it follows that $x\in X_{P,>0}^{\mathrm{af}}$ has $q^P_i(x)>0$ for all $i\in I^P$.
It follows from the following result that we also have $X^{\mathrm{af}}_{B, >0} = X^{\mathrm{af}}_{>0}$.
\begin{lem}
Suppose $x \in X_{P,>0}^{\mathrm{af}}$. Then $\sigma_w^P(x) > 0$ for all $w\in W^P$.
\end{lem}
\begin{proof}
It follows from Theorem~\ref{t:PetIsoLoop}(2) and the definitions that $\sigma_w^P(x) \geq 0$. Suppose $\sigma_w^P(x) = 0$. Let $r_\theta$ denote the reflection in the longest root, and $\pi_P(r_\theta) \in W^P$ be the corresponding minimal length parabolic coset representative.
Proposition 11.2 of \cite{LaSh:QH} states that
$$
\sigma^P_{\pi_P(r_\theta)}\, \sigma_w^P =q_{\eta_P(\theta^\vee - w^{-1}\theta^\vee)} \sigma_{\pi_P(r_\theta w)}^P + q_{\eta_P(\theta^\vee)} \sum_{s_i w < w} a_i^\vee\, \sigma_{s_i w}^P.
$$
We refer the reader to Appendix \ref{s:proof1} and \cite{LaSh:QH} for the notation used here. Applying this repeatedly, we see that for large $\ell$, the product $(\sigma^P_{\pi_P(r_\theta)})^{\ell} \, \sigma_w^P$ is a (positive) combination of quantum Schubert classes which includes a monomial in the $q_i^P$. This contradicts $q_i^P(x) > 0$ for each $i$.
\end{proof}
The remainder of this section will be devoted to the proof of the following proposition.
\begin {prop} The map $\pi^P_{>0}:X^{\mathrm{af}}_{P,>0}\longrightarrow \mathbb R_{>0}^{k}$ is
bijective.
\end{prop}
We follow the proof in type $A$ given in \cite{Rie:QCohPFl}, shortening
somewhat the proof of our Lemma \ref{l:indec} below (Lemma~9.3 in~\cite{Rie:QCohPFl}), by using
a result of Fulton and Woodward
\cite{FuWo:SchubProds}: the quantum
product of Schubert classes is always nonzero.
Fix a point $Q\in (\mathbb R_{>0})^{k}$ and
consider its fiber under $\pi=\pi^P$. Let us define
\begin{equation*}
R_Q:=qH^*(G/P)/(q^P_1-Q_1,\dotsc, q_k^P-Q_k).
\end{equation*}
This is the (possibly non-reduced) coordinate ring of $\pi^{-1}(Q)$.
Note that $R_Q$ is a $|W^P|$-dimensional algebra with basis given
by the (image of the) Schubert basis. We will use the same
notation $\sigma^P_w$ for the image of a Schubert basis element
from $qH^*(G/P)$ in the quotient $R_Q$. The proof of the following result from \cite{Rie:QCohPFl} holds in our situation verbatim.
\begin{lem}[{\cite[Lemma~9.2]{Rie:QCohPFl}}]\label{l:EV}
Suppose $\mu\in R_Q$ is a nonzero simultaneous eigenvector for all
linear operators $R_Q\to R_Q$ which are defined by multiplication by
elements in $R_Q$. Then there exists a point $p\in\pi^{-1}(Q)$ such that (up to
a scalar factor)
\begin{equation*}
\mu=\sum_{w\in W^P}\sigma^P_w(p)\,\sigma^P_{PD(w)}.
\end{equation*}\qed
\end{lem}
Set
\begin{equation*}
\sigma:=\sum_{w\in W^P}\sigma_w^P\in R_Q.
\end{equation*}
Suppose the multiplication operator on $R_Q$ defined by
multiplication by $\sigma$ is given by the matrix
$M_\sigma=(m_{v,w})_{v,w\in W^P}$ with respect to the Schubert
basis. That is,
\begin{equation*}
\sigma\cdot\sigma_v^P=\sum_{w\in W^P} m_{v,w}\sigma^P_w.
\end{equation*}
Then since $Q\in \mathbb R_{>0}^k$ and by positivity of the structure
constants it follows that $M_\sigma$ is a nonnegative matrix.
\begin{lem} [{\cite[Lemma~9.3]{Rie:QCohPFl}}]\label{l:indec}
$M_\sigma$ is an indecomposable matrix.
\end{lem}
\begin{proof}
Suppose indirectly that the matrix $M_\sigma$ is
reducible. Then there exists a nonempty, proper subset $V\subset
W^P$ such that the span of $\{\sigma_v\ | \ v\in V \}$ in $R_Q$ is
invariant under $M_\sigma$. We will derive a contradiction to this
statement.
First let us show that $1\in V$. Suppose not. Since $V\ne\emptyset$
we have a $v\ne 1$ in $V$. Since $1\notin V$, the coefficient of
$\sigma_{1}$ in $\sigma_{w}\cdot\sigma_{v}$ must be zero for all $w\in
W^P$, or equivalently
\begin{equation}\label{e:zero}
\left\langle\,\sigma_w\cdot\sigma_{v}\cdot \sigma_{w_0^P}\,\right\rangle_Q=0
\end{equation}
for all $w\in W^P$. Here by the bracket $\left\langle\,\quad \,\right\rangle_Q$ we mean
$\left\langle\,\quad \,\right\rangle_{\mathbf q}$ evaluated at $Q$.
But this \eqref{e:zero} implies
$\langle\sigma_w\cdot\sigma_{v}\cdot \sigma_{w_0^P}\rangle_\mathbf
q=0$, since the latter is a nonnegative polynomial in the
$q_i^P$'s which evaluated at $Q\in \mathbb R_{>0}^k$ equals $0$.
Therefore $\sigma_{v}\cdot\sigma_{w_0^P}=0$ in $qH^*(G/P)$, by
quantum Poincar\'e duality. This leads to a contradiction,
since by work of W.~Fulton and C.~Woodward
\cite{FuWo:SchubProds} no two Schubert classes in $qH^*(G/P)$ ever
multiply to zero.
So $V$ must contain $1$. Since $V$ is a proper subset of $W^P$ we
can find some $w\notin V$. In particular, $w\ne 1$. It is a
straightforward exercise that given $1\ne w\in W^P$ there exists
$\alpha\in \Delta_+^P$ and $v\in W^P$ such that
\begin{equation*}
w=v r_\alpha, \quad\text{and}\quad \ell(w)=\ell(v)+1.
\end{equation*}
Now $\alpha\in \Delta_+^P$ means there exists $i\in I^P$ such
that $\langle \alpha,\omega_{i}^\vee\rangle\ne 0$. And hence by the
(classical) Chevalley Formula we have that
$\sigma_{s_{i}}\cdot\sigma_v$ has $\sigma_w$ as a summand. But if
$w\notin V$ this implies that also $v\notin V$, since $\sigma\cdot
\sigma_v$ would have summand $\sigma_{s_{i}}\cdot\sigma_v$ which has
summand $\sigma_w$. Note that there are no cancellations with other
terms by positivity of the structure constants.
By this process we can find ever smaller elements of $W^P$ which
do not lie in $V$ until we end up with the identity element, so a
contradiction.
\end{proof}
Given the indecomposable nonnegative matrix $M_\sigma $,
then by Perron-Frobenius theory (see e.g. \cite{Minc:NonnegMat}
Section 1.4) we know the following. \vskip .3cm
\parbox[c]{12cm}{
The matrix $M_\sigma$ has a positive eigenvector $\mu$ which is
unique up to scalar (positive meaning it has positive coefficients
with respect to the standard basis). Its eigenvalue, called the
Perron-Frobenius eigenvalue, is positive, has maximal absolute
value among all eigenvalues of $M_{\sigma}$, and has algebraic
multiplicity $1$. The eigenvector $\mu$ is unique even in the
stronger sense that any nonnegative eigenvector of $M_\sigma$ is a
multiple of $\mu$.} \vskip .3cm
Suppose $\mu$ is this eigenvector chosen normalized such that
$\left<\mu\right>_Q=1$. Then since the eigenspace containing $\mu$
is $1$--dimensional, it follows that $\mu$ is joint eigenvector
for all multiplication operators of $R_Q$. Therefore by
Lemma~\ref{l:EV} there exists a $p_0\in \pi^{-1}(Q)$ such that
\begin{equation*}
\mu=\sum_{w\in W^P}\sigma^P_w(p_0)\, \sigma^P_{PD(w)}.
\end{equation*}
Positivity of $\mu$ implies that $\sigma^P_w(p_0)\in\mathbb R_{>0}$ for all
$w\in W^P$. Of course all of the
$q_i(p_0)=Q_i$ are positive too. Hence $p_0\in X_{P,>0}^{\mathrm{af}}$. Also the point $p_0$
in the fiber over $Q$ with the property that all $\sigma^P_w(p_0)$ are positive is unique.
Therefore
\begin{equation}\label{e:homeo}
X^{\mathrm{af}}_{P,>0}\longrightarrow \mathbb R_{>0}^{k}
\end{equation}
is a bijection.
\section{Proof of Theorem \ref{t:main}.(2)}\label{s:proof2}
We establish Theorem \ref{t:main}(2) for $X^{\mathrm{af}}_{P,>0}$ instead of $X_{P,>0}$. In Proposition \ref{p:closures}, we will we establish the equality $X^{\mathrm{af}}_{P,>0} = X_{P,>0}$.
Since $qH^*(G/P)$ is free over $\mathbb C[q_1^P,\ldots,q_k^P]$, it follows that $\pi^P$ is flat.
Let $Q=\pi^P(p_0)$. Let $R=qH^*(G/P)$ and $I\subset R$ the ideal
$(q_1-Q_1,\dotsc, q_k-Q_k)$.
The Artinian ring $R_Q=R/I$ is isomorphic to the sum of local
rings $R_Q\cong\bigoplus_{x\in (\pi^P)^{-1}(Q)} R_x/IR_x$. And for
$x=p_0$ the local ring $R_{p_0}/IR_{p_0}$ corresponds in $R_Q$ to
the Perron--Frobenius eigenspace of the multiplication operator
$M_\sigma$ from the above proof. Since this is a one-dimensional
eigenspace (with algebraic multiplicity one) we have that
$\dim(R_{p_0}/IR_{p_0})=1$. It follows that the map $\pi^P$ is unramified at the point $p_0$. Thus, for example by \cite[Ex.III.10.3]{Hartshorne}, $\pi^P$ is etale at $p_0$. Since $(\mathbb C^*)^k$ is smooth, it follows that $X_P$ is smooth at $p_0$.
\section{Proofs of Theorem \ref{thm:three} and Theorem \ref{t:main}(1)}
\begin{lem}\label{l:semi}
$X_{\geq 0}$ and $X^\mathrm{af}_{\geq 0}$ are closed subsemigroups of $X$.
\end{lem}
\begin{proof}
For $X_{\geq 0}$ this follows from the fact that $(U^\vee_-)_{> 0}$ is a subsemigroup of $U^\vee_-$, and $X \subset U^\vee_-$ is a subgroup.
For $X^\mathrm{af}_{\geq 0}$, closed-ness follows from the definition. Suppose $x, y \in X^\mathrm{af}_{\geq 0}$.
Then for any affine Schubert class $\xi_w$, we have $$\xi_w(xy) =
\Delta(\xi_w)(x \otimes y) = \sum_{v,u}
c_{v,u}^w \xi_v(x) \otimes \xi_u(y) \geq 0$$ where $\Delta$ denotes the coproduct of $H_*(\operatorname{Gr}_G)$, and $c_{v,u}^w \geq 0$ are nonnegative integers \cite{KuNo:pos}. Thus $xy \in X^\mathrm{af}_{\geq 0}$.
\end{proof}
The first statement of Theorem \ref{thm:three} follows from Corollary \ref{c:AFtoTN} and the following proposition.
\begin{prop} \label{p:connected} The totally positive part and the affine Schubert positive part
of $X$ agree,
\begin{equation*}
X^{\mathrm{af}}_{>0}=X_{>0}.
\end{equation*}
\end{prop}
\begin{proof}
Our proof is identical to the proof of Proposition~12.2 from \cite{Rie:QCohPFl}.
By \cite[Section 5]{Kos:QCoh} or combining Theorem \ref{e:PetIsoLoopHom} with \cite[Proposition 1.9]{Gin:GS}, we see that each $q_i$ is a ratio of `chamber minors' and so $\pi^B$ takes positive values on $X_{>0}$. By Corollary \ref{c:AFtoTN} we have the following commutative diagram
\begin{equation*}
\begin{matrix}X^{\mathrm{af}}_{>0}&\hookrightarrow &X_{>0}\\
\qquad \searrow & & \swarrow\qquad \\
&\mathbb R_{>0}^{n}&
\end{matrix}
\end{equation*}
where the top row is clearly an open inclusion and the maps going
down are restrictions of $\pi^B$. By \eqref{e:homeo} and Section \ref{s:proof2}, the left hand map to $\mathbb R_{>0}^{n}$ is a homeomorphism. It follows from this and elementary point set
topology that $X^{\mathrm{af}}_{>0}$ must be closed inside $X_{>0}$. So
it suffices to show that $X_{>0}$ is connected.
For an arbitrary element $u\in X$ and $t\in \mathbb R$, let
\begin{equation}\label{e:ut}
u_t:=t^{-\rho} u t^{\rho},
\end{equation}
where $t\mapsto t^\rho$ is the one-parameter subgroup of
$T^\vee$ corresponding to the coroot $\rho$ (a coroot relative to $G^\vee$).
Then $u_0=e$ and $u_1=u$, and if $u\in X_{>0}$, then so is $u_t$
for all positive $t$.
Let $u, u'\in X_{>0}$ be two arbitrary points. Consider the
paths
\begin{eqnarray*}
\gamma\ : {[0,1]\to X_{>0} ~,}& \gamma(t) =u u'_t\ \\
\gamma' : {[0,1]\to X_{>0} ~,}& \,\gamma'(t)=u_t u'.
\end{eqnarray*}
Note that these paths lie entirely in $X_{>0}$ since $X_{>0}$
is a semigroup (Lemma \ref{l:semi}). Since $\gamma$ and $\gamma'$ connect $u$
and $u'$, respectively, to $uu'$, it follows that $u$ and $u'$
lie in the same connected component of $X_{>0}$, and we are
done.
\end{proof}
The second statement of Theorem \ref{thm:three} and Theorem \ref{t:main}(1) follow from:
\begin{prop}\label{p:closures}
We have $\overline{X_{> 0}} = X_{\geq 0} =X \cap U^\vee_{-,\geq 0}$. We have $\overline{X^\mathrm{af}_{>0}}=X^\mathrm{af}_{\geq 0}$. Thus
$$X_{P,>0}=X^{\mathrm{af}}_{P,>0}.$$
\end{prop}
\begin{proof}
Suppose $x\in X_{\geq 0}$. Then for any $u\in X_{>0}$, we have $u_t \in X_{>0}$ for all positive $t$,
where $u_t$ is defined in \eqref{e:ut}. The
curve $t\mapsto x(t)=xu_t$ starts at $x(0)=x$ and lies in
$X_{>0}$ for all $t>0$. Therefore $x\in \overline{X_{>0}}$ as desired.
The same proof holds for $X^\mathrm{af}_{>0}$, using Lemma \ref{l:semi} and the fact that $u_t \in X_{>0} = X^\mathrm{af}_{>0}$ (Proposition \ref{p:connected}).
\end{proof}
\section{Proof of Theorem \ref{t:main}(3)}
To define $\Delta_{\geq 0}$, we set $\Delta_i = \xi_{t_{m_i \omega_i^\vee}}$, where $m_i$ is chosen so that $m_i \omega_i^\vee \in Q^\vee$. Then $\Delta_{\geq 0} = (\Delta_1,\ldots,\Delta_{n})$.
It follows from the explicit description \cite{LaSh:QH} of $\pi_P(t_\lambda)$ and $\eta_P(\lambda)$ of Theorem \ref{t:PetIsoLoop} that for each $i$, some power of $q^P_i$ is equal to $\xi_\lambda \xi_\mu^{-1}$ on $X_P$, for certain $\lambda, \mu \in Q^\vee$. Furthermore, the map $$\pi^P_{>0}=(q^P_1,\dotsc, q^P_k):X_{P,>0}\to \mathbb R^{k}_{>0}$$ is related to the map $$\Delta^P_{>0} = (\Delta_{i_1},\Delta_{i_2},\ldots,\Delta_{i_k}): X_{P,>0}\to \mathbb R^{k}_{>0}$$ by a homeomorphism of $\mathbb R^k_{>0}$, where $I^P = \{i_1,i_2,\ldots,i_k\}$.
But $X_{\ge 0}=\bigsqcup X_{P,>0}$, so we have that
\begin{equation*}
\Delta_{\ge 0}: X_{\ge 0}\longrightarrow \mathbb R_{\ge 0}^{n}
\end{equation*}
is bijective. So $\Delta_{\ge 0}$ is continuous and bijective.
Since $\Delta$ is finite it follows that it is closed, that is, takes closed sets to closed
sets. (This holds true also in the Euclidean topology, since the
preimage of a bounded set under a finite map must be bounded, compare
\cite[Section~5.3]{Shafarevich:AG1}). Since $X_{\ge 0}$
is closed in $X$ the restriction $\Delta_{\ge 0}$ of $\Delta$ to $X_{\ge 0}$ is also
closed. Therefore $\Delta_{\ge 0}^{-1}$ is continuous.
\section{Proof of Proposition \ref{P:genBZ}}\label{s:genBZ}
It suffices to prove the Proposition for $G$ of adjoint type. Call a dominant weight $\lambda$ {\it allowable} if it is
a character of the maximal torus of adjoint type $G$.
We note that the tensor product $V = V_\lambda \otimes V_\mu$ of two irreducible representations inherits a {\it tensor Shapovalov form} $\ip{\cdot,\cdot}$ defined by $\ip{v \otimes w, v' \otimes w'} = \ip{v,v'}\ip{w,w'}$. This is again a positive-definite non-degenerate symmetric form on $V_{\lambda,\mathbb R} \otimes V_{\mu,\mathbb R}$ satisfying \eqref{E:adjoint}. It follows from \eqref{E:adjoint} that if $V_\nu, V_{\rho} \subset V$ are irreducible subrepresentations, and $\nu \neq \rho$ then $\ip{v,v'} = 0$ for $v \in V_\nu$ and $v' \in V_\rho$. Thus if the highest-weight representation $V_\nu$ occurs in $V$ with multiplicity one, the restriction of $\ip{\cdot,\cdot}$ from $V$ to $V_\nu$ must be a positive-definite non-degenerate symmetric bilinear form satisfying $\eqref{E:adjoint}$, and thus must be a multiple of the Shapovalov form. By scaling the inclusion $V_\nu \subset V$, we shall always assume that the restricted form is the Shapovalov form. The above comments extend to the case of $n$-fold tensor products.
\subsection{Type $A_n$}
We shall establish the criterion used in Proposition \ref{P:BZ}. First suppose $n$ is even. Let $V_{\omega_i}$ be a fundamental representation, and let $v_{\omega_i}^+ \in V_{\omega_i}$ be the highest weight vector, and $v = \dot w \cdot v_{\omega_i}^+$ an extremal weight vector. The weight space with weight $\dot w \cdot (n+1)\omega_i$ is extremal (and one-dimensional) in $V_{(n+1) \omega_i}$, and $V_{(n+1) \omega_i}$ is an irreducible representation for $PSL_{n+1}(\mathbb C)$. Thus for $y$ as in Proposition~\ref{P:BZ},
$$
\ip{v,y \cdot v^+_{\omega_i}}^{n+1} =
\ip{v^{\otimes {(n+1)}}, y\cdot (v^+_{\omega_i})^{\otimes {(n+1)}}} > 0.
$$
Since $n$ is even, this implies that $\ip{v,y \cdot v_{\omega_i}^+} > 0$.
For odd $n$, let us fix $w \in W$, and consider the set of signs $a_i =
\operatorname{sign}(\ip{\dot w \cdot v_{\omega_i}, y \cdot v_{\omega_i}})$. We want to prove that the $a_i$ are
all $+1$. Note that a sum of (not necessarily distinct) fundamental weights,
$\omega_{i_1}+ \cdots +\omega_{i_k}$, is allowable precisely if it is trivial on the center of $SL_{n+1}$, that is if $i_1 + \cdots + i_k$ is
divisible by $n+1$. Let $(i_1,i_2,\ldots,i_k)$ be such a sequence of indices, for which
$\omega_{i_1}+ \cdots +\omega_{i_k}$ is allowable. Then the
weight $w(\omega_{i_1}+ \cdots +\omega_{i_k})$ is an extremal weight of
the representation $V_{\omega_{i_1}+ \cdots +\omega_{i_k}}$ of $PSL_{n+1}(\mathbb C)$, and we have
\begin{multline*}
\ip{\dot w\cdot v^+_{\omega_{i_1}},y \cdot v_{\omega_{i_1}}^+}
\ip{\dot w\cdot v^+_{\omega_{i_2}},y \cdot v_{\omega_{i_2}}^+}\cdots
\ip{\dot w\cdot v^+_{\omega_{i_k}},y \cdot v^+_{\omega_{i_k}}} \\=
\ip{\dot w\cdot(v^+_{\omega_{i_1}}\otimes \dotsc\otimes v^+_{\omega_{i_k}}), y\cdot (v^+_{\omega_{i_1}}\otimes \dotsc\otimes v^+_{\omega_{i_k}})} > 0.
\end{multline*}
Therefore $a_{i_1}\dotsc a_{i_k}=+1$ if $i_1+ \dotsc + i_k= n+1$. In particular $a_{i} a_1^{n+1-i}=+1$, implying that $a_i=+1$ for even $i$,
and $a_i=a_1$ for odd $i$.
We now show that $a_1 = +1$. Let $V = V_{\omega_1}=\mathbb C^{n+1}$ with standard basis $\{v_1,\dotsc, v_{n+1}\}$, and let $Z= V^{\otimes (n+1)}$.
If we take $v^+_{\omega_1}=v_1$ then the Shapovalov form on $V$ is the standard symmetric bilinear form given by $\ip{v_i,v_j}=\delta_{i,j}$.
Let us consider $U =V_{(n+1)\omega_1} = \operatorname{Sym}^{n+1}(V)$, which occurs with multiplicity $1$ in $Z$ and has standard basis $\{v_{i_1}\dotsc v_{i_{n+1}} \ |\ 1\le i_1\le i_2\le \dotsc\le i_{n+1}\le n+1\ \}$ of symmetrized tensors,
$$
v_{i_1}\dotsc v_{i_{n+1}}=\frac{1}{(n+1)!}\sum_{\sigma\in S_{n+1}} v_{\sigma(i_1)}\otimes v_{\sigma(i_2)}\otimes\cdots\otimes v_{\sigma(i_{n+1})}.
$$
These are clearly orthonormal for the tensor Shapovalov form restricted to $U$, which is the Shapovalov form $\ip{\ ,\ }_U$ of $U$.
We have $\dot w\cdot v_1=v_k$ for some $k$. Consider the vector $z=v_1^n v_k \in U$ which has weight $\dot w \cdot \omega_1 + n\omega_1$. Clearly $\ip{z,x \cdot v^+_{(n+1)\omega_1}}_U=\ip{v_1^n v_k,x \cdot v_1^{n+1}}_U >0$ for all totally positive $x \in U^-_{>0}$, and $z$ lies in a $1$-dimensional weight space of $U$.
Therefore our assumptions imply that
$$
0 < \ip{z, y \cdot v^+_{(n+1)\omega_1}}_U=
\ip{v_1^{n}v_k, y \cdot v_1^{n+1}}_Z=
\ip{v_1, y \cdot v_1}^n \ip{v_k, y \cdot v_1}=\ip{v_k, y \cdot v_1}.
$$
Since $\ip{v_k, y \cdot v_1}_U=\ip{\dot w\cdot v_{\omega_1}^+, y \cdot v_{\omega_1}^+}_U$ this says precisely that $a_1=+1$.
\subsection{Type $B_n$}
The approach we use for the other Dynkin types can also be applied in this case, but we shall proceed using a different approach.
The adjoint group of type $B_n$ is $SO_{2n+1}(\mathbb C)$.
We realize $SO_{2n+1}(\mathbb C)$ as subgroup of $SL_{2n+1}(\mathbb C)$
following Berenstein and Zelevinsky in \cite{BeZe:Chamber} by setting
$$
SO_{2n+1}(\mathbb C)=\{ A\in SL_{2n+1}(\mathbb C) \ | A J A^t=J\},
$$
for the symmetric bilinear form
$$
J=\begin{pmatrix} & & & & 1 \\
& & &-1 &\\
& &\iddots & & \\
&-1 & & & \\
1& & & &
\end{pmatrix}.
$$
Let $\tilde e_i, \tilde f_i$ be the usual Chevalley generators
of $\mathfrak {sl}_{2n+1}$. Then we can take $e_i=\tilde e_i +\tilde e_{2n+1-i}$ and
$f_i=\tilde f_i +\tilde f_{2n+1-i}$ to be Chevalley generators
of $SO_{2n+1}(\mathbb C)$, and we have a corresponding pinning. Let $\tilde T$
denote the maximal torus of diagonal matrices in $SL_{2n+1}$ with
character group $X^*(\tilde T)=\mathbb Z\ip{\tilde \varepsilon_1,\dotsc,\tilde \varepsilon_{2n+1} }/ (\sum \tilde \varepsilon_i)$, where
$\tilde\varepsilon_i(t)$ is the $i$-th diagonal entry of $t$.
The maximal torus $T$ of $SO_{2n+1}(\mathbb C)$ in this embedding looks like
$$
T=\left\{\left. t= \begin{pmatrix} t_1& && & \\
& \ddots & & & &&\\
& &t_n & && &\\
& & & 1 & &&\\
& & & &t_n^{-1} &&\\
& & & &&\ddots &\\
& & & & &&t_1^{-1}
\end{pmatrix}\ \right | \ t_i\in \mathbb C^*
\right \}.
$$
The restriction of characters from $\tilde T$ to $T$ gives a map
$X^*(\tilde T)\to X^*(T)$
whose kernel is precisely generated by the characters $\tilde\varepsilon_i+\tilde\varepsilon_{2n-i+2}$
for $ 1\le i\le n$ and $\tilde\varepsilon_{n+1}$.
By \cite{BeZe:Chamber} the totally
nonnegative part of $SO_{2n+1}(\mathbb C)$
is the intersection of $SO_{2n+1}(\mathbb C)$ with the totally nonnegative
part of $SL_{2n+1}(\mathbb C)$.
Consider an element $y \in U^-\subset SO_{2n+1}(\mathbb C)$ satisfying
the condition \eqref{e:allowable}. We want to show that
$y$ is totally positive as element of $U^-$, or equivalently
by \cite[Corollary~7.2]{BeZe:Chamber}, that $y$ is
totally positive in $\tilde U^-$.
In the Weyl group $\tilde W$ of $SL_{2n+1}$ let $w=(s_1 s_{2n})(s_2 s_{2n-1})\dotsc (s_n s_{n+1}) s_n$.
Then multiplying $w$ with itself $n$ times gives a reduced expression for the longest
element $w_0$ of $\tilde W$. By the Chamber Ansatz of \cite{BeFoZe:TotPos} we can
associate to this reduced expression a set of `chamber minors' which suffice to check
the total positivity of any element of $\tilde U^-\subset SL_{2n+1}(\mathbb C)$.
The chamber minors can be worked out graphically using the pseudo-line arrangement
for the reduced expression. For $w$ the pseudo-line arrangement is illustrated in Figure~\ref{f:firstw}.
We concatenate $n$ copies of this pseudo-line arrangement together
to get the relevant pseudo-line arrangement $w_0$. To every chamber in the arrangement
we associate
a set $J\subset \{1,\dotsc, 2n+1\}$, by recording the numbers of the lines running below the chamber.
We order them, so let $j_1<\dotsc<j_k$ be the elements of $J$, and associate a minor to $J$ by setting
$$
\tilde\Delta_{J}(y):=\Delta_{J}(y^T)= \left<y^T\cdot e_{j_1}\wedge\dotsc\wedge e_{j_k}, v_{\omega_k}^+\right>=
\left<e_{j_1}\wedge\dotsc\wedge e_{j_k},y\cdot v_{\omega_k}^+\right>,
$$
where $\Delta_{J}$ is the `chamber minor' as defined in \cite{BeFoZe:TotPos}. Here
$k=|J|$ and $v_{\omega_k}^+=e_1\wedge\cdots\wedge e_k$ is the highest weight
vector of the irreducible representation $V_{\tilde\omega_k}=\bigwedge^k \mathbb C^{2n+1}$ of $SL_{2n+1}(\mathbb C)$.
\begin{figure}\label{f:firstw}
\begin{tikzpicture}
\draw (0,3.5)-- (1,3.5);
\node [left] at (0,3.5) {$1$};
\node [left] at (0, 3) {$2$};
\draw (0,3)-- (1,3);
\node [left] at (0, 2.5) {$3$};
\draw (0,2.5)-- (2,2.5);
\draw (2,2.5)--(2.5,3);
\draw (2.5,2.5)--(2,3);
\draw (1.5,3)--(2,3);
\draw [dotted] (.5, 1.7) -- (.5,2.2);
\node [left] at (0, 1.5) {$n+1$\ };
\draw (0,1.5)-- (5.5,1.5);
\draw [dotted] (.5, .7) -- (.5,1.3);
\draw (0,.5)--(2,.5);
\draw (0,0)--(1,0);
\node [left] at (0,0) {$2n\ \ \quad$};
\node [left] at (0,.5) {$2n-1$};
\draw (1,3.5) -- (1.5,3);
\draw (1,3) -- (1.5,3.5);
\draw (1,-.5) -- (1.5, 0);
\draw (1,0)-- (1.5, -.5);
\draw (1.5,3.5)--(7.5,3.5);
\draw (1.5,-.5) -- (7.5,-.5);
\draw (0,-.5) -- (1,-.5);
\node [left] at (0,-.5) {$2n+1$};
\draw (1.5,0)--(2,0);
\draw (2,0)--(2.5,.5);
\draw (2.5,0)-- (2,.5);
\draw [dotted] (2.5,0)--(3,0);
\draw [dotted] (2.5,.5)--(3,.5);
\draw [dotted] (2.5,2.5) -- (3,2.5);
\draw [dotted] (2.5,3) -- (3,3);
\draw (5.5,1.5)--(6,2);
\draw (5.5,2)--(6.5,1);
\draw (6.5,1)--(7.5,1);
\draw (6,1)--(7,2);
\draw (6,2)--(6.5,2);
\draw (6.5,2)--(7,1.5);
\draw (7,1.5)--(7.5,1.5);
\draw (5.5,1)--(6,1);
\draw (7,2)--(7.5,2);
\draw [dotted] (6.5,0) -- (7,0);
\draw (7,0) -- (7.5,0);
\draw [dotted] (6.5,3) -- (7,3);
\draw (7,3) -- (7.5,3);
\draw [dotted] (5,2)--(5.5,2);
\draw [dotted] (5,1)--(5.5,1);
\node [right] at (7.5,-.5) {$2n$};
\node [right] at (7.5,0) {$2n-1$};
\node [right] at (7.5,1) {$1$};
\node [right] at (7.5,1.5) {$n+1$};
\node [right] at (7.5,2) {$2n+1$};
\node [right] at (7.5,3) {$3$};
\node [right] at (7.5,3.5) {$2$};
\draw [dotted] (7.7,2.6)--(7.7,2.3);
\draw [dotted] (7.7,.3)--(7.7,.7);
\end{tikzpicture}\caption{Pseudoline arrangement for $w$.}
\end{figure}
By our assumption, $y$ lies in $SO_{2n+1}(\mathbb C)$ and we know that matrix
coefficients of $y$ of a certain type \eqref{e:allowable} are positive. Indeed, a chamber minor
$\tilde\Delta_{J}(y)=\left<e_{j_1}\wedge\cdots\wedge e_{j_k},\ y\cdot v_{\omega_k}^+\right>$ is of this
allowable type precisely if
$v=e_{j_1}\wedge\cdots\wedge e_{j_k}$ lies in a $1$-dimensional weight space
of the restricted representation, $Res^{SL_{2n+1}}_{SO_{2n+1}} V_{\tilde\omega_k}$.
All the weight spaces of fundamental representations $V_{\tilde\omega_k}$
of $SL_{2n+1}(\mathbb C)$ are $1$-dimensional. Furthermore, the weights which
stay non-zero when we restrict to $SO_{2n+1}(\mathbb C)$ all stay distinct. Therefore
their weight spaces stay $1$-dimensional. (Whereas the zero weight space
of $Res^{SL_{2n+1}}_{SO_{2n+1}} V_{\tilde\omega_k}$ becomes potentially
higher dimensional).
Now the weight vector $e_{j_1}\wedge\dotsc\wedge e_{j_k}$ in $V_{\tilde\omega_k}$
has weight $\tilde\varepsilon_{j_1}+\dotsc+\tilde \varepsilon_{j_k}$, which restricts to
a non-zero weight of the torus $T$ of $SO_{2n+1}$ precisely if the set $J$
of indices is `asymmetric' about $n+1$, so if there is some $m\in J$ for which $2n+2-m$
is not in $J$.
The following Claim implies that the chamber minors of our reduced expression
$w_0=w^n$ all have this property. Therefore $\Delta_J(y)>0$ for these minors,
by \eqref{e:allowable}. And therefore $y$ is totally positive, as desired.
\vskip .2cm
\noindent{\it Claim:} Every chamber in the pseudo-line arrangement associated
to the reduced expression $w^n$ of $w_0$ lies between lines labeled $k$
and $2n+2-k$ for some $k$.
\vskip .2cm
\noindent{\it Proof of the Claim:}
The pseudo-line arrangement is made up of $n$ copies of the one in Figure~\ref{f:firstw}.
The $j$-th copy is illustrated in Figure~\ref{f:jthw}.
Any chamber in this part of the
pseudo-line arrangement either lies in between the
lines labeled $j$ and $2n+2-j$, or between
the lines $j+1$ and $2n+1-j$.
This proves the claim.
\begin{figure}\label{f:jthw}
\begin{tikzpicture}
\draw (0,3.5)-- (1,3.5);
\node [left] at (0,3.5) {$j$};
\node [left] at (0, 3) {$j+1$};
\draw (0,3)-- (1,3);
\node [left] at (0, 2.5) {$j+2$};
\draw (0,2.5)-- (2,2.5);
\draw (2,2.5)--(2.5,3);
\draw (2.5,2.5)--(2,3);
\draw (1.5,3)--(2,3);
\draw [dotted] (.5, 1.7) -- (.5,2.2);
\node [left] at (0, 1.5) {$n+1$\ };
\draw (0,1.5)-- (5.5,1.5);
\draw [dotted] (.5, .7) -- (.5,1.3);
\draw (0,.5)--(2,.5);
\draw (0,0)--(1,0);
\node [left] at (0,0) {$2n+1-j$};
\node [left] at (0,.5) {$2n-j$};
\draw (1,3.5) -- (1.5,3);
\draw (1,3) -- (1.5,3.5);
\draw (1,-.5) -- (1.5, 0);
\draw (1,0)-- (1.5, -.5);
\draw (1.5,3.5)--(7.5,3.5);
\draw (1.5,-.5) -- (7.5,-.5);
\draw (0,-.5) -- (1,-.5);
\node [left] at (0,-.5) {$2n+2-j$};
\draw (1.5,0)--(2,0);
\draw (2,0)--(2.5,.5);
\draw (2.5,0)-- (2,.5);
\draw [dotted] (2.5,0)--(3,0);
\draw [dotted] (2.5,.5)--(3,.5);
\draw [dotted] (2.5,2.5) -- (3,2.5);
\draw [dotted] (2.5,3) -- (3,3);
\draw (5.5,1.5)--(6,2);
\draw (5.5,2)--(6.5,1);
\draw (6.5,1)--(7.5,1);
\draw (6,1)--(7,2);
\draw (6,2)--(6.5,2);
\draw (6.5,2)--(7,1.5);
\draw (7,1.5)--(7.5,1.5);
\draw (5.5,1)--(6,1);
\draw (7,2)--(7.5,2);
\draw [dotted] (6.5,0) -- (7,0);
\draw (7,0) -- (7.5,0);
\draw [dotted] (6.5,3) -- (7,3);
\draw (7,3) -- (7.5,3);
\draw [dotted] (5,2)--(5.5,2);
\draw [dotted] (5,1)--(5.5,1);
\node [right] at (7.5,-.5) {$2n+1-j$};
\node [right] at (7.5,0) {$2n-j$};
\node [right] at (7.5,1) {$j$};
\node [right] at (7.5,1.5) {$n+1$};
\node [right] at (7.5,2) {$2n+2-j$};
\node [right] at (7.5,3) {$j+2$};
\node [right] at (7.5,3.5) {$j+1$};
\draw [dotted] (7.7,2.6)--(7.7,2.3);
\draw [dotted] (7.7,.3)--(7.7,.7);
\end{tikzpicture}
\caption{The $j$-th segment of the pseudo-line arrangement for $w_0=w^n$ in the proof for type $B_n$.}
\end{figure}
\subsection{Type $C_n$}
The order of the weight lattice modulo the root lattice (index of connection) is 2. Let us choose a basis $\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_n \in \mathfrak t_\mathbb R^*$ so that the long simple root is $\alpha_1 = 2\varepsilon_1$, and the short simple roots are $\alpha_k = \varepsilon_k - \varepsilon_{k-1}$ for $2 \leq k \leq n$.
Note that $\omega_{n-1} = \varepsilon_{n-1}+\varepsilon_n$ is allowable, while $\omega_n=\varepsilon_n$ is not.
For the fundamental representations $V_{\omega_i}$ for $1\le i\le n-2$, which are not representations of the adjoint group,
we consider $V_{\omega_i + \omega_n} \subset V_{\omega_i} \otimes V_{\omega_n}$.
(Note that since $\omega_n$ is also not allowable and the index of connection is $2$, $\omega_i+\omega_n$ is allowable.)
Then for any $w\in W$ we have
$$
\ip{\dot w\cdot v^+_{\omega_i}, y\cdot v^+_{\omega_i}}\ip{ \dot w \cdot v^+_{\omega_n}, v^+_{\omega_n}}=\ip{\dot w\cdot (v^+_{\omega_i}\otimes v^+_{\omega_n}),y\cdot (v^+_{\omega_i}\otimes v^+_{\omega_n})}>0,
$$
by assumption \eqref{e:allowable} on $y$. It remains to show that $\ip{\dot w \cdot v_{\omega_n},y \cdot v_{\omega_n}}>0$,
since then $\ip{\dot w \cdot v_{\omega_i},y \cdot v_{\omega_i}}>0$ for all $w\in W$ and fundamental weights $\omega_i$,
whereby $y$ has to be totally positive, because of Proposition~\ref{P:BZ}.
We now consider $V = V_{\omega_n}$, which is $2n$-dimensional with weights $\pm \varepsilon_k$ for $1 \leq k \leq n$.
We have the following:
\begin{lem} \label{L:Clemma} \
\begin{enumerate}
\item
The equivalence relation on the weights of $V=V_{\omega_n}$ generated by $\lambda \sim \mu$ if $\lambda + \mu \in W \cdot \omega_{n-1}$ has a single equivalence class.
\item
$\omega_{n-1}$ appears as a weight in $V_{2\omega_n}$ with multiplicity 1. The weight
$\omega_{n-1}$ appears as a weight in $V \otimes V$ with multiplicity 2.
\item
$V_{\omega_{n-1}}$ occurs as an irreducible factor of $V \otimes V$ with multiplicity 1.
\end{enumerate}
\end{lem}
\begin{proof}
We have $W \cdot \omega_{n-1} = \{\pm \varepsilon_i \pm \varepsilon_j \mid 1 \leq i < j \leq n\}$. (1) follows by inspection. The first statement of (2) follows from the fact that $2\omega_n - \omega_{n-1} = \alpha_n$ is a simple root. The second statement of (2) follows by inspection of the weights of $V$. (3) follows from the fact that there are no weights $\mu$ satisfying $2\omega_n > \mu > \omega_{n-1}$ in dominance order.
\end{proof}
We may now proceed as in the proof for $A_n$ for $n$ odd. We consider the inclusion $U = V_{2\omega_n} \subset V \otimes V = Z$ and look at a vector $z \in U$ with weight $\nu = \lambda + \mu \in W \cdot \omega_{n-1}$. We first argue that $z$ can be chosen so that $\ip{z,x\cdot v_{2\omega_n}^+} >0$ for all totally positive $x \in U^-_{>0}$. In \cite[Corollary 7.2]{BeZe:Chamber}, Berenstein and Zelevinsky show that there is an inclusion $Sp_{2n}(\mathbb C) \to SL_{2n}(\mathbb C)$ such that the image of the totally positive part $U^-_{>0}$ of the unipotent of $Sp_{2n}$ lies in the totally nonnegative part of $SL_{2n}$. Now, $V$ is the standard representation of $SL_{2n}$ and contains the irreducible representation $\operatorname{Sym}^2(V)$. The restriction of $\operatorname{Sym}^2(V)$ to $Sp_{2n}(\mathbb C)$ contains the representation $U$, and $v_{2\omega_n}^+$ is exactly the highest-weight vector of $\operatorname{Sym}^2(V)$. By Remark \ref{rem:can}, we can choose weight vectors $z \in \operatorname{Sym}^2(V)$ such that $\ip{z,x\cdot v_{2\omega_n}^+} > 0$ for all $x$ which are totally positive in the unipotent of $SL_{2n}$. It follows that $\ip{z,x\cdot v_{2\omega_n}^+} > 0$ for $x \in U^-_{>0}$.
Under the inclusion $U \subset Z$, the vector $z$ is a linear combination of $v_\lambda \otimes v_\mu$ and $v_\mu \otimes v_\lambda$ by Lemma \ref{L:Clemma}(2). Here $\lambda, \mu \in W \cdot \omega_n$, and if $\lambda = w \cdot \omega_n$, then $v_\lambda = \dot w \cdot v_{\omega_n} \in V$ and similarly for $\mu$.
We have $z=A v_\lambda \otimes v_\mu + B v_\mu \otimes v_\lambda$ for positive $A,B$. Using Lemma \ref{L:Clemma}(3), we obtain that
\begin{align*}
0 &< \ip{z, y \cdot v^+_{2\omega_n}}_U\\
&= \ip{A v_\lambda \otimes v_\mu + B v_\mu \otimes v_\lambda, y \cdot (v^+_{\omega_n} \otimes v^+_{\omega_n})}_Z\\
&=
(A+B) \ip{v_{\lambda},y \cdot v^+_{\omega_n}}_V \,\ip{v_\mu,y \cdot v^+_{\omega_n}}_V
\end{align*}
It follows that $\ip{v_{\lambda},y \cdot v^+_{\omega_n}}$ and $\ip{v_{\mu},y \cdot v^+_{\omega_n}}$ have the same sign. By Lemma \ref{L:Clemma}(1)
$\lambda, \mu$ can be any two weights in $W\cdot\omega_n$ in the arguments above, therefore $\ip{v_{\lambda},y \cdot v^+_{\omega_n}}$ has the same sign
as $\ip{v^+_{\omega_n},y \cdot v^+_{\omega_n}}=1$. This concludes the proof in the $C_n$ case.
\subsection{Type $D_n$}
We take as simple roots $\alpha_1 = \varepsilon_1 + \varepsilon_2$ and $\alpha_k = \varepsilon_k-\varepsilon_{k-1}$ for $2 \leq k \leq n$. Let us consider the (spin) representation $V = V_{\omega_1}$ with highest weight $1/2(\varepsilon_1+\varepsilon_2 + \cdots \varepsilon_n)$. The argument is the same as for $C_n$ (using also Remark \ref{rem:can}), after the following Lemma. Note that $\omega_3$ is allowable.
\begin{lem} \
\begin{enumerate}
\item
The equivalence relation on the weights of $V$ generated by $\lambda \sim \mu$ if $\lambda + \mu \in W \cdot \omega_{3}$ has a single equivalence class.
\item
$\omega_{3}$ appears as a weight in $V_{2\omega_1}$ with multiplicity 1. The weight
$\omega_{3}$ appears as a weight in $V \otimes V$ with multiplicity 2.
\item
$V_{\omega_{3}}$ occurs as an irreducible factor of $V \otimes V$ with multiplicity 1.
\end{enumerate}
\end{lem}
\begin{proof}
The representation $V$ has dimension $2^{n-1}$, with weights the even signed permutations of the vector $(1/2,1/2,\ldots,1/2) \in \mathbb R^n$. The rest of the argument is identical to the proof of Lemma \ref{L:Clemma}.
\end{proof}
\subsection{Type $E_6$}
The index of connection of $E_6$ is 3, which is odd. The proof for $A_n$ with $n$ even can be applied here essentially verbatim.
\subsection{Type $E_7$}
We fix a labelling of the Dynkin diagram by letting $7$ label the minuscule node (at the end of the long leg), and $6$ be the unique node adjacent to $7$. We note that $\omega_6$ is allowable. The argument is the same as for $C_n$ (using also Remark \ref{rem:can}), after the following Lemma.
\begin{lem} \
\begin{enumerate}
\item
The equivalence relation on the weights of $V$ generated by $\lambda \sim \mu$ if $\lambda + \mu \in W \cdot \omega_{6}$ has a single equivalence class.
\item
$\omega_{6}$ appears as a weight in $V_{2\omega_7}$ with multiplicity 1. The weight
$\omega_{6}$ appears as a weight in $V \otimes V$ with multiplicity 2.
\item
$V_{\omega_{6}}$ occurs as an irreducible factor of $V \otimes V$ with multiplicity 1.
\end{enumerate}
\end{lem}
\begin{proof}
Can be verified by computer, which we did using John Stembridge's {\tt coxeter/weyl} package.
\end{proof}
\subsection{Types $E_8$, $F_4$, and $G_2$}
The adjoint group is simply-connected, so there is nothing to prove here.
| 36,993 |
\section{Introduction}
The discovery of neutrino masses \cite{Nakamura:2010zzi} motivates the introduction of right-handed neutrinos into the standard model and, by extension, into the minimal $N=1$ supersymmetric standard model (MSSM) \cite{MSSM1, MSSM2, MSSM3}. Remarkably, this right-handed neutrino extended MSSM can arise from vacua of the $E_{8} \times E_{8}$ heterotic superstring \cite{e1, Lukas:1998yy, Lukas:1999kt, Donagi:1999jp, Donagi:1999ez, e2, e3}. Specifically, smooth compactifications on elliptically fibered ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ Schoen manifolds \cite{Donagi:2003tb, Braun:2004xv} with $SU(4)$ ``extension'' bundles \cite{Braun:2005ux, Braun:2005bw, Braun:2005zv, Braun:2005fk, Ambroso:2008kb, Buchbinder:2002ji, Buchbinder:2002pr} can lead to four-dimensional, $N=1$ supersymmetric theories with exactly the particle spectrum of the MSSM with three families of right-handed neutrino chiral supermultiplets, one per family, and no vector-like pairs or exotic states \cite{Braun:2005nv, lukas}. Furthermore, the theory is invariant under the $SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ gauge group of the standard model.
However, the fact that this ``extended'' MSSM arises from heterotic string theory has important theoretical and phenomenological implications. First, and foremost, is the fact that the low energy gauge group ${\cal{G}}$ must contain one extra $U(1)$ factor. This is due to the fact that the last step in the symmetry breaking sequence $E_{8} \rightarrow Spin(10) \rightarrow {\cal{G}}$\footnote{We will refer to the $SO(10)$ group, as it is commonly called in the model building literature, as $Spin(10)$ to be more mathematically correct.} is accomplished by two ``Wilson lines,'' each corresponding to a generator of the ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ isometry group. Since these Wilson lines are Abelian, they preserve the rank of the gauge group. It follows that the rank 5 $Spin(10)$ group is spontaneously broken to the rank 4 standard model group plus an additional factor of $U(1)$, which can be associated with B-L (baryon minus lepton number). That is, ${\cal{G}}=SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y} \times U(1)_{B-L}$. A renormalization group analysis of this
B-L MSSM theory, including the radiative breaking of both B-L and electroweak symmetry, the associated B-L/EW hierarchy and predictions for the masses of the $Z^{\prime}$ boson, the Higgs bosons and all superpartners was presented in \cite{Ambroso:2009jd, Ambroso:2009sc}. The implications for proton decay, and dark matter, as well as a discussion of the associated cosmic strings, were given in \cite{Ambroso:2010pe} and \cite{Brelidze:2010hf} respectively, while consequences in the neutrino sector were discussed in \cite{Ambroso:2010pe, Mohapatra:1986aw, Ghosh:2010hy, Barger:2010iv}
Meanwhile, supersymmetric theories associated with B-L have had a long, rich history because of the intimate relationship between R-parity (first defined in~\cite{R1,R2}) and B-L: $R=(-1)^{3(B-L) + 2S}$, where $S$ is the spin. The fate of R-parity plays a crucial role in the phenomenology and cosmology of supersymmetry. This relation was originally explored in the context of a global symmetry~\cite{AM} and later expanded to gauged symmetries~\cite{Hayashi} and more minimally in~\cite{Mohapatra:1986aw}; all of which resulted in R-parity violation. This was followed by a medley of works exploring both facets of the fate of R-parity in a variety of contexts, including~\cite{Masiero:1990uj, Kuchimanchi:1993jg, Martin:1996kn, Aulakh:1999cd, Aulakh:2000sn, Feldman:2011ms}.
Recently, this B-L MSSM theory was advanced in a series of papers~\cite{FileviezPerez:2008sx, Barger:2008wn, Everett:2009vy} as the simplest B-L supersymmetric theory, since it only extends the MSSM by the right-handed neutrinos required by anomaly cancellation. An automatic prediction follows: R-parity violation is spontaneously broken in minimal B-L models because only the right-handed sneutrino can break the B-L symmetry in a realistic way. These papers explored many of the consequences of this theory and showed it to be consistent with present experimental data. Furthermore, these papers were recently supplemented by a discussion of the lepton number violating signals that could accompany the B-L MSSM at the LHC~\cite{FileviezPerez:2012mj}. Therefore, from both the top-down and bottom-up point of view, the B-L MSSM theory appears to be very compelling.
The analyses in \cite{Ambroso:2009jd, Ambroso:2009sc, Ambroso:2010pe}, in order to elucidate the mechanism for radiative B-L breaking and the B-L/EW hierarchy, necessarily was carried out over a relatively restricted set of initial parameters. In addition, the basis of generators of $Y$ and $B-L$ is not ``orthogonal'' in the Cartan subalgebra of ${\mathfrak{so}}(10)$. Hence, the associated $U(1)_{Y}$ and $U(1)_{B-L}$ field strengths exhibit ``kinetic mixing'', both initially and at all lower scales. This greatly complicates the RGEs of the gauge couplings and was analyzed only approximately in \cite{Ambroso:2009jd, Ambroso:2009sc, Ambroso:2010pe}. Finally, the running parameters of the B-L MSSM generically experience five mass ``thresholds'', that is, scales where the coefficients of the associated RG beta functions change. In \cite{Ambroso:2009jd, Ambroso:2009sc, Ambroso:2010pe}, this was approximated by only three such masses, the other two being sufficiently close to these that, in a restricted regime of parameter space, they could safely be ignored.
In this paper, we rectify the last two of these problems. The first issue, carrying out the RG analysis for a greatly expanded set of initial parameters, will be analyzed in forthcoming work
\cite{Preparation}. We begin by presenting a detailed mathematical analysis of the Cartan subalgebra of ${\mathfrak{so}}(10)$, deriving the two-dimensional subalgebra that commutes with ${\mathfrak{su}}(3)_{C} \oplus {\mathfrak{su}}(2)_{L}$ and introducing the ``canonical'' basis which spans it. Wilson lines derived from any linearly independent elements $Y_{1}$ and $Y_{2}$ of this subalgebra will spontaneously break $Spin(10) \rightarrow SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y_{1}}\times U(1)_{Y_{2}}$. Many, but not all, such bases will lead to the right-handed neutrino extended MSSM spectrum at low energy. We will restrict our attention to those bases that do-- such as $Y_{Y}$,$Y_{B-L}$ and $Y_{T_{3R}}$,$Y_{B-L}$, as well as to the specific ``non-canonical'' basis introduced and analyzed in Appendix A. We show that, as with the $Y$, B-L generators discussed in
\cite{Ambroso:2009jd,Ambroso:2009sc,Ambroso:2010pe}, the quark/lepton and Higgs superfields in both the canonical and non-canonical bases each arise from different ${\bf 16}$ and ${\bf 10}$ representations of $Spin(10)$ respectively. It follows that the soft supersymmetry breaking terms, as well as the Yukawa couplings, are uncorrelated by their origin in $Spin(10)$ multiplets. This ``liberates'' the initial parameter space of these theories and has important implications for low energy.
Next, we introduce the Killing inner product on the Cartan subalgebra. It is shown that as long as the basis elements $Y_{1}$,$Y_{2}$ are orthogonal, that is, that their Killing bracket vanishes, then kinetic mixing of the associated gauge field strengths at the initial ``unification'' scale will also vanish. Both the canonical and specific non-canonical basis elements, but not the $Y_{Y}$,$Y_{B-L}$ generators of \cite{Ambroso:2009jd, Ambroso:2009sc, Ambroso:2010pe}, satisfy this condition. Henceforth, we restrict our discussion to such orthogonal bases. Using the RG analyses presented in \cite{Babu:1996vt}, we then show that kinetic mixing will continue to vanish at all lower energy-momentum scales if and only if $Tr(Y_{1}Y_{2})$ over the entire matter and Higgs spectrum of the B-L MSSM is zero. The canonical basis is shown to satisfy this condition and, hence, never exhibits kinetic mixing. However, in Appendix A we show that the specific non-canonical basis does not. Furthermore, we prove a theorem that the only orthogonal basis satisfying this condition is precisely the canonical basis, and appropriate multiples thereof. We note in passing that the $Y_{Y}$,$Y_{B-L}$ generators of \cite{Ambroso:2009jd, Ambroso:2009sc, Ambroso:2010pe} also don't have vanishing trace over the B-L MSSM. Thus, in this paper, we identify a unique basis of generators $Y_{T_{3R}}$,$Y_{B-L}$ whose low energy gauge group, $U(1)_{Y_{T_{3R}}}\times U(1)_{Y_{B-L}}$, does not exhibit kinetic mixing at any scale.
For this reason, the bulk of this paper analyzes the mass thresholds, boundary conditions and RG running of the gauge parameters and soft gaugino masses associated with this canonical basis. An important aspect of the analysis is that the Wilson lines associated with each of $Y_{T_{3R}}$,$Y_{B-L}$ need not ``turn on'' at the same scale. Rather, because the inverse radius of the ``hole'' in the Calabi-Yau threefold that they wrap depends on moduli, one such mass scale can precede the other, perhaps by as much as an order of magnitude. Allowing for this, there are then two different ``breaking patterns''. In the first, $Spin(10)$ is broken by the $Y_{B-L}$ Wilson line to an intermediate region containing a left-right type model. This has gauge group $SU(3)_{C} \times SU(2)_{L} \times SU(2)_{R} \times U(1)_{B-L}$ with a specific spectrum that we calculate directly from string theory. At a lower scale, $M_{I}$, the $Y_{T_{3R}}$ Wilson line turns on and the theory becomes the extended MSSM. Conversely, if the $Y_{T_{3R}}$ Wilson line turns on first, then $Spin(10)$ is broken to a Pati-Salam type theory with gauge group $SU(4)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}}$ and an explicit spectrum. Again, we calculate this directly using string theory. At a lower scale, $M_{I}$, the $Y_{B-L}$ Wilson line turns on and the theory becomes the extended MSSM. It is proven that in either case, subject to imposing the experimental values of the gauge couplings at the $Z$-mass, $M_{I}$ can be chosen so as to enforce the unification of all gauge couplings, albeit for different values of $M_{I}$. Of course, if both Wilson lines turn on simultaneously, then $Spin(10)$ is broken immediately to the B-L MSSM. In this case, gauge unification cannot occur.
Using this technology, in the final section we present a detailed discussion of the five mass thresholds, the boundary conditions at these scales, explicit computer analysis' plotting the running of the gauge couplings and a discussion of the gaugino soft masses. This is done independently for both the left-right and Pati-Salam intermediate breaking patterns, imposing the experimental values of the gauge couplings at the $Z$-mass. These plots show the unification of these couplings, and the explicit mass threshold behavior. The unification scale, the unified gauge coupling and the value of $M_{I}$ associated with such unification are evaluated in each case. Finally, we present the results when both Wilson lines are simultaneous and explore the extent to which the gauge parameters ``miss'' unification.
\section{The ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ Wilson Lines}
We begin by searching for the most general $U(1) \times U(1)$ subgroup of $Spin(10)$ that commutes with $SU(3)_{C} \times SU(2)_{L}$. This is most easily carried out using the associated Lie algebra ${\mathfrak{so}}(10)$, whose relevant properties can be found, for example, in \cite{Georgi}.
We will identify the color subalgebra ${\mathfrak{su}(3)_C} \subset {\mathfrak{so}}(10)$ with the $\alpha^{1}$ and $\alpha^{2}$ nodes of the ${\mathfrak{so}}(10)$ Dynkin diagram shown in Figure 1.
\begin{figure}[h]
\begin{center}
\begin{picture}(5,2)(0,.2)
\put(.625,1){\line(1,0){.75}}
\put(1.625,1){\line(1,0){.75}}
\put(2.588,.912){\line(1,-1){.53}}
\put(2.588,1.088){\line(1,1){.53}}
\put(.5,1){\circle{.25}}
\put(1.5,1){\circle{.25}}
\put(2.5,1){\circle{.25}}
\put(3.207,.293){\circle{.25}}
\put(3.207,1.707){\circle{.25}}
\put(.4,.6){$\alpha_1$}
\put(1.4,.6){$\alpha_2$}
\put(2.4,.6){$\alpha_3$}
\put(3.457,1.6){$\alpha_4$}
\put(3.457,.2){$\alpha_5$}
\put(.2,1.3){\color{red} \line(1,0){1.6}}
\put(1.8,1.3){\color{red} \line(0,-1){.9}}
\put(.2,.4){\color{red} \line(1,0){1.6}}
\put(.2,1.3){\color{red} \line(0,-1){.9}}
\put(2.9,1.4){\color{blue} \line(1,0){1}}
\put(3.9,2){\color{blue} \line(0,-1){.6}}
\put(2.9,2){\color{blue} \line(1,0){1}}
\put(2.9,2){\color{blue} \line(0,-1){.6}}
\end{picture}
\end{center}
\label{Figure1}
\caption{\small The Dynkin diagram for $Spin(10)$ with the $\mathfrak{su}(3)_{C}$ and $\mathfrak{su}(2)_{L}$ generators highlighted in red and blue respectively.}
\end{figure}
\noindent The complete set of ${\mathfrak{su}}(3)_C$ roots is given by
\begin{equation}
\alpha^{1}=(1,-1,0,0,0), \quad \alpha^{2}=(0,1,-1,0,0), \quad \beta=(1,0,-1,0,0)
\label{1}
\end{equation}
as well as minus these roots. We denote the associated elements of ${\mathfrak{su}}(3)_C$ by $E_{\pm\alpha^{1}}, E_{\pm \alpha^{2}}$ and $E_{\pm \beta}$ respectively. When added to
${\cal{H}}_{1}=\frac{1}{2}(H_{1}-H_{2})$ and ${\cal{H}}_{2}=\frac{1}{2}(H_{2}-H_{3})$
of the Cartan subalgebra ${\mathfrak{h}}$, these eight elements span ${\mathfrak{su}}(3)_C \subset {\mathfrak{so}}(10)$.
Similarly, we will identify the electroweak subalgebra ${\mathfrak{su}}(2)_{L} \subset {\mathfrak{so}}(10)$ with the $\alpha^{4}$ node of the ${\mathfrak{so}}(10)$ Dynkin diagram in Figure 1. The complete set of roots is given by
\begin{equation}
\alpha^{4}=(0,0,0,1,-1)
\label{2}
\end{equation}
and its minus. Denote the associated elements of ${\mathfrak{su}}(2)_L$ by $E_{\pm\alpha^{4}}$. Added to ${\cal{H}}_{4}=\frac{1}{2}(H_{4}-H_{5})$ of the Cartan subalgebra, these three elements span ${\mathfrak{su}}(2)_L \subset {\mathfrak{so}}(10)$.
To identify the most general $U(1) \times U(1)$ subgroup of $Spin(10)$ that commutes with $SU(3)_{C} \times SU(2)_{L}$, we simply search for the subspace of the five-dimensional Cartan subalgebra ${\mathfrak{h}}$ that commutes with all of the ${\mathfrak{su}}(3)_C \oplus {\mathfrak{su}}(2)_L$ generators listed above. Using the commutation relations
\begin{equation}
[H_{i},H_{j}]=0, \quad [H_{i},E_{\alpha}]=\alpha_{i}E_{\alpha}
\label{3}
\end{equation}
valid for {\it any} root $\alpha$, this is equivalent to solving for the most general element of
${\mathfrak{h}}$ that annihilates $\alpha^{1},\alpha^{2},\beta,\alpha^{4}$. Writing this as
$H_{3\oplus2}={\sum}_{i=1}^{5}a^{i}H_{i}$,
it follows from \eqref{1} and \eqref{2} that
\begin{equation}
H_{3\oplus2}=a(H_{1}+H_{2}+H_{3})+b(H_{4}+H_{5})
\label{4}
\end{equation}
for any real coefficients a and b. That is, the elements of ${\mathfrak{so}}(10)$ that commute with
${\mathfrak{su}}(3)_C \oplus {\mathfrak{su}}(2)_L$ form the two-dimensional subspace ${\mathfrak{h}}_{3\oplus2}$ of the Cartan subalgebra spanned by
\begin{equation}
H_{1}+H_{2}+H_{3}, \quad H_{4}+H_{5}~.
\label{5}
\end{equation}
Of course, any linearly independent basis of ${\mathfrak{h}}_{3\oplus2} \subset {\mathfrak{h}}$ is of potential physical interest. However, the basis \eqref{5} arises naturally in the above calculation. For this reason, and others to be specified below, we will refer to \eqref{5} as the ``canonical'' basis and discuss its properties first and in detail. In Appendix A, we briefly analyze a non-canonical basis and prove a general theorem about such bases.
\subsection{The Canonical Basis \label{CB}}
To identify the physical meaning of each of the canonical generators, it is useful to find their explicit form in the ${\bf 16}$ representation of ${\mathfrak{so}}(10)$ since it contains a complete family of quarks/leptons including the right-handed neutrino. This can be accomplished using standard methods; see, for example, \cite{Georgi,Slansky:1981yr}. Written in a basis in which the ${\bf 16}$ decomposes under ${\mathfrak{su}}(3)_C \oplus {\mathfrak{su}}(2)_L$ as
\begin{equation}
{\bf 16}= (\overline{\bf 3}, {\bf 1})^{\oplus 2} \oplus ({\bf 3},{\bf 2}) \oplus ({\bf 1},{\bf 2})\oplus({\bf 1},{\bf 1})^{\oplus 2}~,
\label{6}
\end{equation}
that is, into the u, d, Q quarks and the L, $\nu$, e leptons respectively,
we find that
\begin{equation}
2(H_{1}+H_{2}+H_{3})= ((-1){\bf 1}_{3})^{\oplus 2} \oplus (1){\bf 1}_{6} \oplus (-3){\bf 1}_{2}
\oplus ((3){\bf 1}_{1})^{\oplus 2}~.
\label{7}
\end{equation}
We have multiplied $H_{1}+H_{2}+H_{3}$ by $2$ to ensure that all diagonal elements are integers.
It follows that one can identify
\begin{equation}
2(H_{1}+H_{2}+H_{3})=3(B-L)~.
\label{8}
\end{equation}
Similarly, written in the same basis we find
\begin{equation}
H_{4}+H_{5}= (-1){\bf 1}_{3} \oplus (1){\bf 1}_{3} \oplus (0){\bf 1}_{6} \oplus (0){\bf 1}_{2}
\oplus (-1){\bf 1}_{1} \oplus (1){\bf 1}_{1}~.
\label{9}
\end{equation}
If hypercharge is defined using the relation $Q=T_{3L}+Y$, then one can identify
\begin{equation}
H_{4}+H_{5}= 2(Y-\frac{1}{2}(B-L))~.
\label{10}
\end{equation}
It is useful to note that
%
\begin{equation}
Y-\frac{1}{2}(B-L)=T_{3R}~,
\label{11}
\end{equation}
where $T_{3R}$ is the diagonal generator of $SU(2)_{R}$. Noting that the complete set of roots associated with the $\alpha^{5}$ node of the ${\mathfrak{so}}(10)$ Dynkin diagram in Figure 1 is
\begin{equation}
\alpha^{5}=(0,0,0,1,1)
\label{12}
\end{equation}
and its minus, one can see immediately that the three elements ${\cal{H}}_{5}=\frac{1}{2}( H_{4}+H_{5})$ and $E_{\pm\alpha^{5}}$ span an $SU(2)$ Lie algebra. It follows from \eqref{11} that this can be identified with ${\mathfrak{su}}(2)_R \subset {\mathfrak{so}}(10)$. Having made these physical identifications, we henceforth denote these generators by
\begin{equation}
Y_{B-L}=2(H_{1}+H_{2}+H_{3}), \quad Y_{T_{3R}}=H_{4}+H_{5} ~.
\label{13}
\end{equation}
Note that their Killing brackets are given by
\begin{equation}
(Y_{B-L}|Y_{B-L})=12, \quad (Y_{T_{3R}}|Y_{T_{3R}})=2, \quad (Y_{B-L}|Y_{T_{3R}})=0
\label{14}
\end{equation}
where we have used the relation $(H_{i}|H_{j})=\delta_{ij}$.
To set our notation, we reiterate that
\begin{equation}
[Y_{B-L}]_{\bf 16} = ((-1){\bf 1}_{3})^{\oplus 2} \oplus (1){\bf 1}_{6} \oplus (-3){\bf 1}_{2}
\oplus ((3){\bf 1}_{1})^{\oplus 2}
\label{15}
\end{equation}
and
\begin{equation}
[Y_{T_{3R}}]_{\bf 16} = (-1){\bf 1}_{3} \oplus (1){\bf 1}_{3} \oplus (0){\bf 1}_{6} \oplus (0){\bf 1}_{2} \oplus (-1){\bf 1}_{1} \oplus (1){\bf 1}_{1}~.
\label{16}
\end{equation}
The Killing bracket is defined on ${\mathfrak{so}}(10)$ and is independent of the representation of the Lie algebra. That being said, the Killing bracket of any two elements $x,y \in {\mathfrak{so}}(10)$ can be evaluated in any representation $R$ using the formula
\begin{equation}
(x|y)=\frac{1}{I_{R}} Tr([x]_{R}[y]_{R}) ~,
\label{17}
\end{equation}
where
\begin{equation}
I_{R}=\frac{d_{R}}{d_{\mathfrak{so}(10)}}C_{2}(R)
\label{18}
\end{equation}
is the Dynkin index, ${d_{\mathfrak{so}(10)}}=45$ and $C_{2}(R)$ is the quadratic Casimir invariant for the representation. Using this relation, one can check the validity of \eqref{15} and \eqref{16}. Note that $C_{2}({\bf 16})=\frac{45}{4}$ and, hence, $I_{{\bf 16}}=4$. Furthermore, from \eqref{15} and \eqref{16} one finds
\begin{eqnarray}
&&Tr([Y_{B-L}]_{\bf 16}[Y_{B-L}]_{\bf 16})=48, \quad Tr( [Y_{T_{3R}}]_{\bf 16} [Y_{T_{3R}}]_{\bf 16})=8, \nonumber \\
&&\qquad \qquad \qquad Tr( [Y_{B-L}]_{\bf 16}[Y_{T_{3R}}]_{\bf 16})=0 ~.
\label{19}
\end{eqnarray}
It then follows from \eqref{17} that $[Y_{B-L}]_{\bf 16}$ and $[Y_{T_{3R}}]_{\bf 16}$ satisfy the Killing relations \eqref{14}, as they must.
In the following, it will be important to know the explicit form of $Y_{B-L}$ and $Y_{T_{3R}}$ in the
${\bf 10}$ representation of ${\mathfrak{so}}(10)$. Written in a basis in which ${\bf 10}$ decomposes under ${\mathfrak{su}}(3)_C \oplus {\mathfrak{su}}(2)_L$ as
\begin{equation}
{\bf 10}= ({\bf 3},{\bf 1})\oplus(\overline{\bf 3}, {\bf 1}) \oplus ({\bf 1},{\bf 2})^{\oplus 2}~,
\label{20}
\end{equation}
that is, into the color triplet $H_{C}, {\bar{H}}_{C}$ and weak doublet $H, {\bar{H}}$ Higgs fields respectively, we find using \cite{Georgi} that
\begin{equation}
[Y_{B-L}]_{\bf 10}=(2){\bf 1}_{3} \oplus (-2){\bf 1}_{3} \oplus (0){\bf 1}_{2}\oplus (0){\bf 1}_{2}
\label{21}
\end{equation}
and
\begin{equation}
[Y_{T_{3R}}]_{\bf 10}=(0){\bf 1}_{3} \oplus (0){\bf 1}_{3} \oplus (1){\bf 1}_{2}\oplus (-1){\bf 1}_{2}~.
\label{22}
\end{equation}
As above, these expressions can be checked by computing their Killing brackets. We find
\begin{eqnarray}
&&Tr([Y_{B-L}]_{\bf 10}[Y_{B-L}]_{\bf 10})=24, \quad Tr( [Y_{T_{3R}}]_{\bf 10} [Y_{T_{3R}}]_{\bf 10})=4, \nonumber \\
&&\qquad \qquad \qquad Tr( [Y_{B-L}]_{\bf 10}[Y_{T_{3R}}]_{\bf 10})=0 ~.
\label{23}
\end{eqnarray}
Noting that $C_{2}({\bf 10})=9$ and, hence, $I_{{\bf 10}}=2$, it follows from \eqref{17} that $[Y_{B-L}]_{\bf 10}$ and $[Y_{T_{3R}}]_{\bf 10}$ indeed satisfy the Killing relations \eqref{14}.
\subsection{Properties of the Canonical Basis \label{PCB}}
There are four fundamental properties possessed by the canonical basis that make it particularly interesting. These are derived in the following.
\subsubsection*{Wilson Lines and the MSSM: \label{MSSM}}
Let us consider the two Wilson lines associated with the canonical basis. As abstract $Spin(10)$ group elements, these are
\begin{equation}
\chi_{B-L} = e^{iY_{B-L}\frac{2\pi}{3}}, \quad \chi_{T_{3R}}=e^{iY_{T_{3R}}\frac{2\pi}{3}} \ .
\label{24}
\end{equation}
Note that $\chi_{B-L}^{3}=\chi_{T_{3R}}^{3}=1$ and, hence, each generates a finite ${\mathbb{Z}}_{3}$ subroup of $Spin(10)$. We identify these with the ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ isometry of the Calabi-Yau threefold $X$. When turned on simultaneously, these Wilson lines spontaneously break
\begin{equation}
Spin(10) \longrightarrow SU(3)_{C} \times SU(2)_{L} \times U(1)_{T_{R}} \times U(1)_{B-L} \ .
\label{24a}
\end{equation}
As discussed previously, the ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ isometry acts equivariantly on the chosen vector bundle $V$ and, hence, the associated sheaf cohomology groups of tensor products of $V$ carry a representation of ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$. To determine the zero modes of the Dirac operator twisted by $V$ and, hence, the low energy spectrum, one takes each $H^{1}(X,U_{R}(V))$, tensors it with the associated representation $R$, and then chooses the invariant subspace
$(H^{1}(X,U_{R}(V)) \otimes R)^{{\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}}$. Let us carry this out for each of the relevant representations of $Spin(10)$.
For $R={\bf 16}$, the associated sheaf cohomology is
\begin{equation}
H^{1}(X,V)=RG^{\oplus3}~,
\label{25}
\end{equation}
where $RG$ is the regular representation of ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ given by
\begin{equation}
RG=1 \oplus \chi_{1} \oplus \chi_{2} \oplus \chi_{1}^{2} \oplus \chi_{2}^{2} \oplus \chi_{1}\chi_{2} \oplus \chi_{1}^{2}\chi_{2} \oplus \chi_{1}\chi_{2}^{2} \oplus \chi_{1}^{2}\chi_{2} ^{2}
\label{26}
\end{equation}
and $\chi_{1}, \chi_{2}$ are the third roots of unity which generate the first and second factors of
${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$. Note that
\begin{equation}
h^{1}(X,V)=27
\label{27}
\end{equation}
and, hence, there are $27$ ${\bf 16}$ representations of $Spin(10)$ in the spectrum prior to turning on the Wilson lines. Choosing the Wilson line generators in the canonical basis \eqref{13}, it follows from \eqref{15} and \eqref{16} that the action of the Wilson lines \eqref{24} on each ${\bf 16}$ is given by
\begin{eqnarray}
{\bf 16}&=& \chi_{T_{3R}}^{2} \cdot \chi_{B-L}^{2} ({\bf{\bar{3}},\bf {1}},-1,-1)\oplus \chi_{T_{3R}} \cdot \chi_{B-L}^{2} ({\bf{\bar{3}},\bf {1}},1,-1) \label{28} \\
&& \oplus 1 \cdot \chi_{B-L} ({\bf{3}},{\bf {2}},0,1) \oplus 1 \cdot 1({\bf{1}},{\bf {2}},0,-3) \oplus \chi_{T_{3R}}^{2} \cdot 1 ({\bf{1}},{\bf {1}},-1,3) \nonumber\\
&& \oplus \chi_{T_{3R}} \cdot 1({\bf{1}},{\bf {1}},1,3)~. \nonumber
\end{eqnarray}
Choosing $\chi_{1}=\chi_{T_{3R}}$ and $\chi_{2}=\chi_{B-L}$ in \eqref{26}, we find that $(H^{1}(X,V) \otimes {\bf 16})^{{\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}}$ consists of {\it three families} of quarks and leptons, each family transforming as
\begin{equation}
Q= (U,D)^T = ({\bf{3}},{\bf {2}},0,\frac{1}{3}), \quad u=({\bf{\bar{3}},\bf {1}},-\frac{1}{2},-\frac{1}{3}), \quad d=({\bf{\bar{3}},\bf {1}},\frac{1}{2},-\frac{1}{3})
\label{29}
\end{equation}
and
\begin{equation}
L=(N,E)^T=({\bf{1}},{\bf {2}},0,-1), \quad \nu=({\bf{1}},{\bf {1}},-\frac{1}{2},1), \quad e=({\bf{1}},{\bf {1}},\frac{1}{2},1)
\label{30}
\end{equation}
under $SU(3)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}} \times U(1)_{B-L}$.
For $R={\bf 10}$ the associated sheaf cohomology is
\begin{equation}
H^{1}(X,\wedge^{2}V)=\chi_{1} \oplus \chi_{1}^{2} \oplus \chi_{1}\chi_{2}^{2} \oplus \chi_{1}^{2}\chi_{2} \ .
\label{31}
\end{equation}
Note that
\begin{equation}
h^{1}(X,\wedge^{2}V)=4
\label{32}
\end{equation}
and, hence, there are 4 ${\bf 10}$ representations of $Spin(10)$ in the spectrum prior to turning on the Wilson lines. Choosing the Wilson line generators in the canonical basis \eqref{13}, it follows from \eqref{21} and \eqref{22} that the action of the Wilson lines \eqref{24} on each ${\bf 10}$ is given by
\begin{eqnarray}
{\bf 10}&= &1 \cdot \chi_{B-L}^{2} ({\bf 3},{\bf 1},0,2) \oplus 1 \cdot \chi_{B-L} ({\bar{\bf 3}},{\bf 1},0,-2) \nonumber \\
&&\oplus \chi_{T_{3R}} \cdot 1({\bf 1},{\bf 2},1,0) \oplus \chi_{T_{3R}}^{2} \cdot 1 ({\bf 1},{\bf 2},-1,0)~.
\label{33}
\end{eqnarray}
Choosing $\chi_{1}=\chi_{T_{3R}}$ and $\chi_{2}=\chi_{B-L}$ in \eqref{31}, we find that $(H^{1}(X,\wedge^{2}V) \otimes {\bf 10})^{{\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}}$ consists of a {\it single pair} of Higgs doublets transforming as
\begin{equation}
H=({\bf 1},{\bf 2},\frac{1}{2},0), \quad \bar{H}= ({\bf 1},{\bf 2},-\frac{1}{2},0)
\label{34}
\end{equation}
under $SU(3)_{C} \times SU(2)_{L} \times U(1)_{T_{R}} \times U(1)_{B-L}$. These results lead to the following important property of the canonical basis.\\
\noindent $\bullet$ {\it When the two Wilson lines corresponding to the canonical basis are turned on simultaneously, the resulting low energy spectrum is precisely that of the MSSM--that is, three families of quark/lepton chiral superfields, each family with a right-handed neutrino supermultiplet, and one pair of Higgs-Higgs conjugate chiral multiplets. There are no vector-like pairs or exotic particles.}\\
The canonical basis exhibits a second, related, property that has important consequences for the the low energy effective Lagrangian. Consider, once again, the $R={\bf 16}$ case and the ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ invariant tensor product of $H^{1}(X,V)$ in \eqref{25},\eqref{26} with the ${\bf 16}$ decomposition in \eqref{28}. Note that, with the exception of $\chi_{B-L}$, $\chi_{T_{3R}}\chi_{B-L}^{2}$ and $\chi_{T_{3R}}^{2}\chi_{B-L}^{2}$ in \eqref{26} which project out all terms in \eqref{28}, each of the remaining six entries in each RG form a ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ invariant with only one term in a ${\bf 16}$. That is, each quark and lepton chiral multiplet in the low energy theory arises from a different ${\bf 16}$ representation of $Spin(10)$. Now consider the $R={\bf 10}$ case. It is easily seen from \eqref{31} and \eqref{33} that, with the exception of $\chi_{T_{3R}}\chi_{B-L}^{2}$ and $\chi_{T_{3R}}^{2}\chi_{B-L}$ in $\eqref{31}$ which project out all terms in \eqref{33}, the remaining two entries $\chi_{T_{3R}}$ and $\chi_{T_{3R}}$ each form a ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ invariant with only a single component of \eqref{33}, the Higgs and Higgs conjugate chiral multiplets respectively. Hence, each arises from a different {\bf 10} representation of $Spin(10)$. This leads to the second important property of the canonical basis. \\
\noindent ${\bullet}$ {\it Since each quark/lepton and Higgs superfield of the low energy Lagrangian arises from a different ${\bf 16}$ and ${\bf 10}$ representation of $Spin(10)$ respectively, the parameters of the effective theory, and specifically the Yukawa couplings and the soft supersymmetry breaking parameters, are uncorrelated by the $Spin(10)$ unification. For example, the soft mass squared parameters of the right-handed sneutrinos need not be universal with the remaining slepton supersymmetry breaking parameters.}
\subsubsection*{The Kinetic Mixing Parameter: }
Prior to turning on the ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ Wilson lines, the conventionally normalized kinetic energy part of the gauge field Lagrangian is $Spin(10)$ invariant and given by
\begin{equation}
{\cal{L}}_{kinetic}=\frac{-1}{2I_{R}}Tr(F^{a}T^{a}_{R})^{2}~,
\label{35}
\end{equation}
where $\{T^{a}_{R}, a=1,\dots.45\}$ is an orthogonal basis of ${\mathfrak{so}}(10)$ in any representation $R$ , each basis element Killing normalized to $\frac{1}{\sqrt{2}}$. In particular, defining
\begin{equation}
T^{1}= Y-\frac{1}{2}(B-L)=\frac{1}{2} Y_{T_{3R}}, \quad T^{2}=\sqrt{\frac{3}{8}}(B-L)=\frac{1}{2\sqrt{6}}Y_{B-L}
\label{36}
\end{equation}
we see from \eqref{14} that
\begin{equation}
(T^{1}|T^{1})=(T^{2}|T^{2})=\frac{1}{2}, \quad (T^{1}|T^{2})=0
\label{37}
\end{equation}
and, hence,
\begin{equation}
{\cal{L}}_{kinetic}=-\frac{1}{4}(F_{\mu\nu}^{1})^{2}-\frac{1}{4}(F_{\mu\nu}^{2})^{2} +\dots ~.
\label{38}
\end{equation}
That is, there is no kinetic mixing term of the form $F_{\mu\nu}^{1} F^{2\mu\nu}$. This is a consequence of the fact that the canonical basis elements $Y_{T_{3R}}$ and $Y_{B-L}$ are Killing orthogonal, and is of little importance while $Spin(10)$ remains unbroken. However, if both Wilson lines are turned on simultaneously, the gauge group is spontaneously broken to $SU(3)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}} \times U(1)_{B-L}$. For general $U(1) \times U(1)$, the two Abelian field strengths can exhibit kinetic mixing; that is,
\begin{equation}
{\cal{L}}_{kinetic}=-\frac{1}{4}((F_{\mu\nu}^{1})^{2}+2\alpha F_{\mu\nu}^{1} F^{2\mu\nu}+(F_{\mu\nu}^{2})^{2} +\dots) ~.
\label{39}
\end{equation}
for some real parameter $\alpha$. However, for $U(1)_{T_{3R}} \times U(1)_{B-L}$ the normalized canonical generators satisfy \eqref{37} and, specifically, are orthogonal in ${\mathfrak{so}}(10)$. It follows that the initial value of $\alpha$ at the unification scale, $M_{u}$, must vanish. This is the third important property of the canonical basis.\\
\noindent $\bullet$ {\it Since the generators of the canonical basis are Killing orthogonal in
${\mathfrak{so}}(10)$, the value of the kinetic field strength mixing parameter $\alpha$ must vanish at the unification scale. That is, $\alpha(M_{u})=0$.}\\
Once the $Spin(10)$ symmetry is broken by both Wilson lines, either by turning them on at the same scale or sequentially, as discussed below, one expects the mixing parameter $\alpha$ to regrow due to radiative corrections. In this case, the Abelian field strengths develop a non-vanishing mixing term which greatly complicates the renormalization group analysis of the low energy effective theory. Radiative kinetic mixing has been discussed by a number of authors, see, for example, \cite{Babu:1996vt,delAguila:1988jz,Holdom:1985ag,Dienes:1996zr,Foot:1991kb, Fonseca:2011vn}. Let us briefly review the analysis. Consider a theory with unspecified $U(1) \times U(1)$ gauge factors. Then, in general, at an arbitrary momentum scale
\begin{equation}
{\cal{L}}_{kinetic}=-\frac{1}{4}((F_{\mu\nu}^{1})^{2}+2\alpha F_{\mu\nu}^{1} F^{2\mu\nu}+(F_{\mu\nu}^{2})^{2} +\dots) ~.
\label{40}
\end{equation}
The associated gauge covariant derivative is given by
\begin{equation}
D=\partial-iT^{1}g_{1}A^{1}-iT^{2}g_{2}A^{2} \ ,
\label{41}
\end{equation}
where we denote the coupling parameters and gauge fields associated with $T^{1}$ and $T^{2}$ by $g_{1},A^{1}$ and $g_{2},A^{2}$ respectively. Defining new gauge fields by $\vec{A}={\cal{O}}\vec{A}^{\prime}$ where
\begin{equation}
{\cal{O}}=\frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 1 \\
-1 & 1
\end{pmatrix}
\label{42}
\end{equation}
diagonalizes the kinetic energy terms to
\begin{equation}
{\cal{L}}_{kinetic}=-\frac{1}{4}((1-\alpha)(F_{\mu\nu}^{\prime 1})^{2}+(1+\alpha)(F_{\mu\nu}^{\prime 2})^{2} +\dots) ~.
\label{43}
\end{equation}
Further rescaling of the gauge fields by $\vec{A}^{\prime}={\cal{D}}^{-\frac{1}{2}}\vec{A}^{\prime\prime}$ with
\begin{equation}
{\cal{D}}^{-\frac{1}{2}}=
\begin{pmatrix}
\frac{1}{\sqrt{1-\alpha}} & 0 \\
0 & \frac{1}{\sqrt{1+\alpha}}
\end{pmatrix}
\label{44}
\end{equation}
leads to a canonically normalized kinetic term
\begin{equation}
{\cal{L}}_{kinetic}=-\frac{1}{4}((F_{\mu\nu}^{\prime \prime1})^{2}+(F_{\mu\nu}^{\prime \prime 2})^{2} +\dots) ~.
\label{45}
\end{equation}
However, the covariant derivative now has off-diagonal gauge couplings
\begin{equation}
D=\partial -i(T^{1},T^{2})
\begin{pmatrix}
\frac{g_{1}}{\sqrt{1-\alpha}} & \frac{g_{1}}{\sqrt{1+\alpha}} \\
\frac{-g_{2}}{\sqrt{1-\alpha}} & \frac{g_{2}}{\sqrt{1+\alpha}}
\end{pmatrix}
\begin{pmatrix}
A^{\prime \prime 1} \\
A^{\prime \prime 2}
\end{pmatrix} \ .
\label{46}
\end{equation}
Note that the four gauge couplings are not independent, being functions of the three parameters $\alpha, g_{1}$ and $g_{2}$ in the original Lagrangian. It is not surprising, therefore, that a further field redefinition will eliminate one of them. The transformation should be orthogonal so as to leave the field strength kinetic term diagonal and canonically normalized. This can be achieved by setting $\vec{A}^{\prime\prime}
={\cal{P}}\vec{\cal{A}}$ where
\begin{equation}
{\cal{P}}=\frac{1}{\sqrt{2}}
\begin{pmatrix}
\sqrt{1-\alpha} & - \sqrt{1+\alpha} \\
\sqrt{1+\alpha} & \sqrt{1-\alpha}
\end{pmatrix} \ .
\label{47}
\end{equation}
We find that the covariant derivative now becomes
\begin{equation}
D=\partial -i(T^{1},T^{2})
\begin{pmatrix}
{\cal{G}}_{1} & {\cal{G}}_{M} \\
0 & {\cal{G}}_{2}
\end{pmatrix}
\begin{pmatrix}
{\cal{A}}^{1} \\
{\cal{A}}^{2}
\end{pmatrix} \ ,
\label{48}
\end{equation}
with
\begin{equation}
{\cal{G}}_{1}=g_{1}, \quad {\cal{G}}_{2}=\frac{g_{2}}{\sqrt{1-\alpha^{2}}}, \quad {\cal{G}}_{M}=\frac{-g_{1}\alpha}{\sqrt{1-\alpha^{2}}} \ .
\label{49}
\end{equation}
Note that in the limit that $\alpha \rightarrow 0$, ${\cal{G}}_{2}=g_{2}$ and ${\cal{G}}_{M}=0$.
The renormalization group equations for the gauge couplings in this ``upper triangular'' realization were given in \cite{Babu:1996vt}. Here, however, it suffices to present the RGE for the off-diagonal coupling ${\cal{G}}_{M}$. It is found to be
\begin{equation}
\frac{d{\cal{G}}_{M}}{dt}=\frac{1}{16\pi^{2}} \beta_{M}
\label{50}
\end{equation}
where
\begin{equation}
\beta_{M}={\cal{G}}_{2}^{2}{\cal{G}}_{M}B_{22}+{\cal{G}}_{M}^{3}B_{11}+2{\cal{G}}_{1}^{2}{\cal{G}}_{M}B_{11}+2{\cal{G}}_{2}{\cal{G}}_{M}^{2}B_{12}+{\cal{G}}_{1}^{2}{\cal{G}}_{2}B_{12}
\label{51}
\end{equation}
and
\begin{equation}
B_{ij}=Tr(T^{i}T^{j}) \ .
\label{52}
\end{equation}
The trace in \eqref{52} is over the entire matter and Higgs spectrum of the MSSM. Note that all of the terms in the $\beta$ function \eqref{51}, with the exception the last term, contain at least one power of ${\cal{G}}_{M}$. If the mixing parameter $\alpha$ and, hence, the off-diagonal coupling ${\cal{G}}_{M}$ vanish at some initial scale, as they will for our canonical basis, then the terms containing ${\cal{G}}_{M}$ will not, by themselves, generate a non-zero mixing parameter at any lower scale. However, a non-vanishing ${\cal{G}}_{M}$ will be generated by the last term. The only exception to this is if the charges $T^{1}$ and $T^{2}$ are such that
\begin{equation}
B_{12}=Tr(T^{1}T^{2})=0 \ .
\label{53}
\end{equation}
Generically, this will not be the case for arbitrary charges of $U(1) \times U(1)$; see Appendix A.3. However, let us break $Spin(10)$ to $U(1)_{T_{3R}}\times U(1)_{B-L}$ with both Wilson lines of the canonical basis. The associated normalized charges $T^{1}$ and $T^{2}$ were presented in \eqref{36} and satisfy
\begin{equation}
(T^{1}|T^{2})=0 \ .
\label{54}
\end{equation}
It then follows from \eqref{17} that
\begin{equation}
Tr([T^{1}]_{R}[T^{2}]_{R})=0
\label{55}
\end{equation}
for any complete ${\mathfrak{so}}(10)$ representation $R$. Recalling that each quark/lepton family with a right-handed neutrino fills out a complete ${\bf 16}$ multiplet, one can conclude that
\begin{equation}
Tr([T^{1}]_{quarks/leptons}[T^{2}]_{quarks/leptons})=0 \ .
\label{56}
\end{equation}
However, in the reduction to the zero-mode spectrum the color triplet Higgs $H_{C}$ and ${\bar{H}}_{C}$ are projected out. Hence, the electroweak Higgs doublets $H$ and ${\bar{H}}$ do not make up a complete ${\bf 10}$ of ${\mathfrak{so}}(10)$. Therefore, the trace of $T^{1}T^{2}$ over the Higgs fields of the MSSM is not guaranteed to vanish. It is straightforward to compute this trace using \eqref{21} and \eqref{22}. If we ignore the color triplet components, then
\begin{equation}
[Y_{T_{3R}}]_{H,{\bar{H}}}= (1){\bf 1}_{2}\oplus (-1){\bf 1}_{2}
\label{57}
\end{equation}
and
\begin{equation}
[Y_{B-L}]_{H,{\bar{H}}}=(0){\bf 1}_{2}\oplus (0){\bf 1}_{2} \ .
\label{58}
\end{equation}
It then follows from \eqref{36} and the $0$-entries in \eqref{58} that
\begin{equation}
Tr([T^{1}]_{H,{\bar{H}}}[T^{2}]_{H,{\bar{H}}})= \frac{1}{4\sqrt{6}}Tr([Y_{T_{3R}}]_{H,{\bar{H}}}[Y_{B-L}]_{H,{\bar{H}}})=0 \ .
\label{59}
\end{equation}
We conclude from \eqref{56} and \eqref{59} that
\begin{equation}
B_{12}=0 \ .
\label{60}
\end{equation}
Therefore, for the canonical basis if the initial value of $\alpha$ and, hence, ${\cal{G}}_{M}$ vanish, then both will remain zero at any lower scale. This is the fourth important property possessed by the canonical basis.\\
\noindent $\bullet$ {\it The generators of the canonical basis are such that $Tr(T^{1}T^{2})=0$ when the trace is performed over the matter and Higgs spectrum of the MSSM. This guarantees that if the original kinetic mixing parameter vanishes, then $\alpha$ and, hence, ${\cal{G}}_{M}$ will remain zero under the RG at any scale. This property of not having kinetic mixing greatly simplifies the renormalization group analysis of the $SU(3)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}}\times U(1)_{B-L}$ low energy theory}.\\
\subsection{Sequential Wilson Line Breaking \label{swb}}
In Subsection \ref{MSSM} we introduced the two Wilson lines $\chi_{B-L}$ and $\chi_{T_{3R}}$ associated with the canonical basis. As abstract $Spin(10)$ group elements, these were given in \eqref{24}. Each generates a finite ${\mathbb{Z}}_{3}$ subroup of $Spin(10)$, together representing the ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ isometry of the Calabi-Yau threefold $X$. When turned on {\it simultaneously}, they spontaneously break $Spin (10)$ to the gauge group $SU(3)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}} \times U(1)_{B-L}$. The associated low energy spectrum was computed in Subsection \ref{MSSM} and found to be exactly that of the MSSM; that is, three families of quarks/leptons, each with a right-handed neutrino supermultiplet, as well as one pair of Higgs-Higgs conjugated superfields.
Since $\pi_{1}(X/({{\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}}))={\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$, there are two independent classes of non-contractible curves in the quotient of $X$. Each Wilson line corresponds to the ${\mathbb{Z}}_{3}$ holonomy group of a flat gauge bundle wrapped around a curve in one of these classes. A ``mass scale'' can be assigned to each Wilson line; namely, the inverse radius of the associated non-contractible curve. This radius will depend on the geometric moduli of the Calabi-Yau threefold, leading to a larger (smaller) mass scale for a smaller (larger) radius curve. It follows that the two Wilson lines can ``turn on'' at different scales, depending on the moduli of the geometry. There are three possibilities-- a) Both Wilson lines have approximately the same scale corresponding to the unification mass. This was the situation discussed in Subsection \ref{MSSM}. b) The $\chi_{B-L}$ Wilson line turns on at the unification mass, followed sequentially at a smaller scale by $\chi_{T_{3R}}$. c) The converse situation, that is, the $\chi_{T_{3R}}$ Wilson line turns on at the unification mass followed by $\chi_{B-L}$ at a lower scale. If the mass scales of the Wilson lines are sufficiently separated, these scenarios can have different low energy precision predictions. In this subsection, we will explore the gauge groups and spectra associated with scenarios b) and c).
\subsubsection*{$ M_{\chi_{B-L}} > M_{\chi_{T_{3R}}}:$}
Recall from \eqref{13} that $Y_{B-L}=2(H_{1}+H_{2}+H_{3})$. By construction, $Y_{B-L}$ commutes with the generators of ${\mathfrak{su}}(3)_C \oplus {\mathfrak{su}}(2)_L$. Furthermore, it is clear that it annihilates the $\alpha^{5}$ root given in \eqref{12} and, hence, also commutes with ${\mathfrak{su}}(2)_R$. It is straightforward to check that these are the only subalgebras of ${\mathfrak{so}}(10)$ that it commutes with. For example, the root associated with the remaining
$\alpha^{3}$ node of the Dynkin diagram in Figure 1 is given by
\begin{equation}
\alpha^{3}=(0,0,1,-1,0) \ .
\label{61}
\end{equation}
This is clearly not annihilated by $Y_{B-L}$. That is, the commutant of $Y_{B-L}$ in ${\mathfrak{so}}(10)$ is ${\mathfrak{su}}(3)_C \oplus {\mathfrak{su}}(2)_L \oplus {\mathfrak{su}}(2)_R$.
It follows that when the $Y_{B-L}$ Wilson line is turned on, it spontaneously breaks
\begin{equation}
Spin(10) \longrightarrow SU(3)_{C} \times SU(2)_{L} \times SU(2)_{R} \times U(1)_{B-L} \ ,
\label{62}
\end{equation}
the so-called left-right model~\cite{Mohapatra:1974gc, Senjanovic:1975rk, Senjanovic:1978ev}.
Let us now determine the zero-mode spectrum associated with this breaking.
First consider the $R={\bf 16}$ representations of $Spin(10)$. The associated sheaf cohomology is given in \eqref{25} where, now, prior to turning on $\chi_{T_{3R}}$
\begin{equation}
RG=1 \oplus 1 \oplus \chi_{B-L} \oplus 1 \oplus \chi_{B-L}^{2} \oplus \chi_{B-L} \oplus \chi_{B-L} \oplus \chi_{B-L}^{2} \oplus \chi_{B-L}^{2} \ .
\label{63}
\end{equation}
Similarly, it follows from \eqref{15} that, prior to turning on $\chi_{T_{3R}}$, the action of $\chi_{B-L}$ on each ${\bf 16}$ representation is given by
\begin{eqnarray}
{\bf 16}&=& \chi_{B-L}^{2} ({\bf{\bar{3}},\bf {1}},{\bf 2},-1) \oplus \chi_{B-L} ({\bf{3}},{\bf {2}},{\bf 1},1)\label{64} \\
&& \oplus 1({\bf{1}},{\bf {2}},{\bf 1},-3) \oplus 1({\bf{1}},{\bf {1}},{\bf 2},3) \ . \nonumber
\end{eqnarray}
We find from \eqref{63} and \eqref{64} that $(H^{1}(X,V) \otimes {\bf 16})^{{\mathbb{Z}}_{3}^{B-L}}$ consists of {\it nine families} of matter multiplets, each family transforming as
\begin{equation}
Q=({\bf 3},{\bf 2},{\bf 1},\frac{1}{3}), \quad Q_{R}=
\begin{pmatrix}
d \\
u
\end{pmatrix}
=({\bar{\bf 3}},{\bf 1}, {\bf 2}, -\frac{1}{3})
\label{65}
\end{equation}
and
\begin{equation}
L=({\bf 1}, {\bf 2},{\bf 1}, -1), \quad L_{R}=
\begin{pmatrix}
e \\
\nu
\end{pmatrix}
=({\bf 1},{\bf 1}, {\bf 2}, 1)
\label{66}
\end{equation}
under $SU(3)_{C} \times SU(2)_{L} \times SU(2)_{R} \times U(1)_{B-L} $.
For $R={\bf 10}$ the associated sheaf cohomology is given in \eqref{31} where, now, prior to turning on $\chi_{T_{3R}}$
\begin{equation}
H^{1}(X,\wedge^{2}V)=1 \oplus 1 \oplus \chi_{B-L}^{2} \oplus \chi_{B-L} \ .
\label{67}
\end{equation}
Similarly, it follows from \eqref{21} that, prior to turning on $\chi_{T_{3R}}$, the action of
$\chi_{B-L}$ on each {\bf 10} representation is given by
\begin{equation}
{\bf 10}= \chi_{B-L}^{2} ({\bf 3}, {\bf 1}, {\bf 1}, 2) \oplus \chi_{B-L} ({\bar{\bf 3}}, {\bf 1}, {\bf 1}, -2) \oplus 1({\bf 1}, {\bf 2}, {\bf 2}, 0) \ .
\label{68}
\end{equation}
Tensoring \eqref{67} and \eqref{68} together and taking the invariant subspace, we find that $(H^{1}(X,\wedge^{2}V) \otimes {\bf 10})^{{\mathbb{Z}}_{3}^{B-L}}$ consists of {\it two pairs} of Higgs-Higgs conjugate multiplets, each transforming as
\begin{equation}
\mathcal{H} =
\begin{pmatrix}
H \\
{\bar{H}}
\end{pmatrix}
=({\bf 1}, {\bf 2}, {\bf 2}, 0)
\label{69}
\end{equation}
and a {\it single pair} of colored Higgs-Higgs conjugate multiplets transforming as
\begin{equation}
H_{C}=({\bf 3}, {\bf 1}, {\bf 1}, \frac{2}{3}), \quad {\bar{H}}_{C}=({\bar{\bf 3}}, {\bf 1}, {\bf 1}, -\frac{2}{3})
\label{70}
\end{equation}
under $SU(3)_{C} \times SU(2)_{L} \times SU(2)_{R} \times U(1)_{B-L} $.
We conclude that in the ``intermediate'' energy region between the unification scale, $M_{u}=M_{\chi_{B-L}}$ , where $\chi_{B-L}$ is turned on and the intermediate scale, which we denote by $M_I = M_{\chi_{T_{3R}}}$, when $\chi_{T_{3R}}$ becomes significant, the effective theory will consist of the zero-modes given in \eqref{65},\eqref{66} and \eqref{69},\eqref{70}. The parameters of their Lagrangian are subject to several constraints. First, note that each $1$ in RG given in \eqref{63} forms a ${\mathbb{Z}}_{3}^{B-L}$ invariant with a pair $L \oplus L_{R}$ in the associated ${\bf 16}$. It follows that the components of each pair, there are nine pairs in total, will have correlated Yukawa and soft supersymmetry breaking parameters in the intermediate region due to the $Spin(10)$ unification symmetry. Second, even though each $Q_{R}$,$L_{R}$ and $\mathcal{H}$ arises from a unique ${\bf 16}$ and ${\bf 10}$ of $Spin(10)$ respectively, their components superfields will have identical Yukawa and soft supersymmetry breaking parameters due to the $SU(2)_{R}$ symmetry.
What happens when $\chi_{T_{3R}}$ is eventually switched on? By construction, this Wilson line commutes with $SU(3)_{C} \times SU(2)_{L} \times U(1)_{B-L} $, but will spontaneously break
\begin{equation}
SU(2)_{R} \longrightarrow U(1)_{T_{3R}} \ .
\label{71}
\end{equation}
Decomposing the $SU(2)_{R}$ component of the ${\bf 16}$ representations with respect to $U(1)_{T_{3R}}$ gives
\begin{eqnarray}
&& Q=({\bf 3},{\bf 2},{\bf 1},1) \rightarrow 1({\bf 3},{\bf 2}, 0,1) \nonumber \\
&& Q_{R}=({\bar{\bf 3}},{\bf 1}, {\bf 2}, -1) \rightarrow \chi_{T_{3R}}^{2}({\bar{\bf 3}},{\bf 1}, -1, -1) \oplus \chi_{T_{3R}}({\bar{\bf 3}},{\bf 1}, 1, -1) \nonumber \\
&& L=({\bf 1}, {\bf 2},{\bf 1}, -3) \rightarrow 1({\bf 1}, {\bf 2},0, -3) \label{72} \\
&& L_{R}=({\bf 1},{\bf 1}, {\bf 2}, 3) \rightarrow \chi_{T_{3R}}^{2}({\bf 1},{\bf 1}, -1, 3) \oplus \chi_{T_{3R}}({\bf 1},{\bf 1}, 1, 3) \nonumber
\end{eqnarray}
where we have explicitly displayed the action of $\chi_{T_{3R}}$. Inserting these expressions into \eqref{64} exactly reproduces the decomposition of the {\bf 16} given in \eqref{28}.
Similarly, the same decomposition of the $SU(2)_{R}$ component of the ${\bf 10}$ representations yields
\begin{eqnarray}
&& \mathcal{H}=({\bf 1}, {\bf 2}, {\bf 2}, 0) \rightarrow \chi_{T_{3R}}({\bf 1}, {\bf 2}, 1, 0) \oplus \chi_{T_{3R}}^{2}({\bf 1}, {\bf 2}, -1, 0) \nonumber \\
&& H_{C}=({\bf 3}, {\bf 1}, {\bf 1}, 2) \rightarrow 1({\bf 3}, {\bf 1}, 0, 2) \label{73} \\
&& {\bar{H}}_{C}=({\bar{\bf 3}}, {\bf 1}, {\bf 1}, -2) \rightarrow 1({\bar{\bf 3}}, {\bf 1}, 0, -2) \nonumber
\end{eqnarray}
Inserting these expressions into \eqref{68} reproduces the decomposition of the {\bf 10} given in \eqref{33}. It follows that turning on the $\chi_{T_{3R}}$ Wilson line spontaneously breaks $SU(3)_{C} \times SU(2)_{L} \times SU(2)_{R} \times U(1)_{B-L} \rightarrow SU(3)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}} \times U(1)_{B-L}$ with exactly the spectrum of the MSSM, as it must. Furthermore, the requirement of ${\mathbb{Z}}_{3}^{R}$ invariance chooses at most one of $L$ or $L_{R}$ from each $L\oplus L_{R}$ pair. Thus, their correlation is broken since at most one of each pair will descend to lower energy. Additionally,
each MSSM field will arise as a ${\mathbb{Z}}_{3}^{R}$ invariant from a different intermediate region multiplet. Hence, the correlation of the component fields of the $SU(2)_{R}$ doublets $Q_{R}$,$L_{R}$ and $\mathcal{H}$ will also be broken, since each doublet contributes at most one of its components to the MSSM.
Finally, we note that since $T^{1}=\frac{1}{2}Y_{T_{3R}}$ is embedded in the non-Abelian subalgebra ${\mathfrak{su}}(2)_{R}$, the associated gauge field strength cannot mix with the Abelian $T^{2}=\frac{1}{2\sqrt{6}}Y_{B-L}$ field strength in the intermediate regime. It follows that at the scale $M_{T_{3R}}$ the mixing parameter $\alpha$ and, hence, ${\cal{G}}_{M}$ must vanish.
Let us now consider the converse situation where $\chi_{T_{3R}}$ is turned on at the unification mass followed sequentially by $\chi_{B-L}$ at a lower scale.
\subsubsection*{$M_{\chi_{T_{3R}}} > M_{\chi_{B-L}}:$}
Recall from \eqref{13} that $Y_{T_{3R}}=H_{4}+H_{5}$. By construction, $Y_{T_{3R}}$ commutes with the generators of ${\mathfrak{su}}(3)_C \oplus {\mathfrak{su}}(2)_L$. However, it does not annihilate the remaining node $\alpha^{3}$ of the Dynkin diagram in Figure 1. Hence, one might conclude that ${\mathfrak{su}}(3)_C \oplus {\mathfrak{su}}(2)_L$ is its largest non-Abelian commutant. However, examining all the remaining roots of ${\mathfrak{so}}(10)$ we find that $Y_{T_{3R}}$ annihilates the non-simple root $\alpha^{0}$ given by
\begin{equation}
\alpha^{0}=(-1,-1,0,0,0) \ ,
\label{74}
\end{equation}
as well as the 2 Weyl reflections of $\alpha^{0}$ around $\alpha^{1}, \alpha^{2}$ and their minuses. These 6 roots, along with the Abelian generator ${\cal{H}}_{0}=\frac{1}{2}(H_{1}+H_{2})$, extend ${\mathfrak{su}}(3)_C $ to the fifteen dimensional subalgebra ${\mathfrak{su}}(4)_{C}$. This is shown in the ``extended'' Dynkin diagram presented in Figure 2.
\begin{figure}[h]
\begin{center}
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\put(1.625,1){\line(1,0){.75}}
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\put(.4,.6){$\alpha_1$}
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\end{center}
\label{fig:2}
\caption{\small The extended Dynkin diagram of $Spin(10)$ with the ${\mathfrak{su}}(4)_{C}\supset \mathfrak{su}(3)_{C}$ and $\mathfrak{su}(2)_{L}$ subgroups highlighted in green and blue respectively. }
\end{figure}\\
Therefore, the commutant of $Y_{T_{3R}}$ is actually the enlarged subalgebra ${\mathfrak{su}}(4) _{C}\oplus {\mathfrak{su}}(2)_L$.
It follows that when the $Y_{T_{3R}}$ Wilson line is turned on, it spontaneously breaks
\begin{equation}
Spin(10) \longrightarrow SU(4)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}} \ ,
\label{75}
\end{equation}
a gauge group closely related to the so-called Pati-Salam model\footnote{Pati-Salam contains the full $SU(2)_R$ group instead of $U(1)_{T_{3R}}$.}~\cite{Pati:1974yy}.
We note that $Y_{B-L}$ does not annihilate $\alpha^{0}$ and, hence, $U(1)_{B-L} \subset SU(4)_{C}$.
Let us now determine the zero-mode spectrum associated with this breaking.
First consider the $R={\bf 16}$ representations of $Spin(10)$. The associated sheaf cohomology is given in \eqref{26} where, now, prior to turning on $\chi_{B-L}$
\begin{equation}
RG=1 \oplus \chi_{T_{3R}} \oplus 1 \oplus \chi_{T_{3R}}^{2} \oplus 1 \oplus \chi_{T_{3R}} \oplus \chi_{T_{3R}}^{2} \oplus \chi_{T_{3R}} \oplus \chi_{T_{3R}}^{2} \ .
\label{76}
\end{equation}
Similarly, it follows from \eqref{16} that, prior to turning on $\chi_{B-L}$, the action of $\chi_{T_{3R}}$ on each ${\bf 16}$ representation is given by
\begin{equation}
{\bf 16}=1({\bf 4},{\bf 2},0) \oplus \chi_{T_{3R}}^{2}({\bar{\bf{4}}},{\bf 1}, -1) \oplus \chi_{T_{3R}}({\bar{\bf{4}}},{\bf 1}, 1) \ .
\label{77}
\end{equation}
We find from \eqref{76} and \eqref{77} that $(H^{1}(X,V) \otimes {\bf 16})^{{\mathbb{Z}}_{3}^{R}}$ consists of {\it nine families} of matter multiplets, each family transforming as
\begin{equation}
\begin{pmatrix}
Q \\
L
\end{pmatrix}
=({\bf 4},{\bf 2},0), \quad
\begin{pmatrix}
u \\
\nu
\end{pmatrix}
=({\bar{\bf{4}}},{\bf 1}, -\frac{1}{2}), \quad
\begin{pmatrix}
d \\
e
\end{pmatrix}
=({\bar{\bf{4}}},{\bf 1}, \frac{1}{2})
\label{78}
\end{equation}
under $SU(4)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}}$.
For $R={\bf 10}$ the associated sheaf cohomology is given in \eqref{31} where, now, prior to turning on $\chi_{B-L}$
\begin{equation}
H^{1}(X,\wedge^{2}V)= \chi_{T_{3R}} \oplus \chi_{T_{3R}}^{2} \oplus \chi_{T_{3R}} \oplus \chi_{T_{3R}}^{2} \ .
\label{79}
\end{equation}
Similarly, it follows from \eqref{22} that, prior to turning on $\chi_{B-L}$, the action of
$\chi_{T_{3R}}$ on each {\bf 10} representation is given by
\begin{equation}
{\bf 10}=1({\bf 6},{\bf 1},0) \oplus \chi_{T_{3R}}({\bf 1},{\bf 2}, 1) \oplus \chi_{T_{3R}}^{2}({\bf 1},{\bf 2}, -1) \ .
\label{80}
\end{equation}
Tensoring \eqref{79} and \eqref{80} together and taking the invariant subspace, we find that $(H^{1}(X,\wedge^{2}V) \otimes {\bf 10})^{{\mathbb{Z}}_{3}^{R}}$ consists of {\it two pairs} of electroweak Higgs-Higgs conjugate multiplets, each transforming as
\begin{equation}
H=({\bf 1},{\bf 2}, \frac{1}{2}), \quad {\bar{H}}=({\bf 1},{\bf 2}, -\frac{1}{2})
\label{81}
\end{equation}
under $SU(4)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}}$.
We conclude that in the ``intermediate'' energy region between the unification scale, $M_{u}=M_{\chi_{T_{3R}}}$, where $\chi_{T_{3R}}$ is turned on and the intermediate scale, $M_I = M_{\chi_{B-L}}$, when $\chi_{B-L}$ becomes significant, the effective theory will consist of the zero-modes given in \eqref{78} and \eqref{81}. Note that even though each matter multiplet in the intermediate region arises from a different ${\bf 16}$ of $Spin(10)$, their component fields have correlated Yukawa and soft supersymmetry breaking parameters due to the $SU(4)_{C}$ symmetry.
What happens when $\chi_{B-L}$ is eventually switched on? By construction, this Wilson line commutes with $ SU(2)_{L} \times U(1)_{T_{3R}} $, but will spontaneously break
\begin{equation}
SU(4)_{C} \longrightarrow SU(3)_{C} \times U(1)_{B-L} \ .
\label{82}
\end{equation}
Decomposing the $SU(4)_{C}$ component of the ${\bf 16}$ representation with respect to $SU(3)_{C} \times U(1)_{B-L}$ gives
\begin{eqnarray}
&&\begin{pmatrix}
Q \\
L
\end{pmatrix}
=({\bf 4},{\bf 2},0) \rightarrow \chi_{B-L}({\bf 3},{\bf 2},0,1) \oplus 1({\bf 1},{\bf 2},0,-3) \nonumber \\
&&\begin{pmatrix}
u \\
\nu
\end{pmatrix}
=({\bar{\bf{4}}},{\bf 1}, -1) \rightarrow \chi_{B-L}^{2}({\bar{\bf 3}},{\bf 1},-1,-1) \oplus 1({\bf 1},{\bf 1},-1,3) \label{83} \\
&&\begin{pmatrix}
d \\
e
\end{pmatrix}
=({\bar{\bf{4}}},{\bf 1}, 1) \rightarrow \chi_{B-L}^{2}({\bar{\bf 3}},{\bf 1},1,-1) \oplus 1({\bf 1},{\bf 1},1,3)\nonumber
\end{eqnarray}
where we have explicitly displayed the $\chi_{B-L}$ representation. Inserting these expressions into \eqref{77} exactly reproduces the decomposition of the {\bf 16} given in \eqref{28}.
Similarly, the same decomposition of the $SU(4)_{C}$ component of the ${\bf 10}$ representations yields
\begin{eqnarray}
&&({\bf 6},{\bf 1},0) \rightarrow \chi_{B-L}^{2}({\bf 3}, {\bf 1},0,2) \oplus \chi_{B-L}({\bar{\bf 3}}, {\bf 1},0,-2) \nonumber \\
&&H=({\bf 1},{\bf 2},1) \rightarrow 1({\bf 1},{\bf 2},1,0) \label{84} \\
&&{\bar{H}}=({\bf 1},{\bf 2},-1) \rightarrow 1({\bf 1},{\bf 2},-1,0) \nonumber
\end{eqnarray}
Inserting these expressions into \eqref{80} reproduces the decomposition of the {\bf 10} given in \eqref{33}. It follows that turning on the $\chi_{B-L}$ Wilson line spontaneously breaks $SU(4)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}} \rightarrow SU(3)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}} \times U(1)_{B-L}$ with exactly the spectrum of the MSSM, as it must. Furthermore, each MSSM field will arise as a ${\mathbb{Z}}_{3}^{B-L}$ invariant from a different intermediate region multiplet. Hence, the correlation between the components of the $SU(4)_{C}$ ${\bf 4}$ matter multiplets is lost. Finally, we note that since $T^{2}=\frac{1}{2\sqrt{6}}Y_{B-L}$ is embedded in the non-Abelian subalgebra ${\mathfrak{su}}(4)_{C}$, the associated field strength cannot mix with the Abelian $T^{1}=\frac{1}{2}Y_{T_{3R}}$ field strength in the intermediate region. It follows that at the scale $M_{\chi_{B-L}}$ the mixing parameter $\alpha$ and, hence, ${\cal{G}}_{M}$ must vanish.\\
Both sequential breaking patterns, specifying the gauge groups and the associated zero-mode spectra, are shown schematically in Figure 3.
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\begin{center}
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\put(7.2,1.9){\color{red}$M_{\chi_{T_{3R}}}=M_I$}
\put(-1.75,2.5){\vector(2,-1){3}}
\put(5.75,2){\vector(-3,-1){3}}
\put(3.5,1.7){$\chi_{T_{3R}}$}
\put(.1,2){$\chi_{T_{B-L}}$}
\put(-1,.5){\color{blue}$\underline{SU(3)_C\times SU(2)_L\times U(1)_{T_{3R}}\times U(1)_{B-L}}$}
\put(-.5,-.5){$L=(\textbf{1},\textbf{2},0,-1)$}
\put(-.5,-1){$e=(\textbf{1},\textbf{1},\frac{1}{2},1)$}
\put(-.5,-1.5){$\nu=(\textbf{1},\textbf{1},-\frac{1}{2},1)$}
\put(-.5,-2){$Q=(\bar{\textbf{3}},\textbf{2},0,\frac{1}{3})$}
\put(-.5,-2.5){$u=(\textbf{3},\textbf{1},-\frac{1}{2},-\frac{1}{3})$}
\put(-.5,-3.0){$d=(\textbf{3},\textbf{1},\frac{1}{2},-\frac{1}{3})$}
\put(-.5,-3.5){$H=(\textbf{1},\textbf{2},\frac{1}{2},0)$}
\put(-.5,-4.0){$\bar H=(\textbf{1},\textbf{2},-\frac{1}{2},0)$}
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\put(-.5,-3.1){\line(-1,0){.2}}
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\put(3.4,-1.5){\oval(.4,.4)[bl]}
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\put(3,-2.9){\oval(.4,.4)[br]}
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\put(-1.4,-3.8){\textbf{10}}
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\put(5.3,-2.5){3 right-handed neutrino}
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\end{picture}
\end{center}
\vspace{5cm}
\caption{\small The two sequential Wilson line breaking patterns of $Spin(10)$. The unification and intermediate masses are specified, as well as the particle spectra in the associated scaling regimes.}
\label{fig:3}
\end{figure}
\section{Gauge Coupling Unification and the Intermediate Mass Scales}
Within the context of the explicit Wilson line breaking scenarios introduced above, we now present the renormalization group analysis of the gauge coupling parameters. These are chosen for discussion since
their RG running does not depend on introducing initial values for the soft breaking parameters. It is clear from the previous section that the RG flow must be integrated over several distinct regimes, each with a different gauge group and multiplet content and, hence, different gauge couplings and beta functions. We begin, therefore, by carefully elucidating all relevant mass scales and the scaling regimes between them.
\subsection{Mass Scales}
From the top down, the important mass scales are the following.
\subsubsection*{The Compactification/Unification Scale $(M_{u})$:} The transition from ten-dimensional string theory to the four-dimensional effective field theory is not ``sharp''. Rather, it occurs over a small interval in energy-momentum, roughly centered around the ``average'' inverse radius of the compactification manifold $X/({\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3})$. This compactifiction scale, $M_{c}$, is clearly dependent on the geometric moduli and, hence, difficult to determine directly from string theory. The $E_{8}$ gauge symmetry of the heterotic string is spontaneously broken to $Spin(10)$ by the $SU(4)$ structure group of the vector bundle $V$. Ignoring any further breaking by Wilson lines, what emerges at the low energy end of the transition region is an effective field theory with unified gauge group $Spin(10)$. We denote this gauge ``unification scale'' by $M_{u}$. If one ignores string ``threshold effects'' due to the transition, then we can identify $M_{c} \simeq M_{u}$. These corrections are expected to be small and, in any case, are not the subject of the present paper. Henceforth, we make this identification.\\
\subsubsection*{The Wilson Line/Intermediate Scale $(M_{I})$:}
If the compactification manifold is relatively ``round'', one expects the inverse radii of the all non-contractible curves in $X/({\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3})$ to be approximately that of the average inverse radius. That is, $M_{\chi_{T_{3R}}} \simeq M_{\chi_{B-L}} \simeq M_{u}$. In this case, both Wilson lines turn on simultaneously and break $Spin(10)$ down to the low energy gauge group $SU(3)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}} \times U(1)_{B-L}$ at the unification scale.
However, if the manifold is in a region of its moduli space for which one of the non-contractable inverse radii is much larger than the other, then the Wilson lines will turn on at different scales. Furthermore, ignoring threshhold effects, it is clear that the larger Wilson line mass should be identified with the unification scale $M_{u}$. In this case, there are two possibilities. First, if $M_{u} \simeq M_{\chi_{B-L}} > M_{\chi_{T_{3R}}}$ then $Spin(10)$ is broken to
$SU(3)_{C} \times SU(2)_{L} \times SU(2)_{R} \times U(1)_{B-L}$ at $M_{c}$. The second case occurs for $M_{u} \simeq M_{\chi_{T_{3R}}} > M_{\chi_{B-L}}$. Now $Spin(10)$ is broken at $M_{u}$ to $SU(4)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}}$.
The smaller of the two Wilson line masses introduces a new mass scale, $M_{I}$, into the analysis. Below this ``intermediate'' scale, independently of whether $M_{I} \simeq M_{\chi_{T_{3R}}}$ or
$M_{I} \simeq M_{\chi_{B-L}}$, the theory is spontaneneously broken to $SU(3)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}} \times U(1)_{B-L}$ with precisely the spectrum of the MSSM with three generations of right-handed neutrino supermultiplets. Of course, in the case where the Wilson line masses are degenerate, the intermediate mass $M_{I}$ is simply identified with the unification scale $M_{u}$.
\subsubsection*{The B-L Breaking Scale $(M_{B-L})$:}
At some scale considerably below $M_{I}$, but above the electroweak scale, the $U(1)_{T_{3R}} \times U(1)_{B-L}$ symmetry must be spontaneously broken to $U(1)_{Y}$. Let us briefly analyze how this breaking takes place, the scale $M_{B-L}$ at which it occurs and the boundary conditions it places on the relevant gauge parameters.
It follows from \eqref{48} and the fact that ${\cal{G}}_{M}=0$ in the canonical basis, that the $U(1)_{T_{3R}} \times U(1)_{B-L}$ part of the covariant derivative is given by
\begin{eqnarray}
\nonumber
D&=&\partial -i(T^{1},T^{2})
\begin{pmatrix}
g_{3R} & 0 \\
0 & g_{BL}
\end{pmatrix}
\begin{pmatrix}
W^{0}_{R} \\
B_{B-L}
\end{pmatrix}\\
&=&\partial -i
\begin{pmatrix}
Y-\frac{1}{2}(B-L), \sqrt{\frac{3}{8}}(B-L)
\end{pmatrix}
\begin{pmatrix}
g_{3R} & 0 \\
0 & g_{BL}
\end{pmatrix}
\begin{pmatrix}
W^{0}_{R} \\
B_{B-L}
\end{pmatrix}.
\label{85}
\end{eqnarray}
The generators $T^{1},T^{2}$ were defined in \eqref{36}, $g_{3R}$ and $g_{BL}$ are the gauge parameters and we denote the gauge bosons associated with $U(1)_{T_{3R}} \times U(1)_{B-L}$ as $W^{0}_{R}$ and $B_{B-L}$ respectively. To simplify notation, it is useful to define
\begin{equation}
g_{BL}^{\prime}=\sqrt{\frac{3}{2}} g_{BL} \ .
\label{86}
\end{equation}
Recall that the matter spectrum of the $SU(3)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}} \times U(1)_{B-L}$ theory contains three right-handed neutrino chiral supermultiplets, one per family, each of which has charge $(-1/2,1)$ under $U(1)_{T_{3R}} \times U(1)_{B-L}$.
The potential of one of the associated sneutrinos, ${\tilde{\nu}}$, can be approximated by
\begin{equation}
V=m_{{\tilde{\nu}}}^{2}|{{\tilde{\nu}}}|^{2}+\frac{1}{8}(g^{\prime 2}_{BL}+g_{3R}^{2})|{\tilde{\nu}}|^{4} \ ,
\label{87}
\end{equation}
where $m_{{\tilde{\nu}}}^{2}$ is the soft supersymmetry breaking squared mass. As shown in detail in \cite{Ambroso:2009jd,Ambroso:2009sc,Ambroso:2010pe}, $m_{{\tilde{\nu}}}^{2}$, although positive at $M_{I}$, can become negative under the RG at a lower mass scale. It follows that potential \eqref{87} develops a vacuum expectation value given by
\begin{equation}
v_{R}=\sqrt{\frac{-8m_{{\tilde{\nu}}}^{2}}{g^{\prime 2}_{BL}+g_{3R}^{2}}} \ ,
\label{88}
\end{equation}
where $\langle {\tilde{\nu}} \rangle=1/\sqrt{2} v_{R}$.
This breaks $U(1)_{T_{3R}} \times U(1)_{B-L}$ to a $U(1)$ subgroup. Although growing over a range, $v_{R}$ ``freezes out'', that is, stops significantly evolving, when the energy momentum becomes smaller than the associated sneutrino mass. This mass, which we denote by
$M_{B-L}$, can be identified with the scale of $U(1)_{T_{3R}} \times U(1)_{B-L}$ breaking.
To analyze this breaking, expand the theory around the sneutrino expectation value.
In the basis $(W_R^0,B_{B-L})$, the matrix for the squared masses of the gauge bosons is
\begin{equation}
M_\mathcal Z =
\begin{pmatrix}
\frac{1}{4} g_{3R} v_R^2
&
-\frac{1}{4} g_{BL}' \, g_{3R} \, v_R^2
\\
-\frac{1}{4} g_{BL}' \, g_{3R} \, v_R^2
&
\frac{1}{4} g_{BL}'^2 v_R^2
\end{pmatrix} .
\label{89}
\end{equation}
This can be diagonalized into the physical states
\begin{align}
&Z_{3R} = \cos \theta_R W_R^0 -\sin \theta_R B_{B-L}, \label{90}
\\
&B = \sin \theta_R W_R^0 +\cos \theta_R B_{B-L} \nonumber
\end{align}
with masses
\begin{eqnarray}
&& M_{Z_{3R}}^2 = \frac{1}{4} \left( g_{3R}^2 + g_{BL}'^2 \right) v_R^2 = 2 | m_{ {\tilde{\nu}}}|^2, \quad M^{2}_{B}=0 \label{91}
\\
&&\qquad \qquad \cos \theta_R = \frac{g_{3R}}{\sqrt{g_{3R}^2 + g_{BL}'^2}} \ . \label{92}
\end{eqnarray}
Note that the non-vanishing $Z_{3R}$ mass is closely associated with the soft supersymmetry breaking mass of the sneutrino. $Z_{3R}$ is generically called a $Z'$ boson and $B$ is the massless gauge boson of the unbroken $U(1)$ gauge symmetry. To determine exactly which symmetry has been broken and which persists, rewrite the covariant derivative \eqref{85} in the physical basis. Using \eqref{36}, \eqref{86},\eqref{90} and \eqref{92}, we find that
\begin{equation}
D=\partial -iYg_{Y} B -i\big(Y\cos^{2}\theta_{R}-\frac{1}{2}(B-L))g_{Z_{3R}}Z_{3R} \ ,
\label{93}
\end{equation}
where
\begin{equation}
g_{Y}= \frac{g^{\prime}_{BL}g_{3R}}{\sqrt{g_{3R}^2 + g_{BL}'^2}}, \quad g_{Z_{3R}}=\sqrt{g_{3R}^{2}+g_{BL}^{'2}} \ .
\label{94}
\end{equation}
Since $B$ is massless, it follows that hypercharge $Y$ is the unbroken gauge charge, that is,
\begin{equation}
U(1)_{T_{3R}} \times U(1)_{B-L} \rightarrow U(1)_{Y}
\label{95}
\end{equation}
and $B$ is the hypercharge boson. The broken $U(1)$ symmetry is then the combination of $Y$ and $B-L$ specified by the rotated charge generator in the last term of \eqref{93}. This couples to the massive $Z_{3R}$ vector boson.
The expression for the hypercharge parameter in \eqref{94} leads to an important boundary condition at the $M_{B-L}$ scale. Defining $\alpha_{i}=g_{i}^{2}/4\pi$, it follows that
\begin{equation}
\alpha_{Y}=\frac{1}{\alpha_{3R}^{-1}+\alpha_{BL}^{\prime~-1}} \ .
\label{96}
\end{equation}
However, this is not the most useful form of this relation. First, one should re-express it in terms of $g_{BL}$ using \eqref{86}. Second, note from \eqref{36} that
\begin{equation}
Y=T^{1}+\sqrt{\frac{2}{3}}T^{2} \ .
\label{97}
\end{equation}
Using the Killing relations in \eqref{37}, one finds $(Y|Y)=5/6$. This is not the canonical normalization of $1/2$ used for the basis elements of ${\mathfrak{so}(10)}$. To be consistent with this normalization, define
\begin{equation}
Y^{\prime}=\sqrt{\frac{3}{5}}Y, \quad g_{1}=\sqrt{\frac{5}{3}}g_{Y} \ .
\label{98}
\end{equation}
Then $(Y^{\prime}|Y^{\prime})=1/2$ and \eqref{96} becomes
\begin{equation}
\alpha_{1}=\frac{5}{3\alpha_{3R}^{-1}+2\alpha_{BL}^{-1}} \ .
\label{99}
\end{equation}
When transitioning through the $M_{B-L}$ mass scale, the $T^{1}$,$T^{2}$ and $Y^{\prime}$ gauge parameters will be related using this boundary condition.
Values of $M_{B-L}$ in \cite{Ambroso:2010pe} are found to between $\sim 5 \times10^2~GeV$ and $10^4 ~GeV$, with the former being more ``typical.''
\subsubsection*{The Supersymmetry Breaking Scale $(M_{SUSY})$:}
Below $M_{B-L}$, the massive $Z_{3R}$ vector superfield--both the gauge boson and its
gaugino--decouple. Furthermore, since the breaking of the extra $U(1)$ gauge factor arises from an expectation value in one of the right-handed sneutrinos, this neutrino supermultiplet--both the right-chiral neutrino and its sneutrino partner--also get a mass of order $M_{B-L}$ and decouple. The resulting theory is exactly the MSSM, that is, gauge group $SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y} $ with three families of quarks/leptons and one Higgs-Higgs conjugate pair, along with the soft breaking interactions. There are also the remaining two right-handed neutrino multiplets but, since they are uncharged under the standard model gauge group, they will not effect the RG running of parameters.
The Higgs contribution to the quadratic potential is given by
\begin{equation}
V_{Higgs}=m^{2}_{H}|H|^{2}+m^{2}_{{\bar{H}}}|{\bar{H}}|^{2}+b(H{\bar{H}}+hc)+\mu^{2}(|H|^{2}
+|{\bar{H}}|^{2}) \ ,
\label{100}
\end{equation}
where $m^{2}_{H}$, $m^{2}_{\bar{H}}$ and $b$ are the supersymmetry breaking Higgs masses, while $\mu$ is the supersymmetric Higgs mass. The beta function for the $H$ soft mass has a significant contribution from the square of the top quark Yukawa coupling. That is,
\begin{equation}
\frac{dm^{2}_{H}}{dt} \simeq \frac{6}{16\pi^{2}}|\lambda_{t}|^{2}\big(m_{{\tilde{Q}}_{3}}^{2}+m_{{\tilde{t}}}^{2}+m^{2}_{H} \big) + \dots \ .
\label{101}
\end{equation}
Potential \eqref{100} can be diagonalized to
\begin{equation}
V_{Higgs}=m^{2}_{H^{\prime}}|H^{\prime}|^{2}+m^{2}_{{\bar{H}}^{\prime}}|{\bar{H}}^{\prime}|^{2}
\label{103}
\end{equation}
where, for sufficiently small $b$,
\begin{equation}
m^{2}_{H^{\prime}} \simeq m^{2}_{H}-\frac{b^{2}}{m^{2}_{\bar{H}} - m^{2}_{H} },
\quad
m^{2}_{\bar{H}^{\prime}} \simeq m^{2}_{\bar{H}} + \frac{b^{2}}{m^{2}_{\bar{H}} - m^{2}_{H} }\ .
\label{104}
\end{equation}
Running down from $M_{B-L}$, the left and right stop masses in \eqref{101} drive $m_{H}^{2}$ toward zero, making $m^{2}_{H^{\prime}}$ negative and signaling the radiative breakdown of electroweak symmetry. Note that the Higgs expectation value continues to grow until one reaches the stop mass threshold. At that point, the stops decouple and the Higgs expectation value is relatively fixed.
A good estimate of the stop decoupling scale is given by\footnote{See~\cite{Gamberini:1989jw} for a more detailed study.}
\begin{equation}
M_{SUSY} \simeq \sqrt{m_{{\tilde{Q}}_{3}}m_{{\tilde{t}}}} \ .
\label{105}
\end{equation}
In addition to this being the scale where radiative Higgs breaking ``freezes out'', it is also a good indicator of the masses of all superpartners. Again due to the large top Yukawa coupling, one finds with approximately universal soft squared masses that all scalar superpartners tend to lie just above \eqref{105}, as shown in \cite{Ambroso:2010pe} for these types of B-L models. Furthermore, the fermionic superpartners lie near or just below the stop threshold. That is, $M_{SUSY}$ is a reasonable estimate of superpartner masses and, therefore, of the scale where supersymmetry is broken. Hence, the subscript $M_{SUSY}$. Finally, we note that the component Higgs and Higgsino fields of the ${\bar{H}}^{\prime}$ supermultiplet also develop masses of order $M_{SUSY}$.
The value of $M_{SUSY}$ is constrained on the low end by non-observation of SUSY particles, and on the high end by fine tuning arguments. A typical value satisfying all constraints is
\begin{eqnarray}
M_{SUSY}=1~TeV \ .
\label{eq:}
\end{eqnarray}
We conclude that at $M_{SUSY}$ all superpartners and the Higgs conjugate supermultiplet decouple. Hence, at lower energy-momentum the theory has the matter and Higgs spectrum of the non-supersymmetric standard model. Although electroweak symmetry has been broken at $M_{SUSY}$, the vacuum expectation value
\begin{equation}
\langle H^{\prime} \rangle
\simeq {\cal{O}}(10^{2}~GeV) << M_{SUSY} \ .
\label{106}
\end{equation}
Therefore, the gauge symmetry below $M_{SUSY}$ remains approximately $SU(3)_{C}\times SU(2)_{L} \times U(1)_{Y}$.
\subsubsection*{The Electroweak Scale $(M_{EW})$:}
At ${\cal{O}}(10^{2}~GeV)$ the Higgs expectation value becomes relevant and spontaneously breaks the gauge group to $SU(3)_{C} \times U(1)_{EM}$. We will identify this scale with the $Z$ boson mass; that is,
\begin{equation}
M_{EW}=M_{Z} \ .
\label{107}
\end{equation}
We will input all experimental data for the gauge parameters, Yukawa couplings and so on at this scale.
\subsection{Scaling Regimes}
From the top down, the scaling regimes are the following.
\subsubsection*{$M_{u} \rightarrow M_{I}$:}
In this regime, the theory has softly broken $N=1$ supersymmetry. The gauge group is either $SU(3)_{C} \times SU(2)_{L} \times SU(2)_{R} \times U(1)_{B-L}$ or $SU(4)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}}$ depending on whether the $\chi_{B-L}$ or $\chi_{T_{3R}}$ Wilson line turns on first. The associated matter and Higgs spectra are the superfields discussed in Subsection \ref{swb} and listed in Figure 3.
\subsubsection*{$M_{I} \rightarrow M_{B-L}$:} The theory remains supersymmetric with soft breaking interactions. Regardless of which breaking pattern occurs above $M_{I}$, in this regime the gauge group is $SU(3)_{C} \times SU(2)_{L} \times U(1)_{T_{3R}} \times U(1)_{B-L}$. The associated spectrum is exactly that of the MSSM with three right-handed neutrino supermultiplets, as discussed in Subsection \ref{swb} and listed in Figure 3.
\subsubsection*{$M_{B-L} \rightarrow M_{SUSY}$:} The theory remains softly broken supersymmetric. However, the gauge group is now reduced to $SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ of the standard model. The spectrum is the same as in the previous regime with the exception that the $Z^{\prime}$ vector supermultiplet as well as one neutrino chiral multiplet have decoupled.
\subsubsection*{$M_{SUSY} \rightarrow M_{Z}$:} In this regime supersymmetry is completely broken with all superpartners and the ${\bar{H}}^{\prime}$ conjugate Higgs integrated out. The gauge group remains $SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ with the spectrum precisely that of the non-supersymmetric standard model.\\
In any of the above regimes, the RGE for each gauge coupling parameter $g_{i}$ is
\begin{equation}
\frac{d\alpha_{i}}{dt}=\frac{b_{i}}{2\pi} \alpha_{i}^{2} \ ,
\label{108}
\end{equation}
where $\alpha_{i}=\frac{g_{i}^{2}}{4\pi}$. In a regime where the gaugino has been integrated out, the coefficients $b_{i}$ are given by~\cite{Jones:1981we}
\begin{equation}
b_{i}=-\frac{11}{3} C_{2}(G_{i})+\frac{1}{3} \sum_{\rm scalars} I_{i}(R)+\frac{2}{3} \sum_{\rm fermions} I_{i}(R) \ ,
\label{109}
\end{equation}
where $C_{2}(G_{i})$ is the second Casimir invariant for the adjoint representation of $G_{i}$ and $ I_{i}(R)$ is the Dynkin index for the representation $R$ of $G_{i}$ defined in \eqref{18}. For a simplified version relevant for supersymmetric theory see~\cite{Martin:1993zk}.
\subsection{Gauge Unification via the $SU(2)_{L} \times SU(2)_{R}$ ``Left-Right'' Model \label{LR}}
When the $\chi_{B-L}$ Wilson line is turned on first, the scaling region between $M_{u}$ and $M_{I}$ is populated by a ``left-right model''; that is, the theory with $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ gauge group and the particle content composed of 9 families of quarks/leptons, two Higgs bi-doublets--each one containing a pair of MSSM-like Higgs doublets--and two colored triplets. Proton decay is highly sensitive to the presence of such colored triplets~\cite{Nath:2006ut}, even at scales close to the GUT scale. However, because these fields do not come from the same {\bf 10} as the MSSM Higgs, their couplings are unknown and could be small enough to avoid the current bounds. Alternatively, this can be viewed as a justification for the Pati-Salam type model or the simultaneous Wilson like scenario, discussed in the next two sections, which do not contain these fields below $M_u$.
One of the advantages of seperating $M_u$ and $M_I$ is the extra freedom granted by the latter, which allows the exact unification of gauge couplings, an observation which has been discussed in situations with similar gauge groups but different particle content~\cite{DeRomeri:2011ie,Malinsky:2005bi}. Specifically, $M_I$ can be chosen independently of $M_{B-L}$ so as to allow exact unification of the gauge couplings. Although not strictly necessary, we will enforce such unification as a way of exactly specifying the low energy theory. Below $M_I$, the left-right model reduces to the gauge group $SU(3)_C \times SU(2)_L \times U(1)_{T_{3R}} \times U(1)_{B-L}$ with the MSSM particle content supplemented by three families of right-handed neutrino chiral multiplets. There are enough boundary conditions at $M_{u}$, $M_{I}$ and $M_{B-L}$ to determine all gauge couplings at $M_{SUSY}$, and, hence, at $M_{Z}$ where they can be compared to the experimental data. The RG analysis requires three things: 1) the boundary conditions for the gauge couplings at $M_{u}$, $M_{I}$, $M_{B-L}$ and $M_{SUSY}$, 2) the beta functions for the gauge couplings between these different scales and 3) the input of the low-energy values of the gauge couplings from experimental data. First consider the boundary conditions. They are given by
\begin{align}
& \alpha_3(M_{u}) = \alpha_2(M_{u}) = \alpha_R(M_{u}) = \alpha_{BL}(M_{u}) \equiv \alpha_u, \label{110}
\\
& ~~\qquad \qquad \alpha_R(M_I) = \alpha_{3R}(M_I) \label{111}
\end{align}
and
\begin{equation}
\alpha_1(M_{B-L}) = \frac{5}{3 \alpha_{3R}^{-1}(M_{B-L}) + 2 \alpha_{BL}^{-1}(M_{B-L})} \ ,
\label{112}
\end{equation}
where $\alpha_3$ and $\alpha_2$ are gauge couplings for $SU(3)_C$ and $SU(2)_L$ respectively. Gauge couplings without a specific boundary condition at a given scale
are simply identical above and below that mass.
Below $M_{u}$, the gauge couplings of the $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ left-right model have the following RG slope factors in the intermediate regime:
\begin{equation}
b_3 = 10, \quad b_2 = 14, \quad b_R = 14, \quad b_{BL} = 6,
\label{113}
\end{equation}
where \eqref{109} has been used.
The RG for the gauge couplings of the $SU(3)_C \times SU(2)_L \times U(1)_{T_{3R}} \times U(1)_{B-L}$ theory between $M_I$ and $M_{B-L}$ has the slope factors:
\begin{equation}
b_3 = -3, \quad b_2 = 1, \quad b_{3R} = 7, \quad b_{BL} = 6.
\label{114}
\end{equation}
Once $U(1)_{T_{3R}} \times U(1)_{B-L}$ breaks to $U(1)_Y$, the RGE slope factors are simply the well-known ones of the MSSM:
\begin{equation}
b_3 = -3, \quad b_2 = 1, \quad b_{1} = \frac{33}{5}.
\label{115}
\end{equation}
Integrating out the superparners and the conjugate Higgs chiral multiplet at $M_{SUSY}$ leaves the familiar standard model result
\begin{equation}
b_3 = -7, \quad b_2 = \frac{19}{16}, \quad b_{1} = \frac{41}{10}.
\label{116}
\end{equation}
Finally, $M_Z$ is the scale where the initial values of the gauge couplings can be inputed from experiment. They are
\begin{equation}
\alpha_1 = 0.017, \quad \quad \alpha_2 = 0.034, \quad \quad \alpha_3 = 0.118.
\label{117}
\end{equation}
Our procedure is to start with the experimental values of $\alpha_1$, $\alpha_2$ and $\alpha_3$ at the $Z$ mass scale and to run them to the SUSY scale using the standard model RGEs. From here, the MSSM RGEs are used up to $M_{B-L}$. Above this scale, gauged hypercharge is replaced by $U(1)_{T_{3R}} \times U(1)_{B-L}$ whose coupling parameters $\alpha_{3R}$ and $\alpha_{BL}$, as yet not determined, are related to $\alpha_1$ by the boundary condition \eqref{99}. However, $\alpha_2$ and $\alpha_3$ remain valid gauge couplings and can be evolved up to the point where they become equal to each other. We will, for specificity, equate the scale of $\alpha_2$ and $\alpha_3$ unification with the unification mass $M_{u}$. Note that this scale is independent of $M_I$ since the additional particle content in the intermediate region fits into complete multiplets of $Spin(10)$. Hence, the value of the $M_{u}$ is completely determined. We denote the unified coupling parameter by $\alpha_{u}$. This quantity, however, is dependent on both the value of $M_{I}$ as well as the boundary conditions at $M_{B-L}$.
The unification of $\alpha_{3}$ and $\alpha_{2}$ supplies a necessary boundary condition for $\alpha_{BL}$ and $\alpha_R$; namely, that they be equal to $\alpha_u$ at $M_{u}$. Now $\alpha_{BL}$ and $\alpha_R$ can be evolved back down to the B-L scale, remembering that $g_{3R}(M_I) = g_R(M_I)$. The values of $\alpha_{BL}$ and $\alpha_R$ at the B-L scale are not independent because of gauge coupling unification and will furthermore depend on $M_I$. However, \eqref{112} can be used to solve for $M_I$.
To procede, we must choose values of $M_{SUSY}$ and $M_{B-L}$. For the former, the typical value already discussed is $1~TeV$. For the latter, \cite{Ambroso:2010pe} shows a range of possible values. In order to distinguish between the SUSY and B-L scales, both for the plots and to avoid any appearance of conflating them, we will first give our results for a rather high $M_{B-L}=10~TeV$. Since such a large $M_{B-L}$ could be invisible to the LHC, and is disfavored by \cite{Ambroso:2010pe}, we will also give results for $M_{B-L}=1~TeV$. An exhaustive analysis of these mass thresholds will be presented in future publications \cite{Preparation}.
Taking the values
\begin{equation}
M_{SUSY} = 1~TeV, \quad M_{B-L} = 10~TeV \ ,
\label{burt1}
\end{equation}
we find that
\begin{eqnarray}
\label{LR.values}
&&\quad \quad M_{u} = 3.0 \times 10^{16}~GeV, \quad M_I = 3.7 \times 10^{15}~GeV \label{burt2} \\
&&\alpha_{u}=0.046, \quad \alpha_{3R}(M_{B-L}) = 0.0179, \quad \alpha_{BL}(M_{B-L})=0.0187 \ . \nonumber
\end{eqnarray}
The associated running coupling parameters are plotted in Figure~\ref{LR.GUT}.
\begin{figure}[h!]
\centering
\includegraphics[scale=.9]{IntermediateUnificationShop4}
\caption{\small One-loop RGE running of the inverse gauge couplings, $\alpha_i^{-1}$ in the case of the left-right model with $M_{B-L}=10~TeV$ with an enlarged image of the intermediate region.}
\label{LR.GUT}
\end{figure}
Similarly, for
\begin{equation}
M_{SUSY} = 1~TeV, \quad M_{B-L} = 1~TeV \ ,
\label{burt1a}
\end{equation}
we find
\begin{eqnarray}
\label{LR.1values}
&&\quad \quad M_{u} = 3.0 \times 10^{16}~GeV, \quad M_I = 3.7 \times 10^{15}~GeV \label{burt2a} \\
&&\alpha_{u}=0.046, \quad \alpha_{3R}(M_{B-L}) = 0.0171, \quad \alpha_{BL}(M_{B-L})=0.0180 \ . \nonumber
\end{eqnarray}
The running coupling parameters are plotted in Figure~\ref{LR.GUT2}. It is worth noting that changing the value of $M_{B-L}$ did not affect any of the running above $M_{B-L}$. That is, $M_u$, $M_I$, and $\alpha_u$ are unchanged.
\begin{figure}[h]
\centering
\includegraphics[scale=.9]{IntermediateUnification1TeVShop4}
\caption{\small One-loop RGE running of the inverse gauge couplings, $\alpha_i^{-1}$ in the case of the left-right model with $M_{B-L}=1~TeV$ with an enlarged image of the intermediate region.}
\label{LR.GUT2}
\end{figure}
\subsection{Gauge Unification via the $SU(4)_{C} \times SU(2)_{L}$ Pati-Salam Type Model}
When the $\chi_{T{3R}}$ Wilson line is turned on first, the scaling region between $M_{u}$ and $M_{I}$ is populated by the Pati-Salam type model; that is, a theory with an $SU(4)_{C} \times SU(2)_L \times U(1)_{T_{3R}}$ gauge group whose particle content again has nine quark/lepton families and two pairs of $H$-${\bar{H}}$ doublets. However, unlike the left-right case, there are no color triplets.
Below $M_I$, the Pati-Salam-like model again reduces to the gauge group $SU(3)_C \times SU(2)_L \times U(1)_{T_{3R}} \times U(1)_{B-L}$ with the MSSM particle content supplemented by three right-handed chiral neutrino supermultiplets. The scenario that follows is much like that in Subsection~\ref{LR}. First, one inputs the boundary conditions at the various transition masses, as well as giving the beta function coefficients for the gauge couplings between these scales. As in the left-right scenario, we will {\it enforce gauge coupling unification} as a way of specifying, and simplifying, the low energy analysis. The unification condition then fixes the value of $M_I$. There remain enough conditions to completely determine all gauge couplings at the low scale.
Specifically, the boundary conditions are
\begin{align}
\alpha_4&(M_u) = \alpha_2(M_u) = \alpha_{3R}(M_u) \equiv \alpha_u, \label{118}
\\
& \alpha_4(M_I) = \alpha_{BL}(M_I) = \alpha_3(M_I) \label{119}
\end{align}
and
\begin{equation}
\alpha_1(M_{B-L}) = \frac{5}{3 \alpha_{3R}^{-1}(M_{B-L}) + 2 \alpha_{BL}^{-1}(M_{B-L})},
\label{120}
\end{equation}
where $\alpha_4$ is the $SU(4)_C$ gauge coupling. Gauge parameters without a specific boundary condition at a given scale
are simply identical above and below that mass.
Using \eqref{109}, the gauge coupling beta function coefficients can be calculated in the scaling regime $M_u \rightarrow M_{I}$. The result is
\begin{equation}
b_4 = 6, \quad b_2 = 14, \quad b_{3R} = 20 \ .
\label{121}
\end{equation}
Since below $M_I$ the theory is identical to the left-right case, all subsequent beta function coefficients are given in \eqref{114}, \eqref{115} and \eqref{116} respectively.
Again, the analysis begins with the experimental values of $\alpha_1$, $\alpha_2$ and $\alpha_3$ at $M_{Z}$ given in \eqref{117}. We then follow the procedure discussed in Subsection~\ref{LR}. One notable difference is that the particle content in the $M_{u} \rightarrow M_{I}$ intermediate region does not contain full multiplets of $Spin(10)$. This shifts the unification scale $M_{u}$ from its value in the previous example, as well as the value of $M_{I}$.
The $\alpha_2$ and $\alpha_3$ gauge couplings continue to be used to fix the unification scale, although, at this scale, $\alpha_3$ is replaced by $\alpha_4$. The coupling $\alpha_{3R}$ can be evolved down from the unification scale and $\alpha_{BL}$ can be scaled from the intermediate mass.
Taking the values
\begin{equation}
M_{SUSY} = 1~TeV, \quad M_{B-L} = 10~TeV \ ,
\label{burt1b}
\end{equation}
we find that
\begin{eqnarray}
&&\quad \quad M_{u} = 1.5 \times 10^{16}~GeV, \quad M_I = 7.4 \times 10^{15}~GeV \label{burt3} \\
&&\alpha_{u}=0.041, \quad \alpha_{3R}(M_{B-L}) = 0.0175, \quad \alpha_{BL}(M_{B-L})=0.0195 \ . \nonumber
\end{eqnarray}
The associated running coupling parameters are plotted in Figure \ref{PS.GUT}.
\begin{figure}[H]
\centering
\includegraphics[scale=.9]{IntermediateUnificationPatiSalamShop4}
\caption{\small One-loop RGE running of the inverse gauge couplings, $\alpha_i^{-1}$ in the case of the Pati-Salam type model with $M_{B-L}=10~TeV$ with an enlarged image of the intermediate region.}
\label{PS.GUT}
\end{figure}
Similarly, for
\begin{equation}
M_{SUSY} = 1~TeV, \quad M_{B-L} = 1~TeV \ ,
\label{burt1c}
\end{equation}
we find that
\begin{eqnarray}
&&\quad \quad M_{u} = 1.5 \times 10^{16}~GeV, \quad M_I = 7.4 \times 10^{15}~GeV \label{burt3c} \\
&&\alpha_{u}=0.041, \quad \alpha_{3R}(M_{SUSY}) = 0.0167, \quad \alpha_{BL}(M_{SUSY})=0.0187 \ . \nonumber
\end{eqnarray}
The running coupling parameters are plotted in Figure \ref{PS.GUT2}. Again, the running above $M_{B-L}$ is not affected by the change in $M_{B-L}$.
\begin{figure}[h]
\centering
\includegraphics[scale=.9]{IntermediateUnificationPatiSalam1TeVShop4}
\caption{\small One-loop RGE running of the inverse gauge couplings, $\alpha_i^{-1}$ in the case of the Pati-Salam type model with $M_{B-L}=1~TeV$ with an enlarged image of the intermediate region.}
\label{PS.GUT2}
\end{figure}
\subsection{Gauge Unification with Simultaneous Wilson Lines}
When both Wilson lines turn on simultaneously, so that $M_I=M_u$, the intermediate region is absent and $Spin(10)$ is immediately broken to $SU(3)_C\times SU(2)_L\times U(1)_{T_{3R}}\times U(1)_{B-L}$ with the MSSM particle content supplemented by three families of right-handed neutrino chiral mulitplets.
Naively, one might try to impose the boundary condition
\begin{align}
& \alpha_3(M_u) = \alpha_2(M_u) = \alpha_{3R}(M_u) = \alpha_{BL}(M_u) \ . \label{122}
\end{align}
However, as we will see below, unlike in the left-right and Pati-Salam cases, this unification condition is inconsistent with the experimental values of $\alpha_{3}$, $\alpha_{2}$ and $\alpha_{1}$ at $M_{Z}$ within the assumptions we have made about the mass thresholds. Hence, we will not input this condition. Rather, we will scale up to $M_{u}$ from the experimental input at $M_{Z}$ and examine to what extent unification is violated. Of course, the boundary condition
\begin{equation}
\alpha_1(M_{B-L}) = \frac{5}{3 \alpha_{3R}^{-1}(M_{B-L}) + 2 \alpha_{BL}^{-1}(M_{B-L})},
\label{123}
\end{equation}
at $M_{B-L}$ continues to hold.
Since the theory is identical to the left-right and Pati-Salam cases below $M_I$, the beta functions in all subsequent scaling regimes are given in \eqref{114}, \eqref{115} and \eqref{116}.
In the previous two sections, the final step of the RG procedure was to solve for $M_I$. In both cases, there was a unique solution for $M_I$ that satisfied the boundary conditions--including gauge coupling unification at $M_{u}$. In the simultaneous Wilson lines case, however, we are fixing $M_I=M_u$ in advance. Hence, if we continue to use the full set of boundary conditions mandated in the previous sections, the system will be overdetermined. Specifically, we find that one cannot simultaneously impose \eqref{122} and \eqref{123} while also matching the low energy experimental input \eqref{117}. To proceed, some boundary condition must be relaxed. Constraint \eqref{122} has the greatest uncertainty due to string threshold effects. Hence, we will no longer impose it. There is no flexibility in the running of $\alpha_3$ and $\alpha_2$, their running and unification being completely determined by the experimental input. However, the low energy value of $\alpha_Y$ along with \eqref{123} can be used to write a relationship between $\alpha_{3R}$ and $\alpha_{BL}$ at $M_{B-L}$, but not fix them. Most choices for these two couplings will lead to neither of them unifying with $\alpha_{3}$, $\alpha_{2}$ at $M_{u}$. However, it is possible to choose one of them so that it indeed unifies at $M_{u}$. In this case, however, the other coupling, calculated from the first using \eqref{123}, will not unify.
And vice, versa.
Let us first demand that $\alpha_{BL}$ unify with $\alpha_3$, $\alpha_2$ at $M_{u}$. Using \eqref{123} to solve for $\alpha_{3R}$ at $M_{B-L}$, we find that $\alpha_{3R}(M_{u})$ will miss unification by $\sim 8\%$. To be precise,
\begin{eqnarray}
\left|\frac{\alpha_{BL}(M_u)-\alpha_{3R}(M_u)}{\alpha_{3R}(M_u)}\right|\approx 8\%.
\label{124}
\end{eqnarray}
As one might expect from the previous sections, this result is unchanged for the range of $M_{B-L}$ from 1 to 10 $TeV$.
Another, potentially instructive, way to think of this procedure is to start with the left-right model of Subsection 2.3 and move $M_I$ continuously up to $M_u$, without changing any of the RG running below $M_I$. Recall that the unification scale of $\alpha_3$ and $\alpha_2$ is independent of $M_I$, since the additional particle content in the intermediate region fits into complete multiplets of $Spin(10)$. Note that $\alpha_{BL}$ is affected in the same way, which means that all three of these couplings will continue to unify at the same scale as we move $M_I$ up toward $M_{u}$. However, $\alpha_{3R}$ will be affected differently because, at the intermediate scale, it changes from a $U(1)_{T_{3R}}$ coupling to an $SU(2)_R$ coupling. Hence, it will not continue to unify with the others as $M_I$ approaches $M_{u}$.
If we demand that $\alpha_{3R}$ unify with $\alpha_3$, $\alpha_2$, and use \eqref{123} to solve for $\alpha_{BL}$ at $M_{B-L}$, We find that $\alpha_{3R}(M_{u})$ will miss unification by $\sim 13\%$. The RG running of the gauge coupling in each of these scenarios is shown in Figure \ref{fig:133}.
\begin{figure}[h]%
\centering
\subfloat[]{\includegraphics[scale=.8]{NoIntermediateUnification4.pdf}}%
\subfloat[]{\includegraphics[scale=.8]{OtherNoIntermediateUnification4.pdf}}%
\caption{\small The gauge couplings do not unify exactly if the two Wilson lines turn on simultaneously. In (a), $\alpha_{BL}$ is chosen to unify exactly. In (b) $\alpha_{3R}$ is chosen to unify exactly. $M_{B-L} = 10~TeV$ in both plots.}%
\label{fig:133}%
\end{figure}
It is interesting to note that exact unification with simultaneous Wilson lines can be achieved by accounting for the fact that all superpartners will not have exactly the same mass, and, therefore, will not all decouple at exactly the same scale, $M_{SUSY}$. It is sufficient to assume that the scale at which the colored superpartners decouple, $M_{SUSY_c}$, is higher than the scale at which the non-colored superpartners decouple, $M_{SUSY_n}$. The beta functions below $M_{SUSY_n}$ and above $M_{SUSY_c}$ are unchanged. In between these two scales, the theory is the MSSM without the colored superpartners. Using \eqref{109}, the beta function coefficients in the regime $M_{SUSY_c} \to M_{SUSY_n}$ are calculated to be
\begin{equation}
b_3 = -7, \quad b_2 = -\frac{1}{2}, \quad b_1 = \frac{11}{2}.
\end{equation}
Choosing $M_{SUSY_n}$ and $M_{B-L}$ and demanding unification of gauge couplings specifies the value of $M_{SUSY_c}$. A specific example is shown in Figure~\ref{fig:sim.uni}, where
\begin{equation}
\label{sim.mass.scales}
M_{SUSY_n} = 500 ~GeV, \quad M_{B-L} = 10~TeV
\end{equation}
is chosen. This yields
\begin{eqnarray}
\label{SWL.values}
&&\quad \quad M_{u} = 8.3 \times 10^{15}~GeV, \quad M_{SUSY_c} = 3.7~TeV \\
&&\alpha_{u}=0.038, \quad \alpha_{3R}(M_{B-L}) = 0.0176, \quad \alpha_{BL}(M_{B-L})=0.0191 \ . \nonumber
\end{eqnarray}
The wino and gluino play a critical role in allowing unification here. Note that the ratio of the non-colored to the colored SUSY scale (and therefore the masses of the wino and gluino) is approximately
\begin{eqnarray}
M_2:M_3\sim M_{SUSY_n}:M_{SUSY_c}\sim1:7
\label{eq:wpe}
\end{eqnarray}
in this case. In the next section we will examine whether or not this can happen in our model.
The output parameters of interests for the three unification models (left-right, Pati-Salam, and simultaneous Wilson lines) are displayed in Table~\ref{tab:param}. The low energy observables, such as $\sin^2 \theta_R$, are too similar between the different cases to be experimentally distinguishable at the LHC.
\begin{figure}[t!]%
\centering
{\includegraphics[scale=.8]{SplitUnification4.pdf}}%
\caption{\small Exact unification for simultaneous Wilson lines, requiring a splitting between the colored and non-colored superpartners.}%
\label{fig:sim.uni}%
\end{figure}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Value & $\ $ LR $\ $ & $\ $ PS $\ $ & SWL
\\
\hline
$M_u$ & $3.0 \times 10^{16}$ & $1.5 \times 10^{16}$ & $8.3\times10^{15}$
\\
$M_I$ & $3.7 \times 10^{15}$ & $7.4 \times 10^{15}$ & $8.3\times10^{15}$
\\
$\alpha_u$ & 0.046 & 0.043 & 0.038
\\
$ \ \sin^2 \theta_R \ $ & 0.61 & 0.63 & 0.61
\\
$g_{Z_{3R}}$ &0.76 & 0.77 & 0.76
\\
\hline
\end{tabular}
\end{center}
\caption{Values of interest specified by gauge coupling unification for the left-right model in the intermediate regime (LR), the Pati-Salam type model in the intermediate regime (PS) and for the simultaneous Wilson lines (SWL). The results do not depend significantly on $M_{B-L}$, but these are evaluated for $M_{B-L} = 10~TeV$.}
\label{tab:param}
\end{table}%
\subsection{Running Gaugino Masses}
One of the predictive consequences of assuming the unification of the gauge couplings at $M_u$ is the unification of the gauginos into a single gaugino field of $Spin(10)$. These individual components therefore have the same mass, $ {M}_{1/2}$, at $M_u$. The RGE for the gaugino mass is \cite{Martin:1993zk}
\begin{equation}
\frac{d}{dt} {M}_{a} = \frac{b_a \alpha_a {M}_a}{2 \pi},
\end{equation}
where the $b_a$ are the same coefficients as for the gauge couplings in the different regimes. The boundary conditions at the scales of interest follow in a similar way from those of the gauge couplings. At $M_u$ in the left-right model,
\begin{eqnarray}
{M}_3(M_u) = {M}_2(M_u) = {M}_R(M_u) = {M}_{BL}(M_u) \equiv {M}_{1/2},
\label{eq:}
\end{eqnarray}
and in the Pati-Salam type model,
\begin{eqnarray}
{M}_4(M_u) = {M}_2(M_u) = {M}_{3R}(M_u) \equiv {M}_{1/2}.
\label{eq:}
\end{eqnarray}
At $M_I$ in the left-right model,
\begin{eqnarray}
\quad & {M}_{3R}(M_I) = {M}_R(M_I),
\label{eq:}
\end{eqnarray}
and in the Pati-Salam type model,
\begin{eqnarray}
\quad & {M}_4(M_I) = {M}_{BL}(M_I) = {M}_{3}(M_I).
\label{eq:}
\end{eqnarray}
Solving the RGE using unification and the boundary conditions at $M_I$ yields that $M_a(\mu)$ is proportional to $\alpha_a(\mu)$.
That is,
\begin{equation}
{M}_a(\mu) = \frac{{M}_{1/2}}{\alpha_u} \alpha_a(\mu) \ .
\label{eq:M.gen}
\end{equation}
Note that we have not discussed the boundary condition at $M_{B-L}$, so this solution is only valid above that scale.
Therefore,
the ratio of the gaugino mass parameters at the $B-L$ scale depends only on the ratio of the gauge couplings. We now turn to the discussion of the hypercharge gaugino mass once $B-L$ is broken.
The relevant mass matrix
in the basis $(\nu, \tilde B_{B-L}, \tilde W_R^0)$, neglecting electroweak effects, is
\begin{equation}
{M}_{\tilde \chi} =
\begin{pmatrix}
0
&
\cos \theta_R M_{Z_{3R}}
&
\sin \theta_R M_{Z_{3R}}
\\
\cos \theta_R M_{Z_{3R}}
&
{M}_{3R}
&
0
\\
\sin \theta_R M_{Z_{3R}}
&
0
&
{M}_{BL},
\end{pmatrix}
\label{gaugino.mass}
\end{equation}
where we remind the reader that $M_{Z_{3R}}$ is the mass of the $Z'$ gauge boson.
It will be useful, motivated by \eqref{99}, to define,
\begin{eqnarray}
\label{eq:HatDef}
\hat M_1 = \left.\frac{M_{1/2}}{\alpha_u}\alpha_1\right|_{M_{B-L}}= \left.\frac{{M}_{1/2}}{\alpha_u} \frac{5}{2\alpha_{BL}^{-1} + 3 \alpha_{3R}^{-1}} \right|_{M_{B-L}}\ .
\end{eqnarray}
Let us now diagonalize \eqref{gaugino.mass}. Using \eqref{eq:M.gen} and \eqref{eq:HatDef}, the gaugino masses ${M}_{3R}$ and ${M}_{BL}$ can be parameterized in terms of $\hat{M}_1$:
\begin{equation}
{M}_{3R} = \hat {M}_1(1 + \epsilon_{3R}),
\quad
{M}_{BL} = \hat {M}_1(1 + \epsilon_{BL}),
\end{equation}
where
\begin{equation}
\label{epsilon}
\epsilon_i = \frac{\alpha_i}{\alpha_1} - 1 \ .
\end{equation}
Note that in any of the unification scenarios, the $\epsilon$ values at the $B-L$ scale will be small since $\alpha_1$ is close to $\alpha_{3R}$ and $\alpha_{BL}$. (For example, in the left-right case, with $M_{B-L}=1~TeV$, $\alpha_1(M_{B-L}) = 0.017$. Using values for $\alpha_{3R}(M_{B-L})$ and $\alpha_{BL}(M_{B-L})$ from \eqref{LR.1values}, we find $\epsilon_{BL}\approx.06$.) Therefore, we can diagonalize \eqref{gaugino.mass} as a perturbative expansion in
$\epsilon$.
The hypercharge gaugino mass is found to be
\begin{equation}
{M}_1 = \hat{M}_1(1 + \cos^2 \theta_R \epsilon_{BL} + \sin^2 \theta_R \epsilon_{3R}+\mathcal O(\epsilon^2)) \ .
\end{equation}
Using \eqref{epsilon} along with \eqref{92}, \eqref{94}, and \eqref{98}, the two terms proportional to $\epsilon$ cancel, leaving,
\begin{equation}
M_1=\hat M_1(1+\mathcal O(\epsilon^2))\simeq \hat M_1
\end{equation}
Therefore, the hypercharge gaugino mass is, to first order in $\epsilon$, equal to ${\hat{M}}_{1}$ at $M_{B-L}$. The consequences are that, when calculated at $M_{SUSY}$, the ratios between the MSSM gauginos are
\begin{equation}
{M}_1:{M}_2:{M}_3 = \alpha_1:\alpha_2:\alpha_3 \sim 1:2:5 \ .
\label{gaugino.ratio}
\end{equation}
This is a prediction that should be visible to the LHC. However, the same prediction would come from other supersymmetric models in which the standard model gauge group unifies into a single gauge group with a single gaugino, so it is not unique to our theory. An important aspect of this conclusion for our model is that it is not consistent with the scenario of exact unification with simultaneous Wilson lines presented in the previous section. According to \eqref{eq:wpe} the ratio ${M}_2:{M}_3$ would need to be approximately $1:7$ for unification in the case of simultaneous Wilson lines, whereas \eqref{gaugino.ratio} predicts it to be $2:5$. We find that if $\alpha_{BL}$ is chosen to unify, $\alpha_{3R}$ will miss uniification by $\sim4\%$. This leads to the following important point:\\
\noindent ${\bullet}$ {\it The predictions for the gaugino mass ratios from gauge unification are inconsistent with the ratio required for exact unification with simultaneous Wilson lines. Therefore unification cannot be achieved naturally with simultaneous Wilson lines in our model.}
\section{Conclusion}
MSSM extensions by an Abelian gauge group with the MSSM particle content plus three right-handed neutrinos can be derived from $E_8 \times E_8$ heterotic string theory and have also been proposed by the model building community as attractive minimal TeV scale options. In the former case, such models arise from ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ Wilson lines breaking $Spin(10)$. Here, a detailed search for the most general $U(1) \times U(1)$ subgroup of $Spin(10)$ consistent with this framework was conducted and a canonical model was found: $U(1)_{B-L} \times U(1)_{T_{3R}}$, that is, baryon minus lepton number and the third component of right-handed isospin. It has the following four appealing and important features:
\begin{enumerate}
\item
As mentioned above, the particle content is simply that of the MSSM supplemented by three right-handed neutrinos while the gauge group has one rank more than the SM gauge group.
\item
Each quark/lepton and Higgs superfield of the low energy Lagrangian descends from a different ${\bf{16}}$ and ${\bf{10}}$ representation of $Spin(10)$ respectively, indicating that Yukawa couplings and soft SUSY breaking masses are uncorrelated at low energy by $Spin(10)$ relations .
\item
At the scale of $Spin(10)$ breaking, there is no kinetic mixing between the two $U(1)$ symmetries since the generators of the canonical basis are Killing orthogonal.
\item
Furthermore, no kinetic mixing is generated through renormalization group effects because the trace of the product of the two $U(1)$ charges over the entire low energy particle content is zero. The physics of kinetic mixing was discussed in detail.
\end{enumerate}
Lastly, there are several predictions associated with the breaking of B-L, among them the spontaneous breaking of R-parity. These features hold at the low scale regardless of whether the Wilson lines turn on simultaneously or at different scales. The latter situation is interesting, however, since the extra freedom associated with the size of the second Wilson line, the intermediate scale, allows for the unification of the gauge couplings. The gauge symmetry in the intermediate regime depends on the order of the Wilson line turn on-- with $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ symmetry (left-right model) in the case of the $B-L$ Wilson line turning on first and $SU(4)_C \times SU(2)_L \times U(1)_{T_{3R}}$ symmetry (Pati-Salam type) for the $T_{3R}$ line turning on first. The features of both spectra are summarized in Figure~3.
The threshold scales and beta functions necessary for a one-loop analysis of the gauge coupling running were carefully outlined for the three possible scenarios: $B-L$ Wilson line first, $T_{3R}$ Wilson line first and simultaneous Wilson lines, with the results displayed in Figures~\ref{LR.GUT},\ref{LR.GUT2}, in Figures~\ref{PS.GUT},\ref{PS.GUT2} and Figures~\ref{fig:133},\ref{fig:sim.uni} respectively. Unification in the simultaneous Wilson line case is possible, but the necessary gaugino masses are not consistent with gaugino unification. Low energy observables are summarized in Table~\ref{tab:param}, although their values are too similar to be used to differentiate between the scenarios. The results of this paper also help to set up a full analysis of the soft mass RGEs in order to investigate boundary conditions consistent with $B-L$ and electroweak symmetry breaking: an analysis which will be conducted in a future publication.
\section*{Appendix A: Non-Canonical Bases}
In this Appendix, we briefly analyze ``non-canonical'' bases and their relevant properties. For specificity, we consider one such basis, before going on to prove an important general theorem.
\subsection*{A.1-- A Non-Canonical Basis}
Define
\begin{equation}
Y_{1}=4Y_{T_{3R}}-2Y_{B-L}, \quad Y_{2}=3Y_{T_{3R}}+Y_{B-L} \ .
\label{84a}
\end{equation}
Note from \eqref{8}, \eqref{10} and \eqref{13} that
\begin{equation}
Y_{1}=8\big (Y-\frac{5}{4}(B-L) \big), \quad Y_{2}=6Y \ .
\label{84b}
\end{equation}
Using \eqref{14}, it follows that $Y_{1}$,$Y_{2}$ satisfy the Killing brackets
\begin{equation}
(Y_{1}|Y_{1})=80, \quad (Y_{2}|Y_{2})=30, \quad (Y_{1}|Y_{2})=0 \ .
\label{84c}
\end{equation}
Hence, they form an orthogonal--but ``non-canonical''--basis of the two-dimensional subspace ${\mathfrak{h}}_{3 \oplus 2}$ of the Cartan subalgebra that commutes with ${\mathfrak{su}}(3)_C \oplus {\mathfrak{su}}(2)_L$.
The explicit form of $Y_{1}$ and $Y_{2}$ in the $\bf{16}$ representation of ${\mathfrak{so}}(10)$ is easily constructed from \eqref{15},\eqref{16}.
We find that
\begin{eqnarray}
&[Y_{1}]_{\bf 16} = (-2){\bf 1}_{3} \oplus (6){\bf 1}_{3} \oplus (-2){\bf 1}_{6} \oplus (6){\bf 1}_{2} \oplus (-10){\bf 1}_{1} \oplus (-2){\bf 1}_{1} \ ,\label{84d} \\ \nonumber \\
&[Y_{2}]_{\bf 16} = (-4){\bf 1}_{3} \oplus (2){\bf 1}_{3} \oplus (1){\bf 1}_{6} \oplus (-3){\bf 1}_{2} \oplus (0){\bf 1}_{1} \oplus (6){\bf 1}_{1} \ . \label{84e}
\end{eqnarray}
Similarly, using \eqref{21},\eqref{22} the explicit form of $Y_{1}$ and $Y_{2}$ in the $\bf{10}$ representation is given by
\begin{eqnarray}
&[Y_{1}]_{\bf 10}=(-4){\bf 1}_{3} \oplus (4){\bf 1}_{3} \oplus (4){\bf 1}_{2}\oplus (-4){\bf 1}_{2} \ ,\label{84f} \\
\nonumber \\
&[Y_{2}]_{\bf 10}=(2){\bf 1}_{3} \oplus (-2){\bf 1}_{3} \oplus (3){\bf 1}_{2}\oplus (-3){\bf 1}_{2} \ .\label{84g}
\end{eqnarray}
\subsection*{A.2-- Properties of the Non-Canonical Basis}
The non-canonical basis \eqref{84a} shares three of the four fundamental properties of the canonical basis. We analyze this as follows.
\subsubsection*{Wilson Lines and the MSSM:}
Consider the two Wilson lines associated with the non-canonical basis. As abstract ${\mathfrak{so}}(10)$ group elements, these are
\begin{equation}
\chi_{1}=e^{iY_{1}\frac{2\pi}{3}}, \quad \chi_{2}=e^{iY_{2}\frac{2\pi}{3}} \ .
\label{84h}
\end{equation}
Note that $\chi_{1}^{3}=\chi_{2}^{3}=1$ and, hence, each generates a finite ${\mathbb{Z}}_{3}$ subgroup of $Spin(10)$. When turned on simultaneously, these Wilson lines spontaneously break
\begin{equation}
Spin(10) \rightarrow SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y_{1}} \times U(1)_{Y_{2}} \ .
\label{84i}
\end{equation}
The ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ isometry acts equivariantly on the chosen vector bundle $V$ and, hence, the associated sheaf cohomology groups of tensor products of $V$ carry a representation of ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$. To determine the zero modes of the Dirac operator twisted by $V$ and, hence, the low energy spectrum, one takes each $H^{1}(X,U_{R}(V))$, tensors it with the associated representation $R$, and then chooses the invariant subspace
$(H^{1}(X,U_{R}(V)) \otimes R)^{{\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}}$.
For $R={\bf{16}}$, the associated sheaf cohomology $H^{1} \big(X,V \big)$ was given in \eqref{25}. The explicit representation of ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ on this
linear space was presented in \eqref{26}. Choosing the Wilson line generators in the non-canonical basis \eqref{84a}, it follows from \eqref{84d} and \eqref{84e} that the action of the Wilson lines \eqref{84h} on each $\bf{16}$ is given by
\begin{eqnarray}
{\bf 16}&=& \chi_{1} \cdot \chi_{2}^{2} ({\bf{\bar{3}},\bf {1}},-2,-4)\oplus 1 \cdot \chi_{2}^{2} ({\bf{\bar{3}},\bf {1}},6, 2) \label{84j} \\
&& \oplus \chi_{1} \cdot \chi_{2} ({\bf{3}},{\bf {2}},-2,1) \oplus 1 \cdot 1({\bf{1}},{\bf {2}},6,-3) \oplus \chi_{1}^{2} \cdot 1 ({\bf{1}},{\bf {1}}, -10, 0) \nonumber\\
&& \oplus \chi_{1} \cdot 1({\bf{1}},{\bf {1}},-2,6)~. \nonumber
\end{eqnarray}
Using this and \eqref{26}, we find that $(H^{1}(X,V) \otimes {\bf 16})^{{\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}}$ consists of {\it three families} of quarks and leptons, each family transforming as
\begin{equation}
Q= ({\bf{3}},{\bf {2}},-2,1), \quad u=({\bf{\bar{3}},\bf {1}},-2,-4), \quad d=({\bf{\bar{3}},\bf {1}},6, 2)
\label{84k}
\end{equation}
and
\begin{equation}
L=({\bf{1}},{\bf {2}},6,-3), \quad \nu=({\bf{1}},{\bf {1}}, -10,0), \quad e=({\bf{1}},{\bf {1}},-2,6)
\label{84l}
\end{equation}
under $SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y_{1}} \times U(1)_{Y_{2}}$.
For $R={\bf 10}$, the associated sheaf cohomology is $H^{1}(X,\wedge^{2}V)$. The explicit representation of ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ on this
linear space was presented in \eqref{31}. Choosing the Wilson line generators in the non-canonical basis \eqref{84a}, it follows from \eqref{84f} and \eqref{84g} that the action of the Wilson lines \eqref{84h} on each $\bf{10}$ is given by
\begin{eqnarray}
{\bf 10}&= &\chi_{1}^{2} \cdot \chi_{2}^{2} ({\bf 3},{\bf 1},-4,2) \oplus \chi_{1} \cdot \chi_{2} ({\bar{\bf 3}},{\bf 1},4,-2) \nonumber \\
&&\oplus \chi_{1} \cdot 1({\bf 1},{\bf 2},4,3) \oplus \chi_{1}^{2} \cdot 1 ({\bf 1},{\bf 2},-4,-3)~.
\label{84m}
\end{eqnarray}
Using this and \eqref{31}, we find that $(H^{1}(X,\wedge^{2}V) \otimes {\bf 10})^{{\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}}$ consists of a {\it single pair} of Higgs doublets transforming as
\begin{equation}
H=({\bf 1},{\bf 2},4,3), \quad \bar{H}= ({\bf 1},{\bf 2},-4,-3)
\label{84n}
\end{equation}
under $SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y_{1}} \times U(1)_{Y_{2}}$. These results lead to the following property of non-canonical basis \eqref{84a}.\\
\noindent $\bullet$ {\it When the two Wilson lines of the non-canonical basis are turned on simultaneously, the resulting low energy spectrum is precisely that of the MSSM--that is, three families of quark/lepton chiral superfields, each family with a right-handed neutrino supermultiplet, and one pair of Higgs-Higgs conjugate chiral multiplets}. \\
This non-canonical basis exhibits a second, related, property.
Consider, once again, the $R={\bf 16}$ case and the ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ invariant tensor product of $H^{1}(X,V)$ in \eqref{25},\eqref{26} with the ${\bf 16}$ decomposition in \eqref{84j}.
Note that each quark and lepton chiral multiplet in the low energy theory arises from a different ${\bf 16}$ representation of $Spin(10)$. Now consider the $R={\bf 10}$ case. It is easily seen from \eqref{31} and \eqref{84m} that
the Higgs and Higgs conjugate chiral multiplets each arise from a different {\bf 10} representation of $Spin(10)$. This leads to the second property of the non-canonical basis. \\
\noindent ${\bullet}$ {\it Since each quark/lepton and Higgs superfield of the low energy Lagrangian arises from a different ${\bf 16}$ and ${\bf 10}$ representation of $Spin(10)$ respectively, the parameters of the effective theory, and specifically the Yukawa couplings and the soft supersymmetry breaking parameters, are uncorrelated by the $Spin(10)$ unification.} \\
Thus, the non-canonical basis \eqref{84a} shares these two important properties with the canonical basis. Now consider the following.
\subsubsection*{The Kinetic Mixing Parameter:}
Prior to turning on the ${\mathbb{Z}}_{3} \times {\mathbb{Z}}_{3}$ Wilson lines, the conventionally normalized kinetic energy part of the gauge field Lagrangian is $Spin(10)$ invariant and given in
\eqref{35}, where $\{T^{a}_{R}, a=1,\dots.45\}$ is an orthogonal basis of ${\mathfrak{so}}(10)$ in any representation $R$ , each basis element Killing normalized to $\frac{1}{\sqrt{2}}$. Defining
\begin{equation}
T^{1}= \sqrt{\frac{2}{5}}\big(Y-\frac{5}{4}(B-L)\big)=\frac{1}{4\sqrt{10}} Y_{1}, \quad T^{2}=\sqrt{\frac{3}{5}}Y=\frac{1}{2\sqrt{15}}Y_{2}
\label{84o}
\end{equation}
we see from \eqref{84c} that
\begin{equation}
(T^{1}|T^{1})=(T^{2}|T^{2})=\frac{1}{2}, \quad (T^{1}|T^{2})=0
\label{84p}
\end{equation}
and, hence,
\begin{equation}
{\cal{L}}_{kinetic}=-\frac{1}{4}(F_{\mu\nu}^{1})^{2}-\frac{1}{4}(F_{\mu\nu}^{2})^{2} +\dots ~.
\label{84q}
\end{equation}
That is, there is no kinetic mixing term of the form $F_{\mu\nu}^{1} F^{2\mu\nu}$. This is a consequence of the fact that the non-canonical basis elements $Y_{1}$ and $Y_{2}$ are Killing orthogonal, and is of little importance while $Spin(10)$ remains unbroken. However, if both Wilson lines are turned on simultaneously, the gauge group is spontaneously broken to $SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y_{1}} \times U(1)_{Y_{2}}$. For general $U(1) \times U(1)$, the two Abelian field strengths can exhibit kinetic mixing; that is,
\begin{equation}
{\cal{L}}_{kinetic}=-\frac{1}{4}((F_{\mu\nu}^{1})^{2}+2\alpha F_{\mu\nu}^{1} F^{2\mu\nu}+(F_{\mu\nu}^{2})^{2} +\dots) ~.
\label{84r}
\end{equation}
for some real parameter $\alpha$. However, for $U(1)_{Y_{1}} \times U(1)_{Y_{2}}$ the normalized canonical generators satisfy \eqref{84p} and, specifically, are orthogonal in ${\mathfrak{so}}(10)$. It follows that the initial value of $\alpha$ at the unification scale, $M_{u}$, must vanish. Hence, there is a third property that the non-canonical basis \eqref{84a} shares with the canonical basis.\\
\noindent $\bullet$ {\it Since the generators of the non-canonical basis are Killing orthogonal in
${\mathfrak{so}}(10)$, the value of the kinetic field strength mixing parameter $\alpha$ must vanish at the unification scale. That is, $\alpha(M_{u})=0$.}\\
Once the $Spin(10)$ symmetry is broken by both Wilson lines, either by turning them on at the same scale or sequentially, one expects the mixing parameter $\alpha$ to regrow due to radiative corrections. In this case, the Abelian field strengths develop a non-vanishing mixing term which greatly complicates the renormalization group analysis of the low energy effective theory. As discussed in detail in Subsection \ref{PCB}, at an arbitrary scale the covariant derivative can be written in an ``upper triangular'' realization given by
\begin{equation}
D=\partial -i(T^{1},T^{2})
\begin{pmatrix}
{\cal{G}}_{1} & {\cal{G}}_{M} \\
0 & {\cal{G}}_{2}
\end{pmatrix}
\begin{pmatrix}
{\cal{A}}^{1} \\
{\cal{A}}^{2}
\end{pmatrix} \ ,
\label{84s}
\end{equation}
with
\begin{equation}
{\cal{G}}_{1}=g_{1}, \quad {\cal{G}}_{2}=\frac{g_{2}}{\sqrt{1-\alpha^{2}}}, \quad {\cal{G}}_{M}=\frac{-g_{1}\alpha}{\sqrt{1-\alpha^{2}}} \ .
\label{84t}
\end{equation}
The RGE for the off-diagonal coupling ${\cal{G}}_{M}$ was presented in \eqref{50},\eqref{51} and \eqref{52}. Recall that if the mixing parameter $\alpha$ and, hence, the off-diagonal coupling ${\cal{G}}_{M}$ vanish at some initial scale, as they will for the non-canonical basis \eqref{84a}, then a non-vanishing ${\cal{G}}_{M}$ will be generated at a lower scale if and only if the charges $T^{1}$ and $T^{2}$ are such that
\begin{equation}
B_{12}=Tr(T^{1}T^{2})\neq0 \ .
\label{84u}
\end{equation}
The trace in \eqref{84u} is over the entire matter and Higgs spectrum of the MSSM.
Let us break $Spin(10)$ to $U(1)_{Y_{1}}\times U(1)_{Y_{2}}$ with both Wilson lines of the non-canonical basis \eqref{84a}. The associated normalized charges $T^{1}$ and $T^{2}$ were presented in \eqref{84o} and satisfy
\begin{equation}
(T^{1}|T^{2})=0 \ .
\label{84v}
\end{equation}
It then follows from \eqref{17} that
\begin{equation}
Tr([T^{1}]_{R}[T^{2}]_{R})=0
\label{84w}
\end{equation}
for any complete ${\mathfrak{so}}(10)$ representation $R$. Recalling that each quark/lepton family with a right-handed neutrino fills out a complete ${\bf 16}$ multiplet, one can conclude that
\begin{equation}
Tr([T^{1}]_{quarks/leptons}[T^{2}]_{quarks/leptons})=0 \ .
\label{84x}
\end{equation}
However, in the reduction to the zero-mode spectrum the color triplet Higgs $H_{C}$ and ${\bar{H}}_{C}$ are projected out. Hence, the electroweak Higgs doublets $H$ and ${\bar{H}}$ do not make up a complete ${\bf 10}$ of ${\mathfrak{so}}(10)$. Therefore, the trace of $T^{1}T^{2}$ over the Higgs fields of the MSSM is not guaranteed to vanish. It is straightforward to compute this trace using \eqref{84f}, \eqref{84g} and \eqref{84o}. If we ignore the color triplet components, then
\begin{eqnarray}
&&[Y_{1}]_{H,{\bar{H}}}= (4){\bf 1}_{2}\oplus (-4){\bf 1}_{2} \label{84y} \\ \nonumber \\
&&[Y_{2}]_{H,{\bar{H}}}=(3){\bf 1}_{2}\oplus (-3){\bf 1}_{2} \ . \label{84z}
\end{eqnarray}
It then follows from \eqref{84o} and \eqref{84y}, \eqref{84z} that
\begin{equation}
Tr([T^{1}]_{H,{\bar{H}}}[T^{2}]_{H,{\bar{H}}})= \frac{1}{8\sqrt{150}}Tr([Y_{1}]_{H,{\bar{H}}}[Y_{2}]_{H,{\bar{H}}})=\frac{\sqrt 6}{5} \ .
\label{84aa}
\end{equation}
We conclude from \eqref{84x} and \eqref{84aa} that
\begin{equation}
B_{12}=\frac{\sqrt 6}{5} \neq 0 \ .
\label{84bb}
\end{equation}
Therefore, for the non-canonical basis \eqref{84a}, although the initial value of $\alpha$ and, hence, ${\cal{G}}_{M}$ vanish, these off-diagonal parameters will re-grow at any lower scale. That is, unlike the case for the canonical basis, here kinetic mixing will re-emerge due to radiative corrections. We conclude that the non-canonical basis \eqref{84a} does not share the fourth property possessed by the canonical basis.\\
\noindent $\bullet$ {\it The generators of the non-canonical basis \eqref{84a} are such that $Tr(T^{1}T^{2}) \neq 0$ when the trace is performed over the matter and Higgs spectrum of the MSSM. Thus, unlike the canonical basis, even though the original kinetic mixing parameter vanishes,
$\alpha$ and, hence, ${\cal{G}}_{M}$ will regrow under the RG at any scale. This property of kinetic mixing greatly complicates the renormalization group analysis of the $SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y_{1}}\times U(1)_{Y_{2}}$ low energy theory}.\\
\subsection*{A.3-- General Theorem About Kinetic Mixing}
In this final subsection, we will prove that this property of basis \eqref{84a} is shared by all orthogonal non-canonical bases. That is, the only basis with the property that kinetic mixing vanishes at all scales is the canonical basis, or any appropriate multiples of this basis. Consider a generic basis of ${\mathfrak{h}}_{3 \oplus 2}$ given by
\begin{equation}
Y_{1}=mY_{T_{3R}}+nY_{B-L}, \quad Y_{2}=pY_{T_{3R}}+qY_{B-L} \ .
\label{84cc}
\end{equation}
Using \eqref{14}, we find that
\begin{eqnarray}
&&(Y_{1}|Y_{1})=2(m^{2}+6n^{2}), \quad (Y_{2}|Y_{2})=2(p^{2}+6q^{2}) \label{84dd} \\
&& \quad \qquad \qquad (Y_{1}|Y_{2})=2(mp+6nq) \ .\label{84ee}
\end{eqnarray}
In order for the initial value of the mixing parameter $\alpha(M_{u})$ to vanish, we want to consider bases for which $(Y_{1}|Y_{2})=0$. Hence, one must choose
\begin{equation}
mp=-6nq \ .
\label{84ff}
\end{equation}
As previously, define the normalized generators
\begin{equation}
T^{1}=\frac{1}{2\sqrt{m^{2}+6n^{2}}}Y_{1}, \quad T^{2}=\frac{1}{2\sqrt{p^{2}+6q^{2}}}Y_{2}
\label{84gg}
\end{equation}
so that
\begin{equation}
(T^{1}|T^{1})=(T^{2}|T^{2})=\frac{1}{2}, \quad (T^{1}|T^{2})=0 \ .
\label{84hh}
\end{equation}
We will, henceforth, make the assumption that the associated Wilson lines lead to exactly the MSSM spectrum with three families of right-handed neutrino supermultiplets with no vector-like pairs or exotic fields. The non-canonical basis \eqref{84a} serves as a proof that, in addition to the canonical basis, there is at least one other basis with this property.
Having guaranteed that the initial mixing parameter $\alpha$ vanishes, we want to decide when this parameter can remain zero at any lower scale under the RG. As discussed above, this will be the case if and only if the charges $T^{1}$ and $T^{2}$ are such that
\begin{equation}
B_{12}=Tr(T^{1}T^{2})=0 \ ,
\label{84ii}
\end{equation}
where the trace is over the entire matter and Higgs spectrum of the MSSM.
The normalized charges $T^{1}$ and $T^{2}$ in \eqref{84gg} satisfy
\begin{equation}
(T^{1}|T^{2})=0
\label{84jj}
\end{equation}
and, hence,
\begin{equation}
Tr([T^{1}]_{R}[T^{2}]_{R})=0
\label{84kk}
\end{equation}
for any complete ${\mathfrak{so}}(10)$ representation $R$. Recalling that each quark/lepton family with a right-handed neutrino fills out a complete ${\bf 16}$ multiplet, one can conclude that
\begin{equation}
Tr([T^{1}]_{quarks/leptons}[T^{2}]_{quarks/leptons})=0 \ .
\label{84ll}
\end{equation}
However, in the reduction to the zero-mode spectrum the color triplet Higgs $H_{C}$ and ${\bar{H}}_{C}$ are projected out. Hence, the electroweak Higgs doublets $H$ and ${\bar{H}}$ do not make up a complete ${\bf 10}$ of ${\mathfrak{so}}(10)$. Therefore, the trace of $T^{1}T^{2}$ over the Higgs fields of the MSSM is not guaranteed to vanish. It is straightforward to compute this trace using \eqref{21}, \eqref{22} and \eqref{84cc}. If we ignore the color triplet components, then
\begin{eqnarray}
&&[Y_{1}]_{H,{\bar{H}}}= (m){\bf 1}_{2}\oplus (-m){\bf 1}_{2} \label{84mm} \\ \nonumber \\
&&[Y_{2}]_{H,{\bar{H}}}=(p){\bf 1}_{2}\oplus (-p){\bf 1}_{2} \ . \label{84nn}
\end{eqnarray}
It follows from \eqref{84gg} and \eqref{84mm}, \eqref{84nn} that
\begin{eqnarray}
&Tr([T^{1}]_{H,{\bar{H}}}[T^{2}]_{H,{\bar{H}}})= \frac{1}{4\sqrt{(m^{2}+6n^{2})(p^{2}+6q^{2})}}Tr([Y_{1}]_{H,{\bar{H}}}[Y_{2}]_{H,{\bar{H}}}) \\
& \quad = \frac{mp}{\sqrt{(m^{2}+6n^{2})(p^{2}+6q^{2})}} \ .
\label{84oo}
\end{eqnarray}
We conclude from \eqref{84ll} and \eqref{84oo} that
\begin{equation}
B_{12}= \frac{mp}{\sqrt{(m^{2}+6n^{2})(p^{2}+6q^{2})}} \ .
\label{84pp}
\end{equation}
Therefore, for $B_{12}$ to satisfy \eqref{84ii} one must choose
\begin{equation}
mp=0 \ .
\label{84qq}
\end{equation}
We conclude that to have kinetic mixing vanish at any energy-momentum scale, a generic basis \eqref{84cc} must satisfy both \eqref{84ff} and \eqref{84qq}. It is straightforward to show that the only bases that satisfy these two constraints is the canonical basis, and appropriate multiples of the canonical basis.\\
\noindent $\bullet$ {\it The only basis of ${\mathfrak{h}}_{3 \oplus 2} \subset {\mathfrak{h}}$ for which $U(1)_{Y_{1}} \times U(1)_{Y_{2}}$ kinetic mixing vanishes at all values of energy-momentum is the canonical basis $Y_{T_{3R}}$, $Y_{B-L}$ and appropriate multiples of this basis.}
\section*{Acknowledgments}
The work of Burt Ovrut and Austin Purves is supported in part by the DOE under contract No. DE-AC02-76-ER-03071. B.A.O. acknowledges partial support from the NSF RTG grant DMS-0636606 and from NSF Grant 555913/14 for International Collaboration.
| 48,039 |
\section{Introduction}
\label{sec:intro}
The properties and evolution of galaxies are known to be strongly linked to their external environment. Massive halos are found to play a key role in galaxy evolution. At low redshifts, it has been found that clusters of galaxies are mostly dominated by early-type galaxies composed of old stellar populations, while low-density environments host typically late-type galaxies with younger and bluer stars, producing the star-formation (SF) - density - distance to cluster centers relations and affecting the morphology of galaxies \citep[e.g.,][]{oemler1974,dressler1980}. We expect that galaxies in dense regions experience various physical processes such as tidal forces, mergers, high-speed interactions, harassment and gas stripping, which in turn contribute to dramatic morphological changes and quenching of star formations \citep[e.g.][]{larson1980,byrd1990}, although, the precise timing and the relative importance of these physical processes are not yet well understood.
The environmental processes which affect galaxy evolution could, directly or indirectly influence the accretion onto the central black hole in galaxies, notably, those with a stellar bulge \citep{magorrian1998}. Both local and large-scale processes which may affect cluster galaxies also have the potential to affect the distribution of gas in the galaxies, and hence may trigger or suppress active galactic nuclei (AGN) activity.
Apart from the role of galaxy group and cluster environment on radio emission of the brightest galaxy of the group, \cite{khosroshahi2017} suggested that the radio luminosity of the brightest group galaxy (BGG) also depends on the group dynamics, in a way that BGGs in groups with a relaxed/virialised morphology are less radio luminous than the BGG with the same stellar mass but in an evolving group. This was supported numerically by a semi-analytic approach \citep{raouf2018}, where they made a prediction of the radio power for the first time. However without an observational constraint reaching high redshift, the numerical models cannot be constrained.
Many radio studies \citep{best2002,barr2003,miller2003,reddy2004} showed an increase in radio-loud AGN activity in galaxy clusters, at a range of redshifts and in both relaxed and merging systems. The radio emission ($<30$ GHz) in galaxies is dominated by synchrotron radiation from accelerating relativistic electrons, with a fraction of free-free emission \citep[e.g.,][]{sadler1989,condon1992,clemens08,tabatabaei2017}. The feedback from supernovae explosions in star-forming galaxies (SFGs) and that from the growth of the central supermassive black hole (SMBH) in AGN are two main sources of acceleration of cosmic electrons.
For the purpose of using radio emission as a proxy for measuring star formation rates (SFRs) or AGN feedback, it is important to estimate which process dominates the radio emission: star formation processes, or SMBH accretion. We follow the method demonstrated in \citep{delvecchio17} who measured the radio excess compared to the total star-formation-based infrared (IR) emission. Objects which exhibit radio excess above what is expected from star formation alone, as calculated from their infrared emission, are deemed AGN, and the rest are SFGs. These populations contribute different percentages to the energy budget. In the radio, this is quantified by calculating the radio luminosity function (RLF). \cite{novak18} studied the 3 GHz VLA-COSMOS RLF and calculated the relative contributions to the RLF from the AGN and SFG populations, down to submicrojansky levels, and found that the RLF can reproduce the observed radio sky brightness very well. AGN and SFGs contribute differently to the RLF, where AGN are known to dominate the bright part of the RLF, and SFGs dominate the faint. In particular, 90\% of the population at the faint end ($< 0.1~\mu$Jy) is linked to SFGs. In clusters of galaxies, \cite{yuan16} who studied the RLF of brightest cluster galaxies (BCGs) up to $z$ = 0.45 found no evolution, and a dominant population of AGN, as most of their BCGs are associated with AGN. \cite{branchesi06} compared clusters at 0.6 $< z < 0.8$ to the local Abell clusters and found very different RLFs. These studies target populations dominated by AGN, and thus probe the high end of the radio luminosity function. The question arises, how much do smaller mass environments, those of groups of galaxies, and their members contribute to the observed radio sky brightness.
In this paper we investigate the population of galaxies inside X-ray galaxy groups in the COSMOS field \citep{gozaliasl19} to quantify their contribution to the RLF at 3 GHz VLA-COSMOS \citep{novak18,smolcic17a}. In Section~\ref{sec:data} we describe the X-ray and radio data used throughout this work. In Section~\ref{sec:lumfun} we focus on methods for deriving the RLF and its evolution through cosmic time. In Section~\ref{sec:evolution} we present and discuss the results on the RLF of group galaxies, and calculate their contribution to the total RLF at 3 GHz. We further separate the galaxies to BGGs and satellites (SGs). We also use the radio excess parameter and the presence of jets/lobes to disentangle AGN and SFGs in the radio, and provide the relative contributions of these populations to the group galaxies (GG) RLF and to the total RLF. This is presented in Section~\ref{sec:agn}. Finally, in Section~\ref{sec:summary} we provide a brief summary. The tables with the results of the analysis can be found in the Appendix.
We assume flat concordance Lambda Cold Dark Matter ($\Lambda$CDM) cosmology defined with a Hubble constant of $H_0=70$\,km\,s$^{-1}$\,Mpc$^{-1}$, dark energy density of $\Omega_{\Lambda}=0.7$, and matter density of $\Omega_\text{m}=0.3$. For the radio spectral energy distribution we assume a simple power law described as $S_{\nu}\propto\nu^{-\alpha}$, where $S_\nu$ is the flux density at frequency $\nu$ and $\alpha$ is the spectral index. If not explicitly stated otherwise, $\alpha=0.7$ is assumed.
\section{The Data}
\label{sec:data}
The Cosmic Evolution Survey (COSMOS) is a deep multi-band survey covering a 2 deg$^2$ area, thus offering a comprehensive data-set to study the evolution of galaxies and galaxy systems. The full definition and survey goals can be found in \cite{scoville07}.
The sample selection for this study is described below. We used radio selected samples of galaxies which have been cross-matched with multi-wavelength optical/near-infrared (NIR) and value-added catalogues in the COSMOS field. The radio data have been selected from the VLA-COSMOS 3~GHz Large Project \citep{smolcic17a}, with a median sensitivity of 2.3~\ensuremath{\mu\text{Jy}\,\text{beam}^{-1}}\ and resolution of $0.75$ arcsec. The cross-correlation of the radio and multiwavelength sources can be found in \cite{smolcic17b}. Only sources within the COSMOS2015 catalogue \citep{laigle16} or with $i$-band counterparts have been given the availability of reliable redshift measurements. The COSMOS2015 catalogue contains the high-quality multiwavelength photometry of $\sim$800\,000 sources across more than 30 bands from near-ultraviolet (NUV) to near-infrared (NIR) through a number of surveys and legacy programs \citep[see][ for detailed description]{laigle16}.
On completion of the visionary Chandra program \citep{elvis09, civano16}, high-resolution imaging across the full COSMOS field became available. Furthermore, more reliable photometric data provided a robust identification of galaxy groups at a higher redshift, thus resulting in a revised catalogue of extended X-ray sources in COSMOS \citep{gozaliasl19} which is obtained by combining both the Chandra and XMM-Newton data for the COSMOS field. We use this galaxy groups catalogue \citep{gozaliasl19} for our analysis. Within the virial radius of these groups, the above selection criteria resulted in a total of 306 objects distributed in the galaxy groups. The brightest group galaxy (BGG) is the most massive and generally the most luminous galaxy in $r$-band within the virial radius associated with the extended X-ray emission. The rest of group galaxies (GGs) are labelled as satellites (SGs).
The details of the sample selection and the accuracy of the group findings using spectroscopic and photometric redshifts can be found in \cite{gozaliasl19}. In Fig.~\ref{fig:z_gg} we present the data for the group galaxies used in our analysis. The spectroscopic redshifts are available for 35\% of our sources and the median accuracy of the photometric redshifts is $\Delta z/(1+z_\text{spec})=0.01 (0.04)$.
To perform our analysis on the radio luminosity function of group galaxies in COSMOS, we cross-match the galaxy group catalogue and the 3 GHz VLA-COSMOS data with radius of $0.8{\arcsec}$. We furthermore use the 3 GHz VLA-COSMOS data presented in \cite{novak18}, who constructed RLFs up to $z \sim$ 5.5, to compare to the total RLF in COSMOS up to $z \sim$ 2.3. Furthermore, \cite{novak18} separated objects in SFGs and AGN, following the radio excess prescription of \cite{delvecchio17}. This method is based on the excess radio emission from what is expected from star formation alone. \cite{delvecchio17} fitted the infrared spectral energy distribution of radio sources at 3 GHz VLA-COSMOS and calculated the contribution of the 3 GHz VLA-COSMOS radio sources to the radio luminosity, by applying a conservative cut. Galaxies which exhibit an excess in radio emission above $3\sigma$ from what is expected from SF alone were deemed AGN, with the rest being SFGs. This method was used to separate the \cite{novak18} sample which we use here for comparison, in SFGs and AGN. Finally, \cite{novak18} described possible biases and uncertainties associated with the data sample selection and thus we refer the reader to this reference.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{N_zcl_groupgalaxies.pdf}
\includegraphics[width=\linewidth]{L1400MHz_zcl_groupgalaxies.pdf}
\caption{Top: Number of sources per redshift. Bin size is 0.1. Bottom: Radio luminosity at 1.4 GHz versus redshift. The redshift plotted is the one of the galaxy groups. The radio luminosity is calculated from the 1.4 GHz flux density for the redshift of the object. Black represents all group galaxies, red is for BGGs, and yellow is for SGs (see Sec.~\ref{sec:data} for clarification on the classification).}
\label{fig:z_gg}
\end{figure}
\section{Methods and analysis}
\label{sec:lumfun}
We describe the process of calculating the RLF for galaxies in groups in COSMOS \citep{gozaliasl19} using the VLA-COSMOS 3~GHz data. We applied a cut in group mass $M_{200c} > 10^{13.3} M_{\odot}$, to account for a difference in the limiting mass of the group catalog with redshift. We further separate group galaxies in BGGs and SGs. We compare to the RLF of the population of SFGs and AGN at 3 GHz VLA-COSMOS in the same redshift bins, and to the total RLF calculated from the 3 GHz data \citep{novak17, smolcic17a}. We fit linear and power-law models to the RLFs of GGs and compare them to the total RLF to obtain the contribution of GGs to the total RLF at 3 GHz VLA-COSMOS, something that has not been shown before in COSMOS.
\subsection{Measuring the radio luminosity function}
\label{sec:lumfun_total}
To obtain the total RLFs, for GGs, SFGs and AGN, we followed the method adopted by \cite{novak17} (see their Sec. 3.1). They computed the maximum observable volume \ensuremath{V_\text{max}}\, for each source \citep{schmidt68} and simultaneously applied completeness corrections that take into account the non-uniform $rms$ noise and the resolution bias \citep[see Sec.~3.1 in][]{novak17}.
Then the RLF is
\begin{equation}
\Phi(L,z) = \frac{1}{\Delta {\rm log} L} \sum_{i=1}^{N} \frac{1}{V_{max,i}}
\label{eq:phi}
\end{equation}
\noindent
where $L$ is the rest-frame luminosity at 1.4 GHz, derived using the radio spectral index of a source calculated between 1.4 GHz \citep{schinnerer10} and 3~GHz \citep{smolcic17a}, and $\Delta log L$ is the width of the luminosity bin. The radio spectral index should remain unchanged between frequencies, and is only available for a quarter of the 3 GHz VLA-COSMOS sample. For the rest of the sources detected only at 3~GHz, we assumed $\alpha=0.7$. The latter corresponds to the average spectral index of the entire 3~GHz population \citep[see Sec.~4 in][]{smolcic17a}. $V_{\rm max}$ is the maximum observable volume given by
\begin{equation}
V_{max,i} = \sum_{z=z_{min}}^{z_{max}} [V(z + \Delta z) - V(z)] C(z),
\label{eq:vmax}
\end{equation}
\noindent
where the sum starts at $z_{min}$ and adds co-moving spherical shells of volume $\Delta V = V(z + \Delta z) - V(z)$ in small redshift steps $\Delta z$ = 0.01 until $z_{max}$. $C(z)$ is the redshift-dependent geometrical and statistical correction factor. This takes into account sample incompleteness. For a thorough description of the biases see Section~6.4 in \cite{novak17}. The correction factor is given by
\begin{equation}
C(z) = \frac{A_{obs}}{41 253 deg^{2}} \times C_{radio}(S_{3 GHz}(z)) \times C_{opt(z)},
\label{eq:cz}
\end{equation}
\noindent
where $A_{obs}$ = 1.77 deg$^{2}$ is the effective unflagged
area observed in the optical to NIR wavelengths, $C_{radio}$ is the
completeness of the radio catalogue as a function of the flux density $S_{3 GHz}$, and $C_{opt}$ is the completeness owing to radio sources
without assigned optical-NIR counterpart. Completeness corrections are shown in \cite{smolcic17a} in their Fig. 16 and Table 2, and in \cite{novak17}, in their Fig. 2.
The redshift bins are large enough not to be affected severely by photometric redshift uncertainty, and follow the selection of \cite{novak17} to allow comparisons. Luminosity bins in each redshift bin span exactly the observed luminosity range of the data. In order to eliminate possible issues due to poorer sampling, the lowest luminosity has a range of values from the faintest observed source to the 5$\sigma$ detection threshold at the upper redshift limit (corresponding to $5\times 2.3~\ensuremath{\mu\text{Jy}\,\text{beam}^{-1}}$ at 3~GHz). The reported luminosity for each RLF is the median luminosity of the sources within the bin. The RLFs for all group galaxies are shown in Fig.~\ref{fig:lfgrid} (black points) and are also listed in Table~\ref{tab:lumfun_vmax}. The RLFs for the BGGs and SGs are also shown in Fig.~\ref{fig:lfgrid} (red squares/yellow stars), and are listed in Tables~\ref{tab:lumfun_bgg}~and~\ref{tab:lumfun_sg}, respectively. The low-$z$ bin in Fig.~\ref{fig:rlf_evol} is split in two halo mass $M_{\rm 200c}$ bins, above and below $10^{13.3}M_{\odot}$, and for our further analysis we use the values above. We note that at $1.6 < z < 2.3$ we have only 2 SGs above our halo-mass cut ($M_{\rm 200c} > 10^{13.3} M_{\odot}$), thus we do not calculate their RLF. This is related to limits in the radio power that are probed at those redshifts, leading to low statistics as well as scarcity of massive groups. As discussed in \cite{novak18}, there is only 5-10\% loss of completeness on the optical/NIR counterparts above $z\sim2$.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{lf_grid.pdf}
\caption{Total radio luminosity functions of galaxies in groups. Black points indicate the RLFs for group galaxies derived using the \ensuremath{V_\text{max}}\ method (see Section~\ref{sec:lumfun_total}). Red squares and yellow stars mark brightest group galaxies and satellites, respectively. The blue and red shaded areas show the $\pm 3\sigma$ ranges of the best-fit evolution for the individual SFG and AGN populations, respectively (outlined in Section~\ref{sec:lumfun_pop}). The black dashed line is the fit to the total RLF at 3 GHz VLA-COSMOS \citep{novak17, smolcic17c}. The black solid line is the fit to the group galaxies RLF, and the red solid line is the best fit for BGGs, as explained in Section~\ref{sec:lumfun_linfit}. We do not show the best fit for SGs for clarity. For the low-$z$ sub-samples ($z <$ 0.4), the halos have been split into massive ($M_{\rm 200c} > 10^{13.3} M_{\odot}$) and low-mass halos ($M_{\rm 200c} < 10^{13.3} M_{\odot}$).}
\label{fig:lfgrid}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=1.\linewidth]{fraction_GGs_sep21.pdf}
\includegraphics[width=1.\linewidth]{fraction_massive_allvla3.pdf}
\caption{Top: Fraction for group galaxies showing their contribution to the total 3 GHz radio luminosity function at different epochs vs radio luminosity at 1.4 GHz. Colours represent different redshift bins. At redshifts $>$ 1, both the low and the high radio emission fraction remains constant. As the universe evolves the fraction of the high radio emission increases dramatically. Bottom: Same as above, but for all massive galaxies ($M_{*} > 10^{11.2}M_{\odot}$) with radio emission at 3 GHz \citep{smolcic17b, laigle16}.}
\label{fig:frac_lum}
\end{figure}
\subsection{The total RLF at 3 GHz VLA-COSMOS}
\label{sec:lumfun_pop}
We use the total RLF derived from the SFG and AGN populations at 3 GHz VLA-COSMOS \citep{novak17, novak18, smolcic17c} to compare to the RLF values derived for the group galaxies in COSMOS. The RLF of the SFG and AGN populations are calculated in the same way and for the same redshift bins as described above, and for the same area coverage as the galaxy groups in COSMOS. To fit the RLF two models are used in literature \citep[e.g.,][]{condon84, sadler02, gruppioni13}, the pure luminosity evolution (PLE) and the pure density evolution (PDE). The RLF is fitted under the assumption that its shape remains unchanged at all observed cosmic times. Only the position of the turnover and normalisation can change with redshift. This corresponds to the translation of the local LF in the $log{L} - log{\Phi}$ plane \citep{condon84} and can be divided into pure luminosity evolution (horizontal shift) and pure density evolution (vertical shift).
To describe an RLF across cosmic time, the local RLF is evolved in luminosity, or density, or both \citep[e.g.][]{condon84}. This is parametrised \citep{novak18} using two free parameters for density evolution ($\alpha_{D}$, $\beta_{D}$), and two for luminosity evolution ($\alpha_{L}$, $\beta_{L}$) to obtain
\begin{equation}
\Phi(L,z, \alpha_L, \beta_L, \alpha_D, \beta_D) = (1+z)^{\alpha_D+z\cdot\beta_D}\times\Phi_{0} \left( \frac{L}{(1+z)^{\alpha_L+z\cdot\beta_L}}
\right),
\label{eq:lfevol_model}
\end{equation}
\noindent
where $\Phi_{0}$ is the local RLF. Since the shape and evolution of the RLF depend on the galaxy population type, \cite{novak17} used a power-law plus log-normal shape of the local RLF for SFGs. They used the combined data from \cite{condon02}, \cite{best05} and \cite{mauch07} to obtain the best fit for the local value
\begin{equation}
\Phi_0^{\text{SF}}(L)=\Phi_\star\left(\frac{L}{L_\star}\right)^{1-\alpha} \exp\left[-\frac{1}{2\sigma^2}\log^2\left(1+\frac{L}{L_\star}\right)\right],
\label{eq:lflocal_sf}
\end{equation}
\noindent
where $\Phi_\star=3.55\times10^{-3}~\text{Mpc}^{-3}\text{dex}^{-1}$, $L_\star=1.85\times10^{21}~\ensuremath{\text{W}\,\text{Hz}^{-1}}$, $\alpha=1.22$, and $\sigma=0.63$.
It was noted by \cite{novak17} that the PDE of SF galaxies would push the densities to very high numbers, thus making them inconsistent with the observed cosmic star formation rate densities. This is a consequence of the fact that our data can constrain only the bright log-normal part of the SF RLF. For AGN it was shown by \cite{smolcic17c} that the PDE and PLE models are similar, mostly because the shape of the RLF does not deviate strongly from a simple power law at the observed luminosities. Considering the above reasoning while also trying to keep the parameter space degeneracy to a minimum, we decided to use only the PLE for our analysis. Thus we adopt the approach of \cite{novak18} who fitted the total RLF for SFG and AGN populations by constructing a four--parameter redshift-dependent pure luminosity evolution model with two parameters for the SFG and AGN populations of the form:
\begin{equation}
\begin{split}
\Phi(&L,z,\alpha_L^{\text{SF}}, \beta_L^{\text{SF}}, \alpha_L^{\text{AGN}}, \beta_L^{\text{AGN}}) =\\ &=\quad\Phi_{0}^{\text{SF}} \left( \frac{L}{(1+z)^{\alpha_L^{\text{SF}}+z\cdot\beta_L^{\text{SF}}}}\right)+
\Phi_{0}^{\text{AGN}} \left( \frac{L}{(1+z)^{\alpha_L^{\text{AGN}}+z\cdot\beta_L^{\text{AGN}}}}\right),
\label{eq:lf_total}
\end{split}
\end{equation}
\noindent
where $\Phi_{0}^{\text{SF}}$ is the local RLF for SFGs as in Eq.~\ref{eq:lflocal_sf}, and for the non-local Universe is a function of the quantity in the parenthesis, $L / (1+z)^{\alpha_L^{\text{SF}}+z\cdot\beta_L^{\text{SF}}}$. $\Phi_{0}^{\text{AGN}}$ is the local RLF for AGN of the form
\begin{equation}
\Phi_0^{\text{AGN}}(L)=\frac{\Phi_\star}{(L_\star / L)^{\alpha} + (L_\star / L)^{\beta}},
\label{eq:lflocal_agn}
\end{equation}
\noindent
where $\Phi_\star=\frac{1}{0.4}10^{-5.5}~\text{Mpc}^{-3}\text{dex}^{-1}$, $L_\star=10^{24.59}~\ensuremath{\text{W}\,\text{Hz}^{-1}}$, $\alpha=-1.27$, and $\beta=-0.49$ \citep{smolcic17c, mauch07}, and for the non-local Universe is a function of the quantity $L / (1+z)^{\alpha_L^{\text{AGN}}+z\cdot\beta_L^{\text{AGN}}}$.
\cite{novak18} used the Markov chain Monte Carlo (MCMC) algorithm, available in the Python package \textsc{emcee} \citep{foreman13}, to perform a multi-variate fit to the data. The redshift dependence of the total evolution parameter $\alpha+z\cdot\beta$ (see Eq.~\ref{eq:lf_total}) is necessary to describe the observations at all redshifts. The best fit values, based on the results of \cite{novak18}, for SFGs are $\alpha_L^{\text{SF}}=3.16$ and $\beta_L^{\text{SF}}=-0.32$, and for AGN are $\alpha_L^{\text{AGN}}=2.88$ and $\beta_L^{\text{AGN}}=-0.84$. The $\alpha_{L}$ and $\beta_{L}$ values for both SFGs and AGN are valid for $z <$ 5.5 and within the redshift range of our sample of group galaxies. We use these values to plot the fit to the RLF for SFGs and AGN in Fig.~\ref{fig:lfgrid}, shown in blue and red lines, respectively. The total RLF (including all SFG and AGN), is shown as a dashed black line in Fig.~\ref{fig:lfgrid}.
\subsection{Fitting the RLF of group galaxies and comparing to the total 3 GHz RLF}
\label{sec:lumfun_linfit}
We fit a linear regression to the radio luminosity function of the group galaxies, and separately of the BGGs and SGs, from redshifts 0.07 to 2.3 of the form
\begin{equation}
y = \Phi_\star / (L_\star^\gamma) * L^\gamma,
\label{eq:plaw}
\end{equation}
\noindent
where $L_{\star}$ is arbitrarily chosen to be $10^{24} ~W ~Hz^{-1}$, and $\Phi_{\star}$ is the density at $L_{\star}$. The results for each fit are given in Table~\ref{tab:lf_fit_par}. The best fit model for the GGs is shown in Fig.~\ref{fig:lfgrid}, where we also compare the radio luminosity function of group galaxies to the total radio luminosity function of radio galaxies in the 3GHz VLA-COSMOS survey, and the radio luminosity function of AGN and the star forming population. We also present the linear fit for BGGs, but exclude the one for SGs for clarity. It is evident that the radio emission due to the star formation over-weighs that from the AGN in lower redshifts, i.e at $z<1$, except at high radio luminosity bins where the AGN contribution is dominating. This behaviour has been described in \cite{novak17, smolcic17a} and agrees with other surveys.
To quantify the contribution of the RLF of group galaxies to the total RLF of the 3 GHz population we divide the power-law of GGs, assuming a fixed slope of $\gamma=-0.75$, with the total RLF from the 3 GHz sample for each redshift bin. This gives the fractional contribution of group galaxies to the total RLF. In the top panel of Fig.~\ref{fig:frac_lum} we plot the fraction with respect to the radio luminosity at 1.4 GHz up to $10^{25} ~W ~Hz^{-1}$. The reason for that is the RLF of 3 GHz VLA-COSMOS observations is not well constrained above that luminosity \citep{novak18}. Furthermore, we should consider that our model might be a good representation of the universe at $L > 10^{25} ~W ~Hz^{-1}$, and the COSMOS is not suited for low-$z$ studies due to the small volume coverage at low redshifts. Because of the different total RLF shapes per redshift bin there is a bump in the curve, as expected. The use of a fixed slope of $\gamma=-0.75$ does not impact our calculations as the value is within the errors for the fitted $\gamma$ values presented in Table~\ref{tab:lf_fit_par}.
We also calculate the RLF of all massive galaxies with $M_{*} > 10^{11.2} M_{\odot}$, which are 3 GHz sources, and compare it to the total RLF. This fraction is shown in the bottom panel of Figure Fig.~\ref{fig:frac_lum}. We immediately see that X-ray groups contribute even more to radio activity at $z < 1$ than massive galaxies in the field do. We discuss this further in the next section.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{lf_evol_param_norm.pdf}
\includegraphics[width=\linewidth]{lf_radio_fraction_param.pdf}
\caption{Top: Radio luminosity function vs redshift for the group galaxies (black circles), brightest group galaxies BGGs (red squares), and satellite galaxies SGs (yellow stars). The low-$z$ redshift bin is split in high ($log_{10}(M_{\rm 200c} / M_{\odot}) > 13.3$) and low ($log_{10}(M_{\rm 200c} / M_{\odot}) < 13.3$) mass halos. Cyan lines show the $\Phi_{*}$ values at log$_{10}(L_{1.4\rm GHz} / W~Hz^{-1}) = 24$ for each redshift bin; these values were placed a dex lower for presentation reasons. Bottom: The relative contribution of GGs to the total radio luminosity, as described in Sec.~\ref{sec:lumfun_linfit}, vs redshift. The values plotted here are the ones reported in Table~\ref{tab:lf_fit_par} for $\gamma = -0.75$. Black circles denote GGs, red squares denote BGGs, and yellow stars denote SGs.}
\label{fig:rlf_evol}
\end{figure}
\section{Evolution of the RLF in galaxy groups}
\label{sec:evolution}
The top panel of Fig.~\ref{fig:frac_lum} shows that the contribution of GGs to the total RLF increases from $z \sim$ 2 to the present day, and in particular for objects above radio luminosities at 10$^{23}\rm W~Hz^{-1}$. This picture suggests an evolutionary scenario for the RLF of galaxy groups. We investigate this further by plotting in Fig.~\ref{fig:rlf_evol} the RLF of GGs, BGGs, and SGs, and their relative contribution to the total 3 GHz RLF as a function of redshift. We see an evolution in the RLF of group galaxies from $z \sim$ 2 to the present day by a factor of 3. This is an interesting trend, which is not observed in the total RLF. As seen by the normalisation value of the total $\Phi_{*}$ for a fixed radio luminosity at $10^{24} \rm W~Hz^{-1}$, shown with cyan, galaxies in the 3 GHz sample display a decrease in their RLF with redshift, while the RLF of GGs below $z<1$ is fairly constant (top panel of Fig.~\ref{fig:rlf_evol}). Interestingly, \cite{smolcic17c} show that a similar trend can be reproduced with galaxies. In their Fig. 1 they present a slight increase in the median values of $M_{*}$ in their radio excess sample above redshift $z=1$, and a depletion of massive galaxies above $z>1$, which we also see in the X-ray groups. At the bottom panel of Fig.~\ref{fig:frac_lum} we see that massive galaxies above $10^{11.2} M_{\odot}$ contribute a large fraction to the total 3 GHz RLF below $z<1$, contrary to the decreasing RLF in the field. This suggests two things: 1) not all massive galaxies are in groups, but those who are remain radio active (Fig.~\ref{fig:frac_lum}); and 2) the increase in the group contribution is because radio activity in groups is nearly constant at $z<1$, while it is declining in the field (Fig.~\ref{fig:rlf_evol}--Top).
The RLF of SGs dominates the RLF of group galaxies up to redshift of $z_{\rm med} \sim$ 1.2, with an overdensity just below $z=1$ (Fig.~\ref{fig:rlf_evol}--Bottom). \cite{scoville13} studied the large-scale structure (LSS) in COSMOS and also report a statistically significant overdensity at $z =$ 0.93. Above $z \sim$ 2 we don't have a large-enough number of SGs to perform a robust analysis. This is linked to the resolution and sensitivity of the Chandra observations, which results in detecting a small number of group galaxies and satellites in the groups above $z \sim$ 1.6 (Fig.~\ref{fig:z_gg}). The SGs contribute up to $\sim$80\% more than the BGGs to the GG RLF below $z_{\rm med} \sim$ 1, as is shown on the bottom panel of Fig.~\ref{fig:rlf_evol}. This highlights how important the identification of SGs within galaxy groups is, and the need for high resolution and sensitivity observations.
The relative contributions of the SGs to the RLF of group galaxies is higher by a factor of 3 than that of BGGs up to $z \sim$ 1. Additionally, BGGs contribute a small amount to the RLF of GGs, as seen by the bottom panel of Fig.~\ref{fig:rlf_evol}, despite being the most massive galaxies of the group. This is a very interesting result highlighting the importance of identifying the member group galaxies within a group and the need for high sensitivity and high resolution observations.
Splitting the low redshift bin in Fig.~\ref{fig:rlf_evol} in low and high halo mass objects allows us to further identify the contributions to the RLF of galaxy groups. Those with halo masses above $10^{13.3}~M_{\odot}$ contribute more to the RLF of GGs than those below, and this contribution is linked to SGs are we mentioned above. The low halo mass points (Table~\ref{tab:lf_fit_par} \& Fig.~\ref{fig:lfgrid}) show a faster turnover as we do not expect to detect many low mass, high luminosity objects.
\cite{yuan16}, who studied brightest cluster galaxies (BCGs), found that RLFs of 7138 BCGs in the range 0.05 $< z <$ 0.45 do not show significant evolution with redshift. This no evolution pattern of BCGs agrees with our results for BGGs in COSMOS. At the bottom panel of Fig.~\ref{fig:rlf_evol} we see that the RLF of BGGs fluctuates slightly with redshift but it is the RLF of satellites that drives the redshift evolution.
\cite{novak18} discuss possible biases which could affect the calculations. These include the assumed shape of the radio SED to be a power law, and the radio excess criterion to be too conservative and thus excluding low-luminosity AGN from the sample. We refer the reader to their discussion (see their Section 3.4). Given that \cite{novak18} were able to reproduce the radio sky brightness with the 3 GHz VLA-COSMOS data, we are confident our results, at least for $L_{\rm 1.4~GHz} < 10^{25}~\rm W~Hz^{-1}~sr^{-1}$, are robust. Furthermore, \cite{novak18} discuss possible biases that affect the RLF of the high luminosity bin, i.e. bright radio but faint in the near-infrared sources ($K$ = 24.5 mag).
We have constrained our sample to halo masses above $10^{13.3} M_{\odot}$, to perform an unbiased analysis. Incidentally, after the halo-mass cut, the remaining group galaxies in our sample are brighter than $K$ = 24.5 mag.
\section{The AGN and SFG contribution to the GG RLF}
\label{sec:agn}
The group galaxy population has a mixture of contributions from AGN and SFGs. To explore how much these population contribute to the GG RLF, we cross-correlate the X-ray galaxy group catalogue with the sample of \cite{vardoulaki21}, which is a value-added catalogue at 3 GHz VLA-COSMOS, which includes 130 FR-type \citep{fr74} and 1818 jet-less compact radio AGN (COM AGN), as well as 7232 SFGs (see Table~\ref{tab:fr_nums}). Radio AGN in the \cite{smolcic17b} sample were selected on the basis of their radio excess, as mentioned above. This criterion, due to the 3$\sigma$ cut applied, excludes several FR-type radio AGN, which were identified in \cite{vardoulaki21} and classified as radio AGN because they exhibit jets/lobes. SFGs are objects which do not display radio excess.
\begin{table}
\label{tab:fr_nums}
\begin{center}
\caption{The AGN and SFGs inside X-ray galaxy groups. Data from \citep{vardoulaki21}, cross-correlated with the X-ray galaxy group catalogue \citep{gozaliasl19}.}
\renewcommand{\arraystretch}{1.5}
\ifonecol \scriptsize \fi
\begin{tabular}[t]{l c c c c c}
\hline\hline
\multicolumn{1}{l}{Number of sources}& \multicolumn{1}{c}{AGN} & \multicolumn{1}{c}{SFGs} \\
\hline\hline
total 3 GHz VLA-COSMOS & 1948 & 7232 \\
same area \& $z$ as X-ray groups & 1038 & 6452 \\
X-ray group members & 131 & 242\\
BGGs & 66 & 40 \\
SGs & 65 & 202\\
\hline
\end{tabular}
\end{center}
\end{table}
To quantify the contribution of these population separately to the group RLF and to the total RLF we calculate their RLF as described in Sec.~\ref{sec:lumfun}, using the $V/V_{\rm max}$ method. All AGN and SFGs are in groups with halo masses $M_{200c} > 10^{13.3} M_{\odot}$. For the purposes of this analysis, we use the 1.4 GHz luminosities \citep{schinnerer10}. The results for the AGN and SFG populations inside galaxy groups are shown in Fig.~\ref{fig:rlf_agn}, where we also plot the RLF of AGN and SFGs from the sample of \cite{novak18} as in Fig.~\ref{fig:lfgrid} and the total RLF at 3 GHz. In order to compare the RLF of AGN and SFGs which are GGs and total RLF, we follow the analysis in Sec.~\ref{sec:lumfun_linfit}. We fit a linear regression and normalise it to $10^{24}\rm ~W ~Hz^{-1}$ by applying Eq.~\ref{eq:plaw} for $\gamma$ = -0.75. The results are shown in Fig.~\ref{fig:rlf_agn}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Phi_L_grid_AGN_sep21.pdf}
\includegraphics[width=\linewidth]{Phi_L_grid_sfg_sep21.pdf}
\caption{Total radio luminosity functions of galaxies in groups, as in Fig.~\ref{fig:lfgrid}, including RLFs for different populations: radio AGN inside galaxy groups as magenta hexagons (top) and SFGs inside galaxy groups as green stars (bottom). To compare to the GG sample, we normalise the fit to the AGN and SFGs inside groups to $L_{1.4 \rm GHz} = 10^{24}~W~Hz^{-1}$ and slope of $\gamma$ = -0.75. For comparison we show the GG sample (black circles for data, and black solid line with a slope $\gamma$ = -0.75 for the fit). The red solid line shows the RLF for all AGN, the blue solid line for all SFGs, and the dotted black line is the total RLF.}
\label{fig:rlf_agn}
\end{figure}
We further calculate and plot the fractional contribution of AGN and SFGs which lie inside groups to the total RLF at 3 GHz (Fig.~\ref{fig:fraction_agn}), by replicating Fig.~\ref{fig:frac_lum}. The fraction was calculated by dividing the RLF of AGN and of SFGs inside galaxy groups to the total 3 GHz RLF. The fractions per redshift bin are curved lines due to the total RLF being curved. We find that there is a significant contribution from group AGN and SFGs at redshifts $z <$ 1, and very little contribution above. We present the values for these fractions in Table~\ref{tab:agn_fraction}.
Additionally, we explore the fractional contribution of the AGN and SFGs inside GGs to the RLF of GGs, shown in Fig.~\ref{fig:fraction_agn} as dashed-dotted lines. The fraction was calculated by dividing the RLF of AGN and SFGs inside galaxy groups to the GG RLF. We get a constant value across all luminosities in Fig.~\ref{fig:fraction_agn} because the divided fits are both linear. The contribution of AGN RLF to the GG RLF is significant at the redshift bin $z_{\rm med}$ = 0.6 of around 45\% and at $z_{\rm med}$ = 0.8 with fraction around 30\%. The fraction in SFGs is around 20\% for $z_{\rm med}$ = 0.6 and $z_{\rm med}$ = 0.8, while at $z_{\rm med}$ = 0.3 it dominates the GG RLF. The excess we see in the fraction of SFGs at $z_{\rm med}$ = 0.3 can be explained by the linear fit being normalised to $10^{24} ~W ~Hz^{-1}$ and forced to have a slope of $\gamma$ = -0.75. For $z >$ 1 the contribution of AGN and SFGs to the GG RLF drops sharply and below 10\%. These findings suggest AGN inside groups significantly contribute to the RLF at redshifts between 0.4 $< z < $ 1.0, and SFGs contribute significantly at 0.07 $< z < $ 0.4.
There are 66 AGN associated with BBGs and 65 with SGs, as shown in Table~\ref{tab:fr_nums}. For SFGs we get 40 BGGs and 202 SGs. Due to the small numbers of sources per bin, we cannot replicate Fig.~\ref{fig:rlf_evol} by splitting the AGN and SFGs RLF inside groups in BGGs and SGs and calculating their RLF. Fig.~\ref{fig:rlf_evol} suggests the evolution of the GG RLF is driven by satellites. Based on our results on from Fig.~\ref{fig:fraction_agn}, at the $z_{\rm med}$ = 0.3 redshift bin, the SG SFGs are responsible for the peak of the GG RLF, while at $z_{\rm med}$ = 0.8 the increase is mainly driven by AGN.
How much of the AGN contribution to the GG RLF comes from extended radio emission, given the capabilities of the 3 GHz VLA-COSMOS survey, is not easy to estimate due to sample size limitations. From Table~\ref{tab:fr_nums} we see that $\sim$ 82\% of AGN inside galaxy groups are jet-less AGN. But in order to robustly answer this question we need to separate FRs and COM AGN inside groups and calculate their RLFs per redshift bin, as above, which we cannot do given the small number of FRs per redshift bin.
To get an idea of how extended the FRs within the AGN sample are, we have a look at the linear projected sizes $D$ of FRs in \cite{vardoulaki21}. The sensitivity and resolution of the 3 GHz VLA-COSMOS survey are 2.3 $\mu$Jy/beam and 0".75, respectively. This means that we are able to resolve and disentangle structures of $\sim$ 6 kpc at $z \sim$ 2. The smallest FR reported in \cite{vardoulaki21} has $D$ = 8.1 kpc at $z = $ 2.467, just above the resolution limit, and the smallest edge-brightened FR has $D$ = 24.3 kpc at $z =$ 1.128, where the lobes are separated by 8 kpc. Inside X-ray galaxy groups, the smallest FR has $D$ = 13.37 kpc at $z$ = 0.38 with the most extended having $D$ = 608.4 kpc and $z$ = 1.168; this is also the most extended object in the \cite{vardoulaki21} FR sample. Future surveys with increased sensitivity and resolution will be able to resolve jets and lobes in AGN which appear compact at 3 GHz VLA-COSMOS. With future observations at larger sky area and improved statistics we will be in a better position to answer this question.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fraction_AGN_GGs_sep21.pdf}
\includegraphics[width=\linewidth]{fraction_sfg_GGs_sep21.pdf}
\caption{Fractional contribution to the total radio luminosity function at different epochs vs radio luminosity at 1.4 GHz for GGs (solid lines as in Fig.~\ref{fig:frac_lum}), and for different populations (dotted lines): AGN (top; labelled AGN v tot) and SFGs (bottom; labelled SFG v tot). Dotted-dashed lines show the fractional contribution of AGN (labelled AGN v GG) and SFG (labelled SFG v GG) to the GG RLF. Different colours represent different redshift bins as in Fig.~\ref{fig:frac_lum}}
\label{fig:fraction_agn}
\end{figure}
\begin{table*}
\begin{center}
\caption{Fractional contribution $f$ of AGN and SFGs RLFs, inside groups, to the GG RLF and to the total RLF at $L_{\rm 1.4~GHz}$ = 10$^{23}$ and $L_{\rm 1.4~GHz}$ = 10$^{25}\rm ~W~Hz^{-1}$, as in Fig.~\ref{fig:fraction_agn}.}
\label{tab:agn_fraction}
\renewcommand{\arraystretch}{1.5}
\ifonecol \scriptsize \fi
\begin{tabular}[t]{l | c c c | c c c }
\hline\hline
\multicolumn{1}{l}{$z_{\rm med}$}& \multicolumn{1}{l}{$f_{\rm AGN-GG}$} & \multicolumn{1}{l}{$f_{\rm AGN-tot23}$} & \multicolumn{1}{l}{$f_{\rm AGN-tot25}$} & \multicolumn{1}{l}{$f_{\rm SFG-GG}$} & \multicolumn{1}{l}{$f_{\rm SFG-tot23}$} & \multicolumn{1}{l}{$f_{\rm SFG-tot25}$}\\
\hline\hline
0.3 & 0.85 & 0.11 & 0.29 & 1.70 & 0.23 & 0.59\\
0.6 & 0.44 & 0.01 & 0.04 & 0.19 & 0.007 & 0.02\\
0.8 & 0.30 & 0.01 & 0.04 & 0.21 & 0.01 & 0.03\\
1.2 & 0.02 & 1$\times10^{-4}$ & 4$\times10^{-4}$ & 0.06 & 2$\times10^{-4}$ & 9$\times10^{-4}$\\
1.9 & 6$\times10^{-6}$ & 1$\times10^{-7}$ & 1$\times10^{-6}$ & 0.01 & 3$\times10^{-5}$ & 2$\times10^{-4}$\\
\hline
\end{tabular}
\end{center}
\end{table*}
\section{Summary and Conclusions}
\label{sec:summary}
We presented a study of radio luminosity functions RLFs of group galaxies in the COSMOS field, based on data from the VLA-COSMOS 3~GHz Large Project \citep{smolcic17a} and the X-ray galaxy groups catalogue \citep{gozaliasl19}.
The X-ray galaxy groups cover halo masses in the range $M_{\rm 200c} = 8 \times 10^{12} - 3 \times 10^{14} M_{\odot}$ and the redshift range 0.07 $< z <$ 2.3. To probe the same group population at all redshifts, we apply a halo-mass cut and only select groups with halo masses $M_{\rm 200c} > 10^{13.3} M_{\odot}$. Furthermore, we applied completeness corrections to the calculation of the RLF \citep{novak17} and all galaxy-group members are brighter than $K$ = 24.5 mag, which allows for an unbiased analysis.
We calculated the RLF of group galaxies based on the $V/V_{\rm max}$ method and compared it to the 3 GHz RLF from \cite{novak18} who fitted the total RLF with pure luminosity evolution models that depend on redshift. The AGN and SFG populations, characterised by the radio excess parameter, were fitted with a Markov chain Monte Carlo algorithm. We fitted the group galaxies' RLFs with linear and power-law model and estimate their contribution to the total RLF. We also studied how much satellites, brightest group galaxies, AGN, and SFGs contribute to the RLF of galaxy groups and to the total 3 GHz RLF.
Our main results are summarised below:
\begin{enumerate}
\item The relative contribution of the group galaxies to the total 3 GHz radio luminosity function in galaxies in the COSMOS field generally decreases with increasing redshift, from 10\% locally, to 1\% at $z >$ 1, with an overdensity just below $z <$ 1, in line with large-scale structure studies of the COSMOS field.
\item We see an evolution in the RLF of GGs from $z \sim$ 2 to the present day by a factor of 3, which is driven mainly by satellite GGs.
\item The RLF of SGs dominates the RLF of group galaxies up to redshift of $z \sim$ 1.2, where we observe a drop in the RLF of both BGGs and SGs almost by a factor of 10.
\item At $z_{\rm med}$ = 0.3, galaxies with halo masses above $10^{13.3}~M_{\odot}$ contribute more to the RLF of GGs than those below, and this contribution is linked to SGs.
\item The SG SFGs are responsible for the peak of the GG RLF at $z_{\rm med}$ = 0.3, while at $z_{\rm med}$ = 0.8 the increase is mainly driven by AGN.
\item The increase in the group contribution is because radio activity in groups is nearly constant at $z<1$, while it is declining in the field. This is not because massive galaxies all get to be in groups, it is because only those in groups remain active.
\end{enumerate}
We showed there is a switch in radio activity at $z=1$ observed in group environment. It is the galaxies inside groups which remain radio active, compared to the ones in the field. This trend can be reproduced with galaxies as shown by \cite{Smolcic2009,smolcic17c}. Despite the quirk of the COSMOS field with the known overdensity at $z$ = 0.9, we used a halo-mass selected sample up to $z = 2.3$ and observed the redshift evolution of the RLF of group galaxies, mainly driven by their satellites. Our a study provides an observational probe for the accuracy of the numerical predictions of the radio emission in galaxies in a group environment.
\begin{acknowledgements}
We would like to thank Aritra Basu for useful discussions. EV acknowledges support by the Carl Zeiss Stiftung with the project code KODAR.
\end{acknowledgements}
| 13,307 |
\section{Introduction}
Let $l$ be a prime number, and let $F$ be a number field. Let
$\sch{G}$ be a reductive algebraic group over $F$, and let $\sigma$ be
an automorphism of order $l$ of $\sch{G}$. D.Treumann and A.Venkatesh
constructed a functorial lift of a mod-$l$ automorphic form for
$\sch{G}^\sigma$ to a mod-$l$ automorphic form for $\sch{G}$ (see
\cite{MR3432583}). They conjectured that the mod-$l$ local functoriality at
ramified places must be realised in Tate cohomology and they defined
the notion of linkage (see \cite[Section 6.3]{MR3432583} for more
details). Among many applications of this set up, we focus on mod-$l$
base change lift from $\sch{G}^\sigma=\sch{{\rm GL}}_n/F$ to
$\sch{G}={\rm Res}_{E/F}\sch{\rm GL}_n/E$, where $E/F$ is a Galois
extension with $[E:F]=l$. Truemann and Venkatesh's conjecture on
linkage in Tate cohomology is verified for local base change of
depth-zero cuspidal representations by N.Ronchetti, and a precise
conjecture in the context of base change of $l$-adic higher depth
cuspidal representations was formulated in \cite[Conjecture 2,
Pg. 266]{MR3551160}. In this article, using Whittaker models and
Rankin-Selberg zeta functions, we prove the conjecture for
${\rm GL}_2$ and ${\rm GL}_3$.
Let us introduce some notations for stating the precise results of
this article. From now, we assume that $F$ is a finite
extension of $\mathbb{Q}_p$ and let $E$ be a finite Galois extension of prime
degree over $F$ with $[E:F]=l$. Let $\pi_F$ be a cuspidal irreducible
smooth $l$-adic representation of ${\rm GL}_n(F)$, and let $\pi_E$ be
a cuspidal smooth $l$-adic representation of ${\rm GL}_n(E)$ such
that $\pi_E$ is the base change of $\pi_F$. Note that there exists an
isomorphism $T:\pi_E\rightarrow \pi_E^\gamma$, where $\pi_E^\gamma$ is
the twist of $\pi_E$ by an element $\gamma\in {\rm Gal}(E/F)$. Let
$\mathcal{L}$ be a ${\rm GL}_n(E)$-invariant lattice in $\pi_E$. We
may (and do) assume that $\mathcal{L}$ is stable under $T$. Then the
semisimplification of the Tate cohomology group
$\widehat{H}^i({\rm Gal}(E/F), \mathcal{L})$, as a ${\rm GL}_n(F)$
representation, is independent of the choice of $\mathcal{L}$ and $T$
\cite[Lemma 5]{MR3551160}. The semisimplification of the Tate
cohomology group $\widehat{H}^0({\rm Gal}(E/F), \mathcal{L})$ is
denoted by $\widehat{H}^0(\pi_E)$. Let $k$ be an algebraically closed
field of characteristic $l$. If $(\tau, V)$ is a representation of a
group $H$, on a $k$-vector space $V$, then $\tau^{(l)}$ is the
Frobenius twist of $\tau$ defined as the representation of $H$ on
$V\otimes_{\rm Frob} k$. If $\pi$ is an $l$-adic representation of a
locally profinite group $H$, with a finitely generated $H$-invariant
lattice $\mathcal{L}$ then $r_l(\pi)$ denotes the semisimplification
of $\mathcal{L}\otimes \overline{\mathbb{F}}_l$. We prove the
following results:
\begin{theorem}\label{intro_n=2_thm}
Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a
finite Galois extension of $F$ with $[E:F] =l$. Assume that $l$ and
$p$ are distinct odd primes. Let $\pi_F$ be an integral $l$-adic
cuspidal representations of ${\rm GL}_2(F)$ and let $\pi_E$ be the
representation of ${\rm GL}_2(E)$ such that $\pi_E$ is the base
change of $\pi_F$. Then
\begin{center}
$\widehat{H}^0(\pi_E) \simeq r_l(\pi_F)^{(l)}.$
\end{center}
\end{theorem}
Note that $\pi_E$ in the above theorem is an integral $l$-adic
cuspidal representation of ${\rm GL}_2(E)$. Using the above result we
show the following theorem.
\begin{theorem}\label{intro_n=3_thm}
Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a
finite Galois extension of $F$ with $[E:F] =l$, where $p$ and $l$ are
distinct primes with $p,l \not =3$, and $l$ is banal for
${\rm GL}_2(F)$. Let $\pi_F$ be an integral $l$-adic cuspidal
representations of ${\rm GL}_3(F)$ and let $\pi_E$ be the base
change of $\pi_F$. Then
\begin{center}
$\widehat{H}^0(\pi_E) \simeq r_l(\pi_F)^{(l)}.$
\end{center}
\end{theorem}
We note some immediate remarks on the hypothesis in Theorem
\ref{intro_n=3_thm}. Similar to the case where $n=2$, the
representation $\pi_E$ is a cuspidal integral $l$-adic representation
of ${\rm GL}_3(E)$. The hypothesis that $l$ is banal for
${\rm GL}_2(F)$ is required in the proof of a vanishing result of
Rankin--Selberg integrals on ${\rm GL}_2(F)$ (the analogue of
\cite[Lemma 3.5]{MR620708} or \cite[6.2.1]{MR1981032}). This banality
condition on $l$-can be removed by using $\gamma$-factors defined over
local Artinian $k$-algebras defined in the work of G.Moss and
N.Matringe in \cite{matringe2022kirillov}. However, the right notion
of Base change over local Artinian $k$-algebras is not clear to the
authors and hence, we use the mild hypothesis that $l$ is banal for
${\rm GL}_2(F)$.
When $F$ is a local function field, the above theorems follow from the
work of T.Feng \cite{feng2020equivariant}. T.Feng uses the
constructions of Lafforgue and Genestier-Lafforgue
\cite{genestier2017chtoucas}. N.Ronchetti also proved the above
results for depth-zero cuspidal representations using compact
induction model. Our methods are very different from the work of
N.Ronchetti and the work of T.Feng. We rely on Rankin--Selberg
integrals and lattices in Whittaker models. We do not require the
explicit construction of cuspidal representations. We use the $l$-adic
local Langlands correspondence and various properties of local
$\epsilon$ and $\gamma$-factors both in $l$-adic and mod-$l$
situations associated to the representations of the $p$-adic group and
the Weil group. The machinery of local $\epsilon$ and $\gamma$-factors
of both $l$-adic and mod-$l$ representations of ${\rm GL}_n(F)$ is
made available by seminal work of D.Helm, G.Moss, N.Matringe and
R.Kurinczuk (see \cite{MR3867634}, \cite{MR3556429}, \cite{MR4311563},
\cite{MR3595906}).
We sketch the proof of the above Theorem for $n=3$. Theorem
(\ref{intro_n=2_thm}) is proved using Kirillov model and using some
results of Vigneras on the set of integral functions in a Kirillov
model being an invariant lattice. The discussion is this paragraph is
valid for any positive integer $n$. Let $(\pi_F, V)$ be a generic
$l$-adic representation of ${\rm GL}_n(F)$, in particular, $V$ is a
$\overline{\mathbb{Q}}_l$-vector space. Let $N_n(F)$ be the group of
unipotent upper triangular matrices in ${\rm GL}_n(F)$. Let
$\Theta:N_n(F)\rightarrow \overline{\mathbb{Q}}_l^\times$ be a
non-degenerate character and we let $W(\pi_F, \Theta)$ to be the
Whittaker model of $\pi_F$. Similar notations for $\pi_E$ are
followed. Let $\pi_E$ be the base change of $\pi_F$. It is easy to
note that (Lemma \ref{inv_whit_model}) $W(\pi_E, \Theta)$ is stable
under the action of ${\rm Gal}(E/F)$ on the space
${\rm Ind}_{N_n(E)}^{{\rm GL}_n(E)}\Theta$. Let $\pi_F$ be an integral
generic $l$-adic representation of ${\rm GL}_n(F)$, and let
$W^0(\pi_F, \Theta)$ be the set of $\overline{\mathbb{Z}}_l$-valued
functions in $W(\pi_F, \Theta)$. It follows from the work of Vigneras
that the subset $W^0(\pi_F, \Theta)$ is a ${\rm GL}_n(F)$ invariant
lattice. Let $K(\pi_F, \Theta)$ be the Kirillov model of $\pi_F$, and
let $K^0(\pi_F, \Theta)$ be the set of
$\overline{\mathbb{Z}}_l$-valued functions in $K(\pi_F,
\Theta)$. Using the result \cite[Corollary 4.3]{matringe2022kirillov}
we get that the restriction map from $W^0(\pi_F, \Theta)$ to
$K^0(\pi_F, \Theta)$ is a bijection. Let $\pi_F$ and $\pi_E$ be
cuspidal representations. It follows from the work of
Treumann--Venkatesh that
$\widehat{H}({\rm Gal}(E/F), K^0(\pi_E, \Theta))$ is equal to
$K(r_l(\pi_F)^{(l)}, \Theta^l)$, as representations of $P_n(F)$, where
$P_n(F)$ is the mirabolic subgroup of ${\rm GL}_n(F)$. Now, to prove
the main theorem it is enough to check the compatibility of the action
of $\pi_E(w)$ on the space
$\widehat{H}^0({\rm Gal}(E/F), K^0(\pi_E, \Theta))$ with the action of
$r_l(\pi_F)^{(l)}$ on $K(r_l(\pi_F)^{(l)}, \Theta^l)$. Such a
verification, as is well known, involves arithmetic properties of the
representations.
We now assume the hypothesis of theorem (\ref{intro_n=3_thm}). To complete
the proof we show that it is enough to prove that
\begin{equation}\label{intro_eq_rk}
I(X, r_l(\pi_E)(w)W,
\sigma(w)W')=I(X, r_l(\pi_F)^{(l)}(w)W, \sigma(w)W'),
\end{equation}
for all
$W\in K^0(\pi_E, \Theta)^{{\rm Gal}(E/F)}\otimes
\overline{\mathbb{F}}_l$ and $W'\in W(\sigma, \Theta^l)$ where
$\sigma$ is an $l$-modular generic representation of ${\rm GL}_2(F)$;
and $I(X, W, W')$ is a mod-$l$ Rankin--Selberg zeta functions written
as a formal power series in the variable $X$ instead of the
traditional $q^s$ \cite[Section 3]{MR3595906}. We transfer the
local Rankin--Selberg zeta functions
$I(X, r_l(\pi_E)(w)W, \sigma(w)W')$ made from integrals on
${\rm GL}_2(F)/N_2(F)$ to Rankin--Selberg zeta functions defined by
integrals on ${\rm GL}_2(E)/N_2(E)$. Then, using local Rankin--Selberg
functional equation, we show that the equality in (\ref{intro_eq_rk})
is equivalent to certain identities of mod-$l$ local
$\gamma$-factors.
We briefly explain the contents of this article. In Section $2$, we
recall various notations, conventions on integral representations,
Whittaker models and Kirillov models. In Section $3$, we collect
various results on local constants both in mod-$l$ and $l$-adic
settings. In Section 4, we put some well known results from $l$-adic
local Langlands correspondence. In Section 5, we recall and set up
some initial results on Tate cohomology on lattices on smooth
representations. In section 6, we prove Theorems (\ref{intro_n=2_thm}) and
(\ref{intro_n=3_thm}).
\section{Notation}\label{x}
\subsection{}\label{16}
Let $K$ be a non-Archimedean local field and let $\mathfrak{o}_K$ be
the ring of integers of $K$. Let $\mathfrak{p}_K$ be the maximal ideal
of $\mathfrak{o}_K$ and let $q_K$ be the cardinality of the residue
field $\mathfrak{o}_K/\mathfrak{p}_K$. Let
$\nu_K: K^\times \rightarrow \mathbb{Z}$ be the normalized
valuation. Let $l$ and $p$ be two distinct odd primes. Let $F$ be a
finite extension of $\mathbb{Q}_p$ and let $E$ be a finite Galois
extension of $F$ with $[E:F] = l$. The prime number $l$ is assumed to be
banal for ${\rm GL}_2(F)$, i.e., the integers $1$, $q_F$ and $q_F^2$ are
distinct modulo $l$. We denote the group ${\rm Gal}(E/F)$ by $\Gamma$.
\subsection{}\label{15}
For any ring $A$, let $M_{r \times s}(A)$ be the $A$-algebra of all
$r \times s$ matrices with entries from $A$. Let
$GL_n(K) \subseteq M_{n \times n}(K)$ be the group of all invertible
$n \times n$ matrices. We denote by $G_n(K)$ the group $GL_n(K)$ and
$G_n$ is equipped with locally compact topology induced from the
local field $K$. For
$r \in \mathbb{Z}$, let
$$ G_n^r(K) = \{g \in G_n(K) : \nu_K(g) = r\}. $$
We set $P_n(K)$, the mirabolic subgroup, defined as the group:
\begin{center}
$\bigg\{
\begin{pmatrix}
A & M\\
0 & 1
\end{pmatrix}
: A \in G_{n-1}(K), M \in M_{(n-1) \times 1}(K)\bigg\}$
\end{center}
Let $B_n(K)$ be the group of all upper triangular matrices and let $N_n(K)$ be
its unipotent radical. We denote by $w_n$, the following matrix of $G_n(K)$ :
\begin{center} $w_n =
\begin{pmatrix}
0 & & & & 1 \\
& & & 1 \\
& & . \\
& . \\
1 & & & & 0
\end{pmatrix}.$
\end{center}
For $r \in \mathbb{Z}$, let $Y_K^r $ denote the coset space
$N_{n-1}(K) \setminus G_{n-1}^r(K)$.
\subsection{}
Fix an algebraic closure $\mathbb{\overline{Q}}_l$ of the field
$\mathbb{Q}_l$. Let $\mathbb{\overline{Z}}_l$ be the integral closure
of $\mathbb{Z}_l$ in $\mathbb{\overline{Q}}_l$ and let
$\mathfrak{P}_l$ be the unique maximal ideal of
$\mathbb{\overline{Z}}_l$. We have
$\mathbb{\overline{Z}}_l/\mathfrak{P}_l \simeq
\mathbb{\overline{F}}_l$. We fix a square root of $q_F$ in
$\overline{\mathbb{Q}}_l$, and it is denoted by $q_F^{1/2}$. The
choice of $q_F^{1/2}$ is required for transferring the complex local
Langlands correspondence to a local $l$-adic Langlands correspondence.
\cite[Chapter 8]{MR2234120}.
\subsection{Smooth representation and Integral representation
(\cite{MR3244922}) :}
Let $G$ be a locally compact and totally disconnected group. A
representation $(\pi,V)$ is said to be {\it smooth} if for every vector $v \in
V$,
the $G$-stabilizer of $v$ is an open subgroup of $G$. All the representations
are assumed to be smooth and the representation spaces are vector spaces
over $R$, where
$R = \mathbb{\overline{Q}}_l$ or $\mathbb{\overline{F}}_l$. A representation
$(\pi,V)$ is called {\it $l$-adic} when $R=\mathbb{\overline{Q}}_l$ and
$(\pi,V)$ is
called {\it $l$-modular} when $R=\mathbb{\overline{F}}_l$. We denote by $\rm
Irr$$(G,R)$, the set of all irreducible $R$-representations of $G$. Let $C_c^\infty(G,R)$ denotes the set of all locally constant and compactly supported functions on $G$ taking values in $R$, where $R = \overline{\mathbb{Q}}_l$ or $\overline{\mathbb{Z}}_l$ or $\overline{\mathbb{F}}_l$.
Let $(\pi,V)$ be an $l$-adic representation of $G$. A {\it lattice} in $V$ is a
free
$\mathbb{\overline{Z}}_l$-module $\mathcal{L}$ such that $\mathcal{L}
\otimes_{\mathbb{\overline{Z}}_l} \mathbb{\overline{Q}}_l = V$. The
representation $(\pi,V)$ is said to {\it integral} if it has finite
length
as a representation of $G$
and there
exist a $G$-invariant lattice $\mathcal{L}$ in $V$. A character is a smooth
one-dimensional representation $\chi : G \longrightarrow R^\times$. We say that
a character $\chi : G \longrightarrow \mathbb{\overline{Q}}_l^\times$ is
integral if it takes values in $\mathbb{\overline{Z}}_l$.
Let $(\pi,V)$ be an integral $l$-adic representation of $G$. Choose a
$G$-invariant lattice $\mathcal{L}$ in $V$. Then the group $G$ acts on
$\mathcal{L}/\mathfrak{P}_l\mathcal{L}$, which is a vector space over
$\mathbb{\overline{F}}_l$. This gives an $l$-modular representation, which
depends on the choice of the $G$-invariant lattice $\mathcal{L}$. By the
Brauer-Nesbitt principal (Theorem 1,\cite{MR2058628}), the semi-simplification
of
$\big(\pi, \mathcal{L}/\mathfrak{P}_l\mathcal{L}\big)$ is independent of the
choice of the $G$-invariant lattice in $V$. We denote the semi-simplification
of the representation $\big(\pi, \mathcal{L}/\mathfrak{P}_l\mathcal{L}\big)$ by
$r_l(\pi)$. The representation $r_l(\pi)$ is called the reduction modulo $l$ of
the $l$-adic representation $\pi$. We say that an $l$-modular representation
$\sigma$ lifts to an integral $l$-adic representation $\pi$ if there exists a
$G$-invariant lattice $\mathcal{L} \subseteq \pi$ such that
$\mathcal{L}/\mathfrak{P}_l\mathcal{L} \simeq \sigma$.
\subsection{Parabolic induction :}\label{ss}
Let $H$ be a closed subgroup of $G$. Let $\rm Ind$$_H^G$ and $\rm ind$$_H^G$ be
the smooth induction functor and compact induction functor respectively. We
follow \cite{MR579172} for the definitions.
Set $G = G_n(K)$, $P = P_n(K)$ and $N = N_n(K)$, where $G_n(K)$, $P_n(K)$ and
$N_n(K)$ are defined in subsection (\ref{15}). Let $\lambda
=(n_1,n_2,....,n_t)$ be an ordered partition of $n$. Let
$Q_\lambda\subseteq G_n(K)$ be the
group of matrices of the form
\begin{center}
$\begin{pmatrix}
A_1 & * & * & * & *\\
& A_2 & * & * & *\\
& & . & * & * \\
& & & . & *\\
& & & & A_t
\end{pmatrix}$,
\end{center}
where $A_i \in G_{n_i}(K)$, for all $1\leq i\leq t$. Then
$Q_\lambda = M_\lambda \ltimes U_\lambda$, where $M_\lambda$ is the
group of block diagonal matrices of the form
\begin{center}
$\begin{pmatrix}
A_1 & & & & \\
& A_2 & & & \\
& & . & & \\
& & & . & \\
& & & & A_t
\end{pmatrix}$, $A_i \in G_{n_i}(K)$,
\end{center}
for all $1\leq i\leq t$
and $U_\lambda$ is the unipotent radical of $Q_\lambda$ consisting of
matrices of the form
\begin{center}
$U_\lambda =
\begin{pmatrix}
I_{n_1} & * & * & * & *\\
& I_{n_2} & * & * & *\\
& & . & * & * \\
& & & . & *\\
& & & & I_{n_t}
\end{pmatrix}$,
\end{center}
where $I_{n_i}$ is the ${n_i} \times {n_i}$ identity matrix.
Let $\sigma$ be an $R$-representation of $M_\lambda$. Then the representation
$\sigma$ is considered as a representation of $Q_\lambda$ by inflation
via the map
$Q_\lambda \rightarrow Q_\lambda/U_\lambda \simeq M_\lambda$. The induced
representation $\rm Ind$$_{Q_\lambda}^G(\sigma)$ is called the parabolic
induction of $\sigma$. We denote the normalized parabolic induction of $\sigma$
corresponding to the partition $\lambda$ by $i_{Q_\lambda}^G(\sigma)$. For
details, see \cite{MR579172}.
\subsubsection{}
Let $\lambda$ be an ordered patition of $n$. Let $\sigma$ be an integral
$l$-adic representation of $M_\lambda$ and let $\mathcal{L}$ be a $G$-invariant
lattice in $\sigma$. Now by (I 9.3, \cite{MR1395151}), the space $\rm
Ind$$_{Q_\lambda}^G(\mathcal{L})$, consisting functions in $\rm
Ind$$_{Q_\lambda}^G(\sigma)$ taking values in $\mathcal{L}$, is a $G$-invariant
lattice in $\rm Ind$$_{Q_\lambda}^G(\sigma)$. Moreover, we have
\begin{center}
$\rm Ind$$_{Q_\lambda}^G(\mathcal{L}
\otimes_{\mathbb{\overline{Z}}_l}\mathbb{\overline{F}}_l)
\simeq \rm Ind$$_{Q_\lambda}^G(\mathcal{L}) \otimes_{\mathbb{\overline{Z}}_l}
\mathbb{\overline{F}}_l$.
\end{center}
Hence parabolic induction commutes with reduction modulo $l$ that is
\begin{center}
$r_l(\rm Ind$$_{Q_\lambda}^G(\sigma)) \simeq [\rm
Ind$$_{Q_\lambda}^G(r_l(\sigma))]$,
\end{center}
where the square bracket denotes the semi-simplification of $\rm
ind$$_{Q_\lambda}^G(r_l(\sigma))$.
\subsection{Cuspidal and Supercuspidal representation :}
Keep the notation as in (\ref{ss}). Let $\pi$ be an irreducible
$R$-representation of $G$. Then $\pi$ is called a cuspidal representation if
for all proper subgroups $Q_\lambda =M_\lambda \ltimes U_\lambda$ of $G$ and
for all irreducible $R$-representations $\sigma$ of $M_\lambda$, we have
\begin{center}
$\rm Hom$$_G(\pi, i_Q^G(\sigma)) = 0$.
\end{center}
The representation $\pi$ is called supercuspidal if for all proper subgroups
$Q_\lambda = M_\lambda \ltimes U_\lambda$ of $G$ and for all irreducible
$R$-representations $\sigma$ of $M_\lambda$, the representation $(\pi,V)$ is
not a subquotient of $i_Q^G(\sigma)$.
\begin{remark}
\rm Let $k$ be an algebraically closed field and let $\pi$ be a
$k$-representation of
$G$. If the characteristic of $k$ is 0 then $\pi$ is cuspidal if and only if
$\pi$ is supercuspidal. But when characteristic of $k$ is $l > 0$, there are
cuspidal representations of $G$ which are not supercuspidal. For details, see
\cite[Section 2.5, Chapter 2]{MR1395151}.
\end{remark}
\subsection{Whittaker Model :}\label{rr}
Let $\psi$ be a non-trivial additive character of $F$. Let $\Theta :N
\rightarrow R$ be a character, defined as follows:
\begin{center}
$\Theta((x_{ij})) = \psi(\sum_{i=1}^{n-1}x_{i,i+1})$
\end{center}
Let $(\pi,V)$ be an irreducible $R$- representation of $G$. Then recall that
\begin{center}
$\rm dim$$_R\big(\rm Hom$$_N(\pi,\Theta)\big) \leq 1$
\end{center}
For the proof, see \cite{MR0425030} when $R=\mathbb{\overline{Q}}_l$ and see
\cite{MR1395151} when $R=\mathbb{\overline{F}}_l$.
An irreducible $R$-representation $(\pi,V)$ of $G$ is called {\it generic} if
\begin{center}
$\rm dim$$_R\big(\rm Hom$$_N(\pi,\Theta)\big) = 1$.
\end{center}
A generic representation is also called a representation of {\it Whittaker
type}.
\subsubsection{}
Let $(\pi,V)$ be a generic $R$-representation of $G$. Using Frobenius
reciprocity, the representation $\pi$ is embedded in the space
${\rm Ind}_N^G(\Theta)$. Let $W$ be a non-zero linear functional in
the space ${\rm Hom}_N(\pi,\Theta)$. Let
$\mathbb{W}(\pi,\Theta) \subset {\rm Ind}_N^G(\Theta)$ be the space
consisting of functions $W_v$, $v \in V$, where
$$ W_v(g) := W\big(\pi(g)v\big), $$
for all $g \in G$. Then the map $v \mapsto W_v$ induces an isomorphism
from $(\pi,V)$ to $\mathbb{W}(\pi,\Theta)$. The space of functions
$\mathbb{W}(\pi,\Theta)$ is called the Whittaker model of $\pi$.
\subsubsection{}
Let $(\pi,V)$ be an integral generic $l$-adic representation of
$G_n(K)$. Consider the space $\mathbb{W}^0(\pi,\Theta)$ consisting of
$W \in \mathbb{W}(\pi,V)$, taking values in
$\mathbb{\overline{Z}}_l$. Using \cite[Theorem 2]{MR2058628}, the
$\overline{\mathbb{Z}}_l$-module $\mathbb{W}^0(\pi,\Theta)$ is a
$G_n(K)$-invariant lattice in $\mathbb{W}(\pi,\Theta)$.
Let $\tau$ be an $l$-modular generic representation of $G_n(K)$ and let $\pi$
be an $l$-adic generic representation of $G_n(K)$. Then the representation
$\pi$ is called a Whittaker lift of $\tau$ if there exists a lattice
$\mathcal{L} \subseteq \mathbb{W}^0(\pi,\Theta)$ such that
$$ \mathcal{L} \otimes_{\overline{\mathbb{Z}}_l} \overline{\mathbb{F}}_l \simeq
\mathbb{W}(\tau, r_l(\Theta)). $$
\subsubsection{}
Now we follow the notations as in subsection (\ref{16}). Choose a generator
$\gamma$ of $\Gamma$. Let $\pi$ be an $R$-representation of $G_n(E)$. The group
$\Gamma =\rm Gal$$(E/F)$ acts on $G_n(E)$ componentwise i.e. for $\gamma \in
\Gamma, g = (a_{ij})_{i,j=1}^n \in G_n(E)$,
\begin{center}
$\gamma . g := (\gamma(a_{ij}))_{i,j =1}^n$.
\end{center}
Let $\pi^\gamma$ be the representation of $G_n(E)$ on $V$, defined by
\begin{center}
$\pi^\gamma (g) := \pi(\gamma.g)$, for all $g \in G_n(E)$.
\end{center}
We say that the representation $\pi$ of $GL_n(E)$ is
$\Gamma$-equivariant if the representations $\pi$ and $\pi^\gamma$ are
isomorphic.
\subsubsection{}\label{sr}
We end this subsection with a lemma concerning the invariant property
of Whitakker model. Let $\Theta_F$ be a character of
$N_n(F)$ as defined in (\ref{rr}), and let
$\Theta_E$ be the character of $N_n(E)$, defined as $\Theta_E =
\Theta_F \circ \rm
Tr$$_{E/F}$, where ${\rm Tr}_{E/F}$ is the trace map of the extension
$E/F$. Now consider the action of $\Gamma$ on the space
${\rm Ind}_{N_n(E)}^{G_n(E)}(\Theta_E)$, given by
$$ (\gamma . f)(g) := f(\gamma^{-1}g) $$
for all $\gamma \in \Gamma$, $g \in G_n(E)$ and
$f \in \rm Ind$$_{N_n(E)}^{G_n(E)}(\Theta_E)$.
\begin{lemma}\label{inv_whit_model}
Let $(\pi, V)$ be a generic $R$-representation of $G_n(E)$ such that
$(\pi, V)$ is $\Gamma$-equivariant. Then the Whittaker model
$\mathbb{W}(\pi, \Theta_E)$ of $\pi$ is invariant under the action
of $\Gamma$.
\end{lemma}
\begin{proof}
Let $W$ be a Whittaker functional on the representation $\pi$. For $v
\in V$, we have
\begin{center}
$W(\pi^\gamma(n)v)= \Theta_E(\gamma n)W(v)= (\Theta_F \circ \rm
Tr$$_{E/F})(\gamma n)W(v) = \Theta_E(n)W(v)$,
\end{center}
for all $n \in N_n(E)$. Thus,
$W$ is also a Whittaker functional for the representation
$(\pi^\gamma, V)$. Let
$W_v \in \mathbb{W}(\pi,\Theta_E)$. Then
$$(\gamma^{-1}.W_v)(g) = W(\pi^\gamma(g)v).$$
From the uniqueness of the Whittaker model, we have $\gamma^{-1}.W_v \in
\mathbb{W}(\pi,\Theta_E)$. Hence the lemma.
\end{proof}
\subsection{Kirillov Model :}
Let $\pi$ be a generic $R$-representation of $G_n(K)$. Following the notations
as in the subsections (\ref{ss})and (\ref{rr}), consider the space
$\mathbb{K}(\pi, \Theta)$ of all elements $W$ restricted to $P$, where $W$
varies over $\mathbb{W}(\pi, \Theta)$. Then $\mathbb{K}(\pi,\Theta)$ is
$P$-invariant. By Frobenius reciprocity, there is a
non-zero (unique upto scalar constant) linear map $A_\pi : V \longrightarrow \rm
Ind$$_N^P(\Theta)$, which is injective and compatiable with the action of $P$.
In fact,
\begin{center}
$A_\pi(V) = \mathbb{K}(\pi, \Theta) \simeq \mathbb{W}(\pi, \Theta) \simeq \pi$.
\end{center}
Moreover,
$\mathcal{K}(\Theta) = {\rm ind}_N^P(\Theta) \subseteq \mathbb{K}(\pi,
\Theta)$ and the equality holds if $\pi$ is cuspidal. The space of all
elements in $\mathbb{K}(\pi,
\Theta)$$\big($ respectively in
$\mathcal{K}(\Theta)\big)$, taking values in
$\mathbb{\overline{Z}}_l$ is denoted by $\mathbb{K}^0(\pi,
\Theta)$$\big($respectively by $\mathcal{K}^0(\Theta)\big)$.
We now recall the Kirillov model for $n=2$ and some of its properties. For
details, see \cite{MR2234120}. Up to isomorphism, any irreducible
representation of
$P_2(K)$, which is not a character, is isomorphic to
\begin{equation}\label{f}
J_\theta := {\rm ind}_{N_2(K)}^{P_2(K)}(\theta),
\end{equation}
for some non-trivial smooth additive character $\theta$ of $N_2(K)$. Two
different non-trivial characters of $N_2(K)$ induce isomorphic
representations of $P_2(K)$. The space (\ref{f}) is identified with
the space of locally constant compactly supported functions on
$K^\times$, to be denoted by $C_c^\infty(K^\times,
\mathbb{\overline{Q}}_l)$. The action of $P_2(K)$ on the space
$C_c^\infty(K^\times, \mathbb{\overline{Q}}_l)$ is as follows :
For $a, y \in K^\times$ and $x \in K$,
\begin{center}
$\bigg[J_\theta
\begin{pmatrix}
a & 0\\
0 & 1
\end{pmatrix}
f\bigg](y) = f(ay)$,\\
$\bigg[J_\theta
\begin{pmatrix}
1 & x\\
0 & 1
\end{pmatrix}
f\bigg](y) = \theta(xy) f(y)$.
\end{center}
For any cuspidal representation of $(\pi, V)$ of $G_2(K)$, we get a model for
the representation $(\pi, V)$ on the space
$C_c^\infty(K^\times,\mathbb{\overline{Q}}_l)$. The action of the group
$G_2(K)$ on $C_c^\infty(K^\times,\mathbb{\overline{Q}}_l)$ is
denoted by $\mathbb{K}_\theta^\pi$; by definition the restriction of
$\mathbb{K}_\theta^\pi$ to $P_2(K)$ is isomorphic to $J_\theta$. The operator
$\mathbb{K}_\theta^\pi(w)$ completely describes the action of $G_2(K)$ on
$C_c^\infty(K^\times, \mathbb{\overline{Q}}_l)$, where
\begin{center}
$w =
\begin{pmatrix}
0 & 1\\
-1 & 0
\end{pmatrix}$
\end{center}
Here, we follow the exposition in \cite[Section 37.3]{MR2234120}). Let
$\chi$ be a smooth character of $K^\times$ and $k$ be an
integer. Define a function $\xi\{\chi, k\}$ in
$C_c^\infty(K^\times, \mathbb{\overline{Q}}_l)$ by
\begin{center}
$\xi\{\chi, k\}(x)$ =
$\begin{cases}
\chi(x) &\text{ if $\nu_K(x)=k$}\\
0 &\text{otherwise.}
\end{cases}$
\end{center}
where $\nu_K$ is a discrete valuation on $K^\times$. Then we have :
\begin{equation}
\mathbb{K}_\theta^\pi(w)\xi\{\chi,k\} =
\epsilon(\chi^{-1}\pi,\theta) \xi \{\chi^{-1}\varpi_\pi, -n(\chi^{-1}\pi,\theta)
- k\}
\end{equation}
\section{Review of Local Constants and Weil-Deligne representations}
\subsection{}\label{17}
Keeping the notation as in Section \ref{x}, we briefly discuss about the Weil
group and its Weil-Deligne representations. For a reference, see \cite[Chapter
7]{MR2234120} and \cite[Chapter 4]{MR0349635}.
We choose a separable algebraic closure $\overline{K}$ of $K$. Let
$\Omega_K$ be the absolute Galois group $\rm Gal$$(\overline{K}/K)$ and Let
$\mathcal{I}_K$ be the inertia subgroup of $\Omega_K$. Let $\mathcal{W}_K$
denote the Weil group of $K$. fix a geometric Frobenius element $\rm Frob$ in
$\mathcal{W}_K$. Then we have $$ \mathcal{W}_K = \mathcal{I}_K \rtimes \rm
Frob^{\mathbb{Z}}. $$
There is a natural Krull topology on the absolute Galois group $\Omega_K$ and
the inertia group $\mathcal{I}_K$, as a subgroup of $\Omega_K$, is equipped
with the subspace topology. Let the fundamental system of neighbourhoods of the
Weil group $\mathcal{W}_K$ be such that each neighbourhood of the identity
$\mathcal{W}_K$ contains an open subgroup of $\mathcal{I}_K$. Then under this
topology, the Weil group $\mathcal{W}_K$ becomes a locally compact and totally
disconnected group. If $K_1/K$ is a finite extension with $K_1 \subseteq
\overline{K}$, then the Weil group $\mathcal{W}_{K_1}$ is considered as a
subgroup of $\mathcal{W}_K$.
An $R$-representation $\rho$ of $\mathcal{W}_K$ is called unramified if $\rho$
is trivial on $I_K$. Let $\nu$ be the unramified character of $\mathcal{W}_K$
which satisfies $\nu(\rm Frob)$ = $q_K^{-1}$. We now define semisimple
Weil-Deligne representations of $\mathcal{W}_K$.
\subsection{Semisimple Weil-Deligne representation :}\label{18}
A Weil-Deligne representation of $\mathcal{W}_K$ is a pair $(\rho, U)$, where
$\rho$ is a finite dimensional representation of $\mathcal{W}_K$ and $U$ is a
nilpotent operator such that $U \in \rm Hom$$_{\mathcal{W}_K}(\nu \rho, \rho)$.
A Weil-Deligne representation $(\rho, U)$ of $\mathcal{W}_K$ is called
semisimple if $\rho$ is semisimple as a representation of $\mathcal{W}_K$. Note
that any semisimple representation $\rho$ of $\mathcal{W}_K$ is considered as a
semisimple Weil-Deligne representation of the form $(\rho, 0)$. For two
Weil-Deligne representations $(\rho,U)$ and $(\rho', U')$ of $\mathcal{W}_K$,
let
\begin{center}
$\rm Hom_D$$((\rho, U), (\rho', U')) = \{f \in \rm
Hom$$_{\mathcal{W}_K}(\rho, \rho') : f \circ U = U' \circ f \}$,
\end{center}
We say that $(\rho, U)$ and $(\rho',
U')$ are isomorphic if there exists a map $f \in \rm
Hom$$_D((\rho, U)$, $(\rho', U'))$ such that $f$ is bijective. Let
$\mathcal{G}^n_{ss}(K)$ be the set of all $n$-dimensional semisimple
Weil-Deligne representations of the Weil group $\mathcal{W}_K$.
\subsection{Local Constants of Weil-Deligne representations :}
Keep the notations as in section (\ref{17}) and (\ref{18}). In this subsection,
we consider the local constants for $l$-adic Weil-Deligne representations of
$\mathcal{W}_K$.
\subsubsection{L-factors :}
Let $(\rho, U)$ be an $l$-adic semisimple Weil-Deligne representation of
$\mathcal{W}_K$. Then the $L$-factor corresponding to $(\rho, U)$ is defined by
\begin{center}
$L(X,(\rho, U))$ = $\rm det((Id-X\rho(Frob))|_{ker(U)^{\mathcal{I}_K}})^{-1}$.
\end{center}
\subsubsection{Local $\epsilon$-factors and $\gamma$-factors :}
Let $\theta_K: K \rightarrow \overline{\mathbb{Q}}_l$ be a non-trivial additive
character and choose a self dual additive Haar measure on $K$ with respect to
$\theta_K$. Let $\rho$ be an $l$-adic representation of
$\mathcal{W}_K$. The epsilon factor $\epsilon(X,\rho,\theta_K)$ of $\rho$,
relative to $\theta_K$ is defined in \cite{MR0349635}. Let $K'/K$ be a finite
extension inside $\overline{K}$. Let $\theta_{K'}$ denotes the character of
$K'$, where $\theta_{K'} = \theta_K \circ {\rm Tr}_{K'/K}$. Then the epsilon
factor satisfies the following properties :
\begin{enumerate}
\item If $\rho_1$ and $\rho_2$ are two $l$-adic representations of
$\mathcal{W}_{K}$, then
$$ \epsilon(X,\rho_1 \oplus \rho_2,\theta_{K}) = \epsilon(X,\rho_1,\theta_{K})
\epsilon(X,\rho_2,\theta_{K}). $$
\item $\rho$ is an $l$-adic representation of $\mathcal{W}_{K'}$,
then
\begin{equation}\label{60}
\frac{\epsilon\big(X, {\rm ind}_{\mathcal{W}_{K'}}^{\mathcal{W}_{K}}(\rho),
\theta_{K}\big)}{\epsilon(X, \rho, \theta_{K'})} = \frac{\epsilon\big(X, {\rm
ind}_{\mathcal{W}_{K'}}^{\mathcal{W}_{K}}(1_{K'}), \theta_{K}\big)}{\epsilon(X,
1_{K'},\theta_{K'})},
\end{equation}
where $1_{K'}$ denotes the trivial character of $\mathcal{W}_{K}$.
\item If $\rho$ is an $l$-adic representation of $\mathcal{W}_{K'}$,
then
\begin{equation}\label{61}
\epsilon\big(X, \rho,\theta_{K}\big) \epsilon\big(q_K^{-1}X^{-1}, \rho^\vee,
\theta_{K}\big) = \rm det\big(\rho(-1)\big),
\end{equation}
where $\rho^\vee$ denotes the dual of the representation $\rho$.
\item For an $l$-adic representation $\rho$ of $\mathcal{W}_K$, there exists an integer $n(\rho,\theta_K)$ for which
$$ \epsilon(X,\rho,\theta_K) = (q_K^{\frac{1}{2}}X)^{n(\rho,\theta_K)}\epsilon(\rho,\theta_K). $$
\end{enumerate}
Now for an $l$-adic semisimple Weil-Deligne representation $(\rho, U)$, the
$\epsilon$-factor is defined as
$$ \epsilon\big(X, (\rho, U), \theta_K\big) = \epsilon(X, \rho,
\theta_K)\frac{L(q_K^{-1}X^{-1},\rho^\vee)}{L(X,\rho)}\frac{L(X,(\rho,U))}{L(q
_K^{-1}X^{-1},(\rho, U)^\vee)}, $$
where $(\rho, U)^\vee = (\rho^\vee, -U^\vee)$. Set
$$ \gamma(X,(\rho, U),\theta_K) =
\epsilon(X,(\rho,U),\theta_K)\frac{L(X,(\rho,U))}{L(q_K^{-1}X^{-1},(\rho,U)
^\vee)}. $$
The element $\gamma(X, (\rho, U), \theta_K)$ is called the $\gamma$-factor of
the Weil-Deligne representation $(\rho, U)$.
Now we state a result \cite[Proposition 5.11]{MR4311563} which concerns the
fact that the $\gamma$-factors are compatible with reduction modulo $l$. For $P
\in \mathbb{\overline{Z}}_l[X]$, we denote by $r_l(P) \in
\mathbb{\overline{F}}_l[X]$ the polynomial obtained by reduction mod-$l$ to the
coefficients of $P$. For $Q \in \mathbb{\overline{Z}}_l[X]$, such that $r_l(Q)
\not= 0$, we set $r_l(P/Q) = r_l(P)/r_l(Q)$.
\begin{proposition}\label{62}
\rm Let $\rho$ be an integral $l$-adic semisimple representation of
$\mathcal{W}_K$, then
$$ r_l\big(\gamma(X,\rho,\theta)\big) = \gamma\big(X, r_l(\rho),\theta\big). $$
\end{proposition}
\subsection{Local constants of \texorpdfstring{$p$}{}-adic
representations:}
We now define the $L$-factors and $\gamma$-factors for irreducible
$R$-representations of $G_n(K)$. For details, see \cite{MR3595906}.
Let $\pi$ be an $R$-representation of Whittaker type of $G_n(K)$ and let $\pi'$
be an $R$-representation of Whittaker type of $G_{n-1}(K)$. Let $W \in
\mathbb{W}(\pi,\Theta)$ and $W' \in \mathbb{W}(\pi',\Theta^{-1})$. The function
$W
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
W'(g)$ is compactly supported on $Y_K^r$ \cite[Proposition 3.3]{MR3595906}.
Then the following integral
$$ c^K_r(W,W') = \int_{Y_K^r} W
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
W'(g) \,dg ,$$
is well defined for all $r \in \mathbb{Z}$, and vanishes for $r << 0$. In
this paper, we deal with base change where two different $p$-adic
fields are involved. So to avoid
confusion, we use the notation $c^K_r(W,W')$ instead of the notation
$c_r(W,W')$ used in \cite[Proposition 3.3]{MR3595906}.
$Y_K^r$. Let $I(X,W,W')$ be the following power series:
$$ I(X,W,W') = \sum_{r \in \mathbb{Z}}c^K_k(W,W') q_K^{r/2} X^r \in
R((X)). $$ Note that $I(X,W,W')$ is a rational function in $X$ (see
\cite[Theorem 3.5]{MR3595906}).
\subsubsection{$L$ -factors :}
Let $\pi$ and $\pi'$ be two $R$-representations of Whittaker type of
$G_n(K)$ and $G_{n-1}(K)$ respectively. Then the $R$-submodule spanned
by $I(X,W,W')$ as $W$ varies in $\mathbb{W}(\pi,\Theta)$ and $W'$
varies in $\mathbb{W}(\pi',\Theta^{-1})$, is a fractional ideal of
$R[X,X^{-1}]$ and it has a unique generator which is an Euler factor
denoted by $L(X,\pi,\pi')$. The generator $L(X,\pi.\pi')$ called the
$L$-factor associated to $\pi$, $\pi'$ and $\Theta$.
\begin{remark}
If $\pi$ and $\pi'$ are $l$-adic representations of Whittaker type
of $G_n(K)$ and $G_{n-1}(K)$ respectively, then
$1/L(X,\pi,\pi') \in \mathbb{\overline{Z}}_l[X]$.
\end{remark}
We conclude this section with a theorem \cite[Theorem 4.3,]{MR3595906} which
describes $L$-factors of cuspidal representations.
\begin{theorem}\label{k}
\rm Let $\pi_1$ and $\pi_2$ be two cuspidal $R$-representations of $G_n(K)$ and
$G_m(K)$ respectively. Then $L(X,\pi_1,\pi_2)$ is equal to 1, except in the
following case : $\pi_1$ is banal in the sense of \cite{MR3178433} and $\pi_2
\simeq \chi \pi_1^\vee$ for some unramified character $\chi$ of $K^\times$.
\end{theorem}
In our proof we only consider the case when $m=n-1$. Then by the above theorem
the $L$-factor $L(X,\pi_1,\pi_2)$ associated with the cuspidal
$R$-representations $\pi_1$ and $\pi_2$ is equal to 1.
\subsubsection{Functional Equations and Local $\gamma$-factors :}\label{w}
Let $\pi$ and $\pi'$ be two $R$-representations of Whittaker type of $G_n(K)$
and $G_{n-1}(K)$ respectively. Then there is an invertible element
$\epsilon(X,\pi,\pi',\Theta)$ in $R[X,X^{-1}]$ such that for all $W \in
\mathbb{W}(\pi,\Theta)$, $W' \in \mathbb{W}(\pi',\Theta^{-1})$, we have the
following functional equation :
$$ \dfrac{I(q_K^{-1}X^{-1},\widetilde{W},
\widetilde{W'})}{L(q_K^{-1}X^{-1}, \widetilde{\pi},
\widetilde{\pi'})} = \omega_{\pi'}(-1)^{n-2} \epsilon(X,
\pi,\pi',\Theta) \dfrac{I(X,W,W')}{L(X,\pi,\pi')}, $$ where
$\omega_{\pi'}$ denotes the central character of the representation
$\pi'$. We call $\epsilon(X,\pi,\pi',\Theta)$ the local
$\epsilon$-factor associated to $\pi$, $\pi'$ and $\Theta$. Moreover,
if $\pi$ and $\pi'$ be $l$-adic representations of Whittaker type of
$G_n(K)$ and $G_{n-1}(K)$ respectively, then the factor
$\epsilon(X,\pi,\pi',\Theta)$ is of the form $cX^k$ for a unit
$c \in \mathbb{\overline{Z}}_l$. In particular, there exists an
integer $n(\pi,\pi',\Theta)$ such that
\begin{equation}\label{degree}
\epsilon(X,\pi,\pi,\Theta) = (q_K^{\frac{1}{2}}X)^{n(\pi,\pi',\Theta)} \epsilon(\pi,\pi',\Theta).
\end{equation}
Now the local $\gamma$-factor associated with
$\pi$, $\pi'$ and $\Theta$ is defined as :
\begin{center}
$\gamma(X,\pi,\pi',\Theta) = \epsilon(X,\pi,\pi',\Theta)
\dfrac{L(q_K^{-1}X^{-1}, \widetilde{\pi},
\widetilde{\pi'})}{L(X,\pi,\pi')}$.
\end{center}
\subsubsection{Compatibility with reduction modulo $l$ :}
Let $\tau$ and $\tau'$ be two $l$-modular
representations of Whittaker type of $G_n(K)$ and $G_{n-1}(K)$ respectively.
Let $\pi$ and $\pi'$ be the respective Whittaker lifts of $\tau$ and $\tau'$.
Then
$$ L(X,\tau,\tau') | r_l\big(L(X,\pi,\pi')\big) $$
and
$$ \gamma\big(X,\tau, \tau',r_l(\Theta)\big) =
r_l\big(\gamma(X,\pi,\pi',\Theta)\big). $$ For details, see
\cite[Section 3.3]{MR3595906}. Now we end this section with a simple
lemma which will be needed later in the proof of Theorem \ref{m}
and \ref{n=3_thm}.
\begin{lemma}\label{i}
\rm Let $E/F$ be a cyclic Galois extension of prime degree $l$ and
assume $l \not= 2$. Let $\rho$ be an $l$-adic
representation of $\mathcal{W}_E$ of even dimension. Then
\begin{center}
$\epsilon(X, \rho, \psi_E) = \epsilon\big(X, {\rm
ind}_{\mathcal{W}_E}^{\mathcal{W}_F}(\rho), \psi_F\big)$.
\end{center}
\end{lemma}
\begin{proof}
Let $\mathcal{C}_{E/F}(\psi) = \dfrac{\epsilon\big(X, {\rm
ind}_{\mathcal{W}_E}^{\mathcal{W}_F}(1_E), \psi_F\big)}{\epsilon(X, 1_E,
\psi_E)}$, where
$1_E$ denotes the trivial character of $\mathcal{W}_E$. Then
$\mathcal{C}_{E/F}(\psi)$ is independent of X (see \cite[Corollary 30.4,
Chapter 7]{MR2234120}). Using the equality (\ref{60}), we get
\begin{center}
$\dfrac{\epsilon\big(X, {\rm ind}_{\mathcal{W}_E}^{\mathcal{W}_F}(\rho),
\psi_F\big)}{\epsilon(X, \rho, \psi_E)}
=(\mathcal{C}_{E/F}\big(\psi)\big)$$^{\rm dim\rho}$.
\end{center}
In view of the functional equation (\ref{61}), we have
\begin{center}
$\mathcal{C}_{E/F}^2(\psi) = \xi_{E/F}(-1)$,
\end{center}
where $\xi_{E/F} = {\rm det}\big({\rm
ind}_{\mathcal{W}_E}^{\mathcal{W}_F}(1_E)\big)$, a quadratic
character of $\mathcal{W}_F$. Since $\xi_{E/F}^l = 1$ and $l
\not=2$, we get that $\xi_{E/F} =
1_F$, the trivial character of $\mathcal{W}_F$. Hence the lemma.
\end{proof}
\section{Local Langlands Correspondence}
\subsection{\texorpdfstring{$l$}{}-adic local Langlands correspondence :}
In this subsection we recall the $l$-adic local Langlands correspondence. For
details, see \cite[Section 35.1, Chapter 8]{MR2234120}. Keep the notation as in
section (\ref{x}).
Recall that Local langlands correspondence over
$\overline{\mathbb{Q}}_l$ gives a bijection
\begin{center}
$\Pi_K : {\rm Irr}\big(GL_n(K),\mathbb{\overline{Q}}_l\big)
\longrightarrow \mathcal{G}_{ss}^n(K)$
\end{center}
such that
$$ \gamma(X, \sigma \times \sigma', \Theta) = \gamma(X, \Pi_K(\sigma) \otimes
\Pi_K(\sigma'), \Theta) $$
and
$$ L(X, \sigma \times \sigma') = L(X, \Pi_K(\sigma) \otimes \Pi_K(\sigma')), $$
for all $\sigma \in \rm
Irr$$(G_n(K),\mathbb{\overline{Q}}_l)$,
$\sigma' \in \rm
Irr$$(G_m(K),\mathbb{\overline{Q}}_l)$. Moreover, the set of all
cuspidal $l$-adic representations of
$GL_n(K)$ is mapped onto the set $n$-dimensional irreducible
$l$-adic representations of the Weyl group
$\mathcal{W}_K$ via the bijection
$\Pi_K$ (see \cite{harris_taylor}, \cite{henniart_une_preuve} or
\cite{scholze_llc}).
\subsection{Local base Change for the extension \texorpdfstring{$E/F$}{}}
Now we recall local base change for a cyclic extension of a $p$-adic
field. The base change operation on irreducible smooth representations
of ${\rm GL}_n(F)$ over complex vector spaces is characterised by
certain character identities (see\cite{MR1007299}). Let us recall the
relation between $l$-adic local Langlands correspondence and local
base change for $GL_n$. Let $\pi_F$ be an $l$-adic cuspidal
representation of $GL_n(F)$. Let
$\rho_F : \mathcal{W}_K \longrightarrow GL_n(\overline{\mathbb{Q}}_l)$
be a semisimple Weil-Deligne representation such that
$\Pi_F(\pi_F) = \rho_F$, where $\Pi_F$ is the $l$-adic local Langlands
correspondence as described in the previous section. Let $\pi_E$ be
the $l$-adic cuspidal representation of $GL_n(E)$ such that
\begin{center}
$\rm Res$$_{\mathcal{W}_E}^{\mathcal{W}_F}\big(\Pi_E(\pi_E)\big) \simeq
\Pi_F(\pi_F)$.
\end{center}
The representation $\pi_E$ is the base change of
$\pi_F$. Note that in this case $\pi_E \simeq
\pi_E^\gamma$, for all $\gamma \in \Gamma$.
\section{Tate Cohomology}\label{50}
In this section, we recall Tate cohomology groups and some useful
results on $\Gamma$- equivariant
$l$-sheaves of $\mathbb{\overline{Z}}_l$-modules on an $l$-space
$X$ equipped with an action of
$\Gamma$. For details, see \cite{MR3432583}.
Fix a generator $\gamma$ of $\Gamma$. Let $M$ ba a
$\mathbb{\overline{Z}}_l[\Gamma]$-module. Let $T_\gamma$ be the
automorphism of $M$ defined by
\begin{center}
$T_\gamma(m) = \gamma . m$, for $\gamma \in \Gamma, m \in M$.
\end{center}
We set $N_\gamma$ to be operator
$$\rm 1 + T_\gamma + T_{\gamma^2} +....+ T_{\gamma^{l-1}},$$
also known as the norm operator. The Tate cohomology groups
$\widehat{H}^0(M)$ and $\widehat{H}^1(M)$ are defined as :
\begin{center}
$\widehat{H}^0(M) = \dfrac{{\rm ker}(\rm id - T_\gamma)}{{\rm
image}(N_\gamma)}$,
$\widehat{H}^1(M) = \dfrac{{\rm ker}(N_\gamma)}{{\rm image}(\rm id -
T_\gamma)}$.
\end{center}
\subsection{}
Let $X$ be an $l$-space with an action of a finite group $<\gamma>$ of
order $l$. Let $\mathcal{F}$ be an $l$-sheaf of
$\mathbb{\overline{F}}_l$ or $\mathbb{\overline{Z}}_l$ -modules on $X$. Write
$\Gamma_c(X;\mathcal{F})$ for the space of compactly supported
sections of $\mathcal{F}$. In particular, if $\mathcal{F}$ is the
constant sheaf with stalk $\mathbb{\overline{F}}_l$ or
$\mathbb{\overline{Z}}_l$, then
$\Gamma_c(X;\mathcal{F}) = C_c^\infty(X,\mathbb{\overline{F}}_l)$ or
$C_c^\infty(X,\mathbb{\overline{Z}}_l)$ respectively. The
assignment $\mathcal{F} \mapsto \Gamma_c(X; \mathcal{F})$ is a
covariant exact functor. If $\mathcal{F}$ is $\gamma$-equivariant,
then $\gamma$ can be regarded as a map of sheaves
$T_\gamma : \mathcal{F}|_{X^\gamma} \rightarrow \mathcal{F}|_{X^\gamma}$ and
the Tate cohomology is defined as :
\begin{center}
$\widehat{H}^0(\mathcal{F}|_{X^\gamma}) := \dfrac{{\rm
ker}(1-T_\gamma)}{{\rm image}(N_\gamma)}$,\\
$\widehat{H}^1(\mathcal{F}|_{X^\gamma}) :=
\dfrac{{\rm ker}(N_\gamma)}{{\rm image}(1-T_\gamma)}$.
\end{center}
A compactly supported section of $\mathcal{F}$ can be restricted to a
compactly supported section of $\mathcal{F}|_{X^\gamma}$. The following result
is often useful in calculating Tate cohomology groups.
\begin{proposition}[\cite{MR3432583}] \label{tr}
The restriction map $\Gamma_c(X;\mathcal{F}) \longrightarrow
\Gamma_c(X^\gamma,\mathcal{F}|_{X^\gamma})$ induces an isomorphism of the
following spaces:
\begin{center}
$\widehat{H}^i(\Gamma_c(X; \mathcal{F})) \longrightarrow \Gamma_c(X^\gamma;
\widehat{H}^i(\mathcal{F}))$.
\end{center}
\end{proposition}
\subsection{Comparison of integrals of smooth functions :}\label{ur}
Keep the notation as in section (\ref{x}). The group
$\Gamma =< \gamma>$ act on the set $Y^r_E$ and hence acts on the space
$C_c^\infty(Y^r_E, \mathbb{\overline{F}}_l)$ as follows:
\begin{center}
$(\gamma . \phi)(x) := \phi(\gamma^{-1}x)$ for all $x \in Y^r_E$,
$\phi \in C_c^\infty(Y^r_E,\mathbb{\overline{F}}_l)$.
\end{center}
We end this section with a proposition comparing the integrals on the spaces
$X_E$ and $X_F$.
\begin{proposition}\label{1}
\rm Let $\phi \in C_c^\infty(Y^r_E, \mathbb{\overline{F}}_l)$ be a
$\Gamma$-invariant function. Then we have
$$\int_{Y^r_E}\phi \,d\mu_E = \int_{Y^r_F}\phi \,d\mu_F .$$
\end{proposition}
\begin{proof}
The group $\Gamma = <\gamma>$ acts on $G_n(E)$. Since $N_n(E)$ is
$\Gamma$-stable, we have the following exact sequence of non-abelian
cohomology \cite[Chapter VII, Appendix]{MR1324577}:
\begin{center}
$0 \longrightarrow H^0(\Gamma,N_n(E)) \longrightarrow H^0(\Gamma,G_n^r(E))
\longrightarrow H^0(\Gamma,Y^r_E) \longrightarrow H^1(\Gamma; N_n(E))
\longrightarrow H^1(\Gamma; G_n(E))$
\end{center}
Since $H^1\big(\Gamma; N_n(E)\big) = 0$, we get from the above exact
sequence that
\begin{equation}\label{t}
(Y_E^r)^\Gamma \simeq Y_F^r.
\end{equation}
Since $Y^r_F$ is closed in $Y^r_E$, we have the following exact sequence
of $\Gamma$-modules:
\begin{equation}\label{s}
0 \longrightarrow C_c^\infty(Y^r_E \setminus Y^r_F,
\mathbb{\overline{F}}_l)\longrightarrow C_c^\infty(Y^r_E,
\mathbb{\overline{F}}_l)\longrightarrow
C_c^\infty(Y^r_F,\mathbb{\overline{F}}_l)\rightarrow 0
\end{equation}
By (\ref{t}), the action of $\Gamma$ on $Y^r_E \setminus Y^r_F$ is free. Then
in view of \cite[Section 3.3]{MR3432583}, there exists a fundamental domain $U$
such that $Y^r_E \setminus Y^r_F = \bigsqcup_{i=0}^{l-1} \gamma^iU$, and this
implies that
\begin{equation}\label{r}
H^1(\Gamma,C_c^\infty(Y^r_E \setminus Y^r_F,\mathbb{\overline{F}}_l)) = 0.
\end{equation}
Using (\ref{s}) and (\ref{r}), we get the following exact sequence :
\begin{center}
$0 \longrightarrow C_c^\infty(Y^r_E \setminus Y^r_F,
\mathbb{\overline{F}}_l)^\Gamma \longrightarrow C_c^\infty(Y^r_E,
\mathbb{\overline{F}}_l)^\Gamma \longrightarrow
C_c^\infty(Y^r_F,\mathbb{\overline{F}}_l)\rightarrow 0$
\end{center}
Again the free action of $\Gamma$ on $Y^r_E \setminus Y^r_F$ gives a
fundamental domain $U$ such that $Y^r_E \setminus Y^r_F = \bigsqcup_{i=0}^{l-1}
\gamma^iU$, and we have
$$ \int_{Y^r_E\setminus Y^r_F}\phi \,d\mu_E = l\sum_{i=0}^{l-1}\int_U\phi
d\mu_E = 0, $$
for all $\phi \in C_c^\infty(Y^r_E \setminus
Y^r_F, \mathbb{\overline{F}}_l)^\Gamma$.
Hence the linear functional $d\mu_E$ induces a $G_n(F)$-invariant linear
functional on $C_c^\infty(Y^r_F, \mathbb{\overline{F}}_l)$ and the proposition
now follows from the uniqueness of Haar measure on $C_c^\infty(Y^r_F,
\mathbb{\overline{F}}_l)$.
\end{proof}
\subsection{Frobenius Twist :}
Let $G$ be a locally compact and totally disconnected group. Let $(\sigma, V)$
be an $l$-adic representation of $G$. Consider the vector space $V^{(l)}$,
where the underlying additive group structure of $V^{(l)}$ is same as that of
$V$ but the scalar action $*$ on $V^{(l)}$ is given by
\begin{center}
$c * v = c^{\frac{1}{l}} . v$ , for all $c \in \mathbb{\overline{F}_\ell}, v
\in V$.
\end{center}
Then the action of $G$ on $V$ induces a representation $\sigma^{(l)}$ of $G$ on
$V^{(l)}$. The representation $(\sigma^{(l)}, V^{(l)})$ is called the Frobenius
twist of the representation $(\sigma,V)$.
We end this subsection with a lemma which will be useful for the main results.
\begin{lemma}\label{ar}
If $(\pi,V_\pi)$, $(\sigma, V_\sigma)$ are two $l$-modular irreducible generic
representations of $G_n(F)$ and $G_{n-1}(F)$ respectively. Then
$$ \gamma(X,\pi,\sigma,\Theta_F)^l =
\gamma(X,\pi^{(l)},\sigma^{(l)},\Theta_F^l). $$
\end{lemma}
\begin{proof}
Let $W_\pi$ be a Whittaker functional on the representation
$\pi$. Then the composite map
$V_\pi \xrightarrow{W_\pi} \overline{\mathbb{F}}_l \xrightarrow{x
\mapsto x^l} \overline{\mathbb{F}}_l$, denoted by $W_{\pi^{(l)}}$,
is a Whittaker functional on the representation $\pi^{(l)}$, as
$$ W_{\pi^{(l)}}(c.v) = W_{\pi}((c^{\frac{1}{l}}v))^l = c W_{\pi^{(l)}}(v) $$
and
$$ W_{\pi^{(l)}}(\pi^{(l)}(n)v) = W_{\pi}(\Theta_F(n)v)^l = \Theta^l_F(n)W_{\pi^{(l)}}(v), $$
for all $v \in V_{\pi}$, $c \in \overline{\mathbb{F}}_l$ and all
$n \in N_n(F)$.
So the Whittaker model $\mathbb{W}(\pi^{(l)},\Theta_F^l)$ consists of
the functions $W_v^l$, where $W_v$ varies in
$\mathbb{W}(\pi,\Theta_F)$. Similarly the Whittaker model
$\mathbb{W}(\sigma^{(l)}, \Theta_F^l)$ of $\sigma^{(l)}$ consists of
the functions $U_v^l$, where $U_v$ varies in
$\mathbb{W}(\sigma,\Theta_F)$. Then by the Rankin-Seilberg functional
equation in the subsection (\ref{w}), we have
\begin{equation}\label{30}
\sum_{r \in \mathbb{Z}}c^F_r(\widetilde{W_v},\widetilde{U_v})^lq_F^{-lr/2}X^{-lr}
= \omega_\sigma(-1)^{n-2} \gamma(X,\pi,\sigma,\Theta_F)^l \sum_{r \in
\mathbb{Z}}c^F_r(W_v,U_v)^lq_F^{lr/2}X^{lr}
\end{equation}
and
\begin{equation}\label{31}
\sum_{r \in \mathbb{Z}}c^F_r(\widetilde{W_v^l},\widetilde{U_v^l})q_F^{-r/2}X^{-r}
= \omega_{\sigma^{(l)}}(-1)^{n-2} \gamma(X,\pi^{(l)},\sigma^{(l)},\Theta_F^l)
\sum_{r \in \mathbb{Z}}c^F_r(W_v^l,U_v^l)q_F^{r/2}X^r.
\end{equation}
Replace $X$ by $X^{1/l}$ to the equation (\ref{30}), we have
\begin{equation}\label{32}
\sum_{r \in \mathbb{Z}}c^F_r(\widetilde{W_v},\widetilde{U_v})^lq_F^{-lr/2}X^{-r}
= \omega_\sigma(-1)^{n-2} \gamma(X,\pi,\sigma,\Theta_F)^l \sum_{r \in
\mathbb{Z}}c^F_r(W_v,U_v)^lq_F^{lr/2}X^{r}.
\end{equation}
Then in view of the equations (\ref{31}) and (\ref{32}), we get
$$ \gamma(X,\pi,\sigma,\Theta_F)^l =
\gamma(X,\pi^{(l)},\sigma^{(l)},\Theta_F^l). $$
Thus we prove the lemma.
\end{proof}
\section{Tate Cohomology of Whittaker lattices}
Let $(\pi, V)$ be an irreducible integral $l$-adic smooth
representation of $G_n(E)$ such that $\pi^\gamma$ is isomorphic to
$\pi$, for all $\gamma\in \Gamma$. Let $W(\pi, \Theta)$ be the
Whittaker model of $\pi$. For $W\in W(\pi, \Theta)$, we recall that
$\gamma.W$ is function given by
$$\gamma.W(g)=W(\gamma^{-1}(g)).$$
Note that $\gamma.W\in W(\pi, \Theta)$ (see Lemma
\ref{inv_whit_model}). Thus, we
define
$$T_\gamma: W(\pi, \Theta)\rightarrow W(\pi, \Theta)$$ by setting
$T_\gamma(W)=\gamma.W$, for all $W\in W(\pi, \Theta)$. The map
$T_\gamma$ gives an isomorphism between $(\pi^\gamma, V)$ and
$(\pi, V)$ as we have
$$T_\gamma(\pi(g)W_v)(h)=\pi(g)W_v(\gamma^{-1}(h))=W_v(\gamma^{-1}(h)g)$$
and
$$[\pi^\gamma(g)T(W_v)](h)=T(W_v)(h\gamma(g))=W_v(\gamma^{-1}(h)g),$$
for all $g, h\in G$. Moreover, the lattice $W^0(\pi, \Theta)$ is a
$T_\gamma$ and $G_n(E)$-invariant lattice. Using the result \cite[Lemma
4]{MR3551160}, the
semisimplification of Tate cohomology is independent of the choice of
lattice and the isomorphism between $\pi$ and $\pi^\gamma$; we choose
$T_\gamma$ using the Whittaker models. We now prove Theorem \ref{intro_n=2_thm}.
\begin{theorem}\label{n=2_thm}
Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a
finite Galois extension of $F$ with $[E:F] =l$. Assume that $l$ and
$p$ are distinct odd primes. Let $\pi_F$ be an integral $l$-adic
cuspidal representations of ${\rm GL}_2(F)$ and let $\pi_E$ be the
representation of ${\rm GL}_2(E)$ such that $\pi_E$ is the base
change of $\pi_F$. Then
\begin{center}
$\widehat{H}^0(\pi_E) \simeq r_l(\pi_F)^{(l)}.$
\end{center}
\end{theorem}
\begin{proof}
Fix a non-trivial additive character $\psi$ of $F$. Let $\Theta_E$,
$\Theta_F$ be defined as in subsection (\ref{sr}). Let
$\big(\mathbb{K}_{\Theta_E}^{\pi_E},
C_c^\infty(E^\times,\mathbb{\overline{Q}}_l)\big)$ be a Kirillov
model of the representation $\pi_E$. By \cite{MR1390835}, the
lattice $C_c^\infty(E^\times,\mathbb{\overline{Z}}_l)$ is stable
under the action of $\mathbb{K}_{\Theta_E}^{\pi_E}(w)$. Recall that
the group $\Gamma$ acts on
$C_c^\infty(E^\times,\mathbb{\overline{Z}}_l)$. We denote by
$\widehat{H}^0(\pi_E)$ the cohomology group
$\widehat{H}^0\big(C_c^\infty(E^\times,\mathbb{\overline{Z}}_l)\big)$. Then
using proposition (\ref{tr}), we have
\begin{center}
$\widehat{H}^0(\pi_E) \simeq C_c^\infty(F^\times,\mathbb{\overline{F}}_l)$.
\end{center}
The space $\widehat{H}^0(\pi_E)$ is isomorphic to
$\rm
ind$$_{N_2(F)}^{P_2(F)}(\Theta_F^l)$ as a representation of
$P_2(F)$; and the induced action of the operator
$\mathbb{K}_{\Theta_E}^{\pi_E}(w)$ on
$\widehat{H}^0(\pi_E)$ is denoted by
$\overline{\mathbb{K}}_{\Theta_E}^{\pi_E}(w)$. The theorem now follows
from
the following claim.\\
\textbf{Claim :} $\overline{\mathbb{K}}_{\Theta_E}^{\pi_E}(w)(f) =
\mathbb{K}_{\Theta_F^l}^{r_l(\pi_F)^{(l)}}(w)(f)$, for all $f \in
C_c^\infty(F^\times,\mathbb{\overline{F}}_l)$.\\
Now for a function $f \in
C_c^\infty(F^\times,\mathbb{\overline{F}}_l)$, any covering of $\rm
supp$$(f)$ by open subsets of $F^\times$ has a finite refinement of
pairwise disjoint open compact subgroups of $F^\times$. So we may
assume that $\rm supp$$(f) \subseteq \varpi^rxU_F^1$, where $r \in
\mathbb{Z}$, $\varpi$ is an uniformizer of $F$ and
$x$ is a unit in
$(\mathfrak{o}_K/\mathfrak{p}_K)^\times$ embedded in
$F^\times$. Then there exists an element $u \in P_2(F)$ such that $\rm
supp$$(u.f) \subseteq U_F^1$. Therefore it is sufficient to prove the
claim for functions
$f \in C_c^\infty(F^\times,\mathbb{\overline{F}}_l)$ with
$\rm supp$$(f) \subseteq U_F^1$, and we have
\begin{center}
$ f = c_\chi\sum_{\chi \in \widehat{U_F^1}}\xi\{\chi,0\}$,
\end{center}
where $c_\chi \in \overline{\mathbb{F}}_l$ and $\widehat{U_F^1}$ is taken to be
the set of smooth characters of $F^\times$ which are trivial on
$(\mathcal{O}_F/\mathcal{P}_F)^\times$ and $\varpi_F$. We now prove the claim
for the function $\xi\{\chi, 0\}$ for $\chi \in \widehat{U_F^1}$. There exists
a character $\chi_0 \in \widehat{U_F^1}$ such that $\chi_0^l = \chi$. Define a
character $\widetilde{\chi}_0$ of $E^\times$ by,
\begin{center}
$\widetilde{\chi}_0(x) = \chi_0(Nr_{E/F}(x))$, for $x \in E^\times$.
\end{center}
We have the following relations :
\begin{equation}\label{b}
\overline{\mathbb{K}}_{\psi_E}^{\pi_E}(w)\xi\{\chi,0\} =
r_l\big(\epsilon(\widetilde{\chi}_0^{-1}\pi_E,\Theta_E)\big)
\xi\bigg\{\chi,\cfrac{-n(\widetilde{\chi}_0^{-1}\pi_E,\Theta_E)}{e(E:F)}\bigg\}
\end{equation}
and
\begin{equation}\label{c}
\overline{\mathbb{K}}_{\psi_F^l}^{r_l(\pi_F)^{(l)}}(w)\xi\{\chi,0\} =
\epsilon(\chi^{-1}r_l(\pi_F)^{(l)},\Theta_F^l) \xi \{\chi,
-n(\chi^{-1}r_l(\pi_F)^{(l)},\Theta_F^l)\},
\end{equation}
where $e(E:F)$ denotes the ramification index of the extension
$E$ over $F$.
Now recall that for a cuspidal integral $l$-representation
$\pi$ of $G_2(K)$, $\Pi_\pi$ is an integral
$2$-dimensional irreducible
$\mathbb{\overline{Q}}_l$ -representation of
$\mathcal{W}_K$ associated by the
$l$-adic local Langlands correspondence. Using base change and Lemma
(\ref{i}), we have :
\begin{center}
$\epsilon(X, \widetilde{\chi}_0\pi_E, \Theta_E) = \epsilon\big(X, (\Pi_{\chi_0}
\otimes \Pi_{\pi_F}) \otimes \rm ind$$_{\mathcal{W}_E}^{\mathcal{W}_F}(1_E),
\Theta_F\big)$.
\end{center}
In view of the Theorem (\ref{k}), the $\epsilon$-factors and $\gamma$-factors
are equal. Then using reduction mod-$l$ (Proposition (\ref{62})) and $l$-adic
local langlands correspondence , we get
\begin{equation}\label{12}
r_l\Big(\epsilon\big(X, (\Pi_{\chi_0} \otimes \Pi_{\pi_F}) \otimes {\rm
ind}_{\mathcal{W}_E}^{\mathcal{W}_F}(1_E), \Theta_F\big)\Big) = \epsilon\big(X,
\chi_0r_l(\pi_F), \Theta_F\big)^l.
\end{equation}
By the lemma (\ref{ar}), we have
\begin{equation}\label{rrr}
r_l\big(\epsilon(X, \widetilde{\chi}_0\pi_E, \Theta_E)\big) = \epsilon\big(X, \chi_0
r_l(\pi_F),\Theta_F\big)^l = \epsilon(X, \chi
r_l(\pi_F)^{(l)},\Theta_F^l\big)
\end{equation}
Now using the relation (\ref{degree}) and comparing the degree of $X$ from (\ref{rrr}), we get
\begin{equation}\label{d}
\cfrac{n(\widetilde{\chi}_0\pi_E,\Theta_E)}{e(E:F)} = n(\chi r_l(\pi_F)^{(l)},
\Theta_F^l)
\end{equation}
and
\begin{equation}\label{e}
r_l\big(\epsilon(\widetilde{\chi_0}\pi_E, \Theta_F)\big) = \epsilon\big(\chi
r_l(\pi_F)^{(l)}, \Theta_F^l\big).
\end{equation}
In view of the equations (\ref{b}), (\ref{c}), (\ref{d}) and (\ref{e}), we have :
\begin{center}
$\overline{\mathbb{K}}_{\psi_E}^{\pi_E}(w)$ $\xi\{\chi,0\} =
\overline{\mathbb{K}}_{\psi_F^l}^{r_l(\pi_F)^{(l)}}(w)$ $\xi\{\chi,0\}$.
\end{center}
Hence we prove the claim, and the theorem follows.
\end{proof}
We now prove a lemma which will be useful in the proof of
Theorem \ref{intro_n=3_thm}.
\begin{lemma}\label{j}
\rm Let $\phi \in \rm ind$$_{N_2(F)}^{G_2(F)}(\Theta_F)$. Then $\phi = 0$ if
and only if $$\int_{Y^r_F}\phi(t)W(t) \,dt = 0 ,$$
for all $r \in \mathbb{Z}$, $W \in \mathbb{W}(\sigma_F, \Theta_F^{-1})$ and for
all irreducible generic representations $\sigma_F$ of $G_2(F)$.
\end{lemma}
\begin{proof}
The above lemma is well known in the case of complex numbers, and see
\cite[Lemma 3.5]{MR620708} for an algebraic proof. In the case of
$W(\overline{\mathbb{F}}_l)$-smooth representations, the above lemma
is due to Helm and Moss \cite{MR3867634}. Here,
$\mathbb{W}(\mathbb{\overline{F}}_l)$ is the ring of witt vectors of
$\mathbb{\overline{F}}_l$. In the
mod-$l$ case, for non-banal primes
$l$, the above lemma does not hold. For banal primes
$l$, we sketch a proof using the work of Helm and Moss. If
$\mathcal{Z}$ is the center of the cateogory of smooth
$\mathbb{W}(\mathbb{\overline{F}}_l)[G_2(F)]$ -modules and let
$\mathfrak{e}$ be a primitive idempotent of
$\mathcal{Z}$, then $\mathfrak{e}\mathcal{Z} \otimes
\mathbb{\overline{F}}_l$ has many nilpotents in general. But when
$l$ is a banal prime, that is, $1$, $q_F$,
$q_F^2$ are distinct mod
$l$, it follows from \cite[Example 13.9]{MR3508741}, that the
$\overline{\mathbb{F}}_l$-algebra
$\mathbb{W}(\mathbb{\overline{F}}_l)$ is reduced. Hence from
\cite[Section 5]{moss_mod_l_nilpotent_gamma}, we get the proof of this lemma.
\end{proof}
Now we prove Theorem \ref{intro_n=3_thm}.
\begin{theorem}\label{n=3_thm}
Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a
finite Galois extension of $F$ with $[E:F] =l$, where $p$ and $l$ are
distinct primes with $p,l \not =3$, and $l$ is banal for
${\rm GL}_2(F)$. Let $\pi_F$ be an integral $l$-adic cuspidal
representations of ${\rm GL}_3(F)$ and let $\pi_E$ be the base
change of $\pi_F$. Then
\begin{center}
$\widehat{H}^0(\pi_E) \simeq r_l(\pi_F)^{(l)}.$
\end{center}
\end{theorem}
\begin{proof}
Consider the Whitaker model $\mathbb{W}(\pi_E, \Theta_E)$ of $\pi_E$. Recall
that the space of functions $W \in \mathbb{W}(\pi_E, \Theta_E)$, taking values
in $\mathbb{\overline{Z}}_l$, is denoted by $\mathbb{W}^0(\pi_E, \Theta_E)$ .
Then by \cite[Theorem 2]{MR2058628}, the $\overline{\mathbb{Z}}_l$-module
$\mathbb{W}^0(\pi_E, \Theta_E)$ is invariant under the action of $G_3(E)$. The
restriction map $W \longmapsto W|_{P_3(E)}$ gives an isomorphism
$\mathbb{W}^0(\pi_E, \Theta_E) \simeq \mathbb{K}^0_E(\pi_E,\Theta_E)$
\cite[Corollary 4.3]{matringe2022kirillov}.
Then we have
\begin{center}
$\widehat{H}^0\big(\mathbb{W}^0(\pi_E, \Theta_E)\big) \simeq \widehat{H}^0
\big(\mathbb{K}_E^0(\pi_E,\Theta_E)\big) \simeq \mathcal{K}_F^0(\Theta_F^l)$.
\end{center}
Now take $W \in \widehat{H}^0 \big(\mathbb{K}_E^0(\pi_E, \Theta_E)\big)$ and
let $\widetilde{W}=\pi_E(w_3)W$ , $W^\dagger= r_l(\pi_F)^{(l)}(w_3)W$. We will
show that $\widetilde{W} = W^\dagger$. But in view of Lemma (\ref{j}) it is
enough to show that
$$ \sum_{r \in \mathbb{Z}} c^F_r\big(\widetilde{W},\sigma_F^{(l)}(w_3)W'\big)
q_F^{-r/2}X^{-r} = \sum_{r \in \mathbb{Z}} c^F_r\big(W^\dagger,
\sigma_F^{(l)}(w_3)W'\big) q_F^{-r/2}X^{-r}, $$
for all $W' \in \mathbb{W}(\sigma, \Theta_F^{-1})$, and for all irreducible
$l$-modular generic representations $\sigma_F$ of $G_2(F)$.
We first consider the case where $\sigma_F$ is a cuspidal representation of
$G_2(F)$. Then there exists an $l$-adic integral cuspidal representation
$\widetilde{\sigma_F}$ of $G_2(F)$ such that $r_l(\widetilde{\sigma_F}) =
\sigma_F$ \cite{MR1395151}. Let $\widetilde{\sigma_E}$ be the base change of
$\widetilde{\sigma_F}$ to $G_2(E)$. Note that $\widetilde{\sigma_E}$ is an
$l$-adic cuspidal representation as $l \not= 2$. By Theorem (\ref{m}), we have
$$ \widehat{H}^0(\widetilde{\sigma_E}) \simeq \sigma_F^{(l)}. $$
Hence, the restriction map induces a natural surjection
$\mathbb{K}^0(\sigma_E,\Theta_E^{-1})^\Gamma \otimes \overline{\mathbb{F}}_l
\longrightarrow \mathcal{K}^0_F(\Theta_F^{-l})$. Let $W' \in
\widehat{H}^0\big(\mathbb{K}_E^0(\widetilde{\sigma_E}, \Theta_E^{-1})\big)$.
Then there exists $\mathcal{S} \in \mathbb{K}_E^0(\pi_E, \Theta_E)^\Gamma$ and
$\mathcal{H'} \in \mathbb{K}_E^0(\widetilde{\sigma_E}, \Theta_E^{-1})^\Gamma$
such that via the composite maps
\begin{equation}\label{75}
\mathbb{K}^0(\pi_E,\Theta_E)^\Gamma \xrightarrow{mod-l}
\mathbb{K}^0(\pi_E,\Theta_E)^\Gamma \otimes \mathbb{\overline{F}}_l
\longrightarrow \widehat{H}^0\big(\mathbb{K}^0(\pi_E,\Theta_E)\big)
\end{equation}
and
\begin{equation}\label{76}
\mathbb{K}^0(\widetilde{\sigma_E},\Theta_E^{-1})^\Gamma \xrightarrow{mod-l}
\mathbb{K}^0(\widetilde{\sigma_E},\Theta_E^{-1})^\Gamma \otimes
\mathbb{\overline{F}}_l
\longrightarrow\widehat{H}^0\big(\mathbb{K}^0(\widetilde{\sigma_E},\Theta_E^{-1
})\big),
\end{equation}
the images of $\mathcal{S}$ and $\mathcal{H'}$ are $W$ and $W'$ respectively.
Now the functional equation in subsection (\ref{w}) gives the following
relations :
\begin{equation}\label{2}
\sum_{r \in \mathbb{Z}}
c^E_r\big(\pi_E(w_3)\mathcal{S},\widetilde{\sigma_E}(w_2)\mathcal{H}'\big)
q_F^{-\frac{r}{2}f}X^{-r}
= \gamma(X, \pi_E, \widetilde{\sigma_E}, \Theta_E) \sum_{r \in \mathbb{Z}}
c^E_r\big(\mathcal{S},\mathcal{H}'\big) q_F^{\frac{r}{2}f}X^{r}
\end{equation}
and
\begin{equation}\label{3}
\sum_{r \in \mathbb{Z}} c^F_r\big(W^\dagger,\sigma_F^{(l)}(w_2)W'\big)
q_F^{-\frac{r}{2}}X^{-r} = \gamma\big(X, r_l(\pi_F)^{(l)}, \sigma_F^{(l)},
\Theta_F^l\big)\sum_{r \in \mathbb{Z}} c^F_r\big(W, W'\big)
q_F^{\frac{r}{2}}X^{r},
\end{equation}
where $f$ is the residue degree of the extension $E$ of $F$. Note that
(\ref{2}) is a functional equation for $l$-adic representations \cite[Corollary
3.11]{MR3595906} and (\ref{3}) is a functional equations for $l$-modular
representations \cite[Corollary 3.11]{MR3595906}. Using Proposition (\ref{1}) ,
we have
\begin{equation}\label{7}
\int_{Y^r_E}W
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
W'(g) \,dg = \int_{Y^r_F}W
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
W'(g) \,dg
\end{equation}
and
\begin{equation}\label{8}
\int_{Y^r_E}\big(\pi_E(w_3)W\big)
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
\big(\widetilde{\sigma_E}(w_2)W'\big)(g) \,dg =
\int_{Y^r_F}\big(\pi_E(w_3)W\big)
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
\big(\widetilde{\sigma_E}(w_2)W'\big)(g) \,dg.
\end{equation}
Under the composite map (\ref{75}) and (\ref{76}), the equation (\ref{2})
becomes
\begin{equation}\label{4}
\sum_{r \in \mathbb{Z}} c^F_r\big(\pi_E(w_3)W,\widetilde{\sigma_E}(w_2)W'\big)
q_F^{-\frac{r}{2}}X^{-r}
= r_l\big(\gamma\big(X, \pi_E, \widetilde{\sigma_E}, \Theta_E\big)\big) \sum_{r
\in \mathbb{Z}} c^F_r\big(W, W'\big) q_F^{\frac{r}{2}}X^{r}.
\end{equation}
In view of the equations (\ref{2}) to (\ref{4}), it remains to show :
$$ r_l\big(\gamma(X, \pi_E, \widetilde{\sigma_E}, \Theta_E)\big) =
\gamma\big(X, r_l(\pi_F)^{(l)}, \sigma_F^{(l)}, \Theta_F^l\big). $$
Now using Lemma (\ref{i}), we have the following relation :
\begin{center}
$\epsilon(X, \pi_E, \widetilde{\sigma_E}, \Theta_E) = \epsilon(X,
\Pi_{\pi_E}\otimes \Pi_{\widetilde{\sigma_E}}, \Theta_E) =\epsilon\big(X, \rm
ind$$_{\mathcal{W}_E}^{\mathcal{W}_F}(\Pi_{\pi_E} \otimes
\Pi_{\widetilde{\sigma_E}}), \Theta_F\big)$
\end{center}
Using the defintion of base change, the above relations turns out to be of the
form :
\begin{center}
$\epsilon\big(X, \rm ind$$_{\mathcal{W}_E}^{\mathcal{W}_F}(\Pi_{\pi_E} \otimes
\Pi_{\widetilde{\sigma_E}}), \Theta_F\big) = \epsilon\big(X, \Pi_{\pi_F}
\otimes \Pi_{\widetilde{\sigma_F}} \otimes \rm
ind$$_{\mathcal{W}_E}^{\mathcal{W}_F}(1_E), \Theta_F\big)$
\end{center}
Now in view of the Theorem (\ref{k}), we have $\epsilon(X, \pi_E,
\widetilde{\sigma_E}, \Theta_E) = \gamma(X, \pi_E, \widetilde{\sigma_E},
\Theta_E)$. Then using $l$-adic local Langlands correspondence and reduction
mod-$l$, we have
\begin{center}
$r_l\Big(\epsilon\big(X, \Pi_{\pi_F} \otimes \Pi_{\widetilde{\sigma_F}} \otimes
\rm ind$$_{\mathcal{W}_E}^{\mathcal{W}_F}(1_E), \Theta_F\big)\Big) =
\epsilon\big(X, r_l(\pi_F), \sigma_F, \Theta_F\big)^l$
\end{center}
Finally by the lemma (\ref{ar}), we get that $\gamma\big(X, r_l(\pi_F),
\sigma_F, \Theta_F\big)^l = \gamma\big(X, r_l(\pi_F)^{(l)}, \sigma_F^{(l)},
\Theta_F^l\big)$.
Next consider $\sigma_F$ to be an irreducible principal series representation
of $G_2(F)$. Suppose $\sigma_F = \rm ind$$_{B_2(F)}^{G_2(F)}(\chi_1 \otimes
\chi_2)$, for some characters $\chi_1, \chi_2$ of $F^\times$. Then there exist
a lift of $\sigma_F$ to an $l$-adic irreducible representation
$\widetilde{\sigma_F}$ which is of the form
$\widetilde{\sigma_F} = \rm ind$$_{B_2(F)}^{G_2(F)}(\widetilde{\chi}_1 \otimes
\widetilde{\chi}_2)$, where $\widetilde{\chi}_1$, $\widetilde{\chi}_2$ are the
characters of $F^\times$, taking values in $\mathbb{\overline{Z}}_l^\times$
such that $r_l(\widetilde{\chi}_j) = \chi_j$ for $j =1, 2$. Let
$\widetilde{\sigma_E}$ be the base change of $\widetilde{\sigma_F}$ to
$G_2(E)$. Then $\widetilde{\sigma_E}$ is an irreducible principal series of the
form $\widetilde{\sigma_E} = \rm ind$$_{B_2(E)}^{G_2(E)}(\eta_1 \otimes
\eta_2)$, where for each $j$, $\eta_j = \widetilde{\chi}_j \circ \rm Nr$ and
$\rm Nr$$ :E^\times \longrightarrow F^\times$ is the usual norm map.
The cyclic group $\Gamma$ acts on the space $\mathbb{W}^0(\widetilde{\sigma_E},
\Theta_E^{-1})$, and we have
$\widehat{H}^0(\widetilde{\sigma_E}) \simeq \sigma_F^{(l)}$. Let $W' \in
\widehat{H}^0\big(\mathbb{K}^0(\widetilde{\sigma_E}, \Theta_E^{-1})\big)$. Then
there exists $\mathcal{S} \in \mathbb{K}^0(\pi_E, \Theta_E)^\Gamma$ and
$\mathcal{H}' \in \mathbb{K}^0_E(\widetilde{\sigma_E},\Theta_E^{-1})^\Gamma$
such that via the composite map (\ref{75}) and (\ref{76}), $\mathcal{S}$ and
$\mathcal{H'}$ is mapped to $W$ and $W'$ respectively. Now using the functional
equation in the subsection (\ref{w}), we have
\begin{equation}\label{5}
\sum_{r \in \mathbb{Z}}
c^E_r\big(\pi_E(w_3)\mathcal{S},\widetilde{\sigma_E}(w_2)\mathcal{H}'\big)
q_F^{-\frac{r}{2}f}X^{-r}
= \gamma(X, \pi_E, \widetilde{\sigma_E}, \Theta_E) \sum_{r \in \mathbb{Z}}
c^E_r\big(\mathcal{S},\mathcal{H}'\big) q_F^{\frac{r}{2}f}X^{r},
\end{equation}
\begin{equation}\label{6}
\sum_{r \in \mathbb{Z}} c^F_r\big(W^\dagger,\sigma_F^{(l)}(w_2)W'\big)
q_F^{-\frac{r}{2}}X^{-r} = \gamma\big(X, r_l(\pi_F)^{(l)}, \sigma_F^{(l)},
\Theta_F^l\big)\sum_{r \in \mathbb{Z}} c^F_r\big(W, W'\big)
q_F^{\frac{r}{2}}X^{r}
\end{equation}
Using Proposition (\ref{1}) , we have
\begin{equation}\label{85}
\int_{Y^r_E}W
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
W'(g) \,dg = \int_{Y^r_F}W
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
W'(g) \,dg
\end{equation}
and
\begin{equation}\label{86}
\int_{Y^r_E}\big(\pi_E(w_3)W\big)
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
\big(\widetilde{\sigma_E}(w_2)W'\big)(g) \,dg =
\int_{Y^r_F}\big(\pi_E(w_3)W\big)
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
\big(\widetilde{\sigma_E}(w_2)W'\big)(g) \,dg.
\end{equation}
Under the maps (\ref{75}) and (\ref{76}), the equation (\ref{5}) turns out to be
\begin{equation}\label{9}
\sum_{r \in \mathbb{Z}} c^F_r\big(\pi_E(w_3)W,\widetilde{\sigma_E}(w_2)W'\big)
q_F^{-\frac{r}{2}}X^{-r}
= r_l\big(\gamma(X, \pi_E, \widetilde{\sigma_E}, \Theta_E)\big) \sum_{r \in
\mathbb{Z}} c^F_r\big(W, W'\big) q_F^{\frac{r}{2}}X^{r}.
\end{equation}
In view of the equations (\ref{5}) to (\ref{86}), we have to show that
$$ r_l\big(\gamma(X, \pi_E, \widetilde{\sigma_E}, \Theta_E)\big) =
\gamma\big(X, r_l(\pi_F)^{(l)}, \sigma_F^{(l)}, \Theta_F^l\big). $$
Now using \cite[Theorem 4.1]{MR3595906}, we have the following factorisation of
local $\gamma$-factors :
\begin{center}
$\gamma(X, \pi_E, \widetilde{\sigma_E}, \Theta_E) = \gamma(X, \pi_E, \eta_1 ,
\Theta_E)$ $\gamma(X, \pi_E, \eta_2 , \Theta_E)$
\end{center}
Now using Lemma (\ref{i}), we have for each $j$
\begin{center}
$\epsilon(X, \pi_E, \eta_j, \Theta_E) = \epsilon(X, \Pi_{\pi_E}\otimes
\Pi_{\eta_j}, \Theta_E) =\epsilon\big(X, \rm
ind$$_{\mathcal{W}_E}^{\mathcal{W}_F}(\Pi_{\pi_E} \otimes \Pi_{\eta_j}),
\Theta_F\big)$
\end{center}
Then using base change, we get
\begin{center}
$\epsilon\big(X, \rm ind$$_{\mathcal{W}_E}^{\mathcal{W}_F}(\Pi_{\pi_E} \otimes
\Pi_{\eta_j}), \Theta_F\big) = \epsilon\big(X, \Pi_{\pi_F} \otimes
\Pi_{\widetilde{\chi}_j} \otimes \rm
ind$$_{\mathcal{W}_E}^{\mathcal{W}_F}(1_E), \Theta_F\big)$
\end{center}
By Theorem (\ref{k}), we have $\gamma(X, \pi_E, \eta_j , \Theta_E) =
\epsilon(X, \pi_E, \eta_j , \Theta_E)$. Then using reduction mod-$l$ and
$l$-adic local Langlands correspondence, we get
\begin{center}
$r_l\Big(\epsilon\big(X, \Pi_{\pi_F} \otimes \Pi_{\widetilde{\chi_j}} \otimes
\rm ind$$_{\mathcal{W}_E}^{\mathcal{W}_F}(1_E), \Theta_F\big)\Big) =
\epsilon\big(X, r_l(\pi_F), \chi_j, \Theta_F\big)^l$.
\end{center}
Now using Lemma (\ref{ar}), we have $\gamma\big(X, r_l(\pi_F), \chi_j,
\Theta_F\big)^l = \gamma\big(X, r_l(\pi_F)^{(l)}, \chi_j^l, \Theta_F^l\big)$,
for all $j$. Finally using again \cite[Theorem 4.1]{MR3595906}, we get
$$ r_l\big(\gamma(X, \pi_E, \widetilde{\sigma_E}, \Theta_E)\big) =
\gamma\big(X, r_l(\pi_F)^{(l)}, \sigma_F^{(l)}, \Theta_F^l\big). $$
Now consider the one remaining case when $\sigma_F$ is a twist of a Steinberg.
Then $\sigma_F$ lifts to an $l$-adic representation $\widetilde{\sigma_F}$ of
$G_2(F)$ which is also a twist of Steinberg. Let $\widetilde{\sigma_E}$ be the
base change of $\widetilde{\sigma_F}$ to $G_2(E)$. Then $\widetilde{\sigma_E}$
is a twist of Steinberg. Now $\sigma_F$ and $\widetilde{\sigma_E}$ are the
quotients of two principal series representations $\tau_F$ and
$\widetilde{\tau_E}$ respectively, where $\widehat{H}^0(\widetilde{\tau_E})
\simeq \tau_F^{(l)}$. Hence we have $\widehat{H}^0(\widetilde{\sigma_E}) \simeq
\sigma_F^{(l)}$.
Now for $W' \in \widehat{H}^0\big(\mathbb{K}^0(\widetilde{\sigma_E},
b\Theta_E^{-1})\big)$, there exists $\mathcal{S} \in \mathbb{K}^0(\pi_E,
\Theta_E)^\Gamma$ and $\mathcal{H}' \in
\mathbb{K}^0_E(\widetilde{\sigma_E},\Theta_E^{-1})^\Gamma$ such that via the
composite map (\ref{75}) and (\ref{76}), $\mathcal{S}$ and $\mathcal{H'}$ is
mapped to $W$ and $W'$ respectively. Now using the functional equation in the
subsection (\ref{w}), we have
\begin{equation}\label{80}
\sum_{r \in \mathbb{Z}}
c^E_r\big(\pi_E(w_3)\mathcal{S},\widetilde{\sigma_E}(w_2)\mathcal{H}'\big)
q_F^{-\frac{r}{2}f}X^{-r}
= \gamma(X, \pi_E, \widetilde{\sigma_E}, \Theta_E) \sum_{r \in \mathbb{Z}}
c^E_r\big(\mathcal{S},\mathcal{H}'\big) q_F^{\frac{r}{2}f}X^{r},
\end{equation}
\begin{equation}\label{81}
\sum_{r \in \mathbb{Z}} c^F_r\big(W^\dagger,\sigma_F^{(l)}(w_2)W'\big)
q_F^{-\frac{r}{2}}X^{-r} = \gamma\big(X, r_l(\pi_F)^{(l)}, \sigma_F^{(l)},
\Theta_F^l\big)\sum_{r \in \mathbb{Z}} c^F_r\big(W, W'\big)
q_F^{\frac{r}{2}}X^{r}
\end{equation}
Using Proposition (\ref{1}) , we have
\begin{equation}\label{82}
\int_{Y^r_E}W
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
W'(g) \,dg = \int_{Y^r_F}W
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
W'(g) \,dg
\end{equation}
and
\begin{equation}\label{83}
\int_{Y^r_E}\big(\pi_E(w_3)W\big)
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
\big(\widetilde{\sigma_E}(w_2)W'\big)(g) \,dg =
\int_{Y^r_F}\big(\pi_E(w_3)W\big)
\begin{pmatrix}
g & 0\\
0 & 1
\end{pmatrix}
\big(\widetilde{\sigma_E}(w_2)W'\big)(g) \,dg.
\end{equation}
Now under the maps (\ref{75}) and (\ref{76}), the equation (\ref{80}) gives
\begin{equation}\label{84}
\sum_{r \in \mathbb{Z}} c^F_r\big(\pi_E(w_3)W,\widetilde{\sigma_E}(w_2)W'\big)
q_F^{-\frac{r}{2}}X^{-r}
= r_l\big(\gamma(X, \pi_E, \widetilde{\sigma_E}, \Theta_E)\big) \sum_{r \in
\mathbb{Z}} c^F_r\big(W, W'\big) q_F^{\frac{r}{2}}X^{r}.
\end{equation}
In view of the equations (\ref{82}) and (\ref{83}), it remains to show that
$$ r_l\big(\gamma(X, \pi_E, \widetilde{\sigma_E}, \Theta_E)\big) =
\gamma\big(X, r_l(\pi_F)^{(l)}, \sigma_F^{(l)}, \Theta_F^l\big). $$
Now by \cite[Remark 3.3]{MR701565}, we have the following equality of gamma
factors :
\begin{center}
$\gamma\big(X, \pi_E,\widetilde{\sigma_E}, \Theta_E\big) = \gamma\big(X,
\widetilde{\sigma_E}, \widetilde{\tau_E}, \Theta_E\big)$.
\end{center}
But for principal series representations $\tau_E$ and $\tau_F$ satisfying
$\widehat{H}^0(\widetilde{\tau_E}) \simeq \tau_F^{(l)}$, we have already proved
that $$ r_l\big(\gamma(X, \pi_E, \widetilde{\tau_E}, \Theta_E)\big) =
\gamma\big(X, r_l(\pi_F)^{(l)}, \tau_F^{(l)}, \Theta_F^l\big). $$
Finally, using \cite[Corollary 4.2]{MR3595906}, we have
$$ r_l\big(\gamma(X, \pi_E, \widetilde{\sigma_E}, \Theta_E)\big) =
\gamma\big(X, r_l(\pi_F)^{(l)}, \sigma_F^{(l)}, \Theta_F^l\big). $$
Hence we prove the theorem.
\end{proof}
\bibliographystyle{amsalpha}
| 32,182 |
\section{Introduction}
\label{sec:intro}
Authors of high performance applications rely on benchmark suites to detect and avoid
program regressions. However, many developers often run benchmarks and interpret their
results in an ad hoc manner with little statistical rigor. This ad hoc interpretation wastes
development time and can lead to misguided decisions that worsen performance.
In this paper, we consider the problem of designing a language- and
platform-agnostic benchmarking methodology that is suitable for continuous
integration (CI) pipelines and manual user workflows. Our methodology
especially focuses on the accommodation of benchmarks whose expected executions
times are short enough that timing measurements are vulnerable to error due to
insufficient system timer accuracy (generally on the order of microseconds or
shorter).
\subsection{Accounting for performance variations}
\label{sec:variations}
Modern hardware and operating systems introduce many confounding factors that complicate a
developer's ability to reason about variations in user space application
performance~\cite{HP5e}.\footnote{A summary of these factors can be found in the
\href{https://github.com/JuliaCI/BenchmarkTools.jl}{\lstinline|BenchmarkTools|}
documentation in the
\href{https://github.com/JuliaCI/BenchmarkTools.jl/blob/4db27210d43abf2c55226366f3a749afe1d64951/doc/linuxtips.md}{docs/linuxtips.md}
file.} Consecutive timing measurements can fluctuate, possibly in a correlated
manner, in ways which depend on a myriad of factors such as environment
temperature, workload, power availability, and network traffic, and operating
system (OS) configuration.
There is a large body of research on system quiescence aiming to identify and
control for individual sources of variation in program run time measurements,
each of which must be ameliorated in its own way. Many factors stem from OS
behavior, including CPU frequency scaling \cite{RHEL6}, address space layout
randomization (ASLR)~\cite{Shacham2004}, virtual memory
management~\cite{Oyama2014,Oyama2016}, differences between CPU privilege
levels~\cite{Zaparanuks2009}, context switches due to interrupt handling~\cite{Tsafrir2007},
activity from system daemons and cluster managers~\cite{Petrini2003}, and suboptimal
process- and thread-level scheduling~\cite{Lozi2016}. Even seemingly irrelevant
configuration parameters like the size of the OS environment can confound experimental
reproducibility by altering the alignment of data in memory~\cite{Mytkowicz2009}. Other
sources of variation come from specific language features or implementation details. For
example, linkers for many languages are free to choose the binary layout of the library or
executable arbitrarily, resulting in non-deterministic memory layouts~\cite{Georges2008}.
This problem is exacerbated in languages like C++, whose compilers introduce arbitrary name
mangling of symbols~\cite{Kalibera2005}. Overzealous compiler optimizations can also
adversely affect the accuracy of hardware counters~\cite{Zaparanuks2009}, or in extreme
cases eliminate key parts of the benchmark as dead code. Yet another example is garbage
collector performance, which is influenced from system parameters such as heap
size~\cite{Blackburn2004}.
\subsection{Statistics of timing measurements are not i.i.d.}
\label{sec:toughstats}
\begin{figure}
\centering
\begin{subfigure}{0.22\textwidth}
\centering
\includegraphics[width=\textwidth]{simple_branchsum_fast}
\caption{Benchmark 1: Unimodal with skew and large outliers}
\end{subfigure}%
~
\begin{subfigure}{0.22\textwidth}
\centering
\includegraphics[width=\textwidth]{bimodal_branchsum}
\caption{Benchmark 2: Bimodal}
\end{subfigure}
\begin{subfigure}{0.22\textwidth}
\centering
\includegraphics[width=\textwidth]{drift_manyallocs_slow}
\caption{Benchmark 3: Drift}
\end{subfigure}
~
\begin{subfigure}{0.22\textwidth}
\centering
\includegraphics[width=\textwidth]{bimodal_drift_sumindex}
\caption{Benchmark 4: Bimodal with drift}
\end{subfigure}
\caption{Variability in the mean benchmark time across multiple trials, showing
that the mean has non-i.i.d., non-normal behavior in four different benchmarks.
Each point represents a mean time computed from trial of 10,000 measurements.
The horizontal axis is the index of the trial, while the vertical axis is time.}
\label{fig:meandistributions}
\vspace{-0.45cm}
\end{figure}
The existence of many sources of performance variation result in timing
measurements that are not necessarily independent and identically distributed
(i.i.d.). As a result, many textbook statistical approaches fail due to
reliance on the central limit theorem, which does not generally hold in the
non-i.i.d. regime. In particular, empirical program timing distributions are
also often heavy-tailed, and hence contain many outliers that distort measures
of central tendency like the mean, and are not captured in others like the
median.
The violation of the central limit theorem can be seen empirically in many Julia benchmarks.
For example, Figure~\ref{fig:meandistributions} shows that none of the four illustrative
benchmarks considered in this paper exhibit normality in the sample mean. Instead, we see
that the mean demonstrates skewed density and outliers in the first benchmark, bimodality in
the second and fourth benchmarks, and upward drift in the third and fourth benchmarks.
Many other authors have also noted the lack of textbook statistical behavior in timing
measurements~\cite{Gil2011,Chen2015,Rehn2015,Barrett2016}. Authors have also noted the poor
stastistical power of standard techniques such as $F$-tests or Student t-tests for benchmark
timings~\cite{Lilja2000,Mytkowicz2009,Kalibera2013,Chen2015,Barrett2016}. Parametric outlier
detection techniques, such as the 3-sigma rule used in benchmarking software like
\lstinline|AndroBench|\cite{Kim2012}, can also fail when applied to non-i.i.d. timing measurements.
There is a lack of consensus over how non-ideal timing measurements should be treated. Some
authors propose automated outlier removal and analyzing the remaining bulk
distribution~\cite{Kim2012}; however, these methods run the risk of fundamentally distorting
the true empirical distribution for the sake of normal analysis. Other authors have proposed
purposely introducing randomness in the form of custom OS kernels
\cite{Tessellation,Akkan2012}, custom compilers providing reproducible~\cite{Georges2008} or
consistently randomized~\cite{Curtsinger2013} binary layouts, or low-variability garbage
collectors~\cite{Huang2004}. Unfortunately, these methods are specific to a single
programming language, implementation, and/or platform. Furthermore, these methods often
require administrative privileges and drastic modifications to the benchmarking environment,
which are impractical to demand from ordinary users.
\subsection{Existing benchmarking methodologies}
\label{sec:existingtools}
While it is impossible to eliminate performance variation
entirely~\cite{Alcocer2015,Barrett2016}, benchmarking methodologies that attempt to account
for both measurement error and external sources of variation do exist. For example, the
Haskell microbenchmarking package \lstinline|criterion|~\cite{criterion} attempts to thwart
error due to timer inaccuracy by growing the number of benchmark executions per timing
measurement as more timing measurements are obtained. After all measurements are taken, a
summary estimate of the benchmark run time is obtained by examining the derivative of the
ordinary least squares regression line at the point of a single evaluation point. There are
three disadvantages to this approach. First, the least squares fit is sensitive to
outliers~\cite{Maronna2006} (though \lstinline|criterion| does warn the user if outliers are
detected). Second, measurements made earlier in this experiment are highly vulnerable to
timer error, since few benchmark repetitions are used. These early measurements can skew the
regression, and hence also skew the final run time estimate. Third, measurements made later in
the experiment can repeat the benchmark more times than are necessary to overcome timer
error, constituting an inefficient use of experiment time.
Another approach focuses on eliminating ``warm-up'', assuming that first few
runs of a benchmark are dominated by transient background events that
eventually vanish and the timing measurements eventually become
i.i.d.~\cite{Kalibera2013}. Their approach is largely platform-agnostic,
recognizes the pitfalls of inter-measurement correlations, and
acknowledges that merely increasing the number of benchmark repetitions is not always a
sufficient strategy to yield i.i.d. samples. However, the assumption (based on
common folklore) that benchmarks exhibit warm-up is often false, as is clear
from Fig.~\ref{fig:meandistributions} and elsewhere~\cite{Barrett2016}:
even Ref.~\cite{Kalibera2013} itself resorts to ad hoc judgment to work
around the lack of a distinct warm-up phase. There is also no reason to believe that even if warm-up were observed, that the post-warm-up timings will be i.i.d.
Furthermore, the authors do not report if their statistical tools generate
correct confidence intervals. The moment-corrected formulae described are
accurate only for near-normal distributions, which is unlikely to hold for the
kinds of distributions we observed in real world statistics. Additionally, the
methodology requires a manual calibration experiment to be run for each
benchmark, compiler, and platform combination. As a result, this method is is
difficult to automate on the scale of Julia's standard library benchmark suite,
which contains over 1300 benchmarks, and is frequently expanded to improve
performance test coverage.
Below, we describe our methodology to benchmarking for detecting performance regressions,
and how it is justified from a microscopic model for variations in timing measurements. To
the best of our knowledge, our work is the first benchmarking methodology that can be fully
automated, is robust in its assumption of non-i.i.d. timing measurement statistics, and
makes efficient use of a limited time budget.
\section{Terms and definitions}
\label{sec:notation}
\begin{itemize}
\item
$P_0$, $P$, $Q_0$, and $Q$ denote \textbf{benchmarkable programs}, each
defined by a tape (sequence) of instructions.
\item
$I^{[i]}_{P}$ is the $i^{\textrm{th}}$ \textbf{instruction} in the tape
defining program $P$.
Instructions are indexed in bracketed superscripts, $\cdot^{[i]}$.
\item
$D^{[i]}_{P}$ is the \textbf{delay instruction} associated with $I^{[i]}_{P}$.
Delay instructions are defined in Sec.~\ref{sec:model}.
\item
$T_i$ is a \textbf{timing measurement}, namely the amount of time taken to
perform $n_i$ \textbf{executions} of a benchmarkable program. This quantity
is directly measurable in an experiment.
\item
$t$ is a \textbf{theoretical execution time}.
$t_{P_0}$ is the minimum time required to perform a single execution of
$P_0$ on a given computer.
\item
\textbf{Estimated quantities} are denoted with a hat, $\hat\cdot$.
For example, $\hat{t}_{P_0}$ is an estimate of the theoretical execution
time $t_{P_0}$.
\item
A benchmark \textbf{experiment} is a recipe for obtaining multiple timing
measurements for a benchmarkable program. Experiments can be executed to
obtain \textbf{trials}. The
$i^{\textrm{th}}$ trial of an experiment is a collection of timing measurements
$T^{\{i\}}_1, \dots T^{\{i\}}_j, \dots T^{\{i\}}_k$. Trial indices are always
written using embraced superscripts, $\cdot^{\{i\}}$.
\item
$\tau$ denotes time quantities that are external to the benchmarkable program:
\begin{itemize}
\item $\tau_{\textrm{budget}}$ is the \textbf{time budget} for an experiment.
\item $\tau_{\textrm{acc}}$ is the \textbf{accuracy} of the system timer, i.e.\ an upper bound on the maximal error in using the system timer to time an experiment.
\item $\tau_{\textrm{prec}}$ is the \textbf{precision} of the system timer, namely the smallest nonzero time interval measurable by the timer.
\end{itemize}
\item
$x_P^{(i)[j]} \tau^{(i)}$ is the \textbf{time delay} due to the
$i^{\textrm{th}}$ \textbf{delay factor} for delay instruction $D^{[j]}$. Specifically,
$\tau^{(i)}$ is the factor's \textbf{time scale} and $x_P^{(i)[j]}$ is the factor's
\textbf{trigger coefficient}, as introduced Sec.~\ref{sec:model}. Delay factors are indexed with parenthesized
superscripts, $\cdot^{(i)}$.
\item
$\epsilon$ is the measurement error due to timer inaccuracy.
\item
$E_m = \frac{T_m}{n_m} - t_{P_0}$ is the total contribution of all delay
factors found in measurement $m$, plus the measurement error $\epsilon$.
\item
$X^{(i)}_P$ is the \textbf{total trigger count} of the $i^{\textrm{th}}$
delay factor during the execution of program $P$.
\item
$\nu$ is an \textbf{oracle function} that, when evaluated at an execution
time $t$, estimates an appropriate $n$ necessary to overcome measurement error
due to
$\tau_{\textrm{acc}}$ and $\tau_{\textrm{prec}}$. The oracle function is described in detail in Sec.~\ref{sec:oracle}.
\end{itemize}
\section{A model for benchmark timing distributions}
\label{sec:model}
We now present a statistical description of how benchmark programs behave
when they are run in serial. Our model deliberately avoids the problematic
assumption that timing measurments are i.i.d. We will use this model later to
justify the design of a new automated experimental procedure.
\subsection{User benchmarks run with uncontrollable delays}
\label{sec:programmodel}
Let $P_0$ be a deterministic benchmark program which consists of an instruction tape
consisting of $k$ instructions:
\begin{equation}
P_0 = \left[I^{[1]}, I^{[2]}, \dots I^{[k]}\right].
\end{equation}
Let $\tau^{[i]}$ be the run time of instruction $I^{[i]}$. Then, the total run
time of $P_0$ can be written $t_{P_0} = \sum_{i=1}^N \tau^{[i]}$.
While a computer may be directed to execute $P_0$, it may not necessarily run the program's
instructions as they are originally provided, since the environment in which $P_0$
runs is vulnerable to the factors described earlier in Sec.~\ref{sec:variations}. Crucially,
these factors only \textit{delay} the completion of the original instructions, rather than
speed them up.\footnote{While there are a very few external factors which might speed up
program execution, such as frequency scaling~\cite{RHEL6}, they can be easily accounted for
by ensuring that power consumption profiles are always set for maximal performance. We
therefore assume that these factors have been accounted for.} Therefore, we call them
\textit{delay factors}; they can be modeled as extra instructions which, when
interleaved with the original instructions, do not change the semantics of
$P_0$, but still add to the program's total run time. Thus, we can define a new
program $P$ which consists of $P_0$'s original instructions interleaved with
additional \textit{delay instructions} $D^{[i]}$:
\begin{equation}
P = \left[I^{[1]}, D^{[1]}, I^{[2]}, D^{[2]}, \dots I^{[k]}, D^{[k]}\right].
\end{equation}
The run time of $P$ can then be written
\begin{equation}
t_P = t_{P_0} + \sum_{i} \tau^{[i]}_D,
\end{equation}
where $\tau^{[i]}_D$ is the execution time of $D^{[i]}$. Since $\tau^{[i]}_D \ge 0$, it
follows that $t_P \ge t_{P_0}$.
The run time of each delay instruction, $\tau^{[i]}_D$, can be further decomposed
into the runtime contributions of individual delay factors. Let us imagine that
each delay factor $j$ can either contribute or not contribute to $D^{[i]}$.
Assuming that each delay factor triggers inside $D^{[i]}$ with constant
probability $p^{[i](j)}$ of taking a fixed time $\tau^{(j)}$, we can then
write:
\begin{equation}
\tau^{[i]}_D = \sum_{i} x_P^{[i](j)} \tau^{(j)},
\end{equation}
where $x_P^{[i](j)}$ is a Bernoulli random variable with success probability $p^{[i](j)}$.
We denote the total number of times the $i^{\textrm{th}}$ delay factor was triggered during
the execution of $P$ as the \textit{trigger count} $X_P^{(i)} = \sum_{j} x_P^{[i](j)}$.
Since the trigger count is a sum of independent Bernoulli random variables with nonidentical
success probabilities, $X_P^{(j)}$ is itself a random variable that follows a Poisson
binomial distribution parameterized by the success probabilities $\left[p^{(1)[j]}, \dots
p^{(k)[j]}\right]$. Our final expression for $t_P$ in terms of these quantities is then:
\begin{align}
t_P &= t_{P_0} + \sum_{i=1}^{k} \tau^{[i]}_D \nonumber \\
&= t_{P_0} + \sum_{i=1}^{k} \sum_{j} x_P^{[i](j)} \tau^{(j)} \nonumber \\
&= t_{P_0} + \sum_{j} X_P^{(j)} \tau^{(j)}.
\end{align}
In summary, our model treats $t_P$ as a random variable whose distribution
depends on the trigger probabilities $p^{[i](j)}$, which are determined by the
combined behavior of the delay factors and the initial benchmark program $P_0$.
\subsection{Repeated benchmark execution is often necessary but not always sufficient}
\label{sec:measuremodel}
As mentioned in Sec.~\ref{sec:existingtools}, experiments which measure program
performance usually incorporate multiple benchmark executions to obtain more
accurate measurements. We now apply our model to show that multiple executions
are necessary to eliminate error due to timer inaccuracy, but are insufficient
to obviate delay factors.
Represent $n$ executions of the program $P_0$ comprised of $k$ instructions as
a single execution of a program $Q_0$, which is the result of concatenating $n$
copies of $P_0$:
\begin{align}
Q_0 &= \left[P_0, P_0, \dots P_0 \right] \nonumber \\
&= \left[I_{P}^{[1]}, \dots I_{P}^{[k]}, I_{P}^{[1]}, \dots I_{P}^{[k]}, I_{P}^{[1]}, \dots I_{P}^{[k]} \right] \nonumber \\
&= \left[I_{Q}^{[1]}, I_{Q}^{[2]}, \dots I_{Q}^{[nk]} \right],
\end{align}
with $I_{P}^{[i]} = I_{Q}^{[i + ck]}$ for $c \in \{0, \dots n - 1\}$. The
subscripts on $I$ denote the program which contains that instruction, with the
0 subsubscript dropped for brevity.
Now interleave delay instructions as before to obtain the program $Q$ that is
actually executed. $Q$ is \textit{not} simply $n$ repetitions of $P$, since the
delay instructions in
$Q$ are not simply copies of the delay instructions in $P$. An observed timing measurement
$T$ of a single execution of $Q$ can be decomposed as:
\begin{align}
T &= t_{Q} + \epsilon \nonumber \\
&= t_{Q_0} + \sum_{j} X_Q^{(j)} \tau^{(j)} + \epsilon \nonumber \\
&= n \, t_{P_0} + \sum_{j} \sum_{i=1}^{nk} x_Q^{[i](j)} \tau^{(j)} + \epsilon,
\end{align}
where $\epsilon$ is the error due to timer inaccuracy (whose magnitude must by definition be smaller than $\tau_\textrm{acc}$).
We may try to determine $t_{P_0}$ from the experimental time as $T/n$, which is
also the gradient of a linear model for $T$ against $n$ when the intercept is
zero. However, our model gives instead:
\vspace{-0.10cm}
\begin{equation}
\frac{T}{n} = t_{P_0} + \frac{\sum_{j} \sum_{i=1}^{nk} x_Q^{[i](j)} \tau^{(j)} + \epsilon}{n}.
\end{equation}
All the terms on the right hand side other than $t_{P_0}$ constitute the
error $E$ in our measurement. For large $n$, the term $\epsilon/n$ arising from
timer inaccuracy becomes negligible, but the behavior of the other term depends
on the specific structure of the delay factors. In the best case, each delay
factor triggers $o(n)$ times, so that $T/n\to t_{P_0}$ as desired. However, in
the worst case, every factor triggers on every instruction, $x_Q^{[i](j)}
= 1$, and the large $n$ behavior of $T/n$ does not reduce to the true run time
$t_{P_0}$, but rather:
\begin{equation}
\label{eq:11}
\lim_{n\to\infty} \frac{T}{n} = t_{P_0} + k \sum_{j} \tau^{(j)}.
\end{equation}
\eqref{eq:11} is a key result of our model: one cannot always reliably
eliminate bias due to external variations simply by executing the benchmark
many times. Whether or not increasing $n$ can render the delay factor term
negligible depends entirely on the distribution of trigger counts $X_Q^{(j)}$,
which are difficult or impossible to control (see Sec.~\ref{sec:variations}).
Therefore, we can only expect that $T/n$ at large $n$ gives us at best an
\textit{overestimate} of the true run time $t_{P_0}$.
\section{An automated procedure for configuring performance experiments}
\label{sec:confexperiment}
In this section, we present an experimental procedure for automatically
selecting useful values of $n$ for a given benchmark program, which can be
justified from our model of serial benchmark execution above. Our procedure
estimates a value for $n$ which primarily minimizes error in timing
measurements and secondarily maximizes the number of measurements obtainable
within a given time budget.
\subsection{An algorithm for estimating the optimal $n$ value}
Given $P_0$ and a total time budget $\tau_{\textrm{budget}}$, we use the automatable
procedure in Alg.~\ref{alg:tuning} for guessing the minimum value of $n$ required to
amortize measurement error due to timer inaccuracy. The algorithm makes use of
an oracle function $\nu$, which is discussed in greater detail below in
Sec.~\ref{sec:oracle}.
\begin{algorithm}
\caption{Estimating $n$, the optimal number of benchmark repetitions required to
minimize timer error and maximize the number of data points obtainable within a
time budget.}
\label{alg:tuning}
\KwIn{$P_0$, $\tau_{\textrm{acc}}$, $\tau_{\textrm{prec}}$, an oracle function $\nu : t \to n$}
\KwOut{$n$}
Let $j = \tau_{\textrm{acc}} / \tau_{\textrm{prec}}$.
For $i \in \{1, \dots j\}$, measure the amount of time it takes to perform $i$
executions of $P_0$, resulting in a collection of timing measurements $T_1, \dots T_j$.
Estimate $t_{P_0}$ as $\hat{t}_{P_0} = \textrm{min}(\frac{T_1}{1}, \dots \frac{T_j}{j})$.
Evaluate $\nu(\hat{t}_{P_0})$ to obtain $n$. Details of $\nu$ are given in Sec.~\ref{sec:oracle}
\end{algorithm}
The upper bound $j$ in Alg.~\ref{alg:tuning} is the ratio of timer accuracy to
timer precision. If each timing measurement consists of more than $j$
repetitions, then the contribution of timer inaccuracy to the total error is
less than $\tau_\textrm{acc} / j = \tau_\textrm{prec}$, and so is too small to
measure. Thus, there is no reason to pick $n > j$. In practice,
$\tau_\textrm{acc}$ need only be an overestimate for the timer accuracy, which
would raise the $n$ determined, but is still an acceptable result.
Alg.~\ref{alg:tuning} need only be applied once per benchmark, since the
estimated $n$ can be cached for use in subsequent experiments on the same
machine. Thus, we consider this algorithm an automated preprocessing step that
does not count against our time budget $\tau_{\textrm{budget}}$. In this
regard, our approach differs significantly from other approaches like
\lstinline|criterion|, which re-determines $n$ every time a benchmark is run.
\subsection{Justifying the minimum estimator}
\label{sec:minimum}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{linear_scan_branchsum}
\caption{Plots of $T/n$ vs.\ $n$ produced by repeated experiments, each
consisting of running Alg.~\ref{alg:tuning} on the \lstinline|branchsum|
benchmark. While each experiment can produce wildly oscillatory curves, the
minimum across all the curves at each $n$ is much smoother and asymptotically
tends toward the same constant value.}
\label{fig:scaling}
\end{figure}
We will now justify Alg.~\ref{alg:tuning}'s use of the minimum to estimate $t_{P_0}$, as
opposed to the more common median or mean.
Consider the total error term for a given timing measurement $E_m = \left(\sum_{i}
X_Q^{(i)} \tau^{(i)} + \epsilon \right)_m / n_m$, such that $T_i/n_i = t_{P_0} +
E_i$. The minimum estimator applied to our timing measurements can then be
written as:
\begin{align}
\hat{t}_{P_0} &= \textrm{min}(\frac{T_1}{1}, \dots \frac{T_j}{j}) \nonumber \\
&= t_{P_0} + \textrm{min}(E_1, \dots E_j).
\end{align}
Thus, $\hat{t}_{P_0}$ is the estimate of $t_{P_0}$ which minimizes the error terms appearing
in our sample.
In the limit where the delay factor time scales are greater than $\tau_{\textrm{acc}}$, the
total error terms will always be positive, such that choosing the smallest timing
measurement will choose the sample with the smallest magnitude of error. If the delay factor
time scales are less than $\tau_{\textrm{acc}}$, choosing the smallest timing measurement
might choose a sample which underestimates $t_{P_0}$ due to negative timer error. In this
case, Alg.~\ref{alg:tuning} will simply pick a larger $n$ than is strictly necessary, which
is still acceptable.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{location_estimators_sumindex}
\caption{The behavior of different location parameters across multiple trials of
the \lstinline|sumindex| benchmark: mean (green filled circles), trimmed mean of
the 5th---95th percentiles (brown filled squares), median (black crosses), and
minimum (blue filled triangles).}
\label{fig:locationmeasures}
\end{figure}
Figs.~\ref{fig:locationmeasures} and \ref{fig:pdfsumindex} provide further
justification for the minimum over other common estimators like the median,
mean, or trimmed mean. Recall from Section~\ref{sec:model} that the error terms
$E_i$ are sampled from a sum of scaled random variables following nonidentical
Poisson binomial distributions. As such, these terms can and do exhibit
multimodal behavior. While estimators like the median and trimmed mean are
known to be robust to outliers~\cite{Maronna2006},
Fig.~\ref{fig:locationmeasures} demonstrates that they still capture bimodality
of the distributions plotted in Fig.~\ref{fig:pdfsumindex}. Thus, these
estimators are undesirable for choosing $n$, since the result could vary
drastically between different executions of Alg.~\ref{alg:tuning}, depending on
which of the estimator's modes was captured in the sample, and hence affect
reproducibility. In contrast, the distribution of the minimum across all
experimental trials is unimodal in all cases we have observed. Thus for our
purposes, the minimum is a unimodal, robust estimator for the location
parameter of a given benchmark's timing distribution.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{kde_pdf_sumindex}
\caption{Kernel density estimates (KDEs) of the probability density functions
(pdfs) across 100 trials of the \lstinline|sumindex| benchmark. Each curve is a
KDE formed from a trial of 10,000 consecutively gathered timing measurements.
Note that the data form two distinct clusters. A cursory investigation did not
reveal any inter-trial correlations that revealed a predictable preference for
which cluster would be observed.}
\label{fig:pdfsumindex}
\end{figure}
\subsection{The oracle function}
\label{sec:oracle}
Our heuristic takes as input an oracle function $\nu(t)$ that maps expected
run times to an optimal number of executions per measurement. While
Alg.~\ref{alg:tuning} does not directly describe $\nu(t)$, appropriate choices
for this function should have the following properties:
\begin{itemize}
\item $\nu(t)$ has a discrete range $\{1, \dots, j\}$.
\item $\nu(t)$ is monotonically decreasing, so that the longer the run time, the fewer repetitions per measurement.
\item $\frac{d\nu}{dt}|_{t \approx \tau_{\textrm{prec}}} \approx 0$,
so that there is only weak dependence on the timer precision parameter,
which may not be accurately known.
\item $\frac{d\nu}{dt}|_{t \approx \tau_{\textrm{acc}}} \approx 0$,
so that there is only weak dependence on the timer accuracy parameter,
which may not be accurately known.
\item $\nu(\tau_{\textrm{prec}}) \approx j$, so that benchmarks that take a short time to run are not repeated more times than necessary to mitigate timer inaccuracy.
\item $\nu(t \ge \tau_{\textrm{acc}}) \approx 1$, so that benchmarks that take a long time to run need not be repeated.
\end{itemize}
There are many functions that satisfy these criteria. One useful example takes
the form of the generalized logistic function:
\begin{equation}
\label{eq:glog}
Y(t) = \floor*{1 + \frac{j - 1}{1 + e^{a (t - b \tau_{\textrm{acc}})}}}
\end{equation}
where reasonable values of $a$ and $b$ are approximately $0.005 < a \tau_{\textrm{prec}} < 0.02$ and $0.4 < b <
0.6$.
In practice, we have found that better results can be achieved by first
approximating $Y(t)$ with a lookup table, then modifying the lookup table based
on empirical observations. This was accomplished by examining many benchmarks
with a variety of known run times at different time scales, seeking for each
run time the smallest $n$ value at which the minimum estimate appears to
converge to a lower bound (e.g.\ around $n = 250$ for the benchmark in
Fig.~\ref{fig:scaling}). Fig.~\ref{fig:oracle} plots both \eqref{eq:glog} and an
empirically obtained lookup table as potential oracle functions.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{oracle}
\caption{Two possible oracle functions for $\nu(t)$ at $\tau_{\textrm{acc}} \approx 1000 \textrm{ns}$, $\tau_\textrm{prec} \approx 1 \textrm{ns}$.
The solid blue curve is an example of an empirically tuned lookup table, while
the dotted black curve is $Y(t)$ from Eq~\ref{eq:glog} with parameters $a =
0.009 / \tau_\textrm{prec}$ and $b = 0.5$.}
\label{fig:oracle}
\end{figure}
\section{Implementation in Julia}
\label{sec:implementation}
The experimental methodology in this paper is implemented in the
\lstinline|BenchmarkTools| Julia
package\footnote{\url{https://github.com/JuliaCI/BenchmarkTools.jl}}. In
addition to the
\lstinline|BaseBenchmarks|\footnote{\url{https://github.com/JuliaCI/BaseBenchmarks.jl}}
and
\lstinline|Nanosoldier|\footnote{\url{https://github.com/JuliaCI/Nanosoldier.jl}}
packages, the \lstinline|BenchmarkTools| package implements the on-demand CI
benchmarking service used by core Julia developers to compare the performance
of proposed language changes with respect to over 1300 benchmarks. Since this
CI benchmarking service began in early 2016, it has caught and prevented the
introduction of dozens of serious performance regressions into Julia's standard
library (defining a serious regression as a $30\%$ or greater increase in a
benchmark's minimum execution time).
The benchmarks referenced in this paper are Julia benchmarks written and
executed using \lstinline|BenchmarkTools|. A brief description of each
benchmark is offered below:
\begin{itemize}
\item The \lstinline|sumindex(a, inds)| benchmark sums over all
\lstinline|a[i]| for all \lstinline|i| in \lstinline|inds|. This test stresses
memory layout via element retrieval.
\item The \lstinline|pushall!(a, b)| benchmark pushes elements from \lstinline|b| into
\lstinline|a| one by one, additionally generating a random number at each iteration (the
random number does not affect the output). This test stresses both random number
generation and periodic reallocation that occurs as part of Julia's dynamic array
resizing algorithm.
\item The \lstinline|branchsum(n)| benchmark loops from \lstinline|1| to \lstinline|n|.
If the loop variable is even, a counter is decremented. Otherwise, an inner loop is
triggered which runs from \lstinline|1| to \lstinline|n|, in which another parity test
is performed on the inner loop variable to determine whether to increment or decrement
the counter. This test stresses periodically costly branching within loop iterations.
\item The \lstinline|manyallocs(n)| allocates an array of \lstinline|n| elements, where
each element is itself an array. The inner array length is determined by a random number
from \lstinline|1| to \lstinline|n|, which is regenerated when each new array is
constructed. However, the random number generator is reseeded before each generation so
that the program is deterministic. This test stresses random number generation and
the frequent allocation of arrays of differing length.
\end{itemize}
The mock benchmark suite referenced in this paper is hosted on GitHub at
\url{https://github.com/jiahao/paper-benchmark}.
\section{Conclusion}
\label{sec:conclusion}
The complexities of modern hardware and software environments produce
variations in benchmark timings, with highly nonideal statistics that
complicate the detection of performance regressions. Timing measurements taken
from real Julia benchmarks confirm the observations of many other authors
showing highly nonideal, even multimodal behavior, exhibited by even the
simplest benchmark codes.
Virtually all timing variations are delays caused by flushing cache lines, task
switching to background OS processes, or similar events. The simple
observation that variations never reduce the run time led us to consider a
straightforward analysis based on a simple model for delays in a serial
instruction pipeline. Our results suggest that using the minimum estimator for
the true run time of a benchmark, rather than the mean or median, is robust to
nonideal statistics and also provides the smallest error. Our model also
revealed some behaviors that challenge conventional wisdom: simply
running a benchmark for longer, or repeating its execution many times, can
render the effects of external variation negligible, even as the error due to
timer inaccuracy is amortized.
Alg.~\ref{alg:tuning} presents an automatable heuristic for selecting the
minimum number of executions of a benchmark per measurement required to defeat
timer error. This strategy has been implemented in the
\lstinline|BenchmarkTools| Julia package, which is employed daily and on demand
as part Julia's continuous integration (CI) pipeline to evaluate the
performance effects of proposed changes to Julia's standard library in a fully
automatic fashion. \lstinline|BenchmarkTools| can also be used to test the
performance of user-authored Julia packages.
\section*{Acknowledgment}
\label{sec:acknowledgement}
We thank the many Julia developers, in particular Andreas Noack (MIT), Steven
G.\ Johnson (MIT) and John M. White (Facebook), for many insightful
discussions.
This research was supported in part by the U.S. Army Research Office under
contract W911NF-13-D-0001, the Intel Science and Technology Center for Big
Data, and DARPA XDATA.
| 10,103 |
\section{Introduction}
\nocite{Pearce}
One of the most striking features of atmospheric vortices, such as tropical
cyclones, is that they often develop a so-called eye; a region of reversed
flow in and around the axis of the vortex. Much has been written about eye
formation, particularly in the context of tropical cyclones, but the key
dynamical processes are still poorly understood \citep{Pearce,Smith,Pearce2}.
Naturally
occurring vortices in the atmosphere are, of course, complicated objects,
whose overall dynamics can be strongly influenced by, for example, planetary
rotation, stratification, latent heat release through moist convection, and
turbulent diffusion. Indeed the structure of eyes in tropical cyclones is
almost certainly heavily influenced by both moist convection and
stratification.
However, the ubiquitous appearance of eyes embedded within large-scale
vortices suggests that the underlying mechanism by which they first form
may be independent (partially if not wholly) of such complexities.
Indeed eye-like structures are observed in other atmospheric vortices such as
tornadoes \citep[][and references therein]{Lugt} or polar lows
\citep{PolarLows}, which
are particularly interesting as they consist of large-scale
convective cyclonic structures observed in high latitudes polar regions.
To put the idea of a simple hydrodynamic mechanism
to the test we consider what is, perhaps, the simplest system in which
eyes may form; that of steady axisymmetric convection in a rotating
Boussinesq fluid. We thus neglect the effects of stratification, and of
moist convection.
Our underlying assumption is that some atmospheric phenomena could be
simple enough to be modelled in a uniform Boussinesq fluid. Indeed, a
recent study by \cite{Guervilly2014} noted that
Boussinesq convection can yield, as in the atmosphere, the formation of
large scale cyclonic vortices.
In this work, we
consider a rotating, cylindrical domain in which
the lower surface is a no-slip boundary, the upper surface stress
free, and the motion driven by a prescribed vertical flux of heat.
In a frame of reference rotating with the lower
boundary, the Coriolis force induces swirl in the convecting fluid,
which in turn sets up an Ekman-like boundary layer on the lower
surface. The primary flow in the vertical plane is then radially
inward near the lower boundary and outward at the upper surface. As
the fluid spirals inward, it carries its angular momentum with it
(subject to some viscous diffusion) and this results in a region of
particularly intense swirl near the axis. The overall flow pattern
is as shown schematically in Figure~\ref{fig1}.
\begin{figure}
\centerline{
\includegraphics[width=0.7\textwidth]{FIGURES/pdffiles/fig1.pdf}
}
\caption{Cartoon showing the global flow pattern in rotating
convection. The motion in the vertical plane consists of the primary
vortex, the eye-wall and the eye, while the azimuthal motion
consists of regions of high angular momentum near the axis and low
(or even negative) angular momentum at larger radii (the vertical axis is
stretched by a coefficient $5$ for readability).}
\label{fig1}
\end{figure}
In the vertical plane the primary vortex has a
clockwise motion, and so has positive azimuthal vorticity. If an eye
forms, however, its motion is anticlockwise in the vertical plane
(Figure~\ref{fig1}), and so the eye is associated with negative azimuthal
vorticity. A key question, therefore, is: where does this negative
vorticity come from? We shall show that it is not generated by
buoyancy, since such forces are locally too weak. Nor does it arise
from so-called vortex tilting, despite the local dominance of this
process, because vortex tilting cannot produce any net azimuthal
vorticity. Rather, the eye acquires its vorticity from the
surrounding fluid by cross-stream diffusion, and this observation
holds the key to eye formation in our simple system.
The region that separates the eye from the primary
vortex is usually called the eyewall, and we shall see that this
thin annular region is filled with intense negative azimuthal
vorticity. So eye formation in our model problem is
really all about the dynamics of creating an eyewall. In this paper
we use numerical experiments to investigate the processes by which
eyes and eyewalls form in our model system. We identify the key
dynamical mechanisms and force balances, and provide a simple
criterion which needs to be met for an eye to form.
\section{Problem Specification and Governing Equations}
We consider the steady flow of a Boussinesq fluid in a rotating,
cylindrical domain of height $H$ and radius $R$, with $R \gg H$. The aspect
ratio is denoted as $\epsilon=H/R$. The flow is described in cylindrical
polar coordinates, $(r,\phi,z)$, where the lower
surface, $z=0$, and the outer radius, $r=R$, are no-slip boundaries. The
upper surface, $z=H$, is impermeable but stress free. The motion is driven
by buoyancy with a fixed upward heat flux maintained between the surfaces
$z=0$ and $z=H$. In static equilibrium there is a uniform temperature
gradient,
${\rm d}{\rm T}_0/{\rm d} z=-\beta$. We decompose ${\rm T}={\rm T}_0(z)+\theta$,
where $\theta$ is the perturbation in temperature from the linear profile.
In order to maintain a constant heat flux the thermal boundary conditions
on the surfaces $z=0$ and $z=H$ are $\partial \theta/\partial z=0$, while
the outer radial boundary is thermally insulating, $\partial
\theta/\partial r=0$.
The flow domain and boundary conditions are summarised in Figure~\ref{fig2}.
\begin{figure}
\centerline{
\includegraphics[width=1\textwidth]{FIGURES/pdffiles/fig2.pdf}
}
\caption{Flow domain and boundary conditions.}
\label{fig2}
\end{figure}
We adopt a frame of reference that rotates with the boundaries.
Denoting $\mbox{\boldmath $\Omega$}$ the background rotation rate, $\mbox{\bf g}$ the gravitational acceleration,
$\nu$ the kinematic viscosity of the fluid,
$\kappa$ its thermal diffusivity, and $\alpha$ its thermal expansion
coefficient, the governing equations are
\begin{multiequations}
\label{eq_NS}
\begin{equation}
\frac{{\rm D} \mbox{\bf u}}{{\rm D} t}
=
- \frac{1}{\rho_0}\, \mbox{\boldmath $\nabla$} p
- 2 \mbox{\boldmath $\Omega$} \times \mbox{\bf u}
+ \nu \, \mbox{\boldmath $\nabla$} ^2 \mbox{\bf u}
-
\alpha
\, \theta \, \mbox{\bf g} \, , \qquad \mbox{\boldmath $\nabla$} \cdot \mbox{\bf u}=0 \, ,
\end{equation}
\singleequation
and
\begin{equation}
\frac{{\rm D} \theta}{{\rm D} t}
=
\kappa
\, \mbox{\boldmath $\nabla$} ^2 \theta
+ \beta \, {\rm u}_z \,,
\label{eq_theta}
\end{equation}
\end{multiequations}
\citep[e.g.][]{Chandra,Drazin}.
We further restrict ourselves to axisymmetric motion, so that we may
decompose the velocity field into poloidal and azimuthal velocity
components, $\mbox{\bf u}_p=({\rm u}_r,0,{\rm u}_z)$ and $\mbox{\bf u}_\phi=(0,{\rm u}_\phi,0)$,
which are separately solenoidal. The azimuthal component of (\ref{eq_NS}$a$)
then becomes an evolution equation for the specific angular momentum in the
rotating frame, $\Gamma=r{\rm u}_\phi$,
\begin{equation}
\frac{{\rm D} \Gamma}{{\rm D} t} = - 2 r \,\Omega\, {\rm u}_r + \nu \mbox{$\nabla_{\star}^{2}$}(\Gamma)\,,
\label{Gammadim}
\end{equation}
where
\[
\mbox{$\nabla_{\star}^{2}$}=r \frac{\partial }{\partial r}\left(\frac{1}{r}\frac{\partial }{\partial r}\right) + \frac{\partial^2 }{\partial z^2}
\]
is the Stokes operator.
Moreover the curl of the poloidal components yields an evolution equation for
the azimuthal vorticity, $\mbox{\boldmath $\omega$}_\phi= \mbox{\boldmath $\nabla$} \times \mbox{\bf u}_p \, ,$
\begin{equation}
\frac{{\rm D}}{{\rm D} t}
\left(\frac{\omega_\phi}{r}\right)=\frac{\partial}{\partial
z}\left(\frac{\Gamma^2}{r^4}\right) + \frac{2\, \Omega}{r} \frac{\partial
{\rm u}_\phi}{\partial z} -\frac{\alpha g}{r}\, \frac{\partial
\theta}{\partial r} + \frac{\nu}{r^2}\mbox{$\nabla_{\star}^{2}$}\left(r \omega_\phi \right)
\,.
\label{omegadim}
\end{equation}
We recognize the curl of the Coriolis, buoyancy and viscous forces on the
right of (\ref{omegadim}). The axial gradient in $\Gamma^2/r^4$ is, perhaps, a
little less familiar as a source of azimuthal vorticity. However, this arises
from the contribution of $\mbox{\boldmath $\nabla$} \times \left( \mbox{\bf u}_\phi \times \mbox{\boldmath $\omega$}_p
\right)\, , $ where $\mbox{\boldmath $\omega$}_p =\mbox{\boldmath $\omega$} - \mbox{\boldmath $\omega$}_\phi \, ,$ to the
vorticity equation and represents the self-advection
(spiralling up) of the poloidal vorticity-lines by axial gradients in swirl
\citep[e.g.][]{Davidson}.
The scalar equations (\ref{Gammadim})
and (\ref{omegadim}) are formally equivalent to (\ref{eq_NS}$a$), with $\Gamma$
and $\omega_\phi$ uniquely determining the instantaneous velocity
distribution. Finally, it is convenient to introduce the Stokes
stream-function, $\psi$, which is defined by $\mbox{\bf u}_p= \mbox{\boldmath $\nabla$} \times
\left[\left(\psi/r\right) \mbox{\bf e}_\phi\right]$ and related to the azimuthal
vorticity by
$
r\omega_\phi=-\mbox{$\nabla_{\star}^{2}$} \psi\,.
$
\section{Global Dynamics}
We perform numerical simulations in the form of an initial value
problem which is run until a steady state is reached. We solve
equations (\ref{eq_theta}), (\ref{Gammadim}) and (\ref{omegadim})
in which the length has been scaled with the height $H$ of the system,
the time with $\Omega ^{-1}\, ,$
and the temperature with $H \beta\,.$
The dimensionless control parameters of the system are the Ekman number
\(
{\rm E}={\nu}/{\Omega H^2}\,,
\)
the Prandtl number
\(
{\rm Pr}={\nu}/{\kappa}\,,
\)
and the Rayleigh number
\(
{\rm Ra}={\alpha g \beta H^4}/{\nu \kappa}\,.
\)
We use second-order finite differences and an implicit second-order
backward differentiation (BDF2) in time. The number of radial and
axial cells is $1000\times500$, and in each simulation grid resolution
studies were undertaken to ensure numerical convergence.
The aspect ratio of the computational
domain is set at $\epsilon=0.1$, a ratio inspired by tropical storms
(for which $H \simeq 10$km, and $R \simeq 100$km). The Ekman number ${\rm E}$
is set to $0.1$, which is a sensible turbulent estimate for tropical
cyclones. The values of $\Pr$ and ${\rm Ra}$ will be varied through this study
to control the strength of the convection.
We shall consider flows in which the local Rossby number
${\rm Ro}={\rm u}_\phi/{\Omega H}\,,$ is of the order
unity or
less at large radius, $r \simeq L \, ,$
but is large near the axis, $r \simeq H \, ,$ which is not
untypical of a tropical cyclone and turns out to be the regime in which an eye and
eyewall form in our numerical simulations. We shall also take a suitably defined
Reynolds number, ${\rm Re} \,,$ to be considerably larger than unity, though not so large that
the laminar flow becomes unsteady. A moderately large Reynolds number also turns
out to be crucial to eye formation.
\begin{figure}
\centerline{
\includegraphics[width=1\textwidth]{FIGURES/pdffiles/fig3.pdf}
}
\caption{Steady state solution in the $(r,z)$-plane for the parameters ${\rm Pr}=0.5$ and ${\rm Ra}=1.5\times 10^{4}$ in (a--d), and for ${\rm Pr}=0.1$ and ${\rm Ra}=2 \times 10^{4}$ in (e--h).
(a,e) The stream-function distribution,
(b,f) the total temperature ${\rm T}={\rm T}_0(z)+\theta$,
(c,g) the azimuthal velocity, ${\rm u}_\phi$,
(d,h) the radial variation of ${\rm Ro}$.}
\label{fig3}
\end{figure}
In order
to focus thoughts, let us start by considering two specific cases:
$\Pr = 0.5\, , \,\,{\rm Ra} = 1.5\times 10^{4}\, ,$ and
$\Pr = 0.1\, , \,\,{\rm Ra} = 2\times 10^{4}\, .$ These two cases are
represented in Fig.~\ref{fig3}.
In both cases, the primary flow in the vertical plane is radially inward near the lower
boundary and outward at the upper surface.
The steady state stream-function distributions are shown in
Fig.~\ref{fig3}a,e.
It is evident that in both
cases an eye
has formed near the axis, but it is much more pronounced in the later case.
Fig.~\ref{fig3}b,f show the corresponding
distributions of total temperature. The poloidal flow sweeps heat towards
axis at low values of $z$,
causing a build-up of heat near the axis with a corresponding cooler
region at larger radii. The resulting negative radial gradient in ${\rm T}$
drives the main poloidal vortex, ensuring that it has positive
azimuthal vorticity in accordance with (\ref{omegadim}).
Fig.~\ref{fig3}c,g present the distributions of the azimuthal velocity.
The Coriolis force induces swirl in the convecting fluid. As the
fluid spirals inward, it carries its angular momentum with it (subject to
some viscous diffusion) and this results in a region of particularly intense
swirl near the axis.
It also shows a substantial region of negative (anti-cyclonic rotation)
at large radius, something that is also observed in tropical cyclones
\citep[e.g.][]{Frank77}. In order to quantify the strength of the azimuthal
flow, we introduce a Rossby number which is a function of radius
${\rm Ro}(r)={\left({\rm u}_\phi\right)_{\rm max}}/{\Omega
H}\,,$ where
$\left({\rm u}_\phi\right)_{\rm max}$ is the maximum value of azimuthal
velocity at any one radial location. The Rossby number as a function of
radius is represented in Fig.~\ref{fig3}d,h. The eye
is obtained when the local Rossby number, is of the order unity at
large radius, $r\simeq L$, but is large near the axis,
$r\simeq H$.
Given the increase of ${\rm Ro}$ near the axis, the
Coriolis force can be neglected in the vicinity of the eye.
Note that, in the second
case (Fig.~\ref{fig3}e,f,g,h), the Rossby number is much larger, and the
eye much more pronounced than in the first case (Fig.~\ref{fig3}a,b,c,d).
To summarise, in both cases, the buoyancy force evidently drives motion in the poloidal plane,
which in turn induces spatial variations in angular momentum,
$\Gamma$, through the Coriolis force, $2 \Omega r{\rm u}_r$, in
(\ref{Gammadim}).
The flow spirals radially inward along the lower
boundary and outward near $z=H$, as shown in Fig.~\ref{fig1}. The Coriolis force then ensures that
the angular momentum, $\Gamma$,
rises as the fluid spirals inward along the bottom boundary, but falls
as it spirals back out along the upper surface towards
$r=L$.
The swirl of the flow as it approaches the axis is thus controled both by
the Ekman number and by the aspect ratio of the domain.
With our choice of the parameters, particularly high levels of azimuthal
velocity built up near the axis, with a correspondingly large value of
${\rm Ro}$ in the vicinity of the eyewall. In some sense, then, the global
flow pattern is both established and shaped by the buoyancy and
Coriolis forces, yet, as we shall discuss, these forces are negligible in
the viscinity of the eye.
\section{Global Versus Local Dynamics}
\subsection{The anatomy of eyewall formation}
\label{ana}
The large value of ${\rm Ro}$ near the axis means that the Coriolis force
is locally negligible in the region where the eye and eyewall form,
and it turns out that this is true also of the buoyancy force in our
Boussinesq simulations. Thus the very forces that establish the global
flow pattern play no significant role in the local dynamics of the
eye. It is worth considering, therefore, the simplified version of
(\ref{Gammadim}) and (\ref{omegadim}) which operate near the axis,
\begin{equation}
\frac{{\rm D} \Gamma}{{\rm D} t} \simeq \nu \mbox{$\nabla_{\star}^{2}$}(\Gamma)\,,
\label{sys1a}
\end{equation}
and
\begin{equation}
\frac{{\rm D}}{{\rm D} t} \left(\frac{\omega_\phi}{r}\right) \simeq
\frac{\partial}{\partial z}\left(\frac{\Gamma^2}{r^4}\right)
+ \frac{\nu}{r^2}\mbox{$\nabla_{\star}^{2}$}\left(r \omega_\phi \right)\,.
\label{sys1b}
\end{equation}
\begin{figure}
\centerline{
\includegraphics[width=0.8\textwidth]{FIGURES/pdffiles/fig4.pdf}
}
\caption{Contours of constant angular momentum (color) superimposed on the
streamlines (black) in the inner quarter of the flow domain
($(r,z)$-plane), for $\Pr=0.1$ and ${\rm Ra}=2\times 10^4$.}
\label{fig4}
\end{figure}
The eye is characterised by anticlockwise motion in the $(r,z)$-plane
($\omega_\phi<0$), in contrast to the global vortex that is clockwise
($\omega_\phi>0$). It is also characterised by low levels
of angular momentum.
A natural question to ask, therefore, is where this
negative azimuthal vorticity comes from. Since $\Gamma$ is small in the eye,
$\omega_\phi/r$ is locally governed by a simple advection diffusion equation
in which the source term is negligible, and so the negative azimuthal
vorticity in the eye has most probably diffused into the eye from the eyewall.
This kind of slow cross-stream diffusion of vorticity into a region of closed
streamlines is familiar from the Prandtl-Batchelor theorem \citep{Batchelor}, and in
this sense the eye is a passive response to the accumulation of negative
$\omega_\phi$ in the eyewall. If this is substantially true, and we shall
see that it is, then the key to eye formation is the generation of significant
levels of negative azimuthal vorticity in the eyewall, and so the central
questions we seek to answer is how, and under what conditions, the eyewall
acquires this negative vorticity.
Given that the Reynolds number is large, it is tempting to consider the
inviscid limit and attribute the growth of negative $\omega_\phi$ to the
first term on the right of (\ref{sys1b}). That is, axial gradients in $\Gamma$
can act as a local source of azimuthal vorticity, and indeed this mechanism
has been invoked by previous authors in the context of tropical cyclones
\citep[e.g.][]{Pearce,Smith,Pearce2}.
The idea is that, in steady state,
if viscous diffusion is ignored in the vicinity of the eyewall, (\ref{sys1a})
and (\ref{sys1b}) locally reduce to
\begin{equation}
\Gamma=\Gamma\left(\psi\right)\, ,
\label{sq1a}
\end{equation}
and
\begin{equation}
\mbox{\bf u} \cdot \mbox{\boldmath $\nabla$}\left(\omega_\phi/r\right)=\frac{\partial}{\partial
z}\left(\frac{\Gamma^2}{r^4}\right)=-2\frac{\Gamma
\Gamma^\prime\left(\psi\right)}{r^3}{\rm u}_r \, .
\label{sq1b}
\end{equation}
To the extent the viscosity can be ignored,
$\Gamma\left(\psi\right)$ increases to a maximum at roughly mid-height,
where $\psi$ is a maximum,
and then drops off as we approach the upper boundary. Thus
$\Gamma^\prime\left(\psi\right)>0$, and so according to (\ref{sq1b})
positive vorticity is induced
as the streamlines curve inward and upward, while negative vorticity is
created after the streamlines turn around and ${\rm u}_r$ reverses sign.
Since the eyewall is associated with the upper region, where the flow is outward,
it is natural to suppose that the negative vorticity in the eyewall arises
from precisely this process.
However, in the case of a Boussinesq fluid, this term cannot produce
any net negative azimuthal vorticity in the eyewall, essentially because
the term on the right of (\ref{sq1b}) is a divergence. To see why this is so,
we rewrite (\ref{sq1b}) as
\begin{equation}
\mbox{\boldmath $\nabla$} \cdot \left[\left(\omega_\phi/r\right)\mbox{\bf u}\right]=\mbox{\boldmath $\nabla$} \cdot
\left[\left(\Gamma^2/r^4\right)\mbox{\bf e}_z\right]\,,
\end{equation}
and integrate this over a control volume in the form of a stream-tube in
the $r-z$ plane composed of two adjacent streamlines that pass through
the eyewall. If the stream-tube within the control volume starts and ends
at a fixed radius somewhat removed from the eyewall, then the right-hand
divergence integrates to zero. The flux of vorticity into the control
volume (the stream-tube) is therefore the same as that leaving.
In short, the azimuthal
vorticity generated in the lower regions where the streamlines curve inward
and upward is exactly counterbalanced by the generation of negative vorticity
in the upper regions where the flow is radially outward. Such a process
cannot result in any negative azimuthal vorticity in the eyewall.
Returning to (\ref{sys1b}) we conclude that the only possible source of net
negative vorticity in the eyewall is the viscous term, and this drives us
to the hypothesis that the negative vorticity in the eyewall has its origins
in the lower boundary layer. That is, negative azimuthal vorticity is
generated at the lower boundary and then advected up into eyewall where it
subsequently acts as the source for a slow cross-stream diffusion of negative
vorticity into the eye. Of course, as the streamlines pass up into, and then
through, the eyewall there is also a generation of first positive and then
negative azimuthal vorticity by the axial gradients in $\Gamma^2/r^4$, but
these two contributions exactly cancel, and so cannot contribute to the
negative vorticity in the eyewall.
The above description is put to the test in Fig.~\ref{fig4} to Fig.~\ref{fig7}.
A more detailed view of the angular momentum
distribution and streamlines in the region adjacent to the eye and eyewall
is provided in Fig.~\ref{fig4}.
In the region to the right of the eye, the contours of
constant angular momentum are roughly aligned with the streamlines,
in accordance with (\ref{sq1a}). Note that, although the contours of
constant $\Gamma$ roughly follow the streamlines, the two sets of contours
are not entirely aligned. This is mostly a result of cross-stream
diffusion, particularly near the eyewall, although it also partially
arises from the (relatively weak) Coriolis force.
\begin{figure}
\centerline{
\includegraphics[width=1.0\textwidth]{FIGURES/pdffiles/fig5.pdf}
}
\caption{Colour maps of the distribution of the
various forces in the azimuthal vorticity equation (\ref{omegadim}) in the inner
part of the flow domain ($(r,z)$-plane): (a) the convective derivative of
azimuthal vorticity; (b) the term associated with axial gradients in
$\Gamma$; (c) diffusion; (d) the Coriolis
term; and (e) buoyancy. Parameters: $\Pr=0.1\, ,\ {\rm Ra}=2\times 10^4$.}
\label{fig5}
\end{figure}
Fig.~\ref{fig5} shows the distribution and relative magnitudes of the various forces in the
azimuthal vorticity equation across the inner part of the flow
domain.
The buoyancy and Coriolis terms, though important
for the large scale dynamics, are locally negligible (panels
\ref{fig5}(d) and \ref{fig5}(e)), while diffusion is largely limited
to the boundary layer, the eyewall, and a region near the axis where
the flow turns around (panel \ref{fig5}c). Note also that
$\partial\left(\Gamma^2/r^4\right)/\partial z$ is small within the eye
(panel \ref{fig5}b) but there are intense regions of equal and
opposite $\partial\left(\Gamma^2/r^4\right)/\partial z$ below the
eyewall, which are matched by corresponding regions of equal and
opposite $\mbox{\bf u} \cdot \mbox{\boldmath $\nabla$}\left(\omega_\phi/r\right)$ in panel
\ref{fig5}a. These figures appear to validate that (\ref{sys1b}) is
the relevant equation in this domain. It is interesting that the very
forces that shape the global flow, i.e.~the Coriolis and buoyancy forces,
play no role in the vicinity of the eye.
This can be further highlighted by introducing
the variable $\tau$, defined as a parametric coordinate along
an iso-$\psi$. It corresponds to the position at a given time $\tau$ for
a particle advected along a streamline; it stems from
\(
\left(\left.{{\rm d} z}/{{\rm d} \tau}\right)\right|_{\psi={\rm cst}}={\rm u}_z\,.
\)
The constant of integration is set such that $\tau= 0$ for the maximum
of $\omega_\phi/r$.
Fig.~\ref{fig6} shows
forces in the azimuthal vorticity equation,
as a function of position on the streamline that passes
through the centre of the eyewall (thick streamline in Fig.~\ref{fig5}).
To first order there is an approximate
balance between the advection of $\omega_\phi/r$ and
$\partial\left(\Gamma^2/r^4\right)/\partial z$, though there is a
significant contribution from
the diffusion of vorticity within the eyewall. Both the Coriolis and
buoyancy terms are completely negligible. In short, the force balance
is that of (\ref{sys1b}).
\begin{figure}
\centerline{
\includegraphics[width=0.8\textwidth]{FIGURES/pdffiles/fig6.pdf}}
\caption{The variation of the terms in the azimuthal vorticity equation
(\ref{omegadim}) with position on the streamline that passes through
the centre of the eyewall. The convective derivative on the left-hand side
of (\ref{omegadim}) is black, the term associated with axial gradients in
$\Gamma$ is red, the viscous term is green, and the Coriolis and buoyancy
terms (blue and light blue) are indistinguishable from the $x$-axis.
Parameters: $\Pr=0.1\, ,\ {\rm Ra}=2\times 10^4$.}
\label{fig6}
\end{figure}
The main features of the eyewall are most clearly seen in Fig.~\ref{fig7}a,
which shows the distribution of azimuthal vorticity, $\omega_\phi/r$,
superimposed on the streamlines. The exceptionally strong levels of
azimuthal vorticity in and around the eyewall is immediately apparent, and
indeed it is tempting to define the eyewall as the outward sloping region
of strong negative azimuthal vorticity which separates the eye from the
primary vortex.
There are two other important features of Fig.~\ref{fig7}a. First, a large
reservoir of negative azimuthal vorticity builds up in the lower boundary
layer, as it must. Second, between the lower boundary and the eyewall there
is a region of intense positive azimuthal vorticity.
Fig.~\ref{fig7}b. shows
the variation of $\omega_\phi/r$ along the streamline that passes through
the centre of the eyewall, as indicated by the thick black line in Fig.~\ref{fig7}a.
As the streamline passes along the bottom boundary
layer, $\omega_\phi/r$ becomes progressively more negative. There is then
a sharp rise in $\omega_\phi/r$ as the streamline pulls out of the boundary
layer and into a region of positive $\partial \Gamma/\partial z$, followed by
a corresponding drop as the streamline passes into the region of negative
$\partial \Gamma/\partial z$. Crucially, the rise and subsequent fall in
$\omega_\phi/r$ caused by axial gradients in angular momentum exactly
cancel, and
so the fluid emerges into the eyewall with the same level of vorticity it
had on leaving the boundary layer.
\begin{figure}
\centerline{
\includegraphics[width=1\textwidth]{FIGURES/pdffiles/fig7.pdf}
}
\caption{(a) Colour map of $\omega_\phi/r$ superimposed on the streamlines
in the inner quarter of the domain ($(r,z)$-plane). (b) The variation of $\omega_\phi/r$ along
the streamline that passes through the centre of the eyewall (as indicated
by the thick black line in a). Parameters: $\Pr=0.1\, , \ {\rm Ra}=2 \times 10^4$.}
\label{fig7}
\end{figure}
Finally we consider in Fig.~\ref{fig8} and Fig.~\ref{fig9} the distribution
of the various contributions to the angular momentum equation
(\ref{Gammadim}).
Fig.~\ref{fig8} shows the variation of these terms along the streamline that
passes through the centre of the eyewall, as indicated by the thick black
line in Figure~\ref{fig5}. As before, the origin for the horizontal axis is taken
to be the innermost point on the streamline. Clearly, within the eyewall
there is a leading-order balance between the advection and diffusion of
angular momentum, in accordance with the approximate equation~\eqref{sys1a}, although
the Coriolis torque is not entirely negligible. However, within the
boundary layer ahead of the eyewall the force balance is quite different,
with the Coriolis and viscous forces being in approximate balance, the
convective growth of angular momentum being small.
A similar impression may be gained from Fig.~\ref{fig9} which shows, for the
inner quarter of the flow domain, the distribution of the various
contributions to equation~\eqref{Gammadim}.
The panel (a)
is the convective derivative of $\Gamma$ on the left of (\ref{Gammadim}), the panel (b)
is the diffusion term on the right, and the panel (c) is the Coriolis torque.
Clearly, the positive Coriolis torque acting near the bottom boundary
layer is largely matched by local cross-stream diffusion, the convective
growth of angular momentum being small. The
negative Coriolis torque acting on the outflow near the upper surface
is mostly balanced by $\mbox{\bf u} \cdot \mbox{\boldmath $\nabla$} \Gamma$, resulting in a fall in $\Gamma$.
Within the eyewall, the force balance is quite different, since there is
a leading-order balance between the advection and
diffusion of angular momentum, and the Coriolis torque is very weak.
This confirms our previous observations in Fig.~\ref{fig5}, and is in
accordance with the approximate equation
(\ref{sys1a}).
To summarise, in this particular simulation the eyewall that separates
the eye from the primary vortex is characterised by high levels of
negative azimuthal vorticity. That vorticity comes not from the term
$\partial(\Gamma^2/r^4)/\partial z$,
despite its local dominance, but rather from the boundary layer at $z =
0$. The eye then acquires its negative vorticity by cross-stream
diffusion from the eyewall, in accordance with the Prandtl-Batchelor
theorem. Although the global flow is driven and shaped by the buoyancy
and Coriolis forces, these play no significant dynamical role in the
vicinity of the eye and eyewall. As we shall see, these dynamical
features characterise all of our simulations that produce eyes.
\begin{figure}
\centerline{
\includegraphics[width=0.8\textwidth]{FIGURES/pdffiles/fig8.pdf}}
\caption{The variation of the various torques in the angular momentum
equation \eqref{Gammadim} with position on the streamline that passes through the
centre of the eyewall. The convective derivative on the left-hand side of
\eqref{Gammadim} is black, the viscous diffusion term is green, and the Coriolis term
is blue.}
\label{fig8}
\end{figure}
\begin{figure}
\centerline{
\includegraphics[width=1\textwidth]{FIGURES/pdffiles/fig9.pdf}
}
\caption{The spatial distribution of the various torques
in the angular momentum equation in the inner
part of the flow domain ($(r,z)$-plane) : (a) the convective
derivative of $\Gamma$, (b) the diffusion term on the right,
and (c) the Coriolis torque. Parameters: $\Pr=0.1\, , \ {\rm Ra}=2\times 10^4$.}
\label{fig9}
\end{figure}
\subsection{A comparison of vortices that do and do not form eyes}
\begin{figure}
\centerline{
\includegraphics[width=1\textwidth]{FIGURES/pdffiles/fig10.pdf}
}
\caption{Colour map of $\omega_\phi/r$ superimposed on the streamlines in
the inner quarter of the domain ($(r,z)$-plane), for ${\rm Pr}=0.3$ and (a)
${\rm Ra}=2\times 10^{3}$, (b) ${\rm Ra}=5.5\times 10^{3}$, (c) ${\rm Ra}=6\times 10^{3}$, (d)
${\rm Ra}=9 \times 10^{3}$, and for ${\rm Pr}=0.1$ and (e)
${\rm Ra}=10^{3}$, (f) ${\rm Ra}=1.7\times 10^{3}$, (g) ${\rm Ra}=2\times 10^{3}$, (h)
${\rm Ra}=5 \times 10^{3}\,.$ The colour code is the same on all graphs.}
\label{fig10}
\end{figure}
\begin{figure}
\centerline{
\includegraphics[width=0.7\textwidth]{FIGURES/pdffiles/fig11.pdf}
}
\caption{The radial variation of ${\rm Ro}$, for the cases with $\Pr=0.1$ and
${\rm Ra}=10^{3}$ (dotted), ${\rm Ra}=2\times 10^{3}$ (dashed), ${\rm Ra}=5\times 10^{3}$
(thin solid) and ${\rm Ra}=2 \times 10^{4}$ (thick solid).}
\label{fig11}
\end{figure}
Let us now compare numerical simulations at given
values of the Prandtl number ($\Pr = 0.3$ and $\Pr = 0.1$) and varying Rayleigh numbers.
Figure~\ref{fig10} shows the stream-function distribution.
Clearly, eyes form in Fig.~\ref{fig10}b,c,d,f,g,h.
From Table~1 and Fig.~\ref{fig11}, we see that the peak value of ${\rm Ro}$ is
of the order of $10$ or larger
in those cases where an eye forms, which is typical of an atmospheric
vortex.
\include{Table_ODD}
The clearest way to distinguish those flows where an eye forms
from those in which it does not is to examine the spatial distribution
of the azimuthal vorticity, shown in Fig.~\ref{fig10}.
In cases with a strong eye (Fig.~\ref{fig10}d,h), the
distribution of $\omega_\phi/r$ is similar to that in Fig.~\ref{fig7}a,
with regions of strong negative vorticity in the boundary layer and
eyewall, and a patch of intense positive vorticity below the eyewall.
The negative eyewall vorticity has its origins in the boundary layer, with
the moderately large Reynolds number allowing this vorticity to be swept
up from the boundary layer into the eyewall.
Note, in particular, that the magnitude of vorticity in the eyewall is similar to that in the boundary
layer.
Turning to Fig.~\ref{fig10}b,c,f,g, which are somewhat
marginal, in the sense that the eye
is small, we see that overall flow pattern is similar, but that the
negative eyewall vorticity is now relatively weak, and in particular
significantly weaker than that in the boundary layer.
It seems likely that the eyewall vorticity is relatively
weak because advection has to compete with cross-stream diffusion as the
poloidal flow tries to sweep the boundary layer vorticity up in the eyewall.
In short, much of the boundary layer vorticity is lost to the surrounding
fluid by diffusion before the fluid reaches the eyewall. Since the eye
acquires its vorticity from the eyewall via cross-steam diffusion, the relative
weakness of the eyewall vorticity explains the weakness of the resulting
eye.
In Fig.~\ref{fig10}a,e, where no eye forms,
no region of intense negative vorticity
forms as the poloidal flow turns and ${\rm u}_r$ changes sign. Since there is
still a reservoir of negative vorticity in the boundary layer, and the basic shape
of the primary vortex is unchanged, we conclude that the Reynolds number
is now too low for the flow to effectively sweep the boundary-layer
vorticity up into the fluid above.
The comparison of Fig.~\ref{fig10}b and Fig.~\ref{fig10}f reveals of
decreasing the Prandtl number yields stronger inertial effects
and thus allows to sustain an eye at lower Rayleigh numbers.
\subsection{A criterion for eye formation}
\label{crit}
The mechanism introduced above works only if the Reynolds number, ${\rm Re}$,
is sufficiently large, so
that the flow can lift the vorticity out of the boundary layer and into the
eyewall before is disperses though viscous diffusion. This suggests that there
is a threshold value in ${\rm Re}$ that must be met in order for an eyewall to form.
We take this Reynolds number to be based on the maximum value of the inward
radial velocity, $| {\rm u}_r |_{\rm max}$, at a radial location just
outside the region containing the eye and eyewall. We (somewhat arbitrarily)
choose the radial location at which $| {\rm u}_r |_{\rm max}$ is
evaluated to be $r=H$, which does in fact lie just outside the eyewall.
Thus ${\rm Re}$ is defined as
\begin{equation}
{\rm Re}=\frac{| {\rm u}_r |_{\rm max} H}{\nu}\,,
\label{Re}
\end{equation}
and is chosen to be larger than unity, though not so large that the flow becomes
unsteady.
The logic behind this specific definition of ${\rm Re}$ is that the poloidal
velocity field in the vicinity of the eyewall is dominated, via the
Biot-Savart law, by the flux of vorticity up through the eyewall. This, in
turn, is related to the flux of negative azimuthal vorticity in the
boundary layer just outside the eyewall region. However, evaluating this
flux by integrating $|\omega_\phi|$ up through this boundary layer from to the point
where $|\omega_\phi|=0$ simply gives $| {\rm u}_r |_{\rm max} \, .$
We conclude, therefore, that $| {\rm u}_r |_{\rm max}$ is the
characteristic poloidal velocity in the vicinity of the eyewall, and that
${\rm Re}$
is therefore a suitable measure of the ratio of advection to diffusion of
azimuthal vorticity in this region.
Note that, since the buoyancy and Coriolis forces are negligible in
the vicinity of the eye and eyewall, the criterion for a transition from
eye formation to no eye cannot be controlled explicitly by ${\rm E}$ or ${\rm Ra}$, but
rather must depend on ${\rm Re}$ only. So, to study the transition in the numerical
experiments, we selected values of ${\rm E}$, ${\rm Pr}$ and ${\rm Ra}$ that yield steady flows with
moderately large Reynolds numbers, as defined by \eqref{Re}, and with ${\rm Ro}$ somewhat
larger than unity in the vicinity of the axis, i.e. those conditions which
are conducive to eye formation.
The values of ${\rm Re}$ obtained from our numerical experiments are
reported in Figure.~\ref{fig12} and some are listed in Table~1.
A reasonably large Reynolds number also turns out to be crucial
to eye formation. There is indeed a critical value of ${\rm Re}$ below
which an eye cannot form. Fig.~\ref{fig12}a shows ${\rm Re}$ as a function of ${\rm Ra}$ and
$\Pr$, with the open symbols representing cases where an eye forms,
and closed symbols those where there is no eye. There is clear
evidence that the two classes of flow are separated by a plateau in
${\rm Re}$.
This is confirmed by the two-dimensional plots of ${\rm Re}$ versus ${\rm Ra}$ and
${\rm Re}$ versus $\Pr$ given in panels (b) and (c). For the model and
parameter regime considered here, the transition occurs at
${\rm Re} \simeq 37$, as indicated by the horizontal grey surface in Fig.~\ref{fig12}a
and the corresponding lines in Fig.~\ref{fig12}b,c.
\begin{figure}
\centerline{
\includegraphics[width=0.9\textwidth]{FIGURES/pdffiles/fig12.pdf}
}
\caption{${\rm Re}$, as defined by (\ref{Re}), as a function of (a) ${\rm Ra}$ and
$\Pr$, (b) ${\rm Ra}$ and (c) $\Pr$. Open symbols represent cases where an eye
forms, and closed symbols those where there is no eye.}
\label{fig12}
\end{figure}
\section{Conclusions}
We considered axisymmetric steady Boussinesq convection. In the vertical
plane the primary vortex has a clockwise motion, and so has
positive azimuthal vorticity. In the elongated and rotating domain
considered here, the flow is characterised by a strong swirl as it
approaches the axis.
We have shown that, in this configuration, for sufficiently vigorous
flows, an eye can form. Its motion is
anticlockwise in the vertical plane, and so the eye is associated
with negative azimuthal vorticity.
The region that separates the eye from the primary vortex, usually called
the eyewall, is characterised by high levels negative azimuthal
vorticity.
We have shown that it is not generated by
buoyancy, since such forces are locally too weak. Nor does it arise from
so-called vortex tilting, despite the local dominance of this process, because
vortex tilting cannot produce any net azimuthal vorticity.
We have shown that this thin annular region is filled with
intense negative azimuthal vorticity, vorticity that has been stripped off
the lower boundary layer.
So that the eye
acquires its vorticity from the surrounding fluid by cross-stream diffusion,
and this observation holds the key to eye formation in our simple system.
\section*{Acknowledgments}
This work was initiated when
PAD visited the ENS in Paris for one month in May 2015.
The authors are gratefull to the ENS for support.
The simulations were performed using HPC resources
from GENCI-IDRIS (Grants 2015-100584 and 2016-100610).
\bibliographystyle{jfm}
| 12,201 |
\section{Introduction}
The currently being developed 5th generation mobile network (5G) has many ambitious goals, like 10Gbps peak data rates, 1ms latency and ultra-reliability. Another high priority is reducing energy consumption, especially to improve battery life of mobile and IoT devices. This goal is crucial for expected omnipresent fleet of remote sensors, monitoring all aspects of our world. Such sensor should be compact, inexpensive and has battery life of order of 10 years, as battery replacement in many applications is economically infeasible. Hence, this is an asymmetric task: the main priority is to reduce energy requirements of the sender, tolerating increased cost at the receiver side. A crucial part of the cost of sending information to the base station is establishing connection - their number and so energy cost can be reduced by buffering information, or can be nearly eliminated if the sensor just transmits the information in time periods precisely scheduled with the base station.
We will discuss approach for reducing energy need of the actual transmission of such buffered data, preferably compressed earlier to reduce its size. The information is encoded in a sequence of symbols as points from a chosen constellation: a discrete set of points in complex (I-Q) plane. This sequence of symbols can be used for time sequence of impulses, or as coefficients for usually orthogonal family of functions (subcarriers), for example in OFDM.
These constellations are often QAM lattices of size up to 64 in LTE. There is assumed uniform probability modulation (UPM) - that every symbol is used with the same frequency. Generally, a stream of symbols having ${p_s}_s$ probability distribution $(\sum_s p_s =1)$ contains asymptotically Shannon entropy $h=\sum_{s=1}^m p_s \lg(1/p_s)$ bits/symbol ($\lg\equiv \log_2$), where $m$ is the size of alphabet. Entropy is indeed maximized for uniform probability distribution $p_s = 1/m$, obtaining $h=\lg(m)$ bits/symbol, which is $\lg(64)=6$ bits/symbol for QAM64.
However, this natural choice of using uniform probability distribution is not always the optimal one. For example when the channel has constraints, like forbidding two successive ones ('11') in Fibonacci coding, then choosing $\Pr(x_{t+1}=0 | x_t=0) = \Pr(x_{t+1}=1 | x_t=0) = 1/2$ is not the optimal way. Instead, we should more often choose '0' symbol, optimally with $\varphi =(\sqrt{5}-1)/2$ probability, as this symbol allows to produce more entropy (information) in the successive step. For a general constraints the optimal probabilities can be found by using Maximal Entropy Random Walk~\cite{d2007}.
Another example of usefulness of nonuniform probability distribution among symbols used for communication are various steganography-watermarking problems, where we want encoding sequence to resemble some common data, for example picture resembling QR codes. Surprisingly, generalization of the Kuznetsov-Tsybakov problem allows such encoding without decoder knowing the used probability distributions (e.g. picture to resemble)~(\cite{KT1, KT}). However, this lack of knowledge makes encoding more expensive. \\
In this paper we will focus on a more basic reason to use nonuniform probability distribution among symbols: that the cost of using various symbols does not have to be the same. Assume $E_s$ is the cost of using symbol $s$, then entropy for a fixed average energy ($E=\sum_s p_s E_s$) is maximized for Boltzmann probability distribution among symbols $\left(\Pr(s)\propto e^{-\beta E_s} \right)$. For example in Morse code $dash$ lasts much longer than $dot$, what comes with higher time and energy cost. Designing a coding with more frequent use of dot ($\Pr(dot) > \Pr(dash)$) would allow to lower average cost per bit. Anther example of nonuniform cost is sending symbol '1' as electric current through a wire, symbol '0' as lack of this current - symbol '1' is more energy costly, hence should be used less frequently. \\
We will focus here on application for wireless communication modulation, where the cost related to emitting a symbol is usually assumed to be proportional to square of its amplitude: $E_x \propto |x|^2$, hence we could improve energy efficiency by more frequent use of low amplitude symbols.
Basic theoretical considerations will be reminded, then used to analyze potential improvements especially for the situation of modulation for wireless technology: to reduce required energy per bit, especially for the purpose of improving battery life of remote sensors. The average amount of bits/symbol (entropy) is maximized for uniform probability distribution (UPM), hence using nonuniform distribution (NPM) means that more symbols are required to write the same message, so the tradeoff of improving energy efficiency (bits per energy unit) is also reducing throughput (bits per symbol).
The use of nonuniform probability distribution of symbols requires a more complex coding scheme, especially from the perspective of error correction (channel coding). Entropy coders allow to work with kind of reversed task: encode a sequence of symbols having some assumed probability distribution into a bit sequence. Switching its encoder and decoder, we can encode a message (a bit sequence) into a sequence of symbols having some chosen probability distribution. Due to low cost, a natural approach would be using a prefix code here, for example $0\to a,\ 10\to b,\ 11\to c$. However, it approximates probabilities with powers of $1/2$ and cannot use probabilities $1/2 < p < 1$, which turn out crucial in the discussed situations. Additionally, its error correction would require some additional protection layer. Hence, a more appropriate recent entropy coding will be discussed for this purpose: tANS coding~(\cite{ANS,ANS1}). While having cost similar to prefix codes (finite state automaton, no multiplication), it operates on nearly accurate probabilities, including $1/2 < p < 1$. Additionally, its processing has an internal state, which can be exploited like the state of convolutional codes~\cite{conv} for error correction purpose - thanks of it encoder does not need to apply another coding layer, saving energy required for this purpose.
\section{Capacity and energy efficiency of nonuniform probability modulation (NPM)}
In this section there will be first reminded why Boltzmann distribution is the optimal choice from the perspective of energy efficiency, then three modulations will be analyzed, first without then with noise.
For better intuition, Shannon entropy is measured in bits: $h=\sum_{s=1}^m p_s \lg(1/p_s)$ bits/symbol ($\lg\equiv \log_2$).
\subsection{Probability distribution maximizing entropy}
Assume $E_s$ is the cost (energy) of using symbol $s$. We want to choose the optimal probability distribution $\{p_s\}_s$ for some fixed average energy $E$:
\begin{equation} \sum_s p_s E_s = E \qquad\qquad\qquad \sum_s p_s =1 \end{equation}
such that Shannon entropy is maximized: $h \ln(2) =- \sum_s p_s \ln(p_s)$.
Using the Lagrange multiplier method for $\lambda$ and $\beta$ parameters:
$$ L= - \sum_s p_s \ln p_s + \lambda \left( \sum_s p_s - 1 \right) + \beta \left(\sum_s p_s E_s - E\right) $$
$$0 = \frac{\partial L}{\partial p_s}= -\ln(p_s) - 1 + \lambda + \beta E_s$$
\begin{equation} p_s = \frac{e^{-\beta E_s}}{e^{1-\lambda}} = \frac{e^{-\beta E_s}}{Z} \end{equation}
where $Z = e^{1-\lambda} = \sum_s e^{-\beta E_s}$ is the normalization factor (called partition function).
The parameter $\beta$ can be determined from average energy:
$$E=\frac{\sum_s E_s e^{-\beta E_s}}{\sum_s e^{-\beta E_s}}$$
As expected, Boltzmann distribution is the optimal way to choose probability distribution of symbols: $p_s \propto e^{-\beta E_s}$. The standard way of evaluating cost of a signal in wireless telecommunication is square of its amplitude: $E_s = |x|^2$. Hence for $x\in\mathbb{R}$ the optimal probability is Gaussian distribution with standard deviation $\sigma^2=E$:
$$\rho_G(x)=\frac{1}{\sqrt{2E\pi}} e^{\frac{-x^2}{2E}}$$
\begin{equation} H_G :=-\int_{-\infty}^{\infty} \rho_G(x) \lg(\rho_G(x)) dx = \frac{1}{2}\lg(2\pi e E) \end{equation}
Let us compare it with uniform distribution, which is usually used in practical modulation schemes. Take a rectangular density function on some $[-a,a]$ range with height $\frac{1}{2a}$ to integrate to 1. Its average energy is $ E=\int_{-a}^a \frac{1}{2a} x^2 dx =\frac{a^2}{3}$, getting $a=\sqrt{3E}$ parameter for a chosen average energy $E$. Now
$$ H_u :=\int_{-a}^{a} \frac{1}{2a} \lg(2a) dx =\lg(2a)=\frac{1}{2}\lg(12E)$$
So the gain of using Gaussian distribution is
\begin{equation} H_G - H_u =\frac{1}{2}\lg(\pi e /6) \approx 0.2546\ \textrm{bits}. \end{equation}
There was used differential entropy (with integrals), which gets natural intuition when approximated with Riemann integration for some quantization step $q$:
$$H=-\int_{-\infty}^{\infty} \rho(x) \lg(\rho(x)) dx \approx -\sum_{k\in\mathbb{Z}} q\rho(kq) \lg(\rho(kq))=$$
$$=-\sum_{k\in\mathbb{Z}} q\rho(kq) \lg(q\rho(kq)) + \sum_{k\in\mathbb{Z}} q\rho(kq) \lg(q)$$
The left hand side term is the standard entropy for probability distribution of quantization with step $q$, the right hand side term is approximately $\lg(1/q)$. So entropy for quantized continuous probability distribution is approximately the differential entropy plus $\lg(1/q)$:
\begin{equation} h_q \approx H + \lg(1/q) \end{equation}
\begin{figure}[t!]
\centering
\includegraphics[width=8cm]{entr.png}\
\begin{center}
\caption{Top: three considered modulations (binary, ternary and hexagonal). Probability of using the zero signal is denoted by $p$, the remaining symbols have equal probability. Middle: dependence of entropy (bits/symbol) from the parameter $p$. It is maximized when all symbols are equally probable (UPM, marked). Bottom: energy efficiency (bits/energy unit) as this entropy divided by average amount of energy used per symbol - it tends to infinity when $p\to 1$, what means communicating by rarely distracting the silence of using the zero-signal. Energy efficiency is improved at cost of reducing throughput, which is proportional to entropy (assuming zero noise). For example for hexagonal modulation, UPM $(p=1/7)$ allows to transmit $\approx 3.275$ bits/energy unit. We can double it by using $p\approx 0.84$ frequency of the zero-signal, at cost of $\approx 2.7$ times lower entropy (throughput). Binary modulation, for example for wire communication, has even larger potential for improvements (e.q. to quadruple efficiency).}
\label{entr}
\end{center}
\end{figure}
\subsection{Three modulations with zero-signal}
As discussed, the gain of using the optimal: Gaussian distribution in contrast to the standard uniform distribution, is $\approx 0.2546$ bits/symbol. Surprisingly, this is an absolute difference - it can bring an arbitrarily large relative difference for low original throughput: low $E$ and sparse quantization (large $q$).
So let us consider 3 basic examples of such modulation, visualized in the top of Fig. \ref{entr}:
a) \emph{binary}: $x\in \{0,1\}$
b) \emph{ternary} $x \in \{-1,0,1\}$
c) \emph{hexagonal} $x\in \{0\} \cup \{e^{i k \pi/3}: k=0,\ldots 5\}$
All of them contain zero-signal: which energy cost is zero (neglecting additional costs). This symbol represents information by using silence. Obviously, other symbols are also required - storing information by choosing moments (or subcarriers) to break this silence. Hexagonal modulation is appropriate for wireless communication. Binary and ternary are less effective, but they can be useful for communication by wire.
For all three cases denote $\Pr(0)=p$ as the probability of using zero signal. The remaining signals have all energy cost $E_x=|x|^2=1$, hence we can assume uniform probability distribution among them (correspondingly: $1-p$, $(1-p)/2$, $(1-p)/6$). The average energy in all three cases is
\begin{equation} E=p\cdot 0 + (1-p)\cdot 1 = 1-p.\end{equation}
The middle plot of Fig. \ref{entr} shows entropy dependence of $p$ for all three cases: average number of bits/symbol. It is maximized for UPM: $p=1/2,\ 1/3$, or $1/7$ correspondingly (marked). However, if we divide entropy by average energy cost of a symbol, getting average bits/energy unit, this energy efficiency $\eta=h/E$ grows to infinity for $p\to 1$, at cost of reduced entropy (throughput).
\subsection{Adding white Gaussian noise}
In real-life scenarios we also need to take noise into consideration: sender adds some redundancy to the transmitted message, then receiver applies forward error correction to repair eventual errors thanks to using this redundancy. The Shannon noisy-channel coding theorem~\cite{shannon} says that capacity of a channel is:
\begin{equation} C =\max_{p_X} I(X;Y) = \max_{p_X} h(Y) - h(Y|X) \end{equation}
Without the maximization, mutual information $I(X;Y)$ determines how many bits/symbol can on average be sent through the channel (including error correction), assuming the sender uses $p_X$ probability distribution among symbols. Capacity $C$ uses probability distribution miximizing the throughput, in analogy to maximizing entropy by UPM in the previously considered noise-free case. In contrast, we will focus on priority of optimizing energy efficiency here:
\begin{equation} \eta = \frac{I(X;Y)}{E}=\frac{I(X;Y)}{\sum_x \Pr(x)E_x}\quad \textrm{bits per energy unit}\end{equation}
\begin{figure}[t!]
\centering
\includegraphics[width=8cm]{dens.png}\
\begin{center}
\caption{Density of $Y=X+Z$ for hexagonal modulation and $N=0.1$ (left) or $N=1$ (right), assuming noise $Z$ from two-dimensional Gaussian distribution.}
\label{dens}
\end{center}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=8cm]{mutinf.png}\
\begin{center}
\caption{Four plots of correspondingly: entropy and efficiency for $N=0.1$, entropy and efficiency for $N=1$ for all three modulations.}
\label{mutinf}
\end{center}
\end{figure}
In the case of modulation, there is usually assumed Gaussian noise:
\begin{equation} Y=X+Z\qquad \textrm{where}\quad Z \sim \mathcal{N}(0,N) \end{equation}
is from Gaussian distribution with $\sigma^2=N$ average energy of noise (variance). We will assume two-dimensional Gaussian noise in complex plane:
$$ \rho_Z(z)=\frac{1}{2\pi N} e^{-\frac{|z|^2}{2N}}$$
Figure \ref{dens} presents examples of $Y=X+Z$ density. After fixing some $X$ value, $Y=X+Z$ is Gaussian distribution around this value, hence $h(Y|X)=h(Z)$. To find capacity, we need to find probability distribution for $X$ to maximize $h(X+Y)$. For a fixed average energy $E$ of $X$, this entropy is maximized for $X\sim \mathcal{N}(0,E)$ Gaussian distribution, getting $Y=X+Y\sim \mathcal{N}(0,E+N)$.
Hence, to optimally exploit the AWGN channel, there should be used NPM with Gaussian probability distribution. Instead, in applications there is used UPM, what as comes with a penalty.\\
Figure \ref{mutinf} presents mutual information and efficiency for two noise levels: $N=0.1$ and $N=1$. Surprisingly, for $N=1$, ternary and hexagonal modulation, throughput is maximized for $p=0$, what means not using the zero-signal. It is caused by the fact that zero-signal is nearly useless for such high level of noise - will be most likely interpreted as a different signal. However, the energy efficiency is maximized at the opposite end: for $p=1$, where for all modulations the limit is
\begin{equation} \lim_{p\to 1} \eta = \frac{1}{N \ln(4)}\approx \frac{0.72135}{N} \end{equation}
The $N=0.1$ case is more realistic. For example for hexagonal modulation, and UPM ($p=1/7$), one can transmit $\approx 2.568$ bits/energy unit. This amount can be doubled by using $p\approx 0.916$ frequency of zero signal, at cost of $\approx 5.1$ times lower mutual information (throughput). Finally $\lim_{p\to 1} \eta \approx 7.2135$, however, with throughput also going to 0.
\section{Reversed entropy coding + channel coding}
The standard entropy coding allows to encode a sequence of symbols using on average approximately Shannon entropy bits per symbol. It translates a sequence of symbols having some assumed probability distribution $\{p_s\}_s$ into a sequence of preferably uncorrelated $\Pr(0)=\Pr(1)=1/2$ bits (to maximize their informational content). To optimally use the assumed probability distribution, symbol of probability $p$ should on average use $\lg(1/p)$ bits of information, which generally does not have to be a natural number. For example $a\to 0,\ b\to 10,\ c\to 11$ is a prefix code optimal for $\Pr(a)=1/2,\ \Pr(b)=\Pr(c)=1/4$ probability distribution.
NPM requires to handle a kind of reversed entropy coding problem: the message to encode is a sequence of bits (preferably uncorrelated $\Pr(0)=\Pr(1)=1/2$ to maximize their content), and we want to translate it into a sequence of symbols of some chosen probability distribution. This purpose can be fulfilled by using entropy coder, but with switched encoder and decoder. For example a prefix code $0\to a,\ 10\to b,\ 11\to c$ can translate an i.i.d. $\Pr(0)=\Pr(1)=1/2$ input sequence of bits into a sequence of symbols with $\Pr(a)=1/2,\ \Pr(b)=1/4,\ \Pr(c)=1/4$ probability distribution.
However, this approach approximates probabilities with powers of 1/2 and cannot handle probability close to 1, useful for example for the discussed zero-signal. Additionally, adding error correction capabilities would require using some additional coding layer. We will discuss using recent tANS coding instead~(\cite{ANS,ANS1}), which has similar processing cost as Huffman coding (finite state automaton, no multiplication), but uses nearly accurate probabilities, including close to $1$. Additionally, it has a history dependent internal state, which can be used for error correction purposes in analogy to convolutional codes.
\subsection{Reversed tANS coding (rtANS)}
Reversing tANS entropy coding, we constructs automaton with $L=2^R$ states, which translates a bit sequence into a sequence of symbols having a chosen probability distribution. For the purpose of this paper (it is usually opposite), we will denote state by $s\in\{L,\ldots,2L-1\}$ and symbol as $x\in\mathcal{A}$. The state $s$ acts as a buffer containing $\lg(s)\in [R,R+1)$ fractional number of bits. Symbol $x$ of probability $1/2^{k_x} \leq p_x < 1/2^{k_x-1}$ modifies the state and produces $k_x$ or $k_x-1$ bits to the bitstream, depending if $s$ is above or below some boundary. $L=4$ state example is presented in Fig. \ref{autom}. In practice there is for example used $L=2048$ states and $|\mathcal{A}|=256$ size alphabet.
The construction of such automaton first chooses quantized probability distribution: $L_x \in \mathbb{N}:\ \sum_x L_x = L$ and $p_x \approx L_x/L$. Then spread symbols: choose $symbol[s]$ for every position $s\in\{L,\ldots,2L-1\}$, such that symbol $x$ was used $L_x$ times.
\begin{figure}[t!]
\centering
\includegraphics[width=8cm]{autom.png}\
\begin{center}
\caption{Example of reversed (encoder and decoder are switched) tANS automaton for $L=4$ states and $\Pr(a)=3/4,\ \Pr(b)=1/4$ probability distribution and its application for a bitstream (bottom). It was generated using $"abaa"$ symbol spread, with $L_a=3,\ L_b=1$ numbers of appearances, corresponding to the assumed probability distribution. Symbol $b$ carries $-\lg(1/4)=2$ bits of information and so the automaton always uses 2 bits for this symbol. In contrast, symbol $a$ carries $\lg(4/3)\approx 0.415$ bits of information. Hence the automaton sometimes uses one bit for this symbol, sometime 0 bits - just accumulating information in the state $s$. The $\rho_s$ denotes probability of using state $s$ by this automaton. Observe that decoding and encoding are performed in opposite directions. }
\label{autom}
\end{center}
\end{figure}
The explanation and details can be found in other sources, here we focus on using for NPM. The reversed tANS (rtANS) encoding (original decoding) is presented as Algorithm \ref{dec0}. Each state determines symbol to produce, and a rule for new state: some base value ($newS$) and the number of bits to read from the bitstream ($nbBits$):
\begin{algorithm}[htbp]
\footnotesize{
\caption{rtANS encoding step, $S=s-L\in\{0,\ldots,L-1\}$}
\label{dec0}
\begin{algorithmic}
\STATE $t = encodingTable[S]$ \quad \COMMENT{ $S\in \{0,..,L-1\}$ is current state }
\STATE useSymbol($t.symbol$) \qquad\qquad \COMMENT{ use or store decoded symbol }
\STATE $s = t.newS + $readBits$(t.nbBits)$ \qquad\qquad \COMMENT{ state transition }
\end{algorithmic}
}
\end{algorithm}
This inexpensive loop would be used by the sender to transform bitstream (message) into stream of symbols of chosen probability distribution. Without getting into details here, the Algorithm \ref{gen} allows to generate the $encodingTable$, what can be done outside the device, and just written in it. This generation requires choosing $symbol[]$ spread function earlier, which determines the coding (has to be the same for decoding and encoding). Sources and benchmarks of various ways of symbol spread it can be found in~\cite{toolkit}.
\begin{algorithm}[htbp]
\footnotesize{
\caption{Preparation for rtANS encoding, $L=2^R$}
\label{gen}
\begin{algorithmic}
\REQUIRE $next[x] = L_x$ \quad\COMMENT{next appearance of symbol $x$}
\FOR {$S=0$ to $L-1$}
\STATE $t.symbol=symbol[S]$ \quad\COMMENT{ symbol is from spread function }
\STATE $s=next[t.symbol]++$
\STATE $t.nbBits = R -\lfloor \lg(s)\rfloor$ \qquad \COMMENT{ number of bits}
\STATE $t.newS = (s << t.nbBits)-L$\qquad\COMMENT{ properly shift $s$ }
\STATE $encodingTable[S]=t$
\ENDFOR
\end{algorithmic}
}
\end{algorithm}
Now let us look at the rtANS decoding step, which will be performed by receiver and is crucial from the point of view of eventual error correction. It is performed in opposite direction then encoding: it starts with the final encoding step (needs to be transmitted) and ends with the initial encoding state (arbitrary, may contain some information). For the discussed purpose encoding can be made in forward direction, decoding in backward. For simplicity we will discuss decoding as in forward direction here.
As in Figure \ref{autom}, we can write decoding step as
\begin{equation} D(s,x)=(s_{sx},B_{sx}) \label{decoding} \end{equation}
where $s$ is the starting state. Symbol $x$ takes us to state $s_{sx}$, and produces $B_{sx}\in\mathcal{A}^*$ bit sequence of length $|B_{sx}|\in\{0,\ldots,\lg(L)\}$ (can be empty). Additionally, $B_{xs}$ are just $|B_{xs}|\approx \lg(1/p_x)$ youngest bits of the state $s$. Optimized implementation and preparation is presented as Algorithms \ref{enc0} and \ref{encprep}:
\begin{spacing}{0.5}
\begin{algorithm}[htbp]
\footnotesize{
\caption{rtANS decoding step for symbol $x$ and state $s=S+L$}
\label{enc0}
\begin{algorithmic}
\STATE $nbBits = (s + nb[x]) >> r$ \quad\qquad \COMMENT{$r=R+1,\ 2^r = 2L$}
\STATE useBits$(s, nbBits)$ \qquad\COMMENT {use $nbBits$ of the youngest bits of $s$}
\STATE $s = decodingTable[start[x] + (s >> nbBits)]$
\end{algorithmic}
}
\end{algorithm}
\end{spacing}
\begin{algorithm}[htbp]
\footnotesize{
\caption{Preparation for rtANS decoding, $L=2^R$, $r=R+1$}
\label{encprep}
\begin{algorithmic}
\REQUIRE $k[x] = R-\lfloor \lg(L_x) \rfloor$ \qquad \COMMENT{$nbBits=k[x]$ or $k[x]-1$}
\REQUIRE $nb[x] = (k[x] << r)-(L_x << k[x])$
\REQUIRE $start[x]= - L_x + \sum_{i < x} L_i$
\REQUIRE $next[x] = L_x$
\FOR {$s=L$ to $2L-1$}
\STATE $x=symbol[s-L]$
\STATE $decodingTable[start[x] + (next[x]++)] = s$;
\ENDFOR
\end{algorithmic}
}
\end{algorithm}
\subsection{Forward error correction}
We will now discuss adding forward error correction: sender adds redundancy in the transmitted data (inexpensive), which is used by receiver to correct eventual errors (relatively expensive). A simple way to add redundancy for rtANS is for example by putting the message in even positions of the bitstream to encode, and 0 in odd positions, getting rate 1/2 code. For simplicity we will focus on this case, but analogously we can get different rational rates, for example 2/3 by putting 0 every 3rd position.
Let us denote $U=\{u_0,\ldots, u_{N-1}\}$ as encoded bitstream, which even positions contain the message, and odd positions are zeros: $\forall_i\ u_{2i+1}=0$. The rtANS has transformed it into the transmitted sequence of symbols $\{x_0,\ldots, x_{n-1}\}$, which was modified by the noise into received $\{y_0,\ldots,y_{n-1}\}$ sequence of symbols. The goal of receiver is to find the closest sequence $X$, for which decoded bitstream has zeros at odd positions.
Let us denote $\{P_0=0,\ P_1 ,\ldots, P_n =N\}$ as positions in $U$ corresponding to successive symbols, and $\{s_0,\ldots,s_n\}$ as states of encoder, such that:
\begin{equation} D(s_t,x_t)=(s_{t+1}, \{U_{P_t},\ U_{P_t+1},\ldots, U_{P_{t+1}-1}\}) \end{equation}
It assumes for simplicity forward decoding. State $s_0$ is initial for decoding (final for encoding, also transmitted), $s_n$ is the final state of decoding (initial for encoding).\\
Error correction in our case has similar inconvenience as for synchronization channels: the relation between symbols and corresponding bit blocks of varying length ($P_t$ sequence) is not known. It is a crucial complication for Viterbi~\cite{viterbi} and BCJR~\cite{bcjr} type of approaches. Fortunately, this is not an issue for sequential decoding~\cite{fano}, also successfully applied for deletion channel~\cite{deletion}.
In sequential decoding we build a tree of corrections: start with the root as known state $s_0$ and expand it, such that each branch in depth $t$ corresponds to one of possible choices of $x_t$. As expanding all possibilities would mean exponential growth, it is crucial to expand only promising looking nodes. In this case, the number of nodes to consider for reasonable parameters is usually a relatively small multiplicity of the length of sequence - tools for analysis can be found in \cite{cortree}.
A depth $t$ node of the tree corresponds to assuming some sequence of symbols $\{x_0,\ldots,x_{t-1}\}$, which corresponds to some $\{u_0,\ldots, u_{T-1}\}$ hypothetical prefix of the bit sequence. The applied redundancy says to consider only nodes fulfilling $\forall_i\ u_{2i+1}=0$, other branches are not expanded. Length $T$ bit sequence has $\lfloor T/2 \rfloor$ such bits verifying that we consider a prefix of a codeword. For an improper correction (node) we can assume that these bits are i.i.d. $\Pr(0)=\Pr(1)=1/2$, so there is $2^{-\lfloor T/2 \rfloor}$ probability of accidentally fulfilling these constraints. While choosing a leaf to expand, Bayes analysis says that probability that a given leaf is the proper one is proportional to:
$$\frac{\Pr(\{x_0,\ldots,x_{t-1}\}|\{y_0,\ldots,y_{t-1}\})}{\Pr(\textrm{accidentially fulfilling the constraints})}\propto$$
\begin{equation} 2^{\lfloor T/2 \rfloor} \prod_{0\leq i < t} (\Pr(x_i) \Pr(y_i|x_i)) \end{equation}
\noindent where $\Pr(x_i)$ is the chosen probability distribution of symbols, $\Pr(y_i|x_i)$ is the assumed model of error, for example Gaussian distribution. In practice there is used weight $w$ as logarithm of the above formula, calculated for new node as weight of its father plus $\Delta w$:
\begin{equation} \Delta w = \lg(\Pr(x_t) \Pr(y_t|x_t)) + \#\{\textrm{new constrained bits}\} \end{equation}
with the number of positions constrained by the rule $u_{2i+1}=0$ in the new bit block corresponding to using symbol $x_t$.
Finally, the correction process after initialization with a root before the first symbol, is a loop of choosing the most promising leaf so far (having largest $w$) and expanding it, until reaching the final position with the proper final state.
The rtANS decoding has convenient property that bits produced in given step are some number of the least significant bits of the state $s$. Knowing the state we can quickly determine the maximal number of such bits to fulfill the $b_{2i+1}=0$ condition, which determines the symbols which could be used in this step.
\section{Conclusions}
While the standard approach is encoding information using uniform probability distribution among some symbols (UPM), we have discussed practical application of non-uniform distributions (NPM). For example capacity of AWGN channel is fulfilled for Gaussian distribution, not the uniform one. Instead of prioritizing on channel capacity, we have focused on energy efficiently here: amount of transmitted bits per energy unit, which can be increased at cost of reduced throughput. It can be practically doubled for hexagonal modulation, or quadrupled for binary modulation, by more frequent use of zero-signal. Example of application is improving battery life of remote sensors.
The discussed solution for coding was tANS entropy coder, which has inexpensive processing cost (finite state automaton, no multiplication), but uses nearly accurate probabilities. Additionally, there was discussed cost-free redundancy addition while this encoding step, for example by inserting zeros at odd positions of the bit sequence. Sequential decoding can be used for error correction of such message. It is slightly more complex than for UPM, but this energy and hardware cost is not paid in the remote sensor.
\bibliographystyle{IEEEtran}
| 8,616 |
\section{Introduction}
A generalisation of the concept of \emph{coset}, from groups to inverse semigroups, was proposed by Schein in \cite{Sch}.
There are three essential ingredients to this generalisation: firstly, cosets are only to be defined for an inverse subsemigroup $L$ of an
inverse semigroup $S$ that is \emph{closed} in the natural partial order on $S$; secondly, an element $s \in S$ will only determine a coset if
$ss^{-1} \in L$; and thirdly, the coset is finally obtained by taking the closure (again with respect to the natural partial order on
$S$) of the subset $Ls$. The details of this construction are presented in Section \ref{intro_cosets}. In fact, Schein takes as his
starting point a characterization of cosets in groups due to Baer \cite{Baer} (see also \cite{Cert}): a subset $C$ of a group $G$ is a coset of some subgroup $H$ of $G$ if and only if $C$ is closed under the ternary operation $(a,b,c) \mapsto ab^{-1}c$ on $G$.
A closed inverse subsemigroup\ $L$ of an inverse semigroup $S$ has \emph{finite index} if and only if there are only finitely many distinct cosets of
$L$ in $S$. In contrast to the situation in group theory, finite index can arise because of the relative paucity of possible
coset representatives satisfying $ss^{-1} \in L$.
For example, in the free inverse monoid ${\it FIM}(a,b)$, the inverse subsemigroup ${\it FIM}(a)$ has finite index. This fact is a
consequence of a remarkable theorem of Margolis and Meakin, characterising the closed inverse submonoids\ in a free inverse monoid ${\it FIM}(X)$ with
$X$ finite:
\begin{theorem}{\cite[Theorem 3.7]{MarMea}}
\label{MMthm}
Let $X$ be a finite set, and let $L$ be a closed inverse submonoid\ of the free inverse monoid ${\it FIM}(X)$. Then the following conditions are equivalent:
\begin{enumerate}
\item $L$ is recognised by a finite inverse automaton,
\item $L$ has finite index in ${\it FIM}(X)$,
\item $L$ is a recognisable subset of ${\it FIM}(X)$,
\item $L$ is a rational subset of ${\it FIM}(X)$,
\item $L$ is finitely generated as a closed inverse submonoid\ of ${\it FIM}(X)$.
\end{enumerate}
\end{theorem}
Condition (e) of Theorem \ref{MMthm} asserts the existence of a finite set $Y \subset {\it FIM}(X)$ such that $L$ is equal to the closed inverse submonoid\ generated by
$Y$. The original statement of the theorem in \cite{MarMea} includes an extra condition related to immersions of finite graphs, which we have omitted.
Our aims in the present paper are to present some basic facts about closed inverse subsemigroups\ of finite index, and to study the relationships between the
conditions given in Theorem \ref{MMthm} when ${\it FIM}(X)$ is replaced by an arbitrary inverse semigroup or monoid.
In Section \ref{intro_cosets} we give an introduction to the concept of cosets in inverse semigroups, and the action of an inverse
semigroup on a set of cosets, based closely on the ideas of Schein \cite{Sch2, Sch}. We establish an \emph{index formula},
relating the indices $[S:K], [S:H]$ and $[H:K]$ for closed inverse subsemigroups $H,K$ of an inverse semigroup $S$ with
$E(S) \subseteq K \subseteq H$ in Theorem \ref{indexformula}, and an analogue of M. Hall's Theorem for groups,
that in a free group of finite rank, there are only finitely many subgroups of a fixed finite index, in Theorem \ref{Hall's theorem}.
Our work based on Theorem \ref{MMthm} occupies section \ref{fg_and_fi}, and is summarised in Theorem \ref{gen_MMthm}.
We show that, in an arbitrary finitey generated inverse monoid $M$, a closed inverse submonoid has finite index if and only if it is recognisable, in which case it is rational and finitely generated as a closed inverse submonoid, but that finite generation (as a
closed inverse submonoid) is a strictly weaker property. This is not a surprise, since any inverse semigroup with a zero is
finitely generated in the closed sense.
The authors thank Mark Lawson and Rick Thomas for very helpful comments, and in
particular for their shrewd scrutiny of Lemma \ref{star-height}.
\section{Cosets of closed inverse subsemigroups}
\label{intro_cosets}
Let $S$ be an inverse semigroup with semilattice of idempotents $E(S)$. Recall that the \emph{natural partial order} on $S$ is defined by
\[ s \leqslant t \Longleftrightarrow \text{there exists} \; e \in E(S) \; \text{such that} \; s=et \,.\]
A subset $A \subseteq S$ is \emph{closed} if, whenever $a \in A$ and $a \leqslant s$, then $s \in A$. The closure $\nobraupset{B}$ of a subset
$B \subseteq S$ is defined as
\[ \nobraupset{B} = \{ s \in S : s \geqslant b \; \text{for some} \; b \in B \}\,.\]
A subset $L$ of $S$ is \emph{full} if $E(S) \subseteq L$.
An {\em atlas} in $S$ is a subset $A \subseteq S$ such that $AA^{-1}A \subseteq A$:
that is, $A$ is closed under the {\em heap} ternary operation $\< a,b,c \> = ab^{-1}c$ (see \cite{Baer}). Since, for all $a \in A$ we have
$\<a,a,a\>=a$, we see that $A$ is an atlas if and only if $AA^{-1}A=A$.
A {\em coset} $C$ in $S$ is a closed atlas: that is, $C$ is both upwards closed in the natural partial order on $S$ and
is closed under the heap operation $\< \dotsm \>$.
Let $X$ be a set and ${\mathscr I}(X)$ its symmetric inverse monoid. Let $\rho : S \rightarrow {\mathscr I}(X)$ be a faithful representation of $S$
on $X$, and write $x(s \rho)$ as $x \lhd s$.
The principal characterisations of cosets that we need are due to Schein:
\begin{theorem}{\cite[Theorem 3.]{Sch}}
\label{cosets}
Let $C$ be a non-empty subset of an inverse semigroup $S$. Then the following are equivalent:
\begin{enumerate}
\item $C$ is a coset,
\item there exists a closed inverse subsemigroup $L$ of $S$ such that, for all $s \in C$, we have $ss^{-1} \in L$ and
$C=\upset{Ls}$.
\item there exists a closed inverse subsemigroup $K$ of $S$ such that, for all $s \in C$, we have $s^{-1}s \in K$ and
$C=\upset{sK}$.
\end{enumerate}
\end{theorem}
\begin{proof}
(a) $\Longrightarrow$ (b): Let $Q=CC^{-1} = \{ ab^{-1} : a,b \in C \}$. Then $Q$ is an inverse subsemigroup of $S$,
since, for all $a,b,c,d \in C$ we have
\begin{itemize}
\item $(ab^{-1})(cd^{-1})=(ab^{-1}c)d^{-1} = \< a,b,c \>d^{-1} \in Q$,
\item $(ab^{-1})^{-1} = ba^{-1} \in Q$.
\end{itemize}
Set $L=\nobraupset{Q}$: then $L$ is a closed inverse subsemigroup of $S$. Let $s \in C$. Obviously $ss^{-1} \in Q \subseteq L$.
Moreover, given any $c \in C$ we have
$c \geqslant c(s^{-1}s) = (cs^{-1})s \in Qs \subseteq Ls$, so that $C \subseteq \upset{Ls}$. Conversely, if $x \in \upset{Ls}$,
we have $x \geqslant us$ for some $u \in L$, with $u \geqslant ab^{-1}$ for some $a,b \in C$. Hence
$x \geqslant us \geqslant ab^{-1}c = \< a,b,c \> \in C$. Since $C$ is closed, $x \in C$ and therefore $\upset{Ls} \subseteq C$.
(b) $\Longrightarrow$ (a): The subset $\upset{Ls}$ is a coset, since it is closed by definition, and if $h_i \in \upset{Ls}$
we have $h_i \geqslant t_is$ for some $t_i \in L$. Then
$\< h_1,h_2,h_3 \> = h_1h_2^{-1}h_3 \geqslant t_1ss^{-1}t_2^{-1}t_3s \in Ls$ since $ss^{-1} \in L$. It follows that
$\upset{Ls}$ is closed under the heap operation $\< \dotsm \>$.
For (a) $\Longleftrightarrow$ (c): we proceed in the same way, with $K = \upset{C^{-1}C}$.
\end{proof}
For the rest of this paper, all cosets will be \emph{right} cosets, of the form $\upset{Ls}$.
\begin{prop}{\cite[Proposition 5.]{Sch}}
\label{cosets_idpt}
A coset $C$ that contains an idempotent $e \in E(S)$ is an inverse subsemigroup of $S$, and in this case $C = \upset{CC^{-1}}$.
\end{prop}
\begin{proof} If $a,b \in C$ then $ab \geqslant aeb = \< a,e,b \> \in C$ and since $C$ is closed, we have $ab \in C$.
Furthermore, $a^{-1} \geqslant ea^{-1}e = \< e,a,e \> \in C$ and so $a^{-1} \in C$. Hence $C$ is an inverse subsemigroup.
Now $ab^{-1} \in CC^{-1}$ and $ab^{-1} \geqslant ab^{-1}e = \< a,b,e \> \in C$. Since $C$ is closed we have
$\upset{CC^{-1}} \subseteq C$. But if $x \in C$ then $x \geqslant xe \in CC^{-1}$ and so $x \in \upset{CC^{-1}}$.
Therefore $C = \upset{CC^{-1}}$.
\end{proof}
Now if $L$ is a closed inverse subsemigroup of $S$, a coset of $L$ is a subset of the form $\upset{Ls}$ where
$ss^{-1} \in L$. Suppose that $C$ is such a coset: then Theorem \ref{cosets} associates to $C$ the closed inverse
subsemigroup $\upset{CC^{-1}}$.
\begin{prop}{\cite[Proposition 6.]{Sch}}
\label{cosetsofL}
Let $L$ be a closed inverse subsemigroup of $S$.
\begin{enumerate}
\item Suppose that $C$ is a coset of $L$. Then $\upset{CC^{-1}}=L$.
\item If $t \in C$ then $tt^{-1} \in L$ and $C = \upset{Lt}$. Hence two cosets of $L$ are either disjoint or they coincide.
\item Two elements $a,b \in S$ belong to the same coset $C$ of $L$ if and only if $ab^{-1} \in L$.
\end{enumerate}
\end{prop}
\begin{proof}
(a) If $c_i \in C$ ($i=1,2$) then there exists $l_i \in L$ such that $c_i \geqslant l_is$.
Hence $c_1c_2^{-1} \geqslant l_1ss^{-1}l_2^{-1} \in L$, and so $CC^{-1} \subseteq L$. Since $L$ is closed,
$\upset{CC^{-1}} \subseteq L$. On the other hand, for any $l \in L$ we have
$l=ll^{-1}l \geqslant lss^{-1}l^{-1}l = (ls)(l^{-1}ls)^{-1} \in CC^{-1}$ and so $L \subseteq \upset{CC^{-1}}$.
(b) If $C = \upset{Ls}$ and $t \in C$ then, for some $l \in L$ we have $t \geqslant ls$. Then
$tt^{-1} \geqslant lss^{-1}l^{-1} \in L$, and since $L$ is closed, $tt^{-1} \in L$. Moreover, if
$u \in \upset{Lt}$ then for some $k \in L$ we have $u \geqslant kt \geqslant kls$ and so $u \in \upset{Ls}$. Hence if
$t \in \upset{Ls}$ then $\upset{Lt} \subseteq \upset{Ls}$. Now
$ls = (ls)(ls)^{-1}t = lss^{-1}l^{-1}t$ and so $l^{-1}ls = l^{-1}lss^{-1}l^{-1}t = ss^{-1}l^{-1}t \in Lt$.
Since $s \geqslant l^{-1}ls$, we deduce that $s \in \upset{Lt}$. Hence $\upset{Ls} \subseteq \upset{Lt}$.
(c) Suppose that $a,b \in \upset{Ls}$. Then for some $k,l \in L$ we have $a \geqslant ks$ and $b \geqslant ls$: hence
$ab^{-1} \geqslant kss^{-1}l^{-1} \in L$ and so $ab^{-1} \in L$. On the other hand, suppose that $ab^{-1} \in L$.
Then $aa^{-1} \geqslant a(b^{-1}b)a^{-1} = (ab^{-1})(ab^{-1})^{-1} \in L$, and similarly $bb^{-1} \in L$. We note that
$a = (aa^{-1})a \in La$ and similarly $b \in Lb$. Then
$a \geqslant a(b^{-1}b) = (ab^{-1})b$ and so $a \in \upset{Lb}$. As in part (b) we deduce that $\upset{La} \subset \upset{Lb}$.
By symmetry $\upset{La} = \upset{Lb}$ and this coset contains $a$ and $b$.
\end{proof}
\begin{example}
\label{Eclosed}
Let $E$ be the semilattice of idempotents of $S$. The property that $E$ is closed is exactly the property that $S$ is
\emph{$E$--unitary}. In this case, for any $s \in S$, we have
\begin{align*}
\upset{Es} &= \{ t \in S : t \geqslant es \; \text{for some} \; e \in E \} \\
&= \{ t \in S : t \geqslant u \leqslant s \; \text{for some} \; u \in S \} \\
&= \{ t \in S : s,t \; \text{have a lower bound in} \; S \}.
\end{align*}
We see that $\upset{Es}$ is precisely the $\sigma$--class of $s$, where $\sigma$ is the minimum group congruence on $S$,
see \cite[Section 2.4]{LwBook}. Hence every element $t \in S$ lies in a coset of $E$, and the set of cosets is in
one-to-one correspondence with the maximum group image $\widehat{S}$ of $S$.
\end{example}
\begin{remark}
Let $L$ be a closed inverse subsemigroup of an inverse semigroup $S$. Then the union $U$, of all the cosets of $L$ is a subset of $S$ but need not be all of $S$, and is not always a subsemigroup of $S$.
\end{remark}
We illustrate this remark in the following example.
\begin{example}
Fix a set $X$ and recall that the \emph{Brandt semigroup} $B_X$ is defined as follows. As a set, we have
\[ B_X = \{ (x,y) : x,y \in X \} \cup \{ 0 \} \]
with
\[ (u,v)(x,y) =
\begin{cases}
(u,y) & \text{if} \; v=x \\
0 & \text{if} \; v \ne x
\end{cases} \]
and $0(x,y) = 0 = (x,y)0$.
The idempotents of $B_X$ are the elements $(x,x)$ for $x \in X$ and $0$. Hence $0 \leqslant (x,y)$ for all $x,y \in X$ and
$(u,v) \leqslant (x,y)$ if and only if $(u,v)=(x,y)$. If a closed inverse semigroup $L$ of $B_X$ contains $(x,y)$ with $x \ne y$
then $(x,y)(x,y)=0 \in L$ and so $L=B_X$. Therefore the only proper closed inverse subsemigroups are the subsemigroups
$E_x = \{ (x,x) \}$ for $x \in X$. An element $(x,y) \in B_X$ then determines the coset
\[ \upset{E_x(x,y)} = \{ (x,y) \} \,.\]
Hence there are $|X|$ distinct cosets of $E_x$ and their union is
\[ U = \{ (x,y) : y \in X \} \,. \]
\end{example}
\begin{prop}
\label{cosetunion}
\leavevmode
\begin{enumerate}
\item Let $L$ be a closed inverse subsemigroup of an inverse semigroup $S$ and let $U$ be the union of all the cosets of $L$ in $S$. Then
$ U= \left\{ s\in S : ss^{-1} \in L \right\} $
and therefore $U=S$ if and only if $L$ is full.
\item $U$ is a closed inverse subsemigroup of $S$ if and only if whenever $e \in E(L)$ and $s \in U$
then $ses^{-1} \in U$ and if,
whenever $s \in S$ with
$ss^{-1}\in L$, then $s^{-1}s \in L \,.$
\end{enumerate}
\end{prop}
\begin{proof}
(a) The coset $\upset{Lu}$ containing $u \in S$ exists if and only if $uu^{-1} \in L$.
(b) Suppose that $L$ satisfies the given conditions. If $s,t \in U$ then $ss^{-1}, tt^{-1} \in L$ and
\[ (st)(st)^{-1} = s(tt^{-1})s^{-1} \in U \]
which implies that $st \in U$, and $s^{-1}s \in L$ which implies that $s^{-1} \in U$. Hence $U$ is an inverse subsemigroup.
Conversely, if $U$ is an inverse subsemigroup and $s \in U$, then $s^{-1} \in U$ which implies that $s^{-1}s \in L$,
and if $s \in U$ and $e \in E(L)$ then $e \in U$ and so $se \in U$ which implies that $(se)(se)^{-1} = ses^{-1} \in U$.
\end{proof}
\subsection{The index formula}
The \textit{index} of the closed inverse subsemigroup $L$ in an inverse semigroup $S$ is the cardinality of the set of right cosets of $L$, and is written $[S:L]$. Note that the mapping $\upset{Ls} \rightarrow \upset{s^{-1}L}$ is a bijection from the set of right cosets to the set of left cosets. A \emph{transversal} to $L$ in $S$ is a choice of one element from each right coset of $L$. For a transversal ${\mathcal T}$, we have the union
$$U = \bigcup_{t \in {\mathcal T}}\,\upset{Lt}, $$
as in Proposition \ref{cosetunion}, and each element $u \in U$ satisfies $u \geqslant ht$ for some $h \in L,\, t \in {\mathcal T}.$
\begin{theorem}\label{indexformula}
Let $S$ be an inverse semigroup and let $H$ and $K$ be two closed inverse subsemigroups of $S$ with $K$ of finite index in $H$ and $H$ of finite index in $S$, and with $K \subseteq H$ and $K$ full in $S.$
Then $K$ has finite index in $S$ and
$$ [ S:K ] = [\, S:H ] [\, H:K ]. $$
\end{theorem}
\begin{proof}
Since $K$ is full in $S$, then so is $H$ and for transversals ${\mathcal T}, {\mathcal U}$ we have
$$S = \bigcup_{t \in {\mathcal T}} \upset{Ht} \quad \text{and} \quad H = \bigcup_{u \in {\mathcal U}} \upset{Ku}.$$
Therefore
$$S = \{ s \in S : s \geqslant ht \quad \textit{for some} \; t \in {\mathcal T}, h \in H \},$$ and
$$H = \{ s \in S : s \geqslant ku \quad \textit{for some} \; u \in {\mathcal U}, k \in K \}.$$
Now if $s \geqslant ht$ and $h \geqslant ku$ then $s \geqslant kut $. Then
$s\in\,\upset{Kut}$ and $\upset{Kut}$ is a coset of $K$ in $S$, since $K$ is full in $S$ and therefore $ (ut)\,(ut)^{-1}\,\in\,K.$\\
Hence
$$ S = \bigcup_{\substack{u \in U \\ t \in {\mathcal T}}} \, \upset{Kut} \,.$$
It remains to show that all the cosets $\upset{Kut}$ are distinct.
Suppose that $\upset {Ku't'} = \upset{Kut}$.
Then by part (c) Proposition \ref{cosetsofL}, $u't't^{-1}u^{-1}\in K$ and so $u't't^{-1}u^{-1} \in H$.
Since $u,u' \in H$ we have
$(u')^{-1}u't't^{-1}u^{-1}u \in H$
and since
$t't^{-1} \geqslant (u')^{-1}u't' t^{-1}u^{-1}u \in H$
and $H$ is closed, then $t't^{-1} \in H .$
This implies that
$ \upset {Ht'}= \upset {Ht}$ and it follows that $t'=t$.
Now
$u't't^{-1}u^{-1} \in K$. Since $t' = t$, then $t't^{-1} \in E(S)$ and so
$u'u^{-1} \geqslant u't't^{-1}u^{-1}$.
But $K$ is closed, so $u'u^{-1} \in K$ and $\upset{Ku'} = \upset {Ku}$. Hence $u'=u$.
Consequently, all the cosets $\upset{Kut}$ are distinct.
\end{proof}
Recall from Example \ref{Eclosed} that the property that $E(S)$ is closed is expressed by saying that $S$ is
$E$--unitary and that in this case, the set of cosets of $E(S)$ is in
one-to-one correspondence with the maximum group image $\widehat{S}$ of $S$.
\begin{prop}
\label{fullfindex}
Let $S$ be an $E$--unitary inverse semigroup. Then:
\begin{enumerate}
\item[(a)] $E(S)$ has finite index if and only if the maximal group image $\widehat{S}$ is finite, and $[ S: E ] = |\widehat{S}|$,
\item[(b)] if $E(S)$ has finite index in $S$ then, for any closed, full, inverse subsemigroup $L$ of $S$ we have
\[ [ S:L ] = |\widehat{S}| \, / \,|\widehat{L}| \,\]
\end{enumerate}
\end{prop}
\begin{proof}
Part (a) follows from our previous discussion. For part (b) we have $E(S) \subseteq L \subseteq S$ and so if the index $[S:E]$ is finite
then so are $[S:L]$ and $[L:E]$ with $[S:E]=[S:L][L:E]$. But now $[S:E]=|\widehat{S}|$ and $[L:E]=|\widehat{L}|$.
\end{proof}
The index formula in Theorem \ref{indexformula} can still be valid when $K$ is not full in $S$ as we show in the following Example.
\begin{example}
\label{clism_of_I3}
We work in the symmetric inverse monoid ${\mathscr I}_3 = {\mathscr I} \left(\{1,2,3\}\right)$, and take $L=stab(1) =\{\sigma \in {\mathscr I} (X):\, 1\sigma =1 \}$,
which is a closed inverse subsemigroup of $ {\mathscr I} (X)$ with $7$ elements. There are $3$ cosets of $L$ in ${\mathscr I}_3$, namely
$$ C_1 = \{\sigma \in {\mathscr I}_3 : 1\sigma =1 \} = L,$$
$$ C_2 = \{\sigma \in {\mathscr I}_3 : 1\sigma =2 \},$$
$$ C_3 = \{\sigma \in {\mathscr I}_3 : 1\sigma =3 \},$$
and so $[{\mathscr I}_3:L]=3\,.$
Now take
\[ K = \,\left\{
\begin{pmatrix}
1 & 2 & 3 \\
1 & 2 & 3\\
\end{pmatrix},
\begin{pmatrix}
1 & 2 & 3 \\
1 & 3 & 2\\
\end{pmatrix}\right\}\,.\]
Then $K$ is a closed inverse subsemigroup of $L.$ The domain of each $\sigma$ in $K$ is $\{1,2,3\}.$ and so the only coset
representatives for $K$ in $L$ are the permutations
\[ \operatorname{id}= \begin{pmatrix}
1 & 2 & 3 \\
1 & 2 & 3\\
\end{pmatrix} \; \text{and}\:\: \;
\sigma = \begin{pmatrix}
1 & 2 & 3 \\
1 & 3 & 2\\
\end{pmatrix}.\]
But these are elements of $K$ and so $\nobraupset{K}=\upset{K \sigma}=K$ and there is just one coset. Hence $[ L:K ]=1$.
Now, we calculate the cosets of $K$ in $ {\mathscr I}_3.$ Each permutation in ${\mathscr I}_3$ is a possible coset representative
for $K$ and these produce three distinct cosets by Proposition \ref{cosetsofL}(3). Hence $[{\mathscr I}_3 :K ]=3$
and in this example
\[ [{\mathscr I}_3:K ] = [{\mathscr I}_3:L ] [ L:K ] \,.\]
\end{example}
The index of a closed inverse subsemigroup\ $L$ of an inverse semigroup $S$ depends on the availability of coset representatives to make cosets, and so on the idempotents of $S$ contained in $L$. In particular, we can have $K \subset L$ but $[S:K| < [S:L]$ as the following Example illustrates.
\begin{example}
\label{index_in_fimxy}
Consider the free inverse monoid ${\it FIM}(x,y)$ and the closed inverse submonoids $K = \nobraupset{ \< x^2 \>}$ and
$H = \nobraupset{\< x^2,y^2 \>}$. As in \cite{MarMea, Munn}, we represent elements of ${\it FIM}(x,y)$ by Munn
trees $(P,w)$ in the Cayley graph $\Gamma(F(x,y),\{x,y\})$ where $F(x,y)$ is the free group on $x,y$.
Consider a coset $\nobraupset{K(P,w)}$. For this to exist, the idempotent $(P,1)=(P,w)(P,w)^{-1}$ must be in $K$ and so as a
subtree of $\Gamma$, $P$ can only involve vertices in $F(x)$ and edges between them. Since $w \in P$ we must have
$w \in F(x)$. It is then easy to see that there are only two cosets, $K$ and $\upset{Kx}$ and so $[{\it FIM}(x,y):K]=2$.
Similarly, $[H:K]=1$.
Now consider a coset $\nobraupset{H(P,w)}$. Now $(P,1) \in H$ and so $P$ must be contained in the subtree of $\Gamma$ spanned by the
vertices of the subgroup $\< x^2,y^2 \> \subset F(x,y)$, and $w \in P$. If $w \in \< x^2,y^2 \>$
then $(P,w) \in H$ and $\nobraupset{H(P,w)}=H$. Otherwise, $w = ux$ or $w=uy$ with $u \in \< x^2,y^2 \>$ and it
follows that there are three cosets of $H$ in ${\it FIM}(x,y)$, namely $H, \upset{Hx}$ and $\upset{Hy}$. Therefore
\[ [{\it FIM}(x,y):H] = 3 > [{\it FIM}(x,y):K] =2 \,.\]
These calculations also follow from results of Margolis and Meakin, see \cite[Lemma 3.2]{MarMea}.
We note that the index formula fails to hold. This does not contradict Theorem \ref{indexformula}, since $K$ is not full in
${\it FIM}(x,y).$
\end{example}
\subsection{Coset actions}
Our next aim is to derive an analogue of Marshall Hall's Theorem (see \cite{MHallJr} and \cite{BaumBook}) that, in a free group of finite rank, there are only finitely many subgroups of a fixed finite index. We first record some preliminary results on actions on cosets:
these results are due to Schein \cite{Sch2} and are presented in \cite[Section 5.8]{HoBook}. We give them here for the reader's convenience.
\begin{lemma}
\label{rt_upset}
For any subset $A$ of $S$ and for any $u \in S$ we have $\upset{Au} = \upset{\nobraupset{A}u}$.
\end{lemma}
\begin{proof}
Since $A \subseteq \upset{A}$ it is clear that $\upset{Au} \subseteq \upset{\upset{A}u}$. On the other hand, if
$x \in \upset{\upset{A}u}$ then for some $a \in A$ we have $y \in S$ with $y \geqslant a$ and $x \geqslant yu$. But then
$x \geqslant au$ and so $x \in \upset{Au}$.
\end{proof}
Let $L$ be a closed inverse subsemigroup of $S$, and let $s \in S$ with $ss^{-1} \in L$ with $C=\upset{Ls}$
Now suppose that $u \in S$ and that $\upset{Cuu^{-1}}=C$. Then we define $C \lhd u = \upset{Cu}$.
\begin{lemma}
\label{eq_action_cond}
The condition $\upset{Cuu^{-1}}=C$ for $C \lhd u$ to be defined is equivalent to the condition that
$suu^{-1}s^{-1} \in L$.
\end{lemma}
\begin{proof}
Since for any $c \in C$ we have $c \geqslant cuu^{-1}$ it is clear that $C \subseteq \upset{Cuu^{-1}}$.
Now suppose that $suu^{-1}s^{-1} \in L$ and that $x \in \upset{Cuu^{-1}} = \upset{Lsuu^{-1}}$ (by
Lemma \ref{rt_upset}). Hence there exists $y \in L$ with
$$x \geqslant ysuu^{-1} = yss^{-1}suu^{-1} = ysuu^{-1}s^{-1}s \in Ls.$$
Therefore $x \in \upset{Ls} = C$ and so $\upset{Cuu^{-1}}=C$. On the other hand, if $\upset{Cuu^{-1}}=C$,
then $\upset{Lsuu^{-1}} \subseteq \upset{Ls}$. Since $ss^{-1} \in L$ we have
$ss^{-1}suu^{-1} = suu^{-1} \in \upset{Ls}$ and so there exists $y \in L$ with $suu^{-1} \geqslant ys$. But then
$suu^{-1}s^{-1} \geqslant yss^{-1} \in L$ and since $L$ is closed we deduce that $suu^{-1}s^{-1} \in L$.
\end{proof}
It follows from Lemma \ref{eq_action_cond} that the condition $suu^{-1}s^{-1} \in L$ does not depend on the
choice of coset representative $s$. This is easy to see directly. If $\upset{Ls} = \upset{Lt}$ then, by
part (c) of Proposition \ref{cosetsofL} we have $st^{-1} \in L$. Then
$$tuu^{-1}t^{-1} \geqslant ts^{-1}suu^{-1}s^{-1}st^{-1} = (st^{-1})^{-1}(suu^{-1}s^{-1})(st^{-1}) \in L$$
and since $L$ is closed, $tuu^{-1}t^{-1} \in L$.
\begin{prop}
\label{transaction}
If $u \in S$ and $\upset{Cuu^{-1}}=C$ then $\upset{Cu} = \upset{Lsu}$ and the rule $C \lhd u = \upset{Cu}$
defines a transitive action of $S$ by partial bijections on the cosets of $L$.
\end{prop}
\begin{proof}
Since $C=\upset{Ls}$, Lemma \ref{rt_upset} implies that $\upset{Cu} = \upset{Lsu}$. To check that $\upset{Lsu}$ is
a coset of $L$, we need to verify that $(su)(su)^{-1} \in L$. But $(su)(su)^{-1} = suu^{-1}s^{-1}$ and so this follows
from Lemma \ref{eq_action_cond}. Moreover, $(Cu) \lhd u^{-1}$ is defined and equal to
$\upset{Cuu^{-1}}=C$, so that the action of $u$ is a partial bijection.
It remains to show that for any $s,t \in S$, the action of $st$ is the same as the action of $s$ followed by the action of $t$ whenever
these are defined. Now the outcome of the actions are certainly the same: for a coset $C$, we have $C \lhd (st) = \upset{Cst}$
and
$$(C \lhd s) \lhd t = \upset{Cs} \lhd t = \upset{\upset{Cs}t} = \upset{Cst}$$
by Lemma \ref{rt_upset}.
The conditions for $C \lhd (st)$, $C \lhd s$, $(C \lhd s) \lhd t$ to be defined are, respectively:
\begin{align}
& \upset{Cstt^{-1}s^{-1}} = C \,, \label{defCst}\\
& \upset{Css^{-1}}=C \,, \label{defCs}\\
& \upset{\upset{Cs}tt^{-1}} = \upset{Cstt^{-1}} = \upset{Cs}. \label{def(Cs)t}
\end{align}
Suppose that \eqref{defCst} holds. Then
$$\upset{Css^{-1}} \subseteq \upset{Cstt^{-1}s^{-1}} = C.$$
But it is clear that $C \subseteq \upset{Css^{-1}}$, and so $\upset{Css^{-1}}=C$ and \eqref{defCs} holds. Now
it is again clear that $\upset{Cs} \subseteq \upset{Cstt^{-1}}$, and
$$\upset{Cstt^{-1}} \subseteq \upset{Cstt^{-1}s^{-1}s} = \upset{\upset{Cstt^{-1}s^{-1}}s} = \upset{Cs}.$$
Therefore \eqref{def(Cs)t} holds.
Now if both \eqref{defCs} and \eqref{def(Cs)t} hold we have
\begin{align*}
\upset{Cstt^{-1}s^{-1}} &= \upset{\upset{Cstt^{-1}}s^{-1}} & \text{by Lemma \ref{rt_upset}} \\
&= \upset{\upset{Cs}s^{-1}} & \text{by \eqref{def(Cs)t}} \\
&= \upset{Css^{-1}} & \text{by Lemma \ref{rt_upset}} \\
&= C & \text{by \eqref{defCs}}.
\end{align*}
and therefore \eqref{defCst} holds.
To show that the action is transitive, consider two cosets $\upset{La}$ and $\upset{Lb}$. Then
$\upset{La} \lhd a^{-1}b$ is defined since $a(a^{-1}b)(a^{-1}b)^{-1}a^{-1}=aa^{-1}bb^{-1} \in L$,
and $\upset{La} \lhd a^{-1}b = \upset{Laa^{-1}b} = \upset{Lb}$, since again $aa^{-1}bb^{-1} \in L$.
\end{proof}
\subsection{Marshall Hall's Theorem for inverse semigroups}
\begin{theorem}
\label{Hall's theorem}
In a finitely generated inverse semigroup $S$ there are at most finitely many distinct closed inverse subsemigroups of a fixed finite index
$d$.
\end{theorem}
\begin{proof}
Suppose that the inverse semigroup $S$ is finitely generated and that the closed inverse subsemigroup $L$ of $S$ has exactly $d$ cosets. We aim to construct an inverse semigroup homomorphism
$$ \phi_{L}: S\longrightarrow {\mathscr I}(D), $$
where $ {\mathscr I}(D) $ is the symmetric inverse monoid on $D=\{1,...,d\}.$
Write the distinct cosets of $L$ as $\upset{L c_{1}},\upset{L c_{2}},...\,,\upset{L c_{d}},$ with $c_{1},c_{2},...,c_{d}\in S,$ and with $\upset{L c_{1}}=L.$ Now take $u \in S.$ If $c_{j}\,u\,u^{-1}\,{c_{j}}^{-1}\in L,$ where $ j\in \{1,...,d\},$ then we can define an action of the element $u \in S$ on the coset $\upset{Lc_j}$ of $L$ as follows:
$$\upset{Lc_{j}} \lhd u = \upset{Lc_{j}u}.$$
By Proposition \ref{transaction}, $\upset{Lc_{j}u}$ is indeed a coset of $L,$ and so $\upset{Lc_{j}u} =\upset{Lc_{k}},$ where $ k\in \{1,...,d\}.$ Then we can write $\upset{Lc_{j}} \lhd u = \upset{Lc_{k}},$ and this action of $u$ induces an action $j \lhd u = k$ of $u$ on $D$, and so we get a homomorphism
$$ \phi_{L}: S\longrightarrow {\mathscr I}(D) . $$
We now claim that different choices of $L$ give us different homomorphisms $\phi_{L}$, or equivalently, that if
$\phi_{L}=\phi_{K}$ then $L=K$.
By Proposition \ref{cosets_idpt}, if $x \in L$ then $L = \upset{Lxx^{-1}}$. By Lemma \ref{eq_action_cond} $L \lhd x$ is defined and is
equal to $\upset{Lx}=L$. Now suppose that $L \lhd y$ is defined and that $\upset{Ly}$ is equal to $L$. By Lemma \ref{eq_action_cond}
we have $\upset{Lyy^{-1}}=L$. Hence $yy^{-1} \in L$, and $y = yy^{-1}y \in \upset{Ly}=L$. It follows that
$\operatorname{stab}(L)=L$ and in the induced action of $S$ on $D$ we have $\operatorname{stab}(1)=L,$ so that $L$ is determined by $\phi_{L}\,.$
Therefore, the number of closed inverse subsemigroups of index $d$ is at most the number of homomorphisms $\phi : S \longrightarrow {\mathscr I} (D)$, and
since $S$ is finitely generated, this number is finite.
\end{proof}
\section{Finite generation and finite index}
\label{fg_and_fi}
In this section, we shall look at the properties of closed inverse submonoids of free inverse monoids considered in Theorem
\ref{MMthm}, and the relationships between these properties when we replace a free inverse monoid by an arbitrary inverse monoid.
Throughout this section, $M$ will be an inverse monoid generated by a finite subset $X$. This means that the smallest inverse submonoid $\< X \>$
of $M$ that contains $X$ is $M$ itself: equivalently, each element of $M$ can be written as a product of elements of $X$ and inverses of elements of $X$, so if we set $A= X \cup X^{-1}$ then each element of $M$ can be written as a product of elements in $A$.
A closed inverse submonoid $L$ of $M$ is said to be \emph{finitely generated as a closed inverse submonoid} if there exists a finite subset $Y \subseteq L$
such that, for each $\ell \in L$ there exists a product $w$ of elements of $Y$ and their inverses such that $\ell \geqslant w$. Equivalently, the
smallest closed inverse submonoid of $M$ that contains $Y$ is $L$.
We remark that in \cite{MarMea} the notation $\< X \>$ is used for the smallest \emph{closed} inverse submonoid of $M$ that
contains $X$. We shall use $\nobraupset{\< X \>}$ for this.
We will need to use some ideas from the theory of finite automata and for background information on this topic we refer to \cite{LwautBook, Pin, SipBook}.
A \emph{deterministic finite state automaton} ${\mathcal A}$ (or just an \emph{automaton} in this section) consists of
\begin{itemize}
\item a finite set $S$ of \emph{states},
\item a finite \emph{input alphabet} $A$,
\item an \emph{initial state} $s_0 \in S$,
\item a partially defined \emph{transition function} $\tau : S \times A \rightarrow S$,
\item a subset $T \subseteq S$ of \emph{final} states.
\end{itemize}
We shall write $s \lhd a$ for $\tau(s,a)$ if $\tau(s,a)$ is defined. Given a word $w = a_1 a_2 \dotsm a_m \in A^*$ we write
$s \lhd w$ for the state $( \dots (s \lhd a_1) \lhd a_2) \lhd \dotsb ) \lhd a_m,$ that is, for the state obtained from $s$ by computing the
succesive outcomes, if all are defined, of the transition function determined by the letters of $w$, with the empty word $\varepsilon$ acting by $s \lhd \varepsilon = s$ for all $s \in S$. We normally think of
an automaton in terms of its \emph{transition diagram}, in which the states are the vertices of a directed graph
and the edge set is $S \times A$, with an edge $(s,a)$ having source $s$ and target $s \lhd a$.
Let $X$ be a finite set, $X^{-1}$ a disjoint set of formal inverses of elements of $X$, and $A=X \cup X^{-1}$ An automaton
${\mathcal A}$ with input alphabet $A$ is called a \emph{dual automaton} if, whenever $s \lhd a = t$ then $t \lhd a^{-1} =s.$ A dual
automaton is called an \emph{inverse automaton} if, for each $a \in A$ the partial function $\tau(-,a): S \rightarrow S$ is injective.
(See \cite[Section 2.1]{LwBook}.)
A word $w \in A^*$ is \emph{accepted} or \emph{recognized} by ${\mathcal A}$ if $s_0 \lhd w$ is defined and $s_0 \lhd w \in T$. The set of all words recognized by
${\mathcal A}$ is the \emph{language} of ${\mathcal A}$:
\[ {\mathscr L}({\mathcal A}) = \{ w \in A^* : s_0 \lhd w \in T \}\,. \]
A language ${\mathscr L}$ is \emph{recognizable} if it is the language recognized by some automaton. The connection between automata and closed inverse subsemigroups of finite index is made, as in \cite{MarMea}, by the coset automaton.
Let $M$ be a finitely generated inverse monoid, generated by $X \subseteq M$, and let $L$ be a closed inverse
submonoid of $M$ of finite index. Since $M$ is generated by $X$, there is a natural monoid homomorphism $\theta: A^* \rightarrow M$. The \emph{coset automaton} ${\mathcal C}={\mathcal C}(M:L)$ is defined as follows:
\begin{itemize}
\item the set of states is the set of cosets of $L$ in $M$,
\item the input alphabet is $A = X \cup X^{-1}$,
\item the initial state is the coset $L$,
\item the transition function is defined by $\tau(\upset{Lt},a) = \upset{Lt(a \theta)}$,
\item the only final state is $L$.
\end{itemize}
By Lemma \ref{eq_action_cond} and Proposition \ref{transaction}, $\upset{Lt} \lhd a$ is defined if and only if
$t(a \theta)(a \theta)^{-1}t^{-1} \in L$.
The following Lemma
occurs as \cite[Lemma 3.2]{MarMea} for the case that $M$ is the free inverse monoid ${\it FIM}(X)$.
\begin{lemma}
\label{coset_aut}
The coset automaton of $L$ in $M$ is an inverse automaton.
The language ${\mathscr L}({\mathcal C}(M:L))$ that it recognizes is
\[ L \theta^{-1} = \{ w \in A^* : w \theta \in L \}\]
and ${\mathcal C}(M:L)$ is the minimal automaton recognizing $L \theta^{-1}$.
\end{lemma}
\begin{proof}
It follows from Proposition \ref{transaction} that ${\mathcal C}(M:L)$ is inverse.
Suppose that $w$ is recognized by ${\mathcal C}(M:L)$. Then $(w \theta)(w\theta)^{-1} = (ww^{-1})\theta \in L$
and $L\upset{w \theta}=L.$ From Proposition \ref{cosetsofL}, we deduce that $w \theta \in L$.
Conversely, suppose that $w = a_{i_1} \dotsc a_{i_m} \in A^*$ and that $s = w \theta \in L$. For $1 \leqslant k \leqslant m$, write
$p_k = a_{i_1} \dotsc a_{i_k}$, $q_k = a_{i_{k+1}} \dotsc a_{i_m}$, so that $w = p_kq_k,$ and take $s_k = p_k\theta,$ so that $s_1 = a_{i_1}\theta$. Then
\[ s_1 s_1^{-1}s = s_1s_1^{-1}s_1(q_2 \theta) = s_1(q_2 \theta) = w \theta =s \]
and so $s_1 s_1^{-1} \geqslant ss^{-1} \in L$. Therefore $s_1 s_1^{-1} \in L$ and $L \lhd a_{i_1} = \upset{Ls_1}$ is defined.
Now suppose that for some $k$ we have that $L \lhd w_k$ is defined and is equal to $\upset{Ls_k}$.
Then
\[ s_{k+1}\,s_{k+1}^{-1}\,s = s_{k+1}\,s_{k+1}^{-1}\,s_{k+1}(q_{k+1} \theta)
= s_{k+1}\,(q_{k+1} \theta) = w \theta = s\]
and so $s_{k+1}\,s_{k+1}^{-1} \geqslant ss^{-1} \in L$ and therefore $s_{k+1}s_{k+1}^{-1} \in L.$
But
\[ s_{k+1}\,s_{k+1}^{-1} = s_k(a_{i_{k+1}}\theta)\,(a_{i_{k+1}}\theta)^{-1}s_k^{-1} \in L \,,\]
and so by Lemma \ref{eq_action_cond}, $\upset{Ls_k} \lhd a_{i_{k+1}}$ is defined and is equal to
$\upset{Ls_k(a_{i_{k+1}}\theta)} = \upset{Ls_{k+1}}$. It follows by induction that
$L \lhd w$ is defined in ${\mathcal C}(M:L)$ and is equal to $\upset{Ls} = L,$ and so $w \in L({\mathcal C}(M:X))$.
Now by a result of Reutenauer \cite[Lemme 1]{Reut}, a connected inverse automaton with one initial and one final state
is minimal.
\end{proof}
The set of \emph{rational} subsets of $M$ is the smallest collection that contains all the finite subsets of $M$
and is closed under finite union, product, and generation of a submonoid. Equivalently, $R \subseteq M$ is a rational subset of $M$ if and only if there exists a recognizable subset $Z \subseteq A^*$ with $Z \theta = R$ (see \cite[section IV.1]{Pin}).
We also recall the notion of \emph{star-height} of a rational set (see \cite[Chapter III]{Ber}).
Let $M$ be a monoid. Define a sequence of subsets $\operatorname{Rat}_h(M),$ with \emph{star-height} $h\geq0,$ recursively as follows:
$$\operatorname{Rat}_{0}(M)=\{X \subseteq M| \,X \text{\,is \,finite\,}\},$$
and $\operatorname{Rat}_{h+1}(M)$ consists of the finite unions of sets of the form $B_{1}B_2 \dotsm B_{m}$ where each $B_{i}$ is either a singleton or $B_{i}=C_{i}^*,$ for some $C_{i}\in \operatorname{Rat}_{h}(M).$ It is well known that $\operatorname{Rat}(M)=\bigcup_{h\geq0} \operatorname{Rat}_{h}(M)$.
A subset $S$ of $M$ is \emph{recognizable}
if there exists a finite monoid $N,$ a monoid homomorphism $\phi: M \rightarrow N$, and a subset $P \subseteq N$ such that
$S = P \phi^{-1}.$ For free monoids $A^*,$ Kleene's Theorem (see for example \cite[Theorem 5.2.1]{LwautBook})
tells us that the rational and recognizable subsets coincide. For finitely generated monoids, we have the following theorem
due to McKnight.
\begin{theorem}
\label{mcknight}
In a finitely generated monoid $M$, every recognizable subset is rational.
\end{theorem}
If $M$ is generated (as an inverse monoid) by $X$, then as above we have a monoid homomorphism $\theta: A^* \rightarrow M$.
We say that a subset $S$ of $M$ is \emph{recognized} by an automaton ${\mathcal A}$ if its full inverse image $S \theta^{-1}$ in $A^*$
is recognized by ${\mathcal A}$. We shall use the Myhill-Nerode Theorem \cite{Myhill, Nerode} to characterize recognizable languages.
Let $K \subseteq A^*$ be a language.
Two words $u,v \in A^*$ are \emph{indistinguishable by} $K$ if, for all $z \in A^*$, $uz \in K$ if and only if $vz \in K$.
We write $u \simeq_K v$ in this case: it is easy to check that $\simeq_K$ is an equivalence relation (indeed, a right congruence)
on $A^*$. The we have:
\begin{theorem}[The Myhill-Nerode Theorem]
\label{nerode Thm}
A language ${\mathscr L}$ is recognizable if and only if the equivalence relation $ \simeq_{{\mathscr L}}$ has finitely many classes.
\end{theorem}
We refer to \cite[Section 9.6]{LwautBook} for more information about, and a sketch proof of this result.
\subsection{Finite index implies finite generation}
In this section, we consider a closed inverse submonoid $L$ that has finite index in a finitely generated inverse monoid
$M$. We shall show that $L$ is finitely generated as a closed inverse submonoid. Our proof differs from that given in \cite[Theorem 3.7]{MarMea}
for the case $M = {\it FIM}(X)$: instead we generalize the approach taken for groups in \cite[Theorem 3.1.4]{BaumBook}. Recall that a transversal to
$L$ in $M$ is a choice of one representative element from each coset of $L$. We always choose the element $1_M$ from the coset $L$
itself. For $s \in S$ we write $\overline{s}$ for the \textit{representative} of the coset that contains $s$ (if it exists), and note the following:
\begin{lemma}\label{delta}
Let ${\mathcal T}$ be a transversal to $L$ in $M$ and define, \,for $r \in {\mathcal T}$ and $s \in M$,
$\delta (r,s)=rs\,{(\overline{rs})}^{-1} \,.$
Then for all $s,t \in M$, with $ss^{-1}, stt^{-1}s^{-1} \in L$,
\begin{enumerate}
\item $\upset{Ls}= \upset{L\overline{s}}$
\item $\overline{\overline{s}t}= \overline{st}$
\item $s \geqslant \delta (1_M,s)\: \overline{s}\,\,.$
\end{enumerate}
\end{lemma}
\begin{proof}
Parts (a) and (c) are clear: part (b) is a special case of Lemma \ref{rt_upset}.
\end{proof}
\begin{theorem}
\label{Schr's lemma}
A closed inverse submonoid of finite index in a finitely generated inverse monoid is finitely generated as a closed inverse submonoid.
\end{theorem}
\begin{proof}
Let $M$ be an inverse monoid generated by a set $X$. We set $A = X \sqcup X^{-1}$: then each $s \in M$ can be expressed as a product $s=a_{1}a_{2} \dotsm a_{n}$
where $a_i \in A$. Suppose that $L$ is a closed inverse subsemigroup of finite index in $M.$ Let ${\mathcal T}$ be a transversal to $L$ in $M$. Given $h \in L,$ we write $h=x_1 x_2 \dotsm x_n$ and consider the prefix $h_i = x_1 x_2 \dotsm x_i$ for
$1 \leqslant i \leqslant n$. Since
\[ h_ih_i^{-1}hh^{-1}= h_ih_i^{-1}h_ix_{i+1} \dotsm x_n h^{-1}= h_ix_{i+1} \dotsm x_nh^{-1} = hh^{-1}, \]
we have $h_ih_i^{-1} \geqslant hh^{-1}$. But $hh^{-1} \in L$ and $L$ is closed, so that $h_ih_i^{-1} \in L$.
Therefore the coset $\upset{Lh_i}$ exists, and so does the representative $\overline{h_i}$. Now
$$h = x_1x_2 \dotsm x_n \geqslant x_1 \cdot \overline{h_1}^{-1} \overline{h_1} \cdot x_{2} \cdot \overline{h_2}^{-1} \overline{h_2} \cdot x_3 \cdot \overline{h_3}^{-1} \dotsm
\overline{h_{n-1}} \cdot x_{n}\,.$$
By part (b) of Lemma \ref{delta} we have $\overline{h_j} = \overline{\overline{h_{j-1}}x_j}$ and so
$$h \geqslant x_1 \cdot \overline{x_1}^{-1} \overline{x_1} \cdot x_{2} \cdot (\overline{\overline{h_1}x_2})^{-1} \cdot \overline{h_2} \cdot x_3 \cdot (\overline{\overline{h_2}x_3})^{-1} \dotsm
\overline{h_{n-1}} \cdot x_{n} \,.$$ting that $1_M=\overline{x_1x_2 \dotsm x_n}$, we have
Now using the elements $\delta(r,s)$ from Lemma \ref{delta}, and no
$$h \geqslant \delta (1_M,x_{1}) \delta (\overline{x_1},x_2) \delta (\overline{h_2},x_3) \dotsm \delta(\overline{h_{n-1}},x_n) \,.$$
Finally, since $\upset{Lrs} = \upset{L\overline{rs}}$ then it follows from Proposition
\ref{cosetsofL}(3) that $\delta(r,s) \in L$. Hence $L$ is generated as a closed inverse submonoid of $M$ by the elements
$\delta(r,x)$ with $r \in {\mathcal T}$ and $x \in A$.
\end{proof}
\subsection{Recognizable closed inverse submonoids}
\begin{theorem}
\label{recog_iff_findex}
Let $L$ be a closed inverse submonoid of a finitely generated inverse monoid $M$. Then the following are equivalent:
\begin{enumerate}
\item[(a)] $L$ is recognized by a
finite inverse automaton,
\item[(b)] $L$ has finite index in $M$,
\item[(c)] $L$ is a recognizable subset of $M$.
\end{enumerate}
\end{theorem}
\begin{proof}
If $L$ has finite index in $M$, then by Lemma \ref{coset_aut}, its coset automaton ${\mathcal C}(M:L)$ is a finite inverse
automaton that recognizes $L$. Conversely, suppose that ${\mathcal A}$ is a finite inverse automaton that recognizes $L$. Again by Lemma \ref{coset_aut}, the coset automaton ${\mathcal C}$ is minimal, and so must be finite. Hence (a) and (b) are equivalent.
If (b) holds, then as in the proof of Theorem \ref{Hall's theorem}, we obtain a homomorphism $M \rightarrow {\mathscr I}(D)$ for which $L$ is te invese image of the stabilizer of $1$. Therefore
(b) implies (c).
We have $M$ generated by $X$, with $A = X \sqcup X^{-1}$, and a monoid homomorphism $\theta : A^* \rightarrow M$.
To prove that (c) implies (b), suppose that $L$ is recognizable and so the language
${\mathscr L}=\{w\in A^*: w \theta \in L\}$ is recognizable. By Theorem \ref{nerode Thm}, the equivalence relation $ \simeq_{{\mathscr L}}$ on $A^*$ has finitely many classes. We claim that if $u \simeq_{{\mathscr L}} v$ and if $\upset{L(u \theta)}$ exists, then $\upset{L(v \theta)}$ exists and
$\upset{L(u \theta)} = \upset{L(v \theta)}$.
Now $(u\theta)\,(u\theta)^{-1} = (uu^{-1})\theta \in L$ and hence $uu^{-1} \in {\mathscr L}$. But by assumption $u \simeq_{{\mathscr L}} v,$ and so $vu^{-1} \in {\mathscr L}$, which implies that $(v\theta)(u\theta)^{-1} \in L.$ By part (c) of Proposition \ref{cosetsofL}, $\upset{L(v \theta)}$ exists and
$\upset{L(v\theta)} = \upset{L(u\theta)}$.
Since $ \simeq_L$ has only finitely many classes, there are only finitely many cosets of $L$ in $M$. Hence
(c) implies (b).
\end{proof}
\subsection{Rational Generation}
In this section we give an automata-based proof of part of \cite[Theorem 3.7]{MarMea}. We adapt the
approach used in \cite[Theorem II]{FrouSakSch} to the proof of the following theorem of Anisimov and Seifert.
\begin{theorem}{\cite[Theorem 3]{AnisSeif}}
\label{anisseif}
A subgroup of a finitely generated group $G$ is a rational subset of $G$ if and only if it is finitely generated.
\end{theorem}
\begin{theorem}
\label{rational gen}
Let $L$ be a closed inverse submonoid of a finitely generated inverse
monoid $M$.
Then $L$ is generated as a closed inverse submonoid by a rational subset if and only if $L$ is
generated as a closed inverse submonoid by a finite subset.
\end{theorem}
\begin{proof}
Since finite sets are rational sets, one half of the theorem is trivial.
So suppose that $L$ is generated (as a closed inverse submonoid) by some rational subset $Y$ of $L$. As above, if
$M$ is generated (as an inverse monoid) by $X$, we take $A = X \cup X^{-1}$,
and let $ \theta$ be the obvious map $A^* \rightarrow M$.
Then $Z = (Y \cup Y^{-1})^*$ is rational and so there exists a rational language
$R$ in $A^*$ such that $R \theta = Z$. The pumping lemma for $R$ then tells us that there exists a constant $C$
such that, if $w \in R$ with $|w| > C$, then
$w=uvz$ with $|uv| \leqslant C, |v| \geqslant 1$, and $uv^iz \in R$ for all $i \geqslant 0$. We set
\[ U = \{ uvu^{-1} : u,v \in A^*, |uv| \leqslant C, (uvu^{-1}) \theta\in L \} \]
and $V = \nobraupset{ \<U \theta \>}$.
Clearly $U$ is finite, and $V \subseteq L$. We claim that $L=V$, and so we shall show that $R \theta \subseteq V$.
We first note that if $w \in R$ and $|w| \leqslant C$ then $w \in U$ (take $u=1,
v=w$) and so $w \theta \in V$. Now suppose that $|w|>C$ but that there exists $n \in L \setminus V$ with $n \geqslant w \theta$
Choose $|w|$ minimal.
The pumping lemma gives $w=uvz$ as above. Since $|uz| < |w|$
it follows that $(uz) \theta \in V$.
Moreover,
\[ (uvu^{-1}) \theta \geqslant (uvzz^{-1}u^{-1}) \theta = (uvz) \theta ((uz)
\theta)^{-1} = (w \theta)\,((uz) \theta)^{-1}. \]
Now $w \theta \in L$ and $(uz) \theta \in V$ : since $L$ is closed,
$(uvu^{-1}) \theta \in L$ and therefore $uvu^{-1} \in U$.
Now
\[ n \geqslant w \theta = (uvz) \theta \geqslant (uvu^{-1}uz) \theta = (uvu^{-1})\theta\,\,
(uz) \theta \in V . \]
Since $V$ is closed, $n \in V$. But this is a contradiction. Hence $L=V$.
\end{proof}
\begin{cor}
\label{rat_imp_fg}
If a closed inverse submonoid $L$ of a finitely generated inverse
monoid $M$ is a rational subset of $M$ then it is
finitely generated as a closed inverse submonoid.
\end{cor}
\begin{proof}
If $L$ is a rational subset of $M$ then it is certainly generated by a rational set, namely $L$ itself.
\end{proof}
However, the converse of Corollary \ref{rat_imp_fg} is not true. We shall use the following Lemma to validate a counterexample
in Example \ref{f2ab}.
\begin{lemma}
\label{star-height}
Let $M$ be a semilattice of groups $G_1 \sqcup G_0$ over the semilattice $1>0$, and suppose that $T$ is a rational subset of
$M$ of star-height $h$. Then $G_1 \cap T$ is a rational subset of $G_1$.
\end{lemma}
\begin{proof}
We proceed by induction on $h$. If $h=0$ then $T$ is finite, and $G_1 \cap T$ is a finite subset of $G_1$ and so is a rational
subset of $G_1$, also of star-height $h_1=0$.
If $h>0,$ then, as in section \ref{fg_and_fi}, $T$ is a finite union $T = S_1 \cup \dotsb \cup S_k$
where each $S_j$ is a product $S_j = R_1 R_2 \dotsm R_{m_j}$ and where each $R_i$ is either a singleton subset of $M$ or $R_i = Q_i^*$ for some rational subset $Q_i$ of $M$ of star-height $h-1$ (see \cite[Chapter III]{Ber}).
Hence
\[ G_1 \cap T = (G_1 \cap S_1) \cup \dotsb \cup (G_1 \cap S_k) \,.\]
Consider the subset $G_1 \cap S_j = G_1 \cap R_1 R_2 \dotsm R_{m_j}$. We claim that
\begin{equation} \label{rat_prod} G_1 \cap R_1 R_2 \dotsm R_{m_j} = (G_1 \cap R_1)(G_1 \cap R_2) \dotsm (G_1 \cap R_{m_j}) \,. \end{equation}
The inclusion $\supseteq$ is clear, and so now we suppose that $g \in G_1$ is a product
$g = r_1 r_2 \dotsm r_{m_j}$ with $r_i \in R_i$. If any $r_i \in G_0$ then $g \in G_0$: hence each $r_i \in G_1$ and so
$g \in (G_1 \cap R_1)(G_1 \cap R_2) \dotsm (G_1 \cap R_{m_j}) \,,$ confirming \eqref{rat_prod}.
The factors on the right of \eqref{rat_prod} are either singleton subsets of $G_1$, or are of the form $G_1 \cap Q_i^*$
where $Q_i$ is a rational subset of $M$ of star-height $h-1$. However, $G_1 \cap Q_i^* = (G_1 \cap Q_i)^*$: the inclusion $G_1 \cap Q_i^* \supseteq (G_1 \cap Q_i)^*$ is again obvious,
and $G_1 \cap Q_i^* \subseteq (G_1 \cap Q_i)^*$ since if $w = x_1 \dots x_m \in Q_i^*$ and some $x_j \in G_0$ then
$w \in G_0$. It follows that if $w \in G_1 \cap Q_i^*$ then $x_j \in G_1$ for all $j$.
Hence $G_1 \cap T$ is a union of subsets, each of which is a product of singleton subsets of $G_1$ and subsets of the form $(G_1 \cap Q_i)^*$ where, by induction
$G_1 \cap Q_i$ is a rational subset of $G_1$ of star-height $h_2 \leqslant h-1$. Therefore $G_1 \cap T$ is a rational subset of $G_1$.
\end{proof}
\begin{cor}\label{rational_clifford}
Let $L = L_1 \sqcup L_0$, where $L_j \subseteq G_j$, be an inverse submonoid of $M$ that is also a rational subset of $M$. Then $L_1$ is a rational subset of $G_1$.
\end{cor}
\begin{proof}
Take $T=L$: then $G_1 \cap L = L_1$ is a rational subset of $G_1$.
\end{proof}
Now, we show that the converse of Corollary \ref{rat_imp_fg} is not true.
\begin{example}
\label{f2ab}
Let $F_2$ be a free group of rank $2$ and consider the semilattice of groups $M = F_2 \sqcup F_2^{ab}$ determined by the abelianisation map
$\alpha : F_2 \rightarrow F_2^{ab}$. The kernel of $\alpha$ is the commutator subgroup
$F_2'$ of $F_2$ and we let $K$ be the closed inverse submonoid $F_2' \sqcup \{ {\mathbf 0} \}$.
\[ \xymatrixcolsep{3pc}
\xymatrix{
F_2' \ar[d] \ar[r] & F_2 \ar[d]^{\alpha}\\
\{ {\mathbf 0} \} \ \ar[r] & F_2^{ab}}
\]
Now $K$ is generated (as a closed inverse submonoid) by $\{ {\mathbf 0} \}$ and so is finitely generated. But $F_2'$ is not
finitely generated as a group (see \cite[Example III.4(4)]{BaumBook}) and so is not a rational subset of $F_2$ by
Theorem \ref{anisseif}. Therefore, by Corollary \ref{rational_clifford}, $K$ is not a rational subset of $M$.
This example also gives us a counterexample to the converse of Theorem \ref{Schr's lemma}: $K$ is
finitely generated as a closed inverse submonoid, but has infinite index in $M$.
\end{example}
\section{Conclusion}
We summarize our findings about the conditions considered by Margolis and Meakin in \cite[Theorem 3.5]{MarMea}.
\begin{theorem}
\label{gen_MMthm}
Let $L$ be a closed inverse submonoid of the finitely generated inverse monoid $M$ and consider the following properties that $L$
might possess:
\begin{enumerate}
\item[(a)] $L$ is recognized by a finite inverse automaton,
\item[(b)] $L$ has finite index in $M$,
\item[(c)] $L$ is a recognizable subset of $M$,
\item[(d)] $L$ is a rational subset of $M$,
\item[(e)] $L$ is finitely generated as a closed inverse submonoid of $M$..
\end{enumerate}
Then properties (a), (b) and (c) are equivalent: each of them implies (d), and (d) implies (e). The latter two implications
are not reversible.
\end{theorem}
\begin{proof}
The equivalence of (a), (b) and (c) was established in Theorem \ref{recog_iff_findex}, and that (d) implies (e)
in Corollary \ref{rat_imp_fg}. The implication that (c) implies (d) is McKinight's Theorem \ref{mcknight}.
Counterexamples for (e) implies (d) and (e) implies (b) are given in Example \ref{f2ab}
\end{proof}
| 20,371 |
\section{Introduction}
Galaxy clusters are composite gravitationally bound systems hosting hundreds of galaxies and large quantities of hot X-ray emitting gas (intracluster medium, ICM). Very early work by \cite{1933AcHPh...6..110Z} on the dynamics of galaxies within clusters revealed that these systems host large quantities of dark matter.
Being the largest virialized structures in the Universe and typically sitting at the knots of the cosmic web, galaxy clusters can be used to trace large scale structure \citep[e.g.][]{1988ARA&A..26..631B}, to measure the abundance of baryonic matter \citep[e.g.][]{2004ApJ...617..879L,2006ApJ...640..691V,2007ApJ...666..147G,2009ApJ...703..982G,2013ApJ...778...14G,2013MNRAS.435.3469H,2013A&A...555A..66L,2016A&A...592A..12E} and to place constraints on cosmological parameters \citep[e.g.][]{2014A&A...571A..20P, 2014MNRAS.440.2077M,2014arXiv1411.8004H,2014A&A...570A..31B,2016arXiv160306522D}.
Measurements of the galaxy cluster mass function in the era of large volume surveys (e.g. Euclid, Dark Energy Survey, eROSITA) are expected to yield cosmological constraints with precision and accuracy at the $\sim 1\%$ level \citep[e.g.][]{2013JCAP...04..022B, 2014MNRAS.441.1769C, Martizzi2014a, 2014MNRAS.439.2485C, 2014A&A...571A..20P, 2016MNRAS.456.2361B}. Accurate measurements of the galaxy cluster mass function require reliable calibrations of scaling relations between the halo mass and observable quantities. Widely used scaling relations are derived from properties measured from clusters detected in the X-ray band and with the thermal Sunyaev-Zel'dovich (SZ) effect \citep[e.g.][]{2010A&A...517A..92A, 2012MNRAS.427.1298H, 2013ApJ...767..116M, 2013ApJ...772...25S, 2013SSRv..177..247G, 2016arXiv160604983D}. Calibration of these scaling relations requires accurate determination of cluster masses.
A variety of estimators exist for measuring cluster masses, including the use of gravitational lensing \citep{2013SSRv..177...75H}, X-ray/SZ data combined with the assumption of hydrostatic equilibrium \citep{2013SSRv..177..119E} and the caustic method \citep{2015arXiv151107872M}. Reconstruction of cluster mass profiles under the assumption of hydrostatic equilibrium and spherical symmetry is one of the most common approaches because of its simplicity. However, differences between mass determinations using X-ray data and gravitational lensing are well documented in the literature: it has been argued that deviations from spherical symmetry \citep[e.g.][]{2010ApJ...713..491M, 2014ApJ...794..136D} are a major source of hydrostatic bias. The neglect of pressure support from non-thermal processes such as turbulence in the ICM, bulk motions, magnetic fields, cosmic ray pressure and electron-ion non-equilibrium in the ICM \citep{2007MNRAS.378..385P, 2009ApJ...705.1129L, 2010ApJ...711.1033Z, 2012ApJ...758...74B, 2013ApJ...771..102F, 2013MNRAS.432..404M, 2015ApJ...808..176A} is also thought to contribute to this bias.
Independent of the method chosen to reconstruct the mass distribution of a cluster, it is important to assess possible biases in the determination of halo masses. If mass determinations for a given cluster sample have been performed with several techniques (e.g. weak lensing and hydrostatic masses from X-ray measurements), observational data can be used to constrain the bias \citep{2015MNRAS.450.3633S}.
On the theoretical side, it is possible to use numerical cosmological simulations of galaxy clusters to directly compare the actual mass profile of a simulated cluster with the one obtained via reconstruction using the methods cited above \cite[e.g.][]{2013ApJ...777..151L}. Driven by the increasing relevance of mass calibration needed for cluster surveys, several groups pursued the goal of constraining non-thermal pressure support to improve hydrostatic mass determinations. In the literature, particular emphasis has been given to the contribution of turbulent and bulk motions induced by active galactic nuclei (AGN) feedback, sloshing, and mergers \citep{2012A&A...544A.103V}. In a series of papers, \cite{2014MNRAS.442..521S}, \cite{2015MNRAS.448.1020S} and \cite{2016MNRAS.455.2936S} developed an analytical model for non-thermal pressure from turbulent and bulk motions in galaxy clusters and used it remove hydrostatic bias; however, this model is based on knowledge of the gas velocity dispersion which is complicated to measure experimentally. The recent paper by \cite{2016arXiv160602293B} used cosmological SPH simulations to characterize hydrostatic bias in galaxy clusters but did not directly implement a method to reconstruct the mass profile. The goal of our paper is to extend this line of research by using a sample of cosmological adaptive mesh refinement (AMR) simulations of galaxy clusters to constrain hydrostatic bias, to quantify the role of non-thermal pressure and to develop a new method for mass reconstruction.
Furthermore, we provide novel insight on the non-thermal pressure in cluster cores in AMR simulations that include AGN feedback.
The paper has the following structure. Section 2 describes the simulation data-set. Section 3 reviews the formalism of hydrostatic mass modeling, describes the analysis procedure and proposes an empirical model for the non-thermal pressure. Section 4 describes our results. Finally, Section 5 summarizes our findings.
\section{Simulation data-set}
\label{sec:simulation_dataset}
We consider a sub-set of 10 cosmological hydrodynamical zoom-in simulations of
galaxy clusters from the sample of \cite{Martizzi2014a}. These simulations were performed with the {\scshape ramses} code \citep{2002A&A...385..337T} and were shown to produce brightest cluster galaxies (BCGs) with realistic mass, sizes and kinematic properties at redshift $z=0$ \citep{Martizzi2014b}, and the host halos have baryon and stellar fractions in broad agreement with observations \citep{Martizzi2014a} and have realistic ICM density and thermal pressure profiles (as we show in Subsection~\ref{sec:measured_quantities}). The zoom-in simulations of the 10 halos considered in this paper have {\it total} masses greater than $10^{14}$~M$_\odot$. Furthermore, neighbouring halos do not have masses larger than half of the total central halo mass within a spherical region of five times the virial radius of the central halo. Half of the clusters in the sample used for this paper are relaxed, based on the criterion described in Subsection~\ref{sec:relaxation}.
\begin{table}
\begin{center}
{\bfseries Cosmological parameters}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\hline
${\rm H_0}$ [km s$^{-1}$Mpc$^{-1}$] & ${\rm \sigma_{\rm 8}}$ & ${\rm n_{\rm s}}$ & $\Omega_\Lambda$ & $\Omega_{\rm m}$ & $\Omega_{\rm b}$ \\
\hline
\hline
70.4 & 0.809 & 0.963 & 0.728 & 0.272 & 0.045 \\
\hline
\hline
\end{tabular}
\caption{Cosmological parameters adopted in our simulations. }\label{tab:cosm_par}
\end{center}
\end{table}
Cosmic structure formation is evolved in the context of the standard $\Lambda$CDM cosmological scenario. The cosmological parameters chosen for our simulations are summarised in Table~\ref{tab:cosm_par}. Cosmological initial conditions have been considered which were computed using the \cite{1998ApJ...496..605E} transfer function and the {\scshape grafic++} code developed by Doug Potter (http://sourceforge.net/projects/grafic/) and
based on the original {\scshape grafic} code \citep{2001ApJS..137....1B}.
The {\scshape ramses} code offers AMR capabilities which is fully exploited to achieve high resolution in our zoom-in simulations. The computational domain is a box size $144$~Mpc/h. We chose an initial level of refinement $\ell=9$ (base mesh size $512^3$), but we allowed for refinement down to a maximum level $\ell_{\rm max}=17$. Grid refinement is implemented using a quasi-Lagrangian approach: when the dark matter or baryonic mass in a cell reaches 8 times the initial mass resolution, it is split into 8 children cells. With these choices, the dark matter particle mass is ${\rm m_{\rm cdm}=1.62\times 10^{8}}$~M$_\odot$/h, while the mass of the baryon resolution element is ${\rm m_{\rm gas}=3.22\times 10^{7}}$~M$_\odot$. The minimum cell size in the zoom-in region is ${\rm \Delta x_{\rm min} = L/2^{\ell_{\rm max}}\simeq 1.07}$ kpc/h. Table~\ref{tab:mass_par} summarizes the particle mass and spatial resolution achieved in our simulations.
\begin{table}
\begin{center}
{\bfseries Mass and spatial resolution}
\begin{tabular}{|c|c|c|}
\hline
\hline
${\rm m_{\rm cdm}}$& ${\rm m_{\rm gas}}$ & ${\rm \Delta x_{\rm min}}$ \\
$[10^{8}$ M$_\odot$/h] & $[10^{7}$ M$_\odot$/h] & [kpc/h] \\
\hline
\hline
$1.62$ & $3.22$ & $1.07$ \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Mass resolution for dark matter particles, gas cells and star particles, and spatial resolution (in physical units) for our simulations. }\label{tab:mass_par}
\end{table}
Our zoom-in simulations include baryons (hydrodynamics), star formation, stellar feedback and AGN feedback. The inviscid equations of hydrodynamics are solved with a second-order unsplit Godunov scheme \citep{2002MNRAS.329L..53B, Teyssier:2006p413, Fromang:2006p400} based on the HLLC Riemann solver and the
MinMod slope limiter \citep{Toro:1994p1151}. We assume a perfect gas equation of state (EOS) with polytropic index $\gamma=5/3$. All the zoom-in runs include sub-grid models for gas cooling which account for H, He and
metals and that use the \cite{1993ApJS...88..253S} cooling function. We directly follow star formation and supernovae feedback (``delayed cooling" scheme, \citealt{2006MNRAS.373.1074S}) and metal enrichment.
The AGN feedback scheme is a modified version of the \cite{2009MNRAS.398...53B} model. Supermassive black holes (SMBHs) are modeled as sink particles and AGN feedback is provided in form of thermal energy injected in a sphere surrounding each SMBH. More details about the AGN feedback scheme can be found in \cite{Martizzi2012a}.
\subsection{Cluster classification by relaxation state}\label{sec:relaxation}
The classification of clusters depending on their relaxation state is critical for our analysis. Reliable mass reconstructions from observational data are usually performed by excluding clusters that are not dynamically relaxed.
In this paper, we also analyze the dynamical state of the simulated clusters using a simple estimator that has been shown to be reliable by \cite{2016MNRAS.tmp.1545C}.
At each cluster-centric distance ${\rm r}$, we define the virial parameter $\eta(<r)$ as
\begin{equation}
{\rm \eta(<r) = \frac{\sigma_{\rm dm}(<r)}{\sigma_{\rm vir}(<r)}},
\end{equation}
where ${\rm \sigma_{\rm dm}(<r)}$ is the velocity dispersion of dark matter particles within radius ${\rm r}$, and ${\rm \sigma_{\rm vir}(r)}$ is the velocity dispersion expected if the system were virialized within the same radius, i.e.
\begin{equation}
{\rm \sigma_{vir}(<r)=\left[\frac{GM_{\rm dm}(<r)}{r}\right]^{1/2}}
\end{equation}
where ${\rm M_{\rm dm}(<r)}$ is the enclosed dark matter mass. Any deviation from $\eta(<r)=1$ implies deviations from virial equilibrium within radius ${\rm r}$.
There are several caveats that one should consider when using such measure of relaxation. First of all, ${\rm \eta(<r)}$ is expected to vary with radius. A cluster may appear to be relaxed within a certain radius ${\rm r_1}$, but not within another radius ${\rm r_2}$.
Figure~\ref{relaxation} shows that ${\rm \eta(<r)}$ varies significantly as a function of radius for all our clusters. In particular, the strongest deviations from virialization are observed in cluster cores (${\rm r<0.1R_{200m}}$) and at intermediate radii, as a consequence of the
presence of non-virialized sub-structure in the cluster. The second caveat is that we have only used dark matter to define ${\rm \eta}$ (for simplicity), but the dynamics in cluster cores might be significantly influenced by the baryonic mass/velocity distribution.
\begin{figure}
\includegraphics[width=.49\textwidth]{./Figures/relaxation.pdf}
\caption{\label{relaxation} Relaxation parameter $\eta(<r)=\sigma(<r)/\sigma_{\rm vir} (<r)$ within a given radius $r$, where $\sigma$. Each line represents the relaxation profile of one of the simulated clusters. $\eta=1$ is achieved only at full virialization.
The shaded area represents $\pm5\%$ deviations from pure virialization. }
\end{figure}
To overcome the limitations of the definition of ${\rm \eta}$, we place a series of restrictive requirements when classifying clusters depending on their dynamical state. We define a cluster as relaxed only if the two following requirements are simultaneously satisfied:
\begin{itemize}
\item ${\rm 0.95\leq \eta(<R_{200m}) \leq 1.05}$.
\item ${\rm 0.90\leq \eta(<0.5R_{200m}) \leq 1.10}$.
\end{itemize}
In other words, a cluster is significantly relaxed only if its virial parameters at ${\rm 0.5R_{200m}}$ and ${\rm R_{\rm 200m}}$ do not deviate significantly from 1. We find that 5 of our halos are relaxed (labeled with IDs from 1 to 5), whereas the other 5 clusters are unrelaxed (labeled with IDs from 6 to 10).
\section{Formalism}\label{sec:formalism}
The mass distribution of a galaxy cluster can be inferred from observational measurements via simplifying assumptions about the state of the cluster. If the cluster ICM is modeled as a fluid in the potential generated by the total mass distribution, then the gas motions are described by Euler's equation:
\begin{equation}\label{eq:euler}
\frac{\rm d {\bf v}}{\rm dt} = {\rm -\nabla \Phi} - \frac{\rm \nabla P}{\rho}.
\end{equation}
where ${\rm {\bf v}} $ is the gas velocity, $P$ is the local pressure, $\rho$ is the gas density and $\Phi$ is the gravitational potential. $\Phi$ is determined by solving Poisson's equation:
\begin{equation}
{\rm \nabla^2\Phi= 4\pi G \rho_{total}}
\end{equation}
where ${\rho_{\rm total}}$ is the total mass density in the cluster (dark matter, stars, ICM, etc.).
Euler's equation can be simplified into a form that leads to a widely used galaxy cluster mass estimator under the assumption of {\it spherical symmetry} and {\it hydrostatic equilibrium} ($\rm d{\bf v}/dt =0$). In this case Equation~\ref{eq:euler} becomes:
\begin{equation}\label{eq:hse}
\frac{\rm d P_{\rm HE}}{\rm dr} = {\rm g_r \rho(r)} = -{\rm \frac{GM_{HE}(<r)\rho(r)}{r^2}},
\end{equation}
where ${\rm r}$ is the cluster-centric distance, ${\rm g_r=-d\Phi/dr}$ is the radial component of the gravitational acceleration, $\rm M_{HE}(<r)$ is the total mass enclosed within radius $r$; we have also introduced $\rm P_{\rm HE}(r)$, the pressure profile required for the cluster to maintain hydrostatic equilibrium.
If the pressure gradient ${\rm dP_{HE}/dr}$ and density profile $\rm \rho(r)$ are known, then Equation~\ref{eq:hse} can be solved for $\rm M_{HE}(<r)$, yielding an estimate of the cluster mass profile as a function of radius. However, $\rm M_{HE}(<r)$ is an accurate estimate of the total mass profile only if hydrostatic equilibrium and spherical symmetry are good approximations. In general, the real mass profile ${\rm M_{real}(<r)\neq M_{HE}(<r)}$, i.e. the hydrostatic mass is a biased estimator. This effect, hydrostatic bias, can be quantified by measuring ${\rm M_{HE}(<r)/M_{real}(<r)}$ in simulated clusters like the ones considered in this paper.
Despite the existence of possible hydrostatic bias, there is an additional bias that derives from the modeling of the source of pressure support in Equation~\ref{eq:hse}. In the most general sense $\rm P_{\rm HE}$ represents the effect of several physical processes that provide pressure support to the cluster against its own gravity. Contribution to $\rm P_{HE}$ may come from thermal pressure of the ICM, $\rm P_{therm}$, turbulent and bulk motions acting as an effective pressure \cite[e.g.][]{2014MNRAS.442..521S} and other non-thermal processes such as magnetic fields, cosmic ray pressure, and electron-ion non-equilibrium in the ICM. To keep our formalism as general as possible, we label these contribution as {\it non-thermal pressure} $\rm P_{nt}$. Under this assumption we have
\begin{equation}
{\rm P_{HE} = P_{therm} + P_{nt}}.
\end{equation}
If $\rm P_{nt}$ is significantly large and is intentionally neglected, then Equation~\ref{eq:hse} will yield a biased mass estimate even if perfect hydrostatic equilibrium holds.
If only thermal pressure is considered for the mass reconstruction, Equation~\ref{eq:hse} yields a commonly used mass estimator:
\begin{equation}\label{eq:mtherm}
{\rm M_{\rm therm} (<r) = -\frac{r^2}{G\rho(r)}\frac{\rm d P_{\rm therm}}{\rm dr} }.
\end{equation}
The thermal pressure profile can be measured or inferred from X-ray and SZ observations, making this estimator easy to apply to observational data-sets. However, this mass estimator is in principle affected by hydrostatic bias and by the neglection of non-thermal pressure; both these biases need to be properly quantified.
\subsection{Measured quantities}\label{sec:measured_quantities}
Cosmological hydrodynamical simulations can help quantify hydrostatic bias and the contribution from non-thermal pressure. Our simulations do not include magnetic fields and cosmic ray physics, therefore the only source of non-thermal pressure that we are able to constrain is from turbulent and bulk motions in the ICM. Distinguishing between the contributions from turbulence and bulk motions is beyond the scope of this paper and we limit our analysis to the net effect of both processes.
Potential caveats and limitations of our simulations are discussed in Section~\ref{sec:conclusion}.
First, we measure the density profile ${\rm \rho (r)}$ and thermal pressure profile ${\rm P_{therm}(r)}$ of each simulated cluster at redshift $z=0$. These profiles are obtained by volume-weighted averaging over spherical shells. The use of volume-weighting smooths out large fluctuations in the profiles that would otherwise be present if mass-weighting were used. Having smoother profiles improves the accuracy of the numerical integration of the differential equations we discuss below.
We also tested mass-weighted averaging, excluding cells whose density is 2$\sigma$ away from the spherically-averaged density at the same radius (similarly to \citealt{2014ApJ...792...25N}), and found that the measured profiles do not differ more than $\sim 1-2\%$ from the volume-weighted profile. After measuring ${\rm \rho (r)}$ and ${\rm P_{therm}(r)}$, Equation~\ref{eq:mtherm} is used to compute the thermal mass profile ${\rm M_{therm}(<r)}$.
Figure~\ref{rho_P_vs_ACCEPT} shows that we achieve satisfactory agreement between the density and thermal pressure profiles in our simulations at redshift $z=0$ and those measured from the ACCEPT cluster sample \citep{2009ApJS..182...12C}.
\begin{figure*}
\includegraphics[width=.99\textwidth]{./Figures/rho_P_vs_ACCEPT.pdf}
\caption{\label{rho_P_vs_ACCEPT} Electron number density (left) and thermal pressure (right) profiles from our simulations (red lines) vs. the observed clusters in the ACCEPT sample (Cavagnolo et al. 2009).}
\end{figure*}
According to Equation~\ref{eq:hse}, the total hydrostatic mass of the simulated clusters can be measured as:
\begin{equation}\label{eq:mass_hse}
{\rm M_{HE}(<r) = -\frac{r^2}{G}g_r},
\end{equation}
where $\rm g_r$ is directly measured from the simulations by projecting the average gravitational acceleration of each AMR cell along the radial direction and subsequently averaging over spherical shells. Due to contribution from non-thermal pressure, ${\rm M_{HE}>M_{therm}}$.
Subsequently, we reconstruct the total pressure required for hydrostatic equilibrium by numerically integrating Equation~\ref{eq:hse}:
\begin{equation}\label{eq:ptot}
{\rm P_{HE} (r) = P_{HE}(0) - \int_0^r \frac{GM_{HE}(<r')\rho(r')}{r'^2}dr'}
\end{equation}
where we imposed the boundary condition ${\rm P_{HE} (0) = P_{therm}(0)+P_{nt}(0)}$, where the central thermal and non-thermal pressures are measured explicitly (Subsection~\ref{sec:ntp-vdisp}). Under the assumption of hydrostatic equilibrium, the non-thermal pressure profile is simply given by ${\rm P_{nt}(r) = P_{HE}(r) - P_{therm}(r)}$.
\subsection{Analytical model for non-thermal pressure}
The procedure described in Subsection~\ref{sec:measured_quantities} by which the non-thermal pressure is measured by subtracting the thermal pressure from the total pressure cannot obviously be applied to real systems where in the most ideal scenario only the temperature (pressure) and density profiles of the ICM are known.
We used our cosmological simulations to calibrate an empirical relation to model the non-thermal pressure as a function of ICM density. We first measure the gas density profile ${\rm \rho(r)}$ and the non-thermal pressure profile ${\rm P_{nt}(r)}$ (see Section~\ref{sec:results}) for all halos. By defining ${\rm x=r/R_{200m}}$, we can express the density
and non-thermal pressure profile as ${\rm \rho(x)}$ and ${\rm P_{nt}(x)}$, respectively. A point in the ${\rm (\rho,P_{\rm nt})}$ space is associated to each halo and radial position ${\rm x}$. We found that values of ${\rm P_{nt}(x)}$ are correlated to values of ${\rm \rho(x)}$ in the ${\rm (\rho,P_{\rm nt})}$ space. Our analytical model is a fit to the
empirical correlation we find in our sample with parameters found by performing a standard $\chi^2$ regression. In this model, the non-thermal pressure is given by:
\begin{eqnarray}\label{eq:model_pnt}
{\rm P}_{\rm nt,model}(r) = 5.388 \times 10^{13} \times \left(\frac{\rm R_{200m}}{1 \hbox{ Mpc}}\right)^3 \times \nonumber \\
\times \left[\frac{\rm \rho(r)}{1 \hbox{ g/cm}^3}\right] \hbox{ erg/cm}^3.
\end{eqnarray}
Figure~\ref{fig:ntp} shows the mean non-thermal pressure profile measured from the sub-sample of 5 relaxed clusters (blue line, with its 1$\sigma$ halo-to-halo scatter) compared to the analytical model from Equation~\ref{eq:model_pnt} (green line). The analytical model is well within the 1$\sigma$ scatter at all radii ${\rm r<R_{200m}}$.
Notice how the non-thermal pressure in Equation~\ref{eq:model_pnt} is expressed as a function of ${\rm R_{200m}}$. In recent papers by \cite{2014ApJ...792...25N}, \cite{2015ApJ...806...68L} and \cite{2016arXiv160501723A}, ${\rm R_{200m}}$ was confirmed to be a natural radius to re-scale density, pressure and temperature profiles for halos of different halo masses and at different redshifts.
In particular, \cite{2014ApJ...792...25N} also provide a fitting formula for the non-thermal pressure in simulations without cooling and star formation that depends on $r/R_{\rm 200m}$ and that provides an excellent fit to simulations up to redshift $z=1.5$. \cite{2016MNRAS.455.2936S} also explore a set of analytical models to accurately describe the total pressure profiles from
simulated clusters. However, our formula is less complicated and can be used to provide a physical interpretation to our results. Since ${\rm R_{200m}=(M_{200m}/4\pi\bar{\rho}_{200m})^{1/3}}$, where ${\rm \bar{\rho}_{200m} = 200\Omega_m\rho_{crit}}$ is the average matter density within ${\rm R_{\rm 200m}}$, we have:
\begin{equation}
{\rm P_{\rm nt,model}(r) \propto M_{200m} \frac{\rho(r)}{\bar{\rho}_{200m}}},
\end{equation}
i.e. the non-thermal pressure scales linearly with local fluctuations of the density with respect to the mean density within ${\rm R_{200m}}$. Of course, this is only an average effect, since we are considering spherically averaged quantities. However, it is important to stress that correlations between local density fluctuations and turbulent velocity fluctuations have already
been discussed in the literature \citep[e.g.][]{2014ApJ...788L..13Z}.
\begin{figure}
\includegraphics[width=.49\textwidth]{./Figures/all_press_r200m_relaxed.pdf}
\caption{\label{fig:ntp} Comparison of the mean non-thermal pressure profile for the relaxed clusters (blue) to the analytical model of Equation~\ref{eq:model_pnt} (green).
The blue shaded region represents the halo-to-halo 1-$\sigma$ scatter. }
\end{figure}
Equation~\ref{eq:model_pnt} can be used for cluster mass reconstruction including non-thermal pressure under the assumption of hydrostatic equilibrium. The resulting mass estimator is:
\begin{equation}\label{eq:mass_theory}
{\rm M_{model} (<r) = -\frac{r^2}{G\rho(r)}\frac{d}{dr}[P_{therm} (r) + P_{nt,model}(r)]}.
\end{equation}
The greatest advantage of the empirical model we propose is that the non-thermal pressure scales linearly with the local ICM density, which can be determined from observations. The dependence on $\rm R_{200m}$ is in practice a factor that allows self-similar scaling of the non-thermal pressure profiles of different clusters. This behaviour is of course only approximate. The dependence ${\rm P_{nt,model}\propto R_{200m}^3}$ is somewhat strong, however ${\rm R_{200m}}$ varies only by a factor $\sim 2$ in the range ${\rm 14 \lesssim \log_{10} (M_{200m}/M_{\odot}) \lesssim 15}$, making the model less sensitive to such variations.
When applying Equation~\ref{eq:model_pnt} to observations, a sufficiently accurate estimate for ${\rm R_{200m}}$ can be obtained by a simple iterative algorithm:
\begin{enumerate}
\item Initial condition. From measured $\rm \rho(r)$, $\rm P_{therm}(r)$, determine ${\rm M_{therm}(<r)}$ and use it to make an initial guess for ${\rm R_{200m}}$.
\item Start iterations. From measured $\rm \rho(r)$, $\rm P_{therm}(r)$ and guess for $\rm R_{200m}$, determine $\rm M_{model} (<r)$.
\item Determine new value for $\rm R_{200m}$ and iterate from point number (ii). Repeat until convergence.
\end{enumerate}
In Section~\ref{sec:results} we will quantitatively asses the quality of this approximation comparing ${\rm M_{model}(<r)}$ to the real mass profile of our simulated clusters ${\rm M_{real}(<r)}$. This comparison will highlight the efficiency of this model at removing biases in the determination of cluster mass.
\begin{figure}
\includegraphics[width=.49\textwidth]{./Figures/central_bc.pdf}
\caption{\label{fig:central_bc} Ratio between central non-thermal pressure and central thermal pressure for the halos in our sample. Halos 1-5 are relaxed, halos 6-10 are not relaxed.}
\end{figure}
\subsection{Non-thermal pressure from ICM velocity dispersion}\label{sec:ntp-vdisp}
In simulations, it is in principle possible to calculate an approximate estimate of the non-thermal pressure as a function of radius as
\begin{equation}\label{eq:pdisp}
{\rm P_{nt}\approx P_{disp}= \rho\sigma_{ICM}^2,}
\end{equation}
where ${\rm \sigma_{ICM}}$ is the velocity dispersion in the gas motions. We used Equation~\ref{eq:pdisp} to estimate an approximate value for the non-thermal pressure in the region ${\rm r < 10}$ kpc, ${\rm P_{\rm nt}(r<10\hbox{ }kpc)}$, and compared its value to the thermal pressure within the same region ${\rm P_{\rm therm}(r<10\hbox{ }kpc)}$.
The result of the comparison is shown in Figure~\ref{fig:central_bc}. For 8 of our halos the central non-thermal pressure is a few \% of the central thermal pressure, for the other two (unrelaxed) clusters the ration increases to $\sim$10 \%.
To assess the robustness of Equation~\ref{eq:pdisp} we also measure the full radial profile ${\rm P_{disp}(r)}$ and use it to perform yet another mass reconstruction:
\begin{equation}
{\rm M_{disp}(<r)=-\frac{r^2}{G\rho(r)}\frac{d}{dr}[P_{therm} (r) + P_{disp}(r)]}
\end{equation}
All the mass estimators defined so far will be compared in the next Section.
\section{Results}\label{sec:results}
\begin{figure*}
\includegraphics[width=.99\textwidth]{./Figures/press_frac_10panel.pdf}
\caption{\label{pressure_fractions} Fraction of pressure contributions with respect to the total pressure as a function of radius, under the assumption of spherical symmetry and hydrostatic equilibrium. Each panel represents one of the 10 halos analysed in this paper. Red solid lines represent the contribution from thermal pressure; blue solid lines represent the contribution from non-thermal pressure; green solid lines represent the empirical formula in Equation~\ref{eq:model_pnt}; magenta solid lines represent the approximate value for the non-thermal pressure ${\rm P_{disp}= \rho\sigma_{ICM}^2}$.}
\end{figure*}
We begin the presentation of our results by showing the ratio of different pressure contributions as a function of radius for our halos at redshift $z=0$ in Figure~\ref{pressure_fractions}. The total pressure ${\rm P_{\rm tot}}$ used as reference in this figure is the one required to maintain hydrostatic equilibrium in each cluster (${\rm P_{HE}}$ in Equation~\ref{eq:ptot}). The thermal pressure profile is directly measured from the simulations. The non-thermal pressure is computed as ${\rm P_{nt}=1-P_{\rm therm}}$. Our results show significant halo-to-halo differences, but a general trend is evident in which thermal pressure dominates in the central regions of the clusters, with non-thermal pressure becoming increasingly relevant at large radii and constituting $\sim 20-40 \%$ of the total pressure. This result is in general agreement with the analysis performed by \cite{2015MNRAS.448.1020S} and \cite{2016arXiv160602293B}. In a few of the clusters there are peaks of ${\rm P_{nt}/P_{HE}}\sim 30-40 \%$ at a distance of a few hundred kpc from the center. Such objects typically have more disturbances in the density/velocity fields.
The green line in Figure~\ref{pressure_fractions} represents the model for non-thermal pressure of Equation~\ref{eq:model_pnt} which aims at approximating the non-thermal pressure profile of the simulated clusters. The model reproduces the general trend of increasing ${\rm P_{nt}/P_{HE}}$ with cluster-centric radius, but is unable to reproduce the local fluctuations of non-thermal pressure seen in the actual profiles. Despite this issue, we stress that the goal of Equation~\ref{eq:model_pnt} is not to provide an exact measure of the non-thermal pressure at a given radius, but rather an empirical formula to improve the results of mass reconstructions of galaxy clusters.
The magenta line in Figure~\ref{pressure_fractions} represents the ratio ${\rm P_{disp}/P_{HE}}$ as a function of radius, where ${\rm P_{disp}=\rho\sigma_{ICM}^2}$ dynamically probes non-thermal pressure. In most clusters, ${\rm P_{disp}}$ agrees with the other estimates of non-thermal pressure only in central regions and it is usually in excess at large radii. Better agreement between ${\rm P_{ nt}}$, ${\rm P_{nt,model}}$ and ${\rm P_{\rm disp}}$ can be appreciated in the relaxed clusters (IDs 1 to 5).
\begin{figure*}
\includegraphics[width=.99\textwidth]{./Figures/mass_profiles_10panel.pdf}
\caption{\label{mass_profiles} Spherically averaged enclosed mass profiles as a function of radius. Each panel represents one of the 10 halos analysed in this paper. Black dashed lines represent the real mass profile directly measured from the simulations; red solid lines represents the mass reconstruction achieved by assuming spherical symmetry, hydrostatic equilibrium and by considering only the thermal pressure; blue solid lines are the mass profiles obtained by assuming spherical symmetry, hydrostatic equilibrium and the contribution from thermal and non-thermal pressure (this profile can be only measured explicitly in simulations); green solid lines are the theoretical mass reconstructions assuming spherical symmetry, hydrostatic equilibrium and the contribution of thermal and non-thermal pressure following the empirical model in Equation~\ref{eq:model_pnt}; magenta solid lines represent mass reconstructions assuming spherical symmetry, hydrostatic equilibrium and the contribution of thermal and non-thermal pressure, with the latter approximated by ${\rm P_{\rm nt}\approx P_{disp}= \rho\sigma_{ICM}^2}$.}
\end{figure*}
The quality of the different approaches to mass reconstruction discussed in Section~\ref{sec:formalism} can be assessed by considering Figure~\ref{mass_profiles} which shows the spherically averaged mass profile ${\rm M_{real}}$ directly measured from each simulated cluster compared to the result of several mass reconstructions techniques. If one assumes spherical symmetry and hydrostatic equilibrium, the `true' hydrostatic mass ${\rm M_{HE}}$ can be estimated by measuring the radial component of the gravity field from the simulations and then using Equation~\ref{eq:mass_hse}; it is important to stress that ${\rm M_{HE}}$ can only be measured from simulations that provide the value of the gravity field. Differences between ${\rm M_{real}}$ and ${\rm M_{HE}}$ represent the combined effect of hydrostatic bias and deviations from spherical symmetry. These differences are typically noticeable in this plot at large radii ${\rm r/R_{200m}>0.6-0.7}$, but also at smaller radii in some of the more dynamically disturbed systems (e.g. halos 8 and 10).
It appears that the assumption of hydrostatic equilibrium yields accurate mass reconstructions for 7 out of 10 clusters in the range ${\rm 0.1<r/R_{200m}<0.7}$.
However, for relaxed clusters, the agreement between ${\rm M_{\rm HE}}$ and ${\rm M_{real}}$ is excellent in the range ${\rm 0.1<r/R_{200m}<1.1}$ (see quantitative comparison in Subsection~\ref{sec:quant}).
The most important comparison in Figure~\ref{mass_profiles} is the one between ${\rm M_{real}}$ and ${\rm M_{therm}}$. The latter is the most basic mass reconstruction that an observer with access to density and thermal pressure measurements can perform, but it neglects the contribution from non-thermal pressure (Equation~\ref{eq:mtherm}). Figure~\ref{mass_profiles} confirms that ${\rm M_{therm}}$ is a biased estimator both with respect to the real mass ${\rm M_{real}}$ and to the total hydrostatic mass ${\rm M_{HE}}$. As expected, ${\rm M_{therm}}$ typically underestimates the mass of the cluster.
Figure~\ref{mass_profiles} shows that the theoretical model of Equation~\ref{eq:mass_theory}, ${\rm M_{model}}$. In this case, the contribution of non-thermal pressure to the equation of hydrostatic equilibrium is taken into account following Equation~\ref{eq:model_pnt}. ${\rm M_{model}}$ provides a more accurate hydrostatic mass reconstruction than ${\rm M_{therm}}$, i.e. ${\rm M_{model}}$ is closer to the `true' hydrostatic mass, ${\rm M_{HE}}$.
\subsection{Quantitative assessment of mass reconstruction biases}\label{sec:quant}
A more quantitative view of the situation is provided by Figure~\ref{mass_bias} that shows the average mass reconstruction bias ${\rm M(<r)/M_{real}(<r)}$ as a function of radius {\itshape for the relaxed clusters}. This plot shows that all the mass estimators we considered are $\sim10\%$ biased at radii ${\rm r<0.2R_{500}}$; within this region all clusters are relatively unrelaxed (see Figure 1), with the implication that hydrostatic equilibrium is a bad approximation. At larger radii, which are more relevant for cosmological applications, ${\rm M_{HE}}$ is the least biased reconstruction with accuracy $\sim 5\%$. This estimator would be useful only if both thermal and non-thermal pressure could be explicitly measured. Unfortunately, this is only feasible in clusters for which high quality data is available \citep[e.g.][]{2013MNRAS.428.2241S, 2015ApJ...806..207U}, but not in general.
Figure~\ref{mass_bias} shows that ${\rm M_{therm}}$ is always underestimating the real mass by $10-20 \%$ at ${\rm r>0.5R_{500}}$. However, ${\rm M_{model}}$, the theoretical model we propose in this paper (Equation~\ref{eq:mass_theory}) is very successful at correcting for the bias in ${\rm M_{therm}}$. On average, ${\rm M_{model}}$ provides a mass reconstruction that is always very close to the `true' hydrostatic mass ${\rm M_{HE}}$. This result is potentially very useful, because it shows that it is possible to weaken at least one source of bias in cluster mass reconstructions: the lack of knowledge on non-thermal pressure from turbulent and bulk motions in the ICM.
\begin{figure}
\includegraphics[width=.49\textwidth]{./Figures/avg_mass_relaxed.pdf}
\caption{\label{mass_bias} Average mass reconstruction bias ${\rm M(<r)/M_{real}(<r)}$ for relaxed clusters as a function of radius for different mass estimators. The solid lines represent the average among the 10 simulated halos and the shaded areas represent the halo-to-halo 1-$\sigma$ scatter.}
\end{figure}
\section{Summary and Discussion}\label{sec:conclusion}
We reviewed the basics of cluster mass reconstruction under the assumption of hydrostatic equilibrium. If deviations from spherical symmetry and contributions from non-thermal pressure are neglected, it is possible to use the equation of hydrostatic equilibrium to yield a reconstruction of a galaxy cluster mass profile. This approach has been widely used in the literature by combining constraints on the ICM thermal pressure from detection of the thermal SZ effect in galaxy clusters \citep{2010A&A...517A..92A, 2013ApJ...768..177S, 2013MNRAS.430.1344O, 2015A&A...583A.111R} and constraints from X-ray observations of galaxy clusters \citep{2010MNRAS.402...65S, 2010PASJ...62..371H, 2013SSRv..177..119E}.
The question of whether additional contribution to pressure support in galaxy clusters has been addressed by many authors \citep{2007MNRAS.378..385P, 2009ApJ...705.1129L, 2010ApJ...711.1033Z, 2012ApJ...758...74B, 2013ApJ...771..102F, 2013MNRAS.432..404M, 2015ApJ...808..176A}. In this paper, we extended this line of research by characterizing the contribution from non-thermal pressure support in a sample of 10 galaxy clusters simulated with the cosmological AMR code {\sc ramses} \citep{2002A&A...385..337T}. With our analysis we were also able to quantify the combined bias of the assumption of hydrostatic equilibrium and spherical symmetry.
We find that the traditional hydrostatic mass reconstruction that only considers of thermal pressure in the ICM underestimated the cluster mass by $10-20 \%$ at radii ${\rm 0.1R_{500}<r<2R_{500}}$, in agreement with recent results in the literature \citep{2016MNRAS.455.2936S, 2016arXiv160602293B}.
The contribution from non-thermal pressure to the support of the ICM against gravity is significant and typically increases with radius, with a maximum contribution of $20-40 \%$ of the total pressure at radii ${\rm R_{500}<r<2R_{500}}$. We showed that adding this contribution is important to remove the bias present in the traditional hydrostatic mass reconstruction method.
One of the outstanding issues for the determination of the contribution from non-thermal pressure in clusters is that it is hard to constrain from observations \citep[however, see][]{2010ApJ...711.1033Z}. In this paper, we use our simulations to calibrate a formula for the non-thermal pressure as a function of ICM density which is used to remove the mass reconstruction bias. The use of such a model for the non-thermal pressure provides an improved hydrostatic cluster mass estimator.
The most important caveat about the method we propose is that even if non-thermal pressure is accounted for, the assumptions of hydrostatic equilibrium and spherical symmetry implicitly carry with them a 5-10 \% mass bias. These effects are more difficult to study and to remove \citep[e.g.][]{2015MNRAS.448.1644S, 2016MNRAS.460..844M} and we defer the analysis of these issues to future work.
An additional caveat is that there are additional sources of pressure support that are not included in our simulations (e.g. magnetic fields and cosmic ray pressure) whose relevance for cluster mass modeling needs to be assessed.
Finally, the spatial resolution $\sim 1$ kpc/h of our simulations does not allow us to explicitly test whether a full the turbulent cascade is achieved. In this case turbulence is expected to dissipate faster than in nature. \cite{2012A&A...544A.103V} performed a detailed analysis of turbulence in simulated clusters and identified resolution effects on scales $\sim 10-20$ kpc. Therefore, we conclude that our results only provide a {\it lower limit} to the contribution of turbulence to pressure balance in clusters and thus a lower limit to the mass bias expected by neglecting this source of pressure support. This issue will be much better addressed by the next generation of cosmological zoom-in simulations which will have $\sim 10$ times better resolution and will include more physical processes.
Fortunately, our simulations have enough resolution to capture the effect of the other large contribution to non-thermal pressure, bulk motions, which typically play an important role at large cluster-centric distances ($r>500$ kpc, \cite{2015MNRAS.448.1020S}).
In conclusion, the approach we propose for mass reconstruction constitutes a significant step forward with respect to traditional methods based on hydrostatic equilibrium. This improvement will be relevant for better mass calibration of cluster scaling relations in the era of large surveys for precision cosmology (e.g. Euclid, LSST, DESI, eROSITA). Re-calibration of cluster scaling relations with the proposed method using archival data (e.g. from HST, ACT, SPT, Planck) and data from future X-ray surveys (eROSITA, ATHENA) will significantly improve the precision of cosmological constraints coming from cluster studies (e.g. cluster mass function).
\section*{Acknowledgments}
D.M. acknowledges support from the Swiss National Science Foundation (SNSF) through the SNSF Early.Postdoc and Advanced.Postdoc Mobility Fellowships. This work was supported by a grant from the Swiss National Supercomputing Centre (CSCS).
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\section[Introduction]{Introduction}\label{sec:intro}
In 1975 John W. \citeauthor{Tukey75}, in his work on mathematics and the picturing of data, proposed a novel way of data description, which evolved into a measure of multivariate centrality named \emph{data depth}. For a data sample, this statistical function determines centrality, or representativeness of an arbitrary point in the data, and thus allows for multivariate ordering of data regarding their centrality. More formally, given a data cloud $\boldsymbol{X}=\{\boldsymbol{x}_1,...,\boldsymbol{x}_n\}$ in $\mathbb{R}^d$, for a point $\boldsymbol{z}$ of the same space, a depth function $D(\boldsymbol{z}|\boldsymbol{X})$ measures how \emph{close} $\boldsymbol{z}$ is located to some (implicitly defined) \emph{center} of $\boldsymbol{X}$. Different concepts of closeness between a point $\boldsymbol{z}$ and a data cloud $\boldsymbol{X}$ suggest a diversity of possibilities to define such a function and a center as its maximizer. Naturally, each depth notion concentrates on a certain aspect of $\boldsymbol{X}$, and thus possesses various theoretical and computational properties.
Many depth notions have arisen during the last several decades
differing in properties and being suitable for various applications.
Mahalanobis \citep{Mahalanobis36}, halfspace \citep{Tukey75}, simplicial volume \citep{Oja83}, simplicial \citep{Liu90}, zonoid \citep{KoshevoyM97}, projection \citep{ZuoS00}, spatial \citep{VardiZ00} depths can be seen as well developed and most widely employed notions of depth function; see \cite{Mosler13} for a recent survey with details on categorization and properties.
Being intrinsically nonparametric, a depth function captures the geometrical features of given data in an affine-invariant way. By that, it appears to be useful for description of data's location, scatter, and shape, allowing for multivariate inference, detection of outliers, ordering of multivariate distributions, and in particular classification, that recently became an important and rapidly developing application of the depth machinery.
While the parameter-free nature of data depth ensures attractive theoretical properties of classifiers, its ability to reflect data topology provides promising predicting results on finite samples.
\subsection{Classification in the depth space}
Consider the following setting for supervised classification: Given a training sample consisting of $q$ classes $\boldsymbol{X}_1,...,\boldsymbol{X}_q$, each containing $n_i$, $i=1,...,q$, observations in $\mathbb{R}^d$. For a new observation $\boldsymbol{x}_0$, a class should be determined, to which it most probably belongs. Depth-based learning started with plug-in type classifiers. \citet{GhoshC05b} construct a depth-based classifier, which, in its na\"{i}ve form, assigns the observation $\boldsymbol{x}_0$ to the class in which it has maximal depth.
They suggest an extension of the classifier, that is consistent w.r.t.~Bayes risk for classes stemming from elliptically symmetric distributions.
Further \citet{DuttaG11,DuttaG12} suggest a robust classifier and a classifier for $L_p$-symmetric distributions, see also \citet{CuiLY08}, \citet{MoslerH06}, and additionally \citet{Joernsten04} for unsupervised classification.
A novel way to perform depth-based classification has been suggested by \citet{LiCAL12}: first map a pair of training classes into a two-dimensional depth space, which is called the $DD$-plot, and then perform classification by selecting a polynomial that minimizes empirical risk. Finding such an optimal polynomial numerically is a very challenging and --- when done appropriately --- computationally involved task, with a solution that in practice can be unstable \citep[see][Section~1.2.2 for examples]{Mozharovskyi15}. In addition,
the $DD$-plot should be rotated and the polynomial training phase should be done twice. Nevertheless, the scheme itself allows to construct optimal classifiers for wider classes of distributions than the elliptical family. Being further developed and applied by \cite{Vencalek11,LangeMM14a,MozharovskyiML15} it proved to be useful in practice, also in the functional setting \citep{MoslerM15,CuestaAlbertosFBOdlF15}.
The general depth-based supervised classification framework implemented in the \proglang{R}-package \pkg{ddalpha} can be described as follows. In the first training phase, each point of the training sample is mapped into the $q$-variate space of its depth values with respect to each of the classes $\boldsymbol{x}_i\mapsto\left(D(\boldsymbol{x}_i|\boldsymbol{X}_1),...,D(\boldsymbol{x}_i|\boldsymbol{X}_q)\right)$. In the second training phase, a low-dimensional classifier, flexible enough to account for the change in data topology due to the depth transform, is employed in the depth space. We suggest to use the $\alpha$-procedure, which is a nonparametric, robust, and computationally efficient separator. When classifying an unknown point $\boldsymbol{x}_0$, the first phase remains unchanged $\left(\boldsymbol{x}_0\mapsto\left(D(\boldsymbol{x}_0|\boldsymbol{X}_1),...,D(\boldsymbol{x}_0|\boldsymbol{X}_q)\right)\right)$, and in the second phase the trained $q$-variate separator assigns the depth-transformed point to one of the classes. Depth notions best reflecting data geometry share the common feature to attain value zero immediately beyond the convex hull of the data cloud. Thus, if such a data depth is used in the first phase, it may happen that $\boldsymbol{x}_0$ is mapped to the origin of the depth space, and thus cannot be readily classified. We call such a point an \emph{outsider} and suggest to apply a special treatment to assign it. If the data is of functional nature, a finitization step based on the \emph{location-slope \mbox{(LS-)} transform} precedes the above described process. Depth transform, $\alpha$-procedure, outsider treatment, and the preceding $LS$-transform constitute the $DD\alpha$-classifier. This together with the depth-calculating machinery constitutes the heart of the \proglang{R}-package \pkg{ddalpha}.
\subsection{The R-package ddalpha}
The \proglang{R}-package \pkg{ddalpha} is a software directed to fuse experience of the applicant with recent theoretical and computational achievements in the area of data depth and depth-based classification. It provides an implementation for exact and approximate computation of seven most reasonable and widely applied depth notions: Mahalanobis, halfspace, zonoid, projection, spatial, simplicial and simplicial volume depths. The variety of depth-calculating procedures includes functions for computation of data depth of one or more points w.r.t.~a data set, construction of the classification-ready $q$-dimensional depth space, visualization of the bivariate depth function for a sample in the form of upper-level contours and of a 3D-surface.
The main feature of the proposed methodology on the $DD$-plot is the $DD\alpha$-classifier, which is an adaptation of the $\alpha$-procedure to the depth space. Except for its efficient and fast implementation, \pkg{ddalpha} suggests other classification techniques that can be employed in the $DD$-plot: the original polynomial separator by \cite{LiCAL12} and the depth-based $k$NN-classifier proposed by \cite{Vencalek11}.
Halfspace, zonoid and simplicial depths
vanish beyond the convex hull of the sample, and thus cause outsiders during classification. For this case, \pkg{ddalpha} offers a number of outsider treatments and a mechanism for their management.
If it is decided to employ the $DD$-classifier, its constituents are to be chosen: data depth, classification technique in the depth space, and, if needed, outsider treatment and aggregation scheme for multi-class classification.
Their parameters, such as type and subset size of the variance-covariance estimator for Mahalanobis and spatial depth, number of approximating directions for halfspace and projection depth or part of simplices for approximating simplicial and simplicial volume depths, degree of polynomial extension for the $\alpha$-procedure or the polynomial classifier, number of nearest neighbors in the depth space or for an outsider treatment, \textit{etc.} must be set.
Rich built-in benchmark procedures allow to estimate the empirical risk and error rates of the $DD$-classifier and the portion of outsiders help in making the decision concerning the settings.
\pkg{ddalpha} possesses tools for immediate classification of functional data in which the measurements are first
brought onto a finite dimensional basis, and then fed to the depth-classifier. In addition, the componentwise classification technique by \cite{DelaigleHB12} is implemented.
Unlike other packages, \pkg{ddalpha} implements under one roof various depth functions and classifiers for multivariate and functional data.
\pkg{ddalpha} is the only package that implements zonoid depth and efficient exact halfspace depth. All depths in the package are implemented for any dimension $d\ge2$; except for the projection depth all implemented algorithms are exact, and supplemented by their approximating versions to deal with the increasing computational burden for large samples and higher dimensions. It also supports user-friendly definitions of depths and classifiers.
In addition, the package contains 50 multivariate and 4 functional ready-to-use classification problems and data generators for a palette of distributions.
Most of the functions of the package are programmed in \proglang{C++}, in order to be fast and efficient.
The package has a module structure, which makes it expandable and allows user-defined custom depth methods and separators.
\pkg{ddalpha} employs \textbf{boost} (package \pkg{BH} \cite{BH}), a well known fast and widely applied library, and resorts to \pkg{Rcpp} \citep{Rcpp} allowing for calls of \proglang{R} functions from \proglang{C++}.
\newpage
\subsection{Existing R-functionality on data depth}
Having proved to be useful in many areas, data depth and its applications find implementation in a number of \proglang{R}-packages: \pkg{aplpack}, \pkg{depth}, \pkg{localdepth}, \pkg{fda.usc}, \pkg{rsdepth}, \pkg{depthTools}, \pkg{MFHD}, \pkg{depth.plot}, \pkg{DepthProc}, \pkg{WMTregions}, \pkg{modQR}, \pkg{OjaNP}.
Functions of \pkg{aplpack} \citep{aplpack} allow to exactly compute bivariate halfspace depth and construct a bagplot.
\proglang{R}-package \pkg{depth} \citep{depth} provides implementation for the exact halfspace depth for $d\le 3$ and for approximate halfspace depth when $d\ge 3$, exact simplicial depth in $\mathbb{R}^2$, and exact simplicial volume depth in any dimension.
\pkg{DepthProc} \citep{DepthProc} calculates Mahalanobis, Euclidean, LP, bivariate regression, and modified band depth and approximates halfspace and projection depth, provides implementation for their local versions w.r.t.~\cite{PaindaveineVB13} and for the corresponding depth median estimators. It also contains functions for depth visualization and produces $DD$-plots.
\cite{AgostinelliR11} proposed local versions for several depth notions. Their \proglang{R}-package \pkg{localdepth} \citep{localdepth} evaluates simplicial, univariate halfspace, Mahalanobis, ellipsoid depths and their localization.
The package \pkg{fda.usc} \citep{fda.usc} provides methods for exploratory and descriptive analysis of functional data. It contains functions for functional data representation, functional outliers detection, functional regression models and analysis of variance model, functional supervised and unsupervised classification. The package calculates the following depth functions for multivariate data: simplicial depth in $\mathbb{R}^2$, halfspace depth (aproximate for $d>3$), Mahalanobis depth, approximate projection depth, likelihood depth; and for functional data: Fraiman and Muniz depth, h-modal depth, random Tukey depth, random projection depth, double random projection depth. In addition the package suggests a number of classifiers acting in the functional depth space, namely maximum depth, polynomial, logistic regression, LDA, QDA, $k$NN and KDA ones.
Several packages have specific purpose or implement only one depth. Among the others are:
\pkg{depthTools} \citep{depthTools}, implementing different statistical tools for the description and analysis of gene expression data based on the modified band depth;
\pkg{depth.plot} \citep{depth.plot}, containing the implementations of spatial depth and spatial ranks and constructing corresponding $DD$-plots;
\pkg{MFHD} \citep{MFHD}, calculating multivariate functional halfspace depth and median for bivarite functional data;
\pkg{rsdepth} \citep{rsdepth}, implementing the ray shooting depth and the corresponding median in $\mathbb{R}^2$;
\pkg{WMTregions} \citep{WMTregions}, computing weighted-mean trimmed regions with zonoid regions as a special case;
\pkg{modQR} \citep{modQR}, calculating multiple-output regression quantiles with halfspace trimmed regions as a special case;
and
\pkg{OjaNP} \citep{OjaNP}, which offers efficient computation of the multivariate Oja median and related statistics.
\subsection{Outline of the article}\label{ssec:introOutline}
To facilitate understanding and keep the presentation solid, the functionality of the \proglang{R}-package \pkg{ddalpha} is illustrated through the article on the same functional data set ``ECG Five Days'' from \cite{UCRArchive}, which is a long ECG time series constituting two classes.
The data set originally contains 890 objects.
We took a subset consisting of 70 objects only (35 from each of the days) which best demonstrates the general and complete aspects of the proposed procedures (\textit{e.g.}, existence of outliers in its bivariate projection or necessity of three features in the $\alpha$-procedure).
\begin{figure}[!h]
\begin{center}
\includegraphics[keepaspectratio=true,width = .495\textwnew, page=1]{functional_data.pdf}
\includegraphics[keepaspectratio=true,width = .495\textwnew, page=2]{functional_data.pdf}
\caption{ECG Five Days data (left) and their derivatives (right).}
\label{fig:funcdata}
\end{center}
\end{figure}
In Section~\ref{sec:functions} functional data are transformed into a finite dimensional space using the $LS$-transform, as it is shown in Figure~\ref{fig:funcdata} presenting the functions and their derivatives.
In this example we choose $L = S = 1$, and thus the $LS$-transform produces the two-dimensional discrete space, where each function is described by the area under the function and under its derivative as it is shown in Figure~\ref{fig:discretespace}, left.
Then in Section~\ref{sec:depth}, the depth is calculated in this two-dimensional space (Figure~\ref{fig:discretespace}, middle) and the $DD$-plot is constructed in Section~\ref{sec:ddclass} (Figure~\ref{fig:discretespace}, right).
The classification is performed by the $DD\alpha$-separator in the $DD$-plot. The steps of the $\alpha$-procedure are illustrated on Figure~\ref{tab:gt}.
Section~\ref{sec:depth} presents a theoretical description of the data depth and the depth notions implemented in the package. In addition, it compares their computation time and performance when employed in the maximum depth classifier.
Section~\ref{sec:ddclass} includes a comprehensive algorithmic description of the $DD\alpha$-classifier with a real-data illustration.
Further, it discusses other classification techniques that can be employed in the $DD$-plot.
The questions whether one should choose a depth that avoids outsiders or should allow for outsiders and classify them separately, and in which way, are considered in Section~\ref{sec:outsiders}.
Section~\ref{sec:functions} addresses the classification of functional data.
In Section~\ref{sec:application}, the basic structure and concepts of the \proglang{R}-package user interface are presented, along with a discussion of their usage for configuring the classifier and examples for calling its functions.
\section[Data depth]{Data depth}\label{sec:depth}
This section regards depth functions.
First (Section~\ref{ssec:depthConcept}), we briefly review the concept of data depth and its fundamental properties.
Then (Section~\ref{ssec:depthNotions}), we give the definitions in their empirical versions
for several depth notions: Mahalanobis, projection, spatial, halfspace, simplicial, simplicial volume, zonoid depths.
For each notion, we shortly discuss relevant computational aspects, leaving motivations, ideas, and details to the corresponding literature and the software manual.
We do not touch the question of computation of depth-trimmed regions for the following reasons: first, for a number of depth notions there exist no algorithms; then, for some depth notions these can be computed using different \proglang{R}-packages, \textit{e.g.} \pkg{WMTregions} for the family of weighted-mean regions including zonoid depth \citep{BazovkinM12} or \pkg{modQR} for multiple-output quantile regression including halfspace depth as a particular case; finally, this is not required in classification.
After having introduced depth notions, we compare the speed of the implemented exact algorithms by means of simulated data (Section~\ref{ssec:depthComputation}).
The section is concluded (Section~\ref{ssec:depthMaxdepth}) by a comparison of error rates of the na\"{i}ve maximum depth classifier, paving a bridge to the more developed $DD$-plot classification which is covered in the following sections.
\subsection{The concept}\label{ssec:depthConcept}
Consider a point ${\boldsymbol z}\in\mathbb{R}^d$ and a data sample $\boldsymbol{X}=(\boldsymbol{x}_1,...,\boldsymbol{x}_n)^\prime$
in the $d$-dimensional Euclidean space, with $\boldsymbol{X}$ being a $(n\times d)$-matrix and $^\prime$ being the transposition operation.
A data depth is a function $D({\boldsymbol z}|\boldsymbol{X}):{\mathbb R}^d\mapsto[0,1]$ that describes how deep, or central, the observation ${\boldsymbol z}$ is located w.r.t.~$\boldsymbol{X}$. In a natural way, it involves some notion of center. This is any point of the space attaining the highest depth value in $\boldsymbol{X}$, and not necessarily a single one. In this view, depth can be seen as a center-outward ordering, \textit{i.e.}~points closer to the center have a higher depth, and those more outlying a smaller one.
The concept of a depth function can be formalized by stating postulates (requirements) it should satisfy. Following \citet{Dyckerhoff04} and \citet{Mosler13}, a \emph{depth function} is a function $D({\boldsymbol z}|\boldsymbol{X}):{\mathbb R}^d\mapsto[0,1]$ that is:
\begin{itemize}
\item[(\emph{D1})] \emph{translation invariant}: $D({\boldsymbol z}+\boldsymbol b|\boldsymbol{X}+\boldsymbol{1}_n \boldsymbol b^\prime)=D({\boldsymbol z}|\boldsymbol{X})$ for all $\boldsymbol b\in{\mathbb R}^d$ (here $\boldsymbol{1}_n = (1,...,1)^\prime$),
\item[(\emph{D2})] \emph{linear invariant}: $D(\boldsymbol A{\boldsymbol z}|\boldsymbol{X}\boldsymbol A^\prime)=D({\boldsymbol z}|\boldsymbol{X})$ for every nonsingular $d\times d$ matrix $\boldsymbol A$,
\item[(\emph{D3})] \emph{zero at infinity}: $\lim_{\|{\boldsymbol z}\|\to\infty}D({\boldsymbol z}|\boldsymbol{X})=0$,
\item[(\emph{D4})] \emph{monotone on rays}: Let ${\boldsymbol z}^*=\argmax_{{\boldsymbol z}\in{\mathbb R}^d}D({\boldsymbol z}|\boldsymbol{X})$, then for all $\,{\boldsymbol r}\in S^{d-1}$ the function $\beta\mapsto D({\boldsymbol z}^*+\beta{\boldsymbol r}|\boldsymbol{X})$ decreases in the weak sense, for $\,\beta>0$,
\item[(\emph{D5})] \emph{upper semicontinuous}: the upper level sets $D_{\alpha}(\boldsymbol{X})=\{{\boldsymbol z}\in{\mathbb R}^d:D({\boldsymbol z}|\boldsymbol{X})\ge\alpha\}$ are closed for all $\,\alpha$.
\end{itemize}
For slightly different postulates see \cite{Liu92} and \cite{ZuoS00}.
The first two properties state that $D(\cdot|\boldsymbol{X})$ is \emph{affine invariant}. $\boldsymbol{A}$ in (D2) can be weakened to isometric linear transformations, which yields an \emph{orthogonal invariant} depth. Taking instead of $\boldsymbol{A}$ some constant $\lambda>0$ gives a \emph{scale invariant} depth function. (D3) ensures that the upper level sets $D_{\alpha}$, $\alpha>0$, are bounded. According to (D4), the upper level sets are starshaped around ${\boldsymbol z}^*$, and $D_{\max_{{\boldsymbol z}\in{\mathbb R}^d}D({\boldsymbol z}|\boldsymbol{X})}(\boldsymbol{X})$ is convex.
(D4) can be strengthened by requiring $D(\cdot|\boldsymbol{X})$ to be a \emph{quasiconcave} function. In this case, the upper level sets are convex for all $\,\alpha>0$. (D5) is a useful technical restriction.
Upper level sets $D_{\alpha}(\boldsymbol{X}) = \{\boldsymbol{x}\in\mathbb{R}^d:D(\boldsymbol{x}|\boldsymbol{X})\ge\alpha\}$
of a depth function are also called \emph{depth-trimmed} or \emph{central regions}. They describe the distribution's location, dispersion, and shape. For given $\boldsymbol{X}$, the sets $D_{\alpha}(\boldsymbol{X})$ constitute a nested family of trimming regions. Note that due to (D1) and (D2) the central regions are affine equivariant, due to (D3) bounded, due to (D5) closed, and due to (D4) star-shaped (respectively convex, if quasiconcaveness of $D(\cdot|\boldsymbol{X})$ is additionally required).
\subsection{Implemented notions}\label{ssec:depthNotions}
The \proglang{R}-package \pkg{ddalpha} implements a number of depths. Below we consider their empirical versions.
For each implemented notion of data depth, the depth surface (left) and depth contours (right) are plotted in Figures~\ref{fig:bivariate_depths1} and~\ref{fig:bivariate_depths2} for bivariate data used in Section~\ref{sec:functions}.
\def.80\textwnew{.80\textwnew}
\def.105\textwnew{.3\textwnew}
\def.05\textwnew{.05\textwnew}
\fboxsep0.1mm
\begin{figure}
\begin{center}
\fbox{\vbox{\hsize=.80\textwnew\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 27mm 6.5mm 28mm 54mm, clip,scale=0.80]{bivariate_depths_no.png}
\hspace{.05\textwnew}\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 10mm 16mm 10mm 20mm, clip, page = 1]{depth_contours_noborder.pdf}\\
{\small Bivariate data}}}
\fbox{\vbox{\hsize=.80\textwnew\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 27mm 6.5mm 28mm 54mm, clip,scale=0.80]{bivariate_depths_mah.png}
\hspace{.05\textwnew}\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 10mm 16mm 10mm 20mm, clip, page = 3]{depth_contours_noborder.pdf}\\
{\small Mahalanobis depth}}}
\fbox{\vbox{\hsize=.80\textwnew\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 27mm 6.5mm 28mm 54mm, clip,scale=0.80]{bivariate_depths_pr.png}
\hspace{.05\textwnew}\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 10mm 16mm 10mm 20mm, clip, page = 4]{depth_contours_noborder.pdf}\\
{\small Projection depth}}}
\fbox{\vbox{\hsize=.80\textwnew\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 27mm 6.5mm 28mm 54mm, clip,scale=0.80]{bivariate_depths_sp.png}
\hspace{.05\textwnew}\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 10mm 16mm 10mm 20mm, clip, page = 5]{depth_contours_noborder.pdf}\\
{\small Spatial depth}}}
\caption{Depth plots and contours of bivariate data.}
\label{fig:bivariate_depths1}
\end{center}
\end{figure}
\begin{figure
\begin{center}
\fbox{\vbox{\hsize=.80\textwnew\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 27mm 6.5mm 28mm 54mm, clip,scale=0.80]{bivariate_depths_zon.png}
\hspace{.05\textwnew}\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 10mm 16mm 10mm 20mm, clip, page = 2]{depth_contours_noborder.pdf}\\
{\small Zonoid depth}}}
\fbox{\vbox{\hsize=.80\textwnew\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 27mm 6.5mm 28mm 54mm, clip,scale=0.80]{bivariate_depths_hs.png}
\hspace{.05\textwnew}\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 10mm 16mm 10mm 20mm, clip, page = 8]{depth_contours_noborder.pdf}\\
{\small Halfspace depth}}}
\fbox{\vbox{\hsize=.80\textwnew\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 27mm 6.5mm 28mm 54mm, clip,scale=0.80]{bivariate_depths_si.png}
\hspace{.05\textwnew}\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 10mm 16mm 10mm 20mm, clip, page = 6]{depth_contours_noborder.pdf}\\
{\small Simplicial depth}}}
\fbox{\vbox{\hsize=.80\textwnew\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 27mm 6.5mm 28mm 54mm, clip,scale=0.80]{bivariate_depths_sv.png}
\hspace{.05\textwnew}\includegraphics[keepaspectratio=true,width = .105\textwnew, trim = 10mm 16mm 10mm 20mm, clip, page = 7]{depth_contours_noborder.pdf}\\
{\small Simplicial volume depth}}}
\caption{Depth plots and contours of bivariate data.}
\label{fig:bivariate_depths2}
\end{center}
\end{figure}
\textbf{\emph{Mahalanobis depth}} is based on an outlyingness measure, \textit{viz.} the Mahalanobis distance \citep{Mahalanobis36} between $\boldsymbol{z}$ and a center of $\boldsymbol{X}$, $\boldsymbol{\mu}(\boldsymbol{X})$ say:
\begin{equation*}
d^2_{Mah}\bigl(\boldsymbol{z};\boldsymbol{\mu}(\boldsymbol{X}),\boldsymbol{\Sigma}(\boldsymbol{X})\bigr)=\bigl(\boldsymbol{z}-\boldsymbol{\mu}(\boldsymbol{X})\bigr)^\prime\boldsymbol{\Sigma}(\boldsymbol{X})^{-1}\bigl(\boldsymbol{z}-\boldsymbol{\mu}(\boldsymbol{X})\bigr).
\end{equation*}
The depth of a point $\boldsymbol{z}$ w.r.t.~$\boldsymbol{X}$ is then defined as \citep{Liu92}
\begin{equation}\label{equ:MahDepth}
D_{Mah}(\boldsymbol{z}|\boldsymbol{X})=\frac{1}{1+d^2_{Mah}\bigl(\boldsymbol{z};\boldsymbol{\mu}(\boldsymbol{X}),\boldsymbol{\Sigma}(\boldsymbol{X})\bigr)},
\end{equation}
where $\boldsymbol{\mu}(\boldsymbol{X})$ and $\boldsymbol{\Sigma}(\boldsymbol{X})$ are appropriate estimates of mean and covariance of $\boldsymbol{X}$.
This depth function obviously satisfies all the above postulates and is quasi-concave, too. It can be regarded as a \emph{parametric depth} as it is defined by a finite number of parameters (namely $\frac{d(d+1)}{2}$). Based on the two first moments, its depth contours are always ellipsoids centered at $\boldsymbol{\mu}(\boldsymbol{X})$, and thus independent of the shape of $\boldsymbol{X}$. If $\boldsymbol{\mu}(\boldsymbol{X})$ and $\boldsymbol{\Sigma}(\boldsymbol{X})$ are chosen to be moment estimates, \textit{i.e.}~ $\boldsymbol{\mu}(\boldsymbol{X})=\frac{1}{n}\boldsymbol{X}^\prime\boldsymbol{1}_n$ being the traditional \emph{average} and $\boldsymbol{\Sigma}(\boldsymbol{X})=\frac{1}{n-1} (\boldsymbol{X}-\boldsymbol{1}_n\boldsymbol{\mu}(\boldsymbol{X})^\prime)^\prime(\boldsymbol{X}-\boldsymbol{1}_n\boldsymbol{\mu}(\boldsymbol{X})^\prime)$ being the \emph{empirical covariance matrix}, the corresponding depth may be sensitive to outliers. A more robust depth is obtained with the \emph{minimum covariance determinant} (MCD) estimator, see \citet{RousseeuwL87}.
Calculation of the Mahalanobis depth consists in estimation of the center vector $\boldsymbol{\mu}(\boldsymbol{X})$ and the inverse of the scatter matrix $\boldsymbol{\Sigma}(\boldsymbol{X})$. In the simplest case of traditional moment estimates the time complexity amounts to $O(nd^2 + d^3)$ only. \citet{RousseeuwD99} develop an efficient algorithm for computing robust MCD estimates.
\textbf{\emph{Projection depth}}, similar to Mahalanobis depth, is based on a measure of outlyingness. See \citet{Stahel81}, \citet{Donoho82}, and also \citet{Liu92}, \citet{ZuoS00}. The worst case outlyingness is obtained by maximizing an outlyingness measure over all univariate projections:
\begin{equation*}
o_{prj}(\boldsymbol{z}|\boldsymbol{X})=\sup_{{\boldsymbol u}\in S^{d-1}}\frac{|\boldsymbol{z}^\prime\boldsymbol{u}-m(\boldsymbol{X}^\prime\boldsymbol{u})|}{\sigma(\boldsymbol{X}^\prime\boldsymbol{u})},
\end{equation*}
with $m(\boldsymbol{y})$ and $\sigma(\boldsymbol{y})$ being any location and scatter estimates of a univariate sample $\boldsymbol{y}$. Taking
$m(\boldsymbol{y})$ as the mean and $\sigma(\boldsymbol{y})$ as the standard deviation
one gets the Mahalanobis outlyingness, due to the projection property \citep{Dyckerhoff04}. In the literature and in practice most often \emph{median}, $med(\boldsymbol{y})=\boldsymbol{y}_{\left(\frac{\lfloor n+1 \rfloor}{2}\right)}$, and \emph{median absolute deviation from the median}, $MAD(\boldsymbol{y})=med(|\boldsymbol{y}-med(\boldsymbol{y})\boldsymbol{1}_n|)$, are used, as they are robust.
Projection depth is then obtained as
\begin{equation}
D_{prj}({\boldsymbol z}|\boldsymbol{X})=\frac{1}{1+o_{prj}(\boldsymbol{z}|\boldsymbol{X})}.
\end{equation}
This depth satisfies all the above postulates and quasiconcavity. By involving the symmetric scale factor $MAD$ its contours are centrally symmetric and thus are not well suited for describing skewed data.
Exact computation of the projection depth is a nontrivial task, which fast becomes intractable for large $n$ and $d$. \citet{LiuZ14} suggest an algorithm \citep[and a MATLAB implementation, see][]{LiuZ15}. In practice one may approximate the projection depth from above by minimizing it over projections on $k$ random lines, which has time complexity $O(knd)$.
It can be shown that finding the exact value is a zero-probability event though.
\textbf{\emph{Spatial depth}} (also $L_1$-depth) is a distance-based depth formulated by \citet{VardiZ00} and \citet{Serfling02}, exploiting the idea of spatial quantiles of \citet{Chaudhuri96} and \citet{Koltchinskii97}.
For a point $\boldsymbol{z}\in\mathbb{R}^d$, it is defined as one minus the length of the average direction from $\boldsymbol{X}$ to $\boldsymbol{z}$:
\begin{equation}\label{equ:sptDepth}
D_{spt}(\boldsymbol{z}|\boldsymbol{X})=1-\Bigl\|\frac{1}{n}\sum_{i=1}^n \boldsymbol{v}\bigl(\boldsymbol{\Sigma}^{-\frac{1}{2}}(\boldsymbol{X})(\boldsymbol{z} - \boldsymbol{x}_i)\bigr)\Bigr\|,
\end{equation}
with $\boldsymbol{v}(\boldsymbol{y})=\frac{\boldsymbol{y}}{\|\boldsymbol{y}\|}$ if $\boldsymbol{y}\neq\boldsymbol{0}$, and $\boldsymbol{v}(\boldsymbol{0})=\boldsymbol{0}$.
The scatter matrix $\boldsymbol{\Sigma}(\boldsymbol{X})$ provides the affine invariance.
Affine invariant spatial depth satisfies all the above postulates, but is not quasiconcave. Its maximum is referred to as the \emph{spatial median}. In the one-dimensional case it coincides with the halfspace depth, defined below.
Spatial depth can be efficiently computed even for large samples amounting in the simplest case to time complexity $O(nd^2 + d^3)$; for calculation of $\boldsymbol{\Sigma}^{-\frac{1}{2}}(\boldsymbol{X})$ see the above discussion of the Mahalanobis depth.
\textbf{\emph{Halfspace depth}} follows the idea of \citet{Tukey75}, see also \citet{DonohoG92}. The Tukey (=halfspace, location) depth of $\boldsymbol{z}$ w.r.t.~$\boldsymbol{X}$ is determined as:
\begin{equation}
D_{hs}(\boldsymbol{z}|\boldsymbol{X}) = \min_{\boldsymbol{u}\in S^{d-1}}\frac{1}{n}\#\{i:\boldsymbol{x}_i^\prime\boldsymbol{u}\le\boldsymbol{z}^\prime\boldsymbol{u};i=1,...,n\}.
\end{equation}
Halfspace depth satisfies all the postulates of a depth function. In addition, it is quasiconcave, and equals zero outside the convex hull of the support of $\boldsymbol{X}$. For any $\boldsymbol{X}$, there exists at least one point having depth not smaller than $\frac{1}{1+d}$ \citep{Mizera02}. For empirical distributions, halfspace depth is a discrete function of ${\boldsymbol z}$, and the set of depth-maximizing locations --- the \emph{halfspace median} --- can consist of more than one point (to obtain a unique median, an average of this deepest trimmed region can be calculated). Halfspace depth determines the empirical distribution uniquely \citep{StruyfR99,Koshevoy02}.
\citet{DyckerhoffM16} develop a family of algorithms (for each $d>1$) possessing time complexity $O(n^{d-1}\log{n})$ and $O(n^d)$ (the last has proven to be computationally more efficient for larger $d$ and small $n$). These algorithms are applicable for moderate $n$ and $d$. For large $n$ or $d$ and (or) if the depth has to be computed many times, approximation by minimizing over projections on random lines can be performed \citep{Dyckerhoff04,CuestaAlbertosNR08}. By that, $D_{hs}(\boldsymbol{z}|\boldsymbol{X})$ is approximated from above with time complexity $O(knd)$, and $D_{hs}(\boldsymbol{X}|\boldsymbol{X})$ with time complexity $O\bigl(kn(d + \log{n})\bigr)$, using $k$ random directions \citep[see also][]{MozharovskyiML15}.
\textbf{\emph{Simplicial depth}} \citep{Liu90}
is defined as the portion of simplices having vertices from $\boldsymbol{X}$ which contain $\boldsymbol{z}$:
\begin{equation}
D_{sim}(\boldsymbol{z}|\boldsymbol{X}) = \frac{1}{{n \choose d+1}}\sum_{1\le i_1<i_2<...<i_{d+1}\le n}I\bigl(\boldsymbol{z}\in\text{conv}(\boldsymbol{x}_{i_1},\boldsymbol{x}_{i_2},...,\boldsymbol{x}_{i_{d+1}})\bigr)
\end{equation}
with $\text{conv}(\mathcal{Y})$ being the convex hull of $\mathcal{Y}$ and $I(\mathcal{Y})$ standing for the indicator function, which equals $1$ if $\mathcal{Y}$ is true and $0$ otherwise.
It satisfies postulates (D1), (D2), (D3), and (D5). The set of depth-maximizing locations is not a singleton, but, different to the halfspace depth, it is not convex (in fact it is not even necessarily connected) and thus simplicial depths fails to satisfy (D4). It characterizes the empirical measure if the data, \textit{i.e.}~the rows of $\boldsymbol{X}$, are in general position, and is, as well as the halfspace depth, due to its nature rather insensitive to outliers, but vanishes beyond the convex hull of the data $\text{conv}(\boldsymbol{X})$.
Exact computation of the simplicial depth has time complexity of $O(n^{d+1}d^3)$. Approximations accounting for a part of simplices can lead to time complexity $O(kd^3)$ only when drawing $k$ random $(d+1)$-tuples from $\boldsymbol{X}$, or reduce real computational burden with the same time complexity, but keeping precision when drawing a constant portion of ${n \choose d+1}$.
For $\mathbb{R}^2$, \cite{RousseeuwR96} proposed an exact efficient algorithm with time complexity $O(n\log n)$ .
\textbf{\emph{Simplicial volume depth}} \citep{Oja83} is defined via the average volume of the simplex with $d$ vertices from $\boldsymbol{X}$ and one being $\boldsymbol{z}$:
\begin{equation}
D_{simv}(\boldsymbol{z}|\boldsymbol{X}) = \frac{1}{1 + \frac{1}{{n \choose d}\sqrt{\text{det}\bigl(\boldsymbol{\Sigma}(\boldsymbol{X})\bigr)}}\sum_{1\le i_1<i_2<...<i_{d}\le n}\text{vol}\bigl(\text{conv}(\boldsymbol{z},\boldsymbol{x}_{i_1},\boldsymbol{x}_{i_2},...,\boldsymbol{x}_{i_{d}})\bigr)}
\end{equation}
with $\text{vol}(\mathcal{Y})$ being the Lebesgue measure of $\mathcal{Y}$.
It satisfies all above postulates, is quasiconcave, determines $\boldsymbol{X}$ uniquely \citep{Koshevoy03}, and has a nonunique median.
Time complexity of the exact computation of the simplicial volume depth amounts to $O(n^d d^3)$, and thus approximations similar to the simplicial depth may be necessary.
\textbf{\emph{Zonoid depth}} has been first introduced by \citet{KoshevoyM97}, see also \citet{Mosler02} for a discussion in detail.
The zonoid depth function is most simply defined by means of depth contours --- the zonoid trimmed regions. The zonoid $\alpha$-trimmed region of an empirical distribution is defined as follows: For $\alpha\in\left[\frac{k}{n},\frac{k+1}{n}\right],\,k=1,...,n-1$ the zonoid region is defined as
\begin{equation*}
Z_{\alpha}(\boldsymbol{X})=\mbox{conv}\Bigl\{\frac{1}{\alpha n}\sum_{j=1}^k{\boldsymbol x}_{i_j}+\Bigl(1-\frac{k}{\alpha n}\Bigr){\boldsymbol x}_{i_{k+1}}:\{i_1,...,i_{k+1}\}\subset \{1,...,n\}\Bigr\},
\end{equation*}
and for $\alpha\in\left[0,\frac{1}{n}\right)$
\begin{equation*}
Z_{\alpha}(\boldsymbol{X})=\mbox{conv}(\boldsymbol{X})
\end{equation*}
Thus, \textit{e.g.}, $Z_{\frac{3}{n}}(\boldsymbol{X})$ is the convex hull of the set of all possible averages involving three points of $\boldsymbol{X}$, and $Z_0(\boldsymbol{X})$ is just the convex hull of $\boldsymbol{X}$.
The zonoid depth of a point ${\boldsymbol z}$ w.r.t.~$\boldsymbol{X}$ is then defined as the largest $\alpha\in[0,1]$ such that $Z_{\alpha}(\boldsymbol{X})$ contains ${\boldsymbol z}$ if ${\boldsymbol z}\in\mbox{conv}(\boldsymbol{X})$ and $0$ otherwise:
\begin{equation}
D_{zon}(\boldsymbol{z}|\boldsymbol{X})=
\sup\{\alpha\in[0,1]:\,\boldsymbol{z}\in Z_{\alpha}(\boldsymbol{X})\},
\end{equation}
where $\sup$ of $\emptyset$ is defined to be 0.
The zonoid depth belongs to the class of weighted-mean depths, see \citet{DyckerhoffM11}. It satisfies all the above postulates and is quasiconcave. As well as halfspace and simplicial depth, zonoid depth vanishes beyond the convex hull of $\boldsymbol{X}$. Its maximum (always equaling $1$) is located at the mean of the data, thus this depth is not robust.
Its exact computation with the algorithm of \citet{DyckerhoffKM96}, based on linear programming and exploiting the idea of Danzig-Wolf decomposition, appears to be fast enough for large $n$ and $d$, not to need approximation.
A common property of the considered above depth notions is that they concentrate on global features of the data ignoring local specifics of sample geometry.
Thus they are unable to reflect multimodality of the underlying distribution. Several depths have been proposed in the literature to overcome this difficulty. Two of them were introduced in the classification context, localized extension of the spatial depth \citep{DuttaG15} and the data potential \citep{PokotyloM16}. They are also implemented in the \proglang{R}-package \pkg{ddalpha}. The performance of these depths and of the classifiers exploiting them depends on the type of the kernel and its bandwidth. While the behaviour of these two notions substantially differs from the seven depth notions mentioned above, we leave them beyond the scope of this article and relegate to the corresponding literature for theoretical and experimental results.
\subsection{Computation time}\label{ssec:depthComputation}
To give insights into the speed of exactly calculating various depth notions we indicate computation times by graphics in Figure~\ref{fig:depth_speed}. On the logarithmic time scale, the lines represent the time (in seconds) needed to compute the depth of a single point, averaged over 50 points w.r.t.~60 samples, varying dimension $d\in\{2, 3, 4, 5\}$ and sample length $n\in\{50,100,250,500,1000\}$.
Due to the fact that computation times of the algorithms do not depend on the particular shape of the data, the data has been drawn from the standard normal distribution. Some of the graphics are incomplete due to excessive time. Projection depth has been approximated using 10\,000\,000$/ n$ random projections, all other depths have been computed exactly. Here we used one kernel of the Intel Core i7-4770 (3.4 GHz) processor having enough physical
memory.
One can see that, for all considered depths and $n\le 1\,000$, computation of the two-dimensional depth never oversteps one second.
For halfspace and simplicial depth this can be explained by the fact that in the bivariate case both depths depend only on the angles between the lines connecting $\boldsymbol z$ with the data points $\boldsymbol x_i$ and the abscissa. Computing these angles and sorting them has a complexity of $O(n\log n)$ which determines the complexity of the bivariate algorithms.
As expected, halfspace, simplicial, and simplicial volume depths, being of combinatorial nature, have exponential time growth in $(n,d)$. Somewhat surprising, zonoid depth being computed by linear programming, seems to be way less sensitive to dimension. One can conclude that in applications with restricted computational resources, halfspace, projection, simplicial and simplicial volume depths may be rather approximated in higher dimensions, while exact algorithms can still be used in the low-dimensional framework, \textit{e.g.} when computing time cuts of multivariate functional depths, or to assess the performance of approximation algorithms.
\begin{figure}[!h]
\begin{center}
\includegraphics[keepaspectratio=true,width = \textwnew, trim = 3mm 2mm 0mm 2mm, clip, page = 1]{depth_speed.pdf}\\
{\footnotesize{\color{red}\LBL} zonoid, {\color{red}\LBD} halfspace, {\color{green}\LBL} Mahalanobis, {\color{green}\LBD} spatial, {\color{black}\LBL} projection, {\color{blue}\LBL} simplicial, {\color{blue}\LBD} simplicial volume}
\caption{Calculation time of various depth functions, on the logarithmic time scale.}
\label{fig:depth_speed}
\end{center}
\end{figure}
\subsection{Maximum depth classifier}\label{ssec:depthMaxdepth}
To demonstrate the differing finite-sample behavior of the above depth notions and to construct a bridge to supervised classification, in this section we compare the depths in the frame of the maximum depth classifier. This is obtained by simply choosing the class in which $\boldsymbol{x}_0$ has the highest depth (breaking ties at random):
\begin{equation}
class(\boldsymbol{x}_0) = \argmax_{i\in\{1,...,q\}}\,D(\boldsymbol{x}_0|\boldsymbol{X_i}).
\end{equation}
\cite{GhoshC05b} have proven that its misclassification rate converges to the optimal Bayes risk if each $\boldsymbol{X}_i,\,i=1,...,q$, is sampled from a unimodal elliptically symmetric distribution having a common nonincreasing density function, a prior probability $\frac{1}{q}$, and differing in location parameter only (location-shift model), for halfspace, simplicial, and projection depths, and under additional assumptions for spatial and simplicial volume depths. Setting $q=2$, and $n=24,50,100,250,500,1000$, $n_i=n/2,\,i=1,2$, we sample $\boldsymbol{X}_i$ from a Student-$t$ distribution with location parameters $\mu_1=[0,0]$, $\mu_2=[1,1]$ and common scale parameter
$\Sigma=\left[ \begin{smallmatrix} 1 & 1 \\ 1 & 4 \\ \end{smallmatrix}\right]$, setting the degrees of freedom to $t=1, 5, 10, \infty$. Average error rates over 250 samples each checked on $1000$ observations are indicated in Figure~\ref{fig:maxdepthgraphs}. The testing observations were sampled inside the convex hull of the training set. The problem of outsiders is addressed in Section~\ref{sec:outsiders}. For $n=1000$, experiments have not been conducted with the simplicial depth due to high computation time.
As expected, with increasing $n$ and $t$ classification error and difference between various depths decrease. As the classes stem from elliptical family, depths accounting explicitly for ellipticity (Mahalanobis and spatial due to covariance matrix), symmetry of the data (projection), and also volume, form the error frontier. On the other hand, except for the projection depth, they are nonrobust and perform poorly for Cauchy distribution. While projection depth, even being approximated, behaves excellent in all the experiments, it may perform poorly if distributions of $\boldsymbol{X}_i$ retain asymmetry due to inability to reflect this.
\begin{figure}[!h]
\begin{center}
\includegraphics[keepaspectratio=true,width = \textwnew, trim = 3mm 2mm 0mm 2mm, clip, page = 1]{maxd_plots.pdf}\\
{\footnotesize{\color{red}\LBL} zonoid, {\color{red}\LBD} halfspace, {\color{green}\LBL} Mahalanobis, {\color{green}\LBD} spatial, {\color{black}\LBL} projection, {\color{blue}\LBL} simplicial, {\color{blue}\LBD} simplicial volume}
\caption{Average error rates of the maximum depth classifier with different data depths. The samples are simulated from the Student-$t$ distribution possessing $1$, $5$, $10$, and $\infty$ degrees of freedom.}
\label{fig:maxdepthgraphs}
\end{center}
\end{figure}
\section[Classification in the $DD$-plot]{Classification in the $DD$-plot}\label{sec:ddclass}
In Section~\ref{ssec:depthMaxdepth}, we have already considered the naive way of depth-based classification --- the maximum depth classifier. Its extension beyond the equal-prior location-shift model, \textit{e.g.} to account for differing shape matrices of the two classes, or unequal prior probabilities, is somewhat cumbersome, \textit{cf.}~\cite{GhoshC05a,CuiLY08}. A simpler way, namely to use the $DD$-plot (or, more general, a $q$-dimensional depth space), has been proposed by~\cite{LiCAL12}. For a training sample consisting of $\boldsymbol{X}_1,...,\boldsymbol{X}_q$, the depth space is constructed by applying the mapping $\mathbb{R}^d\rightarrow[0,1]^q\,:\,\boldsymbol{x}\mapsto\bigl(D(\boldsymbol{x}|\boldsymbol{X}_1),...,D(\boldsymbol{x}|\boldsymbol{X}_q)\bigr)$ to each of the observations. Then the classification is performed in this low-dimensional space of depth-extracted information, which, \textit{e.g.}, for $q=2$ is just a unit square. The core idea of the $DD\alpha$-classifier is the $DD\alpha$-separator, a fast heuristic for the $DD$-plot. This is presented in Section~\ref{ssec:ddclassAlpha}, where we slightly abuse the notation introduced before. This is done in an intuitive way for the sake of understandability and closeness to the implementation. Further, in Section~\ref{ssec:ddclassAlternatives} we discuss application of alternative techniques in the depth space.
\subsection[The DDalpha-separator]{The $DD\alpha$-separator}\label{ssec:ddclassAlpha}
The $DD\alpha$-separator is an extension of the $\alpha$-procedure to the depth space, see \cite{Vasilev03,VasilevL98}, also \cite{LangeM14}. It iteratively synthesizes the space of features, coordinate axes of the depth space or their (polynomial) extensions, choosing features minimizing a two-dimensional empirical risk in each step. The process of space enlargement stops when adding features does not further reduce the empirical risk. Here we give its comprehensive description. The detailed algorithm is stated right below.
Regard the two-class sample illustrated on Figure~\ref{fig:discretespace}, left, representing discretizations of the electrocardiogram curves. Explanation of the data is given in Section~\ref{ssec:introOutline}, we postpone the explanation of the discretization scheme till Section~\ref{sec:functions} and consider a binary classification in the $DD$-plot for the moment. Figure~\ref{fig:discretespace}, middle, represents the depth contours of each class computed using the spatial depth. The $DD$-plot is obtained as a depth mapping $(\boldsymbol{X}_1,\boldsymbol{X}_2)\mapsto\boldsymbol{Z}=\{\boldsymbol{z}_i=(D_{i,1},D_{i,2}),\,i=1,...,n_1+n_2\}$,
when the first class is indexed by $i=1,...,n_1$ and the second by $i=n_1+1,...,n_2$,
and writing $D_{spt}(\boldsymbol{x}_i|\boldsymbol{X}_1)$ (respectively $D_{spt}(\boldsymbol{x}_i|\boldsymbol{X}_2)$) by $D_{i,1}$ (respectively $D_{i,2}$) for shortness. Further, to enable for nonlinear separation in the depth space, but to employ linear discrimination in the synthesized subspaces, the kernel trick is applied. As the $DD\alpha$-separator explicitly works with the dimensions (space axis), a finite-dimensional resulting space is required.
We choose the space extension degree by means of a fast cross-validation,
which is performed over a small range and in the depth space only. The high computation speed of the $DD\alpha$-separator allows for this.
\begin{figure}[!h]
\renewcommand{}{0 17 29 58}
\begin{center}
\includegraphics[keepaspectratio=true,width = .325\textwnew, page = 1, Trim=, clip]{ddaplha_steps_discrete_space.pdf}
\includegraphics[keepaspectratio=true,width = .325\textwnew, page = 4, Trim=, clip]{ddaplha_steps_discrete_space.pdf}
\includegraphics[keepaspectratio=true,width = .325\textwnew, page = 46, Trim=, clip]{ddaplha_steps.pdf}
\caption{The discretized space (left), the depth contours with the separating rule (middle) and the $DD$-plot with the separating line in it (right), using spatial depth. Here
we denote the depth of a point w.r.t.~red and blue classes by $x$ and $y$, respectively.}
\label{fig:discretespace}
\end{center}
\end{figure}
We use polynomial extension of degree $p$, which results in $r = {p + q\choose q} - 1$ dimensions (by default, we choose $p$ among $\{1,2,3\}$ using 10-fold cross-validation); truncated series or another finitized basis of general reproducing kernel Hilbert spaces can be used alternatively. This extended depth space serves as the input to the $DD\alpha$-separator. For $q = 2$, and taking $p = 3$, one gets the extended depth space $\boldsymbol{Z}^{(p)}$ consisting of observations
$\boldsymbol{z}^{(p)}_i=(D_{i,1},~~D_{i,2},~~D_{i,1}^2,~~D_{i,1}\times D_{i,2},~~D_{i,2}^2,~~D_{i,1}^3,~~D_{i,1}^2\times D_{i,2},~~D_{i,1}\times D_{i,2}^2,~~D_{i,2}^3)\in\mathbb{R}^r$.
After initializations, on the \emph{1st step}, the $DD\alpha$-separator starts with choosing the pair of extended properties minimizing the empirical risk.
For this, it searches through all coordinate subspaces $\boldsymbol{Z}^{(k,l)}=\{\boldsymbol{z}^{(k,l)}_i\,|\,\boldsymbol{z}^{(k,l)}_i=(\boldsymbol{z}^{(p)}_{ik},\boldsymbol{z}^{(p)}_{il}),\,i=1,...,n_1+n_2\}$ for all $\,1\le k<l \le r$, \textit{i.e.}~all pairs of coordinate axis of $\boldsymbol{Z}^{(p)}$.
For each of them, the angle $\alpha_1^{(k,l)}$ minimizing the empirical risk is found
\begin{equation}\label{equ:alphaAlpha}
\alpha_1^{(k,l)} \in \argmin_{\alpha\in[0;2\pi)} \Delta^{(k,l)}(\alpha)
\end{equation}
with
\begin{equation}\label{equ:alphaPlane}
\Delta^{(k,l)}(\alpha) = \sum_{i=1}^{n_1} I(\boldsymbol{z}^{(p)}_{ik}\cos\alpha - \boldsymbol{z}^{(p)}_{il}\sin\alpha < 0) + \sum_{i = n_1 + 1}^{n_1 + n_2} I(\boldsymbol{z}^{(p)}_{ik}\cos\alpha - \boldsymbol{z}^{(p)}_{il}\sin\alpha > 0).
\end{equation}
For the regarded example, this is demonstrated in Figure~\ref{tab:gt} by the upper triangle of the considered subspaces. Computationally, it is reasonable to check only those $\alpha$ corresponding to (radial) intervals between points and to choose $\alpha_1^{(k,l)}$ as an average angle between two points from $\boldsymbol{Z}^{(k,l)}$ in case there is a choice, as it is implemented in procedure \textsl{GetMinError}.
Computational demand is further reduced by skipping uninformative pairs, \textit{e.g.}, if one feature is a power of another one and, therefore, the bivariate plot
is collapsed to a line, as shown in Figure~\ref{tab:gt}. Finally, a triplet is chosen:
\begin{equation}\label{equ:alphaRisk}
(\alpha_1^{(k^*,l^*)},k^*,l^*) \in \argmin_{1\le k<l \le r,\,\alpha\in[0;2\pi)} \Delta^{(k,l)}(\alpha),
\end{equation}
\textit{i.e.}~a two-dimensional coordinate subspace $\boldsymbol{Z}^{(k^*,l^*)}$ in which the minimal empirical risk over all such subspaces is achieved, and the corresponding angle $\alpha_1^{(k^*,l^*)}$ minimizing this. Among all the minimizing triplets (there may be several as empirical risk is discrete) it is reasonable to choose $k^*$ and $l^*$ with the smallest polynomial degree, the simplest model. Using $\alpha_1^{(k^*,l^*)}$, $\boldsymbol{Z}^{(k^*,l^*)}$ is convoluted to a real line
\begin{equation}\label{equ:alphaConvolute}
\boldsymbol{z}^{(1^*)}=\{z_i\,|\,z_i=\boldsymbol{z}^{(p)}_{ik^*}\cos\alpha_1^{(k^*,l^*)} - \boldsymbol{z}^{(p)}_{il^*}\sin\alpha_1^{(k^*,l^*)},\,i=1,...,n_1+n_2\},
\end{equation}
--- first feature of the synthesized space.
On each following $s$\emph{-step} ($s\ge 2$), the $DD\alpha$-separator proceeds as follows. The feature, obtained by the convolution on the previous $(s-1)$\emph{-step}, is coupled with each of the extended properties of the depth space, such that a space $\boldsymbol{Z}^{((s-1)^*,k)}=\{\boldsymbol{z}^{((s-1)^*,k)}_i\,|\,\boldsymbol{z}^{((s-1)^*,k)}_i=(\boldsymbol{z}^{((s-1)^*)}_{i},\boldsymbol{z}^{(p)}_{ik})$, $i=1,...,n_1+n_2\}$ is regarded, for all $k$ used in no convolution before. For each $\boldsymbol{Z}^{((s-1)^*,k)}$, $\Delta^{((s - 1)^*,k)}(\alpha_s^{(k)})$ and the corresponding empirical-risk-minimizing angle $\alpha_s^{(k)}$ are obtained using (\ref{equ:alphaAlpha}) and (\ref{equ:alphaPlane}). Out of all considered $k$, the one minimizing $\Delta^{((s - 1)^*,k)}(\alpha_s^{(k)})$ is chosen, as in (\ref{equ:alphaRisk}), and the corresponding $\boldsymbol{Z}^{((s-1)^*,k)}$ is convoluted to $\boldsymbol{z}^{(s^*)}$, as in (\ref{equ:alphaConvolute}).
The second part of Figure~\ref{tab:gt} illustrates a possible second step of the algorithm.
Here we present the algorithm of the $DD\alpha$-separator:
\bigskip
\textbf{The main procedure}
Input: $\tilde{\boldsymbol{X}} = \{\tilde{\boldsymbol{x}}_1,...,\tilde{\boldsymbol{x}}_n\}$, $\tilde{\boldsymbol{x}}_i\in\mathbb{R}^d$,
\\ \phantom{tabInput:s} $\{ y_1,..., y_n\}$, $y_i\in\{-1,1\}$ for all $i=1,...,m=m_{-1}+m_{+1}$.
\begin{enumerate}
\item ${\boldsymbol{X}} = {\tilde{\boldsymbol{X}}}^T = \{{\boldsymbol{x}}_1,...,{\boldsymbol{x}}_d\}$, ${\boldsymbol{x}}_i\in\mathbb{R}^n$.
\item Initialize arrays:
\begin{algenum}
\item array of available properties $\boldsymbol{P} \gets \{1..d\}$;
\item array of constructed features $\boldsymbol{F} \gets \emptyset$;
\item for a feature $f \in \boldsymbol{F}$ denote $f.p$ and $f.\alpha$ the number of the used property and the optimal angle.
\end{algenum}
\item \emph{1st step:} Find the first features:
\begin{algenum}
\item select optimal starting features considering all pairs from $\boldsymbol{P}$:
\item[] $(opt_1, opt_2, e_{min}, \alpha) = \arg\min_{g \in \boldsymbol{G}}g.e$ with \item[]
\mbox{ $\boldsymbol{G} = \{(p_1, p_2, e, \alpha):
(e, \alpha)=\text{\textsl{GetMinError}}({\boldsymbol{x}}_{p_1}, {\boldsymbol{x}}_{p_2}),
p_1, p_2 \in \boldsymbol{P}, p_1 < p_2 \}$}
\item $\boldsymbol{F} \gets \boldsymbol{F} \cup \{(opt_1,0), (opt_2,\alpha)\}$
\item $\boldsymbol{P} \gets \boldsymbol{P} \setminus \{opt_1, opt_2\}$
\item set current feature $f^\prime = {\boldsymbol{x}}_{opt_1}\times\cos(\alpha) + {\boldsymbol{x}}_{opt_2}\times\sin(\alpha)$
\end{algenum}
\item \emph{Following steps:} Search an optimal feature space while empirical error rate decreases\\
{\bf while} $e_{min}\ne 0$ and $\boldsymbol{P}\ne\emptyset$ {\bf do}
\begin{algenum}
\item select next optimal feature considering all properties from $\boldsymbol{P}$:
\item[] $(opt, \tilde{e}_{min}, \alpha) = \arg\min_{g \in \boldsymbol{G}}g.e$ with \item[]
$\boldsymbol{G} = \{(p, e, \alpha):
(e, \alpha)=\text{\textsl{GetMinError}}(f^\prime, {\boldsymbol{x}}_{p}),
p \in \boldsymbol{P} \}$
\item Check if the new feature improves the separation:
\item[] {\bf if} $\tilde{e}_{min}<{e}_{min}$ {\bf then}
\begin{algenum}
\item[] ${e}_{min}=\tilde{e}_{min}$
\item[] $\boldsymbol{F} \gets \boldsymbol{F} \cup (opt, \alpha)$
\item[] $\boldsymbol{P} \gets \boldsymbol{P} \setminus opt$
\item[] update current feature $f^\prime = f^\prime\times\cos(\alpha) + {\boldsymbol{x}}_{opt}\times\sin(\alpha)$
\end{algenum
\item[]{\bf else} \\ \tab {\bf break}
\end{algenum}
\item Get the normal vector of the separating hyperplane:
\begin{algenum}
\item Declare a vector $\boldsymbol{r} \in \mathbb{R}^d$, $\boldsymbol{r}_i=0$ for all $i=1,...,d$. Set $a = 1$.
\item Calculate the vector components as
$\boldsymbol{r}_{{\boldsymbol{F}_i}.p} = \prod_{j=i+1}^{\sharp\boldsymbol{F}} \Bigl(\cos({\boldsymbol{F}_j}.\alpha) \Bigr) \sin({\boldsymbol{F}_i}.\alpha)$:
\item[] {\bf for all} $i\in \{\sharp\boldsymbol{F}..2\}$ {\bf do}
\begin{algenum}
\item[] $\boldsymbol{r}_{{\boldsymbol{F}_i}.p} = a\times\sin({\boldsymbol{F}_i}.\alpha)$
\item[] $a = a\times\cos({\boldsymbol{F}_i}.\alpha)$
\end{algenum
\item[] $\boldsymbol{r}_{\boldsymbol{F}_1.p} = a$
\item Project the points on the ray: $\boldsymbol{p}_i.y = y_i$, $\boldsymbol{p}_i.x = \boldsymbol{r}\cdot\tilde{\boldsymbol{x}}_i$
\item Sort $\boldsymbol{p}$ w.r.t.~$\boldsymbol{p}_{\cdot}.x$ in ascending order.
\item Count the cardinalities before the separation plane
\\ $m_{l-} = \sharp\{i:\boldsymbol{p}_i.y = -1, \boldsymbol{p}_i.x\le0\}$, \\$m_{l+} = \sharp\{i:\boldsymbol{p}_i.y = +1, \boldsymbol{p}_i.x\le0\}$
\item Count the errors
\\$e_{-} = m_{l+} + m_{-} - m_{l-}$, \\$e_{+} = m_{l-} + m_{+} - m_{l+}$
\item {\bf if} $e_{-} > e_{+}$ {\bf then}
\\ \tab $\boldsymbol{r} \gets -\boldsymbol{r}$
\end{algenum}
\end{enumerate}
Output: the normal vector of the separating hyperplane $r$.
\bigskip
\textbf{Procedure \textsl{GetMinError}}
Input: current feature $f\in \mathbb{R}^n$, property $x \in \mathbb{R}^n$.
\begin{enumerate}
\item Obtain angles:
\begin{algenum}
\item Calculate $\alpha_i = \arctan\frac{{x}_{i}}{{f}_{j}}$, $i=1,...,n$, with $\arctan\frac{0}{0}=0$.
\item Aggregate angles into set $\mathcal{A}$. Denote $\mathcal{A}_i.\alpha=\alpha_i$ and $\mathcal{A}_i.y=y_i$ the angle and the pattern of the corresponding point. Set $\mathcal{A}_i.y$ to 0 for the points having both $x_i=0$ and $f_i=0$.
\item Sort $\mathcal{A}$ w.r.t.~$\mathcal{A}_{\cdot}.\alpha$ in ascending order.
\end{algenum}
\item Look for the optimal threshold:
\begin{algenum}
\item Define $i_{opt} = \arg\max_i \left(|\sum_1^{i}\mathcal{A}_i.y| + |\sum_{i+1}^{n}\mathcal{A}_i.y| \right)$ as the place of the optimal threshold
and $e_{min} = n-\max_i \left(|\sum_1^{i}\mathcal{A}_i.y| + |\sum_{i+1}^{n}\mathcal{A}_i.y| \right)$ as the minimal number of incorrectly classified points
\item Define the optimal angle $\alpha_{opt} = \frac{1}{2}(\mathcal{A}_{i_{opt}+1}.\alpha + \mathcal{A}_{i_{opt}+2}.\alpha) - \frac{\pi}{2}$.
\end{algenum}
\end{enumerate}
Output: min error $e_{min}$, optimal angle $\alpha_{opt}$
\bigskip
\input{table_alpha}
From the practical point of view, the routine $DD\alpha$-separator has high computation speed as in each plane it has the complexity of the quick-sort procedure: $O\bigl(\sum_{i=1}^q n_i \log(\sum_{i=1}^q n_i)\bigr)$.
While minimizing empirical risk in two-dimensional coordinate subspaces and due to the choice of efficient for classification features, the $DD\alpha$-separator tends to be \emph{close to} the optimal \emph{risk-minimizing} hyperplane in the extended space.
To a large extent, this explains the performance of the $DD\alpha$-procedure on finite samples.
The robustness of the procedure is twofold: First, \emph{regarding points}, as the depth-space is compact, the outlyingness of the points in it is restricted, and the $DD\alpha$-separator is robust due to its risk-minimizing nature, \textit{i.e.}~by the discrete (zero-or-one) loss function. And second, \emph{regarding features}, the separator is not entirely driven by the exact points' location, but accounts for importance of features of the (extended) depth space. By that, the model complexity is kept low; in practice a few features are selected only, see, \textit{e.g.}, Section~5.2 of~\cite{MozharovskyiML15}.
For theoretical results on the $DD\alpha$-procedure the reader is referred to Section~4 of~\cite{LangeMM14a}. \cite{MozharovskyiML15} provide an extensive comparative empirical study of its performance with a variety of data sets and for different depth notions and outsider treatments, while \cite{LangeMM14b} conduct a simulation study on asymmetric and heavy-tailed distributions.
\subsection[Alternative separators in the $DD$-plot]{Alternative separators in the $DD$-plot}\label{ssec:ddclassAlternatives}
Besides the $DD\alpha$-separator, the package \pkg{ddalpha} allows for two alternative separators in the depth space: a polynomial rule and the $k$-nearest-neighbor ($k$NN) procedure.
When \cite{LiCAL12} introduce the $DD$-classifier, they suggest to use a polynomial of certain degree passing through the origin of the $DD$-plot to separate the two training classes. Based on the fact that by choosing the polynomial order appropriately the empirical risk can be approximated arbitrarily well, they prove the consistency of the $DD$-classifier for a wide range of distributions including some important cases of the elliptically symmetric family. In practice, the minimal error is searched by smoothing the empirical loss with a logistic function and then optimizing the parameter of this function. This strategy has sources of instability such as choice of the smoothing constant and multimodality of the loss function. The authors (partially) solve the last issue by varying the starting point for optimization and multiply running the entire procedure, which increases computation time. For theoretical derivations and implementation details see Sections~4 and~5 of \cite{LiCAL12}. For a simulation comparison of the polynomial rule in the $DD$-plot and the $DD\alpha$-separator see Section~5 of \cite{LangeMM14a}.
In his PhD-thesis, \cite{Vencalek11} suggests to perform the $k$NN classification in the depth space, and proves its consistency for elliptically distributed classes with identical radial densities. For theoretical details and a simulation study see Sections~3.4.3 and~3.7 of \cite{Vencalek11}, respectively. It is worth to notice that the $k$NN-separator has another advantage --- it is directly extendable to more than two classes.
\section[Outsiders]{Outsiders}\label{sec:outsiders}
For a number of depth notions like halfspace, zonoid, or simplicial depth, the depth of a point vanishes beyond the convex hull of the data.
This leads to the problem that new points (to be classified) lying beyond the convex hull of each of the training classes have depth zero w.r.t.~all of them. By that, they are depth-mapped to the origin of the $DD$-plot, and thus cannot be readily classified. We call these points \emph{outsiders} \citep{LangeMM14a}.
\begin{figure}[!h]
\begin{center}
\includegraphics[keepaspectratio=true,width = \textwnew, trim=4mm 15mm 4mm 25mm, clip]{outsiders.pdf}
\caption{Points to be classified (green) in the original (left) and depth (right) space.}
\label{fig:outs_demo}
\end{center}
\end{figure}
Regard Figure~\ref{fig:outs_demo}, where three green points are to be classified. Point ``1'' has positive depth in both classes, and based on its location in the $DD$-plot will be assigned to the less scattered ``red'' class. Point ``2'' has zero depth in the ``red'' class, but a positive one in the more scattered ``blue'' class, to which it will be assigned based on the classification rule in the $DD$-plot. Point ``3'' on the other hand has zero depth w.r.t.~both training classes, and thus classification rule in the $DD$-plot is helpless. Nevertheless, visually it clearly belongs to the ``blue'' class, and most probably would be correctly classified by a very simple classifier, say a poorly tuned $k$NN (\textit{e.g.} 1NN). The suggestion thus is to apply an additional fast classifier to the outsiders.
The \proglang{R}-package \pkg{ddalpha} implements a number of outsider treatments: linear (LDA) and quadratic (QDA) discriminant analysis, $k$NN, maximum depth classifier based on Mahalanobis depth; and additionally random classification or identification of outsiders for statistical analysis or passing to another procedure. For the same experimental setting as in Section~\ref{ssec:depthMaxdepth}, we contrast these treatments in Figure~\ref{fig:outs_graphs}, comparing classification errors on outsiders only. One can see that for the heavy-tailed Cauchy distribution, where classes may be rather mixed, no outsider treatment performs significantly better than random assignment. The situation improves with increasing number of degrees of freedom of the Student-$t$ distribution, with LDA forming the classification error frontier, as the classes differ in location only. On the other hand, with increasing $n_i$, difference between the treatment becomes negligible. For an extensive comparative study of different outsider treatments the reader is referred to \cite{MozharovskyiML15}.
\begin{figure}[!h]
\begin{center}
\includegraphics[keepaspectratio=true,width = \textwnew, trim = 3mm 2mm 0mm 2mm, clip, page = 2]{maxd_plots.pdf}\\
{\small{\color{black}\LBL} number of outsiders, {\color[gray]{0.5}\LBL} random, {\color{blue}\LBL} LDA, {\color[rgb]{1,0.5,0}\LBL} QDA, {\color{green}\LBL} k-NN, {\color{red}\LBL} Mahalanobis max depth}
\caption{Error rates of various outsiders treatment. Only outsiders are classified.}
\label{fig:outs_graphs}
\end{center}
\end{figure}
If outsiders pose a serious problem, one can go for a nowhere-vanishing depth. But in general, the property of generating outsiders should not necessarily be seen as a shortfall, as it allows for additional information when assessing the configured classifier or a data point to be classified. If too many points are identified as outsiders (what can be checked by a validation procedure), this may point onto inappropriate tuning. On the other hand, if outsiders appear extremely rarely in the classification phase (or, \textit{e.g.}, during online learning), an outsider may be an atypical observation not fitting to the data topology in which case one may not want to classify it at all but rather label indicatively.
\section[An extension to functional data]{An extension to functional data}\label{sec:functions}
Similar to Section~\ref{sec:ddclass}, consider a binary classification problem in the space of real valued functions defined on a compact interval, which are continuous and smooth everywhere except for a finite number of points, \textit{i.e.}~given two classes of functions: $\boldsymbol{\mathcal{F}}_1=\{f_1,...,f_{n_1}\}$ and $\boldsymbol{\mathcal{F}}_2=\{f_1,...,f_{n_2}\}$, again indexing observations by $i=1,...,n_1,n_1 + 1,...,n_1 + n_2$ for convenience. (An aggregation scheme extends this binary classification to the multiple one.) The natural extension of the depth-based classification to the functional setting consists in defining a proper depth transform $(\boldsymbol{\mathcal{F}}_1,\boldsymbol{\mathcal{F}}_2)\mapsto\boldsymbol{Z}=\{\boldsymbol{z}_i=(D(f_i|\boldsymbol{\mathcal{F}}_1),D(f_i|\boldsymbol{\mathcal{F}}_2)),\,i=1,...,n_1+n_2\}$ similar to that in Section~\ref{sec:ddclass}. For this, a proper functional depth should be employed \citep[see][and references therein for an overview]{MoslerP12,NietoReyesB16}, followed by the suitable classification technique in the (finite dimensional) depth space. As the functional data depth reduces space dimensionality from infinity to one, the final performance is sensitive to the choice of the depth representation and of the finite-dimensional separator, and thus both constituents should be chosen very carefully. Potentially, this lacks quantitative flexibility because of the finite set of existing components. Nevertheless, in many cases this solution provides satisfactory results; see a comprehensive discussion by \cite{CuestaAlbertosFBOdlF15} with experimental comparisons involving a number of functional depth notions and $q$-dimensional classifiers, as well as their implementation in the \proglang{R}-package \pkg{fda.usc}. Corresponding functional depth procedures can also be used with R-package \pkg{ddalpha}, see Section~\ref{sec:application} for a detailed explanation.
\pkg{ddalpha} suggests two implementations of the strategy of immediate functional data projection onto a finite-dimensional space with further application of a multivariate depth-based classifier: componentwise classification by \cite{DelaigleHB12} and $LS$-transform proposed by \cite{MoslerM15}. Both methodologies allow to control for the quality of classification in a quantitative way (\textit{i.e.}~by tuning parameters) when constructing the multivariate space, which in addition enables consistency derivations. For the first one the reader is referred to the literature; the second one we present right below.
In application, functional data is usually given in a form of discretely observed paths $\boldsymbol{\tilde f}_i = \left[f_{i}(t_{i1}), f_{i}(t_{i2}), ..., f_{i}(t_{iN_i})\right]$, which are the measurements at ordered (time) points $t_{i1}< t_{i2}< ... <t_{iN_i}$, $i=1,...,n_1+n_2$, not necessarily equidistant nor same for all $i$. Fitting these to a basis is avoided as the choice of such a basis turns out to be crucial for classification and thus should better not be independently selected prior to it. Instead, a simple scheme is suggested based on integrating linearly extrapolated data and their derivatives over a chosen number of intervals. Let $\min_i{t_{i1}}=0$ and let $T=\max_i{t_{iN_i}}$, then one obtains the following finite-dimensional transform:
{\footnotesize
\begin{equation}\label{equ:average}
{\hat f}_{i}\mapsto\boldsymbol{x}_{i} = \Bigl[\int_0^{T/L} {\hat f}_{i}(t) dt, \dots, \int_{T(L-1)/L}^{T} {\hat f}_{i}(t) dt, \int_0^{T/S} {\hat f'}_{i}(t) dt, \dots, \int_{T(S-1)/S}^{T} {\hat f'}_{i}(t) dt \Bigr]\,,
\end{equation}
}
with ${\hat f}_{i}(t)$ being the function obtained by connecting the points $(t_{ij},\, f_{i}(t_{ij})), j=1,\dots, N_i$ with line segments and setting ${\hat f}_{i}(t)=f_{i}(t_{i1})$ when $0\le t\le t_{i1}$ and ${\hat f}_{i}(t)=f_{i}(t_{ik_i})$ when $t_{iN_i}\le t\le T$, ${\hat f'}_{i}(t)$ being its derivative, and $L,S\ge 0$, $L+S\ge 2$ being integers. $L$ and $S$ are the numbers of intervals of equivalent length to integrate over the location and the slope of the function, and have to be tuned. One can use intervals of different length or take into account higher-order derivatives (constructed as differences, say), but the suggested way appears to be simple and flexible enough. Moreover it does not introduce any spurious information. The set of considered $LS$-pairs can be chosen on the basis of some prior knowledge about the nature of the functions or just by properly restricting the dimension of the constructed space by $d_{min}\le L + S \le d_{max}$. Cross-validation is then used to choose the best $LS$-pair. \pkg{ddalpha} suggests to reduce the set of cross-validated $LS$-pairs by employing the Vapnik-Chervonenkis bound. The idea behind is that, while being conservative, the bound can still provide insightful ordering of the $LS$-pairs, especially in the case when the empirical risk and the bound have the same order of magnitude.
Given a set of considerable pairs $\mathcal{S}=\{(l_i,s_i)|i=1,...,N_{ls}\}$, for each its element calculate the Vapnik-Chervonenkis bound \citep[see][for this particular derivation]{MoslerM15}
\begin{equation}\label{equ:Vapnik}
b^{VC}_i = \epsilon\left(\boldsymbol{c},\boldsymbol{\mathcal{\hat F}}_1^{(l_i,s_i)},\boldsymbol{\mathcal{\hat F}}_2^{(l_i,s_i)}\right) + \sqrt{\frac{\ln{2\sum_{k=0}^{l_i + s_i - 1}{n_1 + n_2 - 1 \choose k}} - \ln{\eta}}{2(n_1 + n_2)}},
\end{equation}
where $\epsilon\left(\boldsymbol{c},\boldsymbol{\mathcal{\hat F}}_1^{(l_i,s_i)},\boldsymbol{\mathcal{\hat F}}_2^{(l_i,s_i)}\right)$ is the empirical risk achieved by a linear classifier $\boldsymbol{c}$ on the data transformed according to (\ref{equ:average}) with $L=l_i$, $S=s_i$ and $1-\eta$ is the chosen reliability level. In \pkg{ddalpha} we set $\eta=\frac{1}{n_1 + n_2}$, and choose $\boldsymbol{c}$ to be the LDA for its simplicity and speed.
Then a subset $\mathcal{S}^{CV}\subset \mathcal{S}$ is chosen possessing the smallest values of $b^{VC}_i$: $\bigl((l_j,s_j)\in \mathcal{S}^{CV}, (l_k,s_k)\in \mathcal{S}\setminus \mathcal{S}^{CV}\bigr)$ $\Rightarrow$ $(b^{VC}_j < b^{VC}_k)$, and cross-validation is performed over all $(l,s)\in \mathcal{S}^{CV}$. For the subsample referenced in introduction, the functions' levels and slopes are shown in Figure~\ref{fig:funcdata}; the $LS$-representation is selected by reduced cross-validation due to (\ref{equ:average}) having $(L,S)=(1,1)$, and is depicted in Figure~\ref{fig:discretespace}, left.
\section{Usage of the package}\label{sec:application}
The package \pkg{ddalpha} is a structured solution that provides computational machinery for a number of depth functions and classifiers for multivariate and functional data. It also allows for user-defined depth functions and separators in the $DD$-plot (further $DD$-separators). The structure of the package is presented in Figure~\ref{fig:ddalpha_structure}.
\begin{figure}[!h]
\begin{center}
\includegraphics[keepaspectratio=true,width = \textwnew, trim = 30mm 8mm 57mm 61mm, clip, page = 1]{ddalpha_full_structure.pdf}
\caption{The structure of the package}
\label{fig:ddalpha_structure}
\end{center}
\end{figure}
\subsection{Basic functionality}
Primary aims of the package are calculation of data depth and depth-classification.
\textbf{\emph{Data depth}} is calculated by calling
\begin{Code}
depth.<depthName>(x, data, ...),
\end{Code}
where \code{data} is a matrix with each row being a $d$-variate point, and \code{x} is a matrix of objects whose depth is to be calculated. Additional arguments (\code{...}) differ between depth notions. The output of the function is a vector of depths of points from \code{x}.
Most of the depth functions possess both exact and approximative versions that are toggled with parameters \code{exact} and \code{method}, see Table~\ref{tab:depths_impl}.
The exact algorithms of Mahalanobis, spatial, and zonoid depths are very fast and thus exclude the need of approximation.
Mahalanobis and spatial depths use either traditional moment or MCD estimates of mean and covariance matrix.
Methods \code{random} for projection depth and \code{Sunif.1D} for halfspace depth approximate the depth as the minimum univariate depth of the data projected on \code{num.directions} directions uniformly distributed on $S^{d-1}$. The exact algorithms for the halfspace depth implement the framework described in Section~\ref{ssec:depthNotions}, where the dimensionality $k$ of the combinatorial space is specified as follows: $k=1$ for method \code{recursive}, $k=d-2$ for \code{plane} and $k=d-1$ for \code{line}, see additionally~\cite{DyckerhoffM16}.
The second approximating algorithm for projection depth is \code{linearize} --- the Nelder-Mead method for function minimization, taken from \cite{NelderM65} and originally implemented in \proglang{R} by Subhajit Dutta.
For simplicial and simplicial volume depths, parameter \code{k} specifies the number (if \code{k} $>1$) or portion (if $0<$ \code{k} $<1$) of simplices chosen randomly among all possible simplices for approximation.
\newpage
\begin{table}[!th]
\centering
{\small
\begin{tabular}{llll}
\hline
Depth & Exact & Approximative & Parameter\\
\hline
Mahalanobis & \code{moment} & & \code{mah.estimate}\\
& \code{MCD} & & \\
spatial & \code{moment} & & \code{mah.estimate}\\
& \code{MCD} & & \\
& \code{none} & & \\
projection & & \code{random} & \code{method}\\
& & \code{linearize} & \\
halfspace & \code{recursive} & \code{Sunif.1D} & \code{method}\\
& \code{plane} & & \\
& \code{line} & & \\
simplicial & + & + & \code{exact}\\
simplicial volume& + & + & \code{exact}\\
zonoid & + & \\
\hline
\end{tabular}}
\caption{Implemented depth algorithms}\label{tab:depths_impl}
\end{table}
In addition, calculation of the entire $DD$-plot at once is possible by
\begin{Code}
depth.space.<depthName>(data, cardinalities, ...),
\end{Code}
where the matrix \code{data} consists of $q$ stacked training classes, and \code{cardinalities} is a vector containing numbers of objects in each class. The method returns a matrix with $q$ columns representing the depths of each point w.r.t.~each class.
\textbf{\emph{Classification}} can be performed either in two steps --- training the classifier with \code{ddalpha.train} and using it for classification in \code{ddalpha.classify}, or in one step --- by function \code{ddalpha.test(learn, test, ...)} that trains the classifier with \code{learn} sample and checks it on the \code{test} one. Other parameters are the same as for function \code{ddalpha.train} and are described right below.
Function \code{ddalpha.train} is the main function of the package. Its structure is shown on the right part of Figure~\ref{fig:ddalpha_structure}.
\begin{Code}
ddalpha.train(data, depth = "halfspace", separator = "alpha",
outsider.methods = "LDA", outsider.settings = NULL,
aggregation.method = "majority",
use.convex = FALSE,
seed = 0, ...)
\end{Code}
The notion of the depth function and the $DD$-separator are specified with the parameters \code{depth} and \code{separator}, respectively.
Parameter \code{aggregation.method} determines the method applied to aggregate outcomes of binary classifiers during multiclass classification. When \code{"majority"}, $q(q-1)/2$ binary one-against-one classifiers are trained, and for \code{"sequent"}, $q$ binary one-against-all classifiers are taught. During classification, the results are aggregated using the majority voting, where classes with larger proportions in the training sample are preferred when tied (by that implementing both aggregating schemes at once).
Additional parameters of the chosen depth function and $DD$-separator are passed using the dots, and are described in the help sections of the corresponding \proglang{R}-functions.
Also, the function allows to use a pre-calculated $DD$-plot by choosing \code{depth = "ddplot"}.
For each depth function and depth-separator, a validator is implemented --- a special \proglang{R}-function that specifies the default values and checks the received parameters allowing by that definition of custom depths and separators; see Section~\ref{ssec:custom_dsss} for details.
\textbf{\emph{Outsider treatment}} is a supplementary classifier for data that lie outside the convex hulls of all $q$ training classes. It is only needed during classification when the used data depth produces outsiders or obtains zero values in the neighborhood of the data.
Parameter \code{use.convex} of \code{ddalpha.train} indicates whether outsiders should be determined as the points not contained in any of the convex hulls of the classes from the training sample (\code{TRUE}) or those having zero depth w.r.t.~each class from the training sample (\code{FALSE}); the difference is explained by the depth approximation error.
The following methods are available:
\code{"LDA"}, \code{"QDA"} and \code{"kNN"}; affine-invariant $k$NN (\code{"kNNAff"}), \textit{i.e.}~$k$NN with Euclidean distance normalized by the pooled covariance matrix, suited only for binary classification and using aggregation with multiple classes and not accounting for ties, but very fast; maximum Mahalanobis depth classifier (\code{"depth.Mahalanobis"}); equal and proportional randomization (\code{"RandEqual"} and \code{"RandProp"}) and ignoring (\code{"Ignore"}) --- a string ``Ignored'' is returned for the outsiders.
Outsider treatment is set by means of parameters \code{outsider.methods} and \code{outsider.settings} in \code{ddalpha.train}. Multiple methods may be trained and then the particular method is selected in \code{ddalpha.classify} by passing its name to
parameter \code{outsider.method}.
Parameter \code{outsider.methods} of \code{ddalpha.train} accepts a vector of names of basic outsider methods that are applied with the default settings.
Parameter \code{outsider.settings} allows to train a list of outsider treatments, whose elements specify the names of the methods (used in \code{ddalpha.classify} later) and their parameters.
\textbf{\emph{Functional classification}} is performed with functions \code{ddalphaf.train} implementing $LS$-transform \citep{MoslerM15} and \code{compclassf.train} implementing componentwise classification \citep{DelaigleHB12}.
\begin{Code}
ddalphaf.train(dataf, labels,
adc.args = list(instance = "avr",
numFcn = -1,
numDer = -1),
classifier.type = c("ddalpha", "maxdepth",
"knnaff", "lda", "qda"),
cv.complete = FALSE,
maxNumIntervals = min(25, ceiling(length(dataf[[1]]$args)/2)), $ closing dollar, MUST NOT BE PRINTED
seed = 0,
...)
\end{Code}
\begin{Code}
compclassf.train(dataf, labels,
to.equalize = TRUE,
to.reduce = TRUE,
classifier.type = c("ddalpha", "maxdepth",
"knnaff", "lda", "qda"),
...)
\end{Code}
In both functions, \code{dataf} is a list of functional observations, each having two vectors: \code{"args"} for arguments sorted in ascending order and \code{"vals"} for the corresponding functional evaluations;
\code{labels} is a list of class labels of the functional observations;
\code{classifier.type} selects the classifier that separates the finitized data, and additional parameters are passed to this selected classifier with dots.
In the componentwise classification, \code{to.equalize} specifies whether the data is adjusted to have equal (the largest) argument interval, and \code{to.reduce} indicates whether the data has to be projected onto a low-dimensional space via the principal components analysis (PCA) in case its affine dimension after finitization is lower than expected.
(Both parameters are recommended to be set true.)
The $LS$-transform converts functional data into multidimensional ones by averaging over intervals or evaluating values on equally-spaced grid for each function and its derivative on $L$
(respectively $S$) equal nonoverlapping covering intervals.
The dimension of the multivariate space then equals $L+S$.
Parameter \code{adc.args} is a list that specifies: \code{instance} --- the type of discretization of the functions having values \code{"avr"} for averaging over intervals of the same length
and \code{"val"} for taking values on equally-spaced grid;
\code{numFcn} ($L$) is the number of function intervals, and \code{numDer} ($S$) is the number of first-derivative intervals.
The parameters $L$ and $S$ may be set explicitly or may be automatically cross-validated. The cross-validation is turned on by setting \code{numFcn = -1} and \code{numDer = -1}, or by passing a list of \emph{adc.args} objects to \code{adc.args} --- the range of $(L,S)$-pairs to be checked. In the first case all possible pairs of $L$ and $S$ are considered up to the maximal dimension that is set in \code{maxNumIntervals}, while in the latter case only the pairs from the list are considered. The parameter \code{cv.complete} toggles the complete cross-validation; if \code{cv.complete} is set to false the Vapnik-Chervonenkis bound is applied, which enormously accelerates the cross-validation, as described in \cite{MoslerM15} in detail.
The optimal values of $L$ and $S$ are stored in the \emph{ddalphaf} object, that is returned from \code{ddalphaf.train}.
\subsection{Custom depths and separators}\label{ssec:custom_dsss}
As mentioned above, the user can amplify the existing variety by defining his own depth functions and separators. Custom depth functions and separators are defined by implementing three functions: parameters validator, learning, and calculating functions, see Tables~\ref{tab:custom_depth} and~\ref{tab:custom_separator}.
Usage examples are found in the manual of the package \pkg{ddalpha}.
\input{tables_custom}
\emph{Validator} is a nonmandatory function that validates the input parameters and checks if the depth calculating procedure is applicable to the data. All the parameters of a user-defined depth or separator must be returned by a validator as a named list, otherwise they will not be saved in the \code{ddalpha} object.
\textbf{\emph{Definition of a custom depth function}} is done as follows:
The \emph{depth-training function} \code{.<name>_learn(ddalpha)}
calculates any data-based statistics that the depth function needs (\textit{e.g.}, mean and covariance matrix for Mahalanobis depth) and then calculates the depths of the training classes, \textit{e.g.}, by calling for each pattern $i$ the \emph{depth-calculating function} \code{.<name>_depths(ddalpha, objects = ddalpha$patterns[[i]]$points)} that calculates the depth of each point in \code{objects} w.r.t.~each pattern in \code{ddalpha} and returns a matrix with $q$ columns.
The learning function returns a \code{ddalpha} object, where the calculated statistics and parameters are stored. All stored objects, including the parameters returned by the validator, are accessible through the \code{ddalpha} object, on each stage.
After having defined these functions, the user only has to specify \code{depth = "<name>"} in \code{ddalpha.train} and pass the required parameters there. (The functions are then linked via the \code{match.fun} method.)
\textbf{\emph{Definition of a custom separator}} is similar. Recall that there exist binary separators applicable to two classes, and multiclass ones that separate more than two classes at once. In case if the custom method is binary, the package takes care of the voting procedures, and the user only has to implement a method that separates two classes.
The training method for a \emph{binary separator} \code{.<name>_learn(ddalpha, index1, index2, depths1, depths2)} accepts the depths of the objects w.r.t.~two classes and returns a trained classifier.
A \emph{multiclass separator} has to implement another interface: \code{.<name>_learn(ddalpha)}, accessing the depths of the different classes via \code{ddalpha$patterns[[i]]$depths}.
The binary classifier can utilize the whole depth space (\textit{i.e.}~depths w.r.t.~other classes than the two currently under consideration) to get more information like the $\alpha$-separator does, or restrict to the $DD$-plot w.r.t.~the two given classes like the polynomial separator, by accessing \code{depths1} and \code{depths2} matrices.
The \emph{classifying function} \code{.<name>_classify(ddalpha, classifier, objects)} accepts the previously trained \code{classifier} and the depths of the objects that are classified.
For a binary classifier, the indices of the currently classified patterns are accessible as \code{classifier$index1} and \code{classifier$index2}.
A binary classifier shall return a vector with positive values for the objects from the first class, and the multiclass classifier shall assign to each object to be classified the index of the corresponding pattern in \code{ddalpha}.
Similarly to the depth function, the defined separator is accessible by \code{ddalpha.train} by specifying \code{separator = "<name>"}. If a nonbinary method is used, it is important to set \code{aggregation.method = "none"} or (preferred but more complicated) to return \code{ddalpha$methodSeparatorBinary = F} from the validator, otherwise the method will be treated as a binary one, as by default \code{aggregation.method = "majority"}.
\subsection{Additional features}\label{ssec:addFeatures}
A number of additional functions are implemented in the package to facilitate assessing quality and time of classification, handle multimodally distributed classes, and visualize depth statistics.
\textbf{\emph{Benchmark procedures}} implemented in the package allow for estimating expected error rate and training time:
\begin{Code}
ddalpha.test(learn, test, ...)
ddalpha.getErrorRateCV(data, numchunks = 10, ...)
ddalpha.getErrorRatePart(data, size = 0.3, times = 10, ...)
\end{Code}
The first function trains the classifier on the \code{learn} sample, checks it on the \code{test} one, and reports the error rate, the training time and other related values such as the numbers of correctly and incorrectly classified points, number of ignored outsiders, \textit{etc}.
The second function performs a cross-validation procedure over the given data. On each step, every \code{numchunks}\textit{th} observation is removed from the data, the classifier is trained on these data and tested on the removed observations. The procedure is performed until all points are used for testing. Setting \code{numchunks} to $n$ leads to the leave-one-out cross-validation (=jackknife) that is a consistent estimate of the expected error rate. The procedure returns the error rate, \textit{i.e.}~the total number of incorrectly classified objects divided by the total number of objects.
The third function performs a benchmark procedure by partitioning the given data. On each of \code{times} steps, randomly picked \code{size} observations are removed from the data, the classifier is trained on these data and tested on the removed observations.
The outputs of this function are the vector of errors, their mean and standard deviation. Additionally, both functions report mean training time and its standard deviation.
In all three functions, dots denote the additional parameters passed to \code{ddalpha.train}.
Benchmark procedures may be used to \emph{tune the classifier} by setting different values and assessing the error rate.
The function \code{ddalpha.test} is more appropriate for simulated data, while the two others are more suitable for subsampling learning with real data and testing sequences from it.
Analogs of these procedures for a functional setting are present in the package as well:
\begin{Code}
ddalphaf.test(learn, learnlabels, test, testlabels, disc.type, ...)
ddalphaf.getErrorRateCV(dataf, labels, numchunks, disc.type, ...)
ddalphaf.getErrorRatePart(dataf, labels, size, times, disc.type, ...)
\end{Code}
The discretization scheme is chosen with parameter \code{disc.type} setting it to \code{"LS"} or \code{"comp"}.
Note that these procedures are made to assess the error rates and the learning time for a single set of parameters. If the $LS$-transform is used, the parameters $L$ and $S$ shall be explicitly set with \code{adc.args} rather then cross-validated.
\textbf{\emph{Several approaches reflecting multimodality}} of the underlying distribution are implemented in the package.
These methods appear to be useful if the data substantially deviate from elliptical symmetry (\textit{e.g.} having nonconvex or nonconnected support) and the classification based on a global depth fails to achieve close to optimal error rates.
The methods need more complicated and fine parameter tuning, whose detailed description we leave to the corresponding articles.
{\emph{Localized spatial depth}} and a classifier based on it, proposed by \cite{DuttaG15}, can be seen as a $DD$-classifier.
The global spatial depth calculates the average of the unit vectors pointing from the points from $\boldsymbol{X}$ in direction $\boldsymbol{z}$. We rewrite (\ref{equ:sptDepth}) denoting $\boldsymbol{t}_i = \boldsymbol{\Sigma}^{-\frac{1}{2}}(\boldsymbol{X})(\boldsymbol{z} - \boldsymbol{x}_i$)
$$ D_{spt}(\boldsymbol{z}|\boldsymbol{X})=1-\Bigl\|\frac{1}{n}\sum_{i=1}^n \boldsymbol{v}\bigl(\boldsymbol{t}_i\bigr)\Bigr\|.$$
The local version is obtained by kernelizing the distances
$$D_{Lspt}(\boldsymbol{z}|\boldsymbol{X})=\Bigl\|\frac{1}{n}\sum_{i=1}^n K_h(\boldsymbol{t}_i)\Bigr\|-
\Bigl\|\frac{1}{n}\sum_{i=1}^n K_h(\boldsymbol{t}_i) \boldsymbol{v}(\boldsymbol{t}_i)\Bigr\|,$$
with the Gaussian kernel function $K_h(\boldsymbol{x})$.
The bandwidth parameter $h$ defines the localization rate.
(If $h>1$, the depth is multiplied by $h^d$.)
{\emph{The potential-potential (pot-pot) plot}} \citep{PokotyloM16} bears the analogy to the $DD$-plot and thus can be directly used in $DD$-classification as well.
The potential of a class $j$ is defined as a kernel density estimate multiplied by the class's prior probability and is used in the same way as a depth
$$\hat\phi_j(\boldsymbol{x}) = p_j \hat f_j(\boldsymbol{x}) = \frac{1}{n} \sum_{i=1}^{n_j}{K_{\boldsymbol{H}_j}(\boldsymbol{x},\boldsymbol{x}_{ji})},$$
with a Gaussian kernel $K_{\boldsymbol{H}}(\boldsymbol{x})$ and bandwidth matrix $\boldsymbol{H} = h^2\hat{\boldsymbol{\Sigma}}(\boldsymbol{X})$.
The bandwidth parameter $h$ (called \code{kernel.bandwidth} in the package) is separately tuned for each class.
The parameters have to be properly tuned, using the following benchmark procedures:
\begin{Code}
min_error = list(a = NA, error = 1)
for (h in list(c(h_11, h_21), ... , c(h_1k, h_2k)))
{
error = ddalpha.getErrorRateCV(data, numchunks = <nc>,
separator = <sep>, depth = "potential", kernel.bandwidth = h,
pretransform = "NMahMom")
if(error < min_error$error)
min_error = list(a = a, error = error)
}
\end{Code}
{\emph{The depth-based $k$NN}} \citep{PaindaveineVB15} is an affine-invariant version of the $k$-nearest-neighbor procedure. This method is different, in the sense that it is not using the $DD$-plot. It is accessible through functions \code{dknn.train}, \code{dknn.classify} and \code{dknn.classify.trained}. For each point $\boldsymbol{x}_0$ to be classified, data points are appended by their reflection w.r.t.~$\boldsymbol{x}_0$, which results in the extended centrally symmetric data set of size $2n$. Then the depth of each data point is calculated in this extended data cloud, and $\boldsymbol{x}_0$ is assigned to the most representable class among $k$ points with the highest depth value, breaking ties randomly. Each depth notion may be inserted. Training the classifier constitutes in its tuning by the leave-one-out cross-validation. The method is integrated into the benchmark procedures, accessible there by setting \code{separator = "Dknn"}.
\textbf{\emph{Depth visualization}} functions applicable to the two-dimensional data are also implemented in the package.
To visualize a depth function as a three-dimensional landscape, use
\begin{Code}
depth.graph(data, depth_f,
main, xlim, ylim, zlim, xnum, ynum, theta, phi, bold = F, ...)
\end{Code}
The function accepts additional parameters: plot-limiting parameters \code{xlim}, \code{ylim}, \code{zlim} are calculated automatically, parameters \code{xnum}, \code{ynum} control the resolution of the plot, parameters \code{theta} and \code{phi} rotate the plot, and with parameter \code{bold} equal to \code{TRUE} the data points are drawn in bold face.
Depth contours are pictured by the following functions:
\begin{Code}
depth.contours(data, depth,
main, xlab, ylab, drawplot = T, frequency=100, levels = 10, col, ...)
depth.contours.ddalpha(ddalpha,
main, xlab, ylab, drawplot = T, frequency=100, levels = 10)
\end{Code}
Function \code{depth.contours} calculates and draws the depth contours $D_\alpha$ for given \code{data}. Parameter \code{frequency} controls the resolution of the plot, and parameter \code{levels} controls the vector of depth values of $\alpha$ for which the contours are drawn. Note that a single value set as \code{levels} defines either the depth of a single contour ($0< $ \code{levels} $ \le 1$) or the number (as its ceiling) of contours that are equally gridded between zero and maximal depth value (\code{levels} $ >1$). To combine the contours of several data sets or several different depth notions in one plot, parameter \code{drawplot} should be set to \code{FALSE} for all but the first plot and the color should be set individually through \code{col}.
It is also possible to draw depth contours for a previously trained \code{ddalpha} classifier. In this case classes will differ in colors.
Figures~\ref{fig:bivariate_depths1} and~\ref{fig:bivariate_depths2} show depth surface (left) and depth contours (right) for each of the implemented depth notions. The two plots, \textit{e.g.} for Mahalanobis depth, correspond (without additional parameters that orientate the plot) to the calls \code{depth.graph(data, "Mahalanobis")} and \code{depth.contours(data, "Mahalanobis")}.
Another useful function draws the $DD$-plot either from the trained $DD\alpha$-classifier or from the depth space, additionally indicating the separation between the classes:
\begin{Code}
draw.ddplot(ddalpha, depth.space, cardinalities,
main = "DD plot", xlab = "C1", ylab = "C2",
classes = c(1, 2), colors = c("red", "blue", "green"), drawsep = T)
\end{Code}
To facilitate saving the default parameters for the plots and resetting them, which may become annoying when done often, function \code{par(resetPar())} can be used.
\textbf{\emph{Multivariate and functional data sets}} and data generators are included in the package \pkg{ddalpha} to make the empirical comparison of different classifiers and data depths easier.
50 real multivariate binary classification problems were gathered and described by \cite{MozharovskyiML15} and are also available at \url{http://www.wisostat.uni-koeln.de/de/forschung/software-und-daten/data-for-classification/}. The data can be loaded to a separate variable with function \code{variable = getdata("<name>")}.
Class labels are in the last column of each data set.
Functional data sets are accessible through functions \code{dataf.<name>()} and contain four functional data sets and two generators from \cite{CuevasFF07}.
A functional data object contains a list of functional observations, each characterized by two vectors of coordinates, the arguments vector \code{args} and the values vector \code{vals}, and a list of class labels.
Although this format is clear, visualization of such data can be a nontrivial task, which is solved by function \code{plotf}.
\subsection{Tuning the classifier}
Classification performance depends on many aspects: chosen depth function, separator, outsider treatment, and their parameters.
When selecting a depth function, such properties as ability to reflect asymmetry and
shape of the data, robustness, vanishing beyond the convex hull of the data, and computational burden have to be considered.
Depth contours of Mahalanobis depth are elliptically symmetric and those of projection depth are centrally symmetric, thus both are not well suited for skewed data. Contours of spatial depth are also rounded, but fit substantially closer to the data, which can also be said about simplicial volume depth. Being intrinsically nonparametric, halfspace, simplicial, and zonoid depths fit closest to the geometry of the data cloud, but vanish beyond its convex hull, and thus produce outsiders during classification.
All these depths are global and not able to reflect localities possibly present in the data.
Local spatial depth as well as potentials compensate for this by fitting multimodal distributions well, which is bought at the price of computational burden for tuning a parameter due to an application specific criteria.
Halfspace, simplicial, and projection depths are robust, while outlier sensitivity of Mahalanobis and spatial depths depends on the underlying estimate of the covariance matrix.
To obtain their robust versions, the MCD estimator is applied in package \pkg{ddalpha}. Parameter \code{mah.parMcd} used with Mahalanobis and spatial depths corresponds to the portion of the data for which the covariance determinant is minimized. Simplicial volume and zonoid depths, being based on volume and mean, fail to be robust in general as well.
Halfspace, zonoid, and simplicial depths produce outsiders; their depth contours are also not smooth, and the contours of the simplicial depth are even star-shaped. These depths must not be considered if a substantial portion of points lies on the convex hull of the data cloud; in some cases, especially in high dimensions, this may reach 100\%, see also \cite{MozharovskyiML15}.
Most quickly computable are Mahalanobis, spatial, and zonoid depths.
Their calculation speed depends minorly on data dimension and moderately on the size of the data set, while computation time for simplicial, simplicial volume, and exact halfspace depths dramatically increases with the number of points and dimension of the data.
Approximating algorithms balance between calculation speed and precision depending on their parameters.
Random halfspace and projection depths are driven by parameter \code{num.directions}, \textit{i.e.}~the number of directions used in the approximation.
The approximations of simplicial and simplicial volume depths depend on the number of simplices picked, which is set with parameter \code{k}. If a fixed number of simplices \code{k} $>1$ is given the algorithmic complexity is polynomial in $d$ but is independent of $n$, given \code{k}. If a proportion of simplices is given ($0<$ \code{k} $<1$), then the corresponding portion of all simplices is used and the algorithmic complexity is exponential in $n$, but one can assume that the approximation precision is kept on the same level when $n$ changes.
Note that in $\mathbb{R}^2$, the exact efficient algorithm of \cite{RousseeuwR96} is used to calculate simplicial depth.
Based on the empirical study using real data \citep{PokotyloM16}, the classifiers' error rates grow in the following order: $DD\alpha$, polynomial classifier, $k$NN; although $DD\alpha$ and the polynomial classifier provide similar polynomial solutions and $k$NN sometimes delivers good results when the other two fail.
The degree of the $DD\alpha$ and the polynomial classifier and the number of nearest neighbors are automatically cross-validated, but maximal values may be set manually. To gain more insights, depth-transformed data may be plotted (using \code{draw.ddplot}).
The outsider treatment should not be regarded as the one that gives the best separation of the classes in the original space, but rather be seen as a computationally cheap solution for points right beyond their convex hulls.
In functional classification, parameters $L$ and $S$ can be set by the experience-guided applicant or determined automatically by means of cross-validation. The ranges for cross-validation can be based on previous knowledge of the area or conservatively calculated.
Benchmark procedures that we included in the package may be used for empirical parameters' tuning, by iterating the parameters values and estimating the error rates. For example, the following code fragment searches for the separator, depth, and some other parameters, which deliver best classification:
\begin{Code}
min_error = list(error = 1, par = NULL)
for (par in list(par_set_1, ... , par_set_k))
{
error = ddalpha.getErrorRateCV(data, numchunks = <nc>,
separator = par$sep, depth = par$depth,
other_par = par$other_par )
if(error < min_error$error)
min_error = list(error = error, par = par)
}
\end{Code}
\section*{Acknowledgments}
The authors want to thank Karl Mosler for his valuable suggestions that have substantially improved the present work.
The authors would like to express their gratitude to the Cologne Graduate School of Management, Economics and Social Sciences who supported the work of Oleksii Pokotylo and to the Lebesgue Centre of Mathematics who supported the work of Pavlo Mozharovskyi (program PIA-ANR-11-LABX-0020-01).
\input{lit}
\end{document}
| 31,717 |
\section{Introduction}
Statistical and dynamical properties of polymers with nontrivial topology such as ring polymers have attracted much interest in several branches of physics, chemistry and biology. For instance, some fundamental properties of ring polymers in solution were studied many years ago；\cite{Kramers,Zimm-Stockmeyer,Casassa,Semlyen} circular DNA have been found in nature in the 1960s and knotted DNA molecules are synthesized in experiments in the 1980s; \cite{Vinograd,Nature-trefoil,DNAknots,Bates} looped or knotted proteins have been found in nature during the 2000s.\cite{Taylor,Craik} There are many theoretical studies on knotted ring polymers in solution. \cite{Whittington,Micheletti} Due to novel developments in synthetic chemistry, polymers with different topological structures are synthesized in experiments during the last decade such as not only ring polymers but also tadpole (or lasso) polymers, double-ring (or di-cyclic) polymers and even complete bipartite graph polymers. \cite{Tezuka2000,Tezuka2001,Grubbs,Takano05,Takano07,Grayson,Tezuka2010,Tezuka2011,Tezuka2014} Lasso polymers are studied also in the dynamics of protein folding. \cite{Sulkowska} We call polymers with nontrivial topology {\it topological polymers}. \cite{Tezuka-book} In order to characterize topological polymers it is fundamental to study the statistical and dynamical properties such as the mean-square radius of gyration and the diffusion coefficient. They are related to experimental results such as the size exclusion chromatography (SEC) spectrum.
For multiple-ring (or multi-cyclic) polymers hydrodynamic properties are studied theoretically by a perturbative method. \cite{Fukatsu-Kurata} For double-ring polymers they are studied by the Monte-Carlo (MC) simulation of self-avoiding double-polygons.\cite{JCP}
The purpose of the present research is to study fundamental aspects of the statistical and dynamical properties of various topological polymers in solution such as the mean-square radius of gyration and the diffusion coefficient systematically through simulation. We numerically evaluate them for two different models of topological polymers: ideal topological polymers which have no excluded volume and real topological polymers which have excluded volume, and show how they depend on the topology of the polymers. Here, the topological structures of topological polymers are expressed in terms of spatial graphs.
Furthermore, we show that the short-range intrachain correlation is much enhanced for real topological polymers with complex graphs.
We numerically investigate fundamental properties of topological polymers in the following procedures. Firstly, we construct a weighted ensemble of ideal topological polymers by an algorithm which is based on some properties of quaternions. \cite{CDS} By the algorithm we generate $N$-step random walks connecting any given two points, \cite{CDS,UTIHD} where the computational time is linearly proportional to the step number $N$ in spite of the nontrivial constraints on the sub-chains of the graphs. We evaluate the statistical average of some physical quantities over the weighted ensemble.
%
Secondly, we construct an ensemble of conformations of real topological polymers by performing the molecular dynamics of the Kremer-Grest model and evaluate the statistical average of the quantities over the ensemble.
Thirdly, by comparing the results of the real topological polymers with those of the ideal ones, we numerically show that for the mean-square radius of gyration $R_G^2$ and the hydrodynamic radius $R_{H}$ the set of ratios of the quantities among different topological polymers is given by almost the same for ideal and real topological polymers if the valency of each vertex is equal to or less than three in the graphs, as far as for the polymers we have investigated. It agrees with Tezuka's observation that the SEC results are not affected by the excluded volume if each vertex of a topological polymer has less than or equal to three connecting bonds. \cite{Tezuka}
Here we evaluate the hydrodynamic radius $R_{H}$ through Kirkwood's approximation.
\cite{Doi-Edwards,Rubinstein,Teraoka}
It thus follows that the quaternion method for generating ideal topological polymers is practically quite useful for evaluating physical quantities. In fact, we can estimate at least approximately the values of $R_G^2$ or $R_H$ for real topological polymers by making use of the ratios among the corresponding ideal ones, if the valency of vertices is limited up to three. Moreover, we show that the ratio of the gyration radius to the hydrodynamic radius of a topological polymer, $R_G/R_H$, is characteristic to the topology of the polymer. Here, the gyration radius $R_G$ is given by the square root of the mean-square radius of gyration. We remark that the quaternion method has been generalized to a fast algorithm for generating equilateral random polygons through symplectic geometry. \cite{CS,CDSU2016,CDSUprep}
We analytically show that the ratio of the gyration radius to the hydrodynamic radius of a topological polymer, $R_G/R_H$, is characterized by the variance of the probability distribution function of the distance between two segments of the polymer.
In particular, we argue that the ratio decreases if the variance becomes small. Here we call the function the distance distribution function, briefly. It is expressed with the pair distribution function, or the radial distribution function of polymer segments. The mean-square radius of gyration $R_G^2$ and the hydrodynamic radius $R_H$ correspond to the second moment and the inverse moment of the distance distribution function, respectively.
We numerically evaluate the distance distribution function of a topological polymer both for the ideal and real chain models. For the ideal chain model, the numerical plots of the distance distribution function are approximated well by fitted curves of a simple formula. They lead to the numerical estimates of the mean-square radius of gyration and the hydrodynamic radius consistent with those evaluated directly from the chain conformations. In the case of ideal ring polymers, we exactly derive an analytic expression of the pair distribution function, which is consistent with the fitting formula.
For real topological polymers we show that the short-distance intrachain correlation of a topological polymer is much enhanced, i.e. the correlation hole becomes large, if the graph is complex. The exponent of the short-range power-law behavior in the distance distribution function is given by 0.7 for a linear polymer and a ring polymer, while it is given by larger values such as 0.9 and 1.15 for a $\theta$-shaped polymer and a complete bipartite graph $K_{3,3}$ polymer, respectively. We suggest that the estimate to the exponent of the short-distance correlation such as 0.7 for linear and ring chains is consistent with the estimate of exponent $\theta_2$ of the short-range correlation in a self-avoiding walk (SAW) derived by des Cloizeaux with the renormalization group (RG) arguments, \cite{desCloizeaux} as will be shown shortly.
Here we remark that the estimate of $\theta_2$ has been confirmed in the MC simulation of SAW. \cite{UD2013}
In order to describe the intrachain correlations of a SAW, \cite{deGennes,Schaefer} we denote by $p_0(r)$ the probability distribution function of the end-to-end distance $r$ of an $N$-step SAW. It was shown that the large-distance asymptotic behavior of $p_0(r)$ for $r > R_N$ is given by \cite{Fisher,McKenzie}
\begin{equation}
p_0(r) \sim R_N^{-d} A(r/R_N) \exp\left( -(r/R_N)^{\delta} \right) \, ,
\end{equation}
where $R_N= R_0 N^{\nu}$ and $A(y)$ does not change exponentially fast for large $y$, while the short-distance behavior is given by
\begin{equation}
p_0(r) \sim y^g \quad (r < R_N)
\end{equation}
with the exponent given by $g=(\gamma+1 - \alpha - d \nu)/\nu$ through asymptotic analysis \cite{McKenzie} and $g=(\gamma-1)/\nu$ through the RG arguments. \cite{desCloizeaux1974} The asymptotic properties of $p_0(r)$ are studied theoretically \cite{OO81,D85,D86} and numerically. \cite{B81,BC91,BCRF91,Val96,TKC02,CPV02}
In order to study the short-distance intrachain correlation between two segments of a long polymer in a good solvent, let us denote by $p_1(r)$ and $p_2(r)$ the probability distribution function between a middle point and an end point of an $N$-step SAW and that of two middle points, respectively. Then, we define critical exponents $\theta_s$ for $s=0, 1, 2$ as follows. Assuming $p_s(r) = R_N^{-d} F_s(r/R_N)$ we have
\begin{equation}
F_s(y) \sim y^{\theta_s} \quad (r < R_N) \quad \mbox{for} \quad s=0, 1, 2.
\end{equation}
The RG estimates of the exponents for $d=3$ are given by $\theta_0=0.273, \theta_1= 0.46$ and $\theta_2= 0.71$, respectively. \cite{desCloizeaux} They are close to the MC estimates such as
$\theta_0=0.23 \pm 0.02, \theta_1= 0.35 \pm 0.03$ and $\theta_2= 0.74 \pm 0.03$, respectively. \cite{UD2013}
Thus, these estimates of $\theta_2$ are in agreement with the value 0.7 of the exponent
for the short-range power-law behavior of the distance distribution functions of real linear and real ring polymers within errors. Here we remark that most of the pairs of segments in a SAW are given by those between middle points.
The contents of the paper consist of the following. In section 2 we introduce the notation of graphs expressing topological polymers. We explain the quaternion algorithm for generating topological polymers with given graph, and then the Kremer-Grest model of molecular dynamics and give estimates of the relaxation time.
In section 3 we present the data of the mean-square radius of gyration and the hydrodynamic radius both for ideal and real topological polymers. We numerically show that the ratios of the mean-square radius of gyration among such topological polymers having up to trivalent vertices are given by the same values both for ideal and real topological polymers.
We also evaluate the ratio of the gyration radius to the hydrodynamic radius. We argue that the ratio of the root mean-square radius of gyration to the hydrodynamic radius for a topological polymer decreases if the variance of the distance distribution function decreases.
In section 4 we evaluate the distance distribution functions both for ideal and real topological polymers. For ideal ones they are consistent with the exact result for ideal ring polymers. For the real topological polymers we numerically show that short-range intrachain correlation is much enhanced, i.e. the exponent of the short-range behavior in the distance distribution function
becomes large, for topological polymers with complex graphs.
Finally, in section 5 we give concluding remarks.
\section{Topological polymers with various graphs and numerical methods}
\subsection{Definition of topological polymers}
Let us call a polymer of complex topology expressed by a spatial graph $F$
a {\it topological polymer of graph $F$} or a {\it graph $F$ polymer}.
For an illustration, four graphs of topological polymers are given in Fig. \ref{fig:graph}. They are tadpole, $\theta$-shaped, double-ring and complete bipartite $K_{3,3}$ graphs, respectively.
The graph of a tadpole, which we also call a lasso, corresponds to a tadpole polymer. It is given by a polymer of `a ring with a branch' architecture. \cite{Tezuka2001} The graph of a `theta' in Fig. \ref{fig:graph} denotes a $\theta$-shaped curve, which we also call a theta curve. It corresponds to a $\theta$-shaped polymer, which is given by a singly-fused polymer. \cite{Tezuka2001} The graph of a `double ring' corresponds to a double-ring polymer, which we also call a di-cyclic or an 8-shaped polymer. \cite{Tezuka2001}
Here we remark that a complete bipartite graph $K_{3,3}$ gives one of the simplest nonplanar spatial graphs.
\begin{figure}[ht]
\center
\includegraphics[clip,width=7.5cm]{Fig1.png}
\caption{Examples of topological polymers with graphs with their names: `tadpole' for a tadpole or lasso polymer, `theta' for a $\theta$-shaped polymer, `double-ring' for a double-ring, bicyclic or 8-shaped polymer, and `$K_{3,3}$' for a complete bipartite graph $K_{3,3}$ polymer, respectively. }
\label{fig:graph}
\end{figure}
\subsection{Numerical method I: ideal topological polymers}
We shall explain the method of quaternions for generating
a large number of conformations of a topological polymer expressed by a graph. \cite{CDS,UTIHD}
\subsubsection{Hopf map of quaternions}
Let us introduce the basis $\bf i$, $\bf j$ and $\bf k$ of quaternions.
We assume that the square of each basis ${\bf i}, {\bf j}$ and ${\bf k}$ is given by -1:
\begin{equation}
{\bf i}^2 = {\bf j}^2 = {\bf k}^2 = -1
\end{equation}
and they satisfy the anti-commutation relations
\begin{equation}
{\bf i} {\bf j} = - {\bf j} {\bf i} = {\bf k} \, , \quad {\bf j} {\bf k} = - {\bf k} {\bf j} = {\bf i} \, , \quad {\bf k} {\bf i} = - {\bf i} {\bf k} = {\bf j} \, .
\end{equation}
Any quaternion $h$ is expressed in terms of the basis with real coefficients
$w, x, y$ and $z$ as
\begin{equation}
h = w + x \, {\bf i} + y \, {\bf j} + z \, {\bf k} \,. \label{eq:quat}
\end{equation}
If the real part of a quaternion is given by zero,
i.e. $w=0$ in (\ref{eq:quat}), we call it a pure quaternion.
We identify it with a position vector ${\vec r}=(x, y, z)$ in three dimensions.
Under the complex conjugate operation each of the basis ${\bf i}, {\bf j}$ and ${\bf k}$ changes the sign: ${\bf i}^{*} = - {\bf i}$, ${\bf j}^{*} = - {\bf j}$ and ${\bf k}^{*} = - {\bf k}$. Here we remark that the complex conjugate of the product of two quaternions is given by the product of the two complex conjugates in the reversed order: $(h_1 h_2)^{*} = h_2^{*} h_1^{*}$.
We can express any given quaternion $h$ in terms of two complex numbers $u$ and $v$ as
\begin{equation}
h = u + v {\bf j} \,.
\end{equation}
We define the Hopf map by
\begin{equation}
{\rm Hopf}(h) = h^{*} {\bf i} h \, .
\end{equation}
It is instructive to show that the real part of the Hopf map vanishes \
\begin{eqnarray}
h^{*} {\bf i} h
& = & (u + v {\bf j})^{*} {\bf i} (u+ v {\bf j}) \nonumber \\
& = & (|u|^2 - |v|^2) {\bf i} - 2 {\rm Im}\left( u^{*} v\right) {\bf j}
+ 2 {\rm Re}\left(u^{*} v \right) {\bf k} \, .
\end{eqnarray}
Thus, the Hopf map of a quaternion gives a pure quaternion.
For given $N$-dimensional complex vectors ${\vec u}=(u_1, u_2, \ldots, u_N)$ and
${\vec v}=(v_1, v_2, \ldots, v_N)$ we denote by ${\vec h}$ the $N$-dimensional vector of quaternions as follows.
\begin{equation}
{\vec h} = {\vec u} + {\vec v} \, {\bf j}.
\end{equation}
We define the Hopf map for the $N$-dimensional vectors of quaternions
\begin{equation}
{\rm Hopf}({\vec h}) = {\vec h}^{\dagger} {\bf i} {\vec h} \, . \label{eq:vecHopf}
\end{equation}
For a given pair of complex vectors ${\vec u}$ and ${\vec v}$ we define bond vecters ${\vec b}_n=(x_n, y_n, z_n)$ for $n=1, 2, \ldots, N$ by
\begin{equation}
x_n= |u_n|^2 -|v_n|^2 , \, y_n = - 2 {\rm Im}\left( {u}_n^{*} {v_n} \right), \,
z_n = 2 {\rm Re}\left( {u_n}^{*} {v_n} \right) \, . \label{eq:jumpvector}
\end{equation}
The Hopf map (\ref{eq:vecHopf}) corresponds to the sum of the bond vectors
${\vec b}_n$'s.
\subsubsection{Quaternion method for generating random walks with given end-to-end distance}
Suppose that a pair of Gaussian $N$-dimensional complex vectors ${\vec \alpha}$ and ${\vec \beta}$ are given, randomly. We define $N$-dimensional complex vectors ${\vec u}$ and ${\vec v}$ through the Gram-Schmidt method by
\begin{equation}
{\vec u} = N \frac {\vec \alpha} {| {\vec \alpha} |} \, , \quad
{\vec v} = N \frac {{\vec \beta}- ({\vec \beta} \cdot {\vec \alpha}^{*})
{\vec \alpha}/{| {\vec \alpha}|^2}}
{ |{\vec \beta} - ({\vec \beta}\cdot {\vec \alpha}^{*}) {\vec \alpha}/|{\vec \alpha}|^2| } \, .
\label{eq:gram-schmidt}
\end{equation}
They have the same length $N$ and are orthogonal to each other with respect to the standard scalar product among complex vectors: $\langle {\vec \alpha}, {\vec \beta} \rangle = {\vec \alpha} \cdot {\vec \beta}^{*}$. Here ${\vec \alpha}^{*}$ denotes the $N$-dimensional complex vector where each entry is given by the complex conjugate of the corresponding entry of the vector ${\vec \alpha}$: $({\vec \alpha}^{*}) _n= \alpha_n^{*}$, where $\alpha_n$ are the
$n$th component of the vector ${\vec \alpha}$.
For a pair of complex vectors ${\vec u}$ and ${\vec v}$ constructed by (\ref{eq:gram-schmidt}) a series of the three-dimensional vectors ${\vec b}_{n}$ defined by (\ref{eq:jumpvector}) for $n=1, 2, \ldots, n$, gives a random polygon of $n$ segments. \cite{CDS}
In order to construct such a random walk that has the end-to-end distance $R$
we introduce another complex vector ${\vec v}_R$ by
\begin{equation}
{\vec v}_R = \omega {\vec u} + \omega^{'} {\vec v}
\end{equation}
where weights $\omega$ and $\omega^{'}$ are given by
\begin{equation}
\omega = R/2N^2 \, , \quad \omega^{'} = \sqrt{1-R^2/(4 N^4)} \, .
\end{equation}
Here we remark that the vector ${\vec v}_R$ satisfies
\begin{equation}
{\vec u}^{\dagger} {\vec v}_R = R/2 \, .
\end{equation}
We define another quaternion vector ${\vec h}_R$ by
\begin{equation}
{\vec h}_R = {\vec u} + {\vec v}_{R}
\, {\bf j} \, .
\end{equation}
Through the Hopf map it gives an $N$-step random walk with end-to-end distance $R$. We have
\begin{equation}
{\rm Hopf} ({\vec h}_{R}) = R \, {\bf k} \, .
\end{equation}
Thus, a series of the three-dimensional vectors ${\vec b}_{n}$ constructed by (\ref{eq:jumpvector}) with ${\vec v}$ replaced by ${\vec v}_R$ for $n=1, 2, \ldots, n$, gives a random walk of $n$ segments which connects the origin and
the point of $z=R$ on the $z$ axis.
\subsubsection{Method for constructing random configurations of topological polymers}
We construct weighted ensembles of random configurations for the topological polymer of a graph and numerically evaluate the expectation value of a physical quantity by taking the weighted sum for the quantity.
We first generate random configurations for the open chains and closed chains which are part of the given graph, and we attach appropriate weights to the parts of the graph. We determine the weight of the configuration for the whole graph by the product of all the weights to the parts of the graph.
\vskip 12pt
\par \noindent
{\bf Theta curve graph ($\theta$-shaped graph)}
We generate weighted random configurations of a theta-curve graph polymer of $N=3n$ segments as follows. \cite{UTIHD} We first construct random polygons of $2n$ segments by the method of quaternions. Secondly, we take two antipodal points A and B on the polygon such that each of the sub-chains connecting A and B has $n$ segments. Thirdly, we connect the points A and B by an $n$-step random walk with end-to-end distance equal to the distance between the points A and B by making use of the method of quaternions.
To the whole configuration we assign the weight $w_{\rm AB}$ which is given by the $N$-step Gaussian probability density of the end-to-end vector between the points A and B. The expression of the probability density will be given by eq. (\ref{eq:ete-gauss}) in section 4.2, where the length of bond vectors is given by $b=1$.
\vskip 12 pt
\par \noindent
{\bf Complete bipartite graph $K_{3,3}$}
We generate weighted random configurations of the topological polymer of a complete bipartite graph $K_{3,3}$ with $N=9n$ segments as follows.
Firstly, we generate random polygons of $4n$ segments. We take antipodal points A and B on the polygon such that each sub-chain between A and B has $2n$ segments (see Fig. \ref{fig:graph-complete}). We take a point C on one of the sub-chains between A and B such that both the sub-chain between A and C (sub-chain AC) and that of C and B (sub-chain BC) have $n$ segments, respectively. Similarly, we take a middle point D on the other sub-chain between A and B so that sub-chains AD and BD have $n$ segments, respectively. Secondly, we generate an $2n$-step random walk such that it has the end-to-end distance equal to the distance between points A and B, and put it so that it connects A and B. We take the point E on the middle point of it so that sub-chains AE and BE have $n$ segments, respectively. Thirdly, we generate a $2n$-step random walk connecting C and E. We take the middle point $F$ on the walk so that sub-chains CF and EF have $n$ segments, respectively. Finally, we generate an $n$-step random walk DF which connects points D and F.
We associate the weight $w_{\rm AB}$, $w_{\rm CE}$ and $w_{\rm FD}$ for sub-chains AB, CE and FD, respectively. We define the weight for the whole configuration by multiplying them as $w_{\rm total}= w_{\rm AB} w_{\rm CE} w_{\rm FD}$.
\begin{figure}[ht]
\center
\includegraphics[clip,width=7.5cm]{Fig2.png}
\caption{A complete bipartite graph $K_{3,3}$. }
\label{fig:graph-complete}
\end{figure}
\subsection{Numerical Method II: Molecular dynamics simulation}
\subsubsection{The Kremer-Grest model}
A polymer chain in the Kremer-Grest model has both the repulsive Lennard-Jones (LJ) potentials and the finitely extensible nonlinear elongation (FENE) potentials to prevent the bonds from crossing each other.
The LJ potential is given by
\begin{equation}
\label{eq_GeneralLJ}
E_{LJ}\left(r_{ij}\right)=4\epsilon\left(\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^6\right)
\end{equation}
where $r_{ij}$ is the distance between the $i$th and $j$th atoms and we set the LJ parameters $\epsilon$ and $\sigma$ as $\epsilon=1.0$ and $\sigma=1.0$. The term of $r_{ij}^{-12}$ corresponds to the short-range repulsion, and that of $r_{ij}^{-6}$ the long-range attractive interaction. The minimum of the LJ potential is given by $-1$ at $r_{ij}=2^{1/6}\approx1.122$. We introduce cutoff in order to produce a repulsive Lennard-Jones potential as follows.
\begin{equation}
\label{eq_RepulsiveLJ}
E^{'}_{LJ}\left(r_{ij}\right)=\left\{
\begin{array}{ll}
4\left(\left(\frac{1}{r_{ij}}\right)^{12}-\left(\frac{1}{r_{ij}}\right)^6+1/4\right) & \left(r_{ij}<2^{1/6}\right) \\
0 & \left(r_{ij}>2^{1/6}\right) \\
\end{array}
\right.
\end{equation}
where the constant term $1/4$ is added to eliminate the discontinuous jump at $r_{ij}=2^{1/6}$.
The FENE potential between a pair of bonded atoms
\begin{equation}
\label{eq_fene}
E_{F}\left(r_{ij}\right)=-0.5K_0R_0^2\log \left[1-\left(\frac{r_{ij}}{R_0}\right)^2\right]
\end{equation}
is employed to provide a finitely extensible and nonlinear elastic potential, $R_0$ is the maximum extent of the bond. Here we choose $K_0=30.0$ and $R_0=1.5$.
\subsubsection{Relaxation time for the evolution through LAMMPS}
We generate an ensemble of conformations of a topological polymer of graph $F$ by LAMMPS: The initial conformation is given by putting the atoms on the lattice points along the polygonal lines of the given graph $F$ in a cubic lattice. Then by LAMMPS we integrate Newton's equation of motion for the atoms under the repulsive Lennard-Jones and FENE potentials.
The topological type of the conformation of a topological polymer does not change during time evolution. The bonds can hardly cross each other, since the atoms are surrounded by strong barriers which increase with respect to the inverse of $r_{ij}^{12}$ while they are connected with nonlinear elastic springs of finite length.
Let us denote by ${\vec r}_j$ the position vectors of the $j$th segments of a polymer for $j=1, 2, \ldots, N$. We define the correlation between the conformation at time $t$ and that of the initial time $t=0$ by
\begin{equation}
\label{eq_correlation}
\frac{\sum_{i=1}^N \left(\vec{r}_i(t)-\vec{R}_c(t)\right) \, \cdot \, \left(\vec{r}_i(0)-\vec{R}_c(0)\right)}{\sqrt{\sum_{i=1}^N \left(\vec{r}_i(t)-\vec{R}_c(t)\right)^2}\sqrt{\sum_{i=1}^N \left(\vec{r}_i(0)-\vec{R}_c(0)\right)^2}} \, ,
\end{equation}
where the center of mass of the polymer, ${\vec R}_c$, is given by
\begin{equation}
{\vec R}_c = {\frac 1 N} \sum_{j=1}^{N} {\vec r}_j \, .
\end{equation}
We define the relaxation time $\tau$ of the conformational correlation by the number of steps at which the conformational correlation decreases to $1/e$.
We take independent conformations at every $6\tau$ steps.
For example, we have evaluated $\tau=6180$ for a linear polymer of $N=500$ segments,
$\tau= 3800$ for a ring polymer of $N=500$ segments, and $\tau=2430$ for a $K_{3,3}$ graph polymer of $N=492$ segments.
\subsection{Numerical Method III: Distribution function of the distance between two segments}
Let us denote the distance distribution function by $F(r)$ in terms of distance $r$.
Here we recall that it is the probability distribution function of the distance between any given pair of segments of a polymer. The probability that the distance between a given pair of segments in the polymer is larger than $r$ and less than $r + dr$
is given by $F(r) dr$. Suppose that there are $N$ segments in a region of volume $V$.
We denote the average global density by $\rho=N/V$.
In terms of the pair distribution function $\rho g(r)$,
it is expressed as $N F(r) dr = 4 \pi r^2 g(r) dr$.
We numerically evaluate the distance distribution function $F(r)$ as follows. Firstly, we generate an ensemble of random conformations of a polymer with graph $F$. Secondly, for each conformation we choose a pair of segments randomly, and calculate the distance between them. We repeat the procedure several times. Thirdly, we make the histogram of the estimates with the distance between two segments for all conformations in the ensemble.
\section{Mean-square radius of gyration and the hydrodynamic radius}
\subsection{Mean-square radius of gyration for topological polymers}
We define the mean-square radius of gyration for a topological polymer of graph $F$ consisting of $N$ segments by
\begin{equation}
\langle R_G^2 \rangle_F = {\frac 1 N} \sum_{j=1}^{N} \langle ({\vec r}_j - {\vec R}_c)^2 \rangle_F \, .
\end{equation}
Here the symbol $\langle A \rangle_F$ denotes the ensemble average of $A$
over all possible configurations of the topological polymer with graph $F$.
We denote by $R_G(F)$ the square root of the mean-square radius of gyration
$\langle R_G^2 \rangle_F$:
\begin{equation}
R_G(F)= \sqrt{ \langle R_G^2 \rangle_F} \, .
\end{equation}
We also call it the gyration radius of the polymer.
\begin{figure}[ht]
\center
\includegraphics[clip,width=8.0cm]{Fig3.png}
\caption{Mean-square radius of gyration $\langle R_G^2 \rangle_F$ versus the number of segments $N$ for ideal topological polymers with graph $F$ evaluated by the quaternion method for six graphs $F$ such as linear, tadpole (lasso), ring, double-ring, $\theta$-shaped, and complete bipartite graph $K_{3,3}$ polymers, depicted by filled circles, filled diamonds, filled upper triangles, filled lower triangles, filled stars, and filled squares, respectively. Each data point corresponds to the average over $10^4$ samples. }
\label{fig:QRg}
\end{figure}
In Fig. \ref{fig:QRg} we plot against the number of segments $N$ the numerical estimates of the mean-square radius of gyration $\langle R_G^2 \rangle_F$ for ideal topological polymers of graph $F$ for six different graphs. They are evaluated by the quaternion method, and are given in decreasing order for a given number of segments $N$ as follows: those of linear polymers, tadpole (lasso) polymers, ring polymers, double-ring polymers, $\theta$-shaped polymers, and polymers with a complete bipartite graph $K_{3,3}$. Here we remark that the markers in Figures correspond to the same graphs throughout the paper.
We observe in Fig. \ref{fig:QRg} that the estimate of the gyration radius for a double-ring polymer of $N$ segments, depicted by filled lower triangles, is close to that of a $\theta$-shaped polymer of $N$ segments depicted with filled stars. The former is only slightly larger than the latter. For various other graphs the estimates of the mean-square radius of gyration are distinct among the different graphs with same given number of segments $N$.
The estimates of the mean-square radius of gyration for the ideal topological polymers linearly depend on the number of segments $N$, as shown in Fig. \ref{fig:QRg}. They are fitted by the formula
\begin{equation}
\langle R_G^2 \rangle_F = a_0 + a_1 N \, . \label{eq:fit-ideal}
\end{equation}
The best estimates of parameters $a_0$ and $a_1$
in (\ref{eq:fit-ideal}) are listed in Table \ref{tab:ideal-Rg} together with $\chi^2$ values per degree of freedom (DF).
The $\chi^2$ values per DF for the topological polymers with the different graphs are at most 1.1 and are small.
\begin{table}[htbp]
\begin{tabular}{c|ccc}
graph $F$ & $a_0$ & $a_1$ & $\chi^2$/DF \\
\hline
linear & $-0.004 \pm 0.081$ & $0.16605 \pm 0.00043$ & $0.37$ \\
tadpole
(lasso)
& $0.039 \pm 0.059$ & $0.11483 \pm 0.00032$ & $0.43$ \\
ring & $0.084 \pm 0.025$ & $0.08397 \pm 0.00014$ & $0.29$ \\
double-ring & $0.100 \pm 0.020$ & $0.06327 \pm 0.00011$ & $0.33$ \\
$\theta$-shaped & $0.120 \pm 0.037$ & $0.05600 \pm 0.00021$ & $1.13$ \\
complete $K_{3,3}$ & $0.191 \pm 0.022$ & $0.03470 \pm 0.00011$ & $0.09$ \\
\hline
\end{tabular}
\caption{Best estimates of the parameters in eq. (\ref{eq:fit-ideal}) fitted to the estimates of the mean-square radius of gyration $\langle R_G^2 \rangle_F$ for ideal topological polymers of graph $F$ evaluated by the quaternion method and the $\chi^2$ values per degree of freedom (DF). }
\label{tab:ideal-Rg}
\end{table}
\begin{figure}[ht]
\center
\includegraphics[clip,width=8.0cm]{Fig4.png}
\caption{Mean-square radius of gyration $\langle R_G^2 \rangle_F$ versus the number of segments $N$ for real topological polymers with graphs $F$ evaluated by the
molecular dynamics of the Kremer-Grest model.
Each data point corresponds to the average over $5 \times 10^3$ samples. }
\label{fig:LAMMPS_RG}
\end{figure}
For real topological polymers, which have excluded volume, the numerical values of the mean-square radius of gyration are proportional to some power of the number of segments $N$ with scaling exponent $\nu=0.59$, if the number of segments $N$ is large enough. We derive good theoretical curves fitting to the data by applying the following formula:
\begin{equation}
\langle R_G^2 \rangle_F = a_0 + a_1 \, N^{2 \nu} \, . \label{eq:fit-real}
\end{equation}
Here we fix the value of scaling exponent $\nu$ by $\nu=0.59$ when we apply formula (\ref{eq:fit-real}) to the data in order to derive the best estimates of parameters $a_0$ and $a_1$.
We observe in Fig. \ref{fig:LAMMPS_RG} that the estimate of the gyration radius
for a double-ring polymer of $N$ segments is clearly larger than that of a $\theta$-shaped polymer of $N$ segments, for each $N$. The former and the latter are distinct for each segment number $N$. It is in agreement with Tezuka's observation in experiments that if a polymer has such a vertex with four connecting bonds, the average size of the polymer is much enhanced due to the excluded volume effect. \cite{Tezuka} Here we remark that a double-ring polymer indeed has a vertex where four bonds are connected.
We recall that in the case of ideal topological polymers the size of double-ring polymers is close to that of $\theta$-shaped polymers with the same number of segments $N$.
It has been shown in the MC simulation \cite{JCP} that in the case of nonzero finite excluded volume the mean-square radius of gyration for double-ring polymers of $N$ segments is much closer to that of ring polymers of $N$ segments, while in the case of no excluded volume the mean-square radius of gyration for double-ring polymers of $N$ segments is distinctly smaller than that of ring polymers of $N$ segments.
\begin{table}[htbp]
\begin{tabular}{c|ccc}
graph $F$ & $a_0$ & $a_1$ & $\chi^2$/DF \\
\hline
linear & $-3.02 \pm 0.62$ & $0.2568 \pm 0.0019$ & $0.99$ \\
tadpole (lasso) & $-1.42 \pm 0.43$ & $0.1863 \pm 0.0011$ & $1.86$ \\
ring & $-1.57 \pm 0.19$ & $0.13903 \pm 0.00047$ & $1.80$ \\
double-ring & $-0.834 \pm 0.064$ & $0.11813 \pm 0.00016$ & $0.22$ \\
$\theta$-shaped & $-1.13 \pm 0.10$ & $0.09711 \pm 0.00026$ & $2.22$ \\
complete $K_{3,3}$ & $-0.883 \pm 0.017$ & $0.067521 \pm 0.000051$ & $0.28$ \\
\hline
\end{tabular}
\caption{Best estimates of the parameters in eq. (\ref{eq:fit-real}) fitted to the data of the mean-square radius of gyration for topological polymers with excluded volume (the Kremer-Grest model) and the $\chi^2$ values/DF.}
\label{tab:LAMMPS_RG}
\end{table}
We now argue that the ratios among the estimates of the mean-square radius of gyration for ideal topological polymers with graphs $F$ are approximately similar to those of real topological polymers with the same such graphs $F$ consisting of at most trivalent vertices, as far as we have investigated.
Let us first consider the estimates of coefficient $a_1(F)$ in formula (\ref{eq:fit-ideal}) for ideal topological polymers with graphs $F$. If $N$ is large enough, the coefficients $a_1$ determine the ratios among the gyration radii $R_G(F)$ for some graphs $F$. We thus calculate the ratio $a_1(F)/a_1({\rm ring})$ for ideal topological polymers of graph $F$ in order to study the ratios among the gyration radii of topological polymers with different graphs.
They are listed in table \ref{tab:ratio}.
We then consider the estimates of coefficient $a_1(F)$ in formula (\ref{eq:fit-real}) for real topological polymers with graphs $F$. We again calculate the ratios $a_1(F)/a_1({\rm ring})$ for those of real topological polymers. They are listed in the third column of Table \ref{tab:ratio}.
\begin{table}[htbp]
\begin{tabular}{c|cc}
graph $F$ & ratio of $a_1$ (ideal) & ratio of $a_1$ (real) \\
\hline
linear & $1.978 \pm 0.015$ & $1.847 \pm 0.038$ \\
tadpole (lasso) & $1.368 \pm 0.009$ & $1.340 \pm 0.018$ \\
ring & $1.000 \pm 0.003$ & $1.000 \pm 0.007$ \\
double-ring & $0.753 \pm 0.002$ & $0.850 \pm 0.002$ \\
$\theta$-shaped & $0.667 \pm 0.004$ & $0.699 \pm 0.003$ \\
complete $K_{3,3}$ & $0.413 \pm 0.002$ & $0.4857 \pm 0.0005$ \\
\hline
\end{tabular}
\caption{Ratio of coefficients, $a_1(F)/a_1({\rm ring})$, in eqs. (\ref{eq:fit-ideal}) and (\ref{eq:fit-real}) for ideal and real topological polymers with graphs $F$, respectively. }
\label{tab:ratio}
\end{table}
For ideal polymers the ratio $a_1({\rm linear})/a_1({\rm ring})$ is given by 1.98, which is almost equal to 2 with respect to errors, while for real polymers it is given by 1.84. It is clearly smaller than 2 with respect to errors. Thus, the ratio $a_1({\rm linear})/a_1({\rm ring})$ is smaller than 2 for real polymers.
We suggest that for the real linear and ring polymers the ratio $a_1({\rm linear})/a_1({\rm ring})$ is smaller than 2 due to the excluded volume effect. In fact, the mean-square radius of gyration is enhanced through the effective repulsions among segments under the excluded volume. We therefore conclude that the excluded volume is more important in ring polymers than in linear polymers. We suggest that the swelling effect of ring polymers is more significant than in the linear polymers.
For other graphs $F$ such as tadpole and $\theta$-shaped polymers we observe in Table \ref{tab:ratio} that ratio $a_1(F)/a_1({\rm ring})$ is given by almost the same value both for ideal and real topological polymers.
Here we remark that the two graphs have at most three connecting bonds at all vertices. However, in the case of double-ring polymers the ratio is different for ideal and real polymers. The behavior of ratio $a_1(F)/a_1({\rm ring})$ is in agreement with the previous observation
in Figs. \ref{fig:QRg} and \ref{fig:LAMMPS_RG} that the ratio of the mean-square radius of gyration for real double-ring polymers to that of real $\theta$-shaped polymers is clearly larger than that of the corresponding ideal topological polymers.
For the complete bipartite graph $K_{3,3}$ polymer the ratio $a_1(K_{3,3})/a_1({\rm ring})$ for real polymers is larger than that of ideal polymers. It is possible that the number of segments $N=500$ is not large enough to investigate the asymptotic behavior of a polymer with such a complex graph since each sub-chain has only about 55 segments for $N=500$. It is necessary to perform simulation for much larger $N$ in order to study the asymptotic behavior.
The excluded volume plays a significant role in a double-ring polymer, for which one vertex has four valency. \cite{JCP} In fact, the ratio $a_1({\rm double} \, \, {\rm ring})/a_1({\rm ring})$ for a real double-ring polymer is distinctly larger than that of an ideal double-ring polymer. In Table \ref{tab:ratio} the former is given by 0.85 while for the latter given by 0.75. The former is distinctly larger than the latter with respect to errors.
\subsection{Diffusion coefficient via the hydrodynamic radius for a topological polymer}
We evaluate the diffusion coefficients of topological polymers in solution by applying Kirkwood's approximation. \cite{Doi-Edwards,Rubinstein,Teraoka} The diffusion coefficient, $D(F)$, of a topological polymer with graph $F$ consisting of $N$ segments in a good solvent with viscosity $\eta$ at a temperature $T$ is given by
\begin{equation}
D(F) = \frac {k_B T} {6 \pi \eta } {\frac 1 {N^2}} \sum_{n,m=1; n \ne m}^{N}
\langle \frac 1 {|\vec{ r}_m - \vec{r}_n|} \rangle_F \, .
\end{equation}
Here we take the average of the inverse distance between two segments over all pairs of segments of a given topological polymer. Thus, it is useful to introduce the hydrodynamic radius $R_H(F)$ of a topological polymer with graph $F$
of $N$ segments by
\begin{equation}
{\frac 1 {R_H(F)}} = {\frac 1 {N^2}} \sum_{n,m=1; n \neq m}^{N}
\langle \frac 1 {|\vec{ r}_m - \vec{r}_n|} \rangle_F \, . \label{eq:RH}
\end{equation}
In Fig. \ref{fig:QRH} we plot against the number of segments $N$ the square of the hydrodynamic radius, $\left( R_H(F)\right)^2 $, for ideal topological polymers with graphs $F$ evaluated by the quaternion method.
\begin{figure}[ht]
\center
\includegraphics[clip,width=8.0cm]{Fig5.png}
\caption{Hydrodynamic radius squared $\left( R_H(F) \right)^2$ versus the number of segments $N$ for ideal topological polymers of graphs $F$ evaluated by the quaternion method.}
\label{fig:QRH}
\end{figure}
The curves fitted to the data points are given by applying the following formula
\begin{equation}
\left( R_H(F) \right)^2 = a_0 + a_1 N \, . \label{eq:fit-QRH}
\end{equation}
The best estimates of the parameters $a_0$ and $a_1$ are listed in Table \ref{tab:QRH}.
\begin{table}[htbp]
\begin{tabular}{c|ccc}
graph $F$ & $a_0$ & $a_1$ & $\chi^2$/DF \\ \hline
linear & $1.404 \pm 0.085$ & $0.07927 \pm 0.00042$ & $3.96$ \\
tadpole (lasso) & $0.611 \pm 0.033$ & $0.05926 \pm 0.00017$ & $1.34$ \\
ring & $0.193 \pm 0.013$ & $0.053242 \pm 0.000066$ & $0.29$ \\
double-ring & $0.2141 \pm 0.0058$ & $0.042469 \pm 0.000030$ & $0.1$ \\
$\theta$-shaped & $0.211 \pm 0.030$ & $0.03893 \pm 0.00016$ & $1.88$ \\
complete $K_{3,3}$ & $0.246 \pm 0.021$ & $0.02682 \pm 0.00010$ & $0.15$ \\
\hline
\end{tabular}
\caption{
Best estimates of the parameters in eq. (\ref{eq:fit-QRH}) for curves fitted to the hydrodynamic radius $R_H$ of ideal topological polymers versus the number of segments $N$ for various graphs and the $\chi^2$ values/DF. }
\label{tab:QRH}
\end{table}
We observe in Fig. \ref{fig:QRH} that the $N$-dependence of the hydrodynamic radius is well approximated by the fitted curves of formula (\ref{eq:fit-QRH}).
However, the curves fitted to the data are not very good since the $\chi^2$/DF are not very small. In fact, if we apply a simple power of $N$ to the data such as $N^{\nu}$, the estimates of scaling exponent $\nu$ are less than 0.5, as shown in Appendix A.
In experiments the data points of the hydrodynamic radius versus the number of segments for linear polymers in a $\theta$ solvent are fitted by a curve with exponent $\nu=0.484$. \cite{Teraoka,Suda}
We also observe in Fig. \ref{fig:QRH} that the hydrodynamic radius $R_H(F)$ of an ideal double-ring polymer of $N$ segments is close to but slightly larger than
that of an ideal $\theta$-shaped polymer of $N$ segments. It is similar to the case of the mean-square radius of gyration.
In Fig. \ref{fig:LRH} we plot against the number of segments $N$ the hydrodynamic radius for real topological polymers evaluated by performing the molecular dynamical simulation of the Kremer-Grest model through LAMMPS. Here we recall that each molecule has excluded volume in the model.
\begin{figure}[ht]
\center
\includegraphics[clip,width=8.0cm]{Fig6.png}
\caption{Hydrodynamic radius squared $\left( R_H(F) \right)^2$
versus the number of segments $N$ for real topological polymers of graphs $F$
(the Kremer-Grest model). }
\label{fig:LRH}
\end{figure}
The fitted curves are given by applying formula
(\ref{eq:fit-real}) to the data points where we put $\nu=0.565$
\begin{equation}
\left( R_H(F) \right)^2 = a_0 + a_1 N^{1.13} \, . \label{eq:fit-LRH}
\end{equation}
The best estimates of the parameters $a_0$ and $a_1$ are listed in
Table \ref{tab:LRH}. Here, the value of the exponent
$\nu= 0.565$ is slightly smaller than the scaling exponent of SAW:
$\nu_{\rm SAW} = 0.588$.
However, it is also the case in polymer experiments such as $\nu=0.567$. \cite{Teraoka,Bishop}
We observe in Fig. \ref{fig:LRH} that the hydrodynamic radius $R_H$ of real double-ring polymers is approximately close to but distinctly larger than that of ideal $\theta$-shaped polymers. The enhancement due to the excluded volume is not as significant as in the case of the mean-square radius of gyration. However, the ratio of estimates of $a_1$ for double-ring and $\theta$-shaped polymers is clearly larger than 1 as we have $a_1({\rm double} \, \, {\rm ring})/a_1({\rm theta}) = 1.065$.
\begin{table}[htbp]
\begin{tabular}{c|ccc}
graph $F$ & $a_0$ & $a_1$ & $\chi^2$/DF \\ \hline
linear & $2.204 \pm 0.059$ & $0.16292 \pm 0.00022$ & $0.14$ \\
tadpole (lasso) & $1.98 \pm 0.15$ & $0.13578 \pm 0.00045$ & $2.73$ \\
ring & $1.653 \pm 0.075$ & $0.12843 \pm 0.00022$ & $1.01$ \\
double-ring & $1.438 \pm 0.054$ & $0.11193 \pm 0.00017$ & $0.81$ \\
$\theta$-shaped & $1.278 \pm 0.063$ & $0.10511 \pm 0.00019$ & $1.66$ \\
complete $K_{3,3}$ & $0.796 \pm 0.016$ & $0.083379 \pm 0.000058$ & $0.35$ \\
\hline
\end{tabular}
\caption{Best estimates of the parameters in eq.(\ref{eq:fit-LRH}) for the hydrodynamic radius $R_H(F)$ of real topological polymers with graph $F$ of $N$ segments for various graphs $F$. }
\label{tab:LRH}
\end{table}
\subsection{Ratio of the gyration to hydrodynamic radii}
Let us now consider the ratio of the gyration radius $R_G$ to the hydrodynamic radius $R_H$
for topological polymers with graphs $F$. We suggest that the ratio is universal in the sense of the renormalization group arguments, i.e., we expect that it is independent of some details of the polymer model, so that we can compare the estimate of the ratio with other theoretical approaches and experiments. Here we take the analogy of the amplitude ratio of susceptibilities and that of specific heats in critical phenomena. \cite{deGennes}
We numerically evaluated the ratio $R_G/R_H$ for several topological polymers with various different graphs $F$.
For ideal topological polymers of graphs $F$ the numerical estimates of the ratio $R_G/R_H$
are plotted against the number of segments $N$ in Fig. \ref{fig:ratio-Q}.
For real topological polymers with graph $F$ (the Kremer-Grest model)
the estimates of the ratio of the gyration radius
to the hydrodynamic radius, $R_G(F)/R_H(F)$,
are plotted against the number of segments $N$ in Fig. \ref{fig:ratio-L}.
\begin{figure}[ht]
\center
\includegraphics[clip,width=8.5cm]{Fig7.png}
\caption{Ratio of the gyration radius to the hydrodynamic radius, $R_G(F)/R_H(F)$,
versus the number of segments $N$
for ideal topological polymers with graphs $F$ evaluated by the quaternion method}
\label{fig:ratio-Q}
\end{figure}
By comparing Figs. \ref{fig:ratio-Q} and \ref{fig:ratio-L} we observe that the estimates of the ratio of the gyration radius to the hydrodynamic radius are given in the same order for both the ideal and the real topological polymers of graphs $F$. In decreasing order the ratios are given by that of a linear polymer, tadpole (lasso) polymer, ring polymer, double-ring polymer, $\theta$-shaped polymer, and the polymer of a complete bipartite graph $K_{3,3}$, for ideal topological polymers as shown in Fig. \ref{fig:ratio-Q}, and it is also the case for
real topological polymers as shown in Fig. \ref{fig:ratio-L},
\begin{figure}[ht]
\center
\includegraphics[clip,width=8.5cm]{Fig8.png}
\caption{Ratio of the gyration radius to the hydrodynamic radius
$R_G(F)/R_H(F)$ versus the number of segments $N$
for real topological polymers (the Kremer-Grest model). }
\label{fig:ratio-L}
\end{figure}
In each of Figs. \ref{fig:ratio-Q} and \ref{fig:ratio-L} the ratio $R_G(F)/R_H(F)$ increases with respect to the
number of segments $N$ gradually and monotonically.
As the number of segments $N$ becomes very large such as $N=500$,
the ratios approach constant values for all graphs $F$ as shown in Figs. \ref{fig:ratio-Q} and \ref{fig:ratio-L}.
The ratios among topological polymers with the graphs $F$ are given by almost the same values both for the ideal topological polymers and the real topological polymers.
The ratios for topological polymers with various graphs $F$ at $N=500$ are listed in Table \ref{tab:ratioN500}. In each case of graphs $F$ the ratio for the ideal topological polymer
and that of the real topological polymer is given by almost the same values with respect to errors. We suggest that the estimates of the ratio $R_G(F)/R_H(F)$ should be universal since they are given by almost the same value for different models.
\begin{table}[htbp]
\begin{tabular}{c|ccc}
graph $F$ & $N$ & $R_G(F)/R_H(F)$ (ideal) & $R_G(F)/R_H(F)$ (real) \\
\hline
linear & 500 & $1.425 \pm 0.025$ & $1.47 \pm 0.42$ \\
tadpole (lasso) & 500 & $1.380 \pm 0.021$ & $1.354 \pm 0.072$ \\
ring & 500 & $1.253 \pm 0.013$ & $1.205 \pm 0.027$ \\
double-ring & 500 & $1.215 \pm 0.011$ & $1.190 \pm 0.032$ \\
$\theta$-shaped & 500 & $1.194 \pm 0.013$ & $1.112 \pm 0.017$ \\
complete $K_{3,3}$ & 501 & $1.132 \pm 0.031$ & $1.043 \pm 0.014$ \\
\hline
\end{tabular}
\caption{Ratio of the gyration radius to the hydrodynamic radius $R_G/R_H$
versus the number of segments $N$ for ideal and real topological polymers with graph $F$. }
\label{tab:ratioN500}
\end{table}
\subsection{Ratio $R_G/R_H$ in terms of the mean-square deviations of the distance between two segments}
Let us denote by ${\bar r}$ and $\sigma^2$ the mean value and the variance of the distance $r$ between randomly chosen pairs of segments of a polymer, respectively. We now argue that the ratio of the gyration radius to the hydrodynamic radius, $R_G/R_H$, decreases if the standard deviation $\sigma$ becomes small compared with the mean distance ${\bar r}$.
We first recall that the mean-square radius of gyration of a polymer is given by the average of the square distance between two segments over all the pairs
\begin{equation}
R_G^2 = \frac 1 {2N^2} \sum_{j, k=1}^{N} \langle \left( {\vec r}_j - {\vec r}_k \right)^2 \rangle \, .
\end{equation}
Therefore, the mean-square radius of gyration of a polymer is expressed in terms of the mean-square distance between two segments of the polymer, $\langle r^2 \rangle$, as follows.
\begin{equation}
R_G^2 = \langle r^2 \rangle /2 \, .
\end{equation}
Through Kirkwood's approximation the hydrodynamic radius is given by
the average of the inverse distance between two segments over all the pairs of segments
\begin{equation}
R_H^{-1} = \langle 1/r \rangle \, .
\end{equation}
Thus, the gyration radius is given by the root of the mean-square of the distance between two segments, while the hydrodynamic radius by its harmonic mean.
Let us express distance $r$ in terms of the mean distance ${\bar r}$
and the deviation $\Delta r$ as
$r= {\bar r} + \Delta r$.
We define the variance $\sigma^2$ of distance $r$ by
\begin{equation}
\sigma^2 = \langle (\Delta r)^2 \rangle \, .
\end{equation}
The mean-square of distance $r$ is given by
\begin{eqnarray}
\langle r^2 \rangle & = & \langle ({\bar r} + \Delta r)^2 \rangle \nonumber \\
& = & {\bar r}^2 + \sigma^2 \, .
\end{eqnarray}
If the mean distance $\bar r$ is much larger than the standard deviation $\sigma$ of the distance $r$, we can expand the inverse distance as a series of $\Delta r$:
$1/r = 1/{\bar r} - \Delta r/{\bar r}^2 + \cdots $.
We have the following approximation
\begin{equation}
\langle 1/r \rangle = 1/ {\bar r} + \sigma^2/{\bar r}^3 \, .
\end{equation}
The ratio of the gyration radius to the hydrodynamic radius is therefore
approximately given by
\begin{equation}
R_G/R_H = {\frac 1 {\sqrt{2}}} \, \left( 1 + {\frac {\sigma^2} {{\bar r}^2}} \right)^{3/2} \, .
\end{equation}
\section{Pair distribution function of a topological polymer}
\subsection{Pair distribution function}
Let us define the local segment density $\rho({\vec r})$ in terms of
the position vectors ${\vec r}_j$ of the $j$th segments of a polymer consisting
of $N$ segments for $j=1, 2, \ldots, N$, by
\begin{equation}
\rho({\vec r}) = \sum_{j=1}^{N} \delta({\vec r}- {\vec r}_j) \, .
\end{equation}
We call $\langle \rho({\vec r}_1) \rho({\vec r}_2) \rangle$, the statistical average of the product of the local densities at ${\vec r}_1$ and ${\vec r}_2$, the pair distribution function. \cite{Teraoka}
Since the polymer system is homogeneous and isotropic in the ensemble average, we denote the pair distribution function $\langle \rho({\vec r}) \rho({\vec 0}) \rangle$ simply by $\langle \rho(r) \rho(0) \rangle$ with $r=|{\vec r}|$.
It is related to the radial distribution function of polymer segments, $g(r)$, as
$g(r)= \langle \rho(r) \rho(0) \rangle/\rho^2$.
Suppose that we have a polymer of $N$ segments in a region of volume $V$.
We can regard the local segment density at the origin normalized by the average $\rho= N/V$,
i.e. $\rho({\vec 0})/\rho= \rho({\vec 0})/(N/V)$, as the expectation value for the number of a given segment to be found in a unit volume at the origin when we take the ensemble average.
Intuitively, it corresponds to the probability of finding a segment at the origin per volume.
Thus, the average number of segments per volume located at ${\vec r}$ when there is already a segment at the origin ${\vec r}={\vec 0}$ is given by \cite{Teraoka}
\begin{equation}
\langle \rho({\vec r}) \rho({\vec 0}) \rangle/\rho \, .
\end{equation}
\subsection{Exact expression of the pair distribution function of a Gaussian ring polymer}
Let us consider a Gaussian linear chain of $N$ steps with bond length $b$.
For a given ${\vec r}_2$ the probability density for ${\vec r}_1$ is given by
\begin{equation}
G({\vec r}_1, {\vec r}_2; N) = \left( 2 \pi N b^2/3 \right)^{-3/2} \exp\left(- \frac {\left( {\vec r}_1-{\vec r}_2 \right)^2} {2Nb^2/3} \right) \, .
\label{eq:ete-gauss}
\end{equation}
For a Gaussian linear chain of $N$ steps
the pair distribution function is given by \cite{Teraoka}
\begin{eqnarray}
& & \langle \rho({\vec r}) \rho(0) \rangle/\rho = \frac 1 N \int_{0}^{N} dn
\int _0^N G({\vec r}, {\vec 0}; |n-n^{'}| ) dn^{'}
\nonumber \\
& & \quad = \frac 6 {\pi^{3/2} b^2 r} \int_{\sqrt{u}}^{\infty} \left(1- u /x^2 \right) \exp(-x^2) dx
\end{eqnarray}
where $u$ is given by $u=3 r^2/(2Nb^2)$. For small $r$ we have
\begin{equation}
\langle \rho(r) \rho(0) \rangle/\rho \approx 1/r
\label{eq:asymp-linear-small}
\end{equation}
and for large $r$ we have with $R_G^2 = Nb^2/6$
\begin{equation}
\langle \rho(r) \rho(0) \rangle/\rho \approx \frac 1 {r^4} \exp(-r^2/4R_G^2) \, .
\label{eq:asymp-linear-large}
\end{equation}
For the Gaussian ring chain of $N$ steps we formulate the pair distribution function in terms of the probability densities of the two random walks with $n$ and $N-n$ steps connecting the origin and ${\vec r}$ as follows.
\begin{eqnarray}
& & \langle \rho({\vec r}) \rho({\vec 0}) \rangle /\rho \nonumber \\
& & = \frac 1 N \int_0^{N}dn \int_0^{N} dn^{'}
G({\vec r}, {\vec 0}; |n-n^{'}|) G({\vec 0}, {\vec r}; N- |n-n^{'}|) \nonumber \\
& & \quad \times \left(G({\vec 0}, {\vec 0}; N) \right)^{-1} \, .
\label{eq:def-ring}
\end{eqnarray}
Here we remark that the normalization factor in (\ref{eq:def-ring})
is derived as follows.
\begin{eqnarray}
G({\vec 0},{\vec 0}; N) & = &
\int_V d^3{\vec r} \, \left(
{\frac 1 N} \int_0^{N} dn G({\vec 0}, {\vec r}; n) G({\vec r}, {\vec 0}; N-n) \right)
\, . \nonumber \\
\end{eqnarray}
We can show that eq. (\ref{eq:def-ring}) is expressed as
\begin{eqnarray}
& & \langle \rho({\vec r}) \rho({\vec 0}) \rangle /\rho
\nonumber \\
& & \, \, = \int_{0}^{N} G({\vec r}, {\vec 0}; n) G({\vec 0}, {\vec r}; N-n) dn /G({\vec 0},{\vec 0}; N) \, .
\label{eq:pair-polygon}
\end{eqnarray}
By calculating the integral (\ref{eq:pair-polygon}) with respect to variable $n$
we derive exactly an explicit
expression of the pair distribution function of the Gaussian polygon of $N$ steps
\begin{equation}
\langle \rho(r) \rho({0}) \rangle /\rho
=\frac{N} {2 \pi v^2} \, {\frac 1 r} \exp\left(- \frac {r^2} {v^2} \right) \, ,
\label{eq:pair-dist-polygon}
\end{equation}
where $v^2$ is given by
\begin{equation}
v^2 = {N b^2}/{6} \, .
\end{equation}
Here we recall that for random polygons we have $R_G^2= N b^2/12$, and hence we have
$v^2= 2 R_G^2$.
The derivation of (\ref{eq:pair-polygon}) and (\ref{eq:pair-dist-polygon}) will be given in Appendix B.
The exact expression (\ref{eq:pair-dist-polygon}) of the pair distribution function leads to the
integral expression of the static structure factor derived by Casassa. \cite{Casassa}
\subsection{Mean-square radius of gyration and hydrodynamic radius expressed with the distance distribution function}
We recall that the probability that the distance between two given segments
is larger than $r$ and smaller than $r+dr$ is given by $F(r) dr $.
It is related to the pair distribution function as
\begin{equation}
N \, F(r) \, dr = {\frac 1 {\rho}} \langle \rho(r) \rho(0) \rangle \, 4 \pi r^2 dr \, .
\end{equation}
Let us denote by $r_{\rm RMS}$ the square root of the mean-square distance between two segments of a polymer. We define it by
\begin{equation}
\left( r_{\rm RMS} \right)^2 = \int_{0}^{\infty} r^2 \, F(r) \, dr \, . \label{eq:RMS}
\end{equation}
In terms of $r_{\rm RMS}$ we define the normalized distance by
\begin{equation}
x=r/r_{\rm RMS} .
\end{equation}
We then define the probability distribution function $f(x)$ of the normalized distance $x$ by
\begin{equation}
F(r) \, dr =f(r/r_{\rm RMS}) \, dr/r_{\rm RMS} \, .
\end{equation}
That is, we have assumed $F(r) dr = f(x) dx$. We also call $f(x)$ the normalized distance distribution, briefly.
It follows from (\ref{eq:RMS}) that not only the 0th moment but also the second moment of the normalized distance distribution is given by 1.
The mean-square radius of gyration is expressed in terms of the normalized distance distribution as follows.
\begin{eqnarray}
R_G^2 & = & {\frac 1 2} \, \int_{0}^{\infty} r^2 \, F(r) \, dr \nonumber \\
& = & {\frac 1 2} \, r_{\rm RMS}^2 \, \int_{0}^{\infty} x^2 \, f(x) \, dx \, .
\label{eq:Rg}
\end{eqnarray}
Since the second moment of $f(x)$ is given by 1, we have
\begin{equation}
R_G^2 = r_{\rm RMS}^2 /2\, .
\end{equation}
We have $R_G= r_{\rm RMS}/\sqrt{2}$.
The hydrodynamic radius is expressed in terms of the distance distribution function as
\begin{eqnarray}
\frac 1 {R_H} & = & \int_{0}^{\infty} {\frac 1 r} \, F(r) \, dr \nonumber \\
& = & {\frac 1 {r_{\rm RMS}}} \, \int_{0}^{\infty} x^{-1} f(x) dx \, .
\end{eqnarray}
That is, the hydrodynamic radius $R_H$ is given by the harmonic mean of the distance between two segments. of the polymer.
We therefore have the ratio of the gyration radius to the hydrodynamic radius as
\begin{equation}
R_G/R_H = {\frac 1 {\sqrt{2}}} \, \int_{0}^{\infty} x^{-1} f(x) dx \, . \label{eq:ratioRGRH}
\end{equation}
\subsection{Ideal topological polymers}
The data points of the distance distribution function $F(r)$ for an ideal topological polymer of $N=500$ segments with graph $F$ are plotted in Fig. \ref{fig:2pt-Q} against the distance $r$ between two segments of the polymer. Here we consider three graphs $F$, a ring, a theta curve, and a complete bipartite graph $K_{3,3}$.
\begin{figure}[ht]
\center
\includegraphics[clip,width=8.0cm]{Fig9.png}
\caption{Distance distribution function $F(r)$ of the distance $r$ between two segments in an ideal topological polymer of $N=500$ with graph $F$ for three graphs $F$: a complete bipartite graph $K_{3,3}$, a theta curve, and a ring.
Data points for the complete bipartite graph $K_{3,3}$, theta curve and ring correspond to filled squares, filled stars and filled triangles, respectively.
}
\label{fig:2pt-Q}
\end{figure}
The distance distribution function of a complete bipartite graph $K_{3,3}$ polymer has the tallest peak height among the ideal topological polymers of the three graphs. That of a $\theta$-shaped polymer has the next highest, and that of a ring polymer the lowest peak height. Similarly, the peak positions of the distance distribution functions for a complete bipartite graph $K_{3,3}$ polymer, a $\theta$-shaped polymer, and a ring polymer are given by the smallest one, the second smallest one, and the largest one, respectively.
Accordingly, the distance distribution function of
a complete bipartite graph $K_{3,3}$ polymer has the narrowest width among the ideal topological polymers of the three graphs.
\begin{figure}[ht]
\center
\includegraphics[clip,width=8.0cm]{Fig10.png}
\caption{Probability distribution function $f(x)$ of the normalized distance $x$ between any given pair of segments in an ideal topological polymer of $N=500$ with graph $F$. Here the distance between two segments is normalized by the gyration radius of the topological polymer $R_g(F)$.}
\label{fig:2pt-Qnormalized}
\end{figure}
The different shapes of the plots depicted in Fig. \ref{fig:2pt-Q} for the three graphs are mainly due to the different values of the root mean square $r_{\rm RMS}$ for the three graphs. In Fig. \ref{fig:2pt-Qnormalized} the data points of the normalized distance distribution $f(x)$ are plotted against the normalized distance $x=r/r_{\rm RMS}$. The three curves are almost overlapping
and only slightly different from each other. However, such small different behavior may lead to quite different values of the ratio of the gyration radius to the hydrodynamic radius, $R_G/R_H$.
For the Gaussian ring chain, an exact expression of the pair distribution function and therefore the distance distribution function is obtained. The root mean-square distance $r_{\rm RMS}$ is given exactly by
\begin{equation}
r_{\rm RMS}^2 = N b^2/6 .
\end{equation}
In eq. (\ref{eq:pair-dist-polygon}) we have $4 r^2/v^2=r^2/2R_G^2 = (r/r_{\rm RMS} )^2$. We therefore have the exact expression of the normalized distance distribution
\begin{equation}
f(x) = 2 x \exp\left( -x^2 \right) \, . \label{eq:exact-f}
\end{equation}
Here we recall that the normalized distance is given by $x=r/r_{\rm RMS}$.
\begin{table}[htbp]
\begin{tabular}{c|cccc}
graph & $C$ & $\theta$ & $\delta$ & $\chi^2$/DF \\
\hline
ring & $2.0028 \pm 0.0025$ & $1.0044 \pm 0.0020$ & $2.0011 \pm 0.0017$ & 0.96 \\
theta & $2.0796 \pm 0.0056$ & $1.0929 \pm 0.0043$ & $2.0661 \pm 0.0038$ & 2.39 \\
$K_{3,3}$ & $2.1799 \pm 0.0092$ & $1.2096 \pm 0.0070$ & $2.1478 \pm 0.0060$ & 0.97 \\
\hline
\end{tabular}
\caption{Best estimate of the parameters in (\ref{eq:fit-2pt}) for the normalized distance distribution $f(x)$ for an ideal topological polymers of $N$ segments with graph $F$. Here $N=500$. Left-hand sides of (\ref{eq:con0th}) and (\ref{eq:con2nd}) are given by 0.9916 and 0.9985, respectively, for $\theta$-shaped polymers; by 0.9988 and 0.9984 for complete bipartite $K_{3,3}$ graph polymers, respectively.}
\label{tab:2pt}
\end{table}
We shall numerically show that the normalized distance distribution $f(x)$ of an ideal topological polymer with graph $F$ is well approximated by the following
\begin{equation}
f(x) = C x^{\theta} \exp\left(- x^{\delta} \right) \label{eq:fit-2pt} \, .
\end{equation}
Here the three parameters satisfy the two constraints
that the 0th and 2nd moments of $f(x)$ are given by 1
\begin{eqnarray}
{\frac C {\delta}} \Gamma((\theta+1)/\delta) & = & 1 \, , \label{eq:con0th} \\
{\frac C {\delta}} \Gamma((\theta+3)/\delta) & = & 1 \, . \label{eq:con2nd}
\end{eqnarray}
For the Gaussian ring chain case, we have $C=2$, $\theta=1$ and $\delta=2$, exactly. It is clear that they satisfy the constraints (\ref{eq:con0th}) and (\ref{eq:con2nd}).
We now argue that formula (\ref{eq:fit-2pt}) gives good fitted curves to the data.
The best estimates of the parameters $C$, $\theta$ and $\delta$ of eq. (\ref{eq:fit-2pt}) are listed in Table \ref{tab:2pt}. The $\chi^2$ values per DF are rather small for the three graphs: a ring, a $\theta$-shaped curve and a complete bipartite $K_{3,3}$ graph.
For ring polymers the best estimates are close to the exact values: $C=2$, $\theta=1$ and $\delta=2$.
By putting the best estimates in Table \ref{tab:2pt} we evaluate the left hand-sides of (\ref{eq:con0th}) and (\ref{eq:con2nd}). They hold within errors for $\theta$-shaped polymers and complete bipartite $K_{3,3}$ graph polymers as shown in the caption of Table \ref{tab:2pt}.
We thus conclude that formula (\ref{eq:fit-2pt}) gives good fitted curves to the data.
Substituting (\ref{eq:fit-2pt}) to the integral of (\ref{eq:ratioRGRH}) we have
\begin{equation}
R_G/R_H = {\frac 1 {\sqrt{2}}} {\frac C {\delta}} \Gamma({\theta}/{\delta}) \, .
\label{eq:inverse}
\end{equation}
For the Gaussian ring polymer, the ratio $ R_G/R_H$ is exactly given by
\begin{equation}
R_G/R_H = \sqrt{ \frac {\pi} 2} \, . \label{eq:ratio-ring}
\end{equation}
The numerical value $\sqrt{\pi/2} \approx 1.253314$ is rather close to the estimate obtained by the quaternion method presented in Table \ref{tab:ratioN500}.
Putting into (\ref{eq:inverse}) the best estimates of the parameters of (\ref{eq:fit-2pt}) given in Table \ref{tab:2pt} we evaluate the ratio $R_G/R_H$ as follows: 1.24966 for ring polymers, 1.19415 for $\theta$-shaped polymers, and 1.13402 for complete bipartite $K_{3,3}$ graph polymers. They are all consistent with the estimates of $R_G/R_H$ listed in Table \ref{tab:ratioN500}.
\begin{figure}[ht]
\center
\includegraphics[clip,width=8.0cm]{Fig11.png}
\caption{Double-logarithmic plot of the normalized distance distribution $f(x)$ in an ideal topological polymer of $N=500$ with graph $F$.
The data points for a complete bipartite graph $K_{3,3}$ polymer, a $\theta$-shaped polymer, a ring polymer and a linear polymer correspond to filled squares, filled stars, filled triangles, and filled circles, respectively.
}
\label{fig:2pt-Qnormalized-DL}
\end{figure}
In order to investigate the asymptotic behavior the double-logarithmic plots of the normalized distance distribution $f(x)$ for ideal topological polymers of $N=500$ are given for the four graphs $F$ in Fig. \ref{fig:2pt-Qnormalized-DL}. The short-distance behavior of the normalized distance distribution of a linear chain is consistent with the analytical result (\ref{eq:asymp-linear-small}) such as $f(x) \approx x$ for $x \ll 1$. The large-distance behavior of a normalized distance distribution is consistent with the analytic result (\ref{eq:asymp-linear-large}) such as $f(x) \approx \exp(-x ^2/2)$ for $x \gg 1$. For the Gaussian ring chain we have both the short-distance and the large-distance behavior in the exact expression of the normalized distance distribution: $f(x) =2 x \exp(-x^2)$.
\subsection{Real topological polymers}
The estimates of the probability distribution function $f(x)$ of the normalized distance $x$ between two segments in a real topological polymer of $N=300$ with graph $F$ are plotted in Fig. \ref{fig:2pt-realLL} for four graphs: a linear, a ring, a theta curve, and a complete bipartite $K_{3,3}$ graph.
In Fig. \ref{fig:2pt-realLL} the normalized distance distribution of complete bipartite $K_{3,3}$ graph polymers has the highest peak, that of theta curve polymers the second highest peak,and that of ring polymers the third highest peak. Accordingly, the normalized distance distribution of complete bipartite graph $K_{3,3}$ polymers has the narrowest width among the real topological polymers of the three graphs: a ring, a theta curve, and a complete bipartite $K_{3,3}$ graph.
\begin{figure}[ht]
\center
\includegraphics[clip,width=9.0cm]{Fig12.png}
\caption{Probability distribution function $f(x)$ of the normalized distance $x$
between two segments in a real topological polymer (the Kremer-Grest model) of $N=300$ with graph $F$. Here, graphs $F$ correspond to a linear, a ring, a theta curve and a complete bipartite graph $K_{3,3}$ polymer, respectively. }
\label{fig:2pt-realLL}
\end{figure}
\begin{figure}[ht]
\center
\includegraphics[clip,width=8.0cm]{Fig13.png}
\caption{Double logarithmic plot of the probability distribution function $f(x)$ of the normalized distance $x$ between two segments in a real topological polymer (the Kremer-Grest model) of $N=300$ with graph $F$. The fitted lines are given by
$1.6 \, x^{0.7}$, $1.1 \, x^{0.7}$, $1.2 \, x^{0.9}$ and $1.4 \, x^{1.15}$ for a linear, a ring, a theta curve, and a complete bipartite graph $K_{3,3}$ polymer, respectively. }
\label{fig:2pt-normalizedDL}
\end{figure}
The probability distribution function $f(x)$ of the normalized distance $x$ between two segments in a real topological polymer (the Kremer-Grest model) of $N=300$ with graph $F$ is plotted in the double logarithmic scale in Fig. \ref{fig:2pt-normalizedDL} for each of the four graphs such as linear, ring, $\theta$-shaped and complete bipartite $K_{3,3}$ graphs.
The enhancement of the short-distance correlation is characterized by the different values of
the exponent for the short-distance power-law behavior. It is given by $0.7$ for both a linear and a ring polymer, while $0.9$ and $1.15$ for a theta curve polymer and a complete bipartite graph $K_{3,3}$ polymer, respectively, as shown in Fig. \ref{fig:2pt-normalizedDL}.
We suggest that the short-distance exponent of a topological polymer is given by the same value for different numbers of segments $N$. For $N=400$ we have plotted the normalized distance distribution function in a real topological polymer (the Kremer-Grest model) for the four different graphs. We have the same fitted lines for the power-law behavior for the four graphs. That is, the exponent is given by $0.7$ for both a linear and a ring polymer, while it is given by $0.9$ and $1.15$ for a theta curve polymer and a complete bipartite graph $K_{3,3}$ polymer, respectively.
For star polymers the Kratky plot of the static structure factor is calculated for real chains. \cite{TKC02} It seems, however, that the large-$q$ behavior is similar to that of an open chain.
\section{Concluding remarks}
For various polymers with different topological structures we have numerically evaluated the mean-square radius of gyration and the hydrodynamic radius systematically through simulation. We evaluated the two quantities both for ideal and real chain models and show that the ratios of the quantities among different topological types do not depend on the existence of excluded volume if the topological polymers have only up to trivalent vertices, as far as the polymers investigated.
We have shown that the quaternion method for generating ideal topological polymers is practically quite useful for evaluating physical quantities for the topological polymers. In fact, we can estimate at least approximately the values of $R_G^2$ or $R_H$ for real topological polymers by making use of the ratios among the corresponding ideal ones. Furthermore, the quaternion algorithm is quite fast. The computational time is proportional to the number of segments.
For various topological polymers we have evaluated the ratio of the gyration radius to the hydrodynamic radius, $R_G/R_H$, which we expect to be universal from the viewpoint of renormalization group. We have analytically shown that the ratio $R_G/R_H$ of a topological polymer is characterized with the variance of the distance distribution function, i.e. the probability distribution function of the distance between two given segments of the polymer. If the variance decreases, then the ratio $R_G/R_H$ also decreases.
We have shown that the short-range intrachain correlation is much enhanced for real topological polymers expressed by complex graphs. We suggest that the correlation hole becomes large for topological polymers since the excluded-volume effect is enhanced due to the constraints derived from the complex structure of the polymers. The short-range correlation is characterized by the power-law behavior, and the value of the exponent increases as the graph becomes more complex.
It would therefore be interesting to observe the enhanced short-range correlation of topological polymers in the large-$q$ region of the static structure factor through scattering experiments.
\section*{Acknowledgements}
We would like to thank Profs. K. Shimokawa and Y. Tezuka for helpful discussion on topological polymers. The present research is partially supported by the Grant-in-Aid for Scientific Research No. 26310206.
| 23,257 |
\section{Introduction and Main Results}
\subsection{Backgrounds and Motivations}
Throughout the paper,
for positive integers $k$ and $n$, two symbols $[n]$ and ${[n]\choose k}$
stand for the set $\{1,\ldots,n\}$ and the family of all $k$-subsets of $[n]$, respectively.
For a positive integer $s$,
a subset $A$ of $[n]$ is said to be an $s$-stable
subset if $s\leq |i-j|\leq n-s$ for each $i\neq j\in A$. Throughout the paper, the family of all $s$-stable $k$-subsets of $[n]$ is denoted by ${[n]\choose k}_{s}$.
For $n\geq 2k$, the Kneser graph $\operatorname{KG}(n,k)$ is a graph with vertex set consisting of all $k$-subsets of $[n]$ as vertex set and two vertices are adjacent if their corresponding
$k$-sets are disjoint. Kneser~1955~\cite{MR0068536} proved that $\operatorname{KG}(n,k)$ can be properly colored with $n-2k+2$ colors. He also conjectured that this is the best possible, i.e., $\chi(\operatorname{KG}(n,k))\geq n-2k+2$. In~1978, Lov\'asz in a fascinating paper~\cite{MR514625}, using the Borsuk-Ulam theorem, proved this conjecture.
For an integer $d\geq -1$, by the symbol $S^d$, we mean the $d$-dimensional sphere.
For an $x\in S^d$, $H(x)$ is the open hemisphere centered at $x$, i.e.,
$H(x)=\{y\in S^d\;:\; \langle x\, ,y \rangle>0\}$.
There is a famous lemma due to Gale~\cite{MR0085552} which asserts
that for every $k\geq 1$ and every $n\geq 2k$, there is an $n$-set $Z\subset S^{n-2k}$
such that for any $x\in S^{n-2k}$, the open hemisphere $H(x)$ contains at least $k$ points of $Z$. In particular,
if we identify the set $Z$ with $[n]$, then for any $x\in S^{n-2k}$, the open hemisphere $H(x)$ contains some vertex of ${\rm KG}(n,k)$.
Soon after the announcement of the Lov\'asz breakthrough~\cite{MR514625}, B{\'a}r{\'a}ny~\cite{MR514626} presented a
short proof of the Lov\'asz-Kneser theorem based on Gale's lemma.
Next, Schrijver~\cite{MR512648} generalized Gale's lemma by proving that there is an $n$-set $Z\subset S^{n-2k}$ and a suitable identification of $Z$ with $[n]$
such that for any $x\in S^{n-2k}$, the open hemisphere $H(x)$ contains a $2$-stable subset of size at least $k$, i.e.,
$H(x)$ contains some vertex of ${\rm SG}(n,k)$. Using this generalization, Schrijver~\cite{MR512648} found a vertex-critical subgraph $\operatorname{SG}(n,k)$ of $\operatorname{KG}(n,k)$ which has the same chromatic number as $\operatorname{KG}(n,k)$. To be more specific, the Schrijver graph $\operatorname{SG}(n,k)$ is an induced subgraph of $\operatorname{KG}(n,k)$ whose vertex set consists of all $2$-stable $k$-subsets of $[n]$.
\subsection{Main Results}
For an $X=(x_1,\ldots,x_n)\in\{+,-,0\}^n$, an alternating subsequence of $X$ is a subsequence of nonzero terms of $X$ such that each of its two consecutive members have different signs. In other words, $x_{j_1},\ldots,x_{j_m}$ ($1\leq j_1<\cdots<j_m\leq n$) is an alternating subsequence of $X$ if $x_{j_i}\neq 0$ for each $i\in[m]$ and $x_{j_i}\neq x_{j_{i+1}}$ for $i=1,\ldots, m-1$. The length of the longest alternating subsequence of $X$ is denoted by $\operatorname{alt}(X)$. We also define $\operatorname{alt}(0,\ldots,0)=0$. Moreover,
define
$$X^+=\{j:\; x_j=+\}\quad\mbox{ and }\quad X^-=\{j:\; x_j=-\}.$$
Note that, by abuse of notation, we can write $X=(X^+,X^-)$.
Let $V$ be a nonempty finite set of size $n$.
The signed-power set of $V$, denoted $P_s(V)$, is defined as follows;
$$P_s(V)=\left\{(A,B)\ :\ A,B\subseteq V,\, A\cap B=\varnothing\right\}.$$
For two pairs $(A,B)$ and $(C,D)$ in $P_s(V)$, by $(A,B)\subseteq (C,D)$, we mean
$A\subseteq C$ and $B\subseteq D$.
Note that $\left(P_s(V),\subseteq\right)$ is a partial order set (poset).
A signed-increasing property ${\mathcal P}$, is a superset-closed family ${\mathcal P}\subseteq P_s(V)$, i.e., for any $F_1 \in {\mathcal P}$, if $F_1\subseteq F_2\in P_s(V)$, then $F_2\in {\mathcal P}$.
Clearly, for any bijection $\sigma:[n]\longrightarrow V$, the map $X\mapsto X_\sigma=(\sigma(X^+),\sigma(X^-)$ is an identification between $\{+,-,0\}^n$ and $P_s(V)$.
Let $\sigma:[n]\longrightarrow V$ be a bijection and ${\mathcal P}\subseteq P_s(V)$
be a signed-increasing property. Define
$$\operatorname{alt}({\mathcal P},\sigma)=\max\left\{\operatorname{alt}(X):\ X\in\{+,-,0\}^{n} \mbox{ with } \ {X_\sigma}\not\in{\mathcal P}\right\}.$$
Also, define {\it the alternation number of ${\mathcal P}$} to be the following quantity;
$$\operatorname{alt}({\mathcal P})=\displaystyle\min\{\operatorname{alt}({\mathcal P},\sigma): \; \sigma:[n]\longrightarrow V\mbox{ is a bijection}\}.$$
Let $d\geq 0$ be an integer, $S^d$ be the $d$-dimensional sphere, and $Z\subset S^d$ be a finite set.
For an $x\in S^d$, define $Z_x=(Z_x^+,Z_x^-)\in P_s(Z)$ such that $Z_x^+=H(x)\cap Z$ and $Z_x^-=H(-x)\cap Z$.
Now, we are in a position to state the first main result of this paper, which is a generalization of Gale's lemma.
\begin{lemma}\label{galegen}
Let $n$ be a positive integer, $V$ be an $n$-set, and $\sigma:[n]\longrightarrow V$ be a bijection.
Also, let ${\mathcal P}\subseteq P_s(V)$ be a signed-increasing property and set
$d=n-alt({\mathcal P},\sigma)-1$. If $d\neq -1$, then there are a multiset $Z\subset S^{d}$ of size $n$ and a suitable identification of $Z$ with $V$ such that for any $x\in S^{d}$, $Z_x\in{\mathcal P}$. In particular, for $d \geq 1$, $Z$ can be a set.
\end{lemma}
A hypergraph $\mathcal{H}$ is a pair $(V(\mathcal{H}),E(\mathcal{H}))$ where $V(\mathcal{H})$ is a finite nonempty set, called the vertex set of $\mathcal{H}$, and $E(\mathcal{H})$ is a family containing some nonempty distinct subsets of $V(\mathcal{H})$, called the edge set of $\mathcal{H}$. For a set $U\subseteq V(\mathcal{H})$, the induced
subhypergraph $\mathcal{H}[U]$ of $\mathcal{H}$, is a
hypergraph with vertex set $U$ and edge set $\{e\in E(\mathcal{H})\;:\; e\subseteq U\}$.
A graph $G$ is a hypergraph such that each of its edges has cardinality two.
A $t$-coloring of a hypergraph $\mathcal{H}$ is a map $c:V(\mathcal{H})\longrightarrow [t]$
such that for no edge $e\in E(\mathcal{H})$, we have $|c(e)|=1$.
The minimum possible $t$ for which $\mathcal{H}$ admits a $t$-coloring, denoted $\chi(\mathcal{H})$, is called the chromatic number of $\mathcal{H}$.
For a hypergraph $\mathcal{H}$, the Kneser graph $\operatorname{KG}(\mathcal{H})$ is a graph whose vertex set is $E(\mathcal{H})$ and two vertices are adjacent if their corresponding edges are vertex disjoint.
It is known that for any graph $G$, there are several hypergraphs $\mathcal{H}$ such that
$G$ and $\operatorname{KG}(\mathcal{H})$ are isomorphic. Each of such hypergraphs $\mathcal{H}$ is called a Kneser representation of $G$.
Note that if we set $K_n^k=\left([n],{[n]\choose k}\right)$ and $\widetilde{K_n^k}=\left([n],{[n]\choose k}_{2}\right)$, then $\operatorname{KG}(K_n^k)=\operatorname{KG}(n,k)$ and $\operatorname{KG}(\widetilde{K_n^k})=\operatorname{SG}(n,k)$.
The {\it colorability defect} of a hypergraph $\mathcal{H}$, denoted $\operatorname{cd}(\mathcal{H})$,
is the minimum number of vertices which should be excluded so that the induced subhypergraph on the remaining vertices is $2$-colorable.
Dol'nikov~\cite{MR953021} improved Lov\'asz's result~\cite{MR514625} by proving that for any hypergraph $\mathcal{H}$, we have
$\chi(\operatorname{KG}(\mathcal{H}))\geq\operatorname{cd}(\mathcal{H}).$
Let ${\mathcal H}=(V,E)$ be a hypergraph and $\sigma:[n]\longrightarrow V(\mathcal{H})$
be a bijection. Define
$$\operatorname{alt}(\mathcal{H},\sigma)=\max\left\{\operatorname{alt}(X):\; X\in\{+,-,0\}^n\mbox{ s.t. } \max\left(|E(\mathcal{H}[\sigma(X^+)])|,|E(\mathcal{H}[\sigma(X^-)])|\right)=0\right\}$$
and
$$\operatorname{salt}(\mathcal{H},\sigma)=\max\left\{\operatorname{alt}(X):\; X\in\{+,-,0\}^n\mbox{ s.t. } \displaystyle\min\left(\left|E(\mathcal{H}[\sigma(X^+)])\right|,|E(\mathcal{H}[\sigma(X^-)])|\right)=0\right\}.$$
In other words, $\operatorname{alt}(\mathcal{H},\sigma)$ (resp. $\operatorname{salt}(\mathcal{H},\sigma)$) is the maximum possible $\operatorname{alt}(X)$, where $X\in\{+,-,0\}^n$, such that each of (resp. at least one of) $\sigma(X^+)$ and $\sigma(X^+)$ contains no edge of $\mathcal{H}$.
Also, we define
$$\operatorname{alt}(\mathcal{H})=\displaystyle\min_{\sigma}\operatorname{alt}(\mathcal{H},\sigma)\quad \mbox{and} \quad
\operatorname{salt}(\mathcal{H})=\displaystyle\min_{\sigma}\operatorname{salt}(\mathcal{H},\sigma),$$
where the minimum is taken over all bijections $\sigma:[n]\longrightarrow V(\mathcal{H})$.
The present authors, using Tucker's lemma~\cite{MR0020254}, introduced two combinatorial tight lower bounds for the chromatic number of $\operatorname{KG}(\mathcal{H})$ improving Dol'nikov's lower bound.
\begin{alphtheorem}{\rm \cite{2013arXiv1302.5394A}}\label{alihajijctb}
For any hypergraph $\mathcal{H}$, we have
$$\chi(\operatorname{KG}(\mathcal{H}))\geq\displaystyle\max\left(|V(\mathcal{H})|-\operatorname{alt}(\mathcal{H}),|V(\mathcal{H})|-\operatorname{salt}(\mathcal{H})+1\right).$$
\end{alphtheorem}
Let ${\mathcal H}=(V,E)$ be a hypergraph and $\sigma:[n]\longrightarrow V(\mathcal{H})$
be a bijection.
Note that if we set
$${\mathcal P}_1=\left\{(A,B)\in P_s(V):\mbox{ at least one of $A$ and $B$ contains some edge of } {\mathcal H}\right\}$$
and
$${\mathcal P}_2=\left\{(A,B)\in P_s(V):\mbox{ both of $A$ and $B$ contain some edges of } {\mathcal H}\right\},$$
then $\operatorname{alt}({\mathcal P}_1,\sigma)=\operatorname{alt}({\mathcal H},\sigma)$ and $\operatorname{alt}({\mathcal P}_2,\sigma)=\operatorname{salt}({\mathcal H},\sigma)$. Therefore, in view of Lemma~\ref{galegen},
we have the next result.
\begin{corollary}\label{galegencor}
For a hypergraph ${\mathcal H}=(V, E)$ and a bijection $\sigma:[n]\longrightarrow V(\mathcal{H})$,
we have the following assertions.
\begin{itemize}
\item[{\rm a)}]If $d=|V|-\operatorname{alt}({\mathcal H},\sigma)-1$ and $\operatorname{alt}({\mathcal H},\sigma)\neq |V|$, then there are a multiset $Z\subset S^{d}$ of size $|V|$ and a suitable identification of $Z$ with $V$
such that for any $x\in S^{d}$, $H(x)$ or $H(-x)$ contains some edge of ${\mathcal H}$.
In particular, for $d\geq 1$, $Z$ can be a set.
\item[{\rm b)}]If $d=|V|-\operatorname{salt}({\mathcal H},\sigma)-1$ and $\operatorname{salt}({\mathcal H},\sigma)\neq |V|$, then there are a multiset $Z\subset S^{d}$ of size $|V|$ and a suitable identification of $Z$ with $V$
such that for any $x\in S^{d}$, $H(x)$ contains some edge of ${\mathcal H}$.
In particular, for $d\geq 1$, $Z$ can be a set.
\end{itemize}
\end{corollary}
For two positive integers $n$ and $k$, where $n> 2k$,
and for the identity bijection $I:[n]\longrightarrow [n]$, one can see that
$\operatorname{salt}(\widetilde{K_n^k},I)=2k-1$. Therefore, by the second part of
Corollary~\ref{galegencor}, we have the generalization of
Gale's lemma given by Schrijver~\cite{MR512648}:
{\it there is an $n$-subset $Z$ of $S^{n-2k}$ and a suitable identification
of $Z$ with $[n]$ such that
for any $x\in S^{n-2k}$, the hemisphere $H(x)$ contains at least a $2$-stable subset of $[n]$ with size at least $k$.}
Note that for any graph $G$, there are several hypergraphs $\mathcal{H}$
such that $\operatorname{KG}(\mathcal{H})$ and $G$ are isomorphic.
By the help of Lemma~\ref{galegen} and as the second main result of this paper,
we provide two combinatorial approximations for two important topological
lower bounds for the chromatic number of a graph $G$, namely, $ {\rm coind}(B_0(G))+1$ and ${\rm coind}(B(G))+2$, see~\cite{MR2452828}. The quantities ${\rm coind}(B_0(G))$ and ${\rm coind}(B(G))$ are respectively the coindices
of two box complexes ${\rm B}_0(G)$ and ${\rm B}(G)$ which
will be defined in Section~\ref{Preliminaries}. It should be mentioned that
the two inequalities $\chi(G) \geq {\rm coind}(B_0(G))+1$
$\chi(G) \geq {\rm coind}(B(G))+2$
are already proved~\cite{MR2452828} and we restate them in the following
theorem just to emphasize that this theorem is
an improvement of Theorem~\ref{alihajijctb}.
\begin{theorem}\label{coind}
Let $G$ be a graph and $\mathcal{H}$ be a hypergraph such that $\operatorname{KG}(\mathcal{H})$ and
$G$ are isomorphic. Then the following inequalities hold;
\begin{itemize}
\item[{\rm a)}] $\chi(G) \geq {\rm coind}(B_0(G))+1\geq |V(\mathcal{H})|-\operatorname{alt}(\mathcal{H})$,
\item[{\rm b)}] $\chi(G) \geq {\rm coind}(B(G))+2\geq |V(\mathcal{H})|-\operatorname{salt}(\mathcal{H})+1$.
\end{itemize}
\end{theorem}
Note that, in addition to presenting another proof for Theorem~\ref{alihajijctb},
Theorem~\ref{coind} also reveals a new way to compute
parameters ${\rm coind}(B_0(G))$ and ${\rm coind}(B(G))$ for
a graph $G$ as well.
Also, it is worth noting that the results in~\cite{2013arXiv1302.5394A, 2014arXiv1401.0138A, MatchingNew} confirm that we can evaluate
the chromatic number of some family of graphs by computing $\operatorname{alt}(-)$ or $\operatorname{salt}(-)$ for an appropriate choice of a hypergraph, while it seems
that it is~not easy to directly determine
the amounts of ${\rm coind}(B_0(-))+1$ and
${\rm coind}(B(-))+2$ for these graphs by topological methods.\\
\noindent{\bf Remark.} Motivated by Corollary~\ref{galegencor}, we can assign to any hypergraph $\mathcal{H}$ two topological parameters; the
{\it dimension of $\mathcal{H}$} and the {\it strong dimension of $\mathcal{H}$} denoted by ${\rm dim}(\mathcal{H})$ and ${\rm sdim}(\mathcal{H})$, respectively.
The dimension of $\mathcal{H}$ (resp. strong dimension of $\mathcal{H}$) is the
maximum integer $d\geq -1$ such that the vertices of $\mathcal{H}$
can be identified with a multiset $Z\subset S^d$
so that for any $x\in S^d$, at least one of (resp. both of ) open hemispheres $H(x)$ and $H(-x)$ contains some edge of $\mathcal{H}$.
In view of Corollary~\ref{galegencor} and with a proof similar to the one of Theorem~\ref{coind}, one can prove the following corollary.
\begin{corollary}
For a hypergraph $\mathcal{H}$, we have the following inequalities;
\begin{itemize}
\item[{\rm a)}] ${\rm coind}(B_0(\operatorname{KG}(\mathcal{H})))\geq {\rm dim}(\mathcal{H}) \geq |V(\mathcal{H})|-\operatorname{alt}(\mathcal{H})-1$,
\item[{\rm b)}] ${\rm coind}(B(\operatorname{KG}(\mathcal{H})))\geq {\rm sdim}(\mathcal{H}) \geq |V(\mathcal{H})|-\operatorname{salt}(\mathcal{H})-1$.
\end{itemize}
\end{corollary}
This paper is organized as follows.
Section~\ref{Preliminaries} contains a brief review of elementary but essential preliminaries and definitions which will be needed throughout the paper. Section~\ref{proofs} is devoted to the proof of Lemma~\ref{galegen} and Theorem~\ref{coind}. Also, in this section, as an application of the generalization of Gale's lemma (Lemma~\ref{galegen}), we reprove a result by Chen~\cite{JGT21826} about the multichromatic number of stable Kneser graphs.
\section{Topological Preliminaries}\label{Preliminaries}
The following is a brief overview of some topological concepts needed throughout the paper.
We refer the reader to~\cite{MR1988723, MR2279672} for more basic definitions of algebraic topology.
A $\mathbb{Z}_2$-space is a pair $(T,\nu)$, where $T$ is a topological space
and $\nu$ is an {\it involution}, i.e., $\nu:T\longrightarrow T$ is a continuous map such that $\nu^2$ is the identity map.
For an $x\in T$, two points $x$ and $\nu(x)$ are called {\it antipodal}.
The $\mathbb{Z}_2$-space $(T,\nu)$ is called {\it free} if
there is no $x\in T$ such that $\nu(x)=x$.
For instance, one can see that the unit sphere $S^d\subset \mathbb{R}^{d+1}$ with the involution given by the antipodal map $-:x\rightarrow -x$ is free.
For two $\mathbb{Z}_2$-spaces $(T_1,\nu_1)$ and $(T_2,\nu_2)$,
a continuous map $f:T_1\longrightarrow T_2$ is a $\mathbb{Z}_2$-map if $f\circ\nu_1=\nu_2\circ f$.
The existence of such a map is denoted by $(T_1,\nu_1)\stackrel{\mathbb{Z}_2}{\longrightarrow} (T_2,\nu_2)$.
For a $\mathbb{Z}_2$-space $(T,\nu)$, we define the $\mathbb{Z}_2$-index and $\mathbb{Z}_2$-coindex of $(T,\nu)$, respectively, as follows
$${\rm ind}(T,\nu)=\min\{d\geq 0\:\ (T,\nu)\stackrel{\mathbb{Z}_2}{\longrightarrow} (S^d,-)\}$$
and
$${\rm coind}(T,\nu)=\max\{d\geq 0\:\ (S^d,-)\stackrel{\mathbb{Z}_2}{\longrightarrow} (T,\nu)\}.$$
If for any $d\geq 0$, there is no $(T,\nu)\stackrel{\mathbb{Z}_2}{\longrightarrow} (S^d,-)$, then we set ${\rm ind}(T,\nu)=\infty$.
Also, if $(T,\nu)$ is~not free, then ${\rm ind}(T,\nu)={\rm coind}(T,\nu)=\infty$.
For simplicity of notation, when the involution is understood from the context, we speak about $T$ rather than the pair $(T,\nu)$; also,
we set ${\rm ind}(T,\nu)={\rm ind}(T)$ and ${\rm coind}(T,\nu)={\rm coind}(T)$.
Throughout this paper,
we endow the unit sphere $S^d\subset \mathbb{R}^{d+1}$ with the involution given by the antipodal map.
Note that if $T_1\stackrel{\mathbb{Z}_2}{\longrightarrow} T_2$, then ${\rm ind}(T_1)\leq {\rm ind}(T_2)$ and
${\rm coind}(T_1)\leq {\rm coind}(T_2)$. Two $\mathbb{Z}_2$-spaces $T_1$ and $T_2$ are
$\mathbb{Z}_2$-equivalent, denoted $T_1\stackrel{\mathbb{Z}_2}{\longleftrightarrow} T_2$, if $T_1\stackrel{\mathbb{Z}_2}{\longrightarrow} T_2$ and $T_2\stackrel{\mathbb{Z}_2}{\longrightarrow} T_1$. In particular, $\mathbb{Z}_2$-equivalent spaces have the same index and also coindex.
In the following, we introduce the concept of simplicial complex
which provides a bridge between combinatorics and topology. A simplicial complex can be viewed as a combinatorial object, called abstract simplicial complex, or as a topological space, called geometric simplicial complex.
Here we just present the definition of an abstract simplicial complex. However, it should be mentioned that
we can assign a geometric simplicial complex to an abstract simplicial complex, called its geometric realization, and vice versa.
An {\it abstract simplicial complex} is a pair $L=(V,K)$, where $V$ (the vertex set of $L$) is a set and $K\subseteq 2^V$ (the set of simplicial complexes of $L$)
is a hereditary collection of subsets of $V$, i.e., if $A\in K$ and $B\subseteq A$, then $B\in K$.
Any set $A\in K$ is called a complex of $L$.
The geometric realization of an abstract simplicial complex $L$ is denoted by $||L||$.
For two abstract simplicial complexes $L_1=(V_1,K_1)$ and $L_2=(V_2,K_2)$,
a simplicial map $f:L_1\longrightarrow L_2$ is map from $V_1$ into $V_2$ which
preserves the complexes, i.e., if $A\in K_1$, then $f(A)\in K_2$.
A simplicial involution is a simplicial map $\nu: L\longrightarrow L$ such that $\nu^2$
is the identity map.
A $\mathbb{Z}_2$-simplicial complex is a pair $(L,\nu)$ where $L$ is a simplicial complex and $\nu:L\longrightarrow L$ is a simplicial involution.
A simplicial involution $\nu$ and a simplicial complex $(L,\nu)$ is free if there is no face $x$ of $L$ such that $\nu(x)=x$.
For two $\mathbb{Z}_2$-simplicial complexes $(L_1,\nu_1)$ and $(L_2,\nu_2)$,
the map $f:L_1\longrightarrow L_2$ is called a
$\mathbb{Z}_2$-simplicial map if $f$ is a simplicial map and $f\circ\nu_1=\nu_2\circ f$.
The existence of a simplicial map ($\mathbb{Z}_2$-simplicial map) $f:L_1\longrightarrow L_2$ implies the existence of a continuous $\mathbb{Z}_2$-map $||f||:||L_1||\stackrel{\mathbb{Z}_2}{\longrightarrow} ||L_2||$
which is called the geometric realization of $f$. If $||L||$ is a ${\mathbb{Z}_2}$-space, we use ${\rm ind}(L)$ and ${\rm coind}(L)$ for ${\rm ind}(||L||)$ and ${\rm coind}(||L||)$, respectively.
The existence of a homomorphism between two graphs is an important and generally challenging problem in graph theory.
In particular, in general, it is a hard task to determine the chromatic number of a graph $G$.
In the following, we assign some free simplicial $\mathbb{Z}_2$-complexes to graphs
in such a way that graph homomorphisms give rise to $\mathbb{Z}_2$-maps of the corresponding complexes. For a graph $G=(V(G),E(G))$ and
two disjoint subsets $A, B\subseteq V(G)$, define $G[A,B]$ to be the induced bipartite subgraph of $G$ whose parts are $A$ and $B$.\\
\noindent{\bf Box Complex.}
For a graph $G=(V(G),E(G))$ and a subset $A\subseteq V(G)$, set
$${\rm CN}(A)=\{v\in V(G):\ av\in E(G)\ {\rm for\ all\ } a\in A\ \}\subseteq V(G)\setminus A.$$
The {\it box complex} of a graph $G$, $B(G)$, is a free simplicial $\mathbb{Z}_2$- complex with the vertex set $V(G) \uplus V(G) = V(G)\times[2]$ and the following set of simplices
$$\{A\uplus B:\ A,B\subseteq V(G),\ A\cap B=\varnothing,\ G[A,B]\ {\rm is\ complete,\ and}\ {\rm CN}(A)\neq\varnothing \neq {\rm CN}(B) \}.$$
Also, one can consider another box complex $B_0(G)$ with the vertex set $V(G) \uplus V(G) = V(G)\times[2]$ and the following set of simplices
$$\{A\uplus B:\ A,B\subseteq V(G),\ A\cap B=\varnothing,\ G[A,B]\ {\rm is\ complete} \}.$$
An involution on $B(G)$ (resp. $B_0(G)$) is given by interchanging the two copies of $V(G)$;
that is, $(v,1)\rightarrow(v,2)$ and $(v, 2)\rightarrow(v, 1)$, for any $v \in V (G)$. In view of these involutions, one can consider
$||B(G)||$ and $||B_0(G)||$
as free ${\mathbb Z}_2$-spaces. One can check that any graph homomorphism $G\rightarrow H$ implies that there are
two simplicial ${\mathbb Z}_2$-maps $B(G)\stackrel{\mathbb{Z}_2}{\longrightarrow} B(H)$ and $B_0(G)\stackrel{\mathbb{Z}_2}{\longrightarrow} B_0(H)$; and consequently,
${\rm ind}(B(G))\leq {\rm ind}(B(H))$, ${\rm coind}(B(G))\leq {\rm coind}(B(H))$,
${\rm ind}(B_0(G))\leq {\rm ind}(B_0(H))$, and ${\rm coind}(B_0(G))\leq {\rm coind}(B_0(H))$.
One can check that $B(K_n)$ and $B_0(K_n)$ are $\mathbb{Z}_2$-equivalent to $S^{n-2}$ and
$S^{n-1}$, respectively. Hence, $\chi(G)\geq {\rm ind}(B(G))+2\geq {\rm coind}(B(G))+2$ and
$\chi(G)\geq {\rm ind}(B_0(G))+1\geq {\rm coind}(B_0(G))+1$.
Indeed, it is known (see~\cite{MR1988723, MR2279672, MR2452828})
\begin{equation}\label{lbchrom}
\chi(G)\geq {\rm ind}(B(G))+2 \geq {\rm ind}(B_0(G))+1 \geq {\rm coind}(B_0(G))+1 \geq {\rm coind}(B(G))+2.
\end{equation}
\section{Proof of Main Results}\label{proofs}
We should mention that
the following proof is based on an idea similar to that used in an
interesting proof of Ziegler for Gale's lemma~(see page 67 in \cite{MR1988723}).
\noindent{\bf Proof of Lemma~\ref{galegen}.}
For simplicity of notation, assume that $V=\{v_1,\ldots,v_n\}$ where $\sigma(i)=v_i$.
Consider the following curve
$$\gamma=\{(1,t,t^2,\ldots t^{d})\in\mathbb{R}^{d+1}:\ t\in\mathbb{R}\}$$
and set $W=\{w_1,w_2,\ldots,w_n\}$, where $w_i=\gamma(i)$, for $i=1,2,\ldots,n$. Now, let $Z=\{z_1,z_2,\ldots,z_n\}\subseteq S^d$ be a set such that $z_i=(-1)^i{w_i\over ||w_i||}$,
for any $1\leq i\leq n$. Note that if $d\geq 1$, then $Z$ is a set.
Consider the identification between $V$ and $Z$ such that $v_i\in V$ is identified with $z_i$, for any $1\leq i\leq n$.
It can be checked that every hyperplane of $\mathbb{R}^{d+1}$ passing trough the origin intersects
$\gamma$ in no more than $d$ points.
Moreover, if a hyperplane intersects the curve in exactly $d$ points, then the hyperplane cannot be tangent to the curve; and consequently, at each intersection point, the curve passes from one side of the hyperplane to the other side.
Now, we show that for any $y\in S^d$, $Z_{y}\in {\mathcal P}$.
On the contrary, suppose that there is a $y\in S^d$ such that $Z_{y}\not\in {\mathcal P}$.
Let $h$ be the hyperplane passing trough the origin which contains
the boundary of $H(y)$.
We can move this hyperplane continuously to a position such that it still contains the origin and
has exactly $d$ points of $W=\{w_1,w_2,\ldots,w_n\}$ while during this movement no points of $W$ crosses from one side of $h$ to the other side.
Consequently, during the aforementioned movement, no points of $Z=\{z_1,z_2,\ldots,z_n\}$
crosses from one side of $h$ to the other side.
Hence, at each of these intersections, $\gamma$ passes from one side of $h$ to the other side.
Let $h^+$ and $h^-$ be two open half-spaces determined by the hyperplane $h$.
Now consider $X=(x_1,x_2,\ldots,x_n)\in\{+,-,0\}^{n}\setminus\{\boldsymbol{0}\}$ such that
$$
x_i=\left\{
\begin{array}{cl}
0 & {\rm if}\ w_i\ {\rm is\ on}\ h\\
+ & {\rm if}\ w_i\ {\rm is\ in }\ h^+\ {\rm and}\ i\ {\rm is\ even}\\
+ & {\rm if}\ w_i\ {\rm is\ in }\ h^-\ {\rm and}\ i\ {\rm is\ odd}\\
- & {\rm otherwise.}
\end{array}.\right.$$
Assume that $x_{i_1},x_{i_2},\ldots,x_{i_{n-d}}$ are nonzero entries of $X$, where $i_1<i_2<\cdots <i_{n-d}$.
It is easy to check that any two consecutive terms of $x_{i_j}$'s have different signs.
Since $X$ has $n-m=\operatorname{alt}({\mathcal P},\sigma)+1$ nonzero entries, we have $\operatorname{alt}(X)=\operatorname{alt}(-X)=\operatorname{alt}({\mathcal P},\sigma)+1$;
and therefore, both $X_\sigma$ and $(-X)_\sigma$ are in ${\mathcal P}$. Also, one can see that
either $X_\sigma\subseteq Z_y$ or $(-X)_\sigma\subseteq Z_y$. Therefore, since ${\mathcal P}$ is an signed-increasing property, we have $Z_y\in {\mathcal P}$ which is a contradiction.
\hfill$\square$\\
\noindent{\bf Multichromatic Number of Stable Kneser Graphs.}
For positive integers $n,k$, and $s$, the $s$-stable Kneser graph $\operatorname{KG}(n,k)_s$
is an induced subgraph of $\operatorname{KG}(n,k)$ whose vertex set is ${[n]\choose k}_s$. In other words, $\operatorname{KG}(n,k)_s=\operatorname{KG}(\left([n],{[n]\choose k}_s\right))$. The chromatic number of stable Kneser graphs has been studied in several papers~\cite{MR2448565,jonsson,MR2793613}. Meunier~\cite{MR2793613} posed a conjecture about the chromatic number of stable Kneser hypergraphs which is a generalization of a conjecture of Alon, Drewnowski, and
{\L}uczak~\cite{MR2448565}.
In the case of graphs instead of hypergraphs, Meunier's conjecture asserts that the chromatic number of $\operatorname{KG}(n,k)_s$ is $n-s(k-1)$ for $n\geq sk$ and $s\geq 2$. Clearly, in view of Schrijver's result~\cite{MR512648}, this conjecture is true for $s=2$. Moreover,
for $s\geq 4$ and $n$ sufficiently large, Jonsson~\cite{jonsson} gave an affirmative answer to the graph case of Meunier's conjecture.
For two positive integers $m$ and $n$ with $n\geq m$,
an $m$-fold $n$-coloring of a graph $G$ is a homomorphism from $G$ to ${\rm KG}(n,m)$.
The $m^{th}$ multichromatic number of a graph $G$, $\chi_m(G)$, is defined as follows
$$\displaystyle\chi_m(G)=\min\left\{n:\ G\longrightarrow {\rm KG}(n,m)\right\}.$$
Note that $\chi_1(G)=\chi(G)$.
In general, an $m$-fold $n$-coloring of a graph $G$ is called a multicoloring of $G$ with color set $[n]$.
The following conjecture of Stahl~\cite{Stahl1976185} has received a considerable attention in the literature.
\begin{alphconjecture}{\rm (\cite{Stahl1976185})}\label{stahlconj}
For positive integers $m, n$, and $k$ with $n\geq 2k$, we have
$$\chi_m({\rm KG}(n,k))=\lceil{m\over k}\rceil(n-2k)+2m.$$
\end{alphconjecture}
Stahl~\cite{Stahl1998287} proved the accuracy of this Conjecture for $k=2,3$ and arbitrary values of $m$. Chen~\cite{JGT21826} studied the multichromatic number of
$s$-stable Kneser graphs and generalized Schrijver's result.
In what follows, as an application of Lemma~\ref{galegen}, we present another proof of Chen's result.
Note that, as a special case of Chen's result, we have the chromatic number of $s$-stable Kneser graphs provided that $s$ is even.
\begin{alphtheorem}{\rm \cite{JGT21826}}
For positive integers $n, k,$ and $s$ with $n\geq sk$, if $s$ is an even integer and $k\geq m$, then
$\chi_m({\rm KG}(n,k)_s)=n-sk+sm$.
\end{alphtheorem}
\begin{proof}{
It is straightforward to check that
$\chi_m({\rm KG}(n,k)_s)\leq n-sk+sm$.
For a proof of this observation, we refer the reader to~\cite{JGT21826}.
Therefore, it is enough to show $\chi_m({\rm KG}(n,k)_s)\geq n-sk+sm$.
For the set $[n]$,
let ${\mathcal P}={\mathcal P}(n,k,s)\subseteq P_s([n])$ be a signed-increasing property such that
$(A,B)\in {\mathcal P}$ if each of $A$ and $B$ contains at least ${s\over 2}$ pairwise disjoint
$s$-stable $k$-subsets of $[n]$. One can see that $\operatorname{alt}({\mathcal P},I)=sk-1$ where $I:[n]\longrightarrow [n]$ is the identity bijection. Thus, by Lemma~\ref{galegen}, for $d=n-sk$, there exists a multiset $Z\subset S^{d}$ of size $n$ such that under a suitable identification of $Z$ with $V$, for any $x\in S^{d}$, $Z_x\in{\mathcal P}$. In other words, for any $x\in S^{d}$, $H(x)$ contains at least ${s\over 2}$ pairwise disjoint vertices of ${\rm KG}(n,k)_s$.
Now let $c:V({\rm KG}(n,k)_s)\longrightarrow {C\choose m}$ be an $m$-fold $C$-coloring of ${\rm KG}(n,k)_s$.
For each $i\in \{1,2,\ldots,C-sm+1\}$, define
$A_i$ to be a set consisting of all $x\in S^d$ such that $H(x)$ contains some vertex with color $i$.
Furthermore, define $A_{C-sk+2}=S^d\setminus \displaystyle\cup_{i=1}^{C-sm+1}A_i$.
One can check that each $A_i$ contains no pair of
antipodal points, i.e., $A_i\cap (-A_i)=\varnothing$; and also, for $i\in\{1,2,\ldots, C-sm+1\}$,
$A_i$ is an open subset of $S^d$ and $A_{C-sm+2}$
is closed. Now by the Borsuk-Ulam theorem, i.e., for any covering of $S^{d}$ by $d+1$ sets $B_1,\ldots, B_{d+1}$, each $B_i$ open or closed, there exists an
$i$ such that $B_i$ contains a pair of antipodal points. Accordingly, we have $C-sm+2\geq d+2=n-sk+2$ which completes the proof.
}\end{proof}
In what follows, in view of Lemma~\ref{galegen} and with a similar approach as in proof of Proposition~8 of~\cite{MR2279672}, we prove Theorem~\ref{coind}.
\noindent{\bf Proof of Theorem~\ref{coind}.}
Let $\sigma:[n]\longrightarrow V(\mathcal{H})$ be an arbitrary bijection.
To prove the first part, set
$d=|V|-\operatorname{alt}({\mathcal H},\sigma)-1$. In view of inequalities of (\ref{lbchrom}) and the definition of $\operatorname{alt}({\mathcal H})$, it is sufficient to prove that
${\rm coind}(B_0(G))+1\geq |V|-\operatorname{alt}({\mathcal H},\sigma)$. If $d\leq 0$, then one can see that the assertion follows.
Hence, suppose $d\geq 1$. Now in view of Corollary~\ref{galegencor}, there exists an $n$-set $Z\subset S^d$ and an identification of $Z$ with $V$ such that for any $x\in S^{d}$, at least one of open hemispheres $H(x)$ and $H(-x)$ contains some edge of ${\mathcal H}$.
For any vertex $A$ of ${\rm KG}({\mathcal H})$ and any $x\in S^d$, define $D_A(x)$ to be
the smallest distance of a point in $A\subset S^d$ from the set $S^d\setminus H(x)$.
Note that $D_A(x)>0$ if and only if $H(x)$ contains $A$.
Define
$$D(x)=\displaystyle\sum_{A\in E}\left(D_A(x)+D_A(-x)\right).$$
Since,
for any $x\in S^d$, at least one of $H(x)$ and $H(-x)$ contains some edge of ${\mathcal H}$,
we have $D(x)>0$.
Thus, the map
$$f(x)= {1\over D(x)}\left(\sum_{A\in E}D_A(x)||(A,1)||+\sum_{A\in E}D_A(-x)||(A,2)||\right)$$
is a $\mathbb{Z}_2$-map from $S^d$ to $||B_0({\rm KG}({\mathcal H}))||$. It implies ${\rm coind}(B_0(G))\geq d$.\\
\noindent {\rm b)} To prove the second part, set $d=n-\operatorname{salt}({\mathcal H},\sigma)-1$. In view of inequalities of (\ref{lbchrom}) and the definition of $\operatorname{salt}({\mathcal H})$,
it is sufficient to prove that
${\rm coind}(B(G))+2\geq |V|-\operatorname{salt}({\mathcal H},\sigma)+1$. If $d\leq 0$, then one can see that the assertion follows.
Hence, suppose $d\geq 1$. Now
in view of Corollary~\ref{galegencor}, there is an $n$-set $Z\subset S^d$ and an identification of $Z$ with $V$ such that for any $x\in S^{d}$,
$H(x)$ contains some edge of ${\mathcal H}$. Define
$D(x)=\displaystyle\sum_{A\in E}D_A(x).$
For any $x\in S^d$, $H(x)$ has some edge of ${\mathcal H}$ which implies that $D(x)>0$.
Thus, the map
$$f(x)= {1\over 2D(x)}\sum_{A\in E}D_A(x)||(A,1)||+{1\over 2D(-x)}\sum_{A\in E}D_A(-x)||(A,2)||$$
is a $\mathbb{Z}_2$-map from $S^d$ to $||B({\rm KG}({\mathcal H}))||$. This implies ${\rm coind}(B(G))\geq d$.
\hfill$\square$\\
\noindent{\bf Acknowledgement:}
The authors would like to express their deepest gratitude to Professor Carsten~Thomassen for his insightful comments.
They also appreciate the detailed valuable comments of Dr.~Saeed~Shaebani.
The research of Hossein Hajiabolhassan is partially supported by ERC advanced grant GRACOL. A part of this paper was written while Hossein Hajiabolhassan was visiting School of Mathematics, Institute for Research in Fundamental Sciences~(IPM). He acknowledges the support of IPM.
Moreover, they would like to thank Skype for sponsoring their endless conversations in two countries.
\def$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'${$'$}
| 13,040 |
\section{Preliminaries}
In this section we recall some basic notation and results
concerning digraphs and their spectra.
A digraph $G=(V,E)$ consists of a (finite) set
$V=V(G)$ of vertices and a set $E=E(G)$ of arcs (directed edges) between vertices
of $G$. As the initial and final vertices of an arc are not necessarily different, the
digraphs may have \emph{loops} (arcs from a vertex to itself),
and \emph{multiple arcs}, that is, there can be more than one arc from each vertex
to any other. If $a=(u,v)$ is an arc from $u$ to $v$, then vertex $u$ (and arc $a$)
is {\em adjacent to} vertex $v$, and vertex $v$ (and arc $a$) is {\em adjacent from}
$u$. Let $G^+(v)$ and $G^-(v)$ denote the set of arcs adjacent from and to vertex
$v$, respectively. A digraph $G$ is $d$\emph{-regular} if $|G^+(v)|=|G^-(v)|=d$ for all $v\in V$.
In the line digraph $L(G)$ of a digraph $G$, each vertex of $L(G)$ represents an
arc of $G$, that is, $V(L(G))=\{uv|(u,v)\in E(G)\}$; and vertices $uv$ and $wz$ of $L(G)$ are adjacent if and only if $v=w$, namely, when arc $(u,v)$ is adjacent to arc $(w,z)$ in $G$. For $k\geq0$, we consider the sequence of line digraph iterations
$L^0(G)=G,L(G),L^2(G),\ldots,L^k(G)=L(L^{k-1}(G)),\ldots$
It can be easily seen that every vertex of $L^k(G)$ corresponds to a walk
$v_0,v_1,\ldots,v_k$ of length $k$ in $G$, where $(v_{i-1,},v_{i})\in E$ for $i=1,\ldots,k$.
Then, if there is one arc between pairs of vertices and $\mbox{\boldmath $A$}$ is the adjacency matrix of $G$, the $uv$-entry of the power $\mbox{\boldmath $A$}^k$, denoted by $a_{uv}^{(k)}$, is the number of $k$-walks from vertex $u$ to vertex $v$, and the order $n_k$ of $L^k(G)$ turns out to be
\begin{equation}\label{orderL^kG}
n_k=\mbox{\boldmath $j$}\mbox{\boldmath $A$}^k\mbox{\boldmath $j$}^{\top},
\end{equation}
where $\mbox{\boldmath $j$}$ stands for the all-$1$ vector. If there are multiple arcs between pairs of vertices, then the corresponding entry in the matrix is not 1, but the number of these arcs.
If $G$ is a $d$-regular digraph with $n$ vertices then its line
digraph $L^k(G)$ is
$d$-regular with $n_k=d^kn$ vertices.
Recall also that a digraph $G$ is \emph{strongly connected} if there is a
(directed) walk between every
pair of its vertices.
If $G$ is strongly connected, different from a directed cycle, and it has
diameter $D$, then its line digraph $L^k(G)$ has diameter $D+k$. See Fiol, Yebra, and Alegre~\cite{fya84} for more details.
The interest of the line digraph technique is that it allows us to obtain digraphs with small diameter and large connectivity.
For a comparison between the line digraph technique and other techniques to obtain digraphs with minimum diameter see Miller, Slamin, Ryan and Baskoro~\cite{MiSlRyBa13}. Since these techniques are related to the degree/diameter problem, we refer also to the comprehensive survey on this problem by Miller and \v{S}ir\'{a}\v{n}~\cite{ms}.
For the concepts and/or results not presented here, we refer the reader to some of
the basic textbooks and papers on the subject; about digraphs see, for instance,
Chartrand and Lesniak~\cite{cl96} or Diestel~\cite{d10}, and Godsil~\cite{g93} about the quotient graphs.
This paper is organized as follows. In Section~\ref{sec:reg-part}, we recall the
definition of regular partitions and we give some lemmas about them.
In Section~\ref{sec:main-result} we prove our main result.
In Section~\ref{sec:examples}, we give examples in which the sequence on the number
of vertices of iterated line digraphs is increasing, tending to a positive
constant, or tending to zero.
\section{Regular partitions}
\label{sec:reg-part}
Let $G$ be a digraph with adjacency matrix $\mbox{\boldmath $A$}$. A partition $\pi=(V_1,\ldots,
V_m)$ of its vertex set $V$ is called {\em regular} (or {\em equitable})
whenever, for any $i,j=1,\ldots,m$, the {\em intersection numbers}
$b_{ij}(u)=|G^+(u)\cap V_j|$, where $u\in V_i$, do not depend on the vertex $u$ but only on the subsets (usually called {\em classes} or {\em
cells}) $V_i$ and $V_j$. In this case, such numbers are simply written as $b_{ij}$,
and the $m\times m$ matrix $\mbox{\boldmath $B$}=(b_{ij})$ is referred to as the {\em quotient matrix} of $\mbox{\boldmath $A$}$ with respect to $\pi$. This is also represented by the {\em quotient (weighted) digraph} $\pi(G)$ (associated to the partition $\pi$),
with vertices representing the cells, and an arc with weight $b_{ij}$ from vertex
$V_i$ to vertex $V_j$ if and only if $b_{ij}\neq 0$. Of course, if $b_{ii}>0$ for some $i=1,\ldots,m$, the quotient digraph $\pi(G)$ has loops.
The {\em characteristic matrix} of (any) partition $\pi$ is the $n\times m$ matrix
$\mbox{\boldmath $S$}=(s_{ui})$ whose $i$-th column is the characteristic vector of $V_i$, that is, $s_{ui}=1$ if $u\in V_i$, and $s_{ui}=0$ otherwise. In terms of such a matrix,
we have the following characterization of regular partitions.
\begin{Lemma}
Let $G=(V,E)$ be a digraph with adjacency matrix $\mbox{\boldmath $A$}$, and vertex partition $\pi$
with characteristic matrix $\mbox{\boldmath $S$}$. Then $\pi$ is regular if and only if there
exists an $m\times m$ matrix $\mbox{\boldmath $C$}$ such that $\mbox{\boldmath $S$}\mbox{\boldmath $C$}=\mbox{\boldmath $A$}\mbox{\boldmath $S$}$. Moreover, $\mbox{\boldmath $C$}=\mbox{\boldmath $B$}$, the
quotient matrix of $\mbox{\boldmath $A$}$ with respect to $\pi$.
\end{Lemma}
\begin{proof}
Let $\mbox{\boldmath $C$}=(c_{ij})$ be an $m\times m$ matrix. For any fixed $u\in V_i$ and
$j=1,\ldots,m$, we have
$$
(\mbox{\boldmath $S$}\mbox{\boldmath $C$})_{uj} = \sum_{k=1}^m s_{uk}c_{kj}=c_{ij}, \quad
(\mbox{\boldmath $A$}\mbox{\boldmath $S$})_{uj} = \sum_{v\in V} a_{uv} s_{vj}=|G^+(u)\cap V_j|= b_{ij}(u),
$$
and the result follows.
\end{proof}
Most of the results about regular partitions in graphs can be generalized for
regular partitions in digraphs. For instance, using the above lemma it can be proved that all the eigenvalues of the quotient matrix $\mbox{\boldmath $B$}$ are also eigenvalues of $\mbox{\boldmath $A$}$. Moreover, we have the following result.
\begin{Lemma}
Let $G$ be a digraph with adjacency matrix $\mbox{\boldmath $A$}$. Let $\pi=(V_1,\ldots,$ $V_m)$ be a
regular partition of $G$, with quotient matrix $\mbox{\boldmath $B$}$. Then, the number of $k$-walks from each vertex $u\in V_i$ to all vertices of $V_j$ is the $ij$-entry of $\mbox{\boldmath $B$}^{k}$.
\end{Lemma}
\begin{proof}
We use induction. The result is clearly true for $k=0$, since $\mbox{\boldmath $B$}^0=\mbox{\boldmath $I$}$, and for
$k=1$ because of the definition of $\mbox{\boldmath $B$}$. Suppose that the result holds for some $k>1$. Then the set of walks of length $k+1$ from $u\in V_i$ to the vertices of $V_j$ is in bijective correspondence with the set of $k$-walks from $u$ to vertices $v\in V_h$ adjacent to some vertex of $V_j$. Then, the number of such
walks is $\sum_{h=1}^m(\mbox{\boldmath $B$}^{k})_{ih}b_{hj}=(\mbox{\boldmath $B$}^{k+1})_{ij}$, as claimed.
\end{proof}
As a consequence of this lemma, the number of vertices of $L^{k}(G)$ is
\begin{equation}
\label{n_l}
n_{k}=\sum_{i=1}^m |V_i|\sum_{j=1}^m (\mbox{\boldmath $B$}^{k})_{ij}=\mbox{\boldmath $s$}\mbox{\boldmath $B$}^{k}\mbox{\boldmath $j$}^{\top},
\end{equation}
where $\mbox{\boldmath $s$}=(|V_1|,\ldots,|V_m|)$ and $\mbox{\boldmath $j$}=(1,\ldots,1)$.
\section{Main result}
\label{sec:main-result}
In the following result, we obtain a recurrence equation on the number of
vertices $n_k$ of the
$k$-iterated line digraph of a digraph $G$.
\begin{Theorem}
\label{maintheo}
Let $G=(V,E)$ be a digraph on $n$ vertices, and consider a regular partition
$\pi=(V_1,\ldots,V_m)$ with quotient matrix $\mbox{\boldmath $B$}$. Let $m(x)=x^r-\alpha_{r-1} x^{r-1}-\cdots-\alpha_0$ be the minimal polynomial of $\mbox{\boldmath $B$}$. Then, the number of vertices $n_k$ of the $k$-iterated line digraph $L^k(G)$ satisfies the
recurrence
\begin{equation}
\label{recur}
n_k= \alpha_{r-1} n_{k-1}+\cdots+\alpha_{0} n_{k-r},\qquad k=r,r+1,\ldots
\end{equation}
initialized with the values $n_{k}$, for $k=0,1,\ldots,r-1$, given by \eqref{n_l}.
\end{Theorem}
\begin{proof}
Since the polynomial $x^{k-r}m(x)$ annihilates $\mbox{\boldmath $B$}$ for any $k\ge 0$,
we have
$$
\mbox{\boldmath $B$}^k= \alpha_{r-1} \mbox{\boldmath $B$}^{k-1}+\cdots+\alpha_{0} \mbox{\boldmath $B$}^{k-r}.
$$
Then, by \eqref{n_l}, we get the recurrence
\begin{align*}
n_k=\mbox{\boldmath $s$}\mbox{\boldmath $B$}^k\mbox{\boldmath $j$}^{\top} &= \alpha_{r-1} \mbox{\boldmath $s$}\mbox{\boldmath $B$}^{k-1}\mbox{\boldmath $j$}^{\top}+\cdots+\alpha_{0}
\mbox{\boldmath $s$}\mbox{\boldmath $B$}^{k-r}\mbox{\boldmath $j$}^{\top}\\
&=\alpha_{r-1} n_{k-1}+\cdots+\alpha_{0} n_{k-r},
\end{align*}
with the first values $n_{k}$, for $k=0,\ldots, r-1$, given as claimed.
\end{proof}
\section{Examples}
\label{sec:examples}
In what follows, we give examples of the three possible behaviours of the sequence
$n_0, n_1, n_2,\ldots$
Namely, when it is increasing, tending to a positive constant, or tending to zero.
\subsection{Cyclic Kautz digraphs}
\label{sec:ex-CK}
\begin{figure}[t]
\vskip-.5cm
\begin{center}
\includegraphics[width=12cm]{figure1.pdf}
\end{center}
\vskip-2.75cm
\caption{The cyclic Kautz digraph $CK(2,4)$, its quotient $\pi(CK(2,4))$, and
the quotient digraph of $CK(d,4)$.}
\label{fig:quocient-donut-color}
\end{figure}
The {\em cyclic Kautz digraph} $CK(d,\ell)$, introduced by B\"{o}hmov\'{a},
Dalf\'{o}, and Huemer in~\cite{BoDaHu14}, has vertices labeled by all possible sequences $a_1\ldots a_\ell$ with $a_i\in\{0,1,\ldots,d\}$, $a_i\neq a_{i+1}$ for $i=1,\ldots,\ell-1$, and $a_1\neq a_\ell$. Moreover,
there is an arc from vertex $a_1 a_2\ldots a_\ell$ to vertex $a_2 \ldots a_\ell
a_{\ell+1}$, whenever $a_1\neq a_\ell$ and $a_2\neq a_{\ell+1}$. By this definition, we observe that the cyclic Kautz digraph $CK(d,\ell)$ is a subdigraph of the well-known Kautz digraph $K(d,\ell)$, defined in the same way, but without the requirement $a_1\neq a_\ell$.
For example, Figure~\ref{fig:quocient-donut-color}$(a)$ shows the cyclic Kautz
digraph $CK(2,4)$. Notice that, in general, such digraphs are not $d$-regular and, hence, the number of vertices of their iterated line digraphs are not obtained by repeatedly multiplying by $d$. Instead, we can apply our method, as shown next with $CK(2,4)$. This digraph has a regular partition $\pi$ of its vertex set into three classes (each one with 6 vertices): $abcb$ (the second and the last digits are equal), $abab$ (the first and the third digits are equal, and also the second and the last), and $abac$ (the first and the third digits are equal). Then, the quotient matrix of $\pi$ (which in this case coincides with the adjacency matrix of $\pi(CK(2,4))$) is
$$
\mbox{\boldmath $B$}=\left(
\begin{array}{ccc}
0 & 1 & 1 \\
0 & 1 & 1 \\
1 & 0 & 0 \\
\end{array}
\right),
$$
and it has minimal polynomial $m(x)=x^3-x^2-x$. Consequently, by
Theorem~\ref{maintheo}, the number of vertices of $L^k(CK(2,4))$ satisfies the recurrence $n_k=n_{k-1}+n_{k-2}$ for $k\ge 3$. In fact, in this
case, $\mbox{\boldmath $s$}(\mbox{\boldmath $B$}^2-\mbox{\boldmath $B$}-\mbox{\boldmath $I$})\mbox{\boldmath $j$}^{\top}=0$, and the above recurrence applies from $k=2$.
This, together with the initial values $n_0=18$ and $n_1=\mbox{\boldmath $s$}\mbox{\boldmath $B$}\mbox{\boldmath $j$}^{\top}=30$, yields the Fibonacci sequence, $n_2=48, n_3=78, n_4=126\ldots$,
as B\"{o}hmov\'{a}, Dalf\'{o}, and Huemer~\cite{BoDaHu14} proved by using a
combinatorial approach.
Moreover, $n_k$ is also the number of ternary length-2 squarefree words of length
$k+4$ (that is, words on a three-letter alphabet that do not contain an adjacent
repetition of any subword of length $\le 2$); see the sequence A022089 in the On-Line Encyclopedia of Integer Sequences~\cite{Sl}.
In fact our method allows us to generalize this result and, for instance, derive a
formula for the order of $L^k(CK(d,4))$ for any value of the degree $d\ge 2$. To
this end, it is easy to see that a quotient digraph of $CK(d,4)$ for $d>2$ is as
shown in Figure~\ref{fig:quocient-donut-color}$(c)$, where now we have to
distinguish four classes of vertices.
Then, the corresponding quotient matrix is
$$
\mbox{\boldmath $B$}=\left(
\begin{array}{cccc}
1 & d-1 & 0 & 0 \\
0 & 0 & 1 & d-2\\
1 & d-1 & 0 & 0 \\
0 & 0 & 1 & d-2
\end{array}
\right),
$$
and it has minimal polynomial is $m(x)=x^3-(d-1)x^2-x$.
In turn, this leads to the recurrence formula $n_k=(d-1)n_{k-1}+n_{k-2}$, with
initial values $n_0=d^4+d$ and $n_1=d^5-d^4+d^3+2d^2-d$, which are
computed by using \eqref{n_l} with the vector
\begin{align*}
\mbox{\boldmath $s$} & =(|V_1|,|V_2|,|V_3|,|V_4|) \\
& =((d+1)d, (d+1)d(d-1), (d+1)d(d-1), (d+1)d(d-1)(d-2)).
\end{align*}
Solving the recurrence, we get the closed formula
$$
n_k =\frac{2^k d}{\sqrt{\Delta}}
\left(\frac{
(d^2+d)\sqrt{\Delta}-d^3-d-2
}
{(1-d-\sqrt{\Delta})^{k+1}}
+\frac{(d^2+d)\sqrt{\Delta}+d^3+d+2
}
{(1-d+\sqrt{\Delta})^{k+1}}\right),
$$
where $\Delta=d^2-2d+5$ and, hence, $n_k$ is an increasing sequence.
\subsection{Unicyclic digraphs}
\label{sec:ex-unicyclic}
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{figure2.pdf}
\end{center}
\vskip-3.5cm
\caption{The unicyclic digraph $G_{3,2}$ and its quotient digraph.}
\label{fig:unicyclic}
\end{figure}
A unicyclic digraph is a digraph with exactly one (directed) cycle. As usual, we
denote a cycle on $n$ vertices by $C_n$. For example, consider the digraph
$G_{n,d}$, obtained by joining to every vertex of $C_n$ one `out-tree' with $d$
leaves (or `sinks'), as shown in Figure~\ref{fig:unicyclic}$(a)$ for the case
$G_{3,2}$. This digraph has the regular partition $\pi=(V_1,V_2,V_3)$, where $V_1$ is the set of vertices of the cycle, $V_2$ the central vertices of the trees, and $V_3$ the set of leaves. (In the figure $V_1=\{1,2,3\}$, $V_2=\{4,5,6\}$, and $V_3=\{7,8,9,10,11,12\}$).
This partition gives the quotient digraph $\pi(G)$ of
Figure~\ref{fig:unicyclic}$(b)$, and the
quotient matrix
$$
\mbox{\boldmath $B$}=\left(
\begin{array}{ccc}
1 & 1 & 0 \\
0 & 0 & d \\
0 & 0 & 0 \\
\end{array}
\right),
$$
with minimal polynomial $m(x)=x^3-x^2$. Then, by Theorem \ref{maintheo}, the order
of $L^k(G)$ satisfies the recurrence $n_k=n_{k-1}$ for $k\ge 0$, since
$\mbox{\boldmath $s$}(\mbox{\boldmath $B$}^{k}-\mbox{\boldmath $B$}^{k-1})\mbox{\boldmath $j$}^{\top}$ $=0$ for $k=1,2$, where $\mbox{\boldmath $s$}=(n,n,nd)$. Thus, we conclude that all the iterated line digraphs $L^k(G)$ have constant order
$n_k=n_0=n(d+2)$, that is, $n_k$ tends to a positive constant.
(In fact, this is because in this case $L(G)$---and, hence, $L^k(G)$---is
isomorphic to $G$.)
\subsection{Acyclic digraphs}
\label{sec:ex-acyclic}
\begin{figure}[t]
\begin{center}
\includegraphics[width=9cm]{figure3.pdf}
\end{center}
\vskip-8.5cm
\caption{An acyclic digraph and its quotient digraph.}
\label{fig:acyclic}
\end{figure}
Finally, let us consider an example of an acyclic digraph, that is, a digraph
without directed cycles, such as the digraph $G$ of Figure~\ref{fig:acyclic}$(a)$.
Its quotient digraph is depicted in Figure~\ref{fig:acyclic}$(b)$, with quotient
matrix
$$
\mbox{\boldmath $B$}=\left(
\begin{array}{cccccc}
0 & 3 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right),
$$
and minimal polynomial $m(x)=x^5$. This indicates that $n_k=0$ for every $k\ge 5$
(as expected, because $G$ has not walks of length larger than or equal to $5$). Moreover, from \eqref{n_l}, the first values are
$n_0=16$, $n_1=18$, $n_2=15$, $n_3=9$, and $n_4=3$.
\vskip 1cm
\noindent{\large \bf Acknowledgments.} This research is supported by the
{\em Ministerio de Econom\'{\i}a y Competitividad} and the {\em European Regional
Development Fund} under project MTM2011-28800-C02-01, and by the Catalan Government
under project 2014SGR1147.
| 6,523 |
\section{Introduction}
\subsection{Pointwise ergodic families}
Let $\Gamma$ be a countable group with a probability-measure-preserving (pmp) action on a standard probability space $(X,\mu)$. Any probability measure $\zeta$ on $\Gamma$ determines an operator on $L^1(X,\mu)$ defined by
$$\pi_X(\zeta)(f):= \sum_{\gamma\in \Gamma} \zeta(\gamma) f\circ \gamma^{-1},\quad \forall f\in L^1(X,\mu).$$
\begin{defn}
Let ${\mathbb{I}}$ denote either $\RR_{>0}$ or $\NN$. Suppose $\{\zeta_r\}_{r\in {\mathbb{I}}}$ is a family of probability measures on $\Gamma$.
\begin{enumerate}\item If for every pmp action $\Gamma \curvearrowright (X,\mu)$ and every $f\in L^p(X,\mu)$ the functions $ \pi_X(\zeta_r)(f)$ converge pointwise a.e. as $ r\to \infty$ then we say $\{\zeta_r\}_{r\in {\mathbb{I}}}$ is a {\em pointwise convergent family in $L^p$}. If, in addition, the a.e.-pointwise limit of $\pi_X(\zeta_r)(f)$ is the conditional expectation of $f$ on the $\sigma$-algebra of $\Gamma$-invariant Borel sets then $\{\zeta_r\}_{r\in {\mathbb{I}}}$ is a {\em pointwise ergodic family in $L^p$}.
\item We say that $\{\zeta_r\}_{r\in {\mathbb{I}}}$ {\em satisfies the strong $L^p$ maximal inequality} if there is a constant $C_p>0$ depending only on $\{\zeta_r\}_{r\in {\mathbb{I}}}$ such that for every $f \in L^p(X,\mu)$, $\|{\mathbb{M}}[f|\zeta]\|_p \le C_p \|f\|_p$ where ${\mathbb{M}}[f|\zeta] = \sup_{r \in {\mathbb{I}}} \pi_X(\zeta_r)(|f|)$.
\item Similarly, $\{\zeta_r\}_{r\in{\mathbb{I}}}$ is said to {\em satisfy the weak $(1,1)$-type maximal inequality} if there is a constant $C_{1,1}>0$ depending only on $\{\zeta_r\}_{r\in {\mathbb{I}}}$ such that for every $f \in L^1(X,\mu)$ and $t>0$,
$$\mu(\{ x\in X:~ {\mathbb{M}}[f|\zeta](x)\ge t\}) \le \frac{ C_{1,1} \|f\|_1}{t}.$$
\end{enumerate}
It follows from standard interpolation arguments that if $\{\zeta_r\}_{r\in {\mathbb{I}}}$ satisfies the weak (1,1)-type maximal inequality then it also satisfies the strong $L^p$ maximal inequality for all $p>1$. Moreover in that case it satisfies the {\em $L\log(L)$-maximal inequality} which means there is a constant $C_1>0$ depending only on $\{\zeta_r\}_{r\in {\mathbb{I}}}$ such that for every $f \in L\log(L)(X,\mu)$, $\|{\mathbb{M}}[f|\zeta]\|_{1} \le C_1 \|f\|_{L\log L}$ where, for any function $\phi$ of $X$, $\| \phi \|_{L\log L} := \int_X |\phi| \log^+ |\phi |~d\mu$.
\end{defn}
Traditionally, since the time of von-Neumann and Birkhoff, much of the effort in ergodic theory has been devoted to the study of ergodic actions of Abelian and more generally amenable groups (see \cite{OW87}), with the averages studied most often being uniform averages on an asymptotically invariant sequence of sets (a F\o lner sequence). For more on the subject of ergodic theorems on amenable groups we refer to \cite{We03}, the survey \cite{Ne05}, and for a more detailed exposition to \cite{AAB}. We also refer to \cite{Av} for an ergodic theorem pertaining to cross sections of actions of amenable groups.
Our main purpose in the present paper is to give a general new method for constructing probability measures $\zeta_r$ on $\Gamma$ which form pointwise convergent and pointwise ergodic families on general countable groups, which may be amenable or non-amenable. The basic ingredient underlying our approach is the realization that one can utilize the amenable actions of a group in order to construct families of ergodic averages on it. The generality of this approach is underscored by the fact that for any countable group $\Gamma$, the Poisson boundary associated with a generating probability measure is an amenable ergodic action of $\Gamma$ \cite{Z2},
so that any countable group admits such an action.
Some instances of our approach were developed in \cite{BN1} and in \cite{BN2}, and in the present paper, we greatly generalize the constructions used there. We show that pointwise ergodic families on $\Gamma$ can be obtained using $\Gamma$-valued cocycles defined on a pmp amenable equivalence relation, subject to certain natural necessary conditions.
This construction gives rise to a variety of ergodic families on countable groups, some of which will be discussed briefly in \S \ref{sec:example}. To further demonstrate the significance and utility of our approach we note that in \cite{BN4}, the main results of the present paper are crucial to the proof establishing pointwise convergence for geometric averages in Gromov hyperbolic groups, supported in spherical shells. In \cite{BN5}, the methods and results of the present paper are applied to prove pointwise ergodic theorems for Markov groups. More details on these applications will be given in \S 6 below, when the necessary notation and definitions are available.
\subsection{From amenable groups to amenable equivalence
relations}
We proceed to outline our general method for constructing pointwise ergodic sequences for pmp actions of countable groups. Along the way, we will briefly state the main results of the present paper and place them in their natural context.
\subsubsection{Pointwise ergodic theorems for measured equivalence relations} The concept of pointwise ergodic sequence can be generalized from groups to measured equivalence relations. This generalization is very useful, since it is easy to obtain such sequences for amenable measured equivalence relations, and it is then possible to push them forward to groups via cocycles. Before explaining this connection, we explain here how to generalize ergodic theorems to measured equivalence relations. Let then $(B,\nu)$ be a standard probability space and $\cR \subset B \times B$ be a discrete measurable equivalence relation. We assume $\nu$ is $\cR$-invariant: if $c$ denotes counting measure on $B$ then $c \times \nu|_\cR = \nu \times c|_\cR$.
Let $\Omega=\{\omega_i\}_{i\in {\mathbb{I}}}$ be a measurable leafwise collection of probability measures $\omega_i:\cR \to [0,1]$. More precisely, we require that the map $(i, (b,c)) \in {\mathbb{I}} \times \cR \to \omega_i(b,c) \in [0,1]$ is measurable and for a.e. $b\in B$, $\sum_{c \in B} \omega_i(c,b)=1$ where, for convenience, we set $\omega_i(c,b)=0$ if $(c,b) \notin \cR$.
Let $(X,\mu)$ be another standard probability space and $\alpha:\cR \to {\textrm{Aut}}(X,\mu)$ a measurable cocycle, namely $\alpha(a,b)\alpha(b,c)=\alpha(a,c)$ holds for a.e. $a$ and every $b,c$ with $a\cR b \cR c$. We think of $\cR$ as analogous to a group, $\Omega$ as analogous to a family of probability measures on a group and $\alpha$ as analogous to a group action. Given a measurable function $F$ on $B\times X$, define the averages ${\mathbb{A}}[F|\omega_i]$ on $B\times X$ by
$${\mathbb{A}}[F|\omega_i](b,x) = \sum_{c \in B} \omega_i(c,b) F(c, \alpha(c,b)^{-1}x).$$
Let $\cR_\alpha$ be the skew-product relation on $\cR \times X$ defined by $(b,x)\cR_\alpha (c,y) \Leftrightarrow b\cR c$ and $\alpha(c,b)x=y$. We say that $\Omega$ is a pointwise ergodic family in $L^p$ if for every $F \in L^p(B\times X)$, ${\mathbb{A}}[F|\omega_i]$ converges pointwise a.e. as $i\to\infty$ to $\EE[F | \cR_\alpha]$, the conditional expectation of $F$ on the sigma-algebra of $\cR_\alpha$-saturated sets (i.e. those measurable sets which are unions of $\cR_\alpha$-classes modulo null sets).
\subsubsection{Finding pointwise ergodic families} One natural problem that arises is how to find collections $\Omega$ as above. In case $\cR$ is amenable, several criteria for establishing that a given $\Omega$ is a pointwise ergodic family were developed in \cite{BN2}. These criteria are based on a generalization of the notions of doubling, regular, and tempered F\o lner sets in the group to F\o lner subset functions on the equivalence relation. In the case of tempered subset function, for example, \cite{BN2} generalizes Lindenstrauss' pointwise ergodic theorem for tempered Folner sequences on an amenable group \cite{Li01}.
The fact that such an $\Omega$ always exists follows from the fact that $\cR$ can be realized as the orbit-equivalence relation for a $\ZZ$-action \cite{CFW}. It should be noted, however, that this is an abstract existence result and finding an explicit system of probability measures $\Omega$ depends on having an explicit presentation for the relation $\cR$.
\subsubsection{Using cocycles to obtain pointwise ergodic theorems}
Suppose now that we have a measured equivalence relation $\cR$ as above with a pointwise ergodic family $\Omega$. Given a measurable cocycle $\alpha:\cR \to \Gamma$ (where $\Gamma$ is a countable group) we can push $\Omega$ forward to obtain a family of measures on $\Gamma$. But before doing so, let us consider a measure-preserving action of $\Gamma$ on a standard probability space $(X,\mu)$ given by a homomorphism $\beta:\Gamma \to {\textrm{Aut}}(X,\mu)$. Then the composition $\beta\alpha:\cR \to {\textrm{Aut}}(X,\mu)$ is a cocycle. If $f$ is a function on $X$ and $F$ is defined on $B\times X$ by $F(b,x)=f(x)$ then
$$\int {\mathbb{A}}[F|\omega_i](b,x)~d\nu(b) = \int \sum_{c \in B} \omega_i(c,b) f(\beta(\alpha(c,b))^{-1}x)~d\nu(b) = \sum_{\gamma \in \Gamma} \omega^\alpha_i(\gamma) f(\beta(\gamma)^{-1} x)$$
where $\omega^\alpha_i(\gamma) = \int \sum_{c \in B:~\alpha(c,b)=\gamma} \omega_i(c,b) ~d\nu(b)$. From here we would like to conclude that $\pi_X(\omega^\alpha_i)(f)$ converges pointwise a.e. as $i\to\infty$ to $\int \EE[F | \cR_{\beta\alpha}](b,\cdot)~d\nu(b)$. In Theorem \ref{thm:general} below, we obtain this conclusion if $\Omega$ satisfies a strong $L^p$-maximal inequality which allows us to interchange the integral with the limit, $\Omega$ is pointwise ergodic in $L^p$ and $f \in L^p(X)$.
While this shows that $\{\omega^\alpha_i\}_{i\in {\mathbb{I}}}$ is a pointwise convergent family of measures on $\Gamma$, it does not establish that is pointwise ergodic, because we do not know apriori whether the limit $\int \EE[F| \cR_{\beta\alpha}](b,\cdot)~d\nu(b)$ coincides with the conditional expectation of $f$ on the sigma-algebra of $\Gamma$-invariant measurable sets. We say that the cocycle $\alpha:\cR \to \Gamma$ is {\em weakly mixing} if whenever $\beta:\Gamma \to {\textrm{Aut}}(X,\mu)$ is an ergodic action then the skew-product relation $\cR_{\beta\alpha}$ on $B\times X$ is also ergodic. In this case, we can conclude that $\EE[F| \cR_{\beta\alpha}](b,x)= \EE[f|\Gamma](x)$ almost everywhere and therefore $\{\omega^\alpha_i\}_{i\in {\mathbb{I}}}$ is a pointwise ergodic family as desired. Thus our goal now is to find, for a given group $\Gamma$, a weakly mixing cocycle $\alpha$ from an amenable pmp equivalence relation $\cR$ into $\Gamma$ together an appropriate family $\Omega$ of leafwise measures on $\cR$. As we will see, the weakly mixing criterion can be weakened somewhat, a fact that will be important below.
\subsubsection{Equivalence relations from amenable actions: the $II_1$ case} Given a group $\Gamma$, we can construct an amenable pmp equivalence relation $\cR$ with a cocycle $\alpha:\cR \to \Gamma$ from an essentially free amenable action of $\Gamma$. The simplest case occurs when $\Gamma$ admits an essentially free measure-preserving amenable action on a standard probability space $(B,\nu)$. Let $\cR=\{(b,gb):~g\in \Gamma\}$ be the orbit-equivalence relation and $\alpha:\cR \to \Gamma$ be the cocycle $\alpha(gb,b)=g$ which is well-defined a.e. because the action is essentially free. If the action is weakly mixing then $\alpha$ is also weakly mixing. Of course, if $\Gamma$ admits such an action then it must be an amenable group and we can construct the family $\Omega$ from a regular or tempered F\o lner sequence of $\Gamma$. From this construction and \cite[Theorem 2.5]{BN2}, we deduce the pointwise ergodic theorem for regular or tempered F\o lner sequences, in the latter case using an argument modelled after B. Weiss' proof of Lindenstrauss' Theorem \cite{We03}.
\subsubsection{The $II_\infty$ case} The next simplest case occurs when $\Gamma$ admits an essentially free measure-preserving amenable action on a sigma-finite (but infinite) measure space $(B',\nu)$. In this case, let $B \subset B'$ be a measurable subset with $\nu(B)=1$. Let $\cR=\{(b,gb):~g\in \Gamma, b \in B\} \cap (B \times B)$ be the orbit-equivalence relation and $\alpha:\cR \to \Gamma$ be the cocycle $\alpha(gb,b)=g$ which is well-defined a.e. because the action is essentially free. If the action is weakly mixing then $\alpha$ is also weakly mixing. Because the action is amenable, $\cR$ is amenable and so pointwise ergodic families $\Omega$ of $\cR$ do exist although we do not have an explicit general recipe for finding them.
\subsubsection{The $III$-case and the ratio set} Perhaps the most important case occurs when $\Gamma \curvearrowright (\bar{B},\bar{\nu})$ is an essentially free amenable action but $\bar{\nu}$ is not equivalent to an invariant $\sigma$-finite measure. In this case we form the Maharam extension which is an amenable measure-preserving action on a sigma-finite measure space. The details of this construction depend very much on the essential range (also known as the ratio set) of the Radon-Nikodym derivative. To be precise, we say that a real number $t\ge 0$ is in the ratio set if for every $\epsilon>0$ and every subset $A \subset \bar{B}$ with positive measure there exists a subset $A' \subset A$ with positive measure and an element $g \in \Gamma \setminus \{e\}$ such that $gA' \subset A$ and $|\frac{d\bar{\nu}\circ g}{d\bar{\nu}}(a) -t| < \epsilon$ for every $a\in A'$. A similar definition holds for stating that $+\infty$ is in the ratio set. The action is said to be of type $II$ if the ratio set is $\{1\}$, it is type $III_1$ if the ratio set is $[0,\infty]$, it is type $III_\lambda$ if the ratio set is $\{\lambda^n:~n\in \Z\}\cup\{0,\infty\}$ (where $\lambda \in (0,1)$) and it is type $III_0$ if the ratio set is $\{0,1,\infty\}$. There are no other possibilities. It is also known that, after replacing $\bar{\nu}$ with an equivalent measure, we may assume that for every $g \in \Gamma$, $\frac{d\bar{\nu} \circ g}{d\bar{\nu}}(b)$ is contained in the ratio set for a.e. $b$. We say that the action $\Gamma \curvearrowright (\bar{B},\bar{\nu})$ has {\em stable type $III_\lambda$} (for some $\lambda \in (0,1)$) if for every ergodic measure-preserving action $\Gamma \curvearrowright (X,\mu)$ the ratio set of the product action $\Gamma \curvearrowright (\bar{B}\times X, \bar{\nu}\times \mu)$ contains $\{\lambda^n:~n\in \Z\}$ and $\lambda \in (0,1)$ is the largest number with this property. Similarly, the action of $\Gamma$ on $(\bar{B}, \bar{\nu})$ has stable type $III_1$ if the ratio set of the product action is $[0,\infty]$ (for every pmp action $\Gamma \curvearrowright (X,\mu)$). For more details, see \S \ref{sec:type}.
\subsubsection{The $III_1$ case} Suppose that $\Gamma \curvearrowright (\bar{B},\bar{\nu})$ has type $III_1$. Let
$$R(g,b) = \log\left( \frac{d\bar{\nu} \circ g}{d\nu}(b) \right).$$
Let $B' = \bar{B} \times \RR$ and $\nu' = \bar{\nu} \times \theta$ where $\theta$ is the measure on $\R$ defined by $d\theta(t)=e^tdt$. The group $\Gamma$ acts on $(B',\nu')$ by $g(b,t)=(gb,t-R(g,b))$. This action preserves the measure $\nu'$, which is infinite. So we can let $B=\bar{B}\times [0,T] \subset B'$ (for some $T>0$), $\nu$ equal to the restriction of $\nu'$ of $B$ and $\cR$ be the orbit-equivalence relation on $B$ given by: $(b,t)\cR(b',t')$ if there exists $g \in \Gamma$ such that $g(b,t)=(b',t')$. Also let $\alpha:\cR \to \Gamma$ be the cocycle $\alpha(gb,b)=g$. If the action $\Gamma \curvearrowright (\bar{B},\bar{\nu})$ is weakly mixing and stable type $III_1$ then $\alpha$ is weakly mixing. For example, these conditions are satisfied when $\Gamma$ is an irreducible lattice in a connected semi-simple Lie group $G$ without compact factors and $\bar{B}$ is the homogeneous space $G/P$ where $P<G$ is a minimal parabolic subgroup (see \cite{BN2} for details). More generally, if $\Gamma \curvearrowright (\bar{B},\bar{\nu})$ is weakly mixing, type $III_1$ and stable type $III_\lambda$ for some $\lambda \in (0,1]$ then in Theorem \ref{thm:typeIII} we prove that $\alpha$ is weakly mixing relative to a compact group action on $B$ (Definition \ref{defn:compact}). This slightly weaker condition is enough to obtain a pointwise ergodic family on $\Gamma$ from one on $\cR$ (Theorem \ref{thm:general}).
\subsubsection{The $III_\lambda$ case} Suppose, as above, that $\Gamma \curvearrowright (\bar{B},\bar{\nu})$ is an essentially free amenable action and $\bar{\nu}$ is not equivalent to an invariant $\sigma$-finite measure. For $\lambda \in (0,1)$ let
$$R_\lambda(g,b) = \log_\lambda\left( \frac{d\bar{\nu} \circ g}{d\nu}(g) \right).$$
Suppose that $R_\lambda(g,b) \in \Z$ for every $g\in \Gamma$ and a.e. $b$. In this case, we form the discrete Maharam extension. To be precise, let $B' = \bar{B} \times \Z$ and $\nu' = \bar{\nu} \times \theta_\lambda$ where $\theta_\lambda(\{n\})=\lambda^{-n}$. Let $\Gamma$ act on $B'$ by $g(b,t)=(g,b+R_\lambda(g,b))$. This action preserves $\nu'$ and we can define $B, \cR$ and $\alpha$ as in the previous case. If $\Gamma \curvearrowright (\bar{B},\bar{\nu})$ is weakly mixing and stable type $III_\lambda$ then $\alpha$ is also weakly mixing. More generally, if $\Gamma \curvearrowright (\bar{B},\bar{\nu})$ is weakly mixing, type $III_\lambda$ and stable type $III_\tau$ for some $\tau \in (0,1)$ then $\alpha$ is also weakly mixing relative to a certain compact group action (Theorem \ref{previous}) which, as we see from Theorem \ref{thm:general} below, is sufficient for constructing pointwise ergodic families.
\subsubsection{Amenable actions: examples}
We now have the problem of finding weakly mixing cocycles from amenable equivalence relations into $\Gamma$. Theorem \ref{thm:existence} shows that such cocycles exist for any countable group $\Gamma$. However, it is desirable to have such equivalence relations which arise from actions of $\Gamma$ to determine pointwise ergodic theorems in which the family of measures is associated with some geometric structure on the group. Next we discuss a few possibilities.
The first possibility is to choose $(\bar{B},\bar{\nu})$ to be the Poisson boundary $B(\Gamma,\eta)$ with respect to an admissible probability measure $\eta$ on $\Gamma$. It is known that the action $\Gamma$ on $B(\Gamma,\eta)$ is amenable and weakly mixing (in fact doubly-ergodic \cite{Ka}). However, the type and stable type of the action is not well understood in general. There are some exceptions: for example in \cite{INO08} it is proven that the Poisson boundary of a random walk on a Gromov hyperbolic group induced by a nondegenerate measure on $\Gamma$ of finite support is never of type $III_0$.
Another possibility is to choose $(\bar{B},\bar{\nu})$ to be the square of the Poisson boundary $B(\Gamma,\eta)$. Because $\Gamma \curvearrowright B(\Gamma,\eta)$ is amenable and doubly-ergodic \cite{Ka}, this action is amenable and weakly mixing. The type and stable type of this action is not well-understood in general but there are many cases in which it is known to be of type $II_\infty$ (this means that there is an equivalent invariant infinite $\sigma$-finite measure)
If the group $\Gamma$ is word hyperbolic then we may consider the action of $\Gamma$ on its boundary with respect to a Patterson-Sullivan measure. It is well-known that this action is amenable but it is not known whether it is necessarily weakly mixing. In \cite{B2} it is shown that the stable ratio set of this action contains a real number $t \notin \{0,1\}$. It follows that if this action is weakly mixing then it has stable type $III_\lambda$ for some $\lambda \in (0,1]$ and therefore, the methods of this paper apply. It is also known that the square of the Patterson-Sullivan measure on the squared boundary $\partial \Gamma \times \partial \Gamma$ is equivalent to an invariant sigma-finite measure.
If the group $\Gamma$ is an irreducible lattice in a connected semi-simple Lie group $G$ which has trivial center and no compact factors, then we can consider the action of $\Gamma$ on $G/P$ (where $P<G$ is a minimal parabolic subgroup) with respect to a $G$-quasi-invariant measure. In \cite{BN2} it is shown that this action is weakly mixing and has type and stable type $III_1$.
{\bf Organization}.
In \S \ref{sec:mer} we review pointwise ergodic theory for measured equivalence relations. In \S \ref{sec:construct} we prove our first main result, Theorem \ref{thm:general}, which is a general construction of pointwise ergodic families. \S \ref{sec:amenable} reviews type, stable type and the Maharam extension. In \S \ref{sec:typeIII} we prove our second main result, Theorem \ref{thm:typeIII}, which explains how to produce a weakly mixing cocycle relative to a compact group action from a sufficiently nice type $III_1$ action. In \S \ref{sec:example} we illustrate our approach with a variety of examples.
\section{Measured equivalence relations}\label{sec:mer}
Before discussing the construction of pointwise ergodic families for $\Gamma$-actions, we must first introduce the notion of a pointwise ergodic family for a measured equivalence relation. Let $(B,\nu)$ be a standard Borel probability space and $\cR \subset B\times B$ a measurable equivalence relation. Recall that this means that there exists a co-null Borel set $B_0\subset B$ such that the restriction of $\cR$ to $B_0\times B_0$ is a Borel equivalence relation. We assume that $\cR$ is discrete, namely the equivalence classes are countable, almost always, namely on a conull invariant Borel subset $B_0$ . We assume as well that $\nu$ is $\cR$-invariant, i.e., if $\phi:B\to B$ is a measurable automorphism with graph contained in $\cR$ (when restricted to a conull invariant Borel set) then $\phi_*\nu=\nu$.
\subsection{Leafwise pointwise ergodic families on $B$.}
Let ${\mathbb{I}}\in \{\RR_{>0},\NN\}$ be an index set and $\Omega=\{\omega_i\}_{i\in {\mathbb{I}}}$ a measurable collection of leafwise probability measures $\omega_i:\cR \to [0,1]$. More precisely, we assume that each $\omega_i$ is defined on a common conull invariant Borel set $B_0$, where the map $(i, (b,c)) \in {\mathbb{I}} \times \cR \to \omega_i(b,c) \in [0,1]$ is Borel and for every $b\in B_0$, $\sum_{c \in B_0} \omega_i(c,b)=1$ where, for convenience, we set $\omega_i(c,b)=0$ if $(c,b) \notin \cR$.
Let $\alpha:\cR \to {\textrm{Aut}}(X,\mu)$ be a measurable cocycle into the group of measure-preserving automorphisms of a standard Borel probability space $(X,\mu)$. This means that $\alpha(b,c)\alpha(c,d)=\alpha(b,d)$ for every $b,c,d \in B_0$ with $(b,c), (c,d) \in \cR$, and $B_0$ a conull invariant Borel set. Let $\cR_\alpha$ be the equivalence relation on $B_0\times X$ given by $(b,x)\cR_\alpha (c,\alpha(c,b)x)$ (for $(b,c) \in \cR, x\in X$). Given a Borel function $F_0$ on $B_0\times X$, we define its $\Omega$-averages, ${\mathbb{A}}[F_0|\omega_i]$ as a Borel function on $B_0\times X$
$${\mathbb{A}}[F_0|\omega_i](b,x) := \sum_{c\in B_0} \omega_i(c,b) F_0(c, \alpha(c,b)^{-1}x)$$
when this sum is absolutely convergent. We say that $\Omega$ is a {\em pointwise ergodic family in $L^p$} if for every such cocycle $\alpha$, for every $F \in L^p(B\times X,\nu\times \mu)$, choosing some Borel restriction of $F_0$ to $B_0\times X$,
$$\lim_{i\to\infty} {\mathbb{A}}[F_0|\omega_i](b,x) = \EE[F | \cR_\alpha ](b,x)$$
for a.e. $(b,x) \in B\times X$ where $\EE[F| \cR_\alpha]$ denote the conditional expectation of $F$ on the $\sigma$-algebra of $\cR_\alpha$-saturated Borel sets (recall that a set $A \subset B\times X$ is $\cR_\alpha$-saturated if it is a union of $\cR_\alpha$-equivalence classes, up to a null set). Our definition is of course independent of the choice of $F_0$ in the $L^p$-equivalence class of $F$.
We will also need to make use of leafwise maximal inequalities. For this purpose, let ${\mathbb{M}}[F_0| \Omega] : = \sup_{i\in {\mathbb{I}}} {\mathbb{A}}[|F_0| |\omega_i]$, where $F_0$ is Borel on $B_0\times X$. We say that $\Omega$ satisfies the {\em strong $L^p$ maximal inequality} if
${\mathbb{M}}[F_0| \Omega] $ is a measurable function, and there is a constant $C_p>0$ such that $\| {\mathbb{M}}[F_0|\Omega] \|_p \le C_p \|F\|_p$ for every $F \in L^p(B\times X,\nu\times \mu)$, and for every cocycle $\alpha$ as above.
We say that $\Omega$ satisfies the {\em weak (1,1)-type maximal inequality} if there is a constant $C_{1,1}$ such that for every $t>0$ and $F \in L^1(B\times X,\nu\times\mu)$,
$$\nu\times \mu(\{ (b,x):~ {\mathbb{M}}[F_0|\Omega](b,x)\ge t\}) \le \frac{ C_{1,1} \|F\|_1}{t}.$$
Our definitions are of course independent of the choice of $F_0$ in the $L^p$-equivalence class of $F$.
\begin{remark}
In the rest of the paper we will resort to the standard practice of considering just the space $B$, with the relation $\cR$, the cocycle $\alpha$ and the averaging family $\Omega$ being defined and Borel on some unspecified invariant conull Borel set $B_0$. The statements and proofs of the results that we discuss should be interpreted as holding on an (unspecified) conull invariant Borel set, which is abbreviated by saying that the desired property holds almost always, or almost surely, or essentially.
\end{remark}
\subsection{Ergodicity of the extended relation : weak mixing.}
Now suppose $\Gamma$ is a countable group, $\cR\subset B\times B$ is a discrete measurable equivalence relation, and $\alpha:\cR \to \Gamma$ a measurable cocycle (so $\alpha(a,b)\alpha(b,c)=\alpha(a,c)$ almost everywhere). Given a homomorphism $\beta:\Gamma \to {\textrm{Aut}}(X,\mu)$ we can consider $\beta\alpha$ as a cocycle into ${\textrm{Aut}}(X,\mu)$. Suppose we can construct a measurable leafwise systems of probability measures $\Omega$ on $B$ with good averaging properties, namely such that we can establish maximal or pointwise ergodic theorems for the averaging operators ${\mathbb{A}}[\cdot|\omega_i]$ defined above. It is then natural to average these operators over $B$, and thus obtain operators defined on functions on $X$, which are given by averaging over probability measures on $\Gamma$.
This is indeed the path we plan to follow, but note that there is an additional unavoidable difficulty: the equivalence relation $\cR_{\beta\alpha}$ is not necessarily ergodic even if $\Gamma \curvearrowright^\beta (X,\mu)$ is an ergodic action. Because of this problem, even if we prove pointwise convergence, the limit function might not be invariant. To resolve this problem, we need to make additional assumptions on the cocycle, as explained next.
\begin{defn}[Weakly mixing]
We say that a measurable cocycle $\alpha:\cR \to \Gamma$ is {\em weakly mixing} if for every ergodic pmp action $\Gamma \curvearrowright^\beta (X,\mu)$ the induced equivalence relation $\cR_{\beta \alpha}$ on $B\times X$ is ergodic.
\end{defn}
\begin{defn}[Weakly mixing relative to a compact group action]\label{defn:compact}
Let $K$ be a compact group with a nonsingular measurable action $K \curvearrowright (B,\nu)$. We say that a measurable cocycle $\alpha:\cR \to \Gamma$ is {\em weakly mixing relative to the action $K\curvearrowright (B,\nu)$} if for every pmp action $\Gamma \curvearrowright^\beta (X,\mu)$ and every $f \in L^1(X)\subset L^1(B\times X)$,
$$\int \EE[f| {\cR_{\beta \alpha}}] (kb,x)~dk = \EE[f | \Gamma](x)$$
for a.e. $(b,x)$ where $dk$ denotes Haar probability measure on $K$, $\EE[f| {\cR_{\beta \alpha}}]$ is the conditional expectation of $f$ (viewed as an element of $L^1(B\times X)$) on the $\sigma$-algebra of $\cR_{\beta \alpha}$-saturated measurable sets and $\EE[f| \Gamma]$ is the conditional expectation of $f$ on the $\sigma$-algebra of $\Gamma$-invariant sets. For example, if $K$ is the trivial group then this condition implies $\alpha$ is weakly mixing.
\end{defn}
\begin{defn}
We say that an action $K \curvearrowright (B,\nu)$ as above has {\em uniformly bounded RN-derivatives} if there is a constant $C(K)$ such that
$$ \frac{d\nu \circ k}{d\nu}(b) \le C(K)\quad \textrm{ for a.e. } (k,b) \in K \times B.$$
\end{defn}
The condition of weakly-mixing relative to a compact group action is both useful and natural and can be verified in practice in many important situations. The reason for that is its close connection to the notion of type of a non-singular group action, as we will explain further below.
\section{Construction of ergodic averages}\label{sec:construct}
\subsection{Statement of Theorem \ref{thm:general}}
We can now state the main result on the construction of pointwise ergodic families of probability measures on $\Gamma$. It reduces it to the construction of a system of leafwise probability measures $\Omega$ with good averaging properties on an equivalence relation $\cR$ on $B$, provided we also have a weakly-mixing cocycle $\alpha : \cR \to \Gamma$ (relative to a compact group action).
\begin{thm}\label{thm:general}
Let $(B,\nu,\cR)$ be a measured equivalence relation, $\Omega=\{\omega_i\}_{i\in {\mathbb{I}}}$ a measurable family of leafwise probability measures $\omega_i:\cR \to [0,1]$ and $\alpha:\cR \to \Gamma$ be a measurable cocycle into a countable group $\Gamma$. Suppose there is a nonsingular compact group action $K \curvearrowright (B,\nu)$ with uniformly bounded RN-derivatives and $\psi\in L^q(B,\nu)$ ($1<q<\infty$) is a probability density (so $\psi \ge 0$, $\int \psi~d\nu=1$). For $i \in {\mathbb{I}}$, define the probability measure $\zeta_i$ on $\Gamma$ by
$$\zeta_i(\gamma):= \int_B \int_K \sum_{c:~\alpha(c,b)=\gamma} \omega_i(c,kb) \psi(b) ~dkd\nu(b).$$
Let $p>1$ be such that $\frac{1}{p}+\frac{1}{q}=1$.
Then the following hold.
\begin{enumerate}
\item If $\Omega$ satisfies the strong $L^p$ maximal inequality then $\{\zeta_i\}_{i\in{\mathbb{I}}}$ also satisfies the strong $L^p$ maximal inequality.
\item If in addition, $\Omega$ is a pointwise ergodic family in $L^p$, then $\{\zeta_i\}_{i\in{\mathbb{I}}}$ is a pointwise convergent family in $L^p$.
\item If in addition, $\alpha$ is weakly mixing relative to the $K$-action, then $\{\zeta_i\}_{i\in {\mathbb{I}}}$ is a pointwise ergodic family in $L^p$.
\end{enumerate}
Similarly, if $\Omega$ satisfies the weak $(1,1)$-type maximal inequality and $\psi\in L^\infty(B,\nu)$ then $\{\zeta_r\}_{r\in {\mathbb{I}}}$ satisfies the $L \log (L)$ maximal inequality. If in addition, $\Omega$ is a pointwise ergodic family in $L^1$, then $\{\zeta_i\}_{i\in{\mathbb{I}}}$ is a pointwise convergent family in $L\log(L)$. If in addition, $\alpha$ is weakly mixing relative to the $K$-action, then $\{\zeta_i\}_{i\in {\mathbb{I}}}$ is a pointwise ergodic family in $L\log(L)$.
\end{thm}
\subsection{Proof of Theorem \ref{thm:general}}
In this section, we prove Theorem \ref{thm:general}, motivated by the proof of \cite[Theorem 4.2]{BN2}. Let $\Gamma$ be a countable group, $(B,\nu,\cR)$ be a pmp equivalence relation with a leafwise system $\Omega$ of probability measures on the equivalence classes of $\cR$. Let $\alpha:\cR \to \Gamma$ be a measurable cocycle, $K \curvearrowright (B,\nu)$ a nonsingular action of a compact group (with uniformly bounded RN-derivatives), and $\psi \in L^q(B,\nu)$ be a probability density. Let $\Gamma \curvearrowright^\beta (X,\mu)$ be a pmp action and $f \in L^p(X,\mu)$. For convenience, for $g\in \Gamma$ and $x\in X$ we write $gx=\beta(g)x$. We consider $f$ to also be a function of $B\times X$ defined by $f(b,x)=f(x)$. Observe that by definition of the probability measures $\zeta_i$ given in Theorem \ref{thm:general} :
\begin{eqnarray}
\pi_X(\zeta_i)(f)(x)&=& \sum_{\gamma \in \Gamma} \zeta_i(\gamma) f(\gamma^{-1} x)\\
&=& \sum_{\gamma \in \Gamma} f(\gamma^{-1} x) \int_B\int_K\sum_{c:~\alpha(c,b)=\gamma} \omega_i(c,kb) \psi(b) ~dkd\nu(b)\\
&=& \int_B\int_K \sum_{c:~(c,b) \in \cR} \omega_i(c,kb) f(c,\alpha(c,kb)^{-1}x)\psi(b) ~dkd\nu(b)\\
&=& \int_B\int_K {\mathbb{A}}[f | \omega_i](kb,x)\psi(b)~dk d\nu(b). \label{eqn:1}
\end{eqnarray}
Because of this equality, we extend the domain of the operator $\pi_X(\zeta_i)$ from $L^p(X)$ to $L^p(B\times X)$ by setting
\begin{eqnarray}\label{transform}
\Pi(\zeta_i)(F)(x) = \int_B\int_K {\mathbb{A}}[F| \omega_i](kb,x)\psi(b)~dk d\nu(b)
\end{eqnarray}
for any $F \in L^p(B\times X)$.
Thus $\Pi(\zeta_i)$ is an operator from $L^p(B\times X)$ to $L^p(X)$, and $\Pi(\zeta_i)=\pi_X(\zeta_i)$ on the space $L^p(X)$, viewed as the subspace of $L^p(B\times X)$ consisting of functions independent of $b\in B$.
Let us also set ${\mathbb{M}}[F|\zeta] = \sup_{i\in {\mathbb{I}}} \Pi(\zeta_i)(|F|)$ for any $F\in L^p(B\times X)$, and begin the proof by establishing maximal inequalities, as follows.
\begin{thm}\label{thm:maximal}
If $\Omega$ satisfies the strong $L^p$ maximal inequality and $\frac{1}{p} + \frac{1}{q} =1$ then there is a constant $\bar{C}_p>0$ such that for any $F\in L^p(B\times X)$,
$$\| {\mathbb{M}}[F|\zeta] \|_p \le \bar{C}_p \| F\|_p.$$
If $\Omega$ satisfies the $L\log(L)$-maximal inequality and $\psi\in L^\infty(B,\nu)$, then there is a constant $\bar{C}_1>0$ such that for any $F\in L\log L(B\times X)$,
$$\| {\mathbb{M}}[F|\zeta] \|_{1} \le \bar{C}_1 \|F\|_{L\log L}.$$
\end{thm}
\begin{proof}
Without loss of generality, we may assume $F \ge 0$. Let us first consider the case $p>1$ and $\frac{1}{p}+\frac{1}{q}=1$. Because $\Omega$ satisfies the maximal inequality for functions in $L^p(B\times X)$, there is a constant $C_p>0$ such that $\| {\mathbb{M}}[F | \Omega] \|_p \le C_p \|F \|_p$ for every $F\in L^p(B\times X)$.
By (\ref{transform}) and H\"older's inequality,
\begin{eqnarray*}
\|{\mathbb{M}}[F|\zeta]\|_p^p &=& \int_X \left| \sup_{i\in {\mathbb{I}}} \int_B\int_K {\mathbb{A}}[F | \omega_i](kb,x)\psi(b)~dk d\nu(b) \right|^p~d\mu(x)\\
&\le & \int_X \sup_{i\in {\mathbb{I}}}\left( \int_B\int_K {\mathbb{A}}[F | \omega_i](kb,x)^p~dk d\nu(b)\right)\left( \int_B\int_K \psi(b)^q~dkd\nu(b)\right)^{p/q}~d\mu(x)\\
&=& \| \psi\|_q^p \int_X \sup_{i\in {\mathbb{I}}} \int_B\int_K {\mathbb{A}}[F | \omega_i](kb,x)^p~dk d\nu(b)~d\mu(x)\\
&\le&\| \psi\|_q^p \int_X\int_B\int_K {\mathbb{M}}[F | \Omega](kb,x)^p~dk d\nu(b)d\mu(x)\\
&=& \| \psi\|_q^p \int_X\int_B\int_K {\mathbb{M}}[F| \Omega](b,x)^p \frac{d\nu \circ k^{-1}}{d\nu}(b)~dk d\nu(b)d\mu(x)\\
&\le& C(K) \| \psi\|_q^p \int_X\int_B {\mathbb{M}}[F | \Omega](b,x)^p ~ d\nu(b)d\mu(x) = C(K) \| \psi\|_q^p \| {\mathbb{M}}[F | \Omega] \|_p^p\\
&\le& C(K) \| \psi\|_q^p C_p^p \| F\|^p_p = \bar{C}_p \| F\|_p.
\end{eqnarray*}
As to the $L\log L$ results, let us now suppose $F \in L\log L(B \times X)$ and $\psi \in L^\infty(B)$. We assume there is a constant $C_1>0$ (independent of $F$ and the action $\Gamma \curvearrowright (X,\mu)$) such that $\|{\mathbb{M}}[F|\Omega]\|_1 \le C_1\|F\|_{L\log L} = C_1 \|F\|_{L\log L}$ for any $F\in L\log L(B\times X)$. The proof that $\| {\mathbb{M}}[F | \zeta] \|_1 \le C(K)C_1 \|\psi\|_\infty \|F\|_{L\log L}$ is now similar to the proof of the $p>1$ case above.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:general}]
Theorem \ref{thm:maximal} proves the first conclusion, namely the maximal inequalities. To obtain the second conclusion, namely pointwise convergence, without loss of generality we may assume $\Gamma \curvearrowright (X,\mu)$ is ergodic. Suppose now that $F \in L^\infty(B\times X)$. The bounded convergence theorem and the assumption that $\Omega$ is a pointwise ergodic family implies that for a.e. $x\in X$,
\begin{eqnarray*}
\lim_{r\to\infty} \Pi( \zeta_r)(F)(x) &=& \lim_{r\to\infty} \int_B\int_K {\mathbb{A}}[F|\omega_r](kb,x)\psi(b)~dkd\nu(b)\\
&=& \int_B\int_K \lim_{r\to\infty} {\mathbb{A}}[F|\omega_r](kb,x)\psi(b)~dkd\nu(b)\\
&=& \int_B\int_K \EE[F|{\cR_{\beta\alpha}}](kb,x)\psi(b)~dkd\nu(b).
\end{eqnarray*}
In particular, $\{\zeta_r\}_{r\in {\mathbb{I}}}$ is a pointwise convergent sequence in $L^\infty(X) \subset L^\infty(B\times X)$.
Suppose now that $F \in L^p(B\times X)$ for some $p>1$ and $\frac{1}{p}+\frac{1}{q}=1$. We will show that for a.e. $x\in X$,
$$\lim_{r\to\infty} \Pi( \zeta_r)(F)(x) = \int_B\int_K \EE[F|{\cR_{\beta\alpha}}](kb,x)\psi(b)~dkd\nu(b).$$
By replacing $F$ with $F - \EE[F|{\cR_{\beta\alpha}}]$ if necessary, we may assume $\EE[F|{\cR_{\beta\alpha}}]=0$.
Let $\epsilon>0$. Because $L^\infty(B\times X)$ is dense in $L^p(B\times X)$, there exists an element $F' \in L^\infty(B\times X)$ such that $\|F - F'\|_p \le \epsilon$ and $\EE[F'|{\cR_{\beta\alpha}}]=0$. Then for a.e. $x\in X$,
\begin{eqnarray*}
\limsup_{r\to\infty} \left| \Pi( \zeta_r)(F)(x) \right|&\le &
\limsup_{r\to\infty} \left| \Pi(\zeta_r)(F-F')(x)\right| + \lim_{r\to\infty} \left|\Pi(\zeta_r)(F')(x)\right| \\
&=&\limsup_{r\to\infty} \left| \Pi(\zeta_r)(F-F')(x)\right| \le {\mathbb{M}}[F - F'|\zeta](x).
\end{eqnarray*}
Thus if $\tilde{F}:=\limsup_{r\to\infty} \left| \Pi(\zeta_r)(F)\right|$, then
$$\|\tilde{F}\|_p \le \| {\mathbb{M}}[F - F'|\zeta] \|_p \le C'_p \| F-F'\|_p \le C'_p \epsilon $$
for some constant $C'_p>0$ by Theorem \ref{thm:maximal}. Since $\epsilon$ is arbitrary, $\|\tilde{F}\|_p = 0$ which implies
$$\lim_{r\to\infty} \Pi( \zeta_r)(F)(x) = 0= \int_B\int_K \EE[F|{\cR_{\beta\alpha}}](kb,x)\psi(b)~dkd\nu(b)$$
for a.e. $x$ as required. This proves $\{\zeta_r\}_{r\in {\mathbb{I}}}$ is a pointwise convergent sequence in $L^p(X)\subset L^p(B\times X)$.
Finally, to prove the third claim in Theorem \ref{thm:general}, assume $\alpha$ is weakly mixing relative to the $K$-action. Then given $f\in L^p(X)$, because $\iint \EE[f|{\cR_\alpha}](kb,x)\psi(b)~dkd\nu(b) = \EE[f|\Gamma](x)$, $\{\zeta_r\}_{r\in {\mathbb{I}}}$ is a pointwise ergodic sequence in $L^p$. The proof of the case when $f \in L\log L(X)$ and $\psi \in L^\infty(B)$ is similar.
\end{proof}
\section{Amenable actions}\label{sec:amenable}
As noted already, the notion of weakly-mixing of a cocycle relative to a compact group arises naturally when we consider the type of an amenable action, which arises in our context as the obstruction to the ergodicity of the extended relation.
We now proceed to define and discuss the fundamental notions of type and stable type of a non-singular action.
\subsection{The ratio set, type, and stable type of a non-singular action}\label{sec:type}
Let $\Gamma$ be a countable group and $\Gamma \curvearrowright (B,\nu)$ a nonsingular action on a standard probability space. The {\em ratio set} of the action, denoted ${\textrm{RS}}(\Gamma \curvearrowright (B,\nu)) \subset [0,\infty]$, is defined as follows. A real number $r\ge 0$ is in ${\textrm{RS}}(\Gamma \curvearrowright (B,\nu))$ if and only if for every positive measure set $A \subset B$ and $\epsilon>0$ there is a subset $A' \subset A$ of positive measure and an element $g\in \Gamma \setminus\{e\}$ such that
\begin{itemize}
\item $gA' \subset A$,
\item $| \frac{d\nu \circ g}{d\nu}(b)-r| < \epsilon$ for every $b \in A'$.
\end{itemize}
The extended real number $+\infty \in {\textrm{RS}}(\Gamma \curvearrowright (B,\nu))$ if and only if for every positive measure set $A \subset B$ and $n>0$ there is a subset $A' \subset A$ of positive measure and an element $g\in \Gamma \setminus\{e\}$ such that
\begin{itemize}
\item $gA' \subset A$,
\item $ \frac{d\nu \circ g}{d\nu}(b) > n$ for every $b \in A'$.
\end{itemize}
The ratio set is also called the {\em asymptotic range} or {\em asymptotic ratio set}. By Proposition 8.5 of \cite{FM77}, if the action $\Gamma \curvearrowright (B,\nu)$ is ergodic then ${\textrm{RS}}(\Gamma \curvearrowright (B,\nu))$ is a closed subset of $[0,\infty]$. Moreover, ${\textrm{RS}}(\Gamma \curvearrowright (B,\nu)) \setminus \{0,\infty\}$ is a multiplicative subgroup of $\RR_{>0}$. Since
$$\frac{d\nu \circ g^{-1}}{d\nu}(gb) = \left( \frac{d\nu \circ g}{d\nu}(b) \right)^{-1},$$
the number $0$ is in the ratio set if and only if $\infty$ is in the ratio set. So if $\Gamma \curvearrowright (B,\nu)$ is ergodic and non-atomic then the possibilities for the ratio set and the corresponding type classification are:
\begin{displaymath}
\begin{array}{c|cc}
\textrm{ ratio set } & \textrm{ type }\\
\hline
\{1\} & II &\\
\{0,1,\infty\} & III_0&\\
\{0,\lambda^n, \infty: ~n\in \ZZ\} & III_\lambda & (0 < \lambda < 1)\\
\lbrack 0,\infty \rbrack & III_1&
\end{array}\end{displaymath}
For a very readable review, see \cite{KW91}. There is also an extension to general cocycles taking values in an arbitrary locally compact group in section 8 of \cite{FM77}.
Observe that if $\Gamma \curvearrowright (X,\mu)$ is a pmp action then the ratio set of the product action satisfies ${\textrm{RS}}(\Gamma \curvearrowright (B\times X, \nu \times \mu)) \subset {\textrm{RS}}(\Gamma \curvearrowright (B,\nu))$. Therefore, it makes sense to define the {\em stable} ratio set of $\Gamma \curvearrowright (B,\nu)$ by ${\textrm{SRS}}(\Gamma \curvearrowright (B,\nu)) = \cap {\textrm{RS}}(\Gamma \curvearrowright ( B\times X,\nu \times \mu))$ where the intersection is over all pmp actions $G \curvearrowright (X,\mu)$.
We say that $\Gamma \curvearrowright (B,\nu)$ is {\em weakly mixing} if for any ergodic pmp action $\Gamma \curvearrowright (X,\mu)$, the product action $\Gamma \curvearrowright (B\times X,\nu\times\mu)$ is ergodic. If $\nu$ is also non-atomic then the possibilities for the stable ratio set and the corresponding stable type classification are:
\begin{displaymath}
\begin{array}{c|cc}
\textrm{ stable ratio set } & \textrm{ stable type }\\
\hline
\{1\} & II &\\
\{0,1,\infty\} & III_0&\\
\{0,\lambda^n, \infty: ~n\in \ZZ\} & III_\lambda & (0 < \lambda < 1)\\
\lbrack 0,\infty \rbrack & III_1&
\end{array}\end{displaymath}
\subsection{Examples of non-zero stable type}
Let us proceed to give two important examples of these notions. First, consider the case that $\Gamma$ is an irreducible lattice in a connected semisimple Lie group $G$ which has trivial center and no compact factors. Consider the action of $\Gamma$ on $(G/P,\nu)$, where $P<G$ is a minimal parabolic subgroup and $\nu$ is a probability measure in the unique $G$-invariant measure class. As noted in \cite{BN2}, it is well-known that the action is weakly-mixing, and furthermore, it is shown in \cite{BN2} that the action of $\Gamma$ on $G/P$ is stable type $III_1$. Second, in \cite{B2}
it is shown that for the action of a non-elementary word-hyperbolic group on its Gromov boundary, under suitable assumptions the Patterson-Sullivan measure gives rise to a weakly-mixing action which is of stable type $III_\lambda$, with $0 < \lambda \le 1$. A curious example occurs by considering the action of the free group $\FF_r$ of rank $r$ on its Gromov boundary with respect to the usual Patterson-Sullivan measure (see \S \ref{sec:free} for details). This action is type $III_\lambda$ and stable type $III_{\lambda^2}$ where $\lambda = (2r-1)^{-1}$. These are the only results on stable type of which we are aware.
By comparison, in \cite{INO08} it is proven that the Poisson boundary of a random walk on a Gromov hyperbolic group induced by a nondegenerate measure on $\Gamma$ of finite support is never of type $III_0$. In \cite{Su78, Su82}, Sullivan proved that the recurrent part of an action of a discrete conformal group on the sphere $\mathbb{S}^d$ relative to the Lebesgue measure is type $III_1$. Spatzier \cite{Sp87} showed that if $\Gamma$ is the fundamental group of a compact connected negatively curved manifold then the action of $\Gamma$ on the sphere at infinity of the universal cover is also of $III_1$. The types of harmonic measures on free groups were computed by Ramagge and Robertson \cite{RR97} and Okayasu \cite{Ok03}.
\subsection{Maharam Extensions}\label{sec:maharam}
An interesting source of $\sigma$-finite measure-preserving amenable actions is obtained as follows. Let $\Gamma \curvearrowright (B,\nu)$ be a non-singular action on a probability space. Let
$$R(g,b) =\log\left( \frac{ d\nu \circ g}{d\nu}(b)\right).$$
Define a measure $\theta$ on $\RR$ by $d\theta(t) = e^{t} dt$. Then $\Gamma$ acts on $B\times \RR$ by
$$g (b,t) = (gb, t- R(g,b)).$$
This action is called the {\em Maharam extension}. It preserves the $\sigma$-finite measure $\nu\times \theta$. If $\Gamma \curvearrowright (B,\nu)$ is amenable then this action is also amenable.
If the Radon-Nikodym derivative takes values in a proper subgroup of $\RR$ then it is more appropriate to consider the {\em discrete Maharam extension}, which is defined as follows. For $\lambda \in (0,1)$, let
$$R_\lambda(g,b) =\log_\lambda\left( \frac{ d\nu \circ g}{d\nu}(b)\right).$$
Suppose that $R_\lambda(g,b) \in \ZZ$ for every $g\in \Gamma$ and a.e. $b \in B$. Define a measure $\theta_\lambda$ on $\ZZ$ by $\theta_\lambda(\{n\}) = \lambda^{-n}$. Then $\Gamma$ acts on $B\times \ZZ$ by
$$g (b,t) = (gb, t+ R_\lambda(g,b)).$$
This action is called the {\em discrete Maharam extension}. It preserves the $\sigma$-finite measure $\nu\times \theta_\lambda$. If $\Gamma \curvearrowright (B,\nu)$ is amenable then this action is also amenable
\section{Stable type and weakly-mixing relative to a compact group}\label{sec:typeIII}
In this section, we will demonstrate how to utilize the hypothesis that a non-singular action has a non-zero stable type in order to show that its Maharam extension gives rise to an ergodic pmp equivalence relation which is weakly-mixing relative to a compact (in fact, a circle) group.
\subsection
Statement of Theorem \ref{thm:typeIII}}
Suppose $\Gamma \curvearrowright (B,\nu)$ is an essentially free, weakly mixing action of stable type $III_1$. In this case, the Maharam extension $\Gamma \curvearrowright (B\times \RR, \nu\times \theta)$ is weakly mixing. This fact is implied by \cite[Corollary 4.3]{BN2} which is a straightforward application of \cite[Proposition 8.3, Theorem 8]{FM77}. Thus if $B_0 \subset B\times \RR$ is any Borel set with $\nu\times \theta(B_0)=1$ and $\alpha: B_0 \to \Gamma$ is defined by $\alpha(c,b)=\gamma$ where $\gamma b = c$ then $\alpha$ is weakly mixing and therefore, we can apply Theorem \ref{thm:general}.
Since our goal is to utilize amenable actions of $\Gamma$ to construct ergodic averages on $\Gamma$, it becomes necessary to consider non-singular actions with type $III_1$ and stable type $\tau\in (0,1)$. In that case, there arises naturally a non-singular action which is weakly-mixing relative to a compact group $K$, and we will be able to apply Theorem \ref{thm:general} again. We therefore formulate the following result, which will be proved next.
\begin{thm}\label{thm:typeIII}
Let $\Gamma \curvearrowright (B,\nu)$ be a nonsingular essentially free weakly mixing action on a standard probability space of type $III_1$ and stable type $III_\lambda$ for some $\lambda \in (0,1)$. Let $T=-\log(\lambda)$, and $\cR$ be the equivalence relation on $B \times [0,T]$ obtained by restricting the orbit-equivalence relation on the Maharam extension. Thus, $(b,t)\cR(b',t')$ if there exists $\gamma \in \Gamma$ such that $(\gamma b, t- R(\gamma,b)) = (b',t')$. Let $\alpha:\cR \to \Gamma$ be the cocycle $\alpha((b',t'), (b,t))=\gamma$ if $\gamma(b,t)=(b',t')$. Also, let $K =\RR/T\ZZ$ and $K \curvearrowright B\times [0,T]$ be the action $(r + T\ZZ)(b,t) = (b,t+r)$ (where $t+r$ is taken modulo $T$). Then $\alpha$ is weakly mixing relative to this $K$-action.
\end{thm}
We remark that the case where the type of the action is $\lambda\in (0,1)$ and the stable type is $\tau\in (0,1)$ involves the discrete Maharam extension and was considered in \cite{BN2}. For completeness we state this result explicitly.
\begin{thm}\label{previous}\cite[Corollary 5.4]{BN2}.
Let $\Gamma \curvearrowright (B,\nu)$ be a nonsingular essentially free weakly mixing action on a standard probability space of type $III_\lambda$ and stable type $III_\tau$ for some $\lambda,\tau \in (0,1)$. Assume $R_\lambda(g,b)\in \ZZ$ for every $g\in G, b \in B$ and $\tau=\lambda^N$ for some $N\ge 1$. Let $\cR$ be the equivalence relation on $B \times \{0,1,\ldots, N-1\}$ obtained by restricting the orbit-equivalence relation on the discrete Maharam extension. Thus, $(b,t)\cR(b',t')$ if there exists $\gamma \in \Gamma$ such that $(\gamma b, t+ R_\lambda(\gamma,b)) = (b',t')$. Let $\alpha:\cR \to \Gamma$ be the cocycle $\alpha((b',t'), (b,t))=\gamma$ if $\gamma(b,t)=(b',t')$. Also, let $K =\ZZ/N\ZZ$ and $K \curvearrowright B\times \{0,1,\dots, N-1\}$ be the action $(r + T\ZZ)(b,t) = (b,t+r)$ (where $t+r$ is taken modulo $N$). Then $\alpha$ is weakly mixing relative to this $K$-action.
\end{thm}
We remark that the notation in \cite[Corollary 5.4]{BN2} is somewhat different : in that paper $\EE[f|\cI(\tilde{\cR_I})]$ denotes $\EE[f | {\cR_\alpha}]$ and $\theta_{\lambda,I}$ is the probability measure on $\{0,1,\ldots, N-1\}$ given by $\theta_{\lambda,I}(\{j\}) = \frac{\lambda^{-j}}{1+ \lambda^{-1}+\cdots \lambda^{-N+1}}$.
\subsection
Proof of Theorem \ref{thm:typeIII}}
The proof of Theorem \ref{thm:typeIII} utilizes the construction of the Mackey range of the Radon-Nikodym cocycle associated with the measure $\nu\times \mu$ on $B\times X$, via the following result.
\begin{prop}\label{prop:erg}
Suppose $\Gamma \curvearrowright (B,\nu)$ is an essentially free, ergodic action of type $III_1$. Let $\Gamma \curvearrowright (X,\mu)$ be an ergodic pmp action. Suppose that the product action $\Gamma \curvearrowright (B\times X,\nu\times\mu)$ is ergodic and type $III_\tau$ for some $\tau \in (0,1)$. Let $\Gamma$ act on $B \times X \times \RR$ by
$$g(b,x,t) = \left(gb,gx, t -R(g,b) \right)$$
where
$$R(g,b) = \log \left( \frac{d\nu\circ g}{d\nu}(b) \right).$$
Let $T_0=-\log(\tau), I \subset \RR$ be a compact interval, $\widetilde{\cR}_I$ be the equivalence relation on $B \times X \times I$ obtained by restricting the orbit-equivalence relation. Then there is a measurable map $\phi:B \times X \times \RR \to \RR/T_0\ZZ$ defined almost everywhere, and a measurable family of probability measures $\{\eta_z:~ z \in \RR/T_0\ZZ\}$ defined almost everywhere, satisfying
\begin{enumerate}
\item $\phi(b,x,t+t') \equiv \phi(b,x,t) + t' \mod T_0$ for a.e. $(b,x,t)$ and every $t'$;
\item $\phi$ is essentially $\Gamma$-invariant;
\item almost every $\eta_z$ is an ergodic $\widetilde{\cR}_I$-invariant probability measure on $B\times X \times I$;
\item almost every $\eta_z$ is supported on $\phi^{-1}(z)$;
\item $T_0^{-1} \int_0^{T_0} \eta_z ~dz = \nu \times \mu \times \theta_I$ where $\theta_I$ is the restriction of $\theta$ to $I$ normalized to have total mass $1$.
\end{enumerate}
\end{prop}
\begin{proof}
Given any probability space $(U,\eta)$, let ${\mathcal M} (U,\eta)$ denote the associated measure algebra, consisting of equivalence classes of measurable sets modulo null sets. Let $\sigma$ be a probability measure on $\RR$ that is equivalent to Lebesgue measure. Let ${\mathcal M}_\Gamma$ be the collection of all sets $A \in {\mathcal M}(B\times X \times \RR, \nu\times\mu\times \sigma)$ that are essentially $\Gamma$-invariant, modulo null sets. Note that $\RR$ acts on $B\times X \times \RR$ by $(t, (b,x,t')) \mapsto (b,x,t+t')$ and this action commutes with the $\Gamma$-action. By Mackey's point realization theorem (see e.g., \cite[Corollary B.6]{Zi84}), there is a standard Borel probability space $(Z,\kappa)$, a measurable almost everywhere defined map $\phi: B\times X \times \RR \to Z$ and a nonsingular action $\{S_t\}_{t\in \RR}$ of $\RR$ on $(Z,\kappa)$ such that
\begin{enumerate}
\item ${\mathcal M}_\Gamma$ is the pullback under $\phi$ of the measure algebra ${\mathcal M}(Z,\kappa)$;
\item $\phi( g (b,x,t) ) = \phi(b,x,t)$ for a.e. $(b,x,t)$ and every $g\in \Gamma$;
\item $\phi( b,x, t + t') = S_t\phi(b,x,t')$ for a.e. $(b,x,t')$ and every $t \in \RR$.
\end{enumerate}
Because $\Gamma \curvearrowright (B\times X, \nu \times \mu)$ is ergodic, \cite[Proposition 8.1]{FM77} implies that the flow $\{S_t\}_{t\in \RR}$ on $(Z,\kappa)$ is also ergodic. The fact that the action $\Gamma \curvearrowright (B \times X, \nu \times \mu)$ is type $III_\tau$ is equivalent to the statement that the flow $\{S_t\}_{t\in \RR}$ on $(Z,\kappa)$ is isomorphic to the canonical action of $\RR$ on $\RR/T_0\ZZ$ (see \cite[Theorem 8]{FM77}). So without loss of generality, we will assume $Z=\RR/T_0\ZZ$ and $\phi(b,x,t+t') = \phi(b,x,t)+t'$ for a.e. $(b,x,t)$ and every $t'$. The measure $\kappa$ on $Z=\RR/T_0\ZZ$ is quasi-invariant under this action.
Because $\Gamma \curvearrowright (B,\nu)$ is type $III_1$ (and ergodic), it follows that the action $\Gamma \curvearrowright (B\times \RR, \nu \times \theta)$ given by $g (b,t) = (gb, t- R(g,b))$ is ergodic \cite[Theorem 8]{FM77}. For almost every $(b,t)\in B\times \RR$, let $\mu_{(b,t)}$ be the pushforward of $\mu$ under the map $x \mapsto \phi(b,x,t)$. We claim that $\mu_{g(b,t)} = \mu_{(b,t)}~\forall g\in \Gamma$ and almost every $(b,t)$. Indeed, $\mu_{g(b,t)}$ is the pushforward of $\mu$ under the map
$$x \mapsto \phi(gb, x, t- R(g,b)) = \phi( b, g^{-1}x, t- R(g,b) - R(g^{-1},gb)) = \phi(b, g^{-1}x, t)$$
where we have used the fact that $\phi$ is essentially $\Gamma$-invariant and $R(g,b) + R(g^{-1},gb)=0$ by the cocycle equation. Because $\mu$ is $\Gamma$-invariant, this implies the claim: $\mu_{g(b,t)} = \mu_{(b,t)}$ for a.e. $(b,t)$ and every $g\in \Gamma$.
Because the action $\Gamma \curvearrowright (B\times \RR, \nu \times \theta)$ is ergodic, and the map $\mu_{(b,t)}\mapsto \mu_{g(b,t)}$ is essentially invariant, there is a measure $\rho$ on $\RR/T_0\ZZ$ such that $\mu_{(b,t)} = \rho$ for almost every $(b,t)$. We claim that $\rho$ is the Haar probability measure on $\RR/T_0\ZZ$. To see this, note that for almost every $(b,t)$ and every $g\in \Gamma$, $\mu_{(b,t)} = \mu_{(gb,t)} = \rho$. However, $\mu_{(gb,t)}$ is the pushforward of $\mu$ under the map
$$x \mapsto \phi(gb,x,t) = \phi(b,g^{-1}x, t- R(g^{-1},gb)) = \phi(b,g^{-1}x, t) +R(g,b).$$
Because $\mu$ is $\Gamma$-invariant, this is the same as the pushforward of $\mu$ under the map $x \mapsto \phi(b,x,t) + R(g,b)$. In other words, it is the pushforward of $\mu_{(b,t)}$ under the map $t' \mapsto t' +R(g,b)$. Thus we have that $\rho$ is invariant under addition by $R(g,b)$ for a.e. $b\in B$ and $g\in \Gamma$. Because $\Gamma \curvearrowright (B,\nu)$ is type $III_1$, this implies that $\rho$ is $\RR$-invariant. So it is Haar probability measure as claimed.
Let $I \subset \RR$ be a compact interval. Because $\mu_{(b,t)}=\rho$ for a.e. $(b,t)$, it follows $\rho=\phi_*( \nu \times \mu \times \theta_I)$. Let $\{\eta_z:~ z\in \RR/T_0\ZZ\}$ be the decomposition of $\nu\times\mu\times\theta_I$ over $(\RR/T_0\ZZ,\rho)$. This means that
\begin{enumerate}
\item almost $\eta_z$ is a probability measure on $B\times X\times I$ with $\eta_z( \{ (b,x,t):~ \phi(b,x,t)=z\}) = 1$;
\item $\nu\times\mu\times\theta_I = \int \eta_z~d\rho(z)$.
\end{enumerate}
By definition of $\phi$, each $\eta_z$ is $\widetilde{\cR}_I$-invariant and ergodic. This implies the proposition.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:typeIII}]
Recall that $\Gamma \curvearrowright (B,\nu)$ has stable type $III_\lambda$, $\lambda \in (0,1)$ and $T=-\log(\lambda)$. Let $\Gamma \curvearrowright (X,\mu)$ be an ergodic action and $\widetilde{\cR}_I$ is the corresponding equivalence relation on $B\times X \times [0,T]$. So $(b,x,t) \widetilde{\cR}_I (b',x', t')$ if there exists $\gamma \in \Gamma$ such that $(b',x', t')=(\gamma b,\gamma x, t - R(\gamma,b))$.
Because $\Gamma \curvearrowright (B,\nu)$ is stable type $III_\lambda$, the ratio set of $\Gamma \curvearrowright (B\times X,\nu\times\mu)$ contains $\{\lambda^n:~n\in\ZZ\}$. Thus $\lambda = \tau^n$ for some $n\ge 1$, $n \in \ZZ$. Let $T_0=-\log(\tau)$. Because $\Gamma \curvearrowright (B,\nu)$ is weakly-mixing, $\Gamma \curvearrowright (B\times X,\nu \times \mu)$ is ergodic. So Proposition \ref{prop:erg} implies the existence of a map $\phi: B \times X \times \RR \to \RR/T_0\ZZ$ and probability measures $\{\eta_z:~z \in \RR/T_0\ZZ\}$ on $B\times X \times I$ satisfying conditions (1-5) of the proposition. So $\nu\times \mu \times \theta_I = T^{-1} \int_0^T \eta_z~dz$ is the ergodic decomposition of $\nu\times \mu\times \theta_I$ (w.r.t. $\widetilde{\cR}_I$). Thus, for any $f\in L^1(B\times X \times I)$,
$$\EE_{\nu\times\mu\times\theta_I}[ f | {\widetilde{\cR}_I} ] (b,x,t) = \int f ~d\eta_{\phi(b,x,t)}.$$
Because $\phi$ is equivariant with respect to the $\RR$-action, it follows that for a.e. $(b,x)$,
$$\frac{1}{T_0} \int_0^{T_0} \eta_{\phi(b,x,t)} ~dt = \int \eta_z ~d\rho(z) = \nu\times\mu\times\theta_I$$
where $\rho$ is the Haar probability measure on $\RR/T_0\ZZ$. Because $T=n T_0$,
\begin{eqnarray*}
T^{-1} \int_0^T \EE_{\nu\times\mu\times\theta_I}[f| {\widetilde{\cR}_I}|(b,x,t)~dt &=& T^{-1}\int_0^{T} \int f ~d\eta_{\phi(b,x,t)} dt\\
&=& T_0^{-1}\int_0^{T_0} \int f ~d\eta_{\phi(b,x,t)} dt= \int f ~d\nu\times\mu\times\theta_I
\end{eqnarray*}
as required.
\end{proof}
\section{Cocycles and ergodic theorems : Some examples}\label{sec:example}
\subsection{Random walks, the associated cocycle, and ergodic theorems for convolutions}
The construction in the present section is of a weakly-mixing cocycle on an ergodic amenable pmp equivalence relation {\it for an arbitrary countable group}.
\begin{thm}\label{thm:existence}
For any countably infinite group $\Gamma$, there exists a weakly mixing cocycle $\alpha:\cR \to \Gamma$, where $\cR\subset B\times B$ is an amenable discrete ergodic pmp equivalence relation.
\end{thm}
\begin{proof}
Choose a probability measure $\kappa$ on $\Gamma$ whose support generates $\Gamma$. Consider the product space $\Gamma^\ZZ$ with the product topology. Let $\kappa^\ZZ$ be the product measure on $\Gamma^\ZZ$. Let $T: \Gamma^\ZZ \to \Gamma^\ZZ$ be the usual shift action $Tx(n):=x(n+1)$. Let $\cR$ be the equivalence relation on $\Gamma^\ZZ$ determined by the orbits of $\inn{T}\cong \ZZ$, namely $\cR=\{ (x,T^nx):~ x\in \Gamma^\ZZ, n \in \ZZ\}=\cO_\ZZ(S^\ZZ)$. Dye's Theorem \cite{Dy59, Dy63} implies $\cR$ is the hyperfinite $II_1$ equivalence relation (modulo a measure zero set), so that it is amenable \cite{CFW}.
Define $\alpha:\cR\to \Gamma$ by
\begin{displaymath}
\alpha(x,T^nx)= \left\{ \begin{array}{ll}
x(1)x(2)\cdots x(n) & \textrm{ if } n > 0 \\
e & \textrm{ if } n = 0\\
x(0)^{-1}x(-1)^{-1}\cdots x(n+1)^{-1} & \textrm{ if } n <0
\end{array}\right.
\end{displaymath}
$\alpha$ is a measurable cocycle well-defined on an invariant conull measurable set.
To show $\alpha$ is weakly mixing, let $\Gamma \curvearrowright (X,\mu)$ be a pmp action. Define $\tilde{T}:\Gamma^\ZZ \times X \to \Gamma^\ZZ \times X$ by $\tilde{T}(x,y) = (Tx, x(0)^{-1}y)$. Note that the relation $\cR(X) $ is the orbit-equivalence relation of $\tilde{T}$ on $\Gamma^\ZZ\times X$. Therefore, $\cR(X)$ is ergodic if and only if $\tilde{T}$ is ergodic. Kakutani's random ergodic theorem \cite{Ka55} implies $\tilde{T}$ is indeed ergodic.
\end{proof}
What then is the pointwise ergodic theorem arising from our construction above ? To answer this, recall that if $\kappa_1,\kappa_2$ are two probability measures on $\Gamma$ then their convolution $\kappa_1*\kappa_2$ is a probability measure on $\Gamma$ defined by
$$\kappa_1\ast \kappa_2(\{g\}) = \sum_{h \in \Gamma} \kappa_1(\{gh^{-1}\})\kappa_2(\{h\}).$$
Let $\kappa^{*n}$ denote the $n$-fold convolution power of $\kappa$.
\begin{cor}
Let $\kappa$ be a probability measure on $\Gamma$ whose support generates $\Gamma$. Let $\rho_n = \frac{1}{n} \sum_{k=1}^n \kappa^{\ast k}$. Then $\{\rho_n\}_{n=1}^\infty$ is a pointwise ergodic sequence in $L\log L$.
\end{cor}
\begin{proof}
Define $\alpha:\cR \to \Gamma$ as in the proof of Theorem \ref{thm:existence}. Let $\Omega=\{\omega_n\}_{n=1}^\infty$ be the sequence of leafwise probability measures on $\cR$ defined by $\omega_n(x,y) = \frac{1}{n}$ if $y= T^i x $ for some $1\le i \le n$. Because $T$ is ergodic, and these measures are the ergodic averages on $T$-orbits, this sequence satisfies the weak (1,1)-type maximal inequality and is pointwise ergodic in $L^1$, by Birkhoff's and Wiener's Theorem.
By Theorem \ref{thm:general}, if we set
$$\zeta_n(\gamma):= \int_{\Gamma^\ZZ} \sum_{c:~\alpha(c,b)=\gamma} \omega_n(c,b) ~d\kappa^\ZZ(b) $$
then $\{\zeta_n\}_{n=1}^\infty$ is a pointwise ergodic family in $L\log(L)$. To see this, set $K$ equal to the trivial group and $\psi \equiv 1$.
The corollary now follows from a short calculation:
\begin{eqnarray*}
\int \sum_{c:~\alpha(c,b)=\gamma} \omega_n(c,b) ~d\kappa^\ZZ(b) &=& \frac{1}{n} \sum_{i=1}^n \kappa^\ZZ(\{x \in \Gamma^\ZZ:~\alpha(x,T^i x) = \gamma\})\\
&=&\frac{1}{n} \sum_{k=1}^n \kappa^{\ast k}(\{\gamma\}) = \rho_n(\{\gamma\}).
\end{eqnarray*}
\end{proof}
Thus this argument gives a different proof of the pointwise ergodic theorem for the uniform averages of convolution powers, which follows (for example) from the Chacon-Ornstein theorem or the Hopf-Dunford Schwartz theorem on the pointwise convergence of uniform averages of powers of a Markov operator.
\subsection{The free group}
We proceed to demonstrate our approach by obtaining pointwise ergodic theorems from the action of the free group on its boundary and its double-boundary.
\subsubsection{The boundary action}\label{sec:free}
Let $\FF=\FF_r=\langle a_1,\dots ,a_r \rangle$ be the free group of rank $r\ge 2$, and $S=\{a_i^{\pm 1}:~1\le i \le r\}$ a set of free generators. The {\em reduced form} of an element $g\in \FF$ is the unique expression $g=s_1\cdots s_n$ with $s_i \in S$ and $s_{i+1}\ne s_i^{-1}$ for all $i$. Define $|g|:=n$, the length of the reduced form of $g$ and the sphere $S_n=\set{w\,;\, \abs{w}=n}$.
We identify the boundary of $\FF$ with the set of all sequences $\xi=(\xi_1,\xi_2,\ldots) \in S^\NN$ such that $\xi_{i+1} \ne \xi_i^{-1}$ for all $i\ge 1$, and denote it by $\partial \FF$. Let $\nu$ be the probability measure on $\partial \FF$ determined as follows. For every finite sequence $t_1,\ldots, t_n$ with $t_{i+1} \ne t_i^{-1}$ for $1\le i <n$,
$$\nu\Big(\big\{ (\xi_1,\xi_2,\ldots) \in \partial \FF :~ \xi_i=t_i ~\forall 1\le i \le n\big\}\Big) := |S_n|^{-1}=(2r-1)^{-n+1}(2r)^{-1}.$$
By Carath\'eodory's Extension Theorem, this uniquely defines $\nu$ on the Borel sigma-algebra of $\partial \FF$.
The action of $\FF$ on $\partial \FF$ is given by
$$(t_1\cdots t_n)\xi := (t_1,\ldots,t_{n-k},\xi_{k+1},\xi_{k+2}, \ldots)$$ where $t_1,\ldots, t_n \in S$, $g=t_1\cdots t_n$ is in reduced form and $k$ is the largest number $\le n$ such that $\xi_i^{-1} = t_{n+1-i}$ for all $i\le k$. In this case the Radon-Nikodym derivative satisfies
$$\frac{d\nu \circ g}{d\nu}(\xi) = (2r-1)^{2k-n}.$$
Thus if $\lambda=(2r-1)^{-1}$ then $R_\lambda(g,\xi) = 2k-n \in \Z$.
The type of the action $\FF\curvearrowright (\partial \FF, \nu)$ is easily seen to $III_\lambda$ where $\lambda=(2r-1)^{-1}$. It is shown implicitly in \cite[Theorem 4.1]{BN1} that the stable type of $\FF\curvearrowright (\partial \FF, \nu)$ is $III_{\lambda^2}$. In fact, if $\FF^2$ denotes the index 2 subgroup of $\FF$ consisting of all elements $g$ with $|g| \in 2\Z$ then $\FF^2 \curvearrowright (\partial \FF,\nu)$ is type $III_{\lambda^2}$ and stable type $III_{\lambda^2}$. It is also weakly mixing. Indeed, $(\partial \FF,\nu)$ is naturally identified with $P(\FF,\mu)$, the Poisson boundary of the random walk generated by the measure $\mu$ which is uniformly distributed on $S$. By \cite{AL}, the action of any countable group $\Gamma$ on the Poisson boundary $P(\Gamma,\kappa)$ is weakly mixing whenever the measure $\kappa$ is adapted. This shows that $\FF \curvearrowright (\partial \FF,\nu)$ is weakly mixing. Moreover, if $S^2 = \{st:~s, t \in S\}$ then $(\partial \FF,\nu)$ is naturally identified with the Poisson boundary $P(\FF^2, \mu_2)$ where $\mu_2$ is the uniform probability measure on $S^2$. So the action $\FF^2 \curvearrowright (\partial \FF,\nu)$ is also weakly mixing.
Let $\FF$ act on $\partial \FF \times \ZZ$ by $g(b,t) = (gb, t+ R_\lambda(g,b))$. This is the Maharam extension. Although it is possible to use Theorem \ref{previous} to obtain an equivalence relation on $\partial \FF \times \{0,1\}$ and a cocycle, it is more fruitful to consider a slightly different construction. Let $\cR$ be the orbit-equivalence relation restricted to $\partial \FF \times \{0\}$, which we may, for convenience, identify with $\partial \FF$. In other words, $b \cR b'$ if and only if there is an element $g \in \FF$ such that $gb=b'$ and $\frac{d\nu \circ g}{d\nu}(b)=1$. Observe that this is the same as the (synchronous) tail-equivalence relation on $\FF$. In other words, two elements $\xi=(\xi_1,\xi_2,\ldots)$, $\eta=(\eta_1,\eta_2,\ldots) \in \partial \FF$ are $\cR$-equivalent if and only if there is an $m$ such that $\xi_n=\eta_n$ for all $n \ge m$. Also note that if $b\cR (gb)$ then necessarily $g \in \FF^2$. So $\cR$ can also be regarded as the orbit-equivalence relation for the $\FF^2$-action on $\partial \FF \times \ZZ$ restricted to $\partial \FF \times \{0\}$.
Let $\alpha:\cR \to \FF^2$ be the cocycle $\alpha(gb,b)=g$ for $g\in \FF^2, b \in \partial \FF$. This is well-defined almost everywhere because the action of $\FF^2$ is essentially free. Because $\FF^2 \curvearrowright (\partial \FF,\nu)$ has type $III_{\lambda^2}$ and stable type $III_{\lambda^2}$, this cocycle is weakly mixing for $\FF^2$. In other words, if $\FF^2 \curvearrowright (X,\mu)$ is any ergodic pmp action then the equivalence relation $\cR_\alpha$ defined on $\partial \FF \times X$ by $g(b,x)\cR_\alpha (b,x)$ if $\frac{d\nu \circ g}{d\nu}(b)=1$ is ergodic. This is the import of \cite[Theorem 4.1]{BN1}.
Now let $\omega_n: \cR \to [0,1]$ be the leafwise probability measure given by
\begin{displaymath}
\omega_n( gb,b ) = \left\{ \begin{array}{cc}
(2r-2)^{-1}(2r-1)^{-n+1} & \textrm{ if } |g|=2n \\
0 & \textrm{ otherwise} \end{array}\right.
\end{displaymath}
In other words, $\omega_n(\cdot, b)$ is uniformly distributed over the set of all elements of the form $gb$ with $|g|=2n$ and $\frac{d\nu \circ g}{d\nu}(b)=1$. In \cite[Corollary 5.2 \& Proposition 5.3]{BN1} it is shown that $\Omega=\{\omega_n\}_{n=1}^\infty$ is pointwise ergodic in $L^1$ and satisfies the weak (1,1)-type maximal inequality.
It follows from Theorem \ref{thm:general} that if
$$\zeta_n(\gamma):= \int_{\partial \FF} \sum_{c:~\alpha(c,b)=\gamma} \omega_n(c,b) ~d\nu(b)$$
then $\{\zeta_n\}_{n=1}^\infty$ is pointwise ergodic in $L\log(L)$ for $\FF_2$-actions. A short calculation reveal that $\zeta_n$ is uniformly distributed on the sphere of radius $2n$. This yields:
\begin{thm}\label{thm:sphere}
Let $\FF \curvearrowright (X,\mu)$ be a pmp action. Then for any $f \in L\log L(X,\mu)$,
$$|S_{2n}|^{-1} \sum_{|g|=2n} f\circ g$$
converges pointwise a.e as $n\to\infty$ to $\EE[f| \FF^2]$ the conditional expectation of $f$ on the sigma-algebra of $\FF^2$-invariant measurable subsets.
\end{thm}
This theorem was proven earlier in \cite{NS} and \cite{Bu} by different techniques.
\subsubsection{The double boundary action}
Next we analyze the action of $\FF$ on its double boundary $\partial^2 \FF := \partial \FF \times \partial \FF$. To begin, let $\bar{\nu}$ be the probability measure on the double boundary $\partial^2\FF$ defined by
$$d\bar{\nu}(b,c) = (2r)(2r-1)^{2d-1}d\nu(b)d\nu(c)$$
where $d \ge 0$ is the largest integer such that $b_i=c_i$ for all $i\le d$. A short calculation shows that $\bar{\nu}$ is $\FF$-invariant. Because the action $\FF \curvearrowright (\partial \FF,\nu)$ is a strong $\FF$-boundary in the sense of \cite{Ka}, it follows that $\FF \curvearrowright (\partial^2 \FF, \bar{\nu})$ is weakly mixing.
We need to restrict the orbit-equivalence relation on $\partial^2 \FF$ to a fundamental domain. For convenience we choose the domain $D$ equal
to the set of all $(b,c) \in \partial^2 \FF$ such that the geodesic from $b$ to $c$ passes through the identity element. In other words, $b_1 \ne c_1$. Observe that $\bar{\nu}(D)=\left(\frac{2r}{2r-1}\right)\nu \times \nu(D) = 1.$
It is clarifying to view $D,\bar{\nu}$ and the orbit-equivalence relation from a slightly different point of view. To do this, let $\Phi:D \to S^\Z$ be the map
\begin{displaymath}
\Phi(b,c)_i = \left\{\begin{array}{cc}
b_{-i}^{-1} & i <0 \\
c_{i+1} & i \ge 0
\end{array}\right.\end{displaymath}
Note that this is an embedding so we can identify $D$ with its image in $S^\Z$. Moreover $\Phi(D)$ is the set of all sequences $\xi \in S^\Z$ such that $\xi_i \ne \xi_{i+1}^{-1}$ for all $i$. Let $D'$ denote $\Phi(D)$.
Let $\nu' = \Phi_*\bar{\nu}$. Observe that $\nu'$ is a Markov measure. Indeed let $m<n$ be integers and for each $i$ with $m \le i \le n$, let $s_i \in S$ so that $s_i \ne s_{i+1}^{-1}$ for $m \le i < n$. Let $C(s_m,\ldots, s_n)$ be the cylinder set
$$C(s_m,\ldots, s_n)=\{\xi \in D':~ \xi_i = s_i ~\forall m \le i \le n\}.$$
Then $\nu'(C(s_m,\ldots, s_n)) = (2r)^{-1}(2r-1)^{m-n}$. This formula completely determines $\nu'$ by Carath\'eodory's Extension Theorem.
If $g \in \FF$, $(b,c) \in D$ and $g(b,c) = (gb,gc) \in D$ then $(gb)_1 \ne (gc)_1$ implies that we cannot have partial cancellation of $g$ in both products $gb$ and $gc$. In other words, we must either have that $g=(b_1\cdots b_n)^{-1}$ or $g = (c_1 \cdots c_n)^{-1}$ for some $n$. Let us suppose that $g=(b_1\cdots b_n)^{-1}$. Then $gb = (b_{n+1},b_{n+2},\ldots)$, $gc = (b_n^{-1}, b_{n-1}^{-1},\ldots, b_1^{-1},c_1,c_2,\ldots)$ (this uses that $b_1\ne c_1$ since $(b,c)\in D$) and
$\Phi(gb,gc)_i = \Phi(b,c)_{i-n}$ for every $i$. Similarly, if $g= (c_1 \cdots c_n)^{-1}$ then $\Phi(gb,gc)_i = \Phi(b,c)_{i+n}$ for every $i$.
Let $\cR$ be the equivalence relation on $D$ obtained by restricting the orbit-equivalence relation on $\partial^2 \FF$. The previous paragraph implies that if $\cR'$ is the relation on $D'=\Phi(D)$ obtained by pushing forward $\cR$ under $\Phi$ then $\cR'$ is the same as the equivalence relation determined by the shift. To be precise, define $T:D' \to D'$ by $(T\xi)_i = \xi_{i-1}$ for all $i$. Then two sequences $\xi, \eta \in D'$ are $\cR'$-equivalent if and only if there is an $n\in \ZZ$ such that $\xi = T^n\eta$. Note that $\nu'$ is $T$-invariant and therefore $\cR'$-invariant.
Let $\alpha: \cR \to \FF$ be the canonical cocycle given by $\alpha(g(b,c), (b,c)) = g$ for every $g \in \FF, (b,c) \in D$. We can push forward $\alpha$ under $\Phi$ to obtain a cocycle $\alpha': \cR' \to \FF$ given by
\begin{displaymath}
\alpha'( T^n \xi, \xi) = \left\{ \begin{array}{cc}
(\xi_0 \cdots \xi_{n-1})^{-1} & \textrm{ if } n\ge 1 \\
e & \textrm{ if } n=0\\
\xi_n \xi_{n+1}\cdots \xi_{-1} & \textrm{ if } n\le -1
\end{array}\right.\end{displaymath}
Because $\FF \curvearrowright (\partial^2 \FF,\bar{\nu})$ is weakly mixing, it follows that $\alpha$ and therefore $\alpha'$ are weakly mixing cocycles. Let $\omega_n$ be the leafwise measure on $\cR'$ defined by $\omega_n(T^i \xi, \xi) = \frac{1}{n+1}$ if $0\le i \le n$ and $\omega_n = 0$ otherwise. By Birkhoff's ergodic theorem, $\Omega= \{\omega_n\}_{n=1}^\infty$ is pointwise ergodic in $L^1$ and satisfies the weak $(1,1)$-type maximal inequality. It now follows from Theorem \ref{thm:general} that if
$$\zeta_n(\gamma):= \int_{D'} \sum_{\eta:~\alpha'(\eta,\xi)=\gamma} \omega_n(\eta,\xi) ~d\nu'(\xi) $$
then $\{\zeta_n\}_{n=1}^\infty$ is pointwise ergodic in $L\log(L)$. A short calculation shows that
$$\zeta_n = \frac{1}{n+1} \sum_{i=0}^n \sigma_n$$
where $\sigma_n$ is the uniform probability measure on the sphere of radius $n$ in $\FF$. So Theorem \ref{thm:general} implies:
\begin{thm}
Let $\FF \curvearrowright (X,\mu)$ be a pmp action. Then for any $f \in L\log L(X,\mu)$,
$$ \frac{1}{n+1} \sum_{i=0}^n \sigma_n(f)$$
converges pointwise a.e as $n\to\infty$ to $\EE[f| \FF]$ the conditional expectation of $f$ on the sigma-algebra of $\FF$-invariant measurable subsets.
\end{thm}
This theorem was proven earlier in \cite{NS}, \cite{Gr} and \cite{Bu} by different techniques. Of course, it also follows directly from Theorem \ref{thm:sphere}.
\subsection{Hyperbolic groups}
We now turn to briefly describe the application of our approach to establishing ergodic theorems for hyperbolic groups in a geometric setting. Full details will appear in \cite{BN4}, and for simplicity we will mention here only a special case of the results established there.
\begin{thm}\label{thm:hyp}
Suppose $\Gamma$ acts properly discontinuously by isometries on a $CAT(-1)$ space $(X,d_X)$. Suppose there is an $x \in X$ with trivial stabilizer. Define a metric $d$ on $\Gamma$ by $d(g,g'):=d_X(gx,g'x)$. Then there exists a family $\{\kappa_r\}_{r>0}$ of probability measures on $\Gamma$ such that
\begin{enumerate}
\item there is a constant $a>0$ so that each $\kappa_r$ is supported on the annulus $\{g\in \Gamma:~ d(e,g) \in [r-a,r+a]\}$,
\item $\{\kappa_r\}_{r>0}$ is a pointwise ergodic family in $L^p$ for every $p>1$ and in $L\log L$.
\end{enumerate}
\end{thm}
\subsubsection{A brief outline : the equivalence relation, the cocycle, and the weights}
Let $\partial\Gamma$ denote the Gromov boundary of $(\Gamma,d)$. Via the Patterson-Sullivan construction, there is a quasi-conformal probability measure $\nu$ on $\partial\Gamma$. So there are constants $C,{\mathfrak{v}}>0$ such that
$$C^{-1}\exp(-{\mathfrak{v}} h_\xi(g^{-1})) \le \frac{d\nu \circ g}{d\nu}(\xi) \le C \exp(-{\mathfrak{v}} h_\xi(g^{-1}))$$
for every $g\in \Gamma$ and a.e. $\xi \in \partial \Gamma$ where $h_\xi$ is the horofunction determined by $\xi$. To be precise,
$$h_\xi(g)=\lim_{n\to\infty} d(\xi_n,g)-d(\xi_n,e)$$
where $\{\xi_n\}_{n=1}^\infty$ is any sequence in $\Gamma$ which converges to $\xi$. Properties of CAT(-1) spaces ensure that this limit exists \cite{BH99}.
The {\em type} of the action $\Gamma \curvearrowright (\partial \Gamma,\nu)$ encodes the essential range of the Radon-Nikodym derivative, and \cite{B2} it is shown that this type is $III_\lambda$ for some $\lambda \in (0,1]$. If $\lambda \in (0,1)$, set
$$R_\lambda(g,\xi)= - \log_\lambda \left( \frac{d \nu \circ g}{d\nu}(\xi) \right)\,,$$
and set $R_1(g,\xi) = +\log \left( \frac{d \nu \circ g}{d\nu}(\xi) \right)$. Using standard results, it can be shown that if $\lambda \in (0,1)$ then we can choose $\nu$ so that $R_\lambda(g,\xi) \in \ZZ$ for every $g$ and a.e. $\xi$.
In order to handle each case uniformly, set $L=\RR$ if $\lambda=1$ and $L=\ZZ$ if $\lambda \in (0,1)$. Then let $\Gamma$ act on $\partial\Gamma \times L$ by
$$g(\xi,t) = (g \xi, t-R_\lambda(g,\xi)).$$
This action preserves the measure $\nu \times \theta_\lambda$ where $\theta_1$ is the measure on $\RR$ satisfying $d\theta_1(t) = \exp({\mathfrak{v}} t) dt$ and, for $\lambda \in (0,1)$, $\theta_\lambda$ is the measure on $\ZZ$ satisfying $\theta_\lambda(\{n\}) = \lambda^{-n}$.
Given $a,b \in L$, let $[a,b]_L \subset L$ denote the interval $\{a,a+1,\ldots, b\}$ if $L=\ZZ$ and $[a,b] \subset \RR$ if $L=\RR$. Similar considerations apply to open intervals and half-open intervals.
For any real numbers $r, T>0$, and $(\xi,t) \in \partial\Gamma\times [0,T)_L$, let
$$\Gamma_r(\xi,t) = \{g \in \Gamma:~ d(g,e) - h_\xi(g)-t \le r, ~g^{-1}(\xi,t) \in \partial \Gamma \times [0,T)_L \}$$
and
$$\cB^{}_r(\xi,t) := \{ g^{-1} (\xi,t):~ g\in \Gamma_r(\xi,t) \}.$$
$\Gamma_r(\xi,t)$ is approximately equal to the intersection of the ball of radius $r$ centered at the identity with the horoshell $\{g \in \Gamma:~ -t\le h_\xi(g) \le T-t\}$. Of course, $\Gamma_r$ and $\cB_r$ depend on $T$, but we leave this dependence implicit.
The main steps in the proof of Theorem \ref{thm:hyp} are as follows
\begin{enumerate}
\item
The first main technical result is that if $T$ are sufficiently large then the subset family $\cB$ is {\em regular}: there exists a constant $C>0$ such that for every $r>0$ and a.e. $(\xi,t) \in \partial\Gamma \times [0,T)_L$,
$$| \cup_{s\le r} \cB^{-1}_s \cB_r (\xi,t)| \le C |\cB_r(\xi,t)|.$$
The weak (1,1) type maximal inequality for the family of uniform averages on $\cB_r(\xi,t)$ is then established using the general results in \cite{BN2}.
\item
Next we let $\cS^{}_a=\{\cS^{}_{r,a}\}_{r>0}$ be the family of subset functions on $\partial\Gamma\times [0,T)_L$ defined by
$$\cS^{}_{r,a}(\xi,t) := \cB_r(\xi,t) \setminus \cB_{r-a}(\xi,t)$
and observe that $\cS_a$ is also regular if $a,T>0$ are sufficiently large. Therefore the family of uniform averages on $\cS_{r,a}(\xi,t)$ satisfies a weak (1,1) type maximal inequality.
The second main technical result is that $\cS_a$ is {\em asymptotically invariant}. More precisely we let $E$ denote the equivalence relation on $\partial\Gamma \times [0,T)_L$ given by $(\xi,t)E(\xi',t')$ if there exists $g \in \Gamma$ such that $g (\xi,t)=(\xi',t')$. Recall that $[E]$ denotes the full group of $E$ (that is, the group of all Borel isomorphisms on $\partial \Gamma \times [0,T)_L$ with graph contained in $E$), and $\cS_{r,a}$ being asymptotically invariant means that there exists a countable set $\Psi \subset [E]$ which generates the relation $E$ such that
$$\lim_{r \to \infty} \frac{ |\cS_{r,a}(\xi,t) \vartriangle \psi(\cS_{r,a}(\xi,t))|}{|\cS_{r,a}(\xi,t)|} = 0$$
for a.e. $(\xi,t)$ and for every $\psi \in \Psi$.
It now follows from the general results of \cite{BN2} that the uniform averages over $\cS_{r,a}$ form a pointwise ergodic family $\Omega=\{\omega_r\}_{r>0}$ in $L^1$ for the equivalence relation $E$.
\item We let $\alpha:E \to \Gamma$ be the cocycle given by the action of $\Gamma$. First we show that $\Gamma \curvearrowright (\partial\Gamma,\nu)$ is weakly mixing (so $\Gamma \curvearrowright (X\times \partial\Gamma, \mu \times \nu)$ is ergodic). This uses the fact that Poisson boundary actions are weakly mixing \cite{AL, Ka} and that the action on $(\partial\Gamma,\nu)$ is equivalent to a Poisson boundary action \cite{CM07}. From \cite{B2}, it follows that $\Gamma \curvearrowright (\partial\Gamma,\nu)$ has type $III_\rho$ and stable type $III_\tau$ for some $\rho, \tau \in (0,1]$. Therefore $\alpha$ is weakly mixing relative to a compact group action. So we can invoke Theorem \ref{thm:general} and Theorem \ref{thm:typeIII} of the present paper (as well as Theorem \ref{previous} based on \cite{BN2}) and thereby conclude the proof.
\end{enumerate}
| 29,280 |
\section{Introduction}\label{sec:introduction}
We consider a class of stochastic Nash games in which every player
solves a stochastic convex program parametrized by adversarial
strategies. Consider an $N$-person stochastic Nash game in which the
$i$th player solves the parametrized convex problem
\begin{equation}\label{eqn:problem}
\min_{x \in X_i} \EXP{f_i(x_i,x_{-i},\xi_i)},
\end{equation}
where $x_{-i}$ denotes the collection $\{x_j,j\neq i\}$ of decisions of all players other than player $i$.
For each $i$, the vector $\xi_i:\Omega_i \rightarrow \mathbb{R}^{n_i}$ is a random vector with a
probability distribution on some set,
while the function $\EXP{f_i(x_i,x_{-i},\xi_i)}$ is \fy{strongly} convex in $x_i$ for
all $x_{-i} \in \prod_{j \neq i} X_j$. For every $i$, the set $X_i \subseteq\mathbb{R}^{n_i}$ is closed and convex. We focus on the resulting
stochastic variational inequality (VI) and consider the development of
distributed stochastic approximation schemes that rely on adaptive
steplength sequences. Stochastic
approximation techniques have a long tradition. First proposed by Robbins
and Monro~\cite{robbins51sa} for differentiable functions and
Ermoliev~\cite{Ermoliev76,Ermoliev83,Ermoliev88}, significant effort has
been applied towards theoretical and algorithmic examination of such
schemes (cf.~\cite{Borkar08,Kush03}). Yet, there has been markedly
little on the application of such techniques to solution of stochastic
variational inequalities. Exceptions include the work by Jiang and
Xu~\cite{Houyuan08}, and more recently by Koshal~et
al.~\cite{koshal10single}. The latter, in particular, develops a single
timescale stochastic approximation scheme for precisely the class of
problems being studied here viz. monotone stochastic Nash games.
Standard stochastic approximation schemes provide little guidance
regarding the choice of a steplength sequence, apart from requiring that
the sequence, denoted by $\{\gamma_k\}$, satisfies
$ \sum_{k=1}^{\infty} \gamma_k = \infty \mbox{ and }
\sum_{k=1}^{\infty} \gamma_k^2 < \infty.$ This paper is motivated by
the need to develop {\em adaptive} steplength sequences that can be {\em
independently} chosen by players under a limited coordination, while
guaranteeing the overall convergence of the scheme. Adaptive stepsizes
have been effectively used in gradient and subgradient algorithms.
Vrahatis et al. \cite{Vrahatis00} presented a class of gradient
algorithms with adaptive stepsizes for unconstrained minimization.
Spall~\cite{Spall98} developed a
general adaptive SA algorithm based on using a simultaneous perturbation
approach for estimating the Hessian matrix. Cicek et al.
\cite{Zeevi11} considered the Kiefer-Wolfowitz (KW) SA algorithm and
derived general upper bounds on its mean-squared error, together with an adaptive
version of the KW algorithm. {Ram et al. \cite{Ram09} considered distributed stochastic subgradient algorithms for convex optimization problems and studied the effects of stochastic errors
on the convergence of the proposed algorithm. \fy{Lizarraga et al. \cite{Liz08} considered a family of two person Mutil-Plant game and developed Stackelberg-Nash equilibrium conditions based on the Robust Maximum Principle.} More recently, Yousefian et al.
\cite{Farzad1,Farzad2} developed centralized adaptive stepsize SA schemes
for solving stochastic optimization problems and variational inequalities. The main
contribution of the current paper lies in developing a class of \textit{distributed} adaptive \textit{stepsize rules} for SA scheme in which each agent chooses its own stepsizes without any specific information about other agents stepsize policy. This degree of freedom in choosing the stepsizes has not been addressed in the centralized schemes.}
Before proceeding, we briefly motivate the question of distributed
computation of Nash equilibria from two different standpoints: (i) First,
the Nash game can be viewed as a competitive analog of a stochastic
\fy{multi-user} convex optimization problem of the form
$ \min_{\fy{x \in X}} \sum_{i=1}^N \EXP{ f_i(x_i,x_{-i},\xi_i)} .$
Furthermore, under the assumption that equilibria of the associated
stochastic Nash game are efficient, our scheme provides a
distributed framework for computing solutions to this problem. In such a
setting, we may prescribe that players employ stochastic approximation
schemes since the Nash game represents an engineered construct employed
for computing solutions; (ii) A second perspective is one drawn from a
bounded rationality approach towards distributed computation of Nash
equilibria. A fully rational avenue for computing equilibria suggests
that each player employs a best response mapping in updating strategies,
based on what the competing players are doing. Yet, when faced by
computational or time constraints, players may instead take a gradient
step. We work in precisely this regime but allow for flexibility in
terms of the steplengths chosen by the players.
In this paper, we consider the solution of a stochastic Nash game
whose equilibria are completely captured by a stochastic variational
inequality with a strongly monotone mapping. Motivated by the need for
efficient distributed simulation methods for computing solutions to such
problems, we present a distributed scheme in which each player employs
an adaptive rule for prescribing steplengths. Importantly, these rules
can be implemented with relatively little coordination by any given
player and collectively lead to iterates that are shown to converge to
the unique equilibrium in an almost-sure sense.
This paper is organized as follows. In Section \ref{sec:formulation}, we
introduce the formulation of a stochastic Nash games in which every
player solves a stochastic convex problem. In Section
\ref{sec:convergence}, we show the almost-sure convergence of the SA
algorithm under specified assumptions. In Section \ref{sec:recursive
scheme}, motivated by minimizing a suitably defined error bound, we
develop an adaptive steplength stochastic approximation framework in
which every player {\em adaptively} updates his steplength. It is shown
that the choice of adaptive steplength rules can be obtained
independently by each player under a limited coordination. Finally, in
Section \ref{sec:numerical}, we provide some numerical results from a
stochastic flow management game drawn from a communication network
setting.
\textbf{Notation:} Throughout this paper, a vector $x$ is assumed to be
a column vector. We write $x^T$ to denote the transpose of a vector $x$.
$\|x\|$ denotes the Euclidean vector norm, i.e., $\|x\|=\sqrt{x^Tx}$. We
use $\Pi_X(x)$ to denote the Euclidean projection of a vector $x$ on a
set $X$, i.e., $\|x-\Pi_X(x)\|=\min_{y \in X}\|x-y\|$. Vector $g$ is a
\textit{subgradient} of a convex function $f$ with domain dom$f$ at
$\bar x \in \hbox{dom}f$ when $f(\bar x) +g^T(x-\bar x) \leq f(x)$ for
all $x \in \hbox{dom}f.$ The set of all subgradients of $f$ at $\bar x$ is
denoted by $\partial f(\bar x)$. We write \textit{a.s.} as the
abbreviation for ``almost surely'', and use $\EXP{z}$ to denote the expectation of a random variable~$z$.
\section{Problem formulation}\label{sec:formulation}
In this section, we present (sufficient) conditions associated with equilibrium points of
the stochastic Nash game defined by \eqref{eqn:problem}.
The equilibrium conditions of this game
can be characterized by
a stochastic variational inequality problem denoted by VI$(X,F)$, where
\begin{align}\label{eqn:VI_elements}
F(x)\triangleq \left( \begin{array}{ccc}
\nabla _{x_1} \EXP{f_1(x, \xi_1)} \\
\vdots \\ \nabla _{x_N} \EXP{f_N(x, \xi_N)} \end{array} \right), \quad X= \prod_{i=1}^N X_i,
\end{align}
with $x \triangleq (x_1,\ldots,x_N)^T$ and $x_i \in X_i{\subseteq \mathbb{R}^{n_i}}$ for
$i=1,\ldots,N$. Given a set $X \subseteq \mathbb{R}^n$ and a single-valued
mapping $F:X \to \mathbb{R}^n$, then a vector $x^* \in X$ solves a variational inequality VI$(X,F)$, if
\begin{align}\label{eqn:VI}
(x-x^*)^TF(x^*) \geq 0 \hbox{ for all } x \in X.
\end{align}
\fy{Let $n=\sum_{i=1}^N n_i$, and note that when the sets $X_i$ are convex and closed for all
$i$, the set $X \in \mathbb{R}^{n}$ is closed and convex.}
In the context of solving the stochastic variational inequality
VI$(X,F)$ in (\ref{eqn:VI_elements})-(\ref{eqn:VI}), suppose each player employs a stochastic
approximation scheme for given by
\begin{align}
\begin{aligned}
x_{{k+1},i} & =\Pi_{X_i}\left(x_{k,i}-\gamma_{k,i} ( F_i(x_k)+w_{k,i})\right),\cr
w_{k,i} &\triangleq \hat F_i(x_k,\xi_k)-F_i(x_k),
\end{aligned}\label{eqn:algorithm_different}
\end{align}
for all $k\ge 0$ and $i=1,\dots,N$, where
$\gamma_{k,i} >0$ is the stepsize of the $i$th player at iteration $k$,
$x_k =(x_{k,1}\ x_{k,2}\ \ldots \ x_{k,N})^T$,
$\xi_k =(\xi_{k,1}\ \xi_{k,2}\ \ldots \ \xi_{k,N})^T$, \fy{$F_i=\EXP{\nabla _{x_i} f_i(x, \xi_i)}$,} and
\begin{align*}
\hat F(x,\xi)\triangleq \left( \begin{array}{ccc}
\nabla _{x_1} f_1(x, \xi_1) \\
\vdots \\ \nabla _{x_N} {f_N(x , \xi_N)} \end{array} \right), \quad
\xi \triangleq \left( \begin{array}{ccc}
\xi_{1} \\
\vdots \\ \xi_{N} \end{array} \right).
\end{align*}
\fy{Note that in terms of the definition of $w_{k,i}$, $F_i$, and $\hat F_i$, $\EXP{w_{k,i}\mid \mathcal{F}_k}=0.$}
In addition, $x_0 \in X$ is a random
initial vector independent of the random variable $\xi$ and such
that $\EXP{\|x_0\|^2}<\infty$. Note that each player uses its individual
stepsize to update its decision.
\section{A Distributed SA scheme}\label{sec:convergence}
In this section, we present conditions under which algorithm
\eqref{eqn:algorithm_different} converges almost surely to the solution of game
\eqref{eqn:problem} under suitable assumptions on the
mapping.
Also, we develop a distributed variant of a standard
stochastic approximation scheme and provide conditions on the
steplength sequences that lead to almost-sure convergence of the
iterates to the unique solution. Our assumptions include
requirements on the set $X$ and the mapping $F$.
\begin{assumption}\label{assum:different_main}
Assume the following:
\begin{enumerate}
\item[(a)] The sets $X_i \subseteq \mathbb{R}^{n_i}$ are closed and convex.
\item[(b)] \fy{$F(x)$ is {strongly monotone with constant $\eta >0$ and Lipschitz continuous with constant $L$ over the set $X$.}}
\end{enumerate}
\end{assumption}
{\textbf{Remark:} The strong monotonicity is assumed to hold throughout the paper. Although the convergence results may still hold with a weaker assumption, such as strict monotonicity, but the stepsize policy in this paper leverages the strong monotonicity parameter which prescribes a more parametrized stepsize rule. This is the main reason that we assumed the stronger version of monotonicity. In Section \ref{sec:numerical}, we present an example where such an assumption is satisfied.}
Another set of assumptions is for the stepsizes employed by each
player in algorithm \eqref{eqn:algorithm_different}.
\begin{assumption}\label{assum:step_error_sub}
Assume that:
\begin{enumerate}
\item[(a)] The {stepsize sequences are} such that $\gamma_{k,i}>0$ for all $k$ and $i$, with
$\sum_{k=0}^\infty \gamma_{k,i} = \infty$ and $\sum_{k=0}^\infty
\gamma_{k,i}^2 < \infty$.
\item [(b)] There exists a scalar $\beta$ such that $0\le \beta<\frac{\eta}{L}$ and
$\frac{\Gamma_k-\delta_k}{\delta_k}\le \beta$ for all $k\ge 0$, where $\delta_k$ and $\Gamma_k$ are (fixed) positive sequences satisfying $\delta_k \leq \min_{i=1,\ldots,N}{\gamma_{k,i}}$ and $\Gamma_k \geq \max_{i=1,\ldots,N}{\gamma_{k,i}}$ for all $k\ge0$.
\end{enumerate}
\end{assumption}
We let $\mathcal{F}_k$ denote the history of the method up to time $k$, i.e.,
$\mathcal{F}_k=\{x_0,\xi_0,\xi_1,\ldots,\xi_{k-1}\}$ for $k\ge 1$ and $\mathcal{F}_0=\{x_0\}$. {Consider the following assumption on the stochastic errors, $w_k$, of the algorithm.}
\begin{assumption}\label{assum:w_k_bound}
The errors $w_k$ are such that
for some {constant} $\nu >0$,
\[\EXP{\|w_k\|^2 \mid \mathcal{F}_k} \le \nu^2 \qquad \hbox{\textit{a.s.} for all $k\ge0$}.\]
\end{assumption}
We \fy{use the Robbins-Siegmund lemma} in establishing the convergence of
method~(\ref{eqn:algorithm_different}), which can be found
in~\cite{Polyak87} (cf.~Lemma 10, page 49).
\begin{lemma}\label{lemma:probabilistic_bound_polyak}
Let $\{v_k\}$ be a sequence of nonnegative random variables, where
$\EXP{v_0} < \infty$, and let $\{\alpha_k\}$ and $\{\mu_k\}$
be deterministic scalar sequences such that:
\begin{align*}
& \EXP{v_{k+1}|v_0,\ldots, v_k} \leq (1-\alpha_k)v_k+\mu_k
\qquad a.s. \ \hbox{for all }k\ge0, \cr
& 0 \leq \alpha_k \leq 1, \quad\ \mu_k \geq 0, \cr
& \quad\ \sum_{k=0}^\infty \alpha_k =\infty,
\quad\ \sum_{k=0}^\infty \mu_k < \infty,
\quad\ \lim_{k\to\infty}\,\frac{\mu_k}{\alpha_k} = 0.
\end{align*}
Then, $v_k \rightarrow 0$ almost surely.
\end{lemma}
The following lemma provides an error bound for algorithm \eqref{eqn:algorithm_different} {under Assumption \ref{assum:different_main}.}
\begin{lemma}\label{lemma:error_bound_1}
Consider algorithm~\eqref{eqn:algorithm_different}.
Let Assumption \ref{assum:different_main} hold. Then, the following relation holds \fy{a.s.} for all $k \ge 0$:
\begin{align}\label{ineq}
&\EXP{\|x_{k+1}-x^*\|^2\mid\mathcal{F}_k} \leq \Gamma_k^2\EXP{\|w_k\|^2\mid\mathcal{F}_k}\cr
& +(1-2(\eta+L)\delta_k+2L\Gamma_k+L^2\Gamma_k^2)\|x_ k-x^*\|^2.
\end{align}
\end{lemma}
\begin{proof}
By Assumption \ref{assum:different_main}a,
\an{the set $X$ is closed and convex}. Since $F$ is strongly monotone, the existence and uniqueness of the solution to VI$(X,F)$ is guaranteed by Theorem 2.3.3 of~\cite{facchinei02finite}.
Let $x^*$ denote the solution of VI$(X,F)$. From properties of projection operator, we know that a vector $x^*$ solves VI$(X,F)$ problem if and only if $x^*$ satisfies
\[x^*=\Pi_X(x^*-\gamma F(x^*))\qquad\hbox{for any }\gamma>0.\]
\fy{From} algorithm \eqref{eqn:algorithm_different} and the non-expansiveness property of the
projection operator, we have for all $k\ge0$ and~$i$,
\begin{align*}
&\|x_{k+1,i}-x_i^*\|^2 =\|\Pi_{X_i}(x_{k,i}-\gamma_{k,i}(F_i(x_k)+w_{k,i})) \\
& -\Pi_{X_i}(x_i^*-\gamma_{k,i} F_i(x^*))\|^2 \\
& \le \|x_{k,i}-x_i^*-\gamma_{k,i}(F_i(x_k)+w_{k,i}-F_i(x^*))\|^2.
\end{align*}
Taking the expectation conditioned on the past, and using
$\EXP{w_{k,i}\mid \mathcal{F}_k}=0$, we have
\begin{align*}
& \EXP{\|x_{k+1,i}-x_i^*\|^2\mid\mathcal{F}_k} \le \|x_{k,i}-x_i^*\|^2 \cr
&+\gamma_{k,i}^2\|F_i(x_k)-F_i(x^*)\|^2 +\gamma_{k,i}^2\EXP{\|w_{k,i}\|^2\mid \mathcal{F}_k}\cr
&-2\gamma_{k,i} (x_{k,i}-x_i^*)^T(F_i(x_k)-F_i(x^*)).
\end{align*}
Now, by summing the preceding relations over $i$, we have
\begin{align}\label{ineq:main1}
& \EXP{\|x_{k+1}-x^*\|^2\mid\mathcal{F}_k} \le \|x_{k}-x^*\|^2 \cr
&+\underbrace{\sum_{i=1}^N\gamma_{k,i}^2\|F_i(x_k)-F_i(x^*)\|^2}_{\bf Term\,1} +\sum_{i=1}^N\gamma_{k,i}^2
\EXP{\|w_{k,i}\|^2\mid \mathcal{F}_k}\cr
&\underbrace{-2\sum_{i=1}^N\gamma_{k,i} (x_{k,i}-x_i^*)^T(F_i(x_k)-F_i(x^*))}_{\bf Term\,2}.
\end{align}
Next, we estimate Term $1$ and Term $2$ in (\ref{ineq:main1}).
By using the definition of $\Gamma_k$ and by leveraging the Lipschitzian
property of mapping $F$, we obtain
\begin{align}\label{ineq:main2}
\hbox{Term}\,1 \le \Gamma_k^2\|F(x_k)-F(x^*)\|^2\le\Gamma_k^2L^2\|x_k-x^*\|^2.
\end{align}
Adding and subtracting
$-2\sum_{i=1}^N\delta_k(x_{k,i}-x_i^*)^T(F_i(x_k)-F_i(x^*))$ {from} Term $2$, we further obtain
\begin{equation*}
\begin{split}
\hbox{Term}\,2 \le
& -2\delta_k(x_{k}-x^*)^T(F(x_k)-F(x^*))\cr
& -2\sum_{i=1}^N(\gamma_{k,i}-\delta_k)(x_{k,i}-x_i^*)^T{(F_i(x_{k})-F_i(x^*))}.
\end{split}
\end{equation*}
By the Cauchy-Schwartz inequality, \fy{we obtain}
\begin{equation*}
\begin{split}
\hbox{Term}\,2 \le & -2\delta_k(x_{k}-x^*)^T(F(x_k)-F(x^*))\cr
& +2(\gamma_{k,i}-\delta_k)\sum_{i=1}^N\|x_{k,i}-x_i^*\|\|{F_i(x_{k})-F_i(x^*)}\|\cr
\le & -2\delta_k(x_{k}-x^*)^T(F(x_k)-F(x^*))\cr
& +2(\Gamma_k-\delta_k)\|x_{k}-x^*\|\|F(x_{k})-F(x^*)\|,
\end{split}
\end{equation*}
where in the last relation, we use {H\"{o}lder's} inequality.
Invoking the strong monotonicity of the mapping for bounding the first term and by
{utilizing the Lipschitzian property of} the second term of the preceding relation, we have
\begin{equation*}
\begin{split}
\hbox{Term}\,2 \le -2\eta\delta_k\|x_{k}-x^*\|^2 +2(\Gamma_k-\delta_k)L\|x_{k}-x^*\|^2.
\end{split}
\end{equation*}
The desired inequality is obtained by combining relations
(\ref{ineq:main1}) and (\ref{ineq:main2}) with the preceding inequality .
\end{proof}
We next prove that algorithm
\eqref{eqn:algorithm_different} generates a sequence of iterates that
converges \fy{a.s.} to the unique solution of VI$(X,F)$, as seen in the following proposition.
Our proof of this result makes use of Lemma~\ref{lemma:error_bound_1}.
\begin{proposition}[Almost-sure convergence]
\label{prop:rel_bound}
Consider the algorithm \ref{eqn:algorithm_different}. Let Assumption \ref{assum:different_main}, \ref{assum:step_error_sub} and \ref{assum:w_k_bound} hold. Then,
\begin{enumerate}
\item [(a)] The following relation holds \fy{a.s.} for all $k \ge 0$:
\begin{align*}
&\EXP{\|x_{k+1}-x^*\|^2} \leq (1+\beta)^2\delta_k^2\nu^2\cr
& +(1-2(\eta-\beta L)\delta_k+(1+\beta)^2L^2\delta_k^2)\EXP{\|x_ k-x^*\|^2}.
\end{align*}
\item [(b)] The sequence $\{x_k\}$
generated by algorithm (\ref{eqn:algorithm_different}), converges \fy{a.s.}
to the unique solution of VI$(X,F)$.
\end{enumerate}
\end{proposition}
\begin{proof}
(a) \ Assumption \ref{assum:step_error_sub}b implies that $\Gamma_k \leq
(1+\beta)\delta_k$. Combining this with inequality \eqref{ineq}, we obtain
\begin{align*}
&\EXP{\|x_{k+1}-x^*\|^2\mid\mathcal{F}_k} \cr
&\leq (1-2(\eta-\beta L)\delta_k+(1+\beta)^2L^2\delta_k^2)\|x_ k-x^*\|^2\cr
&+(1+\beta)^2\delta_k^2\EXP{\|w_k\|^2\mid\mathcal{F}_k}, \qquad \hbox{for all } k \ge 0.
\end{align*}
Taking expectations in the preceding inequality and using Assumption
\ref{assum:w_k_bound}, we obtain the desired relation.
\noindent (b) \ We {show that the conditions of} Lemma
\ref{lemma:probabilistic_bound_polyak} {are satisfied in order} to
claim almost sure convergence of $x_k$ to $x^*$. Let us define $v_k
\triangleq \|x_{k+1}-x^*\|^2$, $\alpha_k \triangleq 2(\eta-\beta
L)\delta_k-L^2\delta_k^2(1+\beta)^2$, and $\mu_k \triangleq
(1+\beta)^2\delta_k^2\EXP{\|w_k\|^2\mid\mathcal{F}_k}$. Since $\gamma_{k,i}$
tends to zero for any $i=1,\ldots,N$, we {may} conclude that $\delta_k$ goes to zero as $k$ grows.
\an{Recall that} $\alpha_k$ is given by
\[\alpha_k =2(\eta-\beta L)\delta_k\left(1-\frac{(1+\beta)^2L^2\delta_k}{2(\eta-\beta
L)}\right).\]
\an{Due to $\delta_k\to0$, for all} $k$ large enough, say $k>k_1$, we have
\[1-\frac{(1+\beta)^2L^2\delta_k}{2(\eta-\beta L)}>0.\]
Since $\beta < \frac{\eta}{L}$ (Assumption \ref{assum:step_error_sub}b),
it follows $\eta-\beta L>0$. Thus, we have $\alpha_k \geq 0$. Also, for $k$ large enough, say $k>k_2$, we have $\alpha_k \leq 1$. Therefore, when $k>\max\{k_1,k_2\}$ we have $0 \leq \alpha_k \leq 1$. Obviously, $v_k, \mu_k \geq 0$. From Assumption~\ref{assum:step_error_sub}a and
Assumption~\ref{assum:w_k_bound} it follows $\sum_{k}\mu_k < \infty$. We also have
\begin{align*
{\lim_{k \rightarrow \infty}} \frac{\mu_k}{\alpha_k}&={\lim_{k \rightarrow \infty}} \frac{(1+\beta)^2\delta_k^2\EXP{\|w_k\|^2\mid\mathcal{F}_k}}{2(\eta-\beta L)\delta_k\left(1-\frac{(1+\beta)^2L^2\delta_k}{2(\eta-\beta L)}\right)}\cr &= {\lim_{k \rightarrow \infty} }\frac{(1+\beta)^2\delta_k\EXP{\|w_k\|^2\mid\mathcal{F}_k}}{2(\eta-\beta L)}.\end{align*}
Since the term $\EXP{\|w_k\|^2\mid\mathcal{F}_k}$ is bounded by $\nu^2$
(Assumption~\ref{assum:w_k_bound}) and
\an{$\delta_k\to0$, we see} that $\lim_{k \rightarrow
\infty}\frac{\mu_k}{\alpha_k}=0$. Hence, the conditions of Lemma \ref{lemma:probabilistic_bound_polyak} are satisfied, which implies that $x_k$ converges to
the unique solution, $x^*$, almost surely.
\end{proof}
Consider now a special form of algorithm~\eqref{eqn:algorithm_different} corresponding to
the case when all players employ the same stepsize, i.e., $\gamma_{k,i}=\gamma_k$ for all $k$.
Then, the algorithm~\eqref{eqn:algorithm_different} reduces to the following:
\begin{align}
\begin{aligned}
x_{k+1} & =\Pi_{X}\left(x_k-\gamma_k( F(x_k)+w_k)\right),\cr
w_k & \triangleq \hat F(x_k,\xi_k)-F(x_k),
\end{aligned}\label{eqn:algorithm_identical}
\end{align}
for all $k \ge 0$.
Observe that when $\gamma_{k,i}=\gamma_k$ for all $k$, Assumption~\ref{assum:step_error_sub}a
is satisfied when
$\sum_{k=0}^\infty \gamma_k = \infty$ and $\sum_{k=0}^\infty\gamma_k^2 < \infty$.
Assumption~\ref{assum:step_error_sub}b is automatically satisfied
with $\Gamma_k=\delta_k=\gamma_k$ and $\beta=0$.
Hence, as a direct consequence of Proposition \ref{prop:rel_bound}, we have the following corollary.
\begin{corollary}[Identical stepsizes]
\label{corollary:rel_bound}
Consider algorithm (\ref{eqn:algorithm_identical}). Let Assumption \ref{assum:different_main} and
\ref{assum:w_k_bound} hold. Also, let
$\sum_{k=0}^\infty \gamma_k = \infty$ and $\sum_{k=0}^\infty\gamma_k^2 < \infty$.
Then,
\begin{enumerate}
\item [(a)] The following relation holds almost surely:
{\small
\begin{align*}
\EXP{\|x_{k+1}-x^*\|^2} & \leq
(1-2 \eta\gamma_k+L^2\gamma_k^2 )
\EXP{\|x_{k}-x^*\|^2} \cr
&+\gamma_k^2 \nu^2.
\end{align*}}
\item [(b)] The sequence $\{x_k\}$
generated by algorithm (\ref{eqn:algorithm_identical}), converges \fy{a.s.} to the unique solution of VI$(X,F)$.
\end{enumerate}
\end{corollary}
\section{A distributed adaptive steplength SA scheme}\label{sec:recursive scheme}
Stochastic approximation algorithms
require stepsize sequences to be square summable
but not summable. These algorithms provide little advice regarding the choice of
such sequences. One of the most common choices has been the harmonic
steplength rule which takes the form of $\gamma_k=\frac{\theta}{k}$ where $\theta>0$ is a
constant. Although, this choice guarantees almost-sure convergence, it does
not leverage problem parameters. Numerically, it has been observed that
such choices can perform quite poorly in practice. Motivated by this
shortcoming, we present a
distributed adaptive steplength scheme for algorithm
(\ref{eqn:algorithm_different}) which guarantees almost-sure
convergence of $x_k$ to the unique solution of VI$(X,F)$. It is derived
from the minimizer of a suitably defined error bound and leads to a
recursive relation; more specifically, at each step, the
new stepsize is calculated using the stepsize from the preceding iteration and
problem parameters. To begin our analysis, we consider the
result of Proposition \ref{prop:rel_bound}a for all $k \geq 0$:
\begin{align}\label{equ:bound_recursive_01}
&\EXP{\|x_{k+1}-x^*\|^2} \leq (1+\beta)^2\delta_k^2\nu^2\cr
&+(1-2(\eta-\beta L)\delta_k+(1+\beta)^2L^2\delta_k^2)\EXP{\|x_ k-x^*\|^2}.\quad
\end{align}
When the stepsizes are further restricted so that \[0 < \delta_k
\le\frac{\eta-\beta L}{(1+\beta)^2L^2},\] we have
\begin{align*}
1-2(\eta-\beta L)\delta_k +L^2(1+\beta)^2\delta_k^2 \leq 1-(\eta -\beta L) \delta_k.
\end{align*}
Thus, for $0 < \delta_k \le\frac{\eta-\beta L}{(1+\beta)^2L^2}$, from inequality (\ref{equ:bound_recursive_01}) we obtain
{\small \begin{align}\label{rate_nu_est_different}
\EXP{\|x_{k+1}-x^*\|^2}
& \le (1-(\eta-\beta L) \delta_k)\EXP{\|x_k-x^*\|^2}\cr
& + (1+\beta)^2\delta_k^2\nu^2\qquad\hbox{for all }k \geq 0.\quad
\end{align}} \hskip -0.3pc
Let us view the quantity
$\EXP{\|x_{k+1}-x^*\|^2}$ as an error $e_{k+1}$ of the method arising
from the use of the stepsize values $\delta_0,\delta_1,\ldots, \delta_k $. Relation~\eqref{rate_nu_est_different} gives us an estimate of the error of algorithm (\ref{eqn:algorithm_different}). We use this estimate to develop an adaptive stepsize
procedure. Consider the worst case which is the case when ~\eqref{rate_nu_est_different} holds with equality. In this worst case,
the error satisfies the following recursive relation:
\begin{align*}
& e_{k+1} = (1-(\eta-\beta L) \delta_k) e_k+(1+\beta^2)\delta_k^2 \nu^2.
\end{align*}
Let us assume that we want to run the algorithm
(\ref{eqn:algorithm_different}) for a fixed number of iterations, say
{$K$}. The preceding relation shows that $e_{K}$ depends on the stepsize
values up to the $K$th iteration. This motivates us to see the stepsize
parameters as decision variables that can minimize \fy{a suitably defined} error bound of the
algorithm. Thus, the variables are {$\delta_0, \delta_1, \ldots,
\delta_{K-1}$} and the objective function is the error function
$e_K(\delta_0, \delta_1, \ldots, \delta_{K-1})$. We proceed to derive a stepsize rule by minimizing the error $e_{K+1}$;
Importantly, $\delta_{K+1}$ can be shown to be a function of only the most
recent stepsize $\delta_{K}$. We define the real-valued
error function $e_k(\delta_0, \delta_1, \ldots, \delta_{k-1})$ by
the upper bound in \eqref{rate_nu_est_different}:
\begin{align}\label{def:e_k_different}
e_{k+1}(\delta_0,\ldots,\delta_k) \triangleq &(1-(\eta-\beta L) \delta_k) e_k(\delta_0,\ldots,\delta_{k-1})\cr
& +(1+\beta^2)\delta_k^2 \nu^2\qquad\hbox{for all }k \geq 0,
\end{align}
where $e_0$ is a positive scalar, $\eta$ is the strong monotonicity parameter
and $\nu^2$ is the upper bound for the second moments of the error
norms $\|w_k\|$.
Now, let us consider the stepsize sequence $\{\delta^*_k\}$ given by
\begin{align}
& \delta_0^*=\frac{\eta-\beta L}{2(1+\beta)^2\nu^2}\,e_0\label{eqn:gmin0}
\\
& \delta_k^*=\delta_{k-1}^*\left(1-\frac{\eta-\beta L}{2}\delta_{k-1}^*\right)
\quad \hbox{for all }k\ge 1.\label{eqn:gmink}
\end{align}
In what follows, we often abbreviate $e_k(\delta_0,\ldots,\delta_{k-1})$ by $e_k$
whenever this is unambiguous. The next proposition shows that the
lower bound sequence of $\gamma_{k,i}$ given by
(\ref{eqn:gmin0})--(\ref{eqn:gmink}) minimizes the errors $e_k$ over $(0,\frac{\eta -\beta L}{(1+\beta)^2L^2}]^{k}$.
\begin{proposition}\label{prop:rec_results}
Let $e_k(\delta_0,\ldots,\delta_{k-1})$ be defined as in~\eqref{def:e_k_different},
where $e_0>0$ is such that $\,e_0<\frac{2\nu^2}{L^2}$,
and $L$ is the Lipschitz constant of mapping $F$.
Let the sequence $\{\delta^*_k\}$ be given by
\eqref{eqn:gmin0}--\eqref{eqn:gmink}.
Then, the following hold:
\begin{itemize}
\item [(a)] \fy{$e_k(\delta_0^*,\ldots,\delta_k^*) = \frac{2(1+\beta)^2\nu^2}{\eta-\beta L}\,\delta_k^*$ for all $k \geq 0$.}
\item [(b)] For any $k\ge 1$, the vector $(\delta_0^*, \delta_1^*,\ldots,\delta_{k-1}^*)$ is the minimizer of the function $e_k(\delta_0,\ldots,\delta_{k-1})$ over the set
$$\mathbb{G}_k
\triangleq \left\{\alpha \in \mathbb{R}^k : 0< \alpha_j \leq \frac{\eta-\beta L}{(1+\beta)^2L^2}, j =1,\ldots, k\right\},$$
\fy{i.e., for any $k\ge 1$ and $(\delta_0,\ldots,\delta_{k-1})\in \mathbb{G}_k$:}
\begin{align*}
& e_{k}(\delta_0,\ldots,\delta_{k-1})
- e_{k}(\delta_0^*,\ldots,\delta_{k-1}^*)\cr
&\geq (1+\beta)^2\nu^2(\delta_{k-1}-\delta_{k-1}^*)^2.
\end{align*}
\begin{comment}
\item[(c)]
The vector $(\delta^*_{0},\ldots,\delta_{k-1}^*)$
is a stationary point of function $e_k(\delta_0,
\delta_1,\ldots,\delta_{k-1})$ over the set
$\mathbb{G}_k $.\end{comment}
\end{itemize}
\end{proposition}
\begin{proof
(a) \ To show the result, we use induction on $k$.
Trivially, it holds for
$k=0$ from \eqref{eqn:gmin0}. Now, suppose that we have
$e_k(\delta_0^*,\ldots,\delta_{k-1}^*) = \frac{2(1+\beta)^2\nu^2}{\eta-\beta L}\,\delta_k^*$ for some $k$,
and consider the case for $k+1$. From the definition of the error $e_k$
in~\eqref{def:e_k_different} \fy{and the inductive hypothesis, we have
\begin{align*}
e_{k+1}(\delta_0^*,\ldots,\delta_k^*)
&=(1-(\eta-\beta L)\delta_k^*)\frac{2(1+\beta)^2\nu^2}{\eta-\beta L}\,\delta_k^*\cr
&+(1+\beta)^2(\delta_k^*)^2\nu^2\cr
&=\frac{2(1+\beta)^2\nu^2}{\eta-\beta L}\,\delta_k^*\left(1-\frac{\eta-\beta L}{2}\,\delta_k^*\right) \cr
&=\frac{2(1+\beta)^2\nu^2}{\eta-\beta L}\,\delta_{k+1}^*,\end{align*}}
where the last equality follows by the definition of $\delta_{k+1}^*$ in
\eqref{eqn:gmink}. Hence, the result holds for all $k \ge 0$.
\noindent (b) \ {First we need to show that
$(\delta^*_{0},\ldots,\delta_{k-1}^*)\in\mathbb{G}_k$. By the choice
of $e_0$, i.e. $e_0 < \frac{2\nu^2}{L^2}$, we have that $0<\delta_0^*\leq \frac{\eta- \beta L}{(1+\beta)^2 L^2}$.
Using induction, from relations (\ref{eqn:gmin0})--(\ref{eqn:gmink}), it can be shown that $0< \delta_{k}^*<\delta^*_{k-1}$ for all $k\ge1$.
Thus, $(\delta^*_{0},\ldots,\delta_{k-1}^*)\in\mathbb{G}_k$ for all $k\ge1$. }Using induction on $k$, we now show that vector $(\delta_0^*,\delta_1^*,\ldots,\delta_{k-1}^*)$
minimizes the error $e_k$ for all $k\ge1$. From the definition of the error $e_1$ and the relation
\[e_1(\delta_0^*)=\frac{2(1+\beta)^2\nu^2}{\eta-\beta L}\,\delta_1^*\] shown in part (a), we have
\begin{align*}
e_1(\delta_0) -e_1(\delta_0^*)
&=(1-(\eta-\beta L) \delta_0)e_0+(1+\beta)^2\nu^2 \delta_0^2\cr
&-\frac{2(1+\beta)^2\nu^2}{\eta-\beta L}\,\delta_1^*.
\end{align*}
Using $\delta_1^*=\delta_0^*\left(1-\frac{\eta-\beta L}{2}\,\delta_0^*\right)$, we obtain
\begin{align*}
e_1(\delta_0) -e_1(\delta_0^*)
&=(1-(\eta-\beta L) \gamma_0)e_0+(1+\beta)^2\nu^2 \delta_0^2\cr
&-\frac{2(1+\beta)^2\nu^2}{\eta-\beta L}\,\delta_0^*+(1+\beta)^2\nu^2(\delta_0^*)^2.
\end{align*}
where the last equality follows from $e_0=\frac{2(1+\beta)^2\nu^2}{\eta-\beta L}\,\delta_0^*.$
Thus, we have
\begin{align*}
e_1(\delta_0) -e_1(\delta_0^*)
& = (1+\beta)^2\nu^2\left(-2\delta_0\delta_0^*+\delta_0^2+(\delta_0^*)^2 \right)\cr
& =(1+\beta)^2\nu^2\left(\delta_0-\delta_0^*\right)^2,
\end{align*}
and the inductive hypothesis holds for $k = 1$.
Now, suppose that $e_k(\delta_0,\ldots,\delta_{k-1})
\geq e_k(\delta_0^*,\ldots,\delta_{k-1}^*)$ holds for some $k$ and
any $(\delta_0,\ldots,\delta_{k-1}) \in\mathbb{G}_k$, and we \fy{need} to show that
$e_{k+1}(\delta_0,\ldots,\delta_k) \geq e_{k+1}(\delta_0^*,\ldots,\delta_k^*)$
holds for all $(\delta_0,\ldots,\delta_k) \in\mathbb{G}_{k+1}$.
To simplify the notation,
we use $e_{k+1}^*$ to denote the error $e_{k+1}$ evaluated at
$(\delta_{0}^*,\delta_1^*,\ldots,\delta_k^*)$, and $e_{k+1}$ when
evaluating at an arbitrary vector
$(\delta_0,\delta_1,\ldots,\delta_k)\in\mathbb{G}_{k+1}$.
Using (\ref{def:e_k_different}) and part (a), we have
\begin{align*}
e_{k+1}- e_{k+1}^*
& = (1-(\eta-\beta L) \delta_k)e_k
+(1+\beta)^2\nu^2\delta_k^2\cr
& -\frac{2(1+\beta)^2\nu^2}{\eta-\beta L} \delta_{k+1}^*.
\end{align*}
Under the inductive hypothesis, we have $e_k\ge e_k^*$. It can be shown easily that when $(\delta_0,\delta_1,\ldots,\delta_k)\in \mathbb{G}_k$, we have $0<1-(\eta-\beta L)\delta_k< 1$.
Using this, the relation $e_k^*=\frac{2(1+\beta)^2\nu^2}{\eta-\beta L}\gamma_k^*$
of part (a), and the definition of $\delta_{k+1}^*$, we obtain
\begin{align*}
e_{k+1} - e_{k+1}^*
& \geq (1-(\eta-\beta L) \delta_k)\frac{2(1+\beta)^2\nu^2}{\eta-\beta L}\delta_k^*\cr
&+(1+\beta)^2\nu^2\delta_k^2 \cr
&-\frac{2(1+\beta)^2\nu^2}{\eta-\beta L} \delta_k^*\left(1-\frac{\eta-\beta L}{2}\delta_k^*\right)
\cr
&=(1+\beta)^2 \nu^2(\delta_k-\delta_k^*)^2.
\end{align*}
Hence, {$e_{k} - e_{k}^*\ge
(1+\beta)^2\nu^2(\delta_{k-1}-\delta_{k-1}^*)^2$ holds} for all $k\ge1$ and all
$(\delta_0,\ldots,\delta_{k-1})\in\mathbb{G}_k$.
\begin{comment}
\noindent (c) \ {In proof of part (b), we showed that $(\delta^*_{0},\ldots,\delta_{k-1}^*)\in\mathbb{G}_k$.}
From the definition of $e_k$ in~\eqref{def:e_k_different}, we can see that
\begin{align}\label{eqn:recursive}
&e_{k+1}(\gamma_0,\ldots,\gamma_{k}) = e_0 \prod_{i=0}^{k}(1-(\eta-\beta L) \delta_{i})\cr
&+(1+\beta)^2\nu^2\sum_{i=0}^{k-1}\left(\delta_{i}^2\prod_{j=i+1}^{k}(1-(\eta-\beta L)
\delta_{j})\right)\cr
&+(1+\beta)^2\nu^2\delta_k^2.
\end{align}
We now use induction on
$k$. For $k=1$, we have
\[\frac{\partial e_1}{\partial \delta_{0}}=-(\eta-\beta L) e_0+2\delta_0(1+\beta)^2\nu^2.\]
Thus, the derivative of $e_1$ at $\delta_0^*=\frac{\eta-\beta L}{2(1+\beta)^2\nu^2}\, e_0$ is zero,
which satisfies $0<\delta_0^*\leq \frac{\eta - \beta L}{(1+\beta)^2 L^2}$ by the choice of $e_0$.
Hence, $\delta_0^*=\frac{\eta- \beta L}{2(1+\beta)^2\nu^2}\, e_0$ is the stationary point of $e_1$
over the entire real line. Suppose that for $k\ge1$, the vector
$(\delta_0,\ldots,\delta_{k-1})$ is the minimizer of $e_k$ over the set $G_k$.
Consider the case of $k+1$. In this part of proof, we abbreviate $(1+\beta)\nu$ by $\nu_\beta$. The partial derivative of $e_{k+1}$
with respect to $\delta_{\ell}$ is given by
\begin{align*}
&\frac{\partial e_{k+1}}{\partial \delta_{\ell}}
=-(\eta-\beta L) e_0\prod_{i=0,i\neq
\ell}^{k}(1-(\eta-\beta L) \delta_{i})\cr
&-(\eta-\beta L) \nu_\beta^2
\sum_{i=0}^{\ell-1}
\left(\delta_{i}^2\prod_{j=i+1, j\neq \ell}^{k}(1-(\eta-\beta L)
\delta_{j})\right)\cr
& +2\nu_\beta^2\delta_{\ell}
\prod_{i=\ell+1}^{k}(1-(\eta-\beta L) \delta_{i}),
\end{align*}
where $0\le \ell\le k-1.$ Now, by factoring out the common term
$\prod_{i=\ell+1}^{k}(1-(\eta-\beta L)\delta_{i})$,
we obtain
\begin{align}\label{eqn:one}
\frac{\partial e_{k+1}}{\partial \delta_{\ell}}
=&\left[-(\eta-\beta L) ( e_0 \prod_{i=0}^{\ell-1}(1-(\eta-\beta L) \delta_{i})\right.\nonumber\\
& \left. + \nu_\beta^2\sum_{i=0}^{\ell-2}\left(\delta_{i}^2\prod_{j=i+1}^{\ell-1}(1-(\eta-\beta L)
\delta_{j})\right)\right.\nonumber\\
& \left. +\nu_\beta^2\delta_{\ell-1}^2 ) +2\nu_\beta^2\delta_{\ell}\right]\prod_{i=\ell+1}^{k}(1-(\eta-\beta L)\delta_{i}).
\end{align}
By combining relations~\eqref{eqn:recursive} and~\eqref{eqn:one}, we obtain
for all $\ell=0,\ldots,k-1,$
\begin{align*}
\frac{\partial e_{k+1}}{\partial \delta_{\ell}}
=&\left(-(\eta-\beta L) e_\ell(\delta_0,\ldots,\delta_{\ell-1}\right)\cr
&+2\nu_\beta^2\delta_{\ell})\prod_{i=\ell+1}^{k}(1-(\eta-\beta L)\delta_{i}),
\end{align*}
where for $\ell=0$, we assume $e_\ell(\delta_0,\ldots,\delta_{\ell-1})=e_0$.
By part (a), we have \[-(\eta-\beta L) e_\ell(\delta_0^*,\ldots,\delta_{\ell-1}^*)
+2\nu_\beta^2\delta_{\ell}^*=0,\] which shows that
$\frac{\partial e_{k+1}}{\partial \delta_{\ell}}$ vanishes at
$(\delta_{0}^*,\ldots,\delta_k^*)\in\mathbb{G}_{k+1}$ for all $\ell=0,\ldots,k-1$. Now, consider the partial derivative of $e_{k+1}$
with respect to $\delta_k$, for which we have
\begin{align*}
\frac{\partial e_{k+1}}{\partial \delta_k} =&-(\eta-\beta L )e_0\prod_{i=0}^{k-1}(1-(\eta-\beta L) \delta_{i})\cr
&-(\eta-\beta L) \nu_\beta^2
\sum_{i=0}^{k-2}\left(\delta_{i}^2\prod_{j=i+1 }^{k-1}(1-(\eta-\beta L) \delta_{j})\right)\cr
&-(\eta-\beta L) \nu_\beta^2\gamma_{k-1,min}^2+2\nu_\beta^2\delta_k.
\end{align*}
Using relation~\eqref{eqn:recursive},
\[\frac{\partial e_{k+1}}{\partial \delta_k}
=-(\eta-\beta L) e_{k}(\delta_0,\ldots,\delta_{k-1})+2\nu_\beta^2\delta_k.\]
By part (a), we have
\[-(\eta-\beta L) e_k(\delta_0^*,\ldots,\delta_{k-1}^*)+2\nu_\beta^2\delta_k^*=0,\] which shows that
$\frac{\partial e_{k+1}}{\partial \delta_k}$ vanishes at $(\delta_{0}^*,\ldots,\delta_k^*)\in\mathbb{G}_{k+1}.$
Hence, by induction we showed that
$(\delta_{0}^*,\ldots,\delta_k^*)$ is a stationary point
of $e_{k+1}$ in the set $\mathbb{G}_{k+1}$.
\end{comment}
\end{proof}
We have just provided an analysis in terms of the lower bound
sequence $\{\delta_{k}\}$. We can conduct a similar analysis for $\{\Gamma_{k}$\}
and obtain the
corresponding adaptive stepsize scheme using the following relation:
\begin{align*}
&\EXP{\|x_{k+1}-x^*\|^2} \leq \Gamma_k^2\nu^2 \cr & +(1-\frac{2(\eta+L)}{1+\beta}\Gamma_k+2L\Gamma_k
+L^2\Gamma_k^2)\EXP{\|x_ k-x^*\|^2}.
\end{align*}
When $0 < \Gamma_k
\le\frac{\eta-\beta L}{(1+\beta)L^2}$, we have
\begin{align}\label{rate_nu_est_different_max}
\EXP{\|x_{k+1}-x^*\|^2}
& \le (1-\frac{(\eta-\beta L)}{1+\beta} \Gamma_k)\EXP{\|x_k-x^*\|^2}\cr
&+ \Gamma_k^2\nu^2\qquad\hbox{for all }k \geq 0.
\end{align}
Using relation (\ref{rate_nu_est_different_max}) and following similar approach in Proposition \ref{prop:rec_results}, we obtain the sequence $\{\Gamma^*_{k}\}$ given by
\begin{align}
& \Gamma_{0}^*=\frac{\eta-\beta L}{2(1+\beta)\nu^2}\,e_0\label{eqn:gmax0}
\\
& \Gamma_k^*=\Gamma_{k-1}^*\left(1-\frac{\eta-\beta L}{2(1+\beta)}\Gamma_{k-1}^*\right)\qquad \hbox{for all } k \ge 1.
\label{eqn:gmaxk}
\end{align}
\fy{Note that the adaptive stepsize sequence given by (\ref{eqn:gmax0})--(\ref{eqn:gmaxk}) converges to zero and moreover, it is not summable but squared summable (cf. \cite{Farzad1}, Proposition $3$).}
In the following lemma, we derive a relation between two recursive sequences,
which we use later to obtain our main recursive stepsize scheme.
\begin{lemma}\label{lemma:two-rec}
Suppose that sequences
$\{\lambda_k\}$ and $\{\gamma_k\}$ are given with the following recursive equations
for all $k\geq 0$,
\begin{align}
\lambda_{k+1}& =\lambda_{k}(1-\lambda_{k}), \hbox{ and } \gamma_{k+1} & =\gamma_{k}(1-c\gamma_{k}), \notag
\end{align}
where $\lambda_0=c\gamma_0$, $0<\gamma_0<\frac{1}{c}$, and $c>0$.
Then for all $k \geq 0$,
\[\lambda_{k}=c\gamma_k.\]
\end{lemma}
\begin{proof}
We use induction on $k$. For $k=0$, the relation holds since $\lambda_0=c\gamma_0$.
Suppose that for some $k \geq 0$ the relation holds. Then, we have
\begin{align}
\gamma_{k+1}=\gamma_{k}(1-c\gamma_{k}) \quad
& \Rightarrow \quad c\gamma_{k+1}=c\gamma_{k}(1-c\gamma_{k})\cr
& \Rightarrow \quad c\gamma_{k+1}=\lambda_{k}(1-\lambda_{k})\cr
& \Rightarrow \quad \gamma_{k+1}=\lambda_{k+1}.
\end{align}
Hence, the result holds for $k+1$ implying that the result holds for all $k \ge 0$.
\end{proof}
Next, we show a relation for the sequences $\{\delta_k^*\}$ and $\{\Gamma_k^*\}$.
\begin{lemma}\label{lemma:min_maxm_relation}
Suppose that sequences $\{\delta_k^*\}$ and $\{\Gamma_k^*\}$
are given by relations \eqref{eqn:gmin0}--\eqref{eqn:gmink} and \eqref{eqn:gmax0}--\eqref{eqn:gmaxk} and $e_0 < \frac{2 \nu^2}{L^2}$. Then for all $k \geq 0$, {$\Gamma_k^*=(1+\beta)\delta_k^*$.}
\end{lemma}
\begin{proof}
Suppose that $\{\lambda_k\}$ is defined by $
\lambda_{k+1}=\lambda_{k}(1-\lambda_{k})$, for all$k\geq 0$,
where $\lambda_0=\frac{(\eta-\beta L)^2}{4(1+\beta)^2\nu^2}\,e_0$.
{In what follows,} we apply Lemma \ref{lemma:two-rec} twice to obtain
the result. {By the definition of $\lambda_0$ and $\delta_0^*$, we have
that
$\lambda_0=\frac{(\eta-\beta L)}{2}\delta_0^*$. Also, using $e_0
<\frac{2\nu^2}{L^2}$ and definition of $\lambda_0$, we obtain
\[\lambda_0 =\frac{(\eta- \beta L)^2}{4(1+\beta)^2\nu^2}e_0 <
\frac{(\eta- \beta L)^2}{2(1+\beta)^2L^2}\leq
\frac{\eta^2}{2L^2}<1.\] Therefore, the conditions of Lemma
\ref{lemma:two-rec} hold for sequences
$\{\lambda_k\}$ and $\{\delta_k^*\}$.
Hence, Lemma \ref{lemma:two-rec} yields that for all $k \geq 0$,
\begin{align*}
& \lambda _k = \frac{(\eta-\beta L)}{2}\delta_k^*.
\end{align*}
Similarly, invoking Lemma \ref{lemma:two-rec} again, we have $\lambda _k =\frac{(\eta-\beta L)}{2(1+\beta)}\Gamma_k^*$.
Therefore, from the two preceding relations, we can conclude the desired relation.}
Therefore, for all $k \geq 0$, $\Gamma_k^*=(1+\beta)\delta_k^*$.
\end{proof}
The earlier set of results are essentially adaptive rules for
determining the upper and lower bound of stepsize sequences, i.e.
$\{\delta_k^*\}$ and $\{\Gamma_k^*\}$. The
next proposition proposes recursive stepsize schemes for each player of game \eqref{eqn:problem}.
\begin{proposition}{[Distributed adaptive steplength SA rules]}\label{prop:DASA}
Suppose that Assumption \ref{assum:different_main} and \ref{assum:w_k_bound} hold. Assume that set $X$ is bounded, i.e. there exists a positive constant $D \triangleq \max_{x,y \in X}\|x-y\|$. Suppose that the stepsizes for any player $i=1, \ldots,N$ are given by the following recursive equations
Suppose that Assumption \ref{assum:different_main} and \ref{assum:w_k_bound} hold. Assume that set $X$ is bounded, i.e. there exists a positive constant $D \triangleq \max_{x,y \in X}\|x-y\|$. Suppose that the stepsizes for any player $i=1, \ldots,N$ are given by the following recursive equations
\begin{align}
& \gamma_{0,i}=r_i\frac{c}{(1+\frac{\eta-2c}{L})^2\nu^2}\,D^2\label{eqn:gi0}
\\
& \gamma_{k,i}=\gamma_{k-1,i}\left(1-\frac{c}{r_i}\gamma_{k-1,i}\right)
\quad \hbox{for all }k\ge 1.\label{eqn:gik}
\end{align}
where $r_i$ is an arbitrary parameter associated with $i$th player such
that $r_i \in [1, 1+\frac{\eta-2c}{L}]$, $c$ is an arbitrary fixed
constant $0<c < \frac{\eta}{2}$, $L$ is the Lipschitz constant of
mapping $F$, and $\nu$ is the upper bound given by Assumption
\ref{assum:w_k_bound} such that $D < \sqrt{2}\frac{\nu}{L}$. Then,
{the following hold:}
\begin{itemize}
\item [(a)] \an{$\frac{\gamma_{k,i}}{r_i}=\frac{\gamma_{k,j}}{r_j}$ for any $i,j=1,\ldots, N$ and $k \geq 0$.}
\item [(b)] Assumption \ref{assum:step_error_sub}b holds with $\beta=\frac{\eta -2c}{L}$, $\delta_k=\delta_k^*$, $\Gamma_k=\Gamma_k^*$, and $e_0=D^2$, where $\delta_k^*$ and $\Gamma_k^*$ are given by \eqref{eqn:gmin0}--\eqref{eqn:gmink} and \eqref{eqn:gmax0}--\eqref{eqn:gmaxk} respectively.
\item [(c)] The sequence $\{x_k\}$
generated by algorithm (\ref{eqn:algorithm_different}) converges \fy{a.s.}
to the unique solution of stochastic VI$(X,F)$.
\item [(d)] The results of Proposition \ref{prop:rec_results} hold for $\delta_k^*$ when $e_0=D^2$.
\end{itemize}
\end{proposition}
\begin{proof}
\noindent (a) \
Consider the sequence $\{\lambda_k\}$ given by
\begin{align*}
& \lambda_0=\frac{c^2}{(1+\frac{\eta-2c}{L})^2\nu^2}\,D^2,
\\
& \lambda_{k+1}=\lambda_k(1-\lambda_k),
\quad \hbox{for all }k\ge 1.
\end{align*}
Since for any ${i=1,\ldots, N}$, we have $\lambda_0=\frac{c}{r_i}\, \gamma_{0,i}$, using Lemma \ref{lemma:two-rec}, we obtain that for any $1 \leq i \leq N$ and $k \geq 0$,
\[\lambda_k=\frac{c}{r_i}\, \gamma_{k,i}.\]
Therefore, for any $1 \leq i,j \leq N$, we obtain the desired relation in part (a).
\noindent (b) \ First we show that $\delta_k^*$ and $\Gamma_k^*$
are well defined. Consider the relation of part (a). Let $k\ge 0$ be
arbitrarily fixed. If $\gamma_{k,i}>\gamma_{k,j}$ for some $i \neq j$, then we
have $r_{i}>r_{j}.$ Therefore, the minimum possible $\gamma_{k,i}$ {is
obtained} with $r_i=1$ and the maximum possible $\gamma_{k,i}$ {is
obtained} with $r_i=1+\frac{\eta-2c}{L}$. Now, consider
\eqref{eqn:gi0}--\eqref{eqn:gik}. If, $r_i=1$, and
$D^2$ is replaced by $e_0$, and $c$ by $\frac{\eta-\beta L}{2}$, we get the
same recursive sequence defined by
\eqref{eqn:gmin0}--\eqref{eqn:gmink}. Therefore, since the
minimum possible $\gamma_{k,i}$ {is achieved} when $r_i=1$, we
conclude that $\delta_k^* \leq \min_{i=1,\ldots,N} \gamma_{k,i}$ for
any $k\ge 0$. This shows that $\delta_k^*$ is well-defined in
the context of Assumption \ref{assum:step_error_sub}b.
Similarly, it can be shown that $\Gamma_k^*$ is also
well-defined in the context of Assumption
\ref{assum:step_error_sub}b. Now, Lemma~\ref{lemma:min_maxm_relation} implies that
$\Gamma_k^*=(1+\frac{\eta-2c}{L})\delta_k^*$ for any $k \geq 0$,
which shows that {Assumption} \ref{assum:step_error_sub}b is satisfied since $\beta=\frac{\eta-2c}{L}$ and $0<c<\frac{\eta}{2}$.
\noindent (c) In view of Proposition \ref{prop:rel_bound}, to show the
almost-sure convergence, it suffices to show that Assumption
\ref{assum:step_error_sub} holds. Part (b) implies that Assumption
\ref{assum:step_error_sub}b holds for the specified choices. Since
$\gamma_{k,i}$ is a recursive sequence for each $i$, Assumption
\ref{assum:step_error_sub}a holds using Proposition 3 in \cite{Farzad1}.
\noindent (d) Since $D < \sqrt{2}\frac{\nu}{L}$, it follows that $e_0 <\frac{2\n^2}{L^2}$, which shows that the conditions of Proposition \ref{prop:rec_results} are satisfied.
\end{proof}
\fy{
\section{Numerical results}\label{sec:numerical}
In this section, we report the results of our numerical experiments on a stochastic bandwidth-sharing problem in
communication networks (Sec. \ref{sec:traffic}). We
compare the performance of the distributed adaptive stepsize SA
scheme (DASA) given by (\ref{eqn:gi0})--(\ref{eqn:gik}) with that of
SA schemes with harmonic stepsize sequences (HSA), where agents use
the stepsize $\frac{\theta}{k}$ at iteration $k$. More precisely, we
consider three different values of the parameter $\theta$, i.e.,
$\theta = 0.1$, $1$, and $10$. This diversity of choices allows us to observe
the sensitivity of the HSA scheme to different settings of the
parameters.
\subsection{A bandwidth-sharing problem in computer networks}\label{sec:traffic}
We consider a communication network where users compete for the
bandwidth. Such a problem can be captured by an optimization framework (cf. \cite{Cho05}). Motivated by this model, we consider a network
with $16$ nodes, $20$ links and $5$ users. Figure \ref{fig:network}
shows the configuration of this network. \begin{figure}[htb]
\begin{center}
\includegraphics[scale=.30]{images/network2v05.pdf}
\caption{The network}
\label{fig:network}
\end{center}
\vspace{-0.1in}
\end{figure} Users have access to different routes as shown in Figure
\ref{fig:network}. For example, user $1$ can access routes $1$, $2$, and $3$.
Each user is characterized by a cost function. Additionally, there is a
congestion cost function that depends on the aggregate flow. More
specifically, the cost function
user $i$ with flow rate (bandwidth) $x_i$ is defined by
\[f_i(x_i,\xi_i)\triangleq -\sum_{r \in \mathcal{R}(i)} \xi_{i}(r)\log(1+x_i(r)),\]
for $i=1,\ldots, 5$, where $x\triangleq (x_1; \ldots; x_5)$ is the flow decision vector of the users, $\xi \triangleq (\xi_1; \ldots; \xi_5)$ is a random parameter corresponding to the different users,
$\mathcal{R}(i)=\{1,2,\ldots,n_i\}$ is the set of routes assigned to the $i$-th user, $x_{i}(r)$ and $\xi_{i}(r)$
are the $r$-th element of the decision vector $x_i$ and the random vector $\xi_i$, respectively.
We assume that
$\xi_i(r)$ is drawn from a uniform distribution for each $i$ and $r$ and the links have limited capacities given by $b$.
We may define the routing matrix $A$ that describes the relation between set of routes $\mathcal{R}=\{1,2,\ldots,9\}$ and set of links $\mathcal{L}=\{1,2,\ldots,20\}$. Assume that $A_{lr}=1$ if route $r \in \mathcal{R}$ goes through link $l \in \mathcal{L}$ and $A_{lr}=0$ otherwise. Using this matrix, the capacity constraints of the links can be described by $Ax \leq b$.
We formulate this model as a stochastic optimization problem given by
\begin{align}\label{eqn:network-prob}
\displaystyle \mbox{minimize} \qquad & \sum_{i=1}^N \EXP{f_i(x_i,\xi_i)} +c(x) \\
\mbox{subject to} \qquad& Ax \leq b, \hbox{ and } x \geq 0,\notag
\end{align}
where $c(x)$ is the network congestion cost.
We consider this cost of the form $c(x)=\|Ax\|^2$.
Problem (\ref{eqn:network-prob}) is a convex optimization problem and the optimality conditions can be stated as a variational inequality given by $\nabla f(x^*)^T(x-x^*) \geq 0$, where $f(x) \triangleq \sum_{i=1}^N \EXP{f_i(x_i,\xi_i)} +c(x)$. Using our notation in Sec. \ref{sec:formulation}, we have $$F(x)=-\left(
\frac{\bar \xi_1(1)}{1+x_1(1)};\ldots;
\frac{\bar\xi_5(2)}{1+x_5(2)} \right)+2A^TAx,$$
where $\bar \xi_i(r_i) \triangleq \EXP{\xi_i(r_i)}$ for any $i=1,\ldots,5$ and $r_i=1,\ldots,n_i$. It can be shown
that the mapping $F$ is strongly monotone and Lipschitz with specified parameters (cf. \cite{Farzad03}). We solve the bandwidth-sharing problem for $12$
different settings of parameters shown in Table
\ref{tab:network_errors1}. We consider $4$ parameters in our model that
scale the problem. Here, $m_b$ denotes the multiplier of the capacity
vector $b$, $m_c$ denotes the multiplier of the congestion cost function
$c(x)$, and $m_\xi$ and $d_\xi$ are two multipliers that parametrize the
random variable $\xi$. $S(i)$
denotes the $i$-th setting of parameters. For each of these $4$
parameters, we consider $3$ settings where one parameter changes and
other parameters are fixed. This allows us to observe the sensitivity of
the algorithms with respect to each of these parameters.
\begin{table}[htb]
\vspace{-0.05in}
\tiny
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
-&S$(i)$ & $m_b$ & $m_c$ & $m_\xi$ & $d_\xi$
\\
\hline
\hline
{$m_b$} &1 & 1 & 1 & 5 & 2
\\
\hbox{ }& 2 & 0.1 & 1 & 5 & 2
\\
\hbox{ }& 3 & 0.01 & 1 & 5 & 2 \\
\hline
$m_c$ &4 & 0.1 & 2 & 2 & 1
\\
\hbox{ }& 5 & 0.1 & 1 & 2 & 1
\\
\hbox{ }& 6 & 0.1 & 0.5 & 2 & 1 \\
\hline
$m_\xi$ &7 & 1 & 1 & 1 & 5
\\
\hbox{ }& 8 & 1 & 1 & 2 & 5
\\
\hbox{ }& 9 & 1 & 1 & 5 & 5 \\
\hline
$d_\xi$ &10 & 1 & 0.01 & 1 & 1
\\
\hbox{ }& 11 & 1 & 0.01 & 1 & 2
\\
\hbox{ }& 12 & 1 & 0.01 & 1 & 5\\
\hline
\end{tabular}
\caption{Parameter settings}
\label{tab:network_errors1}
\vspace{-0.1in}
\end{table}
The SA algorithms are terminated after $4000$ iterates.
To measure the error of the schemes, we run each scheme $25$ times and
then compute the mean squared error (MSE) using the metric $\frac{1}{25}\sum_{i=1}^{25}\|x_k^i-x^*\|^2$ for any $k=1,\ldots,4000$, where $i$ denotes the $i$-th sample.
Table \ref{tab:Traffic2} and \ref{tab:Traffic3} show the $90\%$ confidence intervals (CIs) of the error for the DASA and HSA schemes.
\begin{table}[htb]
\vspace{-0.05in}
\tiny
\centering
\begin{tabular}{|c|c||c||c|}
\hline
-&S$(i)$ & DASA - $90\%$ CI & HSA with $\theta=0.1$- $90\%$ CI
\\
\hline
\hline
{$m_b$} &1 & [$2.97 $e${-6}$,$4.66 $e${-6}$] & [$1.52 $e${-6}$,$2.37 $e${-6}$]
\\
\hbox{ }&2 & [$2.97 $e${-6}$,$4.66 $e${-6}$] & [$1.52 $e${-6}$,$2.37 $e${-6}$]
\\
\hbox{ }&3& [$1.15 $e${-7}$,$3.04 $e${-7}$] & [$2.12 $e${-8}$,$4.92 $e${-8}$]
\\
\hline
$m_c$ &4 & [$4.39 $e${-7}$,$6.55 $e${-7}$] & [$1.33$e${-6}$,$1.80 $e${-6}$]
\\
\hbox{ }&5 & [$1.29 $e${-6}$,$1.97 $e${-6}$] & [$9.00$e${-6}$,$1.20 $e${-5}$]
\\
\hbox{ }&6 & [$3.44 $e${-6}$,$5.36 $e${-6}$] & [$2.26$e${-4}$,$2.53 $e${-4}$]
\\
\hline
$m_\xi$ &7& [$4.29 $e${-5}$,$6.40 $e${-5}$] & [$7.92 $e${-5}$,$1.49 $e${-4}$]
\\
\hbox{ }&8 &[$3.18 $e${-5}$,$4.83 $e${-5}$] & [$3.46 $e${-5}$,$6.07 $e${-5}$]
\\
\hbox{ }&9 & [$1.83 $e${-5}$,$2.88 $e${-5}$] & [$6.12 $e${-6}$,$9.99 $e${-6}$]
\\
\hline
$d_\xi$ &10 &[$3.82 $e${-4}$,$5.91 $e${-4}$] & [$2.86 $e${+1}$,$2.86 $e${+1}$]
\\
\hbox{ }&11 & [$9.81 $e${-4}$,$1.44 $e${-3}$] & [$2.86 $e${+1}$,$2.86 $e${+1}$]
\\
\hbox{ }&12 &[$6.26 $e${-3}$,$8.44 $e${-3}$] & [$2.85 $e${+1}$,$2.86 $e${+1}$]
\\
\hline
\end{tabular}
\caption{$90\%$ CIs for DASA and HSA schemes -- Part I}
\label{tab:Traffic2}
\vspace{-0.1in}
\end{table}
\begin{table}[htb]
\vspace{-0.05in}
\tiny
\centering
\begin{tabular}{|c|c||c||c|}
\hline
-&S$(i)$ & HSA with $\theta=1$ - $90\%$ CI & HSA with $\theta=10$ - $90\%$ CI
\\
\hline
\hline
{$m_b$} &1 & [$1.70 $e${-6}$,$2.97 $e${-6}$]& [$1.33 $e${-5}$,$1.81 $e${-5}$]
\\
\hbox{ }&2 & [$1.70 $e${-6}$,$2.97 $e${-6}$]& [$1.33 $e${-5}$,$1.81 $e${-5}$]
\\
\hbox{ }&3 & [$4.66 $e${-8}$,$1.17 $e${-7}$]& [$8.07 $e${-7}$,$2.43 $e${-6}$]
\\
\hline
$m_c$ &4 & [$4.71 $e${-7}$,$8.75 $e${-7}$]& [$3.84 $e${-6}$,$5.38 $e${-6}$]
\\
\hbox{ }&5 & [$7.88 $e${-7}$,$1.36 $e${-6}$]& [$5.61 $e${-6}$,$7.98 $e${-6}$]
\\
\hbox{ }&6 & [$1.25 $e${-6}$,$1.99 $e${-6}$]& [$7.34 $e${-6}$,$1.12 $e${-5}$]
\\
\hline
$m_\xi$ &7 & [$2.83 $e${-5}$,$4.75 $e${-5}$]& [$1.84 $e${-4}$,$2.75 $e${-4}$]
\\
\hbox{ }&8 & [$1.97 $e${-5}$,$3.39 $e${-5}$]& [$1.40 $e${-4}$,$1.99 $e${-4}$]
\\
\hbox{ }&9 & [$1.06 $e${-5}$,$1.85 $e${-5}$]& [$8.33 $e${-5}$,$1.13 $e${-4}$]
\\
\hline
$d_\xi$ &10 & [$5.50 $e${-1}$,$5.70 $e${-1}$]& [$7.23 $e${-5}$,$9.64 $e${-5}$]
\\
\hbox{ }&11& [$5.45 $e${-1}$,$5.85 $e${-1}$]& [$2.85 $e${-4}$,$3.80 $e${-4}$]
\\
\hbox{ }&12 & [$5.47 $e${-1}$,$6.44 $e${-1}$]& [$1.77 $e${-3}$,$2.36 $e${-3}$]
\\
\hline
\end{tabular}
\caption{$90\%$ CIs for DASA and HSA schemes -- Part II}
\label{tab:Traffic3}
\vspace{-0.1in}
\end{table}
\textbf{Insights:}
We observe that DASA scheme performs favorably and is far more robust in
comparison with the HSA schemes with different choice of $\theta$.
Importantly, in most of the settings, DASA stands close to the HSA
scheme with the minimum MSE. Note that when $\theta=1$ or $\theta=10$,
the stepsize $\frac{\theta}{k}$ is not within the interval
$(0,\frac{\eta-\beta L}{(1+\beta)^2L^2}]$ for small $k$ and is not
feasible in the sense of Prop. \ref{prop:rec_results}.
Comparing the performance of each HSA scheme in different
settings, we observe that HSA schemes are fairly sensitive to
the choice of parameters. For example, HSA with $\theta=0.1$
performs very well in settings S$(1)$, S$(2)$, and S$(3)$, while
its performance deteriorates in settings S$(10)$, S$(11)$, and
S$(12)$. A similar discussion holds for other two HSA schemes. A
good instance of this argument is shown in Figure \ref{fig:traffic_all_s4} and \ref{fig:traffic_all_s11}.
\begin{figure}[htb]
\centering
\label{fig:traffic_prob4}\includegraphics[scale=.33]{images/traffic_sim04_all.pdf}
\caption{DASA vs. HSA schemes -- Setting S($4$)}
\label{fig:traffic_all_s4}
\end{figure}
\vspace{-.1in}
\begin{figure}[htb]
\centering
\label{fig:traffic_prob11}\includegraphics[scale=.33]{images/traffic_sim11_all.pdf}
\caption{DASA vs. HSA schemes -- Setting S($11$)}
\label{fig:traffic_all_s11}
\end{figure}}
\vspace{-.1in}
\section{Concluding remarks}
We considered distributed monotone stochastic Nash games where each player minimizes a convex function
on a closed convex set. We first formulated the problem as a stochastic
VI and then showed that under suitable conditions, for a strongly monotone and Lipschitz mapping, the SA scheme guarantees almost-sure convergence to the solution.
Next, motivated by the naive stepsize choices of SA algorithm, we proposed a class of distributed adaptive steplength rules where each player can choose his own stepsize independent of
the other players from a specified range. We showed that this scheme provides
almost-sure convergence and also minimizes a suitably defined error bound of the SA algorithm.
Numerical experiments, reported in Section \ref{sec:numerical} confirm this conclusion.
\bibliographystyle{IEEEtran}
| 23,688 |
\section{Introduction}
The Dissipative Particle Dynamics (DPD) simulation method~\cite{hoogerbrugge1992simulating, koelman1993dynamic} was originally developed to cover much longer length and time scales than in conventional atomistic Molecular Dynamics (MD) simulations using a mesoscopic description of the simulated system~\cite{espanol1995statistical} due to soft forces acting between large particles made of clusters of atoms. Since its introduction about two decades ago, several improvements and generalizations~\cite{groot1997dissipative, espanol1997dissipative, flekkoy1999molecular, pagonabarraga2001dissipative, warren2003vapor, espanol2003smoothed, nikunen2003would, maiti2004bead, jakobsen2005constant, travis2007new} have been proposed, making DPD method one of the mostly used coarse-grained approach in soft matter simulations. Examples of such studies are microphase separation of block copolymers~\cite{groot1998dynamic, qian2005computer}, polymeric surfactants in solution~\cite{ryjkina2002molecular, prinsen2002mesoscale}, colloidal suspensions~\cite{whittle2010dynamic}, and the structural and rheological behavior of biological membranes~\cite{shillcock2005tension, de2009effect}. For many of these complex systems mentioned above, the electrostatic interactions play a vital role behind the key phenomena and dynamical processes. The inclusion of electrostatic interactions in DPD simulations is important to capture the long-range interactions at mesoscopic level of material description for many systems such as the conformational properties of polyelectrolyte brushes~\cite{ibergay2009electrostatic, ibergay2010mesoscale, yan2009influence} and the formation of membrane-DNA complexes in biological systems~\cite{gao2009effects, gao2010communications}.
\par
However, when electrostatic interactions are incorporated in DPD method, the mesoscopic DPD particles carrying opposite charges tend to form artificial aggregates due to the soft nature of the conservative interactions (forces) in DPD simulations. In order to avoid a collapse of oppositely charged particles onto each other, Groot~\cite{groot2003electrostatic} smeared out the local point charge into grids around each DPD particle. He adopted a variant of particle-particle particle-mesh (PPPM) approach, which was originally introduced for systems with electrostatic heterogeneities~\cite{beckers1998iterative}, to treat separately the near and far field interactions between charge distributions. As the charge density distributions in the simulated system are affected by hydrodynamic flow~\cite{pagonabarraga2010recent}, this method provides a natural coupling between electrostatics and fluid motion. Simulation results~\cite{ibergay2009electrostatic, yan2009influence, gao2010communications, groot2003electrostatic} demonstrated that this method is reasonably efficient in capturing important features of electrostatic interactions at mesoscopic level.
\par
In an alternative approach, Gonz\'alez-Melchor~\emph{et al.}~\cite{gonzalez2006electrostatic} adopted traditional Ewald summation method and Slater-type charge density distribution in DPD simulations. This method allows a standard approach to calculate electrostatic energy and force of charge density distributions. Although the inclusion of charge density distribution does not directly increase the computational cost in the Ewald summation itself, this method becomes computationally more demanding than the one adopted by Groot~\cite{groot2003electrostatic}.
\par
In order to improve the efficiency and accuracy in treating electrostatic interactions, we suggest here an alternative approach that allows fast calculation of electrostatic energy and force in DPD simulations. The ENUF method, an abbreviation for the Ewald summation using Non-Uniform FFT (Fast Fourier Transform) (NFFT) technique, was recently suggested in our group~\cite{hedman2006ewald, hedman2006thesis} and showed excellent computational efficiency in atomistic MD simulations. In our current work, we implement the ENUF in the DPD method. The ENUF-DPD method is initially applied on simple electrolyte system as a typical model case to optimize the ENUF-DPD parameters and investigate corresponding computational complexity. With suitable parameters, we adopt the ENUF-DPD method to study the dependence of polyelectrolyte conformations on ionic strength and counterion valency of added salts for illustration.
\par
This paper is organized as follows: Sec.~\ref{sec:dpdmethod} contains a brief introduction to the DPD method. Detailed algorithm of the ENUF method is given in Sec.~\ref{sec:enuf}. Secs.~\ref{sec:enufdpd} and~\ref{sec:validation} describe the implementation of ENUF in DPD method and the exploration of suitable parameters for ENUF-DPD method in a simple model electrolyte system. In Sec.~\ref{sec:pe}, the ENUF-DPD method is further validated in studying polyelectrolyte conformations upon addition of salts with multivalent counterions. Finally, main concluding remarks are given in Sec.~\ref{sec:conclusion}.
\section{The DPD Method}\label{sec:dpdmethod}
The DPD method, originally introduced by Hoogerbrugge and Koelman in $1992$~\cite{hoogerbrugge1992simulating}, is a mesoscopic and particle-based simulation method based on a set of pairwise forces. One important conceptual difference between DPD and conventional MD is the use of coarse-graining (CG) procedure allowing a mapping of several atoms or molecules from the real atomistic system onto larger DPD particles. After scaling up the size of the system, the DPD method can capture the hydrodynamic behavior in very large complex systems up to microsecond range and beyond. Like in MD simulations, the time evolutions of DPD particles are governed by Newton's equations of motion
\begin{eqnarray} \label{eq01}
\frac{d\mathbf{r}_i}{dt}=\mathbf{v}_i\,,\qquad m_i\frac{d\mathbf{v}_i}{dt}=\mathbf{f}_i\,,
\end{eqnarray} where $\mathbf{r}_i$, $\mathbf{v}_i$ and $\mathbf{f}_i$ denote the coordinate, velocity, and the total force acting on particle $i$, respectively. The total force, between any pair of DPD particles $i$ and $j$, is normally composed of three different pairwise additive forces: the conservative force $\mathbf{F}_{ij}^{C}$, the dissipative force $\mathbf{F}_{ij}^D$, and the random force $\mathbf{F}_{ij}^R$,
\begin{eqnarray} \label{eq02}
\mathbf{f}_{ij}&=&\sum_{i\neq{j}}(\mathbf{F}_{ij}^{C}+\mathbf{F}_{ij}^{D}+\mathbf{F}_{ij}^{R})\,,
\end{eqnarray} with
\begin{eqnarray} \label{eq0305}
\mathbf{F}_{ij}^C&=&\alpha_{ij}\omega^C(r_{ij})\mathbf{\hat{r}}_{ij}\,,\\
\mathbf{F}_{ij}^D&=&-\gamma\omega^D(r_{ij})(\mathbf{v}_{ij}\cdot\mathbf{\hat{r}}_{ij})\mathbf{\hat{r}}_{ij}\,,\\
\mathbf{F}_{ij}^R&=&\sigma\omega^R(r_{ij})\theta_{ij}\mathbf{\hat{r}}_{ij}\,,
\end{eqnarray} where $\mathbf{r}_{ij}=\mathbf{r}_i-\mathbf{r}_j$, $r_{ij}=|\,\mathbf{r}_{ij}|$, $\mathbf{\hat{r}}_{ij}=\mathbf{r}_{ij}/r_{ij}$, and $\mathbf{v}_{ij}=\mathbf{v}_i-\mathbf{v}_j$. The parameters $\alpha_{ij}$, $\gamma$, and $\sigma$ determine the strength of the conservative, dissipative, and random forces, respectively. $\theta_{ij}$ is a randomly fluctuating variable, with zero mean and unit variance.
\par
The pairwise conservative force is written in terms of a weight function $\omega^C(r_{ij})$, where $\omega^C(r_{ij})=1-r_{ij}/R_c$ is chosen for $r_{ij}\leq R_c$ and $\omega^C(r_{ij})= 0$ for $r_{ij}>R_c$ such that the conservative force is soft and repulsive. The unit of length $R_c$ is related to the volume of DPD particles. In our simulations, we adopt the CG scheme~\cite{alsunaidi2004liquid} with $N_m=4$ and $\rho=4$, in which the former parameter means $4$ water molecules being coarse-grained into one DPD particle and the latter means there are $4$ DPD particles in the volume of $R_c^3$. With this particular scheme, the length unit $R_c$ is given as $R_c=3.107\sqrt[3]{\rho N_m}=7.829\textrm{\AA}$. The conservative interaction parameters between different types of DPD particles are determined by $\alpha_{ij}\approx \alpha_{ii}+2.05\chi$ with $\alpha_{ii}=78.67k_BT$, in which $\chi$ is the Flory-Huggins parameter between different types of DPD particles.
\par
Unlike the conservative force, two weight functions $\omega^D(r_{ij})$ and $\omega^R(r_{ij})$ for dissipative and random forces, respectively, are coupled together to form a thermostat. According to Espa\~nol and Warren~\cite{espanol1995statistical}, the relationship between two functions is described as
\begin{eqnarray} \label{eq06}
\left\{\begin{array}{l}
\omega^D\left(r\right)=\left[\omega^R\left(r\right)\right]^2 \\
\sigma^2=2\gamma k_BT
\end{array}\right.\,.
\end{eqnarray} This precise relationship between dissipative and random forces is determined by the fluctuation-dissipation theorem. We adopt a simple choice of $\omega^D(r)$ due to Groot and Warren~\cite{groot1997dissipative}
\begin{eqnarray} \label{eq07}
\omega^D\left(r\right)=\left[\omega^R\left(r\right)\right]^2=\left\{\begin{array}{ll}
\left(1-r/R_c\right)^2&\left(r\leq R_c\right) \\
0&\left(r>R_c\right)
\end{array}\right.\,.
\end{eqnarray}
\par
For polymer and surfactant molecules, the intramolecular interactions between bonded particles are described by harmonic springs
\begin{eqnarray}
\mathbf{F}_i^S = - \sum_j K^S (r_{ij}-r_{eq})\mathbf{\hat{r}}_{ij}\,,
\end{eqnarray} where $K^S$ is the spring constant and $r_{eq}$ is the equilibrium bond length.
In traditional DPD method, charge density distributions are usually adopted instead of point charges to avoid the divergence of electrostatic interactions at $r=0$. In our implementation, a Slater-type charge density distribution with the form of
\begin{eqnarray} \label{eq26}
\rho_e(r)&=&\frac{q}{\pi\lambda_e^3}e^{\frac{-2r}{\lambda_e}}
\end{eqnarray} is adopted, in which $\lambda_e$ is the decay length of charge $q$. The integration of Eq.~\ref{eq26} over the whole space gives the total charge $q$.
\par
A modified version of velocity-Verlet algorithm~\cite{groot1997dissipative} is used to integrate the equations of motion. For easy numerical handling, we choose the cut-off radius, the particle mass, and $k_BT$ as the units of the simulating system, \emph{i.e.}, $R_c=m=k_BT=1$. As a consequence, the unit of time $\tau$ is expressed as $\tau=R_{c}\sqrt{m/k_BT}=1$. All the related parameters used in our DPD simulations are listed in Table~\ref{table:dpd-parameter}.
\section{The ENUF method}\label{sec:enuf}
\subsection{The Ewald summation method}
Consider a system composed of $N$ charged particles, each one carrying the partial charge $q_i$ at position $\textbf{r}_i$ in a cubic cell with the volume $V=L^3$. Overall charge neutrality is assumed in the simulations. For simplicity, only the simple charge-charge interaction is considered, interactions between dipoles and multipoles are omitted in our current scheme. Charges interact with each other according to the Coulomb's law, and the total electrostatic energy can be written as
\begin{eqnarray} \label{eq15}
\mathbf{U}^E(\textbf{r}^N)&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\sum_{\textbf{n}}^{\dag}\sum_{i}\sum_{j>i} \frac{q_iq_j}{|\textbf{r}_{ij}+\textbf{n}L|}\,,
\end{eqnarray} where $\textbf{n}=(n_x,n_y,n_z)$, and $n_x$, $n_y$, and $n_z$ are integer numbers. The sum over $\textbf{n}$ takes into account the periodic images, and the \dag{} symbol indicates that the self-interaction terms are omitted when $\textbf{n}=0$. The variables $\epsilon_0$ and $\epsilon_r$ are the permittivity of vacuum and the dielectric constant of water at room temperature, respectively.
\par
In the Ewald summation method, the electrostatic energy, as shown in Eq.~\ref{eq15}, is decomposed into real space and reciprocal space contributions~\cite{frenkel1996understanding}. With such decomposition, both real and reciprocal space contributions are short-range and Eq.~\ref{eq15} can be rewritten as
\begin{eqnarray} \label{eq16}
\mathbf{U}^E(\textbf{r}^N)&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\bigg\{\sum_{\textbf{n}}^{\dag}\sum_{j>i}\frac{q_iq_j}{\textbf{r}_{ij}+\textbf{n}L}\textrm{erfc}\big(\alpha
|\textbf{r}_{ij}+\textbf{n}L|\big) \nonumber \\
& & +\frac{2\pi}{V}\sum_{\textbf{n}\neq 0}\frac{e^{-|\textbf{n}|^{2}/4\alpha ^2}}{|\textbf{n}|^2}S(\textbf{n}) S(-\textbf{n}) -\frac{\alpha}{\sqrt{\pi}}\sum_j^N q_j^2\bigg\}\,,
\end{eqnarray} with
\begin{eqnarray}\label{eq17}
S(\textbf{n})=\sum_{i=1}^{N}q_ie^{-\imath \textbf{n}\cdot\textbf{r}_i} \quad \textrm{and} \quad \textbf{n}=\frac{2\pi}{L}(n_x,n_y,n_z)\,.
\end{eqnarray} The first, second, and last terms in the bracket on the right-hand side of Eq.~\ref{eq16} correspond to the electrostatic energies from real space, reciprocal space and self-interaction parts, respectively. $\alpha$ is the Ewald convergence parameter and determines the relative convergence rate between real and reciprocal space summations. $n$ is the magnitude of the reciprocal vector $\textbf{n}$. Choosing suitable parameters, the complexity of the Ewald summation method is reduced from $\mathcal{O}(N^{2})$ to $\mathcal{O}(N^{3/2})$ with considerably accuracy and efficiency~\cite{toukmaji1996ewald}.
\par
As the number of charged particles in the simulated system grows, it is convenient to combine the calculation of the short-range conservative force with the calculation of the real space summations of electrostatic interactions. The cut-off for conservative force calculations should be the same as the cut-off for real space electrostatic interactions. With such combination and suitable value of $\alpha$, the summation of real space electrostatic energy between two charged particles extends no longer than the cut-off distance, and can be expressed as
\begin{eqnarray}\label{eq31}
\mathbf{U}^{E,R}&=& \frac{1}{4\pi\epsilon_0\epsilon_r}\sum_i \sum_{j>i} \frac{q_iq_j}{r}\textrm{erfc}(\alpha r)\,.
\end{eqnarray} The real space electrostatic force on particle $i$ is the negative of the derivative of the potential energy $\mathbf{U}^{E,R}$ respect to its position $\textbf{r}_i$. A common form of the real space electrostatic force is described as
\begin{eqnarray}\label{eq32}
\mathbf{F}_i^{E,R}&=& -\nabla_i\mathbf{U}^{E,R} \nonumber\\
&=& \frac{1}{4\pi\epsilon_0\epsilon_r}\bigg\{\sum_{j\neq i}\frac{q_iq_j}{r^2} \textrm{erfc}(\alpha r)+\frac{2\alpha}{\sqrt{\pi}}e^{-\alpha^2 r^2}\bigg\}\,.
\end{eqnarray} Thus the real space electrostatic energy and force can be calculated together with conservative force, but then the computation of reciprocal space summations of electrostatic interactions becomes the more time-consuming part. The introduction of the FFT technique reduces the computational complexity of Ewald summation to $\mathcal{O}(N\log N)$ by treating reciprocal space summations with FFT technique~\cite{hedman2006ewald, toukmaji1996ewald, hockney1981computer, darden1993particle, luty1994comparison, essmann1995smooth}.
\subsection{Basic features of the FFT technique}
Initially for a finite number of given Fourier coefficients $\hat{\boldsymbol{f_k}}\in\boldsymbol{C}$ with $\boldsymbol{k}\in I_M$, we wish to evaluate the trigonometric polynomial
\begin{eqnarray}\label{eq08}
f(\boldsymbol{x})&=&\sum_{\boldsymbol{k}\in I_M}\hat{\boldsymbol{f_k}}e^{-2\pi \imath \boldsymbol{k}\cdot\boldsymbol{x}}
\end{eqnarray} at each of the $N$ given nonequispaced points
\begin{eqnarray*}
\boldsymbol{X}=\big\{ \boldsymbol{x}_j\in\boldsymbol{D}^d:j=0,1,\ldots,N-1\big\}
\end{eqnarray*}
which are randomly localized in $d$-dimensional domain
\begin{eqnarray*}
\boldsymbol{D}^d = \big\{\boldsymbol{x} = (x_t)_{t = 0,1,\ldots,d-1}:-\frac{1}{2}\leq x_t\leq\frac{1}{2}\big\}\,.
\end{eqnarray*}
The space of the $d$-variable function $f\in\boldsymbol{D}^d$ is restricted to the space of $d$-variable trigonometric polynomials $\left(e^{-2\pi \imath \boldsymbol{k}}:\boldsymbol{k}\in I_M \right)$ with degree $M_t$($t=0,1,\ldots,d-1$) in the $t$-th dimension. The possible frequencies $\boldsymbol{k}$ are collected in the multi index set $I_M$ with
\begin{eqnarray} \label{eq09}
I_M&=&\big\{\boldsymbol{k}=(k_t)_{t=0,1,\ldots,d-1}\in Z^d:-\frac{M_t}{2}\leq k_t\leq \frac{M_t}{2}\big\}\,.
\end{eqnarray} The dimension of the function space or the total number of points in the index set is $M_{\Pi}=\Pi_{t=0}^{d-1}M_t$.
\par
With these in prior definitions, the trigonometric polynomials for the $N$ given points can be described by
\begin{eqnarray} \label{eq10}
f_j=f(\boldsymbol{x}_j)=\sum_{\boldsymbol{k}\in I_M}\hat{\boldsymbol{f_k}}e^{-2\pi \imath \boldsymbol{k}\cdot\boldsymbol{x}_j}\qquad (j=0,1,\ldots,N-1)\,.
\end{eqnarray} Using the matrix-vector notation, all trigonometric polynomials can be rewritten as
\begin{eqnarray} \label{eq11}
\boldsymbol{f = A}\hat{\boldsymbol{f}}\,,
\end{eqnarray} where
\begin{eqnarray}\label{eq12}
\boldsymbol{f}&=&(f_j)_{j=0,1,\ldots,N-1}\,, \\ \nonumber
\boldsymbol{A}&=&(e^{-2\pi\imath\boldsymbol{k}\cdot\boldsymbol{x}_j})_{j=0,1,\ldots,N-1;\,\boldsymbol{k}\in I_M}\,,\\ \nonumber
\hat{\boldsymbol{f}}&=&(\hat{f}_{\boldsymbol{k}})_{\boldsymbol{k}\in I_M}\,.
\end{eqnarray}
\par
In the following implementations, the related matrix-vector products are the conjugated form
\begin{eqnarray} \label{eq13}
\boldsymbol{f}=\bar{\boldsymbol{A}}\hat{\boldsymbol{f}}\,, \qquad
\boldsymbol{f}_j=\sum_{\boldsymbol{k} \in I_M} \hat{\boldsymbol{f_k}} e^{2\pi \imath \boldsymbol{k} \cdot \boldsymbol{x}_j}\,,
\end{eqnarray} and the transposed form
\begin{eqnarray} \label{eq14}
\hat{\boldsymbol{f}}=\boldsymbol{A}^T\boldsymbol{f}\,, \qquad
\hat{\boldsymbol{f_k}}=\sum_{j=0}^{N-1}\boldsymbol{f}_je^{-2\pi \imath \boldsymbol{k}\cdot\boldsymbol{x}_j}\,,
\end{eqnarray} in which the matrix $\bar{\boldsymbol{A}}$ and $\boldsymbol{A}^T$ are the conjugated and transposed complex of the matrix $\boldsymbol{A}$, respectively. With given Fourier coefficients $\hat{\boldsymbol{f}}$, the Fourier samples $\boldsymbol{f}$ can be transformed with suitable FFT techniques in both directions.
\subsection{Detailed description of the ENUF method}
The ENUF method, which combines traditional Ewald summation method with the Non-Uniform fast Fourier transform technique, is a novel and fast method to calculate electrostatic interactions. The NFFT~\cite{press1989fast} is a generalization of the FFT technique~\cite{cooley1965algorithm}. The basic idea of NFFT is to combine the standard FFT and linear combinations of window functions which are well localized in both space and frequency domain. The controlled approximations using a cut-off in the frequency domain and a limited number of terms in the space domain result in an aliasing error and a truncation error, respectively. The aliasing error is controlled by the over-sampling factor $\sigma_s$, and the truncation error is controlled by the number of terms $p$. As described in Refs.~\cite{hedman2006ewald, hedman2006thesis}, NFFT only makes approximations on the reciprocal space part of Ewald summation. Hence in the following, we show the detailed procedure to calculate the reciprocal space summations of electrostatic energy and force with NFFT technique.
\par
Recasting the reciprocal space electrostatic energy in terms of Fourier components, the second term of Eq.~\ref{eq16} can be rewritten as
\begin{eqnarray} \label{eq18}
\mathbf{U}^{E,K}&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{2\pi}{V}\sum_{\textbf{n}\neq 0}\frac{e^{-|\textbf{n}|^{2}/4\alpha^2}}{|\textbf{n}|^2}S(\textbf{n})S(-\textbf{n}) \nonumber\\
&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{1}{2\pi L}\sum_{\textbf{n}\neq0} \frac{e^{-(\pi n)^2/(\alpha L)^2}}{n^2}S(\textbf{n})S(-\textbf{n})\,.
\end{eqnarray} For fixed vector $\textbf{n}$, the structure factor $S(\textbf{n})$ is just a complex number. With normalized locations, $\textbf{x}_j=\textbf{r}_j/L$, the structure factor can be expressed with
\begin{eqnarray} \label{eq19}
S(\textbf{n})&=&\sum_{j=1}^{N}q_je^{-2\pi \imath \textbf{n}\cdot\textbf{x}_j}\,.
\end{eqnarray}
It should be noted that the structure factor in Eq.~\ref{eq19} and the transposed FFT form in Eq.~\ref{eq14} have similar structures.
Suppose that $q_j$ is substituting $\boldsymbol{f}_j$, the structure factor $S(\textbf{n})$ is then a three-dimensional case of transposed FFT form.
By viewing the structure factor $S(\textbf{n})$ as a trigonometric polynomial $\hat{\boldsymbol{f}}_{\textbf{n}}$, the reciprocal space electrostatic energy can be rewritten as
\begin{eqnarray} \label{eq20}
\mathbf{U}^{E,K}&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{1}{2\pi L}\sum_{\textbf{n}\neq 0} \frac{e^{-(\pi n)^2/(\alpha L)^2}}{n^2}|\hat{\boldsymbol{f}}_{\textbf{n}}|^2\,.
\end{eqnarray}
The reciprocal space electrostatic energy is approximated by a linear combination of window functions sampled at nonequidistant $M_{\Pi}$ grids.
These grids are then input to the transposed FFT, with which one can calculate each component of the structure factor $S(\textbf{n})$, and hence the reciprocal space summations of electrostatic energy.
\par
Alternatively, the reciprocal space electrostatic energy in Eq.~\ref{eq18} can be expressed by the real and imaginary parts of the structure factor $S(\textbf{n})$ as
\begin{eqnarray} \label{eq21}
\mathbf{U}^{E,K}&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{1}{2\pi L}\sum_{\textbf{n}\neq 0} \frac{e^{-(\pi n)^2/(\alpha L)^2}}{n^2}\bigg\{Re\big(S(\textbf{n})\big)^2 +Im\big(S(\textbf{n})\big)^2 \bigg\}\nonumber \\
&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{1}{2\pi L}\sum_{\textbf{n}\neq 0} \frac{e^{-(\pi n)^2/(\alpha L)^2}}{n^2}\bigg\{|\sum_iq_i\cos(\frac{2\pi}{L}\textbf{n} \cdot\textbf{r}_i)|^2\nonumber \\
&&{}+|\sum_iq_i\sin(\frac{2\pi}{L}\textbf{n}\cdot\textbf{r}_i)|^2\bigg\}\,.
\end{eqnarray} Similarly, the reciprocal space electrostatic force on particle $i$ is the negative derivative of the potential energy $\mathbf{U}^{E,K}$ respect to its position $\textbf{r}_i$ and is described as
\begin{eqnarray} \label{eq22}
\mathbf{F}_{i}^{E,K}&=&-\nabla_i\mathbf{U}^{E,K}\nonumber\\
&=&-\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{1}{2\pi L}\sum_{\textbf{n}\neq0} \frac{e^{-(\pi n)^2/(\alpha L)^2}}{n^2}\left(\frac{4\pi q_i}{L}\textbf{n}\right) \nonumber \\
&&{}\bigg\{-\sin(\frac{2\pi}{L}\textbf{n}\cdot\textbf{r}_i)\sum_j q_j\cos(\frac{2\pi}{L}\textbf{n}\cdot\textbf{r}_j)\nonumber\\
&&{}+\cos(\frac{2\pi}{L}\textbf{n}\cdot\textbf{r}_i)\sum_j q_j\sin(\frac{2\pi}{L}\textbf{n}\cdot\textbf{r}_j)\bigg\} \nonumber \\
&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{2q_i}{L^2}\sum_{\textbf{n}\neq0} \textbf{n}\frac{e^{-(\pi n)^2/(\alpha L)^2}}{n^2}\bigg\{\sin(\frac{2\pi}{L}\textbf{n} \cdot\textbf{r}_i)Re\big(S(\textbf{n})\big)\nonumber \\
&&{}+\cos(\frac{2\pi}{L}\textbf{n}\cdot\textbf{r}_i)Im\big(S(\textbf{n})\big)\bigg\}\,.
\end{eqnarray} Since the structure factor $S(\textbf{n})$ is a complex number, the expression in the bracket of Eq.~\ref{eq22} can be written as the imaginary part of a product
\begin{eqnarray}\label{eq23}
\sin(\frac{2\pi}{L}\textbf{n}\cdot\textbf{r}_i)Re\big(S(\textbf{n})\big) + \cos(\frac{2\pi}{L}\textbf{n}\cdot\textbf{r}_i)Im\big(S(\textbf{n})\big) &=& Im\bigg\{e^{\frac{2\pi}{L}\imath\textbf{n}\cdot\textbf{r}_i}S(\textbf{n})\bigg\}\,.
\end{eqnarray} Following this expression, Eq.~\ref{eq22} can be expressed with
\begin{eqnarray} \label{eq24}
\mathbf{F}_{i}^{E,K}&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{2q_i}{L^2} \sum_{\textbf{n}\neq0}\textbf{n}\frac{e^{-(\pi n)^2/(\alpha L)^2}}{n^2} Im\bigg\{e^{\frac{2\pi}{L}\imath\textbf{n}\cdot\textbf{r}_i}S(\textbf{n})\bigg\} \nonumber\\
&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{2q_i}{L^2} Im\bigg\{\sum_{\textbf{n}\neq0}\textbf{n} \frac{e^{-(\pi n)^2/(\alpha L)^2}}{n^2}S(\textbf{n})e^{\frac{2\pi}{L}\imath\textbf{n}\cdot\textbf{r}_i}\bigg\}\nonumber\\
&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{2q_i}{L^2}Im\bigg\{\sum_{\textbf{n}\neq0} \hat{\textbf{g}}_{\textbf{n}}
e^{2\pi \imath\textbf{n}\cdot\textbf{x}_i}
\bigg\}\,,
\end{eqnarray} in which $\hat{\textbf{g}}_{\textbf{n}}=\textbf{n}\frac{e^{-(\pi n)^2/(\alpha L)^2}}{n^2}S(\textbf{n})$ with $\textbf{n}\neq0$. Again, Eq.~\ref{eq24} can be considered as a three-dimensional case of conjugated FFT form. Assuming $\textbf{n}\in I_M$ and $\hat{\textbf{g}}_0=0$, we can reformulate Eq.~\ref{eq24} into Fourier terms
\begin{eqnarray} \label{eq25}
\mathbf{F}_{i}^{E,K}&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{2q_i}{L^2} Im\bigg\{\sum_{\textbf{n}\in I_M}\hat{\textbf{g}}_{\textbf{n}}e^{2\pi \imath \textbf{n} \cdot\textbf{x}_i}\bigg\}\nonumber\\
&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{2q_i}{L^2}Im(\textbf{g}_i)\,.
\end{eqnarray} Thus, one can calculate the reciprocal space electrostatic force on particle $i$ using conjugated FFT based on the $S(\textbf{n})$ obtained from transposed FFT.
\par
From the equations shown above, it is clear that by using suitable NFFT techniques, the reciprocal space summations of both electrostatic energy and force can be calculated from ENUF method. The main features of ENUF method are listed in Table~\ref{table:algorithm}.
\section{The ENUF-DPD method}\label{sec:enufdpd}
In the traditional DPD formulation, the conservative force $\mathbf{F}_{ij}^C$ between interacting particles $i$ and $j$ is a short-range repulsive force, modeling the soft nature of neutral DPD particles. The electrostatic interactions between charged particles are long-range and conservative. In the ENUF-DPD simulations, the long-range electrostatic forces and short-range conservative forces are combined together to determine the thermodynamic behavior of the simulated systems~\cite{groot1997dissipative}.
\par
When the electrostatic interactions are included in DPD method, the main problem is that DPD particles with opposite charges show a tendency to collapse onto each other, forming artificial ionic clusters due to the soft nature of short-range repulsive interactions between DPD particles. In order to avoid this, the point charges at the center of DPD particles should be replaced by charge density distributions meshed around particles. Groot~\cite{groot2003electrostatic} firstly smeared out the local point charges around regular grids, and then adopted a variant of PPPM method to solve the near field and far field equations on grids instead of using FFT technique. Gonz\'alez-Melchor~\emph{et al.}~\cite{gonzalez2006electrostatic} directly adopted the Slater-type charge density distribution and traditional Ewald summation method to calculate the electrostatic interactions in DPD simulations. In our ENUF-DPD method, similar Slater-type charge density distribution and Gaussian type window functions in NFFT are adopted, respectively, to calculate real and reciprocal space contributions of electrostatic interactions.
\par
The detailed procedure of deducing electrostatic energy and force between two Slater-type charge density distributions are described as follows. The electrostatic potential field $\phi(r)$ generated by the Slater-type charge density distribution $\rho_e(r)$ can be obtained by solving the Poisson's equation
\begin{eqnarray} \label{eq:poisson_eq}
\nabla^2\phi(r) = -\frac{1}{\epsilon_0\epsilon_r}\rho_e(r)\,,
\end{eqnarray} In spherical coordinates, the Poisson's equation becomes
\begin{eqnarray} \label{eq:poisson_sph}
\frac{1}{r^2}\frac{\partial}{\partial r}\big(r^2\frac{\partial }{\partial r}\phi(r)\big) &=& -\frac{1}{\epsilon_0\epsilon_r}\rho_e(r) \,.
\end{eqnarray} For the Slater-type charge density distribution in Eq.~\ref{eq26}, we define parameter $c=-\frac{2}{\lambda_e}$ for the convenience of integration. Multiply $r^2$ at both sides of Eq.~\ref{eq:poisson_sph} and then integrate this equation, we can get
\begin{eqnarray} \label{eq:poisson_intea}
r^2\frac{\partial}{\partial r}\phi(r)&=&
-\frac{1}{\epsilon_0\epsilon_r}\frac{q}{\pi\lambda_e^3}
\int_0^r(r')^2e^{cr'}dr \nonumber\\
&=&-\frac{1}{\epsilon_0\epsilon_r}\frac{q}{\pi\lambda_e^3}\Big[\big(\frac{r^2}{c}-\frac{2r}{c^2} +\frac{2}{c^3}\big)e^{cr}-\frac{2}{c^3}\Big] \,.
\end{eqnarray} By dividing $r^2$ at both sides of Eq.~\ref{eq:poisson_intea}, the potential field can be integrated analytically to give
\begin{eqnarray}\label{eq:potential_field}
\phi(r)&=&-\frac{1}{\epsilon_0\epsilon_r}\frac{q}{\pi\lambda_e^3}\int\Big[\big(\frac{1}{c}
-\frac{2}{c^2r}+\frac{2}{c^3r^2}\big)e^{cr}-\frac{2}{c^3r^2}\Big]dr \nonumber\\
&=&-\frac{1}{\epsilon_0\epsilon_r}\frac{q}{\pi\lambda_e^3}
\Big[\big(\int\frac{1}{c}e^{cr}dr\big)+\big(\int(\frac{2}{c^3r^2}-\frac{2}{c^2r})e^{cr}dr\big)
-\big(\int\frac{2}{c^3r^2}dr\big)\Big] \nonumber\\
&=&-\frac{1}{\epsilon_0\epsilon_r}\frac{q}{\pi\lambda_e^3}
\Big[\frac{1}{c^2}e^{cr}-\frac{2}{c^3r}e^{cr}+\frac{2}{c^3r}\Big] \,.
\end{eqnarray} With the definition of $c=-\frac{2}{\lambda_e}$, we can reformulate Eq.~\ref{eq:potential_field} as
\begin{eqnarray}
\phi(r)&=&-\frac{1}{\epsilon_0\epsilon_r}\frac{q}{\pi\lambda_e^3}\big(\frac{\lambda_e^2}{4}e^{\frac{-2r}{\lambda_e}}+\frac{\lambda_e^3}{4r}e^{\frac{-2r}{\lambda_e}}-\frac{\lambda_e^3}{4r} \big) \nonumber\\
&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{q}{r}\big(1-(1+\frac{r}{\lambda_e})e^{\frac{-2r}{\lambda_e}}\big)\,.
\end{eqnarray}
\par
The electrostatic energy between interacting particles $i$ and $j$ is the product of the charge of particle $i$ and the potential field generated by particle $j$ at position $r_i$
\begin{eqnarray}\label{eq:coulomb_energy}
\mathbf{U}_{ij}^{E,DPD}(r_{ij})&=&q_i\phi_j(r_i) \nonumber\\
&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{q_iq_j}{r_{ij}}
\big(1-(1+\frac{r_{ij}}{\lambda_e})e^{\frac{-2r_{ij}}{\lambda_e}}\big)\,.
\end{eqnarray} By defining dimensionless parameters $r^*=r/R_c$ as the reduced center-to-center distance between two charged DPD particles and $\beta=R_c/\lambda_e$, respectively, the reduced electrostatic energy between two Slater-type charge density distributions is given by
\begin{eqnarray}\label{eq27}
\mathbf{U}_{ij}^{E,DPD}(r_{ij}^*)&=&\frac{1}{4\pi\epsilon_0\epsilon_r}
\frac{q_iq_j}{R_cr_{ij}^*}\bigg\{1-\big(1+\beta r_{ij}^*\big)e^{-2\beta r_{ij}^*}\bigg\}\,.
\end{eqnarray}
\par
The electrostatic force on charged particle $i$ is
\begin{eqnarray}\label{eq:electro_force}
\mathbf{F}_{i}^{E,DPD}(r_{ij})&=&-\nabla_i\mathbf{U}_{ij}^{E,DPD}(r_{ij})\nonumber\\
&=&-\frac{q_iq_j}{4\pi\epsilon_0\epsilon_r}\bigg\{\nabla_i\Big[\frac{1}{r_{ij}} \big(1-(1+\frac{r_{ij}}{\lambda_e})e^{\frac{-2r_{ij}}{\lambda_e}}\big)\Big]\bigg\} \nonumber \\
&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{q_iq_j}{(r_{ij})^2}\bigg\{1-e^{\frac{-2r_{ij}}{\lambda_e}}-\frac{2r_{ij}}{\lambda_e}e^{\frac{-2r_{ij}}{\lambda_e}}-\frac{2r_{ij}^2}{\lambda_e^2}e^{\frac{-2r_{ij}}{\lambda_e}}
\bigg\}\,.
\end{eqnarray} With two dimensionless parameters $\beta$ and $r^*$, the magnitude of the reduced electrostatic force is
\begin{eqnarray}\label{eq28}
\mathbf{F}_{i}^{E,DPD}(r_{ij}^*)
&=&\frac{1}{4\pi\epsilon_0\epsilon_r}\frac{q_iq_j}{(R_cr_{ij}^*)^2}
\bigg\{1-\Big(1+2\beta r_{ij}^*(1+\beta r_{ij}^*)\Big)e^{-2\beta r_{ij}^*}\bigg\}\,.
\end{eqnarray} Comparing Eq.~\ref{eq15} with Eq.~\ref{eq27}, we can find that the electrostatic energy between two charges is scaled with correction factor $B_1=1-(1+\beta r^*)e^{-2\beta r^*}$ when the Slater-type charge density distributions are introduced in DPD simulations. Similarly, the electrostatic force between two charged particles is scaled with correction factor $B_2=1-\Big(1+2\beta r^*(1+\beta r^*)\Big)e^{-2\beta r^*}$ in DPD simulations.
\par
In the limit of $r_{ij}^* \to 0$, the reduced electrostatic energy and force between two charge density distributions are described by $\lim\limits_{r_{ij}^* \to 0}\mathbf{U}_{ij}^{E,DPD}(r_{ij}^*) = \frac{1}{4\pi\epsilon_0\epsilon_r} \frac{q_iq_j}{R_c}\beta$ and $\lim\limits_{r_{ij}^*\to 0}\mathbf{F}_{ij}^{E,DPD}(r_{ij}^*) = 0$, respectively. It is clear that the adoption of Slater-type charge density distributions in DPD simulations removes the divergence of electrostatic interactions at $r_{ij}^* =0$, which means that both electrostatic energy and force between two charged particles are finite quantities. By matching the electrostatic interactions between two charge density distributions at $r_{ij}^*=0$ with previous work~\cite{groot2003electrostatic} gives $\beta=1.125$. From the relation of $\beta=R_c/\lambda_e$, we can get $\lambda_e = 6.954~\textrm{\AA}$, which is consistent with the electrostatic smearing radii used in Gonz\'alez-Melchor's work~\cite{gonzalez2006electrostatic}.
\par
Fig.~\ref{figure:fig1} shows the representation of the reduced electrostatic energy and corresponding force with respect to the distance between two charged DPD particles. Both the electrostatic energy and force are calculated using ENUF-DPD method and Ewald summation method with Slater-type charge density distributions and reference parameters. For comparison, we also include the standard Coulombic potential and corresponding force, both of which do diverge at $r=0$. The electrostatic energy and force calculated using ENUF-DPD method are almost the same as those calculated using Ewald summation method with reference parameters. The positions of the maximum value of the electrostatic energy in both methods are basically the same, but the maximum value calculated from ENUF-DPD method is slightly smaller than that from the standard Ewald summation method within an acceptable statistical error. Comparing with the standard Coulombic potential and corresponding force, we find that both ENUF-DPD and Ewald summation methods can give indistinguishable energy and force differences at $r\geq 3.0R_c$. Hence, the ENUF method can capture the essential character of electrostatic interactions, as well as the Ewald summation method, at mesoscopic level~\cite{gonzalez2006electrostatic}.
\par
Combining the electrostatic force $\mathbf{F}_{ij}^{E,DPD}$ and the soft repulsive force $\mathbf{F}_{ij}^C$ gives the total conservative force $\mathbf{F}_{ij}^{C*}$ between interacting particles $i$ and $j$ in DPD simulations. The total conservative force $\mathbf{F}_{ij}^{C*}$, together with dissipative force $\mathbf{F}_{ij}^D$ and random force $\mathbf{F}_{ij}^R$, as well as the intramolecular bonding force $\mathbf{F}_i^S$ for polymers and surfactants, act on DPD particles and evolve toward equilibrium conditions before taking statistical analysis. The details of the update scheme for ENUF-DPD method in single integration step are shown in Table~\ref{table:algorithm}.
\section{The choice of ENUF-DPD parameters}\label{sec:validation}
As the number of charged DPD particles in the simulated system grows, the calculation of the reciprocal space electrostatic interactions will become the most time-consuming part. Using the suitable parameters in ENUF-DPD method assures that the time to calculate the real space summations is approximately the same as the time to calculate the reciprocal space summations, thereby reducing the total computational time. Herein we try to explore the ENUF-DPD related parameters and get a set of suitable parameters for following applications.
\par
The implementation of ENUF-DPD method uses the Ewald convergence parameter $\alpha$, required accuracy $\delta (\ll 1)$, and two cut-offs ($r_c$ for real space and $n_c$ for reciprocal space). These parameters are correlated with each other with the following two conditions
\begin{align}\label{eq29}
&e^{-\pi^2|\textbf{n}|^2/(\alpha L)^2}\leq\delta \Longrightarrow n_c\geq \frac{\alpha L}{\pi}\sqrt{-\log(\delta)}\Longrightarrow n_c\propto L\propto N^{1/3}\,, \nonumber \\
&\textrm{erfc}(\alpha r_c)\approx e^{-\alpha^2r_c^2}\leq\delta\Longrightarrow r_c\approx \frac{\pi n_c}{\alpha^2L}\,.
\end{align} With required $\delta$, it is more convenient to pick a suitable value for $n_c$. Then one can determine $\alpha$ and $r_c$ directly from Eq.~\ref{eq29}. However, due to the fact that $n_c$ should be integer and $r_c$ should be a suitable value for the cell-link list update scheme in DPD simulations, we adopt another procedure to get suitable parameters.
\par
First, we choose suitable $\delta$. It has been demonstrated that $\delta = 5.0\times10^{-5}$ is enough to keep acceptable accuracy in ENUF method~\cite{hedman2006ewald}. In DPD simulations, due to the soft repulsive feature of the conservative force $\mathbf{F}^C$ in Eq.~\ref{eq0305}, we adopt $\delta = 1.0\times10^{-4}$ in our ENUF-DPD method.
\par
Secondly, we determine suitable $r_c$ and $\alpha$. Gonz\'alez-Melchor~\emph{et al.}~\cite{gonzalez2006electrostatic} adopted $1.08 R_c$ and $3.0R_c$, respectively, as electrostatic smearing radii and real space cut-off for the calculation of electrostatic interactions with Ewald summation method. In our ENUF-DPD method, as specified in Eqs.~\ref{eq27} and~\ref{eq28}, electrostatic energy $\mathbf{U}^{E,DPD}$ and force $\mathbf{F}^{E,DPD}$ are scaled with two correction factors, $B_1$ and $B_2$, respectively, both of which are $r$-dependent. This means that the reciprocal space summations of electrostatic energy $\mathbf{U}^{E,K,DPD}$ and force $\mathbf{F}^{E,K,DPD}$ are also scaled with corresponding correction factors. It should be noted that what we obtain from FFT is the total influences of other particles on particle $i$. It is difficult to differentiate individual contribution since each corresponding correction factor is related to the relative distance between interacting particles. But if we choose suitable $r_c$, beyond which two correction factors $B_1$ and $B_2$ approximate to $1.0$, the total reciprocal space summations of electrostatic energy $B_1\mathbf{U}^{E,K,DPD}$ and force $B_2\mathbf{F}^{E,K,DPD}$ can be approximately expressed by $\mathbf{U}^{E,K,DPD}$ and $\mathbf{F}^{E,K,DPD}$, respectively. Such approximation enables us to adopt directly the FFT results as reciprocal space summations. In Fig.~\ref{figure:fig2}, we plot two correction factors $B_1$ and $B_2$ with respect to the distance $r$. It is clearly shown that both $B_1$ and $B_2$ approximate to 1.0 when $r \ge 3.0 R_c$. Hence in our simulations, $r_c = 3.0 R_c$ is taken as the cut-off for real space summations of electrostatic interactions. With such adoption, both $B_1$ and $B_2$ are only applied on real space summations of electrostatic interactions within cut-off $r_c =3.0 R_c$.
\par
Since the Fourier-based Ewald methods utilize the FFT technique to evaluate the reciprocal part summations, it is more appropriate to choose suitable $\alpha$, with which we can minimize the total computational time in calculating electrostatic interactions. A large value of $\alpha$ means that a small value of $r_c$ is used for rapid convergence in real space summations, but the reciprocal space calculations will be the more time-consuming part and vice versa. The choice of $\alpha$ is system-dependent and related to the trade-offs between accuracy and computational speed. Based on Eq.~\ref{eq29} and above determined $r_c=3.0R_c$, we deduce that $\alpha\geq 0.12~\textrm{\AA}^{-1}$. Although the electrostatic energy is invariant to the choice of $\alpha$, the $\alpha$ value indeed affects the total time in calculating electrostatic interactions. In order to find a suitable value for $\alpha$, we carry out our first simulation to evaluate the Madelung constant ($M$) of a face-centered cubic (FCC) lattice. The crystal structure is composed of $4000$ charged particles, half of which are cations with net charge $+1$ and the other half are anions carrying net charge $-1$. All ions are located on regular lattice of FCC structure. The electrostatic energy for the FCC crystal structure is calculated within single step and then used to determine Madelung constant $M$. Simulation details are listed in Table~\ref{table:test-detail}. By comparing our calculated $M$ values with the theoretical value in Ref.~\cite{lu2003computer}, we find that for a wide range of $\alpha$ values the calculated $M$ coincides with literature value. The lowest acceptable value, $\alpha=0.20~\textrm{\AA}^{-1}$, is then used in the following simulations to minimize the computational effort.
\par
Finally, with Eq.~\ref{eq29} and above described parameters, we can choose suitable value for $n_c$. For NFFT technique, there are two other parameters, $\sigma_s$ and $p$, controlling the approximation errors. It has been shown that for a fixed over-sampling factor $\sigma_s>1$, the error decays exponentially with $p$~\cite{hedman2006ewald, dutt1993fast}. In atomistic MD simulations, it has been shown that $\sigma_s=2$ is adequate to take enough samples~\cite{hedman2006ewald, hedman2006thesis}. Hence $\sigma_s=2$ is used in our ENUF-DPD simulations to keep a good accuracy.
\par
Since ENUF approximates only the reciprocal space summations of the standard Ewald summation, it is reasonable to expect that both the ENUF and the Ewald summation methods should behave in the same way at mesoscopic level. We then perform the second set of simulations on bulk electrolyte system to determine parameters, $n_c$ and $p$. The volume for each simulation is $V=(10R_c)^3$ with the total number of charged DPD particles $N=4000$, in which $2000$ charged DPD particles represent cations with net charge $+1$ and the same number of DPD particles carrying net charge $-1$ that represent the anions. All simulations are equilibrated for $1\times 10^4$ steps and then another $2\times10^4$ steps to take statistical average for the following analysis. Detailed simulation information are listed in Table~\ref{table:test-detail}.
\par
In Fig.~\ref{figure:fig3}, we present the relative errors of electrostatic energies calculated using ENUF-DPD and Ewald summation methods with explored parameters $n_c$ and $p$, as well as Ewald summation method with reference parameters. It is clear that as $n_c$ increases, the relative errors for electrostatic energy from Ewald summation method converge to $1.0\times10^{-4}$, which is the acceptable accuracy we set at first. For the ENUF-DPD method, the relative electrostatic energy errors generally decrease with the increase of $p$. When $p=1$, the relative errors from EUNF-DPD method converge to $2\times 10^{-3}$ for large $n_c$, indicating that the ENUF-DPD method with $p=1$ is too crude due to the fact that $p$ is not adequate to provide enough sampling terms in simulations. By increasing $p$, the relative errors reduce rapidly with the increase of $n_c$. In Fig.~\ref{figure:fig4}(a), we show three cases of the electrostatic energy errors calculated from ENUF-DPD method with $p=2$ and different $n_c$ values, as well as those from the Ewald summation method with same simulation parameters. It is obvious that for parameters $n_c\geq7$, the electrostatic energy errors, calculated from both ENUF-DPD and Ewald summation methods, fluctuate within $1\times10^{-4}$. This means that the electrostatic energy errors from ENUF-DPD and Ewald summation method with $p=2$ and $n_c=7$ are expected to be practically negligible in comparison with those calculated from traditional Ewald summation method with reference parameters, which gives the most precise electrostatic energies. Similar tendencies are also observed in the electrostatic force errors calculated from ENUF-DPD and Ewald summation methods with the same set of parameters, as shown in Fig.~\ref{figure:fig4}(b). In Fig.~\ref{figure:fig4}(c) we show the maximum errors in electrostatic forces. It is clear that as $p=2$, the increase of $n_c$ will enable ENUF-DPD method to be more and more similar to Ewald summation method. With larger $p$ values, such as $p=3$, we could further increase the accuracy of electrostatic interactions in ENUF-DPD method, but at the same time the total computational time in treating electrostatic interactions increases. By compromising the accuracy and computational speed in the ENUF-DPD method, we adopt $p=2$ and $n_c=7$ in following simulations.
\par
Now we perform the third set of simulations to study the structural properties of the electrolyte solution using Ewald summation method with reference parameters, ENUF-DPD and Ewald summation methods with above determined parameters, respectively. The system consists of $N=4000$ DPD particles in a simulation box of volume $V=(10R_c)^3$. The solvents mimicking water at room temperature are represented by $3736$ neutral DPD particles, and $132$ particles representing ions with net charge $+1$ and the same number of particles with net charge $-1$ representing counterions are also included in the simulations. Mapping these quantities to real units, the simulated system corresponds to dilute electrolyte solution with a salt concentration about $0.6$ M, which is consistent with the simulation condition in Refs.~\cite{groot2003electrostatic, gonzalez2006electrostatic}. Detailed simulation information can be found in Table~\ref{table:test-detail}.
\par
The structural properties of neutral solvents and charged particles are determined by the radial distribution functions (RDFs). The RDFs of neutral solvent-solvent, equal sign ions, and unequal sign ions are calculated from traditional Ewald summation with reference parameters, the Ewald summation and ENUF-DPD method with above determined parameters, as shown in Fig.~\ref{figure:fig5}. It is clear that the RDFs for the same pair particles calculated from three methods show similar tendencies. A general observation is that there is no ionic cluster formation at distance close to $r=0$. Furthermore, we also find that the RDFs between charged particles satisfy $g_{+-}(r)g_{++/--}(r)=g^2_{00}(r)$, where $+$, $-$, and $0$ correspond to positive charged, negative charged, and neutral DPD particles in simulations. This relationship between three RDFs implies that the structures between charged particles are related to the effective electrostatic interparticle potentials, as observed in Ref.~\cite{groot2003electrostatic}.
\par
Another important property of the ENUF-DPD method is its complexity in treating electrostatic interactions. Theoretically, the complexity for treating electrostatic interactions with FFT related technique is $\mathcal{O}(N\log N)$~\cite{toukmaji1996ewald}. For NFFT, the computational complexity is $\mathcal{O}(M_{\Pi}\log M_{\Pi} + \log(N/\delta))$, where $M_{\Pi}$ is the total number of points in the index set, and $\delta$ is the desired accuracy and also a function of $p$ for fixed over-sampling factor $\sigma_s$ in NFFT~\cite{press1989fast, dutt1993fast}. Combining the definition of $M_{\Pi}$ and the relationship in Eq.~\ref{eq29}, we can get $M_{\Pi}\propto n_c^3\propto N$, hence the theoretical complexity of ENUF-DPD method is $\mathcal{O}(N\log N+\log(N/\delta))$.
\par
The scaling behavior of the ENUF-DPD method, as well as the Ewald summation method, including its traditional version and the one with suitable parameters we have explored above, are estimated by varying the number of charged DPD particles in simulations. In each simulation, the total number of DPD particles is $N=32000$ with the volume $V=(20R_c)^3$. Initially, all DPD particles are neutral, and corresponding computational time is taken as the benchmark for DPD simulations without electrostatic interactions. Then we perform a number of different simulations increasing the number of charged particles up to $24000$ while keeping the simulation system neutral and the total number of DPD particles fixed. The detailed simulation information can be found in Table~\ref{table:test-detail}. Fig.~\ref{figure:fig6} shows the averaged run time per $10^3$ steps of DPD simulation as function of the number of charged particles. The averaged execution time per $10^3$ steps is the net time in calculating the electrostatic energy and force in each DPD simulation. It reveals that the original Ewald summation method with reference parameters could generate accurate electrostatic energy and force, but its computational complexity is $\mathcal{O}(N^2)$. With our above determined suitable parameters, the scaling behavior of Ewald summation method is reduced to $\mathcal{O}(N^{3/2})$ in general. Although the parameters for Ewald summation method in our simulations are a little different from the parameters used by Gonz\'alez-Melchor~\emph{et al.}~\cite{gonzalez2006electrostatic}, the computational complexities are described by similar scaling behavior $\mathcal{O}(N^{3/2})$, which has also been verified by Ibergay~\emph{et al.}~\cite{ibergay2009electrostatic}. The ENUF-DPD method scales as $\mathcal{O}(N\log N)$, which is in line with the scaling behavior of PPPM method adopted in Groot's work~\cite{ibergay2009electrostatic, groot2003electrostatic}.
\par
The ENUF-DPD method with above explored parameters shows an excellent computational efficiency and good accuracy in treating electrostatic interactions. In the current study of electrolyte solution and the range of the number of ion pairs investigated here, the ENUF-DPD method performs clearly much faster than the Ewald summation method, and shows similar $\mathcal{O}(N\log N)$ scaling behavior as PPPM method~\cite{ibergay2009electrostatic, groot2003electrostatic} at mesoscopic level.
\section{Conformation of polyelectrolyte in solution}\label{sec:pe}
The above implemented ENUF-DPD method has all capabilities of ordinary DPD, but includes applications where electrostatic interactions are essential but previously inaccessible. One key example is the polyelectrolyte conformation. Electrostatic interactions between charged particles on polyelectrolyte lead to rich conformation of polyelectrolyte qualitatively different from those of uncharged polymers~\cite{dobrynin2005theory, jusufi2009colloquium}. In this section, we use the ENUF-DPD method to study the charge fraction of polyelectrolyte on the conformational behavior of single polyelectrolyte molecule.
\par
Nine charge fractions, defined by $f=N_q / N_p$, in which $N_q$ and $N_p$ are, respectively, the number of charged particles and the total number of particles on polyelectrolyte, are considered. In our simulations, $N_p=48$ is used. Although this generic polyelectrolyte model is much smaller than the real ones, the essential physical effects are still captured~\cite{dobrynin2005theory, de1995precipitation, liu2003polyelectrolyte, kirwan2004imaging}. Going from a neutral polymer chain to a fully ionized polyelectrolyte, $f$ takes $0.0$, $0.125$, $0.25$, $0.375$, $0.5$, $0.625$, $0.75$, $0.875$ and $1.0$, which correspond to $N_q=0$, $6$, $12$, $18$, $24$, $30$, $36$, $42$ and $48$ charged particles on polyelectrolyte, respectively. Each charged particle on polyelectrolyte is characterized by net charge $-1$. Counterions carrying net charge $+1$ are added to preserve charge neutrality of simulation systems. The equilibrium bond distance between bonded particles is set to $r_{eq}=0.7R_c$ and the spring constant is taken as $K^S=64.0k_BT$. The conservative interaction parameters between different types of DPD particles are obtained through $\alpha_{ij} \approx \alpha_{ii}+2.05\chi$ with $\alpha_{ii}=78.67k_BT$. The $\chi$ parameter between polyelectrolyte and solvents is set to $0.87$, which is rescaled from Refs.~\cite{groot2003electrostatic, gonzalez2006electrostatic}.
\par
All simulations are performed in a simulation cell with volume $V = (30R_c)^3$. The total density is fixed at $\rho=4$, and hence the total number of particles in the system is $N=108000$ in all cases. All simulations are equilibrated in $5 \times 10^4$ time steps, and then $2.5 \times 10^5$ time step simulations are further performed to collect statistical data. For a fully ionized polyelectrolyte, additional simulations with larger volume $V = (40R_c)^3$ are also performed. No differences beyond statistical uncertainties are found between the two sets of simulations. In the following discussion, all simulation results are calculated from systems with volume $V = (30R_c)^3$. Simulation details are listed in Table~\ref{table:test-detail}.
\par
The averaged radius of gyration $<$$R_g$$>$ of polyelectrolyte, as function of corresponding charge fraction $f$, are shown in Fig.~\ref{figure:fig7}. Experimentally, it is well known that a neutral polymer in solution has the smallest radius of gyration~\cite{stigter1995theory, roiter2005afm, liao2006counterion}. When partial groups on polyelectrolyte are ionized, such as weakly charged polyelectrolyte in solution with adjustable pH values, the $<$$R_g$$>$ of polyelectrolyte increases with increasing degree of ionization of polyelectrolyte~\cite{roiter2005afm}. For fully ionized polyelectrolyte, the value of $<$$R_g$$>$ is approximately $1.31$ times larger than that of neutral polymer, which is consistent with molecular simulation results~\cite{gonzalez2006electrostatic} and experimental observations~\cite{roiter2005afm}. Typical conformations of polyelectrolyte with charge fraction $f=0.0$, $0.25$, $0.5$, $0.75$ and $1.0$ are shown in Fig.~\ref{figure:fig8}. These simulation results are qualitatively consistent with experimental observations~\cite{kirwan2004imaging, roiter2005afm} and theoretical predictions~\cite{stigter1995theory, liao2006counterion} for weakly charged polyelectrolyte.
\par
With the increase of charge fraction $f$ on polyelectrolyte, the RDFs between charged particles on polyelectrolyte and monovalent counterions are also enhanced, as shown in Fig.~\ref{figure:fig9}(a). In Fig.~\ref{figure:fig9}(b), we show the intramolecular pair correlation functions between charged particles of polyelectrolyte. A quantitative measure of the structure is found by analyzing the intramolecular pair correlation functions. For neutral polymer and polyelectrolyte with various charge fractions, the intramolecular correlations in initial zone $r/R_c < 1$ are dominated by particle-particle repulsions. In the regime $r/R_c >1$, two striking tendencies are shown in Fig.~\ref{figure:fig9}(b). For polyelectrolyte with small charge fraction, \emph{i.e.}, $f < 0.25$, we observe a small scaling-like domain and then followed by a terminal correlation range. In contrast, polyelectrolyte with large charge faction shows a scaling behavior over the entire range. The slope of the fitting in Fig.~\ref{figure:fig9}(b) is $-1.92$, which is in good agreement with Groot's results~\cite{groot2003electrostatic}.
\par
When salts are added into solution, both ionic strength and valency of multivalent counterions of added salts can severely influence the conformational properties of polyelectrolyte due to the strong correlations between multivalent counterions and polyelectrolyte. This behavior is usually specified as the overcharging phenomenon that occurs in many biological and synthetic polyelectrolytes~\cite{sanders2005structure}. Herein, the ENUF-DPD method is further adopted to investigate the effects of ionic strength and counterion valency of added salts on the conformation behavior of fully ionized polyelectrolyte.
\par
The number of multivalent counterions of added salts, $N_c$, is determined by $\theta=qN_c/N_p$, where $\theta$ is the ratio between the total charge of multivalent counterions of added salts and that of polyelectrolyte, and $q$ is the valency of added salts. In our simulations, various $\theta$ values, together with $q = 1$, $2$, and $3$, are selected to consider the dependence of fully ionized polyelectrolyte conformation on the ionic strength and valency of multivalent counterions of added salts. All simulations are equilibrated in $5 \times 10^4$ time steps, and $2.5 \times 10^5$ time step simulations are further performed to collect statistical data. Detailed simulation information are listed in Table~\ref{table:test-detail}.
\par
Fig.~\ref{figure:fig10} shows the dependence of $<$$R_g$$>$ on $\theta$ and valency of multivalent counterions of added salts. In the absence of added salts, polyelectrolyte adopts an extended conformation, owing to the electrostatic repulsions between charged particles of polyelectrolyte. Upon addition of salts, these repulsions are screened and hence $<$$R_g$$>$ decreases. For monovalent counterions ($q = 1$), $<$$R_g$$>$ gradually decreases. By contrast, stronger decreases in $<$$R_g$$>$ are observed in the cases of added salts with divalent ($q = 2$) and trivalent counterions ($q = 3$), occurring at considerably lower ionic strength. This reflects the conformational collapse of polyelectrolyte in solution with multivalent counterions, which has been observed in experiments~\cite{liu2003polyelectrolyte, roiter2010single} and predicted by coarse-grained molecular dynamic simulations~\cite{hsiao2006salt}.
\par
The smallest value of $<$$R_g$$>$ occurs near $c_Z$, \emph{i.e.}, the (Z:$1$) salt concentration at which the total charge of the Z-valent counterions of added salts neutralizes the bare polyelectrolyte. At the same time, polyelectrolyte shows compact conformation. Accordingly, this compact state occurs at a salt concentration that decreases with the increase of corresponding counterion valency, which is consistent with the two-state model~\cite{solis2000collapse}. In addition, multivalent counterions with higher valency are strongly correlated with polyelectrolyte, which can be specified by corresponding pair correlation functions. The RDFs among polyelectrolyte, monovalent counterions of polyelectrolyte, and multivalent counterions of added salts at $\theta=1.0$, are calculated and shown in Fig.~\ref{figure:fig11}. In solutions, monovalent counterions of polyelectrolyte and multivalent counterions of added salts show different condensation abilities on polyelectrolyte. For ($1$:$1$) salt, two kinds of counterions show similar tendencies due to the same amount of net charge on them. With the increase of counterion valency of added salts, counterions with different valency show competition in condensating the polyelectrolyte. The peak of RDF between trivalent counterions and polyelectrolyte is much higher than that of other counterions, implying the strong condensation between trivalent counterions and polyelectrolyte. The strong condensation induced by electrostatic correlations decreases the osmotic pressure, and hence leads to the collapse of polyelectrolyte~\cite{mei2006collapse}. Typical conformations of polyelectrolyte, as well as added salt with the counterion valency $q=1$, $2$, and $3$, are shown in Fig.~\ref{figure:fig12}.
\par
A striking effect occurs once the salt concentration is increased beyond $c_Z$. Polyelectrolyte starts to swell, in close analogy with the redissolution observed for multichain aggregates~\cite{de1995precipitation}. Comparing with conformation of polyelectrolyte in ($1$:$1$) salt, which exhibits a slow, monotonic decrease of $<$$R_g$$>$ with the increase of salt concentrations, the slight swelling behavior of polyelectrolyte in the presence of multivalent counterions emphasizes the important role of counterion valency~\cite{liu2003polyelectrolyte, roiter2010single}.
\par
Concerning the effects of ionic strength and valency of added salts on polyelectrolyte conformation, the decay of correlations between charge particles can be specified by the Debye screening length. The addition of salts with multivalent counterions leads to a short Debye screening length~\cite{yan2009dissipative}, demonstrating that the electrostatic interactions between charge particles separated larger than specific distance become screened and hence are no longer long-range. It is very likely that a finite cut-off for electrostatic interactions, or a screened interaction potential between charge particles, can be used in handling electrostatic interactions in these cases. This topic is beyond the scope of present work, deserving a special attention and consideration in future.
\par
The other detailed application of the ENUF-DPD method is the specific binding structures of dendrimers on amphiphilic membranes~\cite{yonglei}. We construct mutually consistent coarse-grained models for dendrimers and lipid molecules, which can properly describe the conformation of charged dendrimers and the surface tension of amphiphilic membranes, respectively. Systematic simulations are performed and simulation results reveal that the permeability of dendrimers across membranes is enhanced upon increasing dendrimer sizes. The negative curvature of amphiphilic membrane formed in dendrimer-membrane complexes is related to dendrimer concentration. Higher dendrimer concentration together with the synergistic effect between charged dendrimers can also enhance the permeability of dendrimers across amphiphilic membranes. Detailed descriptions of this work are shown in Ref.~\cite{yonglei}.
\section{Conclusion}\label{sec:conclusion}
The ENUF method, which combines the traditional Ewald summation method with NFFT technique to calculate the electrostatic interactions in MD simulations, is incorporated in DPD method. The ENUF-DPD method is applied on simple model electrolyte systems to explore suitable parameters. With required accuracy parameter $\delta=1.0\times10^{-4}$ and cut-off $r_c=3.0R_c$ for real space summations of electrostatic interactions, we find that the Ewald convergence parameter $\alpha=0.20~\textrm{\AA}^{-1}$ can generate accurate Madelung constant for FCC lattice structure and keep considerable accuracy. Simulation results reveal that the ENUF-DPD method with approximation parameter $p=2$ in NFFT and cut-off $n_c=7$ for reciprocal space summations of electrostatic interactions can well describe the electrostatic energy and force, as well as the Ewald summation method does. The computational complexity of ENUF-DPD method is approximately described as $\mathcal{O}(N\log N)$, which shows remarkably better efficiency than the traditional Ewald summation method with acceptable accuracy in treating long-range electrostatic interactions between charged particles at mesoscopic level.
\par
The ENUF-DPD method is further validated by investigating the influence of charge fraction of polyelectrolyte on corresponding conformational properties. Meanwhile, the dependence of the conformations of fully ionized polyelectrolyte on ionic strength and valency of added salts are also studied. These applications, together with a separately published research work on the formation of dendrimer-membrane complexes, show that the ENUF-DPD method is very robust and can be used to study charged complex systems at mesoscopic level.
\section{Acknowledgments}
We gratefully acknowledge financial support from the Swedish Science Council (VR) and generous computing time allocation from SNIC. This work is subsidized by the National Basic Research Program of China ($973$ Program, $2012$CB$821500$), and supported by National Science Foundation of China ($21025416$, $50930001$).
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\section{Introduction}
\indent Antonov in 1962 \cite{Antonov} discovered an instability that later became known as Antonov's instability or gravothermal catastrophe \cite{Bell-Wood}. It signified the beginning of the study of thermodynamics of self-gravitating systems \cite{Padman,Chavanis,Katz,Chavanis2} and the nowadays rapidly developing field of statistical mechanics with long-range interactions \cite{Bell, Dauxois}. \\
\indent In the original formulation of Antonov's instability \cite{Antonov,Bell-Wood}, the system consists of particles (stars) bound by a spherical shell with insulating and perfectly reflecting walls. The particles interact only with Newtonian gravity (no cosmological constant or any relativistic effects are present) and the number of particles and the energy of the system are constant, i.e. the system is studied in the microcanonical ensemble. It was found that the entropy of the system has no global maximum and that for $ER/GM^2 < -0.335$ no equilibrium states exist. For $ER/GM^2 > -0.335$ there exist local entropy maxima, i.e. metastable states, only for density contrast below the critical value $(\rho_0/\rho_R)_{cr} = 709$, where $\rho_0$, $\rho_R$ are the central and edge densities, respectively. \\
\indent In gravity the ensembles are not equivalent. In the microcanonical ensemble, stable equilibria with negative specific heat do exist. This negative specific heat region is replaced by a phase transition in the canonical ensemble \cite{Padman}. The Antonov system in the canonical ensemble (with no cosmological constant), is studied in detail by Chavanis \cite{Chavanis}. He found that the free energy does not
have a global minimum and equilibria exist only for temperatures greater than the critical dimensionless temperature $(RT/GM)_{cr} = 0.40$ which corresponds to dimensionless inverse temperature $(GM\beta/R)_{cr} = 2.52$. These equilibria are stable only for density contrast less than the critical value $(\rho_0/\rho_R)_{cr} = 32.1$ (see Ref. \cite{Katz2}, as well). \\
\indent We studied the gravothermal instability with a cosmological constant $\Lambda$ in the microcanonical ensemble in Ref. \cite{we}. It was found that for a fixed energy, the maximum radius (which equals $R_{cr} = (-0.335/E)GM^2$ with no $\Lambda$) is increasing with increasing $\Lambda$, while for $\Lambda > 0$ there appears a second critical radius which is decreasing and the two radii merge at some marginal $\Lambda$. This was identified as a reentrant behaviour (see Ref.
\cite{reentrant1,reentrant2,reentrant3,reentrant4} for details on reentrant phase transitions in statistical systems with long-range interactions). In the present work, the system is placed in a heat bath and the walls are non-insulating, i.e. we study the canonical ensemble. In contrast to the microcanonical one, we find that the critical radius $R_{cr}$, for a fixed temperature and mass, is decreasing with increasing cosmological constant. In addition the instability occurs for $R<R_{cr}$ in the canonical ensemble, in contrast to the microcanonical one. This is evident in case $\Lambda = 0$, since then there exist no equilibria if $GM\beta/R > 2.52 \Rightarrow R < GM\beta/2.52$ for a fixed temperature and mass. A reentrant phase transition appears in the canonical ensemble, when one examines the system with respect to the temperature for a fixed radius. There appear two critical temperatures for $\Lambda > 0$. For the intermediate temperature values, no thermodynamic equilibria are possible.
\section{Analysis and results}
\indent In the Newtonian limit, the Poisson equation is modified \cite{Wald,Axenides} in the presence of a cosmological constant $\Lambda$ as:
\begin{equation}\label{eq:PoisL}
\nabla^2 \phi = 4\pi G \rho - 8\pi G \rho_\Lambda
\end{equation}
with $\rho_\Lambda = \frac{\Lambda c^2}{8\pi G}$. Thus, the gravitational potential can be decomposed to two parts
\begin{equation}
\phi = \phi_N + \phi_\Lambda
\end{equation}
where $\phi_N$ is the Newtonian and $\phi_\Lambda$ the cosmological potential, that are given by
\begin{equation}\label{eq:potentials}
\phi_N = -G\int{\frac{\rho(r')}{|\vec{r}-\vec{r}\, '|}d^3\vec{r}\, '} \; , \;
\phi_\Lambda = -\frac{4\pi G}{3}\rho_\Lambda r^2
\end{equation}
For convenience we call the Newtonian limit of de Sitter ($\rho_\Lambda>0$) and Anti-de Sitter ($\rho_\Lambda < 0$) spaces \cite{NewtonHook} just dS and AdS, respectively. \\
\indent We consider a self-gravitating gas of $N$ particles with unity mass inside a spherical non-insulating shell inside a heat bath and restrict only to spherical symmetric configurations. We work in the mean field approximation, for which the $N$-body distribution function is replaced by the one body distribution function $f(\vec{r},\vec{\upsilon})$. The Boltzmann entropy is defined as $S = -k\int f\ln f d^3\vec{r} d^3\vec{p}$. The Helmholtz free energy is equal to $F = E - TS$. Equivalently one can work with the Massieu function \cite{Chavanis,Katz2} $J = -F/T$ that gives
\begin{equation}
J = S-\frac{1}{T}E
\end{equation}
The maximization of $J$ with constant $T$ and the maximization of $S$ with constant $E$ (and constant $M$ in both cases), with respect to perturbations $\delta\rho$, are the same \cite{Chavanis} to first order in $\delta\rho$ , i.e. give the same equilibria, described by the Maxwell-Boltzmann distribution function
\begin{equation}
f = \left( \frac{\beta}{2\pi}\right)^{\frac{3}{2}}\rho(r)e^{-\frac{1}{2}\beta\upsilon^2}
\end{equation}
where
\begin{equation}\label{eq:rho}
\rho(r) = \rho_0 e^{-\beta(\phi - \phi(0))}
\end{equation}
This is proved in \cite{Chavanis} without a cosmological constant, and holds in the presence of $\Lambda$ as well, with the difference being that $\phi$ does not satisfy Poisson equation now, but equation (\ref{eq:PoisL}). The two ensembles have the same equilibria. However, the second variations of $J$ and $S$ are different. Therefore, what is different in the two ensembles is the turning point at which an instability sets in. This causes a great qualitative difference for the two ensembles. \\
\indent Introducing the dimensionless variables $y = \beta(\phi - \phi (0))$, $x = r\sqrt{4\pi G \rho_0\beta}$ and $\lambda = 2\rho_\Lambda/\rho_0$, and using equation (\ref{eq:rho}), equation (\ref{eq:PoisL}) becomes
\begin{equation}\label{eq:emdenL}
\frac{1}{x^2}\frac{d}{dx}\left( x^2\frac{d}{dx}y\right) = e^{-y} - \lambda
\end{equation}
called the Emden-$\Lambda$ equation. Let us call $z = R\sqrt{4\pi G \rho_0\beta}$ the value of $x$ at $R$. In order to generate the series of equilibria needed to study the stability of the system, the Emden-$\Lambda$ equation has to be solved with initial conditions $y(0)=y'(0)=0$, keeping $M$ constant and for various values of the parameters $\rho_\Lambda$, $\beta$, $\rho_0$. This is a rather complicated problem, since, unlike $\Lambda = 0$ case, while solving for various $z$, mass is not automatically preserved, because of the mass scale $M_\Lambda = \rho_\Lambda\frac{4}{3}\pi R^3$ that $\Lambda$ introduces. A suitable $\lambda$ value has to be chosen at each $z$.
We define the dimensionless mass
\begin{equation}\label{eq:mDEF}
m \equiv \frac{M}{2M_\Lambda} = \frac{3}{8\pi}\frac{M}{\rho_\Lambda R^3} = \frac{\bar{\rho}}{2\rho_\Lambda}
\end{equation}
where $\bar{\rho}$ is the mean density of matter. Calling $z = R\sqrt{4\pi G \rho_0\beta}$ the value of $x$ at $R$, equation (\ref{eq:mDEF}) can also be written as $m = 3 B/\lambda z^2$
where $B = GM\beta/R$ is the dimensionless inverse temperature. It can be calculated by integrating the Emden-$\Lambda$ equation, to get:
\begin{equation}\label{eq:beta}
B(z) = z y'(z) + \frac{1}{3}\lambda z^2
\end{equation}
We developed an algorithm to solve equation (\ref{eq:emdenL}) for various values of $\lambda$, $z$ keeping $m$ fixed.
From equation (\ref{eq:mDEF}) it is clear that solving for various fixed $m$ can be interpreted as solving for various $\rho_\Lambda$ and/or $R$ for a fixed $M$. \\
\indent We find that in dS case there exist multiple series of equilibria for a given cosmological constant, while in AdS case there is just one series likewise $\Lambda = 0$ case. Studying the series $\beta(\frac{\rho_0}{\rho_R})$ for both AdS and dS, we determined the critical values of radii and temperature beyond which there are no equilibria. These critical values are plotted in Figures \ref{fig:Rcr} and \ref{fig:Tcr}, with respect to $\rho_\Lambda$, where the unshaded region is the region of instability in both figures. \\
\indent In Figure \ref{fig:Rcr} the radius denoted $R_A$ is the minimum radius for which thermodynamic equilibria exist in the canonical ensemble. $R_H$ is the radius of the homogeneous solution which exists only in dS and is defined by equation $\rho = 2\rho_\Lambda = const$, i.e.
$
R_H = (3M/(8\pi\rho_\Lambda))^\frac{1}{3}
$.
In the shaded region of Figure \ref{fig:Rcr} there can always be found metastable states. For AdS case ($\rho_\Lambda < 0$), these have always monotonically decreasing density ($\rho_0 > \rho_R$) and suffer a transition to unstable equilibria at a critical density contrast $(\rho_0/\rho_R)_{cr}$ whose value depends on $\rho_\Lambda$. For dS case and for $R < R_H$, the metastable states suffer a transition to unstable equilibria likewise AdS, but the density is not necessarily monotonic, neither does $\rho_0 > \rho_R$ always hold. For each $\rho_\Lambda$ there exists a tower of solutions with qualitative (and quantitative of course) different density functions. In the region $R>R_H$, all metastable states have $\rho_0 < \rho_R$. The ones, whose density is a monotonic function of $R$ do not suffer any transition to unstable equilibria, while all others series have a turning point. \\
\indent In Figure \ref{fig:Tcr} we see that in AdS the critical temperature is decreasing for increasing cosmological constant, while in dS there appears a reentrant phase transition. For small positive values of $\rho_\Lambda$, compared to the mean density $\bar{\rho}$, the system cannot stabilize for low temperatures. It needs higher temperature values to obtain a significant pressure gradient to balance gravity. As $\rho_\Lambda$ is increasing, the cosmological repelling force increases, enforcing the total outward pushing force and hence, enabling the system to stabilize at lower temperatures. Point $A$, denotes the marginal value $\rho_\Lambda^{min} = \bar{\rho}/4$ for which a dynamically static equilibrium is allowed. This limiting configuration corresponds to a singular solution at which all mass is concentrated at the edge. It can easily be calculated: $GM/2R^2 = \frac{8\pi G}{3}\rho_\Lambda^{\min} R$ $\Rightarrow$ $\rho_\Lambda^{min} = 3M/16\pi R^3$ $=$ $\bar{\rho}/4$. For $\rho_\Lambda > \rho_\Lambda^{min}$, solutions with $T=0$ (static dynamical equilibria) do exist and as $\rho_\Lambda$ increases the mass is allowed to be closer to the center. All thermodynamic equilibria below the lower critical branch (low temperatures) have increasing density ($\rho_0 < \rho_R$). For fixed $R$, $M$ and a fixed $\rho_\Lambda$ with $\bar{\rho}/4 < \rho_\Lambda < \bar{\rho}/2$, as we increase the temperature beginning from zero, we pass from a region of metastable states with $\rho_0 < \rho_R$ to a region of no equilibria, at a first critical temperature $T_1$. We have to increase the temperature even more to reach a new region of metastable states at a second critical temperature $T_2$. For $T > T_2$ the metastable states have decreasing density $\rho_0 > \rho_R$ if $R<R_H$. This is a reentrant phase transition similar to the ones found in other statistical systems with long-range interactions \cite{reentrant1}.
\begin{figure}[t]
\begin{minipage}{18pc}
\includegraphics[width=18pc]{Rcr_canonical.eps}
\caption{\label{fig:Rcr}The critical radius w.r.t. $\rho_\Lambda$ for fixed $M$, $\beta$.}
\end{minipage}\hspace{2pc}%
\begin{minipage}{18pc}
\includegraphics[width=18pc]{Tcr_canonical.eps}
\caption{\label{fig:Tcr}The critical temperature w.r.t. $\rho_\Lambda$ for fixed $M$, $R$.}
\end{minipage}
\end{figure}
\section*{References}
| 3,772 |
\subsection{Supplementary Material}
\textbf{Comparing asymmetric and symmetric application of bias.} Using a QPC on a different device, we explore the effect of bias application on the phase response. Rather than using bias spectroscopy, we measure how the response changes as a function of frequency, dependent on the relative potential of the source and drain reservoirs. Asymmetric application, where one reservoir is grounded (as in the text), is shown in Fig. 5 (a), where the phase response has been normalised to that at zero bias. As in the text, peaks and troughs occur about zero bias. In contrast, symmetric application, where the bias is split across the two ohmics, is shown in (b). This time, there is no discernible difference in the response about zero bias. Taking a cut at 408 MHz in (c), we can see a global slope in the phase response as a function of asymmetric bias, due to the constant change in the capacitive coupling to the reservoirs. This does not occur for symmetric bias, whose equivalent schematic is shown in (d), analogous to that presented in Fig 3.(d).
\begin{figure}
\includegraphics[width=8.6cm]{Supp.pdf}
\caption{Using a QPC on a seperate device, we explore the phase response a a function of frequency for (a) asymmetric and (b) symmetric applications of bias, normalised to the response at zero bias. The colour scale remains the same between the two plots. Asymmetric application reveals peaks and troughs similar to our measurement, while symmetric application does not. (c) Taking a cut at 408 MHz, we observe a global slope in the phase response about the device for asymmetric application (blue), but not for symmetric application (red). (d) The schematic of reservoir potentials for symmetric bias, analogous to that presented for asymmetric bias in Fig. 3(d).}
\end{figure}
\end{document}
| 432 |
\section{Introduction}
Video surveillance for traffic control and security plays a significant role in current public transportation systems. The task of vehicle reID often undergoes intensive changes in appearance and background. Captured images in different datasets by different cameras is a primary cause of such variations. Usually, datasets differ form each other regarding lightings, viewpoints and backgrounds, even the resolution, etc. As shown in Fig.\ref{fig1}, images in VeRi-776 are brighter and have more viewpoints than images in VehicleID. And images in VehicleID have higher resolution than images in VeRi-776. Besides that, it could not contain all cases in real scenario for every dataset, which makes different datasets form their own unique style and causes the domain bias among datsets. For reID \cite{wu20193D}, most existing works follow the supervised learning paradigm which always trains the reID model using the images in the target dataset first to adapt the style of the target dataset \cite{wang2017effective}\cite{Wu2018}\cite{wang2015robust}\cite{wu2018deep}. However, it is observed that, when the well-trained reID model is tested on other dataset without fine-tuning, there is always a severe performance drop due to the domain bias.
\begin{figure}
\centering
\includegraphics[width=8.5cm]{Figure1.pdf}
\caption{Illustration of the datasets bias between VeRi-776 and VehicleID. The VeRi-776 and VehicleID present different styles,e.g., distinct lightings, backgrounds, viewpoints, resolutions etc.} \label{fig1}
\end{figure}
There are few studies on vehicle reID about the cross-domain adaptation. And only a few methods exploit unlabeled target data for unsupervised person reID modelling \cite{DengW}\cite{WangJ}\cite{WeiL}. However, some of them \cite{wu2019cycle} need extra information about source domain while training, such as attribute labels and spatio-temporal labels, which are not existing on some datasets. And there are only several methods exploiting unsupervised learning \cite{Wang2016Iterative,AMultiview} without any labels, for instance, SPGAN \cite{DengW} and PTGAN \cite{WeiL}. SPGAN is designed for person reID that integrates a SiaNet with CycleGAN \cite{KimT} and it does not need any additional labels during training. However, though SPGAN is effective on the person transfer task, it causes deformation and color distortion in vehicle transfer task in our experiment. PTGAN is composed of PSPNet \cite{ZhaoH} and CycleGAN to learn the style of target domain and maintain the identity information of source domain. In order to keep the identity information, PSPNet is utilized to segment the person images first. As we all know, it needs pre-trained segment model for PSPNet, which increases the complexity of the training stage.
To sump up, this paper proposes an end-to-end image-to-image translation network for the vehicle datasets, which named VTGAN. To preserve the identity information of images from source domain and learn style of images from target domain, for every generator in VTGAN, it is composed of a content encoder, a style encoder and a decoder. An attention model is proposed in the content encoder to preserve the identity information from the source domain. And the style encoder is designed to learn the style of the target domain with the style loss. Furthermore, VTGAN does not need any labels and paired images during the translation procedure, which is closer to the real scenario. To better adapt the target domain (unlabeled), ATTNet is designed for vehicle reID with the generated images obtained from the stage of translation. It has better generalization ability through the proposed attention structure to focus on the foreground and neglect the background information of the input image as much as possible during training procedure. In summary, our contributions can be summarized into two aspects:
1) We propose VTGAN to generate the images which have the style of target domain and preserve identity information of source domain. It is an efficient unsupervised learning model and works by transferring content and style between different domains separately.
2) ATTNet is presented to train the generated images, which is based on attention structure and could extract more distinctive cues while suppressing background for vehicle reID task.
\section{Method}
\subsection{Overview}
Our ultimate goal is to perform vehicle reID model in an unknown target domain for which are not labeled directly. Hence, we introduce a two-step vehicle reID method based on Generative Adversarial Network (GAN). The first step is to transfer the style between source domain and target domain. In this step, the VTGAN is proposed to generate images which have the style of target domain and preserve the identity information of source domain. After generating the style transferred images, in the second step, we design a multi-task network with the attention structure to obtain more discriminative features for vehicle reID.
\subsection{VTGAN}
\begin{figure}[ht]
\centering
\includegraphics[width=8.5cm]{VTGAN.jpg}
\caption{The structure of VTGAN. VTGAN contains two mapping functions: $G:X{\rightarrow}Y$ and $F:Y{\rightarrow}X$, and associated adversarial discriminators $D_{T}$ and $D_{S}$. $L_{const}$ and $L_{style}$ represents cycle consistency loss and gram loss which are employed to further regularize the mappings (best viewed in color).} \label{fig2}
\end{figure}
VTGAN is designed to transfer the style between source domain and target domain in the case of preserving the identity information of images from source domain. As illustrated in Fig.\ref{fig2}, VTGAN consists of generators ${G,F}$, and domain discriminators ${D_{S}, D_{T}}$ for both domains. For each generator in VTGAN, it contains content encoder ${E^c}$, style encoder ${E^s}$ and decoder ${De}$ three components. ${E^c}$ is designed to preserve the identity information from images of source domain through the proposed attention model, which could extract the foreground while suppressing background. And to learn the style of target domain, the ${E^s}$ with the gram loss is added to the translation network. At last, the decoder $De$ embeds the output of ${E^c}$ and $E^s$ to generate the translated image.
\subsubsection{Content Encoder}
In order to keep the identity information from source domain, the attention model is designed to assign higher scores of visual attention to the region of interest while suppressing background in the content encoder.
\begin{figure}[ht]
\centering
\includegraphics[width=7cm]{Encoder.jpg}
\caption{The illustration of proposed attention structure.} \label{fig3}
\end{figure}
As shown in Fig.\ref{fig3}, we denote the input feature map of attention model as $f$. In this work, a simple feature fusion structure is utilized to generate the $f$. We fuse the every output of the ResBlock to form $f$, which can be formulated as $f=[f_{r1}, f_{r2}, ..., f_{r9}]$, where $f_{ri}$ is the $ith$ feature map generated by the $ith$ ResBlock. $i\in [1,9]$ and $[\cdot]$ denotes the concatenation operation. For the feature vector $f_{i,j}\in{\Re^C}$ of the feature map at the spatial location $(i,j)$, we can calculate its corresponding attention mask $a_{i,j}$ by
\begin{equation}
a_{i,j}=Sigmod(FC(f_{i,j};W_{a}))
\end{equation}
where $FC$ is the Full Connected layer (FC) to learn a mapping function in the attention module and $W_{a}$ are the weights of the FC. The final attention mask $\alpha=[a_{i,j}]$ is a probability map obtained using a Sigmod layer. The scores represent the probability of foreground in the input image.
And after the attention model, a mask $a$ is generated, which has high scores for foreground and low scores for background. Hence, the attended feature map $f_{c}$ is computed by element-wise product of the attention mask and the input feature map, which could be described as follows:
\begin{equation}
f_{c(i,j)}=a_{i,j}\otimes{f_{i,j}}
\end{equation}
where $(i,j)$ is the spatial location in mask $a$ or feature map $f_{c}$. And $\otimes$ is performed in an element-wise product.
\subsubsection{Style Encoder}
Besides the content branch, there is a branch to learn style of target domain. In this branch, different with the $E^c_g$ and $E^c_f$, the style network $E^s_g$ and $E^s_f$ do not contain the attention model. To learn the style of the target domain, $E^s_g$ is designed with the gram loss to output the style features $f_{s}$ which has similar distribution of the target domain $Y$. The gram loss could be formulated as follows:
\begin{equation}
\begin{split}
L_{style}= {\frac{1}{NM}{{(T(x)-A(y))^2}}}+{\frac{1}{NM}{{(T(y)-A(x))^2}}}
\end{split}
\end{equation}
where $N$ is the number of feature maps, $M$ is calculated by $width\times{height}$, $width$ and $height$ represent the width and height of images. $T(x)$, $T(y)$, $A(y)$ and $A(x)$ are the gram matrix of output features $E_{g}^s(x)$, $E_{f}^s(y)$, $E_{g}^s(y)$ and $E_{f}^s(x)$, respectively.
\subsubsection{Decoder Network}
For the decoder network, it is composed of two deconvolution layers and a convolutional layer to output the generated images $G(I)$. The input of the decoder network is the combination of $f_{c}$ and $f_{s}$ which represent the content features and style features, respectively. In this paper, we employ a concatenate layer to fuse $f_{c}$ and $f_{s}$ and a global skip connection structure to make training faster and resulting model generalizes better, which could be expressed as:
\begin{equation}
G(I)=tanh(conv(deconv(deconv([f_{c}, f_{s}])+f_{e2})))
\end{equation}
where $[.]$ represents the concatenate layer. And $f_{e2}$ represents the feature map generated by the second stride convolution blocks. $f_{c}$ and $f_{s}$ are the output of content encoder and style encoder, respectively.
\subsubsection{Loss function}
We formulate the loss function in VTGAN as a combination of adversarial loss, content loss and style loss:
\begin{equation}
L=L_{GAN}+\lambda_{1}L_{id}+\lambda_{2}L_{style}
\end{equation}
where the $\lambda_{1}$ and $\lambda_{2}$ control the relative importance of three objectives. The style loss $L_{style}$ could be calculated by Eq.(3). VTGAN utilizes the target domain identity constraint as an auxiliary for image-image translation. Target domain identity constraint was introduced by \cite{TaigmanY} to regularize the generator to be the identity matrix on samples from target domain, written as:
\begin{equation}
\begin{split}
L_{id}= E_{y\sim p_{data}(y)}||F(y)- y||_1 +E_{x\sim p_{data}(x)}||G(x)- x||_1
\end{split}
\end{equation}
For $L_{GAN}$, it consists of three parts which two adversarial losses and a cycle consistency loss. VTGAN applies adversarial losses to both mapping functions. For the generator $F$ and its discriminator $D_{T}$, the objective could be expressed as:
\begin{equation}
\begin{split}
L_{T}(F,D_T,X,Y)= & E_{x\sim p_{data}(x)}[(D_T(x))^2] \\ & + E_{y\sim p_{data}(y)}[||D_{T}(F(y))-1||_1]
\end{split}
\end{equation}
where, $X$ and $Y$ represent the source domain and target domain, respectively. $p_{data}(x)$ and $p_{data}(y)$ denote the sample distributions in the source and target domain. The objective of generator $G$ and discriminator $D_{S}$ also could be built. Besides, the VTGAN requires $F(G(x))\approx x$ and $G(F(y))\approx y$ when it learns the mapping of $F$ and $G$. So the cycle consistency loss is employed in VTGAN which could make the network more stable. The cycle consistency loss could be defined as:
\begin{equation}
\begin{split}
L_{cyc}(F,G,X,Y)= & E_{x\sim p_{data}(x)}[||F(G(x))- x||_1] \\ & +E_{y\sim p_{data}(y)}[||G(F(y))- y||_1]
\end{split}
\end{equation}
\subsection{ATTNet}
As we all know, the diversity background is a big factor for the problem of cross domain. And in order to make the reID model adapt to the target domain, we are facing a condition that it is better to focus on the vehicle images and neglect the background when we train the feature learning model. Hence, a two-stream reID network with attention structure is designed in this paper.
\begin{figure*}
\center
\includegraphics[width=13cm]{ATTNet.jpg}
\caption{The structure of ATTNet.} \label{fig4}
\center
\end{figure*}
As shown in Fig.\ref{fig4}, the input images are obtained from the image generation module and are divided into positive and negative sample pairs. For one branch, the input image is fed into five ResNet Blocks \cite{HeK} to output the feature maps $f_{r}$ with the size of $7\times 7\times 2048$. Then they are passed into a Global Average Pooling (GAP) layer to obtain the feature map $f_{g}$. $f_{g}$ is utilized to generate the mask $M$ through the proposed attention structure. Given the feature map $f_{r}$, its attention map is computed as $M = Softmax(Conv(f_{g}))$, where the one $Conv$ operator is $1\times 1$ convolution. After obtaining the attention map $M$, the attended feature map could be calculated by $f_m = f_{g}\otimes M$. The operator $\otimes$ is performed in an element-wise product. Then the attended feature map $f_{m}$ will be fed into the subsequent structure. A shortcut connection architecture is introduced to embed the input of the attention network directly to its output with an element-wise sum layer, which could be described as $f_s = f_{g} + f_{m}$. In this way, both the original feature map and the attended feature map are combined to form features $f_{a}$ and utilized as input to the subsequent structure. And after two FC layers, we could obtain the feature $f_{d}$. At last, a skip connection structure is utilized to integrate $f_{g}$ and $f_{d}$ by the concatenate layer to obtain more discriminative features for identification task and verification task, which could be described as $f_{a} = [f_{d}, f_{g}]$.
\section{Experiments}
\subsection{Datasets}
VeRi-776 \cite{LiuX'} is a large-scale urban surveillance vehicle dataset for reID. This dataset contains over 50,000 images of 776 vehicles with identity annotations, camera geo-locations, image timestamps, vehicle types and colors information. In this paper, 37,781 images of 576 vehicles are employed as a train set. VehicleID \cite{LiuH} is a surveillance dataset from real-world, which contains 26267 vehicles and 221763 images in total. From the original testing data, four subsets, which contain 800, 1600, 2400 and 3200 vehicles, are extracted for vehicle search in different scales.
\subsection{Implementation Details}
For VTGAN, we train the model in the tensorflow \cite{AbadiM} and the learning rate is set to 0.0002. Note that, we do not utilize any label notation during the learning procedure. The min-batch size of the proposed method is 16 and epoch is set to 6. During the testing procedure, we employ the Generator $G$ for VeRi-776 $\to$ VehicleID translation and the Generator $F$ for VehicleID $\to$ VeRi-776 translation. The translated images are utilized for training reID models. For ATTNet, We implement the proposed vehicle re-id model in the Matconvnet \cite{VedaldiA} framework. We utilize stochastic gradient descent with a momentum of $\mu=0.0005$ during the training procedure. The batch size is set to 16. The learning rate of the first 50 epoch is set to 0.1, and the last 5 to 0.01.
\subsection{Evaluation}
\subsubsection{Comparison Methods}
There are really little methods about the vehicle reID of cross domain. So in this paper, we only discuss several methods and test them on VeRi-776 and VehicleID. Direct Transfer means directly applying the model trained by images from source domain on the target domain. CycleGAN \cite{KimT}, SPGAN \cite{DengW} and VTGAN are employed to translate images from source domain to target domain, and then the generated images are utilized to train reID model. Baseline \cite{ZhengZ'} denotes the compared training network of reID. ATTNet is our proposed network.
\subsubsection{Comparison of generated images}
To demonstrate the effectiveness of our proposed style transform model, the VehicleID and VeRi-776 are utilized to train the VTGAN. And CycleGAN and SPGAN are taken as compared methods. Fig.\ref{fig5} is the comparison results, which the source domain is VeRi-776, and target domain is VehicleID. For each group, the first row is the original images in VeRi-776. The second and third rows are generated by CycleGAN and SPGAN, respectively. The last row is generated by the proposed VTGAN.
\begin{figure}[ht]
\centering
\includegraphics[width=6.5cm]{Figure2.pdf}
\caption{The effect of the generated images. The first row is original images. The generated images using CycleGAN, SPGAN lie in the second row and third row respectively. The last row are generated images by VTGAN.} \label{fig5}
\end{figure}
From the Fig.5, we could find that, most images generated by CycleGAN are distorted seriously when transfer images from VeRi-776 to VehicleID. And though the SPGAN works better than the CycleGAN, the generated images also have evident deformation. However, for VTGAN, not only is the vehicle color and type information completely preserved, but also learns the style of the target dataset. As we can see from Fig.5, generated images by VTGAN have higher resolution and become darker, which learns from VehicleID.
\subsubsection{The impact of Image-Image Translation}
Firstly, we utilize CycleGAN to translate labeled images from the source domain to the target domain then train the baseline reID model with translated images in a supervised way. As shown in Table.\ref{tab1}, when trained on VeRi-776 training set using the baseline method and tested on VehicleID different testing sets, rank-1 accuracy improves from 35\% to 39.39\%, 30.42\% to 32.97\%, 27.28\% to 28.44\% and 25.41\% to 26.38\%, respectively. Through such an image-level domain adaptation method, effective domain adaptation baselines can be learned. From the Fig. \ref{fig5}, we could find that, though some of generated images by CycleGAN are distorted, the performance of reID model trained by generated images is improved. This illustrates methods of image-image translation have learned the important style information from the target domain, which could narrow-down the domain gap to a certain degree.
\subsubsection{The impact of VTGAN}
To verify the effectiveness of the proposed VTGAN, we conduct several experiments which training sets are images generated from different image translation methods. As shown in Table.\ref{tab1}, on VehicleID, compared with $CycleGAN+Baseline$, the gains of $VTGAN+Baseline$ are 5.05\%, 6\%, 6.66\% and 5.79\% in rank-1 of different test sets, respectively. Though SPGAN has better performance in the stage image-to-image translation than CycleGAN, it also causes deformation and color distortion in real scenario (see Fig.5). Hence, compared with $SPGAN+Baseline$, for different size of test sets on VehicleID, $VTGAN+Baseline$ has 1.57\%, 1.51\%, 1.56\% and 1.72\% improvements in mAP, respectively. All of these could demonstrate that the structure of VTGAN is more stable and could generate suitable samples for training in the target domain.
\subsubsection{The impact of ATTNet}
To further improve re-ID performance on target dataset, we propose ATTNet. Fig.\ref{fig6} is CMC resutls on VehicleID of different methods. As shown in Fig.\ref{fig6}, compared to methods with baseline reID model, not only original images but also generated images, methods using ATTNet have better performance. For instance, from the Table.\ref{tab1}, we could find that, compared with $Direct\ Transfer+Baseline$, $Direct \ Transfer+ATTNet$ has 8.26\%, 9.05\%, 8.67\%, and 7.99\% improvements in rank-1 of different test sets when the model is trained on VeRi-776 and tested on VehicleID. Besides, it is obvious that compared with the baseline methods, the reID model using the ATTNet have significant improvement for every image translation method. This demonstrates that the reID model trained by the proposed ATTNet can better adapt to cross-domain task than the baseline method.
\begin{table*}[htbp]
\center
\scriptsize
\setlength{\belowcaptionskip}{10pt}
\caption{Comparison of various domain adaptation methods over Baseline model and ATTNet-reID model on VehicleID. }\label{tab1}
\begin{tabular}{|C{2.6cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|}
\hline
\multirow{2}*{Methods} & \multicolumn{3}{c|}{Test size = 800} & \multicolumn{3}{c|}{Test size = 1600}& \multicolumn{3}{c|}{Test size = 2400} & \multicolumn{3}{c|}{Test size = 3200} \\
\cline{2-13} & mAP(\%) & Rank1(\%) & Rank5(\%) & mAP(\%) & Rank1(\%) & Rank5(\%) & mAP(\%) & Rank1(\%) & Rank5(\%) & mAP(\%) & Rank1(\%) & Rank5(\%) \\
\hline
Direct Transfer + Baseline & 40.05 &35.00 &56.68 &34.90 &30.42 &48.85 &31.65 &27.28& 44.49 & 29.57 & 25.41 &42.11\\
CycleGAN + Baseline & 44.24 &39.39 &60.10 &37.68 &32.97 &53.16 &33.17 &28.44 &47.92 & 30.73 & 26.38 &43.84\\
SPGAN + Baseline &48.27 &42.87& 66.55 &42.51 &37.46 &58.97 &38.41 &33.54 &53.68 & 35.04 & 30.45 &49.13\\
VTGAN + Baseline & 49.53 &44.44 &66.74 &43.90 &38.97 &59.93 &40.07 &35.10 &56.29 & 36.86 & 32.17 & 51.63\\
\hline
Direct Transfer + ATTNet & 47.97 &43.26 &62.93 &43.94 &39.47 &58.51 &40.42 &35.95 &54.34 & 37.60 & 33.40 & 50.55\\
CycleGAN + ATTNet & 46.96 &42.68 &60.72 &43.27 &38.88 &57.44 &39.39 &35.09 &53.05 & 37.05 & 33.07 & 49.38\\
SPGAN + ATTNet & 52.72 &48.25 &67.20 &48.01 &43.44 &63.04 &44.17 &39.51 &59.05 & 41.05 & 36.75 & 54.63\\
VTGAN + ATTNet & 54.01 &49.48 &68.66 &49.72 &45.18 &63.99 &45.18 &40.71 &59.02 & 42.94 & 38.72 & 55.87\\
\hline
\end{tabular}
\center
\end{table*}
\begin{figure}[ht]
\centerline{
\subfloat[Test size=800]{\includegraphics[width=1.7in,height=1.3in]{R_800_C.pdf}}
\subfloat[Test size=1600]{\includegraphics[width=1.71in,height=1.31in]{R_1600_C.pdf}}
}
\centerline{
\subfloat[Test size=2400]{\includegraphics[width=1.7in,height=1.3in]{R_2400_C.pdf}}
\subfloat[Test size=3200]{\includegraphics[width=1.7in,height=1.3in]{R_3200_C.pdf}}
}
\caption{The CMC curves of different methods on VehicleID. (a) The result tested on the set with 800 vehicles. (b) The result tested on the set with 1600 vehicles. (c) The result tested on the set with 2400 vehicles. (d) The result tested on the set with 3200 vehicles.} \label{fig6}
\end{figure}
\section{Conclusion}
In this paper, we propose a vehicle reID framework based on GAN, which includes the VTGAN and ATTNet for domain adaptation. The VTGAN is designed to generate the vehicle images which preserve the identity information of source domain and learn the style of target domain. The ATTNet is proposed to train the reID model with generated images. It can be observed from the results that both the VTGAN and ATTNet achieve good results. What's more, it is obvious that existing datasets usually contain several viewpoints of vehicle images. It is also a limit for reID task in new domain. Hence, in our future studies, we would focus on using the GAN to generate the various viewpoints of vehicle images to expand the dataset and improve the performance of reID model.
\bibliographystyle{IEEEbib}
| 6,614 |
\section{Introduction}
\label{sec.intro}
{Continual learning} (CL) learns a sequence of tasks $<$$1, 2, ..., k, ...$$>$ incrementally. Each task $k$ has its dataset $\mathcal{D}_k=\{(\boldsymbol{x}_k^i, y_k^i)_{i=1}^{n_k}\}$, where $\boldsymbol{x}_k^i \in \boldsymbol{X}$ is a data sample in task $k$ and $y_k^i \in \boldsymbol{Y}_k$ is the class label of $\boldsymbol{x}_k^i$ and $\boldsymbol{Y}_k$ is the set of classes of task $k$. The key challenge of CL is \textit{catastrophic forgetting} (CF)~\cite{McCloskey1989}, which refers to the situation where the learning of a new task may significantly change the network weights learned for old tasks, degrading the model accuracy for old tasks. Researchers have mainly worked on two CL problems: \textit{class incremental/continual learning} (CIL) and \textit{task incremental/continual learning} (TIL)~\citep{Dhar2019CVPR,van2019three}.
The main difference between CIL and TIL is that in TIL, the task-id $k$ is provided for each test sample $\boldsymbol{x}$ during testing so that only the model for task $k$ is used to classify $\boldsymbol{x}$, while in CIL, the task-id $k$ for each test sample is not provided.
Existing CL techniques focus on either CIL or TIL~\citep{Parisi2019continual}. In general, CIL methods are designed to function without given task-id to perform class prediction over all classes in the tasks learned so far, and thus tend to forget the previous task performance due to plasticity for the new task. TIL methods are designed to function with a task-id for class prediction within the task. They are more stable to retain previous within-task knowledge, but incompetent if task-id is unknown (see Sec.~\ref{experiment}).
This paper proposes a novel and unified method called CLOM (\textit{C}ontinual \textit{L}earning based on \textit{O}OD detection and Task \textit{M}asking) to solve both problems by overcoming the limitations of CIL and TIL methods.
CLOM has two key mechanisms: (1) a task mask mechanism for protecting each task model to overcome CF, and (2) a learning method for building a model for each task based on out-of-distribution (OOD) detection.
The task mask mechanism is inspired by \textit{hard attention} in the TIL system HAT~\cite{Serra2018overcoming}.
The OOD detection based learning method for building each task model is quite different from the classic supervised learning used in existing TIL systems. It is, in fact, the key novelty and enabler for CLOM to work effectively for CIL.
OOD detection is stated as follows~\cite{bulusu2020anomalous}: Given a training set of $n$ classes, called the \textit{in-distribution} (IND) training data, we want to build a model that can assign correct classes to IND test data and reject or detect OOD test data that do not belong to any of the $n$ IND training classes. The OOD rejection capability makes a TIL model effective for CIL problem because during testing, if a test sample does not belong to any of the classes of a task, it will be rejected by the model of the task. Thus, only the task that the test sample belongs to will accept it and classify it to one of its classes. No task-id is needed. The OOD detection algorithm in CLOM is inspired by several recent advances in self-supervised learning, data augmentation~\cite{he2020momentum}, contrastive learning~\cite{oord2018representation,chen2020simple}, and their applications to OOD detection~\cite{golan2018deep,hendrycks2019using,tack2020csi}.
{The main contributions of this work are as follows:
\begin{itemize}
\item It proposes a novel CL method CLOM, which is essentially a TIL system in training, but solves both TIL and CIL problems in testing. Existing methods mainly solve either TIL or CIL problem, but are weak on the other problem.
\item CLOM uses task masks to protect the model for each task to prevent CF and OOD detection to build each task model, which to our knowledge has not been done before. More importantly, since the task masks can prevent CF,
{continual learning performance gets better as OOD models get better too.}
\item CLOM needn't to save any replay data. If some past data is saved for output calibration, it performs even better.
\end{itemize}}
Experimental results show that CLOM improves state-of-the-art baselines by large margins.~The average TIL/CIL accuracy of CLOM over six different experiments is 87.6/67.9\% while that of the best baseline is only 82.4/55.0\%.
\section{Related Work}
Many approaches have been proposed to deal with CF in CL. Using regularization~\cite{kirkpatrick2017overcoming} and knowledge distillation~\cite{Li2016LwF} to minimize the change to previous models are two popular approaches~\cite{Jung2016less,Camoriano2017incremental,Fernando2017pathnet,Rannen2017encoder,Seff2017continual,zenke2017continual,Kemker2018fearnet,ritter2018online,schwarz2018progress,xu2018reinforced,castro2018end,Dhar2019CVPR,hu2019overcoming,lee2019overcoming,Liu2020}.
Memorizing some old examples and using them to adjust the old models in learning a new task is another popular approach (called \textit{replay})~\cite{Rusu2016,Lopez2017gradient,Rebuffi2017,Chaudhry2019ICLR,Cyprien2019episodic, hou2019learning,wu2019large,rolnick2019neurIPS, NEURIPS2020_b704ea2c_derpp,zhao2020maintaining,rajasegaran2020adaptive,Liu2020AANets}.
Several systems learn to generate pseudo training data of old tasks and use them to jointly train the new task, called \textit{pseudo-replay}~\cite{Gepperth2016bio,Kamra2017deep,Shin2017continual,wu2018memory,Seff2017continual,wu2018memory,Kemker2018fearnet,hu2019overcoming,hayes2019remind,Rostami2019ijcai,ostapenko2019learning}. CLOM differs from these approaches {as it does not replay any old task data to prevent forgetting and can function with/without saving some old data.} {\textit{Parameter isolation} is yet another popular approach, which makes different subsets (which may overlap) of the model parameters dedicated to different tasks using masks~\cite{Fernando2017pathnet,Serra2018overcoming,ke2020continual} or finding a sub-network for each task by pruning \cite{Mallya2017packnet, supsup2020,NEURIPS2019_3b220b43compact}. CLOM uses parameter isolation, but it differs from these approaches as it combines the idea of parameter isolation and OOD detection, which can solve both TIL and CIL problems effectively.}
There are also many other approaches, e.g., a network of experts~\cite{Aljundi2016expert} and generalized CL~\cite{mi2020generalized}, etc. {PASS~\cite{Zhu_2021_CVPR_pass} uses data rotation and regularizes them.
Co$^2$L~\cite{Cha_2021_ICCV_co2l} is a replay method that uses contrastive loss on old samples. CLOM also uses rotation and constrative loss, but its CF handling is based on masks. None of the existing methods uses OOD detection.}
CLOM is a TIL method that can also solve CIL problems. Many TIL systems exist~\cite{Fernando2017pathnet}, e.g., GEM~\cite{Lopez2017gradient}, A-GEM~\cite{Chaudhry2019ICLR}, UCL~\cite{ahn2019neurIPS}, ADP~\cite{Yoon2020Scalable}, CCLL~\cite{singh2020calibrating}, Orthog-Subspace~\cite{chaudhry2020continual}, HyperNet~\cite{von2019continual}, PackNet~\cite{Mallya2017packnet}, CPG~\cite{NEURIPS2019_3b220b43compact}, SupSup~\cite{supsup2020}, HAT~\cite{Serra2018overcoming}, and CAT~\cite{ke2020continual}.
GEM is a replay based method and UCL is a regularization based method. A-GEM~\citep{Chaudhry2019ICLR} improves GEM's efficiency. {ADP decomposes parameters into shared and adaptive parts to construct an order robust TIL system.} CCLL uses task-adaptive calibration on convolution layers. Orthog-Subspace learns each task in subspaces orthogonal to each other. {HyperNet initializes task-specific parameters conditioned on task-id. PackNet, CPG and SupSup find an isolated sub-network for each task and use it at inference.
HAT and CAT protect previous tasks by masking the important parameters.
Our CLOM also uses this general approach, but its model building for each task is based on OOD detection, which has not been used by existing TIL methods. It also performs well in the CIL setting (see Sec.~\ref{resultcomparison}). }
Related work on out-of-distribution (OOD) detection (also called open set detection) is also extensive.
Excellent surveys include~\cite{bulusu2020anomalous,geng2020recent}.
The model building part of CLOM is inspired by the latest method in~\cite{tack2020csi} based on data augmentation~\cite{he2020momentum} and contrastive learning~\cite{chen2020simple}.
\section{Proposed CLOM Technique}
\label{sec.CLOM}
As mentioned earlier, CLOM is a TIL system that solves both TIL and CIL problems.
It takes the parameter isolation approach for TIL. For each task $k$ (task-id), a model is built $f(h(\boldsymbol{x}, k))$ for the task, {where $h$ and $f$ are the feature extractor and the task specific classifier, respectively,}
and $f(h(\boldsymbol{x}, k))$ is the output of the neural network for task $k$.
{We omit task-id $k$ in $f$ to simplify notation.}
{In learning each task $k$, a task mask is also trained at the same time to protect the learned model of the task.
In testing for TIL, given the test sample $\boldsymbol{x}$ with task-id $k$ provided, CLOM uses the model for task $k$ to classify $\boldsymbol{x}$,
\begin{align}
\hat{y} = \argmax
f(h(\boldsymbol{x}, k))
\label{TCLeq}
\end{align}
For CIL (no task-id for each test sample $\boldsymbol{x}$), CLOM uses the model of every task and predicts the class $\hat{y}$ using
\begin{align}
\hat{y} = \argmax
\bigoplus_{1\leq k \leq t} f(h(\boldsymbol{x}, k)),
\label{CLOMeq}
\end{align}
where
$\bigoplus$ is the concatenation over the output space and
$t$ is the last task that has been learned. Eq.~\ref{CLOMeq} essentially chooses the class with the highest classification score among the classes of all tasks learned so far. This works because the OOD detection model for a task will give very low score for a test sample that does not belong to the task. {An overview of the prediction is illustrated in Fig.~\ref{diagram}(a).}
}
Note that in fact any CIL model can be used as a TIL model if task-id is provided. The conversion is made by selecting the corresponding task classification heads in testing. However, a conversion from a TIL method to a CIL method is not obvious as the system requires task-id in testing.
Some attempts have been made to predict the task-id in order to make a TIL method applicable to CIL problems.
iTAML~\cite{rajasegaran2020itaml} requires the test samples to come in batches and each batch must be from a single task. This may not be practical as test samples usually come one by one. CCG~\cite{abati2020conditional} builds a separate network to predict the task-id, which is also prone to forgetting. Expert Gate~\cite{Aljundi2016expert} constructs a separate autoencoder for each task.
Our CLOM classifies one test sample at a time and does not need to {construct another network for task-id prediction.}
{In the following subsections, we present (1) how to build an OOD model for a task and use it to make a prediction, and (2) how to learn task masks to protect each model. An overview of the process is illustrated in Fig. \ref{diagram}(b).}
\subsection{Learning Each Task as OOD Detection} \label{unifiedappr}
\begin{figure}
\centering
\includegraphics[width=3in]{diagram-20211203.pdf}
\caption{(a) Overview of TIL and CIL prediction. For TIL prediction, we consider only the output heads of the given task. For CIL prediction, we obtain all the outputs from task $1$ to current task $t$, and choose the label over the concatenated output.
(b) Training overview for a generic backbone. Each CNN includes a convolution layer with batch normalization, activation, or pooling depending on the configuration. In our experiments, we use AlexNet~\cite{NIPS2012_c399862d} and ResNet-18~\cite{he2016deep}. We feed-forward augmented batch $\tilde{\mathcal{B}}$ consisting of rotated images of different views ($\boldsymbol{x}^{1}_{0}, \boldsymbol{x}^{1}_{90}, \cdots$). We first train feature extractor by using contrastive loss (step 1). At each layer, binary mask is multiplied to the output of each convolution layer to learn important parameters of current task $t$. After training feature extractor, we fine-tune the OOD classifier (step 2).
}
\label{diagram}
\end{figure}
We borrow the latest OOD ideas based on contrastive learning~\cite{chen2020simple, he2020momentum} and data augmentation due to their excellent performance~\cite{tack2020csi}.
Since this section focuses on how to learn a single task based on OOD detection, we omit the task-id unless necessary. The OOD training process {is similar to that of contrastive learning}. It consists of two steps: 1) learning {the feature representation by} the composite $g \circ h$, where $h$ is a feature extractor and $g$ is a projection to contrastive representation, and 2) learning a linear classifier $f$ mapping the feature representation of $h$ to the label space. {In the following, we describe the training process: contrastive learning for feature representation learning (1), and OOD classifier building (2). We then explain how to make a prediction based on an ensemble method for both TIL and CIL settings, and how to further improve prediction using some saved past data.}
\subsubsection{Contrastive Loss for Feature Learning.}
{This is step 1 in Fig.~\ref{diagram}(b).} Supervised contrastive learning is used to try to repel data of different classes and align data of the same class more closely to make it easier to classify them. A key operation is data augmentation via transformations.
{Given a batch of $N$ samples, each sample $\boldsymbol{x}$ is first duplicated and each version then goes through \textit{three initial augmentations}
(also see Data Augmentation in Sec.~\ref{sec:traindetails}) to generate two different views $\boldsymbol{x}^{1}$ and $\boldsymbol{x}^{2}$ (they keep the same class label as $\boldsymbol{x}$).}
Denote the augmented batch by $\mathcal{B}$, which now has $2N$ samples. {In~\cite{hendrycks2019using,tack2020csi}}, it was shown that using image rotations is effective in learning OOD detection models because such rotations can effectively serve as out-of-distribution (OOD) training data.
For each augmented sample $\boldsymbol{x} \in \mathcal{B}$ with class $y$ of a task, we rotate $\boldsymbol{x}$ by $90^{\circ}, 180^{\circ}, 270^{\circ}$ to create three images, which are assigned \textit{three new classes} $y_1, y_2$, and $y_3$, respectively.
{This results in a larger augmented batch $\tilde{\mathcal{B}}$. Since we generate three new images from each $\boldsymbol{x}$,
the size of $\tilde{\mathcal{B}}$ is $8N$. For each original class, we now have 4 classes. For a sample $\boldsymbol{x} \in \tilde{\mathcal{B}}$, let $\mathcal{\tilde{B}}(\boldsymbol{x}) = \mathcal{\tilde{B}} \backslash \{ \boldsymbol{x} \}$
and
let $P(\boldsymbol{x}) \subset \tilde{\mathcal{B}} \backslash \{ \boldsymbol{x} \}$
be a set consisting of the data of the same class as $\boldsymbol{x}$ distinct from $\boldsymbol{x}$.
The contrastive representation of a sample $\boldsymbol{x}$ is $\boldsymbol{z}_{x} = g(h(\boldsymbol{x}, t)) / \| g(h(\boldsymbol{x}, t)) \|$, where $t$ is the current task.
In learning, we minimize the supervised contrastive loss~\cite{khosla2020supervised} of task $t$.
\begin{align}
\mathcal{L}_{c}
&= \frac{1}{8N} \sum_{ \boldsymbol{x} \in \tilde{\mathcal{B}}} \frac{-1}{| P(\boldsymbol{x}) |} \times \nonumber \\
& \sum_{\boldsymbol{p} \in P(\boldsymbol{x})} \log{ \frac{ \text{exp}( \boldsymbol{z}_{\boldsymbol{x}} \cdot \boldsymbol{z}_{\boldsymbol{p}} / \tau)}{\sum_{\boldsymbol{x}' \in \tilde{\mathcal{B}}(\boldsymbol{x}) } \text{exp}( \boldsymbol{z}_{\boldsymbol{x}} \cdot \boldsymbol{z}_{\boldsymbol{x}'} / \tau) } } \label{modsupclr}
\end{align}
where
$\tau$ is a scalar temperature, $\cdot$ is dot product, and $\times$ is multiplication.
The loss is reduced by repelling $\boldsymbol{z}$ of different classes and aligning $\boldsymbol{z}$ of the same class more closely.
$\mathcal{L}_{c}$ basically trains a feature extractor with good
representations for learning an OOD classifier.}
\subsubsection{Learning the Classifier.}
{This is step 2 in Fig.\ref{diagram}(b)}. {Given the feature extractor $h$ trained with the loss in Eq.~\ref{modsupclr}, we {\textit{freeze $h$} and} only \textit{fine-tune} the linear classifier $f$, which is trained to predict the classes of task $t$ \textit{and} the augmented rotation classes.} $f$ maps the feature representation to {the label space in} $\mathcal{R}^{4|\mathcal{C}^{t}|}$, where $4$ is the number of rotation classes including the original data with $0^{\circ}$ rotation and $|\mathcal{C}^{t}|$ is the number of {original} classes in task $t$. We minimize the cross-entropy loss,
\begin{align}
\mathcal{L}_{\text{ft}} = - \frac{1}{|\tilde{\mathcal{B}} |} \sum_{(\boldsymbol{x}, y) \in \tilde{\mathcal{B}}}
\log \tilde{p}(y | \boldsymbol{x}, t)
\label{3obj}
\end{align}
where $\text{ft}$ indicates fine-tune,
and
\begin{align}
\tilde{p}(y | \boldsymbol{x}, t) = \text{softmax} \left( f(h(\boldsymbol{x}, t))
\right) \label{probrotation}
\end{align}
where
$f(h(\boldsymbol{x}, t)) \in \mathcal{
R}^{4|\mathcal{C}^{t}|}$. The output $f(h(\boldsymbol{x}, t))$
includes the rotation classes. The linear classifier is trained to predict the original \textit{and} the rotation classes.
\subsubsection{Ensemble Class Prediction.} \label{ensemblesection}
{
We describe how to predict a label $y \in \mathcal{C}^{t}$ (TIL) and $y \in \mathcal{C}$ (CIL) ($\mathcal{C}$ is the set of original classes of all tasks)
We assume all tasks have been learned and their models are protected by masks, which we discuss in the next subsection.
We discuss the prediction of class label $y$ for a test sample $\boldsymbol{x}$ in the TIL setting first. Note that the network $f\circ h$ in Eq.~\ref{probrotation} returns logits for rotation classes (including the original task classes). Note also for each original class label $j_k \in \mathcal{C}^{k}$ (original classes) of a task $k$, we created three additional rotation classes. For class $j_k$, the classifier $f$ will produce four output values from its four rotation class logits, i.e., $f_{j_k,0}(h(\boldsymbol{x_0}, k))$, $f_{j_k,90}(h(\boldsymbol{x_{90}}, k))$, $f_{j_k,180}(h(\boldsymbol{x_{180}}, k))$, and $f_{j_k,270}(h(\boldsymbol{x_{270}}, k))$, where 0, 90, 180, and 270 represent $0^{\circ}, 90^{\circ}, 180^{\circ}$, and $270^{\circ}$ rotations respectively and $\boldsymbol{x}_0$ is the original $\boldsymbol{x}$.
We compute an ensemble output $f_{j_k}(h(\boldsymbol{x},k))$ for each class $j_k \in \mathcal{C}^{k}$ of task $k$,
\begin{align}
f_{j_k}(h(\boldsymbol{x},k))= \frac{1}{4} \sum_{\text{deg}} f_{j_k,\text{deg}} (h(\boldsymbol{x}_{\text{deg}}, k)) \label{ensemblelogit}.
\end{align}
The final TIL class prediction is made as follows (note, in TIL, task-id $k$ is provided in testing),
\begin{align}
\hat{y} = \argmax_{j_k \in \mathcal{C}^{k}} \Big\{ f_{j_k}(h(\boldsymbol{x},k)) \Big\} \label{ensembleclass}
\end{align}
We use Eq.~\ref{CLOMeq} to make the CIL class prediction, where {the final format} $f(h(\boldsymbol{x},k))$ for task $k$ is the following vector:
\begin{align}
f(h(\boldsymbol{x}, k)) = \left[ f_{1}(h(\boldsymbol{x}, k), \cdots , f_{|\mathcal{C}^{k}|}(h(\boldsymbol{x}, k)) \right]
\label{eq:afterensemble}
\end{align}
Our method so far memorizes no training samples and it already outperforms baselines (see Sec.~\ref{resultcomparison}).
}
\subsubsection{Output Calibration with Memory.}
\label{sec.calibration}
{The proposed method can make incorrect CIL prediction even with a perfect OOD model (rejecting every test sample that does not belong any class of the task). This happens because the task models are trained independently, and the outputs of different tasks may have different magnitudes. We use output calibration to ensure that the outputs are of similar magnitudes to be comparable by using some saved examples in a memory $\mathcal{M}$ with limited budget.}
At each task $k$, we store a fraction of the validation data $\{ (\boldsymbol{x}, y)\}$ into $\mathcal{M}_k$ for output calibration and update the memory $\mathcal{M} \leftarrow update(\mathcal{M}, \mathcal{M}_k)$ {by maintaining an equal number of samples per class}. We will detail the memory budget in Sec.~\ref{ablation}. Basically, we save the same number as the existing replay-based TIL/CIL methods.
After training the network $f$ for the current task $t$,
we freeze the model and use the saved data in $\mathcal{M}$ to find the scaling and shifting parameters $(\boldsymbol{\sigma}, \boldsymbol{\mu}) \in \mathcal{R}^t \times \mathcal{R}^t$ to calibrate the after-ensemble classification output $f(h(\boldsymbol{x},k))$ (Eq.~\ref{eq:afterensemble}) (i.e., using $\sigma_{k} f(h (\boldsymbol{x}, k)) + \mu_k)$ for each task $k$ by minimizing the cross-entropy loss,
\begin{align}
\mathcal{L}_{\text{cal}}
= - \frac{1}{| \mathcal{M} |} \sum_{(\boldsymbol{x}, y) \in \mathcal{M}}
\log{ p(y| \boldsymbol{x}) }
\end{align}
where $\mathcal{C}$ is the set of all classes of all tasks seen so far, and $p(c|\boldsymbol{x})$ is computed using softmax,
\begin{align}
\text{softmax} \Big[ \sigma_{1} f(h (\boldsymbol{x}, 1)) + \mu_1 ;
\cdots ;
\sigma_{t} f(h (\boldsymbol{x}, t)) + \mu_t \Big]
\label{eq:scaling}
\end{align}
Clearly, parameters $\sigma_k$ and $\mu_k$ do not change classification within task $k$ (TIL), but calibrates the outputs such that the ensemble outputs from all tasks are in comparable magnitudes.
For CIL inference at the test time, we make prediction by (following Eq.~\ref{CLOMeq}),
\begin{align}
\hat{y} = \argmax
\bigoplus_{1 \le k \le t}
\left[ \sigma_{k} f(h(\boldsymbol{x}, k)) + \mu_{k} \right]
\label{calibratedpred}
\end{align}
\subsection{Protecting OOD Model of Each Task Using Masks}
\label{sec.hat}
We now discuss the task mask mechanism for protecting the OOD model of each task to deal with CF.
In learning the OOD model for each task, CLOM at the same time also trains a \textit{mask} or \textit{hard attention} for each layer. To protect
{the shared feature extractor}
from previous tasks, their masks are used to block those important neurons so that the new task learning will not interfere with the parameters learned for previous tasks.
The main idea is to use sigmoid to approximate a 0-1 step function as \textit{hard attention} to mask (or block) or unmask (unblock) the information flow to protect the parameters learned for each previous task.
The hard attention (mask) at layer $l$ and task $t$ is defined as
\begin{align}
\boldsymbol{a}_{l}^{t} = u\left( s \boldsymbol{e}_{l}^{t} \right) \label{attn}
\end{align}
where $u$ is the sigmoid function, $s$ is a scalar, and $\boldsymbol{e}_{l}^{t}$ is a \textit{learnable} embedding {of the task-id input $t$}.
The attention is element-wise multiplied to the output $\boldsymbol{h}_{l}$ of layer $l$ as
\begin{align}
\boldsymbol{h}_{l}' = \boldsymbol{a}_{l}^{t} \otimes \boldsymbol{h}_{l}
\end{align}
{as depicted in Fig.~\ref{diagram}(b).} The sigmoid function $u$ converges to a 0-1 step function {as $s$ goes to infinity}. Since the true step function is not differentiable, a fairly large $s$ is chosen to achieve a differentiable pseudo step function based on sigmoid (see Appendix~\ref{hyperparam} for choice of $s$). The pseudo binary value of the attention determines how much information can flow forward and backward between adjacent layers.
Denote $\boldsymbol{h}_{l} = \text{ReLU}(\boldsymbol{W}_{l} \boldsymbol{h}_{l-1} + \boldsymbol{b}_{l})$, where $\text{ReLU}$ is the rectifier function. For units (neurons) of attention $\boldsymbol{a}_{l}^{t}$ with zero values, we can freely change the corresponding parameters in $\boldsymbol{W}_{l}$ and $\boldsymbol{b}_l$ without affecting the output $\boldsymbol{h}_{l}'$. For units of attention with non-zero values, changing the parameters will affect the output $\boldsymbol{h}_{l}'$ for which we need to protect from gradient flow in backpropagation to prevent forgetting.
Specifically, during training the new task $t$, we update parameters according to the attention so that the important parameters for
{past tasks ($1, ..., t-1$) are unmodified. Denote the accumulated attentions (masks) of all past tasks by
\begin{align}
\boldsymbol{a}_{l}^{<t} = \max ( \boldsymbol{a}_{l}^{< t-1}, \boldsymbol{a}_{l}^{t-1} )
\end{align}
where $\boldsymbol{a}_{l}^{<t}$ is the hard attentions of layer $l$ of all previous tasks, and $\max$ is an element-wise maximum\footnote{Some parameters from different tasks can be shared, which means some hard attention masks can be shared.} and $\boldsymbol{a}^{0}_{l}$ is a zero vector. Then the modified gradient is the following,
\begin{align}
\nabla w_{ij, l}' = \left( 1 - \min \left( a_{i, l}^{< t}, a_{j, l-1}^{< t} \right) \right) \nabla w_{ij, l} \label{hatupdate}
\end{align}
where $a^{< t}_{i, l}$ indicates $i$'th unit of $\boldsymbol{a}^{< t}_{l}$ and $l=1, ..., L-1$}.
This reduces the gradient if the corresponding units' attentions at layers $l$ and $l-1$ are non-zero\footnote{{By construction, if $\boldsymbol{a}_{l}^{t}$ becomes $\boldsymbol{1}$ for all layers, the gradients are zero and the network is at maximum capacity. However, the network capacity can increase by adding more parameters.}}.
We do not apply hard attention on the last layer $L$ because it is a task-specific layer.
To encourage sparsity in $\boldsymbol{a}_{l}^{t}$ and parameter sharing with $\boldsymbol{a}_{l}^{<t}$,
a regularization ($\mathcal{L}_{r}$) for attention at task $t$ is defined as
\begin{align}
\mathcal{L}_{r} = \lambda_{t} \frac{\sum_{l}\sum_{i} a_{i, l}^{t}\left( 1 - a_{i, l}^{< t} \right)}{\sum_{l}\sum_{i} \left( 1 - a_{i, l}^{< t} \right)}
\end{align}
where
$\lambda_t$ is a scalar hyperparameter.
For flexibility, we denote $\lambda_t$ for each task $t$. However, in practice, we use the same $\lambda_t$ for all $t \neq 1$.
The final objective to be minimized for task $t$ with hard attention is (see Fig.~\ref{diagram}(b))
\begin{align}
\mathcal{L} = \mathcal{L}_{c} + \mathcal{L}_r \label{1obj}
\end{align}
where $\mathcal{L}_{c}$ is the contrastive loss function (Eq.~\ref{modsupclr}).
By protecting important parameters from changing during training,
the neural network effectively alleviates CF.
\section{Experiments} \label{experiment}
\textbf{Evaluation Datasets:} Four image classification CL benchmark datasets are used in our experiments.
(1) \textbf{MNIST}
:\footnote{http://yann.lecun.com/exdb/mnist/
}
handwritten digits of 10 classes (digits) with 60,000 examples for training and 10,000 examples for testing.
(2) \textbf{CIFAR-10}~\citep{Krizhevsky2009learning}:\footnote{https://www.cs.toronto.edu/~kriz/cifar.html} 60,000 32x32 color images of 10 classes with 50,000 for training and 10,000 for testing.
(3)~\textbf{CIFAR-100}~\citep{Krizhevsky2009learning}:\footnote{https://www.cs.toronto.edu/~kriz/cifar.html} 60,000 32x32 color images of 100 classes with 500 images per class for training and 100 per class for testing.~(4)~\textbf{Tiny-ImageNet}~\cite{Le2015TinyIV}:\footnote{http://tiny-imagenet.herokuapp.com} 120,000 64x64 color images of 200 classes with 500 images per class for training and 50 images per class for validation, and 50 images per class for testing.
Since the test data has no labels in this dataset, we use the validation data as the test data as in \cite{Liu2020}.
\textbf{Baseline Systems:}
We compare our CLOM with both the classic and the most recent state-of-the-art
CIL and TIL methods.
We also include CLOM(-c), which is CLOM without calibration (which already outperforms the baselines).
For CIL baselines, we consider seven \textit{replay} methods, \textbf{LwF.R} (replay version of LwF~\cite{Li2016LwF} with better results~\cite{Liu_2020_CVPR}), \textbf{iCaRL} \cite{Rebuffi2017}, \textbf{BiC} \cite{wu2019large}, \textbf{A-RPS}~\cite{rajasegaran2020adaptive},
\textbf{Mnemonics} \cite{Liu_2020_CVPR}, \textbf{DER++} \cite{NEURIPS2020_b704ea2c_derpp}, and \textbf{Co$^2$L} \cite{Cha_2021_ICCV_co2l};
one {\textit{pseudo-replay}} method \textbf{CCG} \cite{abati2020conditional}\footnote{iTAML~\cite{rajasegaran2020itaml} is not included as they require a batch of test data from the same task to predict the task-id. When each batch has only one test sample, which is our setting, it is very weak. For example, iTAML TIL/CIL accuracy is only 35.2\%/33.5\% on CIFAR100 10 tasks. Expert Gate (EG)~\cite{Aljundi2016expert} is also weak. For example, its TIL/CIL accuracy is 87.2/43.2 on MNIST 5 tasks. Both iTAML and EG are much weaker than many baselines. };
one \textit{orthogonal projection} method
\textbf{OWM}~\cite{zeng2019continuous}; and a \textit{multi-classifier} method
\textbf{MUC}~\cite{Liu2020}; and \textit{a prototype augmentation method} \textbf{PASS}~\cite{Zhu_2021_CVPR_pass}. OWM, MUC, and PASS \textit{do not save any samples} from previous tasks.
TIL baselines include \textbf{HAT}~\cite{Serra2018overcoming}, \textbf{HyperNet}~\cite{von2019continual}, and \textbf{SupSup}~\cite{supsup2020}.
As noted earlier, CIL methods can also be used for TIL. In fact, TIL methods may be used for CIL too, but the results are very poor. We include them in our comparison
\subsection{Training Details}
\label{sec:traindetails}
For all experiments, we use 10\%
of training data as the validation set to grid-search for good hyperparameters.
For minimizing the contrastive loss, we use LARS \cite{you2017large} for 700 epochs with initial learning rate 0.1. We linearly increase the learning rate by 0.1 per epoch for the first 10 epochs until 1.0 and then decay it by
cosine scheduler \cite{loshchilov2016sgdr} after 10 epochs without restart as in~\cite{chen2020simple, tack2020csi}. For fine-tuning the classifier $f$, we use SGD for 100 epochs with learning rate 0.1 and reduce the learning rate by 0.1 at 60, 75, and 90 epochs. The full set of hyperparameters is given in Appendix~\ref{hyperparam}.
We follow the recent baselines (A-RPS, DER++, PASS and Co$^2$L) and use the same class split and backbone architecture for both CLOM and baselines.
For MNIST and CIFAR-10, we split 10 classes into 5 tasks where each task has 2 classes in consecutive order. We save 20 random samples per class from the validation set for output calibration. This number is commonly used in replay methods~\cite{Rebuffi2017,wu2019large,rajasegaran2020adaptive,Liu_2020_CVPR}. MNIST consists of single channel images of size 1x28x28. Since the contrastive learning \cite{chen2020simple} relies on color changes, we copy the channel to make 3-channels. For MNIST and CIFAR-10, we use AlexNet-like architecture \cite{NIPS2012_c399862d} and ResNet-18 \cite{he2016deep} respectively for both CLOM and baselines.
For CIFAR-100, we conduct two experiments. We split 100 classes into 10 tasks and 20 tasks where each task has 10 and 5 classes, respectively, in consecutive order. We use 2000 memory budget as in \cite{Rebuffi2017}, saving 20 random samples per class from the validation set for output calibration. We use the same ResNet-18 structure for CLOM and baselines, but we increase the number of channels twice to learn more tasks.
For Tiny-ImageNet, we follow \cite{Liu2020} and resize the original images of size 3x64x64 to 3x32x32 so that the same ResNet-18 of CIFAR-100 experiment setting can be used. We split 200 classes into 5 tasks (40 classes per task) and 10 tasks (20 classes per task) in consecutive order, respectively.
To have the same memory budget of 2000 as for CIFAR-100, we save 10 random samples per class from the validation set for output calibration.
\textbf{Data Augmentation.} {For baselines, we use data augmentations used in their original papers. For CLOM, following \citep{chen2020simple, tack2020csi}, we use \textit{three initial augmentations} (see Sec.~\ref{unifiedappr}) (i.e., \textit{horizontal flip}, \textit{color change} (\textit{color jitter} and \textit{grayscale}), and \textit{Inception crop} \cite{inception}) and four \textit{rotations} (see Sec.~\ref{unifiedappr}). Specific details about these transformations are given in Appendix~\ref{aug_details}.}
\begin{table*}[t]
\centering
\resizebox{2.1\columnwidth}{!}{
\begin{tabular}{l c c c c c c c c c c c c | c c}
&&&&&&&&&&&&&&\\[-1.1em]
\toprule
\multirow{2}{*}{Method} & \multicolumn{2}{c}{MNIST-5T} & \multicolumn{2}{c}{CIFAR10-5T} & \multicolumn{2}{c}{CIFAR100-10T} & \multicolumn{2}{c}{CIFAR100-20T} & \multicolumn{2}{c}{T-ImageNet-5T} & \multicolumn{2}{c}{T-ImageNet-10T} & \multicolumn{2}{|c}{Average}\\
{} & TIL & CIL & TIL & CIL & TIL & CIL & TIL & CIL & TIL & CIL & TIL & CIL & TIL & CIL \\
&&&&&&&&&&&&&&\\[-1em]
\midrule
\multicolumn{13}{c}{\textbf{CIL Systems}} & \multicolumn{2}{|c}{} \\
\hline
OWM & 99.7\scalebox{0.8}{$\pm$0.03} & 95.8\scalebox{0.8}{$\pm$0.13} & 85.2\scalebox{0.8}{$\pm$0.17} & 51.7\scalebox{0.8}{$\pm$0.06} & 59.9\scalebox{0.8}{$\pm$0.84} & 29.0\scalebox{0.8}{$\pm$0.72} & 65.4\scalebox{0.8}{$\pm$0.07} & 24.2\scalebox{0.8}{$\pm$0.11} & 22.4\scalebox{0.8}{$\pm$0.87} & 10.0\scalebox{0.8}{$\pm$0.55} & 28.1\scalebox{0.8}{$\pm$0.55} & 8.6\scalebox{0.8}{$\pm$0.42} & 60.1 & 36.5 \Tstrut \\
MUC & 99.9\scalebox{0.8}{$\pm$0.02} & 74.6\scalebox{0.8}{$\pm$0.45} & 95.2\scalebox{0.8}{$\pm$0.24} & 53.6\scalebox{0.8}{$\pm$0.95} & 76.9\scalebox{0.8}{$\pm$1.27} & 30.0\scalebox{0.8}{$\pm$1.37} & 73.7\scalebox{0.8}{$\pm$1.27} & 14.4\scalebox{0.8}{$\pm$0.93} & 55.8\scalebox{0.8}{$\pm$0.26} & 33.6\scalebox{0.8}{$\pm$0.18} & 47.2\scalebox{0.8}{$\pm$0.22} & 17.4\scalebox{0.8}{$\pm$0.17} & 74.8 & 37.3 \\
PASS$^{\dagger}$& 99.5\scalebox{0.8}{$\pm$0.14} & 76.6\scalebox{0.8}{$\pm$1.67} & 83.8\scalebox{0.8}{$\pm$0.68} & 47.3\scalebox{0.8}{$\pm$0.97} & 72.4\scalebox{0.8}{$\pm$1.23} & 36.8\scalebox{0.8}{$\pm$1.64} & 76.9\scalebox{0.8}{$\pm$0.77} & 25.3\scalebox{0.8}{$\pm$0.81} & 50.3\scalebox{0.8}{$\pm$1.97} & 28.9\scalebox{0.8}{$\pm$1.36} & 47.6\scalebox{0.8}{$\pm$0.38} & 18.7\scalebox{0.8}{$\pm$0.58} & 71.8 & 38.9 \\
LwF.R & \textbf{99.9}\scalebox{0.8}{$\pm$0.09} & 85.0\scalebox{0.8}{$\pm$3.05} & 95.2\scalebox{0.8}{$\pm$0.30} & 54.7\scalebox{0.8}{$\pm$1.18} & 86.2\scalebox{0.8}{$\pm$1.00} & 45.3\scalebox{0.8}{$\pm$0.75} & 89.0\scalebox{0.8}{$\pm$0.45} & 44.3\scalebox{0.8}{$\pm$0.46} & 56.4\scalebox{0.8}{$\pm$0.48} & 32.2\scalebox{0.8}{$\pm$0.50} & 55.3\scalebox{0.8}{$\pm$0.35} & 24.3\scalebox{0.8}{$\pm$0.26} & 80.3 & 47.6 \\
iCaRL$^*$ & \textbf{99.9}\scalebox{0.8}{$\pm$0.09} & 96.0\scalebox{0.8}{$\pm$0.42} & 94.9\scalebox{0.8}{$\pm$0.34} & 63.4\scalebox{0.8}{$\pm$1.11} & 84.2\scalebox{0.8}{$\pm$1.04} & 51.4\scalebox{0.8}{$\pm$0.99} & 85.7\scalebox{0.8}{$\pm$0.68} & 47.8\scalebox{0.8}{$\pm$0.48} & 54.3\scalebox{0.8}{$\pm$0.59} & 37.0\scalebox{0.8}{$\pm$0.41} & 52.7\scalebox{0.8}{$\pm$0.37} & 28.3\scalebox{0.8}{$\pm$0.18} & 78.6 & 54.0 \\
Mnemonics$^{\dagger *}$ & \textbf{99.9}\scalebox{0.8}{$\pm$0.03} & 96.3\scalebox{0.8}{$\pm$0.36} & 94.5\scalebox{0.8}{$\pm$0.46} & 64.1\scalebox{0.8}{$\pm$1.47} & 82.3\scalebox{0.8}{$\pm$0.30} & 51.0\scalebox{0.8}{$\pm$0.34} & 86.2\scalebox{0.8}{$\pm$0.46} & 47.6\scalebox{0.8}{$\pm$0.74} & 54.8\scalebox{0.8}{$\pm$0.16} & 37.1\scalebox{0.8}{$\pm$0.46} & 52.9\scalebox{0.8}{$\pm$0.66} & 28.5\scalebox{0.8}{$\pm$0.72} & 78.5 & 54.1 \\
BiC & \textbf{99.9}\scalebox{0.8}{$\pm$0.04} & 85.1\scalebox{0.8}{$\pm$1.84} & 91.1\scalebox{0.8}{$\pm$0.82} & 57.1\scalebox{0.8}{$\pm$1.09} & 87.6\scalebox{0.8}{$\pm$0.28} & 51.3\scalebox{0.8}{$\pm$0.59} & 90.3\scalebox{0.8}{$\pm$0.26} & 40.1\scalebox{0.8}{$\pm$0.77} & 44.7\scalebox{0.8}{$\pm$0.71} & 20.2\scalebox{0.8}{$\pm$0.31} & 50.3\scalebox{0.8}{$\pm$0.65} & 21.2\scalebox{0.8}{$\pm$0.46} & 77.3 & 45.8 \\
DER++ & 99.7\scalebox{0.8}{$\pm$0.08} & 95.3\scalebox{0.8}{$\pm$0.69} & 92.2\scalebox{0.8}{$\pm$0.48} & 66.0\scalebox{0.8}{$\pm$1.27} & 84.2\scalebox{0.8}{$\pm$0.47} & 55.3\scalebox{0.8}{$\pm$0.10} & 86.6\scalebox{0.8}{$\pm$0.50} & 46.6\scalebox{0.8}{$\pm$1.44} & 58.0\scalebox{0.8}{$\pm$0.52} & 36.0\scalebox{0.8}{$\pm$0.42} & 59.7\scalebox{0.8}{$\pm$0.6} & 30.5\scalebox{0.8}{$\pm$0.30} & 80.1 & 55.0 \\
A-RPS & & & & & & 60.8 & & 53.5 & & & & \\
CCG & & \textbf{97.3} & & 70.1 & & & & & & & & \\
Co$^2$L & & & 93.4 & 65.6 & & & & & & & & \\
\hline
\multicolumn{13}{c}{\textbf{TIL Systems}} & \multicolumn{2}{|c}{} \Tstrut\Bstrut \\
\hline
HAT & \textbf{99.9}\scalebox{0.8}{$\pm$0.02} & 81.9\scalebox{0.8}{$\pm$3.73} & 96.7\scalebox{0.8}{$\pm$0.18} & 62.7\scalebox{0.8}{$\pm$1.46} & 84.0\scalebox{0.8}{$\pm$0.23} & 41.1\scalebox{0.8}{$\pm$0.93} & 85.0\scalebox{0.8}{$\pm$0.85} & 26.0\scalebox{0.8}{$\pm$0.83} & 61.2\scalebox{0.8}{$\pm$0.72} & 38.5\scalebox{0.8}{$\pm$1.85} & 63.8\scalebox{0.8}{$\pm$0.41} & 29.8\scalebox{0.8}{$\pm$0.65} & 81.8 & 46.6 \Tstrut \\
HyperNet & 99.7\scalebox{0.8}{$\pm$0.05} & 49.1\scalebox{0.8}{$\pm$5.52} & 94.9\scalebox{0.8}{$\pm$0.54} & 47.4\scalebox{0.8}{$\pm$5.78} & 77.3\scalebox{0.8}{$\pm$0.45} & 29.7\scalebox{0.8}{$\pm$2.19} & 83.0\scalebox{0.8}{$\pm$0.60} & 19.4\scalebox{0.8}{$\pm$1.44} & 23.8\scalebox{0.8}{$\pm$1.21} & 8.8\scalebox{0.8}{$\pm$0.98} & 27.8\scalebox{0.8}{$\pm$0.86} & 5.8\scalebox{0.8}{$\pm$0.56} & 67.8 & 26.7 \\
SupSup & 99.6\scalebox{0.8}{$\pm$0.09} & 19.5\scalebox{0.8}{$\pm$0.15} & 95.3\scalebox{0.8}{$\pm$0.27} & 26.2\scalebox{0.8}{$\pm$0.46} & 85.2\scalebox{0.8}{$\pm$0.25} & 33.1\scalebox{0.8}{$\pm$0.47} & 88.8\scalebox{0.8}{$\pm$0.18} & 12.3\scalebox{0.8}{$\pm$0.30} & 61.0\scalebox{0.8}{$\pm$0.62} & 36.9\scalebox{0.8}{$\pm$0.57} & 64.4\scalebox{0.8}{$\pm$0.20} & 27.0\scalebox{0.8}{$\pm$0.45} & 82.4 & 25.8 \\
\hline
CLOM(-c) & \textbf{99.9}\scalebox{0.8}{$\pm$0.00} & 94.4\scalebox{0.8}{$\pm$0.26} & \textbf{98.7}\scalebox{0.8}{$\pm$0.06} & 87.8\scalebox{0.8}{$\pm$0.71} & \textbf{92.0}\scalebox{0.8}{$\pm$0.37} & 63.3\scalebox{0.8}{$\pm$1.00} & \textbf{94.3}\scalebox{0.8}{$\pm$0.06} & 54.6\scalebox{0.8}{$\pm$0.92} & \textbf{68.4}\scalebox{0.8}{$\pm$0.16} & 45.7\scalebox{0.8}{$\pm$0.26} & \textbf{72.4}\scalebox{0.8}{$\pm$0.21} & 47.1\scalebox{0.8}{$\pm$0.18} & 87.6 & 65.5 \Tstrut \\
CLOM & \textbf{99.9}\scalebox{0.8}{$\pm$0.00} & 96.9\scalebox{0.8}{$\pm$0.30} & \textbf{98.7}\scalebox{0.8}{$\pm$0.06} & \textbf{88.0}\scalebox{0.8}{$\pm$0.48} & \textbf{92.0}\scalebox{0.8}{$\pm$0.37} & \textbf{65.2}\scalebox{0.8}{$\pm$0.71} & \textbf{94.3}\scalebox{0.8}{$\pm$0.06} & \textbf{58.0}\scalebox{0.8}{$\pm$0.45} & \textbf{68.4}\scalebox{0.8}{$\pm$0.16} & \textbf{51.7}\scalebox{0.8}{$\pm$0.37} & \textbf{72.4}\scalebox{0.8}{$\pm$0.21} & \textbf{47.6}\scalebox{0.8}{$\pm$0.32} & \textbf{87.6} & \textbf{67.9} \\
\bottomrule
\end{tabular}
}
\caption{
Average accuracy over all classes after the last task is learned. -xT: x number of tasks. {$\dagger$: In their original paper, PASS and Mnemonics use the first half of classes to pre-train before CL. Their results are 50.1\% and 53.5\% on CIFAR100-10T respectively, but they are still lower than CLOM without pre-training. In our experiments, no pre-training is used for fairness.} $^{*}$: \textbf{iCaRL} and \textbf{Mnemonics} give both the final average accuracy as here and the \textit{average incremental accuracy} in the original papers. We report the \textit{average incremental accuracy} and \textit{network size} in Appendix~\ref{avg_inc_acc} and \ref{param_size}, respectively. The last two columns show the average TIL and CIL accuracy of each method over all datasets.
}
\label{Tab:maintable}
\end{table*}
\subsection{Results and Comparative Analysis} \label{resultcomparison}
As in existing works, we evaluate each method by two metrics: \textit{average classification accuracy} on all classes after training the last task, and \textit{average forgetting rate}~\cite{Liu_2020_CVPR}, $F^{t} = \frac{1}{t-1}\sum_{j=1}^{t-1} A_{j}^{\text{init}} - A_{j}^{t}$,
where $A_{j}^{\text{init}}$ is the $j$'th task's accuracy of the network right after the $j$'th task is learned and $A_{j}^{t}$ is the accuracy of the network on the $j$'th task data after learning the last task $t$. We report the forgetting rate after the final task $t$. Our results are averages of 5 random runs.
We present the main experiment results in Tab.~\ref{Tab:maintable}. The last two columns give the average TIL/CIL results of each system/row. For A-RPS, CCG, and Co$^2$L, we copy the results from their original papers as their codes are not released to the public or the public code cannot run on our system. The rows are grouped by CIL and TIL methods.
\textbf{CIL Results Comparison}. {Tab. \ref{Tab:maintable} shows that CLOM and CLOM(-c) achieve much higher CIL accuracy except for MNIST for which CLOM is slightly weaker than CCG by 0.4\%, but CLOM's result on CIFAR10-5T is about 18\% greater than CCG.}
For other datasets, CLOM improves by similar margins. {This is in contrast to the baseline TIL systems that are incompetent at the CIL setting when classes are predicted using Eq. \ref{CLOMeq}.}
Even \textit{without} calibration,
CLOM(-c) already outperforms all the baselines by large margins.
\textbf{TIL Results Comparison}.
{The gains by CLOM and CLOM(-c) over the baselines are also great in the TIL setting.}
CLOM and CLOM(-c) are the same as the output calibration does not affect TIL performance. For the two large datasets CIFAR100 and T-ImageNet, CLOM gains by large margins. {
This is due to contrastive learning and the OOD model. The replay based CIL methods (LwF.R, iCaRL, Mnemonics, BiC, and DER++) perform reasonably in the TIL setting, but our CLOM and CLOM(-c) are much better due to task masks which can protect previous models better with little CF.}
\begin{figure}[!ht]
\includegraphics[width=3.3in]{forget.pdf}
\caption{Average forgetting rate (\%) in the TIL setting as CLOM is a TIL system.
The lower the value, the better the method is. {CIL/TIL systems are shaded in blue/red, respectively (\textit{best viewed in color}).} A negative value indicates the task accuracy has increased from the initial accuracy.}
\label{forgetplot}
\end{figure}
\textbf{Comparison of Forgetting Rate}. Fig. \ref{forgetplot} shows the average forgetting rate of each method in the TIL setting. The CIL systems suffer from more forgetting as they are not designed for the TIL setting, which results in lower TIL accuracy (Tab. \ref{Tab:maintable}).
The TIL systems are highly {effective at preserving previous within-task knowledge}. This results in higher TIL accuracy on large dataset such as T-ImageNet, but they collapse when task-id is not provided (the CIL setting)
as shown in Tab. \ref{Tab:maintable}.
{CLOM is robust to forgetting as a TIL system and it also functions well without task-id.}
We report only the forgetting rate in the TIL setting because our CLOM is essentially a TIL method and not a CIL system by design. The degrading CIL accuracy of CLOM is mainly because the OOD model for each task is not perfect.
\subsection{Ablation Studies} \label{ablation}
\begin{table}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{lcccccccc}
\toprule
& \multicolumn{4}{c}{CIFAR10-5T} & \multicolumn{4}{c}{CIFAR100-10T} \\
& \multicolumn{1}{l}{AUC} & \multicolumn{1}{l}{TaskDR} & \multicolumn{1}{c}{TIL} & \multicolumn{1}{l}{CIL} & \multicolumn{1}{l}{AUC} & \multicolumn{1}{l}{TaskDR} & \multicolumn{1}{c}{TIL} & \multicolumn{1}{l}{CIL} \\
\midrule
SupSup & 78.9 & 26.2 & 95.3 & 26.2 & 76.7 & 34.3 & 85.2 & 33.1 \\
SupSup (OOD in CLOM) & 88.9 & 82.3 & 97.2 & 81.5 & 84.9 & 63.7 & 90.0 & 62.1 \\
\midrule
CLOM (ODIN) & 82.9 & 63.3 & 96.7 & 62.9 & 77.9 & 43.0 & 84.0 & 41.3 \\
CLOM & 92.2 & 88.5 & 98.7 & 88.0 & 85.0 & 66.8 & 92.0 & 65.2 \\
\midrule
CLOM (w/o OOD) & 90.3 & 83.9 & 98.1 & 83.3 & 82.6 & 59.5 & 89.8 & 57.5 \\
\bottomrule
\end{tabular}
}
\caption{{TIL and CIL results improve with better OOD detection. Column AUC gives the average AUC score for the OOD detection method as used within each system on the left. Column TaskDR gives \textit{task detection rate}. TIL and CIL results are average accuracy values. {SupSup and CLOM variants are calibrated with 20 samples per class.}}
}
\label{betterood}
\end{table}
{\textbf{Better OOD for better continual learning.} We show that (1) an existing CL model can be improved by a good OOD model and (2) CLOM's results will deteriorate if a weaker OOD model is applied.
To isolate effect of OOD detection on changes in CIL performance, we further define \textit{task detection} and \textit{task detection rate}. For a test sample from a class of task $j$, if it is predicted to a class of task $m$ and $j=m$, the task detection is correct for this test instance. The \textit{task detection rate} $\sum_{\mathbf{x} \in \mathcal{D}^{\text{test}} } 1_{j = m}/N$, where $N$ is the total number of test instances in $\mathcal{D}^{\text{test}}$, is the rate of correct task detection. We measure the performance of OOD detection using AUC (Area Under the ROC Curve) averaged over all tasks. AUC is the main measure used in OOD detection papers. We conduct experiments on CIFAR10-5T and CIFAR100-10T.
For (1), we use the TIL baseline SupSup as it displays a strong TIL performance and is robust to forgetting like CLOM. We replace SupSup’s task learner with the OOD model in CLOM. Tab.~\ref{betterood} shows that the OOD method in CLOM improves SupSup (SupSup (OOD in CLOM)) greatly. It shows that our approach is applicable to different TIL systems.
For (2), we replace CLOM’s OOD method with a weaker OOD method ODIN~\cite{liang2018enhancing}. We see in Tab.~\ref{betterood} that task detection rate, TIL, and CIL results all drop markedly with ODIN (CLOM (ODIN)).
}
{\textbf{CLOM without OOD detection.} In this case, CLOM uses contrastive learning and data augmentation,
but does not use the rotation classes in classification. Note that the rotation classes are basically regarded as OOD data in training and for OOD detection in testing. CLOM (w/o OOD) in Tab.~\ref{betterood} represents this CLOM variant. We see that CLOM (w/o OOD) is much weaker than the full CLOM. This indicates that the improved results of CLOM over baselines are not only due to contrastive learning and data augmentation but also significantly due to OOD detection.}
\begin{table}
\parbox{.50\linewidth}{
\centering
\resizebox{.50\columnwidth}{!}{
\begin{tabular}{lcc}
\toprule
\multicolumn{1}{c}{$|\mathcal{M}|$} & (a) & (b) \\
\midrule
0 & 63.3\scalebox{0.8}{$\pm$1.00} & 54.6\scalebox{0.8}{$\pm$0.92} \\
5 & 64.9\scalebox{0.8}{$\pm$0.67} & 57.7\scalebox{0.8}{$\pm$0.50} \\
10 & 65.0\scalebox{0.8}{$\pm$0.71} & 57.8\scalebox{0.8}{$\pm$0.53} \\
15 & 65.1\scalebox{0.8}{$\pm$0.71} & 57.9\scalebox{0.8}{$\pm$0.44} \\
20 & 65.2\scalebox{0.8}{$\pm$0.71} & 58.0\scalebox{0.8}{$\pm$0.45} \\
\bottomrule
\end{tabular}
}
}
\hfill
\parbox{.46\linewidth}{
\centering
\resizebox{.46\columnwidth}{!}{
\begin{tabular}{lccc}
\toprule
s & $F^{5}$ & AUC & CIL \\
\midrule
1 & 48.6 & 58.8 & 10.0 \\
100 & 13.3 & 82.7 & 67.7 \\
300 & 8.2 & 83.3 & 72.0 \\
500 & 0.2 & 91.8 & 87.2 \\
700 & 0.1 & 92.2 & 88.0 \\
\bottomrule
\end{tabular}
}
}
\caption{(\textit{Left}) shows changes of accuracy with the number of samples saved per class for output calibration. (a) and (b) are CIFAR100-10T and CIFAR100-20T, respectively. $|\mathcal{M}|=k$ indicates $k$ samples are saved per class. {(\textit{Right}) shows that weaker forgetting mechanism results in larger forgetting and lower AUC, thus lower CIL. For $s=1$, the pseudo-step function becomes the standard sigmoid, thus parameters are hardly protected. $F^{5}$ is the forgetting rate over 5 tasks.}}
\label{memory_and_s}
\end{table}
\textbf{Effect of the number of saved samples for calibration.}
Tab. \ref{memory_and_s} (left) reveals that the output calibration is still effective with a small number of saved samples per class ($|\mathcal{M}|$). For both CIFAR100-10T and CIFAR100-20T, {CLOM achieves competitive performance by using only 5 samples per class.} The accuracy improves and the variance decreases with the number of saved samples.
{\textbf{Effect of $s$ in Eq.~\ref{attn} on forgetting of CLOM.}
We need to use a strong forgetting mechanism for CLOM to be functional. Using CIFAR10-5T, we show how CLOM performs with different values of $s$ in hard attention or masking. The larger $s$ value, the stronger protection is used. Tab.~\ref{memory_and_s} (right) shows that average AUC and CIL decrease as the forgetting rate increases. This also supports the result in Tab.~\ref{betterood} that SupSup improves greatly with the OOD method in CLOM as it is also robust to forgetting. PASS and Co$^2$L underperform despite they also use rotation or constrastive loss as their forgetting mechanisms are weak.
}
\begin{table}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{lcccc}
\toprule
\multicolumn{1}{}{} & \multicolumn{2}{c}{CIFAR10-5T} & \multicolumn{2}{c}{CIFAR100-10T} \\
Aug. & TIL & CIL & TIL & CIL \\
\midrule
Hflip & (93.1, 95.3) & (49.1, 72.7) & (77.6, 84.0) & (31.1, 47.0) \\
Color & (91.7, 94.6) & (50.9, 70.2) & (67.2, 77.4) & (28.7, 41.8) \\
Crop & (96.1, 97.3) & (58.4, 79.4) & (84.1, 89.3) & (41.6, 60.3) \\
All & (97.6, 98.7) & (74.0, 88.0) & (88.1, 92.0) & (50.2, 65.2) \\
\bottomrule
\end{tabular}
}
\caption{Accuracy of CLOM variants when a single or all augmentations are applied.
Hflip: horizontal flip; Color: color jitter and grayscale; Crop: Inception~\cite{inception}. (num1, num2): accuracy without and with rotation.
}
\label{augmentation}
\end{table}
\textbf{Effect of data augmentations.}
{For data augmentation, we use three initial augmentations (i.e., \textit{horizontal flip}, \textit{color change} (color jitter and grayscale), \textit{Inception crop} \cite{inception}), which are commonly used in contrastive learning to build a single model. We additionally use rotation for OOD data in training. To evaluate the contribution of each augmentation
when task models are trained sequentially,
we train CLOM using one augmentation. We do not report their effects on forgetting as we experience rarely any forgetting (Fig. \ref{forgetplot} and Tab. \ref{memory_and_s}). Tab. \ref{augmentation} shows that the performance is lower when only a single augmentation is applied.
When all augmentations are applied, the TIL/CIL accuracies are higher. The rotation always improves the result when it is combined with other augmentations.
More importantly, when we use crop and rotation, we achieve higher CIL accuracy (79.4/60.3\% for CIFAR10-5T/CIFAR100-10T) than we use all augmentations without rotation (74.0/50.2\%).
This shows the efficacy of rotation in our system.}
\section{Conclusions}
This paper proposed a novel continual learning (CL) method called CLOM based on OOD detection and task masking that can perform both task incremental learning (TIL) and class incremental learning (CIL). Regardless whether it is used for TIL or CIL in testing, the training process is the same, which is an advantage over existing CL systems as they focus on either CIL or TIL and have limitations on the other problem. Experimental results showed that CLOM outperforms both state-of-the-art TIL and CIL methods by very large margins.
In our future work, we will study ways to improve efficiency and also accuracy.
\section*{Acknowledgments}
{\color{black}Gyuhak Kim, Sepideh Esmaeilpour and Bing Liu were supported in part by two National Science Foundation (NSF) grants (IIS-1910424 and IIS-1838770), a DARPA Contract HR001120C0023, a KDDI research contract, and a Northrop Grumman research gift.}
\section{Introduction}
\label{sec.intro}
{Continual learning} (CL) learns a sequence of tasks $<$$1, 2, ..., k, ...$$>$ incrementally. Each task $k$ has its dataset $\mathcal{D}_k=\{(\boldsymbol{x}_k^i, y_k^i)_{i=1}^{n_k}\}$, where $\boldsymbol{x}_k^i \in \boldsymbol{X}$ is a data sample in task $k$ and $y_k^i \in \boldsymbol{Y}_k$ is the class label of $\boldsymbol{x}_k^i$ and $\boldsymbol{Y}_k$ is the set of classes of task $k$. The key challenge of CL is \textit{catastrophic forgetting} (CF)~\cite{McCloskey1989}, which refers to the situation where the learning of a new task may significantly change the network weights learned for old tasks, degrading the model accuracy for old tasks. Researchers have mainly worked on two CL problems: \textit{class incremental/continual learning} (CIL) and \textit{task incremental/continual learning} (TIL)~\citep{Dhar2019CVPR,van2019three}.
The main difference between CIL and TIL is that in TIL, the task-id $k$ is provided for each test sample $\boldsymbol{x}$ during testing so that only the model for task $k$ is used to classify $\boldsymbol{x}$, while in CIL, the task-id $k$ for each test sample is not provided.
Existing CL techniques focus on either CIL or TIL~\citep{Parisi2019continual}. In general, CIL methods are designed to function without given task-id to perform class prediction over all classes in the tasks learned so far, and thus tend to forget the previous task performance due to plasticity for the new task. TIL methods are designed to function with a task-id for class prediction within the task. They are more stable to retain previous within-task knowledge, but incompetent if task-id is unknown (see Sec.~\ref{experiment}).
This paper proposes a novel and unified method called CLOM (\textit{C}ontinual \textit{L}earning based on \textit{O}OD detection and Task \textit{M}asking) to solve both problems by overcoming the limitations of CIL and TIL methods.
CLOM has two key mechanisms: (1) a task mask mechanism for protecting each task model to overcome CF, and (2) a learning method for building a model for each task based on out-of-distribution (OOD) detection.
The task mask mechanism is inspired by \textit{hard attention} in the TIL system HAT~\cite{Serra2018overcoming}.
The OOD detection based learning method for building each task model is quite different from the classic supervised learning used in existing TIL systems. It is, in fact, the key novelty and enabler for CLOM to work effectively for CIL.
OOD detection is stated as follows~\cite{bulusu2020anomalous}: Given a training set of $n$ classes, called the \textit{in-distribution} (IND) training data, we want to build a model that can assign correct classes to IND test data and reject or detect OOD test data that do not belong to any of the $n$ IND training classes. The OOD rejection capability makes a TIL model effective for CIL problem because during testing, if a test sample does not belong to any of the classes of a task, it will be rejected by the model of the task. Thus, only the task that the test sample belongs to will accept it and classify it to one of its classes. No task-id is needed. The OOD detection algorithm in CLOM is inspired by several recent advances in self-supervised learning, data augmentation~\cite{he2020momentum}, contrastive learning~\cite{oord2018representation,chen2020simple}, and their applications to OOD detection~\cite{golan2018deep,hendrycks2019using,tack2020csi}.
{The main contributions of this work are as follows:
\begin{itemize}
\item It proposes a novel CL method CLOM, which is essentially a TIL system in training, but solves both TIL and CIL problems in testing. Existing methods mainly solve either TIL or CIL problem, but are weak on the other problem.
\item CLOM uses task masks to protect the model for each task to prevent CF and OOD detection to build each task model, which to our knowledge has not been done before. More importantly, since the task masks can prevent CF,
{continual learning performance gets better as OOD models get better too.}
\item CLOM needn't to save any replay data. If some past data is saved for output calibration, it performs even better.
\end{itemize}}
Experimental results show that CLOM improves state-of-the-art baselines by large margins.~The average TIL/CIL accuracy of CLOM over six different experiments is 87.6/67.9\% while that of the best baseline is only 82.4/55.0\%.
\section{Related Work}
Many approaches have been proposed to deal with CF in CL. Using regularization~\cite{kirkpatrick2017overcoming} and knowledge distillation~\cite{Li2016LwF} to minimize the change to previous models are two popular approaches~\cite{Jung2016less,Camoriano2017incremental,Fernando2017pathnet,Rannen2017encoder,Seff2017continual,zenke2017continual,Kemker2018fearnet,ritter2018online,schwarz2018progress,xu2018reinforced,castro2018end,Dhar2019CVPR,hu2019overcoming,lee2019overcoming,Liu2020}.
Memorizing some old examples and using them to adjust the old models in learning a new task is another popular approach (called \textit{replay})~\cite{Rusu2016,Lopez2017gradient,Rebuffi2017,Chaudhry2019ICLR,Cyprien2019episodic, hou2019learning,wu2019large,rolnick2019neurIPS, NEURIPS2020_b704ea2c_derpp,zhao2020maintaining,rajasegaran2020adaptive,Liu2020AANets}.
Several systems learn to generate pseudo training data of old tasks and use them to jointly train the new task, called \textit{pseudo-replay}~\cite{Gepperth2016bio,Kamra2017deep,Shin2017continual,wu2018memory,Seff2017continual,wu2018memory,Kemker2018fearnet,hu2019overcoming,hayes2019remind,Rostami2019ijcai,ostapenko2019learning}. CLOM differs from these approaches {as it does not replay any old task data to prevent forgetting and can function with/without saving some old data.} {\textit{Parameter isolation} is yet another popular approach, which makes different subsets (which may overlap) of the model parameters dedicated to different tasks using masks~\cite{Fernando2017pathnet,Serra2018overcoming,ke2020continual} or finding a sub-network for each task by pruning \cite{Mallya2017packnet, supsup2020,NEURIPS2019_3b220b43compact}. CLOM uses parameter isolation, but it differs from these approaches as it combines the idea of parameter isolation and OOD detection, which can solve both TIL and CIL problems effectively.}
There are also many other approaches, e.g., a network of experts~\cite{Aljundi2016expert} and generalized CL~\cite{mi2020generalized}, etc. {PASS~\cite{Zhu_2021_CVPR_pass} uses data rotation and regularizes them.
Co$^2$L~\cite{Cha_2021_ICCV_co2l} is a replay method that uses contrastive loss on old samples. CLOM also uses rotation and constrative loss, but its CF handling is based on masks. None of the existing methods uses OOD detection.}
CLOM is a TIL method that can also solve CIL problems. Many TIL systems exist~\cite{Fernando2017pathnet}, e.g., GEM~\cite{Lopez2017gradient}, A-GEM~\cite{Chaudhry2019ICLR}, UCL~\cite{ahn2019neurIPS}, ADP~\cite{Yoon2020Scalable}, CCLL~\cite{singh2020calibrating}, Orthog-Subspace~\cite{chaudhry2020continual}, HyperNet~\cite{von2019continual}, PackNet~\cite{Mallya2017packnet}, CPG~\cite{NEURIPS2019_3b220b43compact}, SupSup~\cite{supsup2020}, HAT~\cite{Serra2018overcoming}, and CAT~\cite{ke2020continual}.
GEM is a replay based method and UCL is a regularization based method. A-GEM~\citep{Chaudhry2019ICLR} improves GEM's efficiency. {ADP decomposes parameters into shared and adaptive parts to construct an order robust TIL system.} CCLL uses task-adaptive calibration on convolution layers. Orthog-Subspace learns each task in subspaces orthogonal to each other. {HyperNet initializes task-specific parameters conditioned on task-id. PackNet, CPG and SupSup find an isolated sub-network for each task and use it at inference.
HAT and CAT protect previous tasks by masking the important parameters.
Our CLOM also uses this general approach, but its model building for each task is based on OOD detection, which has not been used by existing TIL methods. It also performs well in the CIL setting (see Sec.~\ref{resultcomparison}). }
Related work on out-of-distribution (OOD) detection (also called open set detection) is also extensive.
Excellent surveys include~\cite{bulusu2020anomalous,geng2020recent}.
The model building part of CLOM is inspired by the latest method in~\cite{tack2020csi} based on data augmentation~\cite{he2020momentum} and contrastive learning~\cite{chen2020simple}.
\section{Proposed CLOM Technique}
\label{sec.CLOM}
As mentioned earlier, CLOM is a TIL system that solves both TIL and CIL problems.
It takes the parameter isolation approach for TIL. For each task $k$ (task-id), a model is built $f(h(\boldsymbol{x}, k))$ for the task, {where $h$ and $f$ are the feature extractor and the task specific classifier, respectively,}
and $f(h(\boldsymbol{x}, k))$ is the output of the neural network for task $k$.
{We omit task-id $k$ in $f$ to simplify notation.}
{In learning each task $k$, a task mask is also trained at the same time to protect the learned model of the task.
In testing for TIL, given the test sample $\boldsymbol{x}$ with task-id $k$ provided, CLOM uses the model for task $k$ to classify $\boldsymbol{x}$,
\begin{align}
\hat{y} = \argmax
f(h(\boldsymbol{x}, k))
\label{TCLeq}
\end{align}
For CIL (no task-id for each test sample $\boldsymbol{x}$), CLOM uses the model of every task and predicts the class $\hat{y}$ using
\begin{align}
\hat{y} = \argmax
\bigoplus_{1\leq k \leq t} f(h(\boldsymbol{x}, k)),
\label{CLOMeq}
\end{align}
where
$\bigoplus$ is the concatenation over the output space and
$t$ is the last task that has been learned. Eq.~\ref{CLOMeq} essentially chooses the class with the highest classification score among the classes of all tasks learned so far. This works because the OOD detection model for a task will give very low score for a test sample that does not belong to the task. {An overview of the prediction is illustrated in Fig.~\ref{diagram}(a).}
}
Note that in fact any CIL model can be used as a TIL model if task-id is provided. The conversion is made by selecting the corresponding task classification heads in testing. However, a conversion from a TIL method to a CIL method is not obvious as the system requires task-id in testing.
Some attempts have been made to predict the task-id in order to make a TIL method applicable to CIL problems.
iTAML~\cite{rajasegaran2020itaml} requires the test samples to come in batches and each batch must be from a single task. This may not be practical as test samples usually come one by one. CCG~\cite{abati2020conditional} builds a separate network to predict the task-id, which is also prone to forgetting. Expert Gate~\cite{Aljundi2016expert} constructs a separate autoencoder for each task.
Our CLOM classifies one test sample at a time and does not need to {construct another network for task-id prediction.}
{In the following subsections, we present (1) how to build an OOD model for a task and use it to make a prediction, and (2) how to learn task masks to protect each model. An overview of the process is illustrated in Fig. \ref{diagram}(b).}
\subsection{Learning Each Task as OOD Detection} \label{unifiedappr}
\begin{figure}
\centering
\includegraphics[width=3in]{diagram-20211203.pdf}
\caption{(a) Overview of TIL and CIL prediction. For TIL prediction, we consider only the output heads of the given task. For CIL prediction, we obtain all the outputs from task $1$ to current task $t$, and choose the label over the concatenated output.
(b) Training overview for a generic backbone. Each CNN includes a convolution layer with batch normalization, activation, or pooling depending on the configuration. In our experiments, we use AlexNet~\cite{NIPS2012_c399862d} and ResNet-18~\cite{he2016deep}. We feed-forward augmented batch $\tilde{\mathcal{B}}$ consisting of rotated images of different views ($\boldsymbol{x}^{1}_{0}, \boldsymbol{x}^{1}_{90}, \cdots$). We first train feature extractor by using contrastive loss (step 1). At each layer, binary mask is multiplied to the output of each convolution layer to learn important parameters of current task $t$. After training feature extractor, we fine-tune the OOD classifier (step 2).
}
\label{diagram}
\end{figure}
We borrow the latest OOD ideas based on contrastive learning~\cite{chen2020simple, he2020momentum} and data augmentation due to their excellent performance~\cite{tack2020csi}.
Since this section focuses on how to learn a single task based on OOD detection, we omit the task-id unless necessary. The OOD training process {is similar to that of contrastive learning}. It consists of two steps: 1) learning {the feature representation by} the composite $g \circ h$, where $h$ is a feature extractor and $g$ is a projection to contrastive representation, and 2) learning a linear classifier $f$ mapping the feature representation of $h$ to the label space. {In the following, we describe the training process: contrastive learning for feature representation learning (1), and OOD classifier building (2). We then explain how to make a prediction based on an ensemble method for both TIL and CIL settings, and how to further improve prediction using some saved past data.}
\subsubsection{Contrastive Loss for Feature Learning.}
{This is step 1 in Fig.~\ref{diagram}(b).} Supervised contrastive learning is used to try to repel data of different classes and align data of the same class more closely to make it easier to classify them. A key operation is data augmentation via transformations.
{Given a batch of $N$ samples, each sample $\boldsymbol{x}$ is first duplicated and each version then goes through \textit{three initial augmentations}
(also see Data Augmentation in Sec.~\ref{sec:traindetails}) to generate two different views $\boldsymbol{x}^{1}$ and $\boldsymbol{x}^{2}$ (they keep the same class label as $\boldsymbol{x}$).}
Denote the augmented batch by $\mathcal{B}$, which now has $2N$ samples. {In~\cite{hendrycks2019using,tack2020csi}}, it was shown that using image rotations is effective in learning OOD detection models because such rotations can effectively serve as out-of-distribution (OOD) training data.
For each augmented sample $\boldsymbol{x} \in \mathcal{B}$ with class $y$ of a task, we rotate $\boldsymbol{x}$ by $90^{\circ}, 180^{\circ}, 270^{\circ}$ to create three images, which are assigned \textit{three new classes} $y_1, y_2$, and $y_3$, respectively.
{This results in a larger augmented batch $\tilde{\mathcal{B}}$. Since we generate three new images from each $\boldsymbol{x}$,
the size of $\tilde{\mathcal{B}}$ is $8N$. For each original class, we now have 4 classes. For a sample $\boldsymbol{x} \in \tilde{\mathcal{B}}$, let $\mathcal{\tilde{B}}(\boldsymbol{x}) = \mathcal{\tilde{B}} \backslash \{ \boldsymbol{x} \}$
and
let $P(\boldsymbol{x}) \subset \tilde{\mathcal{B}} \backslash \{ \boldsymbol{x} \}$
be a set consisting of the data of the same class as $\boldsymbol{x}$ distinct from $\boldsymbol{x}$.
The contrastive representation of a sample $\boldsymbol{x}$ is $\boldsymbol{z}_{x} = g(h(\boldsymbol{x}, t)) / \| g(h(\boldsymbol{x}, t)) \|$, where $t$ is the current task.
In learning, we minimize the supervised contrastive loss~\cite{khosla2020supervised} of task $t$.
\begin{align}
\mathcal{L}_{c}
&= \frac{1}{8N} \sum_{ \boldsymbol{x} \in \tilde{\mathcal{B}}} \frac{-1}{| P(\boldsymbol{x}) |} \times \nonumber \\
& \sum_{\boldsymbol{p} \in P(\boldsymbol{x})} \log{ \frac{ \text{exp}( \boldsymbol{z}_{\boldsymbol{x}} \cdot \boldsymbol{z}_{\boldsymbol{p}} / \tau)}{\sum_{\boldsymbol{x}' \in \tilde{\mathcal{B}}(\boldsymbol{x}) } \text{exp}( \boldsymbol{z}_{\boldsymbol{x}} \cdot \boldsymbol{z}_{\boldsymbol{x}'} / \tau) } } \label{modsupclr}
\end{align}
where
$\tau$ is a scalar temperature, $\cdot$ is dot product, and $\times$ is multiplication.
The loss is reduced by repelling $\boldsymbol{z}$ of different classes and aligning $\boldsymbol{z}$ of the same class more closely.
$\mathcal{L}_{c}$ basically trains a feature extractor with good
representations for learning an OOD classifier.}
\subsubsection{Learning the Classifier.}
{This is step 2 in Fig.\ref{diagram}(b)}. {Given the feature extractor $h$ trained with the loss in Eq.~\ref{modsupclr}, we {\textit{freeze $h$} and} only \textit{fine-tune} the linear classifier $f$, which is trained to predict the classes of task $t$ \textit{and} the augmented rotation classes.} $f$ maps the feature representation to {the label space in} $\mathcal{R}^{4|\mathcal{C}^{t}|}$, where $4$ is the number of rotation classes including the original data with $0^{\circ}$ rotation and $|\mathcal{C}^{t}|$ is the number of {original} classes in task $t$. We minimize the cross-entropy loss,
\begin{align}
\mathcal{L}_{\text{ft}} = - \frac{1}{|\tilde{\mathcal{B}} |} \sum_{(\boldsymbol{x}, y) \in \tilde{\mathcal{B}}}
\log \tilde{p}(y | \boldsymbol{x}, t)
\label{3obj}
\end{align}
where $\text{ft}$ indicates fine-tune,
and
\begin{align}
\tilde{p}(y | \boldsymbol{x}, t) = \text{softmax} \left( f(h(\boldsymbol{x}, t))
\right) \label{probrotation}
\end{align}
where
$f(h(\boldsymbol{x}, t)) \in \mathcal{
R}^{4|\mathcal{C}^{t}|}$. The output $f(h(\boldsymbol{x}, t))$
includes the rotation classes. The linear classifier is trained to predict the original \textit{and} the rotation classes.
\subsubsection{Ensemble Class Prediction.} \label{ensemblesection}
{
We describe how to predict a label $y \in \mathcal{C}^{t}$ (TIL) and $y \in \mathcal{C}$ (CIL) ($\mathcal{C}$ is the set of original classes of all tasks)
We assume all tasks have been learned and their models are protected by masks, which we discuss in the next subsection.
We discuss the prediction of class label $y$ for a test sample $\boldsymbol{x}$ in the TIL setting first. Note that the network $f\circ h$ in Eq.~\ref{probrotation} returns logits for rotation classes (including the original task classes). Note also for each original class label $j_k \in \mathcal{C}^{k}$ (original classes) of a task $k$, we created three additional rotation classes. For class $j_k$, the classifier $f$ will produce four output values from its four rotation class logits, i.e., $f_{j_k,0}(h(\boldsymbol{x_0}, k))$, $f_{j_k,90}(h(\boldsymbol{x_{90}}, k))$, $f_{j_k,180}(h(\boldsymbol{x_{180}}, k))$, and $f_{j_k,270}(h(\boldsymbol{x_{270}}, k))$, where 0, 90, 180, and 270 represent $0^{\circ}, 90^{\circ}, 180^{\circ}$, and $270^{\circ}$ rotations respectively and $\boldsymbol{x}_0$ is the original $\boldsymbol{x}$.
We compute an ensemble output $f_{j_k}(h(\boldsymbol{x},k))$ for each class $j_k \in \mathcal{C}^{k}$ of task $k$,
\begin{align}
f_{j_k}(h(\boldsymbol{x},k))= \frac{1}{4} \sum_{\text{deg}} f_{j_k,\text{deg}} (h(\boldsymbol{x}_{\text{deg}}, k)) \label{ensemblelogit}.
\end{align}
The final TIL class prediction is made as follows (note, in TIL, task-id $k$ is provided in testing),
\begin{align}
\hat{y} = \argmax_{j_k \in \mathcal{C}^{k}} \Big\{ f_{j_k}(h(\boldsymbol{x},k)) \Big\} \label{ensembleclass}
\end{align}
We use Eq.~\ref{CLOMeq} to make the CIL class prediction, where {the final format} $f(h(\boldsymbol{x},k))$ for task $k$ is the following vector:
\begin{align}
f(h(\boldsymbol{x}, k)) = \left[ f_{1}(h(\boldsymbol{x}, k), \cdots , f_{|\mathcal{C}^{k}|}(h(\boldsymbol{x}, k)) \right]
\label{eq:afterensemble}
\end{align}
Our method so far memorizes no training samples and it already outperforms baselines (see Sec.~\ref{resultcomparison}).
}
\subsubsection{Output Calibration with Memory.}
\label{sec.calibration}
{The proposed method can make incorrect CIL prediction even with a perfect OOD model (rejecting every test sample that does not belong any class of the task). This happens because the task models are trained independently, and the outputs of different tasks may have different magnitudes. We use output calibration to ensure that the outputs are of similar magnitudes to be comparable by using some saved examples in a memory $\mathcal{M}$ with limited budget.}
At each task $k$, we store a fraction of the validation data $\{ (\boldsymbol{x}, y)\}$ into $\mathcal{M}_k$ for output calibration and update the memory $\mathcal{M} \leftarrow update(\mathcal{M}, \mathcal{M}_k)$ {by maintaining an equal number of samples per class}. We will detail the memory budget in Sec.~\ref{ablation}. Basically, we save the same number as the existing replay-based TIL/CIL methods.
After training the network $f$ for the current task $t$,
we freeze the model and use the saved data in $\mathcal{M}$ to find the scaling and shifting parameters $(\boldsymbol{\sigma}, \boldsymbol{\mu}) \in \mathcal{R}^t \times \mathcal{R}^t$ to calibrate the after-ensemble classification output $f(h(\boldsymbol{x},k))$ (Eq.~\ref{eq:afterensemble}) (i.e., using $\sigma_{k} f(h (\boldsymbol{x}, k)) + \mu_k)$ for each task $k$ by minimizing the cross-entropy loss,
\begin{align}
\mathcal{L}_{\text{cal}}
= - \frac{1}{| \mathcal{M} |} \sum_{(\boldsymbol{x}, y) \in \mathcal{M}}
\log{ p(y| \boldsymbol{x}) }
\end{align}
where $\mathcal{C}$ is the set of all classes of all tasks seen so far, and $p(c|\boldsymbol{x})$ is computed using softmax,
\begin{align}
\text{softmax} \Big[ \sigma_{1} f(h (\boldsymbol{x}, 1)) + \mu_1 ;
\cdots ;
\sigma_{t} f(h (\boldsymbol{x}, t)) + \mu_t \Big]
\label{eq:scaling}
\end{align}
Clearly, parameters $\sigma_k$ and $\mu_k$ do not change classification within task $k$ (TIL), but calibrates the outputs such that the ensemble outputs from all tasks are in comparable magnitudes.
For CIL inference at the test time, we make prediction by (following Eq.~\ref{CLOMeq}),
\begin{align}
\hat{y} = \argmax
\bigoplus_{1 \le k \le t}
\left[ \sigma_{k} f(h(\boldsymbol{x}, k)) + \mu_{k} \right]
\label{calibratedpred}
\end{align}
\subsection{Protecting OOD Model of Each Task Using Masks}
\label{sec.hat}
We now discuss the task mask mechanism for protecting the OOD model of each task to deal with CF.
In learning the OOD model for each task, CLOM at the same time also trains a \textit{mask} or \textit{hard attention} for each layer. To protect
{the shared feature extractor}
from previous tasks, their masks are used to block those important neurons so that the new task learning will not interfere with the parameters learned for previous tasks.
The main idea is to use sigmoid to approximate a 0-1 step function as \textit{hard attention} to mask (or block) or unmask (unblock) the information flow to protect the parameters learned for each previous task.
The hard attention (mask) at layer $l$ and task $t$ is defined as
\begin{align}
\boldsymbol{a}_{l}^{t} = u\left( s \boldsymbol{e}_{l}^{t} \right) \label{attn}
\end{align}
where $u$ is the sigmoid function, $s$ is a scalar, and $\boldsymbol{e}_{l}^{t}$ is a \textit{learnable} embedding {of the task-id input $t$}.
The attention is element-wise multiplied to the output $\boldsymbol{h}_{l}$ of layer $l$ as
\begin{align}
\boldsymbol{h}_{l}' = \boldsymbol{a}_{l}^{t} \otimes \boldsymbol{h}_{l}
\end{align}
{as depicted in Fig.~\ref{diagram}(b).} The sigmoid function $u$ converges to a 0-1 step function {as $s$ goes to infinity}. Since the true step function is not differentiable, a fairly large $s$ is chosen to achieve a differentiable pseudo step function based on sigmoid (see Appendix~\ref{hyperparam} for choice of $s$). The pseudo binary value of the attention determines how much information can flow forward and backward between adjacent layers.
Denote $\boldsymbol{h}_{l} = \text{ReLU}(\boldsymbol{W}_{l} \boldsymbol{h}_{l-1} + \boldsymbol{b}_{l})$, where $\text{ReLU}$ is the rectifier function. For units (neurons) of attention $\boldsymbol{a}_{l}^{t}$ with zero values, we can freely change the corresponding parameters in $\boldsymbol{W}_{l}$ and $\boldsymbol{b}_l$ without affecting the output $\boldsymbol{h}_{l}'$. For units of attention with non-zero values, changing the parameters will affect the output $\boldsymbol{h}_{l}'$ for which we need to protect from gradient flow in backpropagation to prevent forgetting.
Specifically, during training the new task $t$, we update parameters according to the attention so that the important parameters for
{past tasks ($1, ..., t-1$) are unmodified. Denote the accumulated attentions (masks) of all past tasks by
\begin{align}
\boldsymbol{a}_{l}^{<t} = \max ( \boldsymbol{a}_{l}^{< t-1}, \boldsymbol{a}_{l}^{t-1} )
\end{align}
where $\boldsymbol{a}_{l}^{<t}$ is the hard attentions of layer $l$ of all previous tasks, and $\max$ is an element-wise maximum\footnote{Some parameters from different tasks can be shared, which means some hard attention masks can be shared.} and $\boldsymbol{a}^{0}_{l}$ is a zero vector. Then the modified gradient is the following,
\begin{align}
\nabla w_{ij, l}' = \left( 1 - \min \left( a_{i, l}^{< t}, a_{j, l-1}^{< t} \right) \right) \nabla w_{ij, l} \label{hatupdate}
\end{align}
where $a^{< t}_{i, l}$ indicates $i$'th unit of $\boldsymbol{a}^{< t}_{l}$ and $l=1, ..., L-1$}.
This reduces the gradient if the corresponding units' attentions at layers $l$ and $l-1$ are non-zero\footnote{{By construction, if $\boldsymbol{a}_{l}^{t}$ becomes $\boldsymbol{1}$ for all layers, the gradients are zero and the network is at maximum capacity. However, the network capacity can increase by adding more parameters.}}.
We do not apply hard attention on the last layer $L$ because it is a task-specific layer.
To encourage sparsity in $\boldsymbol{a}_{l}^{t}$ and parameter sharing with $\boldsymbol{a}_{l}^{<t}$,
a regularization ($\mathcal{L}_{r}$) for attention at task $t$ is defined as
\begin{align}
\mathcal{L}_{r} = \lambda_{t} \frac{\sum_{l}\sum_{i} a_{i, l}^{t}\left( 1 - a_{i, l}^{< t} \right)}{\sum_{l}\sum_{i} \left( 1 - a_{i, l}^{< t} \right)}
\end{align}
where
$\lambda_t$ is a scalar hyperparameter.
For flexibility, we denote $\lambda_t$ for each task $t$. However, in practice, we use the same $\lambda_t$ for all $t \neq 1$.
The final objective to be minimized for task $t$ with hard attention is (see Fig.~\ref{diagram}(b))
\begin{align}
\mathcal{L} = \mathcal{L}_{c} + \mathcal{L}_r \label{1obj}
\end{align}
where $\mathcal{L}_{c}$ is the contrastive loss function (Eq.~\ref{modsupclr}).
By protecting important parameters from changing during training,
the neural network effectively alleviates CF.
\section{Experiments} \label{experiment}
\textbf{Evaluation Datasets:} Four image classification CL benchmark datasets are used in our experiments.
(1) \textbf{MNIST}
:\footnote{http://yann.lecun.com/exdb/mnist/
}
handwritten digits of 10 classes (digits) with 60,000 examples for training and 10,000 examples for testing.
(2) \textbf{CIFAR-10}~\citep{Krizhevsky2009learning}:\footnote{https://www.cs.toronto.edu/~kriz/cifar.html} 60,000 32x32 color images of 10 classes with 50,000 for training and 10,000 for testing.
(3)~\textbf{CIFAR-100}~\citep{Krizhevsky2009learning}:\footnote{https://www.cs.toronto.edu/~kriz/cifar.html} 60,000 32x32 color images of 100 classes with 500 images per class for training and 100 per class for testing.~(4)~\textbf{Tiny-ImageNet}~\cite{Le2015TinyIV}:\footnote{http://tiny-imagenet.herokuapp.com} 120,000 64x64 color images of 200 classes with 500 images per class for training and 50 images per class for validation, and 50 images per class for testing.
Since the test data has no labels in this dataset, we use the validation data as the test data as in \cite{Liu2020}.
\textbf{Baseline Systems:}
We compare our CLOM with both the classic and the most recent state-of-the-art
CIL and TIL methods.
We also include CLOM(-c), which is CLOM without calibration (which already outperforms the baselines).
For CIL baselines, we consider seven \textit{replay} methods, \textbf{LwF.R} (replay version of LwF~\cite{Li2016LwF} with better results~\cite{Liu_2020_CVPR}), \textbf{iCaRL} \cite{Rebuffi2017}, \textbf{BiC} \cite{wu2019large}, \textbf{A-RPS}~\cite{rajasegaran2020adaptive},
\textbf{Mnemonics} \cite{Liu_2020_CVPR}, \textbf{DER++} \cite{NEURIPS2020_b704ea2c_derpp}, and \textbf{Co$^2$L} \cite{Cha_2021_ICCV_co2l};
one {\textit{pseudo-replay}} method \textbf{CCG} \cite{abati2020conditional}\footnote{iTAML~\cite{rajasegaran2020itaml} is not included as they require a batch of test data from the same task to predict the task-id. When each batch has only one test sample, which is our setting, it is very weak. For example, iTAML TIL/CIL accuracy is only 35.2\%/33.5\% on CIFAR100 10 tasks. Expert Gate (EG)~\cite{Aljundi2016expert} is also weak. For example, its TIL/CIL accuracy is 87.2/43.2 on MNIST 5 tasks. Both iTAML and EG are much weaker than many baselines. };
one \textit{orthogonal projection} method
\textbf{OWM}~\cite{zeng2019continuous}; and a \textit{multi-classifier} method
\textbf{MUC}~\cite{Liu2020}; and \textit{a prototype augmentation method} \textbf{PASS}~\cite{Zhu_2021_CVPR_pass}. OWM, MUC, and PASS \textit{do not save any samples} from previous tasks.
TIL baselines include \textbf{HAT}~\cite{Serra2018overcoming}, \textbf{HyperNet}~\cite{von2019continual}, and \textbf{SupSup}~\cite{supsup2020}.
As noted earlier, CIL methods can also be used for TIL. In fact, TIL methods may be used for CIL too, but the results are very poor. We include them in our comparison
\subsection{Training Details}
\label{sec:traindetails}
For all experiments, we use 10\%
of training data as the validation set to grid-search for good hyperparameters.
For minimizing the contrastive loss, we use LARS \cite{you2017large} for 700 epochs with initial learning rate 0.1. We linearly increase the learning rate by 0.1 per epoch for the first 10 epochs until 1.0 and then decay it by
cosine scheduler \cite{loshchilov2016sgdr} after 10 epochs without restart as in~\cite{chen2020simple, tack2020csi}. For fine-tuning the classifier $f$, we use SGD for 100 epochs with learning rate 0.1 and reduce the learning rate by 0.1 at 60, 75, and 90 epochs. The full set of hyperparameters is given in Appendix~\ref{hyperparam}.
We follow the recent baselines (A-RPS, DER++, PASS and Co$^2$L) and use the same class split and backbone architecture for both CLOM and baselines.
For MNIST and CIFAR-10, we split 10 classes into 5 tasks where each task has 2 classes in consecutive order. We save 20 random samples per class from the validation set for output calibration. This number is commonly used in replay methods~\cite{Rebuffi2017,wu2019large,rajasegaran2020adaptive,Liu_2020_CVPR}. MNIST consists of single channel images of size 1x28x28. Since the contrastive learning \cite{chen2020simple} relies on color changes, we copy the channel to make 3-channels. For MNIST and CIFAR-10, we use AlexNet-like architecture \cite{NIPS2012_c399862d} and ResNet-18 \cite{he2016deep} respectively for both CLOM and baselines.
For CIFAR-100, we conduct two experiments. We split 100 classes into 10 tasks and 20 tasks where each task has 10 and 5 classes, respectively, in consecutive order. We use 2000 memory budget as in \cite{Rebuffi2017}, saving 20 random samples per class from the validation set for output calibration. We use the same ResNet-18 structure for CLOM and baselines, but we increase the number of channels twice to learn more tasks.
For Tiny-ImageNet, we follow \cite{Liu2020} and resize the original images of size 3x64x64 to 3x32x32 so that the same ResNet-18 of CIFAR-100 experiment setting can be used. We split 200 classes into 5 tasks (40 classes per task) and 10 tasks (20 classes per task) in consecutive order, respectively.
To have the same memory budget of 2000 as for CIFAR-100, we save 10 random samples per class from the validation set for output calibration.
\textbf{Data Augmentation.} {For baselines, we use data augmentations used in their original papers. For CLOM, following \citep{chen2020simple, tack2020csi}, we use \textit{three initial augmentations} (see Sec.~\ref{unifiedappr}) (i.e., \textit{horizontal flip}, \textit{color change} (\textit{color jitter} and \textit{grayscale}), and \textit{Inception crop} \cite{inception}) and four \textit{rotations} (see Sec.~\ref{unifiedappr}). Specific details about these transformations are given in Appendix~\ref{aug_details}.}
\begin{table*}[t]
\centering
\resizebox{2.1\columnwidth}{!}{
\begin{tabular}{l c c c c c c c c c c c c | c c}
&&&&&&&&&&&&&&\\[-1.1em]
\toprule
\multirow{2}{*}{Method} & \multicolumn{2}{c}{MNIST-5T} & \multicolumn{2}{c}{CIFAR10-5T} & \multicolumn{2}{c}{CIFAR100-10T} & \multicolumn{2}{c}{CIFAR100-20T} & \multicolumn{2}{c}{T-ImageNet-5T} & \multicolumn{2}{c}{T-ImageNet-10T} & \multicolumn{2}{|c}{Average}\\
{} & TIL & CIL & TIL & CIL & TIL & CIL & TIL & CIL & TIL & CIL & TIL & CIL & TIL & CIL \\
&&&&&&&&&&&&&&\\[-1em]
\midrule
\multicolumn{13}{c}{\textbf{CIL Systems}} & \multicolumn{2}{|c}{} \\
\hline
OWM & 99.7\scalebox{0.8}{$\pm$0.03} & 95.8\scalebox{0.8}{$\pm$0.13} & 85.2\scalebox{0.8}{$\pm$0.17} & 51.7\scalebox{0.8}{$\pm$0.06} & 59.9\scalebox{0.8}{$\pm$0.84} & 29.0\scalebox{0.8}{$\pm$0.72} & 65.4\scalebox{0.8}{$\pm$0.07} & 24.2\scalebox{0.8}{$\pm$0.11} & 22.4\scalebox{0.8}{$\pm$0.87} & 10.0\scalebox{0.8}{$\pm$0.55} & 28.1\scalebox{0.8}{$\pm$0.55} & 8.6\scalebox{0.8}{$\pm$0.42} & 60.1 & 36.5 \Tstrut \\
MUC & 99.9\scalebox{0.8}{$\pm$0.02} & 74.6\scalebox{0.8}{$\pm$0.45} & 95.2\scalebox{0.8}{$\pm$0.24} & 53.6\scalebox{0.8}{$\pm$0.95} & 76.9\scalebox{0.8}{$\pm$1.27} & 30.0\scalebox{0.8}{$\pm$1.37} & 73.7\scalebox{0.8}{$\pm$1.27} & 14.4\scalebox{0.8}{$\pm$0.93} & 55.8\scalebox{0.8}{$\pm$0.26} & 33.6\scalebox{0.8}{$\pm$0.18} & 47.2\scalebox{0.8}{$\pm$0.22} & 17.4\scalebox{0.8}{$\pm$0.17} & 74.8 & 37.3 \\
PASS$^{\dagger}$& 99.5\scalebox{0.8}{$\pm$0.14} & 76.6\scalebox{0.8}{$\pm$1.67} & 83.8\scalebox{0.8}{$\pm$0.68} & 47.3\scalebox{0.8}{$\pm$0.97} & 72.4\scalebox{0.8}{$\pm$1.23} & 36.8\scalebox{0.8}{$\pm$1.64} & 76.9\scalebox{0.8}{$\pm$0.77} & 25.3\scalebox{0.8}{$\pm$0.81} & 50.3\scalebox{0.8}{$\pm$1.97} & 28.9\scalebox{0.8}{$\pm$1.36} & 47.6\scalebox{0.8}{$\pm$0.38} & 18.7\scalebox{0.8}{$\pm$0.58} & 71.8 & 38.9 \\
LwF.R & \textbf{99.9}\scalebox{0.8}{$\pm$0.09} & 85.0\scalebox{0.8}{$\pm$3.05} & 95.2\scalebox{0.8}{$\pm$0.30} & 54.7\scalebox{0.8}{$\pm$1.18} & 86.2\scalebox{0.8}{$\pm$1.00} & 45.3\scalebox{0.8}{$\pm$0.75} & 89.0\scalebox{0.8}{$\pm$0.45} & 44.3\scalebox{0.8}{$\pm$0.46} & 56.4\scalebox{0.8}{$\pm$0.48} & 32.2\scalebox{0.8}{$\pm$0.50} & 55.3\scalebox{0.8}{$\pm$0.35} & 24.3\scalebox{0.8}{$\pm$0.26} & 80.3 & 47.6 \\
iCaRL$^*$ & \textbf{99.9}\scalebox{0.8}{$\pm$0.09} & 96.0\scalebox{0.8}{$\pm$0.42} & 94.9\scalebox{0.8}{$\pm$0.34} & 63.4\scalebox{0.8}{$\pm$1.11} & 84.2\scalebox{0.8}{$\pm$1.04} & 51.4\scalebox{0.8}{$\pm$0.99} & 85.7\scalebox{0.8}{$\pm$0.68} & 47.8\scalebox{0.8}{$\pm$0.48} & 54.3\scalebox{0.8}{$\pm$0.59} & 37.0\scalebox{0.8}{$\pm$0.41} & 52.7\scalebox{0.8}{$\pm$0.37} & 28.3\scalebox{0.8}{$\pm$0.18} & 78.6 & 54.0 \\
Mnemonics$^{\dagger *}$ & \textbf{99.9}\scalebox{0.8}{$\pm$0.03} & 96.3\scalebox{0.8}{$\pm$0.36} & 94.5\scalebox{0.8}{$\pm$0.46} & 64.1\scalebox{0.8}{$\pm$1.47} & 82.3\scalebox{0.8}{$\pm$0.30} & 51.0\scalebox{0.8}{$\pm$0.34} & 86.2\scalebox{0.8}{$\pm$0.46} & 47.6\scalebox{0.8}{$\pm$0.74} & 54.8\scalebox{0.8}{$\pm$0.16} & 37.1\scalebox{0.8}{$\pm$0.46} & 52.9\scalebox{0.8}{$\pm$0.66} & 28.5\scalebox{0.8}{$\pm$0.72} & 78.5 & 54.1 \\
BiC & \textbf{99.9}\scalebox{0.8}{$\pm$0.04} & 85.1\scalebox{0.8}{$\pm$1.84} & 91.1\scalebox{0.8}{$\pm$0.82} & 57.1\scalebox{0.8}{$\pm$1.09} & 87.6\scalebox{0.8}{$\pm$0.28} & 51.3\scalebox{0.8}{$\pm$0.59} & 90.3\scalebox{0.8}{$\pm$0.26} & 40.1\scalebox{0.8}{$\pm$0.77} & 44.7\scalebox{0.8}{$\pm$0.71} & 20.2\scalebox{0.8}{$\pm$0.31} & 50.3\scalebox{0.8}{$\pm$0.65} & 21.2\scalebox{0.8}{$\pm$0.46} & 77.3 & 45.8 \\
DER++ & 99.7\scalebox{0.8}{$\pm$0.08} & 95.3\scalebox{0.8}{$\pm$0.69} & 92.2\scalebox{0.8}{$\pm$0.48} & 66.0\scalebox{0.8}{$\pm$1.27} & 84.2\scalebox{0.8}{$\pm$0.47} & 55.3\scalebox{0.8}{$\pm$0.10} & 86.6\scalebox{0.8}{$\pm$0.50} & 46.6\scalebox{0.8}{$\pm$1.44} & 58.0\scalebox{0.8}{$\pm$0.52} & 36.0\scalebox{0.8}{$\pm$0.42} & 59.7\scalebox{0.8}{$\pm$0.6} & 30.5\scalebox{0.8}{$\pm$0.30} & 80.1 & 55.0 \\
A-RPS & & & & & & 60.8 & & 53.5 & & & & \\
CCG & & \textbf{97.3} & & 70.1 & & & & & & & & \\
Co$^2$L & & & 93.4 & 65.6 & & & & & & & & \\
\hline
\multicolumn{13}{c}{\textbf{TIL Systems}} & \multicolumn{2}{|c}{} \Tstrut\Bstrut \\
\hline
HAT & \textbf{99.9}\scalebox{0.8}{$\pm$0.02} & 81.9\scalebox{0.8}{$\pm$3.73} & 96.7\scalebox{0.8}{$\pm$0.18} & 62.7\scalebox{0.8}{$\pm$1.46} & 84.0\scalebox{0.8}{$\pm$0.23} & 41.1\scalebox{0.8}{$\pm$0.93} & 85.0\scalebox{0.8}{$\pm$0.85} & 26.0\scalebox{0.8}{$\pm$0.83} & 61.2\scalebox{0.8}{$\pm$0.72} & 38.5\scalebox{0.8}{$\pm$1.85} & 63.8\scalebox{0.8}{$\pm$0.41} & 29.8\scalebox{0.8}{$\pm$0.65} & 81.8 & 46.6 \Tstrut \\
HyperNet & 99.7\scalebox{0.8}{$\pm$0.05} & 49.1\scalebox{0.8}{$\pm$5.52} & 94.9\scalebox{0.8}{$\pm$0.54} & 47.4\scalebox{0.8}{$\pm$5.78} & 77.3\scalebox{0.8}{$\pm$0.45} & 29.7\scalebox{0.8}{$\pm$2.19} & 83.0\scalebox{0.8}{$\pm$0.60} & 19.4\scalebox{0.8}{$\pm$1.44} & 23.8\scalebox{0.8}{$\pm$1.21} & 8.8\scalebox{0.8}{$\pm$0.98} & 27.8\scalebox{0.8}{$\pm$0.86} & 5.8\scalebox{0.8}{$\pm$0.56} & 67.8 & 26.7 \\
SupSup & 99.6\scalebox{0.8}{$\pm$0.09} & 19.5\scalebox{0.8}{$\pm$0.15} & 95.3\scalebox{0.8}{$\pm$0.27} & 26.2\scalebox{0.8}{$\pm$0.46} & 85.2\scalebox{0.8}{$\pm$0.25} & 33.1\scalebox{0.8}{$\pm$0.47} & 88.8\scalebox{0.8}{$\pm$0.18} & 12.3\scalebox{0.8}{$\pm$0.30} & 61.0\scalebox{0.8}{$\pm$0.62} & 36.9\scalebox{0.8}{$\pm$0.57} & 64.4\scalebox{0.8}{$\pm$0.20} & 27.0\scalebox{0.8}{$\pm$0.45} & 82.4 & 25.8 \\
\hline
CLOM(-c) & \textbf{99.9}\scalebox{0.8}{$\pm$0.00} & 94.4\scalebox{0.8}{$\pm$0.26} & \textbf{98.7}\scalebox{0.8}{$\pm$0.06} & 87.8\scalebox{0.8}{$\pm$0.71} & \textbf{92.0}\scalebox{0.8}{$\pm$0.37} & 63.3\scalebox{0.8}{$\pm$1.00} & \textbf{94.3}\scalebox{0.8}{$\pm$0.06} & 54.6\scalebox{0.8}{$\pm$0.92} & \textbf{68.4}\scalebox{0.8}{$\pm$0.16} & 45.7\scalebox{0.8}{$\pm$0.26} & \textbf{72.4}\scalebox{0.8}{$\pm$0.21} & 47.1\scalebox{0.8}{$\pm$0.18} & 87.6 & 65.5 \Tstrut \\
CLOM & \textbf{99.9}\scalebox{0.8}{$\pm$0.00} & 96.9\scalebox{0.8}{$\pm$0.30} & \textbf{98.7}\scalebox{0.8}{$\pm$0.06} & \textbf{88.0}\scalebox{0.8}{$\pm$0.48} & \textbf{92.0}\scalebox{0.8}{$\pm$0.37} & \textbf{65.2}\scalebox{0.8}{$\pm$0.71} & \textbf{94.3}\scalebox{0.8}{$\pm$0.06} & \textbf{58.0}\scalebox{0.8}{$\pm$0.45} & \textbf{68.4}\scalebox{0.8}{$\pm$0.16} & \textbf{51.7}\scalebox{0.8}{$\pm$0.37} & \textbf{72.4}\scalebox{0.8}{$\pm$0.21} & \textbf{47.6}\scalebox{0.8}{$\pm$0.32} & \textbf{87.6} & \textbf{67.9} \\
\bottomrule
\end{tabular}
}
\caption{
Average accuracy over all classes after the last task is learned. -xT: x number of tasks. {$\dagger$: In their original paper, PASS and Mnemonics use the first half of classes to pre-train before CL. Their results are 50.1\% and 53.5\% on CIFAR100-10T respectively, but they are still lower than CLOM without pre-training. In our experiments, no pre-training is used for fairness.} $^{*}$: \textbf{iCaRL} and \textbf{Mnemonics} give both the final average accuracy as here and the \textit{average incremental accuracy} in the original papers. We report the \textit{average incremental accuracy} and \textit{network size} in Appendix~\ref{avg_inc_acc} and \ref{param_size}, respectively. The last two columns show the average TIL and CIL accuracy of each method over all datasets.
}
\label{Tab:maintable}
\end{table*}
\subsection{Results and Comparative Analysis} \label{resultcomparison}
As in existing works, we evaluate each method by two metrics: \textit{average classification accuracy} on all classes after training the last task, and \textit{average forgetting rate}~\cite{Liu_2020_CVPR}, $F^{t} = \frac{1}{t-1}\sum_{j=1}^{t-1} A_{j}^{\text{init}} - A_{j}^{t}$,
where $A_{j}^{\text{init}}$ is the $j$'th task's accuracy of the network right after the $j$'th task is learned and $A_{j}^{t}$ is the accuracy of the network on the $j$'th task data after learning the last task $t$. We report the forgetting rate after the final task $t$. Our results are averages of 5 random runs.
We present the main experiment results in Tab.~\ref{Tab:maintable}. The last two columns give the average TIL/CIL results of each system/row. For A-RPS, CCG, and Co$^2$L, we copy the results from their original papers as their codes are not released to the public or the public code cannot run on our system. The rows are grouped by CIL and TIL methods.
\textbf{CIL Results Comparison}. {Tab. \ref{Tab:maintable} shows that CLOM and CLOM(-c) achieve much higher CIL accuracy except for MNIST for which CLOM is slightly weaker than CCG by 0.4\%, but CLOM's result on CIFAR10-5T is about 18\% greater than CCG.}
For other datasets, CLOM improves by similar margins. {This is in contrast to the baseline TIL systems that are incompetent at the CIL setting when classes are predicted using Eq. \ref{CLOMeq}.}
Even \textit{without} calibration,
CLOM(-c) already outperforms all the baselines by large margins.
\textbf{TIL Results Comparison}.
{The gains by CLOM and CLOM(-c) over the baselines are also great in the TIL setting.}
CLOM and CLOM(-c) are the same as the output calibration does not affect TIL performance. For the two large datasets CIFAR100 and T-ImageNet, CLOM gains by large margins. {
This is due to contrastive learning and the OOD model. The replay based CIL methods (LwF.R, iCaRL, Mnemonics, BiC, and DER++) perform reasonably in the TIL setting, but our CLOM and CLOM(-c) are much better due to task masks which can protect previous models better with little CF.}
\begin{figure}[!ht]
\includegraphics[width=3.3in]{forget.pdf}
\caption{Average forgetting rate (\%) in the TIL setting as CLOM is a TIL system.
The lower the value, the better the method is. {CIL/TIL systems are shaded in blue/red, respectively (\textit{best viewed in color}).} A negative value indicates the task accuracy has increased from the initial accuracy.}
\label{forgetplot}
\end{figure}
\textbf{Comparison of Forgetting Rate}. Fig. \ref{forgetplot} shows the average forgetting rate of each method in the TIL setting. The CIL systems suffer from more forgetting as they are not designed for the TIL setting, which results in lower TIL accuracy (Tab. \ref{Tab:maintable}).
The TIL systems are highly {effective at preserving previous within-task knowledge}. This results in higher TIL accuracy on large dataset such as T-ImageNet, but they collapse when task-id is not provided (the CIL setting)
as shown in Tab. \ref{Tab:maintable}.
{CLOM is robust to forgetting as a TIL system and it also functions well without task-id.}
We report only the forgetting rate in the TIL setting because our CLOM is essentially a TIL method and not a CIL system by design. The degrading CIL accuracy of CLOM is mainly because the OOD model for each task is not perfect.
\subsection{Ablation Studies} \label{ablation}
\begin{table}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{lcccccccc}
\toprule
& \multicolumn{4}{c}{CIFAR10-5T} & \multicolumn{4}{c}{CIFAR100-10T} \\
& \multicolumn{1}{l}{AUC} & \multicolumn{1}{l}{TaskDR} & \multicolumn{1}{c}{TIL} & \multicolumn{1}{l}{CIL} & \multicolumn{1}{l}{AUC} & \multicolumn{1}{l}{TaskDR} & \multicolumn{1}{c}{TIL} & \multicolumn{1}{l}{CIL} \\
\midrule
SupSup & 78.9 & 26.2 & 95.3 & 26.2 & 76.7 & 34.3 & 85.2 & 33.1 \\
SupSup (OOD in CLOM) & 88.9 & 82.3 & 97.2 & 81.5 & 84.9 & 63.7 & 90.0 & 62.1 \\
\midrule
CLOM (ODIN) & 82.9 & 63.3 & 96.7 & 62.9 & 77.9 & 43.0 & 84.0 & 41.3 \\
CLOM & 92.2 & 88.5 & 98.7 & 88.0 & 85.0 & 66.8 & 92.0 & 65.2 \\
\midrule
CLOM (w/o OOD) & 90.3 & 83.9 & 98.1 & 83.3 & 82.6 & 59.5 & 89.8 & 57.5 \\
\bottomrule
\end{tabular}
}
\caption{{TIL and CIL results improve with better OOD detection. Column AUC gives the average AUC score for the OOD detection method as used within each system on the left. Column TaskDR gives \textit{task detection rate}. TIL and CIL results are average accuracy values. {SupSup and CLOM variants are calibrated with 20 samples per class.}}
}
\label{betterood}
\end{table}
{\textbf{Better OOD for better continual learning.} We show that (1) an existing CL model can be improved by a good OOD model and (2) CLOM's results will deteriorate if a weaker OOD model is applied.
To isolate effect of OOD detection on changes in CIL performance, we further define \textit{task detection} and \textit{task detection rate}. For a test sample from a class of task $j$, if it is predicted to a class of task $m$ and $j=m$, the task detection is correct for this test instance. The \textit{task detection rate} $\sum_{\mathbf{x} \in \mathcal{D}^{\text{test}} } 1_{j = m}/N$, where $N$ is the total number of test instances in $\mathcal{D}^{\text{test}}$, is the rate of correct task detection. We measure the performance of OOD detection using AUC (Area Under the ROC Curve) averaged over all tasks. AUC is the main measure used in OOD detection papers. We conduct experiments on CIFAR10-5T and CIFAR100-10T.
For (1), we use the TIL baseline SupSup as it displays a strong TIL performance and is robust to forgetting like CLOM. We replace SupSup’s task learner with the OOD model in CLOM. Tab.~\ref{betterood} shows that the OOD method in CLOM improves SupSup (SupSup (OOD in CLOM)) greatly. It shows that our approach is applicable to different TIL systems.
For (2), we replace CLOM’s OOD method with a weaker OOD method ODIN~\cite{liang2018enhancing}. We see in Tab.~\ref{betterood} that task detection rate, TIL, and CIL results all drop markedly with ODIN (CLOM (ODIN)).
}
{\textbf{CLOM without OOD detection.} In this case, CLOM uses contrastive learning and data augmentation,
but does not use the rotation classes in classification. Note that the rotation classes are basically regarded as OOD data in training and for OOD detection in testing. CLOM (w/o OOD) in Tab.~\ref{betterood} represents this CLOM variant. We see that CLOM (w/o OOD) is much weaker than the full CLOM. This indicates that the improved results of CLOM over baselines are not only due to contrastive learning and data augmentation but also significantly due to OOD detection.}
\begin{table}
\parbox{.50\linewidth}{
\centering
\resizebox{.50\columnwidth}{!}{
\begin{tabular}{lcc}
\toprule
\multicolumn{1}{c}{$|\mathcal{M}|$} & (a) & (b) \\
\midrule
0 & 63.3\scalebox{0.8}{$\pm$1.00} & 54.6\scalebox{0.8}{$\pm$0.92} \\
5 & 64.9\scalebox{0.8}{$\pm$0.67} & 57.7\scalebox{0.8}{$\pm$0.50} \\
10 & 65.0\scalebox{0.8}{$\pm$0.71} & 57.8\scalebox{0.8}{$\pm$0.53} \\
15 & 65.1\scalebox{0.8}{$\pm$0.71} & 57.9\scalebox{0.8}{$\pm$0.44} \\
20 & 65.2\scalebox{0.8}{$\pm$0.71} & 58.0\scalebox{0.8}{$\pm$0.45} \\
\bottomrule
\end{tabular}
}
}
\hfill
\parbox{.46\linewidth}{
\centering
\resizebox{.46\columnwidth}{!}{
\begin{tabular}{lccc}
\toprule
s & $F^{5}$ & AUC & CIL \\
\midrule
1 & 48.6 & 58.8 & 10.0 \\
100 & 13.3 & 82.7 & 67.7 \\
300 & 8.2 & 83.3 & 72.0 \\
500 & 0.2 & 91.8 & 87.2 \\
700 & 0.1 & 92.2 & 88.0 \\
\bottomrule
\end{tabular}
}
}
\caption{(\textit{Left}) shows changes of accuracy with the number of samples saved per class for output calibration. (a) and (b) are CIFAR100-10T and CIFAR100-20T, respectively. $|\mathcal{M}|=k$ indicates $k$ samples are saved per class. {(\textit{Right}) shows that weaker forgetting mechanism results in larger forgetting and lower AUC, thus lower CIL. For $s=1$, the pseudo-step function becomes the standard sigmoid, thus parameters are hardly protected. $F^{5}$ is the forgetting rate over 5 tasks.}}
\label{memory_and_s}
\end{table}
\textbf{Effect of the number of saved samples for calibration.}
Tab. \ref{memory_and_s} (left) reveals that the output calibration is still effective with a small number of saved samples per class ($|\mathcal{M}|$). For both CIFAR100-10T and CIFAR100-20T, {CLOM achieves competitive performance by using only 5 samples per class.} The accuracy improves and the variance decreases with the number of saved samples.
{\textbf{Effect of $s$ in Eq.~\ref{attn} on forgetting of CLOM.}
We need to use a strong forgetting mechanism for CLOM to be functional. Using CIFAR10-5T, we show how CLOM performs with different values of $s$ in hard attention or masking. The larger $s$ value, the stronger protection is used. Tab.~\ref{memory_and_s} (right) shows that average AUC and CIL decrease as the forgetting rate increases. This also supports the result in Tab.~\ref{betterood} that SupSup improves greatly with the OOD method in CLOM as it is also robust to forgetting. PASS and Co$^2$L underperform despite they also use rotation or constrastive loss as their forgetting mechanisms are weak.
}
\begin{table}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{lcccc}
\toprule
\multicolumn{1}{}{} & \multicolumn{2}{c}{CIFAR10-5T} & \multicolumn{2}{c}{CIFAR100-10T} \\
Aug. & TIL & CIL & TIL & CIL \\
\midrule
Hflip & (93.1, 95.3) & (49.1, 72.7) & (77.6, 84.0) & (31.1, 47.0) \\
Color & (91.7, 94.6) & (50.9, 70.2) & (67.2, 77.4) & (28.7, 41.8) \\
Crop & (96.1, 97.3) & (58.4, 79.4) & (84.1, 89.3) & (41.6, 60.3) \\
All & (97.6, 98.7) & (74.0, 88.0) & (88.1, 92.0) & (50.2, 65.2) \\
\bottomrule
\end{tabular}
}
\caption{Accuracy of CLOM variants when a single or all augmentations are applied.
Hflip: horizontal flip; Color: color jitter and grayscale; Crop: Inception~\cite{inception}. (num1, num2): accuracy without and with rotation.
}
\label{augmentation}
\end{table}
\textbf{Effect of data augmentations.}
{For data augmentation, we use three initial augmentations (i.e., \textit{horizontal flip}, \textit{color change} (color jitter and grayscale), \textit{Inception crop} \cite{inception}), which are commonly used in contrastive learning to build a single model. We additionally use rotation for OOD data in training. To evaluate the contribution of each augmentation
when task models are trained sequentially,
we train CLOM using one augmentation. We do not report their effects on forgetting as we experience rarely any forgetting (Fig. \ref{forgetplot} and Tab. \ref{memory_and_s}). Tab. \ref{augmentation} shows that the performance is lower when only a single augmentation is applied.
When all augmentations are applied, the TIL/CIL accuracies are higher. The rotation always improves the result when it is combined with other augmentations.
More importantly, when we use crop and rotation, we achieve higher CIL accuracy (79.4/60.3\% for CIFAR10-5T/CIFAR100-10T) than we use all augmentations without rotation (74.0/50.2\%).
This shows the efficacy of rotation in our system.}
\section{Conclusions}
This paper proposed a novel continual learning (CL) method called CLOM based on OOD detection and task masking that can perform both task incremental learning (TIL) and class incremental learning (CIL). Regardless whether it is used for TIL or CIL in testing, the training process is the same, which is an advantage over existing CL systems as they focus on either CIL or TIL and have limitations on the other problem. Experimental results showed that CLOM outperforms both state-of-the-art TIL and CIL methods by very large margins.
In our future work, we will study ways to improve efficiency and also accuracy.
\section*{Acknowledgments}
{\color{black}Gyuhak Kim, Sepideh Esmaeilpour and Bing Liu were supported in part by two National Science Foundation (NSF) grants (IIS-1910424 and IIS-1838770), a DARPA Contract HR001120C0023, a KDDI research contract, and a Northrop Grumman research gift.}
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\section*{Acknowledgments}
\begin{sloppypar}
The successful installation, commissioning, and operation of the Pierre
Auger Observatory would not have been possible without the strong
commitment and effort from the technical and administrative staff in
Malarg\"ue. We are very grateful to the following agencies and
organizations for financial support:
\end{sloppypar}
\begin{sloppypar}
Argentina -- Comisi\'on Nacional de Energ\'\i{}a At\'omica; Agencia Nacional de
Promoci\'on Cient\'\i{}fica y Tecnol\'ogica (ANPCyT); Consejo Nacional de
Investigaciones Cient\'\i{}ficas y T\'ecnicas (CONICET); Gobierno de la
Provincia de Mendoza; Municipalidad de Malarg\"ue; NDM Holdings and Valle
Las Le\~nas; in gratitude for their continuing cooperation over land
access; Australia -- the Australian Research Council; Brazil -- Conselho
Nacional de Desenvolvimento Cient\'\i{}fico e Tecnol\'ogico (CNPq);
Financiadora de Estudos e Projetos (FINEP); Funda\c{c}\~ao de Amparo \`a
Pesquisa do Estado de Rio de Janeiro (FAPERJ); S\~ao Paulo Research
Foundation (FAPESP) Grants No.~2019/10151-2, No.~2010/07359-6 and
No.~1999/05404-3; Minist\'erio da Ci\^encia, Tecnologia, Inova\c{c}\~oes e
Comunica\c{c}\~oes (MCTIC); Czech Republic -- Grant No.~MSMT CR LTT18004,
LM2015038, LM2018102, CZ.02.1.01/0.0/0.0/16{\textunderscore}013/0001402,
CZ.02.1.01/0.0/0.0/18{\textunderscore}046/0016010 and
CZ.02.1.01/0.0/0.0/17{\textunderscore}049/0008422; France -- Centre de Calcul
IN2P3/CNRS; Centre National de la Recherche Scientifique (CNRS); Conseil
R\'egional Ile-de-France; D\'epartement Physique Nucl\'eaire et Corpusculaire
(PNC-IN2P3/CNRS); D\'epartement Sciences de l'Univers (SDU-INSU/CNRS);
Institut Lagrange de Paris (ILP) Grant No.~LABEX ANR-10-LABX-63 within
the Investissements d'Avenir Programme Grant No.~ANR-11-IDEX-0004-02;
Germany -- Bundesministerium f\"ur Bildung und Forschung (BMBF); Deutsche
Forschungsgemeinschaft (DFG); Finanzministerium Baden-W\"urttemberg;
Helmholtz Alliance for Astroparticle Physics (HAP);
Helmholtz-Gemeinschaft Deutscher Forschungszentren (HGF); Ministerium
f\"ur Innovation, Wissenschaft und Forschung des Landes
Nordrhein-Westfalen; Ministerium f\"ur Wissenschaft, Forschung und Kunst
des Landes Baden-W\"urttemberg; Italy -- Istituto Nazionale di Fisica
Nucleare (INFN); Istituto Nazionale di Astrofisica (INAF); Ministero
dell'Istruzione, dell'Universit\'a e della Ricerca (MIUR); CETEMPS Center
of Excellence; Ministero degli Affari Esteri (MAE); M\'exico -- Consejo
Nacional de Ciencia y Tecnolog\'\i{}a (CONACYT) No.~167733; Universidad
Nacional Aut\'onoma de M\'exico (UNAM); PAPIIT DGAPA-UNAM; The Netherlands
-- Ministry of Education, Culture and Science; Netherlands Organisation
for Scientific Research (NWO); Dutch national e-infrastructure with the
support of SURF Cooperative; Poland -Ministry of Science and Higher
Education, grant No.~DIR/WK/2018/11; National Science Centre, Grants
No.~2013/08/M/ST9/00322, No.~2016/23/B/ST9/01635 and No.~HARMONIA
5--2013/10/M/ST9/00062, UMO-2016/22/M/ST9/00198; Portugal -- Portuguese
national funds and FEDER funds within Programa Operacional Factores de
Competitividade through Funda\c{c}\~ao para a Ci\^encia e a Tecnologia
(COMPETE); Romania -- Romanian Ministry of Education and Research, the
Program Nucleu within MCI (PN19150201/16N/2019 and PN19060102) and
project PN-III-P1-1.2-PCCDI-2017-0839/19PCCDI/2018 within PNCDI III;
Slovenia -- Slovenian Research Agency, grants P1-0031, P1-0385, I0-0033,
N1-0111; Spain -- Ministerio de Econom\'\i{}a, Industria y Competitividad
(FPA2017-85114-P and FPA2017-85197-P), Xunta de Galicia (ED431C
2017/07), Junta de Andaluc\'\i{}a (SOMM17/6104/UGR), Feder Funds, RENATA Red
Nacional Tem\'atica de Astropart\'\i{}culas (FPA2015-68783-REDT) and Mar\'\i{}a de
Maeztu Unit of Excellence (MDM-2016-0692); USA -- Department of Energy,
Contracts No.~DE-AC02-07CH11359, No.~DE-FR02-04ER41300,
No.~DE-FG02-99ER41107 and No.~DE-SC0011689; National Science Foundation,
Grant No.~0450696; The Grainger Foundation; Marie Curie-IRSES/EPLANET;
European Particle Physics Latin American Network; and UNESCO.
\end{sloppypar}
\section{\label{sec:into}Introduction}
Although the first cosmic rays having energies above $10^{19}~$eV were detected nearly 60 years ago~\cite{LSR1961,Linsley1963}, the question of their origin remains unanswered. In this paper we report a measurement of the energy spectrum of ultra-high energy cosmic rays (UHECRs) of unprecedented precision using data from the Pierre Auger Observatory. Accurate knowledge of the cosmic-ray flux as a function of energy is required to help discriminate between competing models of cosmic-ray origin. As a result of earlier work with the HiRes instrument~\cite{HiRes}, the Pierre Auger Observatory~\cite{Abraham:2008ru} and the Telescope Array~\cite{AbuZayyad:2012ru}, two spectral features
were
identified beyond reasonable doubt (see,
e.g.,~\cite{LSS2011,Watson_2014,KT2014,MV2015,DFS2017,ABDP2018} for recent reviews). These are a hardening of the spectrum at about $5{\times} 10^{18}$~eV (\textit{the ankle}) and a strong suppression of the flux at an energy about a decade higher. The results reported here are based on 215,030 events with energies above $2.5{\times}10^{18}$~eV.
The present measurement, together with recent observations of anisotropies in the arrival directions of cosmic rays on large angular scales above $8{\times} 10^{18}$~eV~\cite{Aab:2017tyv} and on intermediate angular scales above $3.9{\times} 10^{19}$~eV~\cite{Aab:2018chp}, and inferences on the mass composition~\cite{Aab:2014aea,Aab:2017cgk}, provide essential data against which to test phenomenological models of cosmic-ray origin. As part of a broad study of directional anisotropies, the large number of events used in the present analysis allows examination of the energy spectrum as a function of declination as reported below.
The determination of the flux of cosmic rays is a non-trivial exercise at any energy. It has long been recognised that $\simeq 70$ to 80\% of the energy carried by the primary particle is dissipated in the atmosphere through ionisation loss and thus, with ground detectors alone, one must resort to models of shower development to infer the primary energy. This is difficult as a quantitative knowledge of hadronic processes in the cascade is required. While at about $10^{17}$~eV the centre-of-mass energies encountered in collisions of primary cosmic rays with air nuclei are comparable to those
achieved at the Large Hadron Collider, details of the interactions of pions, which are key to the development of the cascade, are lacking, and the presence of unknown processes is also possible. Furthermore one has to make an assumption about the primary mass. Both conjectures lead to systematic uncertainties that are difficult, if not impossible, to assess. To counter these issues, methods using light produced by showers as they cross the atmosphere have been developed. In principle, this allows a calorimetric estimate of the energy. Pioneering work in the USSR in the 1950s~\cite{Nesterova} led to the use of Cherenkov radiation for this purpose, and this approach has been successfully adopted at the Tunka~\cite{Tunka} and Yakutsk~\cite{Yakutsk} arrays. The detection of fluorescence radiation, first achieved in Japan~\cite{Hara:1969} and, slightly later, in the USA~\cite{Bergeson:1977nw}, has been exploited particularly effectively in the Fly's Eye and HiRes projects to achieve the same objective. The Cherenkov method is less useful at the highest energies as the forward-beaming of the light necessitates the deployment of a large number of detectors while the isotropic emission of the fluorescence radiation enables showers to be observed at distances of $\simeq 30$~km from a single station. For both methods, the on-time is limited to moonless nights, and an accurate understanding of the aerosol content of the atmosphere is needed.
The Pierre Auger Collaboration introduced the concept of a hybrid observatory in which the bulk of the events used for spectrum determination is obtained with an array of detectors deployed on the ground and the integral of the longitudinal profile, measured using a fluorescence detector, is used to calibrate a shower-size estimate made with the ground array. This hybrid approach has led to a substantial improvement in the accuracy of reconstruction of fluorescence events and to a calorimetric estimate of the energy of the primary particles for events recorded during periods when the fluorescence detector cannot be operated. The hybrid approach has also been adopted by the Telescope Array Collaboration~\cite{AbuZayyad:2012ru}.
A consistent aim of the Auger Collaboration has been to make the derivation of the energy spectrum as free of assumptions about hadronic physics and the primary composition as possible. The extent to which this has been achieved can be judged from the details set out below. After a brief introduction in Sec.~\ref{sec:Auger} to relevant features of the Observatory and the data-set, the method of estimation of energy is discussed in Sec.~\ref{sec:rec}. In Sec.~\ref{sec:EnSp}, the approach to deriving the energy spectrum is described, including the procedure for evaluating the exposure and for unfolding the resolution effects, as well as a detailed discussion of the associated uncertainties and of the main spectral features. A search for any dependence of the energy spectrum on declination is discussed in Sec.~\ref{sec:declination}, while a comparison with previous works is given in Sec.~\ref{sec:ta}. The results from the measurement of the energy spectrum are summarized in the concluding Sec.~\ref{sec:discussion}.
\section{\label{sec:Auger} The Pierre Auger Observatory and the data sets}
\subsection{\label{sec:observatory} The Observatory}
The Pierre Auger Observatory is sited close to the city of Malarg\"ue, Argentina, at a latitude of 35.2$^{\circ}$ S with a mean atmospheric overburden of 875\,g/cm$^2$. A detailed description of the instrument has been published~\cite{ThePierreAuger:2015rma}, and only brief remarks concerning features relevant to the data discussed in this paper are given.
The surface detector (SD) array comprises about 1600 water-Cherenkov detectors laid out on a 1500~m triangular grid, covering an area of about 3000 km$^{2}$. Each SD has a surface area of 10 m$^{2}$ and a height of 1.2 m, holding 12 tonnes of ultra-pure water viewed by 3 ${\times} 9"$ photomultipliers (PMTs). The signals from the PMTs are digitised using 40 MHz 10-bit Flash Analog to Digital Converters (FADCs). Data collection is achieved in real time by searching on-line for temporal and spatial coincidences at a minimum of three locations. When this occurs, FADC data from the PMTs are acquired from which the pulse amplitude and time of detection of signals is obtained. The SD array is operated with a duty cycle close to $100\%$.
The array is over-looked from four locations, each having six Schmidt telescopes designed to detect fluorescence light emitted from shower excitations of atmospheric nitrogen. In each telescope, a camera with 440 hexagonal PMTs is used to collect light from a 13~m$^{2}$ mirror. These instruments, which form the fluorescence detector (FD), are operated on clear
nights with low background illumination
with an on-time of $\simeq 15\%$.
Atmospheric conditions at the site of the Observatory must be known for the reconstruction of the showers. Accordingly, comprehensive monitoring of the atmosphere, particularly of the aerosol content and the cloud cover, is undertaken as described in~\cite{ThePierreAuger:2015rma}. Weather stations are located close to the sites of the fluorescence telescopes.
Before the Global Data Assimilation system was adopted~\cite{Abreu:2012zg}, an extensive series of balloon flights was made to measure the humidity, temperature and pressure in the atmosphere as a function of altitude.
\subsection{\label{sec:data} The data sets}
The data set used for the measurement of the energy spectrum consists of extensive air showers (EAS) recorded by the SD array. EAS detected simultaneously by the SD and the FD play a key role in this work. Dubbed hybrid events, they are pivotal in the determination of the energy of the much more numerous SD events~\cite{Verzi2013}. We use here SD events with zenith angle $\theta<60^\circ$, as the reconstruction of showers at larger angles requires a different method due to an asymmetry induced in the distribution of the shower particles by the geomagnetic field and geometrical effects (see \cite{AugerHASRec}). A brief description of the reconstruction of SD and hybrid events is given in~\cite{FDNIM2010}: a more detailed description is in ~\cite{AugerRecoPaper}. We outline here features relevant to the present analysis.
The reconstruction of the SD events is used to determine the EAS geometry (impact point of the shower axis and arrival direction) as well as a shower-size estimator. To achieve this, the amplitude and the start-time of the signals, recorded at individual SD stations and quantified in terms of their response to a muon travelling vertically and centrally through it (a vertical equivalent muon or VEM), are used. The arrival direction is determined to about $1^\circ$ from the relative arrival times of these signals. The impact point and the shower-size estimator are in turn derived by fitting the signal amplitudes to a lateral distribution function (LDF) that decreases monotonically with distance from the shower axis. The shower-size estimator adopted is the signal at 1000~m from the axis, $S(1000)$. For the grid spacing of 1500~m, 1000~m is the optimal distance to minimize the uncertainties of the signal due to the imperfect knowledge of the functional form of the LDF in individual events~\cite{NKW2007}. The combined statistical and systematic uncertainty decreases from 15\% at a shower size of 10~VEM to 5\% at the highest shower sizes. The uncertainty on the impact point is of order 50~m. $S(1000)$ is influenced by changes in atmospheric conditions that affect shower development~\cite{AugerJINST2017}, and by the geomagnetic field that impacts on the shower particle-density~\cite{AugerJCAP2011}. Therefore, before using the shower-size estimator in the calibration procedure (Sec.~\ref{sec:rec}), corrections of order 2\% and 1\% for the atmospheric and geomagnetic effects, respectively, are made.
For the analysis in this paper, the SD reconstruction is carried out only for events in which the detector with the highest signal is surrounded by a hexagon of six stations that are fully operational. This requirement not only ensures adequate sampling of the shower but also allows evaluation of the aperture of the SD in a purely geometrical manner in the regime where the array is fully efficient~\cite{Abraham:2010zz}. As shown in Sec.~\ref{sec:EnSp}, such a regime is attained for events with $\theta<60^\circ$ at an energy $2.5{\times} 10^{18}$~eV. With these selection criteria, the SD data set used below consists of 215,030 events recorded between 1 January 2004 and 31 August 2018.
For hybrid events the reconstruction procedure exploits the amplitude and timing of the signals detected by each PMT in each telescope as well as additional timing information from the SD station with the highest signal. Combining the timing information from FD and SD improves the directional precision to $\simeq 0.6^\circ$~\cite{ThePierreAuger:2015rma}. Hybrid reconstruction provides in addition the longitudinal profile from which the depth of the shower maximum ($X_{\rm max}$) and the primary energy are extracted. The light signals in the FD PMTs are converted to the energy deposited in sequential depths in the atmosphere, taking into account the fluorescence and Cherenkov light contributions~\cite{MUFDRec} and their attenuation due to scattering. The longitudinal profile of the energy deposit is reconstructed by means of a fit to a modified Gaisser-Hillas profile~\cite{AugerLR2019}.
Integration of the longitudinal profile yields a calorimetric measure of the ionisation loss in the atmosphere which is supplemented by the addition of the undetected energy, or ``invisible energy'', carried into the ground by muons and neutrinos. We denote the sum of these two contributions, our estimate of the energy carried by the incoming primary particle, as $E_{\mathrm{FD}}$. The invisible-energy correction is estimated with a data-driven analysis and is about 14\% at $2.5 {\times} 10^{18}~$eV falling to about 12\% at $10^{20}~$eV~\cite{InvisibleEnergy2019}. The resolution of $E_\text{FD}~$ is 7.4\% at $2.5 {\times} 10^{18}~$eV and worsens with energy to 8.6\% at $6 {\times} 10^{19}~$eV. It is obtained by taking into account all uncorrelated uncertainties between different showers. In addition to the statistical uncertainty arising from the fit to the longitudinal profile, this resolution includes uncertainties in the detector response, in the models of the state of the atmosphere, and in the expected fluctuations from the invisible energy which, parameterized as a function of the calorimetric energy, is assumed to be identical for any primary of same energy. All the uncorrelated uncertainties are addressed in~\cite{DawsonICRC2019} with further details given in~\cite{MUFDRec}.
We note that at higher energies the showers are detected, on average, at larger distances from the FD telescopes because the detection and reconstruction efficiency at larger distances increases with energy.
This causes a worsening of the energy resolution because of the interplay between the uncertainty from the aerosols increasing with energy and the uncertainty from photoelectrons decreasing with energy.
The hybrid trigger efficiency, i.e. the probability of detecting a fluorescence event in coincidence with at least one triggered SD station, is 100\% at energies greater than $10^{18}$~eV, independent of the mass of the nuclear primaries~\cite{ExpoHybrid2011}. The hybrid data set used for the calibration of the SD events comprises 3,338 events with $E>3{\times} 10^{18}$~eV collected between 1 January 2004 and 31 December 2017. Other criteria for event selection are detailed in Sec.~\ref{sec:rec}.
\section{\label{sec:rec} Energy Estimation From Events Recorded by the Surface Array}
The energy calibration of the SD shower-size estimator against the energy derived from measurements with the FD is a two-step process. For a cosmic ray of a given energy, the value of $S(1000)$ depends on zenith angle because of the different atmospheric depths crossed by the corresponding shower. As detailed in Sec.~\ref{sec:SDrec}, we first correct for such an attenuation effect by using the Constant Intensity Cut (CIC) method~\cite{Hersil1961}. The calibration is then made between the corrected shower-size estimator, denoted by $S_{38}$, and the energy measured by the FD in hybrid events, $E_{\mathrm{FD}}$: the procedure to obtain the SD energy, $E_{\mathrm{SD}}$, is explained in Sec.~\ref{sec:EnCalib}. The systematic uncertainties associated with the SD energy scale thus obtained are described in Sec.~\ref{sec:EnScale}. Finally, the estimation of $E_{\mathrm{SD}}$ from $E_{\mathrm{FD}}$ allows us to derive the resolution, $\sigma_\textrm{SD}(E)$, as well as the bias, $b_\textrm{SD}(E)$, down to energies below which the detector is not fully efficient. We explain in Sec.~\ref{sec:SDResponse} the method used to measure $b_\textrm{SD}(E)$ and $\sigma_\textrm{SD}(E)$, from which we build the resolution function for the SD to be used for the unfolding of the spectrum.
\subsection{\label{sec:SDrec} From $S(1000)$ to $S_{38}$}
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{intensityS1000.pdf}
\caption{\small{Integral intensity above $S(1000)$ thresholds, for different zenithal ranges of equal exposure.}}
\label{fig:IvsS}
\end{figure}
For a fixed energy, $S(1000)$ depends on the zenith angle $\theta$ because, once it has passed the depth of shower maximum, a shower is attenuated as it traverses the atmosphere. The intensity of cosmic rays, defined here as the number of events per steradian above some $S(1000)$ threshold, is thus dependent on zenith angle as can be seen in Fig.~\ref{fig:IvsS}.
Given the highly isotropic flux, the intensity is expected to be $\theta$-independent after correction for the attenuation. Deviations from a constant behavior can thus be interpreted as being due to attenuation alone. Based on this principle, an empirical procedure, the so-called CIC method, is used to determine the attenuation curve as function of $\theta$ and therefore a $\theta$-independent shower-size estimator ($S_{38}$). It can be thought of as being the $S(1000)$ that a shower would have produced had it arrived at $38^\circ$, the median angle from the zenith. The small anisotropies in the arrival directions and the zenithal dependence of the resolution on $S_{38}$ do not alter the validity of the CIC method in the energy range considered here, as shown in Appendix~\ref{app:sin2}.
In practice, a histogram of the data is first built in $\cos^2{\theta}$ to ensure equal exposure; then the events are ordered by $S(1000)$ in each bin. For an intensity high enough to guarantee full efficiency, the set of $S(1000)$ values, each corresponding to the $N$th largest signal in the associated $\cos^2{\theta}$ bin, provides an empirical estimate of the attenuation curve. Because the mass of each cosmic-ray particle cannot be determined on an event-by-event basis, the attenuation curve inferred in this way is an effective one, given the different species that contribute at each intensity threshold. The resulting data points are fitted with a third-degree polynomial, $S(1000)=S_{38}(1+ax+bx^2+cx^3)$, where $x=\cos^2{\theta}-\cos^2{38^\circ}$. Fits are shown in the top panel of Fig.~\ref{fig:CIC} for three different intensity thresholds corresponding to $I_1=2.91{\times}10^{4}~$sr$^{-1}$, $I_2=4.56{\times}10^{3}~$sr$^{-1}$ and $I_3=6.46{\times}10^{2}~$sr$^{-1}$ at 38$^\circ$.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{cic.pdf} \includegraphics[width=0.5\textwidth]{attenuation.pdf}
\caption{\small{Top: $S(1000)$ attenuation as a function of $\sec{\theta}$, as derived from the CIC method, for different intensity thresholds (see text). Bottom: Same attenuation curves, normalised to 1 at $\theta=38^\circ$ (note that $\sec{38^\circ}\approx 1.269$), to exhibit the differences for the three different intensity thresholds. The intensity thresholds are $I_1=2.91{\times}10^{4}~$sr$^{-1}$, $I_2=4.56{\times}10^{3}~$sr$^{-1}$ and $I_3=6.46{\times}10^{2}~$sr$^{-1}$ at 38$^\circ$. Anticipating the conversion from intensity to energy, these correspond roughly to $3{\times} 10^{18}$~eV, $8{\times} 10^{18}$~eV and $2{\times} 10^{19}$~eV, respectively.}}
\label{fig:CIC}
\end{figure}
The attenuation is plotted as a function of $\sec{\theta}$ to exhibit the dependence on the thickness of atmosphere traversed. The uncertainties in each data point follow from the number of events above the selected $S(1000)$ values. The $N$th largest signal in each bin is a realization of a random variable distributed as an order-statistic variable where the total number of ordered events in the $\cos^2{\theta}$ bin is itself a Poisson random variable. Within a precision better than 1\%, the standard deviation of the random variable can be approximated through a straight-forward Poisson propagation of uncertainties, namely $\Delta S(N)\simeq (S(N+\sqrt{N})-S(N-\sqrt{N}))/2$. The number of bins is adapted to the available number of events for each intensity threshold, from 27 for $I_1$ so as to guarantee a resolution on the number of events of 1\% in each bin, to 8 for $I_3$ so as to guarantee a resolution of 4\%.
The curves shown in Fig.~\ref{fig:CIC} are largely shaped by the electromagnetic contribution to $S(1000)$ which, once the shower development has passed its maximum, decreases with the zenith angle because of attenuation in the increased thickness of atmosphere. The muonic component starts to dominate at large angles, which explains the flattening of the curves. In the bottom panel, the curves are normalized to 1 at $38^\circ$ to exhibit the differences for the selected intensity thresholds. Some dependence with the intensity thresholds, and thus with the energy thresholds, is observed at high zenith angles: high-energy showers appear more attenuated than low energy ones. This results from the interplay between the mass composition and the muonic-to-electromagnetic signal ratio at ground level. A comprehensive interpretation of these curves is however not addressed here.
The energy dependence in the CIC curves that is observed is accounted for by introducing an empirical dependence in terms of $y=\log_{10}(S_{38}/40~\textrm{VEM})$ in the coefficients $a$, $b$ and $c$ through a second-order polynomial in $y$. The polynomial coefficients derived are shown in Table~\ref{tab:cic_param}. They relate to $S_{38}$ values ranging from 15~VEM to 120~VEM. Outside these bounds, the coefficients are set to their values at 15 and 120~VEM. This is because below 15~VEM, the isotropy is not expected anymore due to the decreasing efficiency, while above 120~VEM, the number of events is low and there is the possibility of localized anisotropies.
\begin{table}[h]
\caption{Coefficients of the second-order polynomial in terms of $y=\log_{10}(S_{38}/40~\textrm{VEM})$ for the CIC parameters $a$, $b$ and $c$.}
\label{tab:cic_param}
\begin{ruledtabular}
\begin{tabular}{l c c c}
& $~y_0~$ & $~y_1~$ & $~y_2~$ \\
\colrule
$~a~$ & $0.952$ & $0.06$ & $-0.37$ \\
$~b~$ & $-1.64$ & $-0.42$ & $0.09$ \\
$~c~$ & $-0.9$ & $-0.04$ & $1.3$
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{\label{sec:EnCalib} From $S_{38}$ to $E_{\mathrm{SD}}$}
The shower-size estimator, $S_{38}$, is converted into energy through a calibration with $E_{\mathrm{FD}}$ by making use of a subset of SD events, selected as described in Sec.~\ref{sec:Auger}, which have triggered the FD independently. For the analysis, we apply several selection criteria to guarantee a precise estimation of $E_{\mathrm{FD}}$ as well as fiducial cuts to minimise the biases in the mass distribution of the cosmic rays introduced by the field of view of the FD telescopes.
The first set of cuts aims to select time periods during which data-taking and atmospheric conditions are suitable for collecting high-quality data~\cite{Aab:2014}. We require a high-quality calibration of the gains of the PMTs of the FD and that the vertical aerosol optical depth is measured within 1 hour of the time of the event,
with its value integrated up to 3 km above the ground being less than 0.1.
Moreover, measurements from detectors installed at the Observatory to monitor atmospheric conditions~\cite{ThePierreAuger:2015rma} are used to select only those events detected by telescopes without clouds within their fields of view. Next, a set of quality cuts are applied to ensure a precise reconstruction of the energy deposit~\cite{Aab:2014}.
We select events with a total track length of at least 200 ${\rm g/cm^2}$, requiring that any gap in the profile of the deposited energy be less than 20\% of the total track
length and we reject events with an uncertainty in the reconstructed calorimetric energy larger than 20\%. We transform the $\chi^2$ into a
variable with zero mean and unit variance, $z=\left(\chi^2-n_{\rm dof}\right)/\sqrt{2n_{\rm dof}}$ with $n_{\rm dof}$ the number of degrees of freedom, and require that the $z$ values be less than 3.
Finally, the fiducial cuts are defined by an appropriate selection of the lower and upper depth boundaries to enclose the bulk of the $X_{\rm max}$ distribution and by requiring that the maximum accepted uncertainty in $X_{\rm max}$ is $40~{\rm g/cm^2}$ and that the minimum viewing angle of light in the telescope is $20^\circ$~\cite{Aab:2014}. This limit is set to reduce contamination by Cherenkov radiation. A final cut is applied to $E_{\mathrm{FD}}$: it must be greater than $3{\times} 10^{18}$~eV to ensure that the SD is operating in the regime of full efficiency (see Sec.~\ref{sec:raw}).
After applying these cuts, a data set of 3,338 hybrid events is available for the calibration process. With the current sensitivity of our $X_{\rm max}$ measurements in this energy range, a constant elongation rate (that is, a single logarithmic dependence of $X_{\rm max}$ with energy) is observed~\cite{Aab:2014}. In this case, a single power law dependence of $S_{38}$ with energy is expected from Monte-Carlo simulations. We thus describe the correlation between $S_{38}$ and $E_\textrm{FD}$, shown in Fig.~\ref{fig:EnCalib}, by a power law function,
\begin{equation}
E_\textrm{FD} = A~{S_{38}}^{B},
\label{eqn:ECalib}
\end{equation}
where $A$ and $B$ are fitted to data. In this manner the correlation captured through this power-law relationship is fairly averaged over the underlying mass distribution, and thus provides the calibration of the mass-dependent $S_{38}$ parameter in terms of energy in an unbiased way over the covered energy range. Due to the limited number of events in the FD data set at the highest energies, deviations from the inferred power law cannot be fully investigated currently. We note however that any indication for a strong change of elongation rate cannot be inferred at the highest energies from our SD-based indirect measurement reported in~\cite{Aab:2017cgk}.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{ECalib.pdf}
\caption{\small{Correlation between the SD shower-size estimator, $S_{38}$, and the reconstructed FD energy, $E_{\mathrm{FD}}$, for the selected 3,338 hybrid events used in the fit. The uncertainties indicated by the error bars are described in the text. The solid line is the best fit of the power-law dependence $E_\text{FD}~$$=A\,{S_{38}}^B$ to the data.
The reduced deviance of the fit, whose calculation is detailed in Appendix~\ref{app:ES38fit}, is shown in the bottom-right corner.}}
\label{fig:EnCalib}
\end{figure}
The correlation fit is carried out using a tailored maximum-likelihood method allowing various effects of experimental origin to be taken into account~\cite{Dembinski:2015wqa}. The probability density function entering the likelihood procedure, detailed in Appendix~\ref{app:ES38fit}, is built by folding the cosmic-ray flux, observed with the effective aperture of the FD, with the resolution functions of the FD and of the SD. Note that to avoid the need to model accurately the cosmic-ray flux observed through the effective aperture of the telescopes (and thus to rely on mass assumptions), the observed distribution of events passing the cuts described above is used in this probability density function.
The uncertainties in the FD energies are estimated, on an event-by-event basis, by adding in quadrature all uncertainties in the FD energy measurement which are uncorrelated shower-by-shower (see~\cite{DawsonICRC2019} for details). The uncertainties in $S_{38}$ are also estimated on an event-by-event basis considering the event-by-event contribution arising from the reconstruction accuracy of $S(1000)$. The error arising from the determination of the zenith angle is negligible. The contribution from shower-to-shower fluctuations to the uncertainty in $E_{\rm SD}$ is parameterized as a relative error in $S_{38}$ with $0.13 - 0.08x + 0.03x^2$ where $x = \log_{10}(E/\mathrm{eV}) - 18.5$. It is obtained by subtracting in quadrature the contribution of the uncertainty in $S_{38}$ from the SD energy resolution. The latter, as detailed in the following, is measured from data and the resulting shower-to-shower fluctuations are free from any reliance on mass assumption and model simulations.
The best fit parameters are $A=(1.86 {\pm} 0.03){\times} 10^{17}$~eV and $B=1.031 {\pm} 0.004$ and the correlation coefficient between the parameters is $\rho = -0.98$. The resulting calibration curve is shown as the red line in Fig.~\ref{fig:EnCalib}. The goodness of the fit is provided by the value of the reduced deviance, namely $D/n_\mathrm{dof}=3419/3336$. The statistical uncertainty on the SD energies obtained propagating the fit errors on $A$ and $B$ is 0.4 \% at $3 {\times} 10^{18}$ eV, increasing up to 1\% at the highest energies. The most energetic event used in the calibration is detected at all four fluorescence sites. Its energy is $(8.5 {\pm} 0.4){\times} 10^{19}$ eV, obtained from a weighted average of the four calorimetric energies and using the resulting energy to evaluate the invisible energy correction~\cite{InvisibleEnergy2019}. It has a depth of shower maximum of $(763 {\pm} 8)~{\rm g/cm^2}$, which is typical/close to the average for a shower of this energy~\cite{Aab:2014}. The energy estimated from $S_{38}=354~$VEM is $(7.9 {\pm} 0.6){\times} 10^{19}$~eV.
\subsection{\label{sec:EnScale} $E_{\mathrm{SD}}$: systematic uncertainties}
The calibration constants $A$ and $B$ are used to estimate the energy for the bulk of SD events: $E_{\mathrm{SD}}\equiv A{S_{38}}^B$. They define the SD energy scale. The uncertainties in the FD energies are estimated, on an event-by-event basis, by adding in quadrature all uncertainties in the FD energy measurement which are correlated shower-by-shower~\cite{Verzi2013}.
The contribution from the fluorescence yield is $3.6\%$ and is obtained by propagating the uncertainties in the high-precision measurement performed in the AIRFLY experiment of the absolute yield~\cite{FY-Airfly_AbsYield} and of the wavelength spectrum and quenching parameters~\cite{FY-Airfly_spectrum, FY-Airfly_T_h}. The uncertainty coming from the characterization of the atmosphere ranges from 3.4\% (low energies) to 6.2\% (high energies). It is dominated by the uncertainty associated with the aerosols in the atmosphere and includes a minor contribution related to the molecular properties of the atmosphere. The largest correlated uncertainty, associated with the calibration of the FD, amounts to 9.9\%. It includes a 9\% uncertainty in the absolute calibration of the telescopes and other minor contributions related to the relative response of the telescopes at different wavelengths and relative changes with time of the gain of the PMTs. The uncertainty in the reconstruction of the energy deposit ranges from 6.5\% to 5.6\% (decreasing with energy) and accounts for the uncertainty associated with the modelling of the light spread away from the image axis and with the extrapolation of the modified Gaisser-Hillas profile beyond the field of view of the telescopes. The uncertainty associated with the invisible energy is 1.5\%. The invisible energy is inferred from data through an analysis that exploits the sensitivity of the water-Cherenkov detectors to muons and minimizes the uncertainties related to the assumptions on hadronic interaction models and mass composition~\cite{InvisibleEnergy2019}.
\begin{table}[h]
\caption{Calibration parameters in three different zenithal ranges. $N$ is the number of events selected in each range.}
\label{tab:SDCalib_zenith}
\vspace{0.1cm}
\begin{ruledtabular}
\begin{tabular}{l c c c}
& $0^\circ < \theta < 30^\circ$
& $30^\circ < \theta < 45^\circ$
& $45^\circ < \theta < 60^\circ$ \\
\colrule
$N$ & 435 & 1641 & 1262 \\
$A/10^{17}\,\text{eV}$ & $1.89 {\pm} 0.08$
& $1.86 {\pm} 0.04$
& $1.83 {\pm} 0.04$ \\
$B$ & $1.029 {\pm} 0.012$
& $1.030 {\pm} 0.006$
& $1.034 {\pm} 0.006$
\end{tabular}
\end{ruledtabular}
\end{table}
We have performed several tests aimed at assessing the robustness of the analysis that returns the calibration coefficients $A$ and $B$. The correlation fit was repeated selecting events in three different zenithal ranges. The obtained calibration parameters are reported in Table~\ref{tab:SDCalib_zenith}. The calibration curves are within one standard deviation of the average one reported above, resulting in energies within 1\% of the average ones. Other tests performed using looser selection criteria for the FD events give similar results. By contrast, determining the energy scale in different time periods leads to some deviation of the calibration curves with respect to the average one. Although such variations are partly accounted for in the FD calibration uncertainties, we conservatively propagate these uncertainties into a 5\% uncertainty on the SD energy scale.
The total systematic uncertainty in the energy scale is obtained by adding in quadrature all of the uncertainties detailed above, together with the contribution arising from the statistical uncertainty in the calibration parameters. The total is about 14\% and it is almost energy independent as a consequence of the energy independence of the uncertainty in the FD calibration, which makes the dominant contribution.
\subsection{\label{sec:SDResponse} {$E_{\mathrm{SD}}$: resolution and bias} }
Our final aim is to estimate the energy spectrum above $2.5{\times} 10^{18}~$eV. Still it is important to characterize the energies below this threshold because the finite resolution on the energies induces bin-to-bin migration effects that affect the spectrum. In this energy range, below full efficiency of the SD, systematic effects enter into play on the energy estimate. While the FD quality and fiducial cuts still guarantee the detection of showers without bias towards one particular mass in that energy range, this is no longer the case for the SD due to the higher efficiency of shower detection for heavier primary nuclei~\cite{Abraham:2010zz}. Hence the distribution of $S_{38}$ below $3{\times} 10^{18}~$eV may no longer be fairly averaged over the underlying mass distribution, and a bias on $E_{\mathrm{SD}}$ may result from the extrapolation of the calibration procedure, in addition to the trigger effects that favor positive fluctuations of $S_{38}$ at a fixed energy over negative ones. In this section, we determine these quantities, denoted as $\sigma_\textrm{SD}(E,\theta)/E$ for the resolution and as $b_\textrm{SD}(E,\theta)$ for the bias, in a data-driven way. These measurements allow us to characterize the SD resolution function that will be used in several steps of the analysis presented in the next sections. This, denoted as
$\kappa(E_\textrm{SD}|E;\theta)$,
is the conditional p.d.f. for the measured energy
$E_\textrm{SD}$
given that the true value is $E$. It is normalized such that the event is observed at any reconstructed energy, that is,
$\int\mathrm{d} E_\textrm{SD}~\kappa(E_\textrm{SD}|E;\theta)=1$.
In the energy range of interest, we adopt a Gaussian curve, namely:
\begin{widetext}
\begin{equation}
\label{eqn:kappa}
\kappa(E_\textrm{SD}|E;\theta) = \frac{1}{\sqrt{2\pi}\sigma_\textrm{SD}(E,\theta)} \exp{\left[-\frac{(E_\textrm{SD}-E(1+b_\textrm{SD}(E,\theta)))^2}{2\sigma^2_\textrm{SD}(E,\theta)}\right]}.
\end{equation}
\end{widetext}
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{ESD_EFD_distributions.pdf}
\caption{\small{Ratio distribution of the SD energy, $E_{\mathrm{SD}}$, to the FD energy, $E_{\mathrm{FD}}$, from the selected data sample, for three energy ranges. The distributions are all normalized to unity to better underline the difference in their shape. The total number of events for each distribution is 2367, 1261 and 186 from the lower to the higher energy bin, respectively.
}}
\label{fig:hist_resolution}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{ESD_Resolution.pdf}
\caption{\small{Resolution of the SD as a function of energy. The measurements with their statistical uncertainties are shown with points and error bars. The fitted parameterization
is depicted with the continuous line and its
statistical uncertainty is shown as a shaded band. The FD resolution is also shown for reference (dotted-dashed line).}}
\label{fig:sd_resolution}
\end{figure}
The estimation of $b_\textrm{SD}(E,\theta)$ and $\sigma_\textrm{SD}(E,\theta)$ is obtained by analyzing the $E_\textrm{SD}/E_\textrm{FD}$ histograms as a function of $E_\textrm{FD}$, extending here the $E_\textrm{FD}$ range down to $10^{18}~$eV. For Gaussian-distributed $E_\text{FD}~$ and $E_\text{SD}~$ variables, the $E_\textrm{SD}/E_\textrm{FD}$ variable follows a Gaussian ratio distribution. For a FD resolution function with no bias and a known resolution parameter, the searched $b_{\textrm{SD}}(E,\theta)$ and $\sigma_{\textrm{SD}}(E,\theta)$ are then obtained from the data. The overall FD energy resolution is $\sigma_{\textrm{FD}}(E)/E\simeq 7.4\%$.
In comparison to the number reported in Sec.~\ref{sec:data}, $\sigma_{\textrm{FD}}(E)/E$ is here almost constant over the whole energy range because it takes into account that, at the highest energies, the same shower is detected from different FD sites. In these cases, the energy used in analyses is the mean of the reconstructed energies (weighted by uncertainties) from the two (or more) measurements. This accounts for the improvement in the statistical error.
Examples of measured and fitted distributions of $E_\textrm{SD}/E_\textrm{FD}$ are shown in Fig.~\ref{fig:hist_resolution} for three energy ranges: the resulting SD energy resolution is $\sigma_{\textrm{SD}}(E)/E = (21.5 \pm 0.4)\%$, $(18.2 \pm 0.4)\%$ and $(10.0 \pm 0.8)\%$ between $10^{18}$ and $10^{18.1}$~eV, $10^{18.4}$ and $10^{18.5}$~eV, $10^{19}$ and $10^{19.1}$~eV, respectively. The parameter $\sigma_{\textrm{SD}}(E)/E$ is shown in Fig.~\ref{fig:sd_resolution} as a function of $E$: the resolution is $\simeq 20\%$ at $2{\times} 10^{18}~$eV and tends smoothly to $\simeq 7\%$ above $2{\times} 10^{19}~$eV. Note that no significant zenithal dependence has been observed. The bias parameter $b_{\textrm{SD}}(E,\theta)$ is illustrated in Fig.~\ref{fig:sd_bias} as a function of the zenith angle for four different energy ranges. The net result of the analysis is a bias larger than $10\%$ at $10^{18}~$eV, going smoothly to zero in the regime of full efficiency.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{ESD_bias_zenith.pdf}
\caption{\small{Relative bias parameters of the SD as a function of the zenith angle, for four different energy ranges. The results of the fit of the $E_{\rm SD}/E_{\rm FD}$ distributions with the statistical uncertainties are shown with symbols and error bars, while the fitted parameterization is shown with lines.}}
\label{fig:sd_bias}
\end{figure}
Note that the selection effects inherent in the FD field of view induce different samplings of hybrid and SD showers with respect to shower age at a fixed zenith angle and at a fixed energy. These selection cuts also impact the zenithal distribution of the showers.
Potentially, the hybrid sample may thus not be a fair sample of the bulk of SD events. This may lead to some misestimation of the SD resolution determined in the data-driven manner presented above. We have checked, using end-to-end Monte-Carlo simulations of the Observatory operating in the hybrid mode, that the particular quality and fiducial cuts used to select the hybrid sample do not introduce significant distortions to the measurements of $\sigma_{\textrm{SD}}(E)$ shown in Fig.~\ref{fig:sd_resolution}: the ratio between the hybrid and SD standard deviations of the reconstructed energy histograms remain within 10\% (low energies) and 5\% (high energies) whatever the assumption on the mass composition. There is thus a considerable benefit in relying on the hybrid measurements,to avoid any reliance on mass assumptions when determining the bias and resolution factors.
From the measurements, a convenient parameterization of the resolution is
\begin{equation}
\label{eqn:resolution}
\frac{\sigma_{\mathrm{SD}}(E)}{E} = \sigma_{0}+\sigma_{1}\exp{(-\frac{E}{E_\sigma})},
\end{equation}
where the values of the parameters are obtained from a fit to the data: $\sigma_0=0.078$, $\sigma_1=0.16$, and $E_\sigma=6.6{\times} 10^{18}~$eV. The function and its statistical uncertainty from the fit are shown in Fig.~\ref{fig:sd_resolution}. It is worth noting that this parameterization accounts for both the detector resolution and the shower-to-shower fluctuations. Finally, a detailed study of the systematic uncertainties on this parameterization leads to an overall relative uncertainty of about 10\% at $10^{18}$~eV and increasing with energy to about 17\% at the highest energies. It accounts for the selection effects inherent to the FD field of view previously addressed, the uncertainty in the FD resolution and the statistical uncertainty in the fitted parameterization.
The bias, also parameterized as a function of the energy, includes an additional angular dependence:
\begin{equation}
\label{eqn:bias}
b_{\mathrm{SD}}(E,\theta)
= \left(b_0+b_1 \exp{(-\lambda_b(\cos{\theta}-0.5))}\right)\log_{10}{\left(\frac{E_*}{E}\right)},
\end{equation}
for $\log_{10}{(E/\text{eV})}\leq\log_{10}{(E_*/\text{eV})}=18.4$, and $b_{\mathrm{SD}}=0$ otherwise. Here, $b_0=0.20$, $b_1=0.59$ and $\lambda_b=10.0$. The parameters are obtained in a two steps process: we first perform a fit to extract the zenith-angle dependence in different energy intervals prior to determining the energy dependence of the parameters. Examples of the results of the fit to the data are shown in Fig.~\ref{fig:sd_bias}. The relative uncertainty in these parameters is estimated to be within 15\%, considering the largest uncertainties of the data points displayed in the figure. This is a conservative estimate compared to that obtained from the fit, but this enables us to account for systematic changes that would have occurred had we chosen another functional shape for the parameterization.
The two parameterizations of equations~\eqref{eqn:resolution} and ~\eqref{eqn:bias} are sufficient to characterize the Gaussian resolution function of the SD in the energy range discussed here.
\section{\label{sec:EnSp} Determination of the energy spectrum}
In this section, we describe the measurement of the energy spectrum, $J(E)$. Over parts of the energy range, we will describe it using $J(E) \propto E^{-\gamma}$, where $\gamma$ is the spectral index. In Sec.~\ref{sec:raw}, we present the initial estimate of the energy spectrum, dubbed the ``raw spectrum'', after explaining how we determine the SD efficiency, the exposure and the energy threshold for the measurement. In Sec.~\ref{sec:unfolded}, we describe the procedure used to correct the raw spectrum for detector effects, which also allows us to infer the spectral characteristics. The study of potential systematic effects is summarised in Sec.~\ref{sec:syst}, prior to a discussion of the features of the spectrum in Sec.~\ref{sec:features}.
\begin{figure*}[t]
\centering
\includegraphics[width=0.49\textwidth]{Jraw_vs_Energy.pdf}
\includegraphics[width=0.49\textwidth]{E3Jraw_Energy.pdf}
\caption{\small{Left: Raw energy spectrum $J_i^{\mathrm{raw}}$. The error bars represent statistical uncertainties. The number of events in each logarithmic bin of energy is shown above the points. Right: Raw energy spectrum scaled by the cube of the energy.}}
\label{fig:RawSpectrum}
\end{figure*}
\subsection{\label{sec:raw} The raw spectrum}
An initial estimation of the differential energy spectrum is made by counting the number of observed events, $N_i$, in differential bins (centered at energy $E_i$, with width $\Delta E_i$) and dividing by the exposure of the array, $\mathcal{E}$,
\begin{equation}
J^{\mathrm{raw}}_i=\frac{N_i}{\mathcal{E}~\Delta E_i}.
\label{eqn:Jraw}
\end{equation}
The bin sizes, $\Delta E_i$, are selected to be of equal size in the logarithm of the energy, such that the width, $\Delta \log_{10}E_i=0.1$, corresponds approximately to the energy resolution in the lowest energy bin. The latter is chosen to start at $10^{18.4}$~eV, as this is the energy above which the acceptance of the SD array becomes purely geometric and thus independent of the mass and energy of the primary particle. Consequently, in this regime of full efficiency, the calculation of $\mathcal{E}$ reduces to a geometrical problem dependent only on the acceptance angle, surface area and live-time of the array.
The studies to determine the energy above which the acceptance saturates are described in detail in~\cite{Abraham:2010zz}. Most notably, we have exploited the events detected in hybrid mode as this has a lower threshold than the SD. Assuming that the detection probabilities of the SD and FD detectors are independent, the SD efficiency as a function of energy and zenith angle, $\epsilon(E,\theta)$, has been estimated from the fraction of hybrid events that also satisfy the SD trigger conditions. Above $10^{18}$~eV, the form of the detection efficiency (which will be used in the unfolding procedure described in Sec. \ref{sec:unfolded}) can be represented by an error function:
\begin{equation}
\label{eqn:trigeff}
\epsilon(E,\theta)=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\log_{10}\left(E/\text{eV}\right)-p_0\left(\theta\right)}{p_1}\right)\right].
\end{equation}
where $p_1=0.373$ and $p_0(\theta) = 18.63 - 3.18\cos^2\theta + 4.38\cos^4\theta - 1.87\cos^6\theta$.
For energies above $E_\textrm{sat}=2.5{\times} 10^{18}~$eV, the detection efficiency becomes larger than 97\% and the exposure, $\mathcal{E}$, is then obtained from the integration of the aperture of the array over the observation time~\cite{Abraham:2010zz}. The aperture, $\mathcal{A}$, is in turn obtained as the effective area under zenith angle $\theta$, $A_0\cos\theta$, integrated over the solid angle
$\Omega$ within which the showers are observed. $A_0$ is well-defined as a consequence of the hexagonal structure of the layout of the array combined with the confinement criterion described in Sec.~\ref{sec:data}. Each station that has six adjacent, data-taking neighbors, contributes a cell of area $A_\text{cell} = 1.95~$km$^2$;
the corresponding aperture for showers with $\theta\leq 60^\circ$ is $\mathcal{A_\text{cell}} = 4.59~$km$^2$~sr. The number of active cells, $n_\mathrm{cell}(t)$, is monitored second-by-second. The array aperture is then given, second-by-second, by the product of $\mathcal{A_\text{cell}}$ by $n_\mathrm{cell}(t)$. Finally, the exposure is calculated as the product of the array aperture by the number of live seconds in the period under study, excluding the time intervals during which the operation of the array is not sufficiently stable~\cite{Abraham:2010zz}. This results in a duty cycle larger than 95\%.
Between 1 January 2004 and 31 August 2018 an exposure $(60{,}400 \pm 1{,}810)$ \,km$^2$\,sr\,yr was achieved. The resulting raw spectrum, $J_i^{\mathrm{raw}}$, is shown in Fig.~\ref{fig:RawSpectrum}, left panel.
The energies in the x-axis correspond to the ones defined by the center of the logarithmic bins ($10^{18.45}, 10^{18.55}, \cdots$~eV).
The number of events $N_i$ used to derive the flux for each energy bin is also indicated. Upper limits are at the $90\%$ confidence level.
The spectrum looks like a rapidly falling power law in energy with an overall spectral index of about $3$. To better display deviations from this function we also show, in the right panel, the same spectrum with the intensity scaled by the cube of the energy: the well-known ankle and suppression features are clearly visible at $\approx 5{\times}10^{18}$~eV and $\approx 5{\times}10^{19}$~eV, respectively.
\subsection{\label{sec:unfolded} The unfolded spectrum}
The raw spectrum is only an approximate measurement of the energy spectrum, $J(E)$, because of the distortions induced on its shape by the finite energy resolution. This causes events to migrate between energy bins: as the observed spectrum is steep, the migration happens especially from lower to higher energy bins, in a way that depends on the resolution function (see Sec.~\ref{sec:SDResponse}, Eq.~\eqref{eqn:kappa}). The shape at the lowest energies is in addition affected by the form of the detection efficiency (see Sec.~\ref{sec:raw}, Eq.~\eqref{eqn:trigeff}) in the range where the array is not fully efficient as events whose true energy is below $E_\textrm{sat}$ might be reconstructed with an energy above that.
To derive $J(E)$ we use a bin-by-bin correction approach~\cite{Cowan}, where we first fold the detector effects into a model of the energy spectrum and then compare the expected spectrum thus obtained with that observed so as to get the unfolding corrections. The detector effects are taken into account through the following relationship,
\begin{equation}
\label{eqn:Jfolded}
J^{\mathrm{raw}}(E_\textrm{SD};\mathbf{s})=\frac{\int\mathrm{d}\Omega \cos{\theta}\int\mathrm{d} E \epsilon(E,\theta)J(E;\mathbf{s})\kappa(E_\textrm{SD}|E;\theta)}{\int\mathrm{d}\Omega \cos{\theta}}
\end{equation}
where $\mathbf{s}$ is the set of parameters that characterizes the model. The model is used to calculate the number of events in each energy bin, $\mu_i(\mathbf{s})=\mathcal{E}\int_{\Delta E_i} \mathrm{d}E~J(E;\mathbf{s})$.
The bin-to-bin migrations of events, induced by the finite resolution through Eq.~\eqref{eqn:Jfolded}, is accounted for by calculating the number of events expected between $E_i$ and $E_i+\Delta E_i$, $\nu_i(\mathbf{s})$, through the introduction of a matrix that depends only on the SD response function obtained from the knowledge of the $\kappa(E_\textrm{SD}|E)$ and $\epsilon(E)$ functions.
To estimate $\mu_i$ and $\nu_i$, we use a likelihood procedure, aimed at deriving the set of parameters $\mathbf{s}_0$ allowing the best match between the observed number of events, $N_i$, and the expected one, $\nu_i$. Once the best-fit parameters are derived, the correction factors to be applied to the observed spectrum, $c_i$, are obtained from the estimates of $\mu_i$ and $\nu_i$ as $c_i=\mu_i/\nu_i$. More details about the likelihood procedure, the elements used to build the matrix and the calculation of the $c_i$ coefficients are provided in Appendix~\ref{app:unfold}.
Guided by the raw spectrum, we infer the possible functional form for $J(E;\mathbf{s})$ by choosing parametric shapes naturally reproducing the main characteristics visible in Fig.~\ref{fig:RawSpectrum}. As a first step, we set out to reproduce a rapid change in slope (the ankle) followed by a slow suppression of the intensity at high energies. To do so, we use the 6-parameter function:
\begin{eqnarray}
J(E;\mathbf{s})&=&J_0\left(\frac{E}{E_{0}}\right)^{-\gamma_1}\left[1+\left(\frac{E}{E_{12}}\right)^{\frac{1}{\omega_{12}}}\right]^{(\gamma_1-\gamma_2)\omega_{12}} \nonumber \\
&{\times}&\frac{1}{1+(E/E_{\mathrm{s}})^{\Delta\gamma}}.
\label{eqn:J1}
\end{eqnarray}
In addition to the normalization, $J_0$, and to the arbitrary reference energy $E_0$ fixed to $10^{18.5}$~eV, the two parameters $\gamma_1$ and $\gamma_2$ approach the spectral indices around the energy $E_{12}$, identified with the energy of the ankle. The parameter $E_{\mathrm{s}}$ marks the suppression energy around which the spectral index slowly evolves from $\gamma_2$ to $\gamma_2+\Delta\gamma$.
More precisely, it is the energy at which the flux is one half of the value obtained extrapolating the power law after the ankle.
It is worth noting that the rate of change of the spectral index around the ankle is here determined by the parameter $\omega_{12}$ fixed at 0.05, which is the minimal value adopted to describe the transition given the size of the energy intervals.\footnote{With $\omega$=0.05, the transition between the two spectral indexes is roughly completed in $\Delta \log_{10}E=0.1$.}
Unlike a model forcing the change in spectral index to be infinitely sharp, such a choice of transition also makes it possible, subsequently, to test the speed of transition by leaving the parameters free.
We have used this function (Eq.~\eqref{eqn:J1}) to describe our data for over a decade. However, we find that with the exposure now accumulated, it no longer provides a satisfactory fit, with a deviance $D/n_\mathrm{dof}=35.6/14$. A more careful inspection of Fig.~\ref{fig:RawSpectrum} suggests a more complex structure in the region of suppression, with a series of power laws rather than a slow suppression. Consequently, we adopt as a second step a functional form describing a succession of power laws with smooth breaks:
\begin{eqnarray}
J(E;\mathbf{s})&=&J_0\left(\frac{E}{E_{0}}\right)^{-\gamma_1}\prod_{i=1}^3\left[1+\left(\frac{E}{E_{ij}}\right)^{\frac{1}{\omega_{ij}}}\right]^{(\gamma_i-\gamma_j)\omega_{ij}}
\label{eqn:J2}
\end{eqnarray}
with $j=i+1$. This functional shape is routinely used to characterize the cosmic-ray spectrum at lower energies (see~\cite{Lipari2018} and references therein).
\begin{figure}[b]
\centering
\includegraphics[width=0.5\textwidth]{UnfCorr.pdf}
\caption{\small{ {Unfolding correction factor applied to the measured spectrum to account for the detector effects as a function of the cosmic-ray energy.
}}}
\label{fig:UnfoldingFactors}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=0.49\textwidth]{J_vs_E.pdf}
\includegraphics[width=0.49\textwidth]{J_fit.pdf}
\caption{\small{Left: Energy spectrum. The error bars represent statistical uncertainties. Right: Energy spectrum scaled by $E^{3}$ and fitted with the function given by Eq.~\eqref{eqn:J2} with $\omega_{ij}=0.05$ (solid line). The shaded band indicates the statistical uncertainty of the fit.}}
\label{fig:Spectrum}
\end{figure*}
The parameters $E_{23}$ and $E_{34}$ mark the transition energies between $\gamma_2$ and $\gamma_3$, and $\gamma_3$ and $\gamma_4$ respectively. The values of the $\omega_{ij}$ parameters are fixed, as previously, at the minimal value of 0.05. In total, this model has 8 free parameters and leads to a deviance of $D/n_\mathrm{dof}=17.0/12$. That this model better matches the data than the previous one is further evidenced by the likelihood ratio between these models which allows a rejection of Eq.~\eqref{eqn:J1} with $3.9\sigma$ confidence whose calculation is detailed in Appendix~\ref{app:unfold}.
As a third step, we release the parameters $\omega_{ij}$ one by one, two by two and all three of them so as to test our sensitivity to the speed of the transitions.
Free parameters are only adopted as additions if the improvement to the fit is better than $2\sigma$. Such a procedure is expected to result in a uniform distribution of $\chi^2$ probability for the best-fit models, as exemplified in~\cite{Biasuzzi2019}.
For every tested model, the increase in test statistics is insufficient to pass the $2\sigma$ threshold.
The adoption of Eq.~\eqref{eqn:J2} yields the coefficients $c_i$ shown as the black points in Fig.~\ref{fig:UnfoldingFactors} together with their statistical uncertainty.
To be complete, we also show with a curve the coefficients calculated in sliding energy windows, to explain the behavior of the $c_i$ points.
This curve is determined on the one hand by the succession of power laws modeled by $J(E,\mathbf{s}_0)$, and on the other hand by the response function. The observed changes in curvature result from the interplay between the changes in spectral indices occurring in fairly narrow energy windows (fixed by the parameters $\omega_{ij}=0.05$) and the variations in the response function. At high energy,
the coefficients tend towards a constant as a consequence of the approximately constancy of the resolution, because in such a regime, the distortions induced by the effects of finite resolution result in a simple multiplicative factor for a spectrum in power law. Overall, the correction factors are observed to be close to 1 over the whole energy range with a mild energy dependence. This is a consequence of the quality of the resolution achieved.
We use the coefficients to correct the observed number of events to obtain the differential intensities as $J_i=c_i J_i^{\mathrm{raw}}$. This is shown in the left panel of Fig.~\ref{fig:Spectrum}. The values of the differential intensities, together the detected and corrected number of events in each energy bin are given in Appendix~\ref{app:spectrumdata}. The magnitude of the effect of the forward-folding procedure can be appreciated from the following summary: above $2.5 {\times} 10^{18}~$eV, where there are 215,030 events in the raw spectrum, there are 201,976 in the unfolded spectrum; the corresponding numbers above $5 {\times} 10^{19}~$eV and $10^{20}~$eV are 278 and 269, and 15 and 14, respectively. Above $5{\times}10^{19}$~eV ($10^{20}$~eV), the integrated intensity of cosmic rays is $\left( 4.5 \pm 0.3 \right) {\times} 10^{-3}$~km$^{-2}$~yr$^{-1}$~sr$^{-1}$
($\left( 2.4 ^{+0.9}_{-0.6} \right) {\times} 10^{-4}$~km$^{-2}$~yr$^{-1}$~sr$^{-1}$).
\begin{table}[h]
\caption{Best-fit parameters, with statistical and systematic uncertainties, for the energy spectrum measured at the Pierre Auger Observatory.}
\label{tab:pars}
\begin{ruledtabular}
\begin{tabular}{l c}
parameter & value $\pm \sigma_{\mathrm{stat.}} \pm \sigma_{\mathrm{sys.}}$ \\
\colrule
&\\[-1.0em]
$J_{0}$ [km$^{-2}$sr$^{-1}$yr$^{-1}$eV$^{-1}$] & $ (1.315 \pm 0.004 \pm 0.400) {\times} 10^{-18}$ \\
$\gamma_1$ & $3.29 \pm 0.02 \pm 0.10$\\
$\gamma_2$ & $2.51 \pm 0.03 \pm 0.05$ \\
$\gamma_3$ & $3.05 \pm 0.05 \pm 0.10$ \\
$\gamma_4$ & $5.1 \pm 0.3 \pm 0.1$ \\
$E_{12}$ [eV] (ankle) & $\left (5.0 \pm 0.1 \pm 0.8 \right) {\times} 10^{18}$ \\
$E_{23}$ [eV] & $\left (13 \pm 1 \pm 2 \right) {\times} 10^{18}$ \\
$E_{34}$ [eV] (suppression) & $\left (46 \pm 3 \pm 6 \right) {\times} 10^{18}$ \\
\hline
$D/n_\mathrm{dof}$ & $17.0 / 12$
\end{tabular}
\end{ruledtabular}
\end{table}
In the right panel of Fig.~\ref{fig:Spectrum}, the fitted function $J(E,\mathbf{s}_0)$, scaled by $E^3$ to better appreciate the fine structures, is shown as the solid line overlaid on the data points of the final estimate of the spectrum. The characteristics of the spectrum are given in Table~\ref{tab:pars}, with both statistical and systematic uncertainties (for which a comprehensive discussion is given in the next section). These characteristics are further discussed in Sec.~\ref{sec:features}.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{Systematics.pdf}
\caption{\small{Top panel: Systematic uncertainty in the energy spectrum as a function of the cosmic-ray energy (dash-dotted red line). The other lines represent the contributions of the different sources as detailed in the text: energy scale (continuous black), exposure (blue), $S(1000)$ (dotted black), unfolding procedure (gray). The contributions of the latter three are zoomed in the bottom panel.
}}
\label{fig:Syst}
\end{figure}
\subsection{\label{sec:syst} Systematic uncertainties}
There are several sources of systematic uncertainties which affect the measurement of the energy spectrum, as illustrated in Fig.~\ref{fig:Syst}.
The systematic uncertainty in the energy scale gives the largest contribution to the overall uncertainty.
As described in Sec.~\ref{sec:EnScale}, it amounts to about $14\%$ and is obtained by adding in quadrature
all the systematic uncertainties in the FD energy estimation and the contribution arising from the statistical uncertainty in the calibration parameters.
As the effect is dominated by the uncertainty in the calibration of the FD telescopes, the $14\%$ is almost energy independent. Therefore
it has been propagated into the energy spectrum by changing the energy of all events by $\pm 14\%$ and then calculating
a new estimation of the raw energy spectrum through Eq.~\eqref{eqn:Jraw} and repeating the forward-folding procedure.
When considering the resolution, the bias and the detection efficiency in the parameterization of the response function, the energy scale is shifted by $\pm 14\%$. The uncertainty in the energy scale
translates into an energy-dependent uncertainty in the flux shown by a continuous black line in Fig.~\ref{fig:Syst}, top panel.
It amounts to $\simeq 30$ to $40\%$ around $2.5{\times}10^{18}$~eV, decreasing to $25\%$ around $10^{19}$~eV, and increasing again to $60\%$ at the highest energies.
A small contribution comes from the unfolding procedure. It stems from different sub-components: $(i)$ the functional form of the energy spectrum assumed, $(ii)$ the uncertainty in the bias and resolution parameterization determined in Sec.~\ref{sec:SDResponse} and $(iii)$ the uncertainty in the detection efficiency determined in Sec.~\ref{sec:raw}. The impact of contribution $(i)$ has been conservatively evaluated by comparing the output of the unfolding assuming Eq.~\eqref{eqn:J1} and Eq.~\eqref{eqn:J2}
and it is less than $1\%$ at all energies.
That of contribution $(ii)$ remains within $2\%$ and is maximal at the highest and lowest energies, while the one of contribution $(iii)$ is
estimated propagating the statistical uncertainty in the
fit function that parametrizes the detection efficiency (Eq.~\eqref{eqn:trigeff}) and it is
within $\simeq$ $1\%$ below $4{\times}10^{18}$~eV and negligible above. The statistical uncertainties in the unfolding correction factors also contribute to the total systematic uncertainties in the flux and are taken into account. The overall systematic uncertainties due to unfolding are shown as a gray line in both panels of Fig.~\ref{fig:Syst} and are at maximum of $2\%$ at the lowest energies.
A third source is related to the global uncertainty of $3\%$ in the estimation of the integrated SD exposure~\cite{Abraham:2010zz}. This uncertainty, constant with energy, is shown as the blue line in both panels of Fig.~\ref{fig:Syst}.
A further component is related to the use of an average functional form for the LDF. The departure of this parameterized LDF from the actual one is source of a systematic uncertainty in $S(1000)$. This can be estimated using a subset of high quality events for which the slope of the LDF~\cite{AugerRecoPaper} can be measured on an event by--event basis. The impact of this systematic uncertainty on the spectrum (shown as a black dotted line in Fig.~\ref{fig:Syst}) is around $2\%$ at $2.5 {\times} 10^{18}$~eV, decreasing to $-3\%$ at $10^{19}$~eV, before rising again to $3\%$ above $\simeq$ $3{\times}10^{19}$~eV.
Other sources of systematic uncertainty have been investigated and are negligible.
We have performed several tests to assess the
robustness of the measurement. The spectrum, scaled by $E^3$, is shown in top panel of Fig.~\ref{fig:J_vs_theta} for three zenith angle intervals. Each interval is of equal size in $\sin^2{\theta}$ such that the exposure is the same, one third of the total one. The ratio of the three spectra to the results of the fit performed in the full field of view presented in Sec.~\ref{sec:unfolded} is shown in the bottom panel of the same figure.
The three estimates of the spectrum are in statistical agreement. In the region below
$2 {\times} 10^{19}$~eV, where there are large numbers of events, the dependence on zenith angle is below 5\%. This is a robust demonstration of the efficacy of our methods.
We have also searched for systematic effects that might be seasonal to test the effectiveness of the corrections applied to $S(1000)$ to account for the influence
of the changes in atmospheric temperature and pressure on the shower structure~\cite{AugerJINST2017}, and also searched for temporal effects as the data have been collected over a period of 14 years.
Such tests have been performed by keeping the energy calibration curve determined in the full data taking period, as the systematic uncertainty associated with a non-perfect monitoring in time
of the calibration of the FD telescopes is included in the overall $\pm 14\%$ uncertainty in the energy scale.
The integral intensities above 10$^{19}$ eV for the four seasons are (0.271, 0.279, 0.269, 0.272)$\pm$0.004 km$^{-2}$ sr$^{-1}$ yr$^{-1}$ for winter, spring, summer and autumn respectively. The largest deviation with respect to the average of 0.273 $\pm$ 0.002 km$^{-2}$ sr$^{-1}$ yr$^{-1}$ is around 2\% for spring. To look for long term effects we have divided the data into 5 sub--samples of equal number of events ordered in time. The integrated intensities above $10^{19}$~eV (corresponding to 16737 raw events) are (0.258, 0.272, 0.280, 0.280, 0.275)$\pm$0.005 km$^{-2}$ sr$^{-1}$ yr$^{-1}$, with a maximum deviation of 5\% with respect to the average value ($=$ 0.273 $\pm$ 0.002 km$^{-2}$ sr$^{-1}$ yr$^{-1}$). The largest deviation is in the first period (Jan 2004 -- Nov 2008) when the array was still under construction.
The total systematic uncertainty, which is dominated by the uncertainty on the energy scale, is obtained by the quadratic sum of the described contributions and is depicted as a dashed red line in Fig.~\ref{fig:Syst}.
The systematic uncertainties on the spectral parameters are also obtained adding in quadrature all the contributions above described, and are shown in Table~\ref{tab:pars}. The uncertainties in the energy of the features ($E_{ij}$) and in the normalization parameter ($J_{0}$) are dominated by the uncertainty in the energy scale. On the other hand, those on the spectral indexes are also impacted by the other sources of systematic uncertainties.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{J_3zenith_vs_E_res.pdf}
\caption{\small{Top panel: energy spectrum scaled by $E^3$ in three zenithal ranges of equal exposure. The solid line shows the results of the fit in the full f.o.v. presented in Sec.~\ref{sec:unfolded}.
Bottom panel: relative difference between the spectra in the three zenithal ranges and the fitted spectrum in the full f.o.v..
An artificial shift of $\pm 3.5\%$ is applied to the energies in the $x-$axis for the spectra obtained with the most and less inclined
showers to make easier to identify the different data points.
}}
\label{fig:J_vs_theta}
\end{figure}
\subsection{\label{sec:features} Discussion of the spectral features}
The unfolded spectrum shown in Fig.~\ref{fig:Spectrum} can be described using four power laws as detailed in Table~\ref{tab:pars} and equation~\eqref{eqn:J2}. The well-known features of the ankle and the steepening are very clearly evident. The spectral index, $\gamma_3$, used to describe the new feature identified above $1.3 {\times} 10^{19}$~eV, differs from the index at lower energies, $\gamma_2$, by $\approx 4\sigma$ and from that in the highest energy region, $\gamma_4$, by $\approx 5\sigma$.
The representation of our data, and similar sets of spectral data, using spectral indices is long-established although, of course, it is unlikely that Nature generates exact power laws. Furthermore these quantities are not usually derived from phenomenologically-based predictions. Rather it is customary to compare measurements to such outputs on a point-by-point basis (e.g.\ \cite{PhysRevD.74.043005,Aab_2017}). Accordingly, the data of Fig.~\ref{fig:Spectrum} are listed in Table~\ref{tab:Jdata}.
An alternative manner of presentation of the data is shown in Fig.~\ref{fig:GammasVsEnergy} where spectral indices have been computed over small ranges of energy (each point is computed for 3 bins at low energies growing to 6 at the highest energies). The impact of the unfolding procedure is most clearly seen at the lowest energies (where the energy resolution is less good): the effect of the unfolding procedure is to sharpen the ankle feature. It is also clear from Fig.~\ref{fig:GammasVsEnergy} that slopes are constant only over narrow ranges of energy, one of which embraces the new feature starting just beyond $10^{19}$~eV. Above $\approx 5 {\times} 10^{19}$~eV, where the spectrum begins to soften sharply, it appears that $\gamma$ rises steadily up to the highest energies observed. However, as beyond this energy there are only 278 events, an understanding of the detailed behaviour of the slope with energy must await further exposure.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{SpectralIndex.pdf}
\caption{\small{Evolution of the spectral index as a function of energy. The spectral indices are derived from power-law fits to the raw and unfolded spectra performed over sliding windows in energy. Each slope is calculated using bins
$\Delta\log_{10}E=0.05$. The width of the sliding windows are 3 bins at the lower energies and, to reduce the statistical fluctuations, are increased to 6 bins at the highest energies.
}
}
\label{fig:GammasVsEnergy}
\end{figure}
\section{\label{sec:declination} The declination dependence of the energy spectrum}
\begin{figure*}[t]
\centering
\includegraphics[width=0.49\textwidth]{spectra_3decbands.pdf}
\includegraphics[width=0.49\textwidth]{ratiospectra_3decbands.pdf}
\caption{\small{Left: Energy spectra in three declination bands of equal exposure. Right: Ratio of the declination-band spectra to that of the full field-of-view. The horizontal lines show the expectation from the observed dipole~\cite{AugerAnis2018}. An artificial shift of $\pm 5\%$ is applied to the energies in the $x-$axis of the northernmost/southernmost declination spectra to make it easier to identify the different data points.}}
\label{fig:DeclinationSpectrum}
\end{figure*}
In the previous section, the energy spectrum was estimated over the entire field of view, using the local horizon and zenith at the Observatory site to define the local zenithal and azimuth angles $(\theta,\varphi)$. Alternatively, we can make use of the fixed equatorial coordinates, right ascension and declination $(\alpha,\delta)$, aligned with the equator and poles of the Earth, for the same purpose. The wide range of declinations covered by using events with zenith angles up to $60^\circ$, from $\delta=-90^\circ$ to $\delta\simeq +24.8^\circ$ (covering 71\% of the sky), allows a search for dependencies of the energy spectrum on declination. We present below the determination of the energy spectrum in three declination bands and discuss the results.
For each declination band under consideration, labelled as $k$, the energy spectrum is estimated as
\begin{equation}
J_{ik}=\frac{N_{ik}c_{ik}}{\mathcal{E}_k~\Delta E_i},
\label{eqn:J}
\end{equation}
where $N_{ik}$ and $c_{ik}$ stand for the number of events and the correction factors in the energy bin $\Delta E_i$ and in the declination band considered $k$, and $\mathcal{E}_k$ is the exposure restricted to the declination band $k$. For this study, the observed part of the sky is divided into declination bands with equal exposure, $\mathcal{E}_k=\mathcal{E}/3$. The correction factors are inferred from a forward-folding procedure identical to that described in section~\ref{sec:EnSp}, except that the response matrix is adapted to each declination band (for details see Appendix~\ref{app:unfold}).
The intervals in declination that guarantee that the exposure of the bands are each $\mathcal{E}/3$ are determined by integrating the directional exposure function, $\omega(\delta)$, derived in Appendix~\ref{app:direxp}, over the declination so as to satisfy
\begin{equation}
\label{eqn:exposure_domega}
\frac{\int_{\delta_{k-1}}^{\delta_k} \mathrm{d}\delta\cos{\delta}~\omega(\delta)}{\int_{\delta_0}^{\delta_3} \mathrm{d}\delta\cos{\delta}~\omega(\delta)}=\frac{1}{3},
\end{equation}
where $\delta_0=-\pi/2$ and $\delta_3=+24.8^\circ$. Numerically, it is found that $\delta_1=-42.5^\circ$ and $\delta_2=-17.3^\circ$.
The resulting spectra (scaled by $E^3$) are shown in the left panel of Fig.~\ref{fig:DeclinationSpectrum}. For reference, the best fit of the spectrum obtained in section~\ref{sec:unfolded} is shown as the black line. No strong dependence of the fluxes on declination is observed.
\begin{table}[h]
\caption{Integral intensity above $8{\times} 10^{18}~$eV in the three declination bands considered.}
\label{tab:decintensity}
\begin{ruledtabular}
\begin{tabular}{l c}
declination band & integral intensity [km$^{-2}$~yr$^{-1}$~sr$^{-1}$] \\
\colrule
$-90.0^\circ\leq\delta < -42.5^\circ$ & $(4.17\pm0.04){\times}10^{-1}$ \\
$-42.5^\circ\leq\delta < -17.3^\circ$ & $(4.11\pm0.04){\times}10^{-1}$ \\
$-17.3^\circ\leq\delta < +24.8^\circ$ & $(4.11\pm0.04){\times}10^{-1}$
\end{tabular}
\end{ruledtabular}
\end{table}
To examine small differences, a ratio plot is shown in the right panel by taking the energy spectrum observed in the whole field of view as the reference.
A weighted-average over wider energy bins is performed to avoid large statistical fluctuations preventing an accurate visual appreciation. For each energy, the data points are observed to be in statistical agreement with each other. Note that the same conclusions hold when analyzing data in terms of integral intensities, as evidenced for instance in table~\ref{tab:decintensity} above $8{\times}10^{18}~$eV. Similar statistical agreements are found above other energy thresholds. Hence this analysis provides no evidence for a strong declination dependence of the energy spectrum.
A 4.6\% first-harmonic variation in the flux in right ascension has been observed in the energy bins above $8{\times} 10^{18}~$eV shown in the right panel of Fig.~\ref{fig:DeclinationSpectrum}~\cite{AugerAnis2018}. It is thus worth relating the data points reported here to these measurements that are interpreted as dipole anisotropies. The technical details to establish these relationships are given in Appendix~\ref{app:direxp}.
The energy-dependent lines drawn in Fig.~\ref{fig:DeclinationSpectrum}-right show the different ratios of intensity expected from the dipolar patterns in each declination band relative to that across the whole field of view. The corresponding data points are observed, within uncertainties, to be in fair agreement with these expectations.
Overall, there is thus no significant variation of the spectrum as a function of the declination in the field of view scrutinized here. A trend for a small declination dependence, with the flux being higher in the Southern hemisphere, is observed consistent with the dipolar patterns reported in~\cite{AugerAnis2018}. At the highest energies, the event numbers are still too small to identify any increase or decrease of the flux with the declinations in our field of view.
\section{\label{sec:ta} Comparison with other measurements}
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{augerta_fullsky.pdf}
\caption{\small{Comparison between the $E^3$-scaled spectrum derived in this work and the one derived at the Telescope Array.}}
\label{fig:AugerTA}
\end{figure}
Currently, the Telescope Array (TA) is the leading experiment dedicated to observing UHECRs in the northern hemisphere. As already pointed out, TA is also a hybrid detector making use of a 700~km$^2$ array of SD scintillators overlooked by fluorescence telescopes located at three sites. Although the techniques for assigning energies to events are similar, there are differences as to how the primary energies are derived, which result in differences in the spectral estimates, as can be appreciated in Fig.~\ref{fig:AugerTA} where the $E^3$-scaled spectrum derived in this work and the one derived by the TA Collaboration~\cite{IvanovICRC19} are shown.
A useful way to appraise such differences is to make a comparison of the observations at the position of the ankle. Given the lack of anisotropy in this energy range, this spectral feature must be quasi-invariant with respect to direction on the sky. The energy at the ankle measured using the TA data is found to be $(4.9\pm0.1~(\mathrm{stat.})){\times} 10^{18}~$eV, with an uncertainty of 21\% in the energy scale~\cite{TA-escale}
in good agreement with the one reported here ($(5.0\pm0.1~(\mathrm{stat.})\pm0.8~(\mathrm{sys.})){\times} 10^{18}~$eV).
Consistency between the two spectra can be obtained in the ankle-energy region up to $\simeq 10^{19}~$eV by rescaling the energies by $+5.2\%$ for Auger and $-5.2\%$ for TA. The factors are smaller than the current systematic uncertainties in the energy scale of both experiments. These values encompass the different fluorescence yields adopted by the two Collaborations, the uncertainties in the absolute calibration of the fluorescence telescopes, the influence of the atmospheric transmission used in the reconstruction, the uncertainties in the shower reconstruction, and the uncertainties in the correction factor for the invisible energy. It is worth noting that better agreement can be obtained if the same models are adopted for the fluorescence yield and for the invisible energy correction. Detailed discussions on these matters can be found in~\cite{VIT2018}.
However, even after the rescaling, differences persist above $\simeq 10^{19}~$eV. At such high energies, anisotropies might increase in size and induce differences in the energy spectra detected in the northern and southern hemispheres. To disentangle possible anisotropy issues from systematic effects, a detailed scrutiny of the spectra in the declination range accessible to both observatories has been carried out~\cite{VerziUHECR16}. A further empirical, energy-dependent, systematic shift of $+10\%$ ($-10\%$) per decade for Auger (TA) is required to bring the spectra into agreement. A comprehensive search for energy-dependent systematic uncertainties in the energies has resulted in possible non-linearities in this decade amounting to $\pm 3\%$ for Auger and $(-0.3\pm9)\%$ for TA, which are insufficient to explain the observed effect~\cite{IvanovUHECR18}. A joint effort is underway to understand further the sources of the observed differences and to study their impact on the spectral features~\cite{DelignyICRC19}.
\section{Summary}
\label{sec:discussion}
We have presented a measurement of the energy spectrum of cosmic rays for energies above $2.5{\times}10^{18}~$eV based on 215,030 events recorded with zenith angles below $60^\circ$. The corresponding exposure of 60,400 km$^2$~yr~sr, calculated in a purely geometrical manner, is independent of any assumption on unknown hadronic physics or primary mass composition. This measurement relies on estimates of the energies that are similarly independent of such assumptions. This includes the analysis that minimizes the model/mass dependence of the invisible energy estimation as presented in~\cite{InvisibleEnergy2019}.
In the same manner, the flux correction for detector effects is evaluated using a data-driven analysis.
Thus the approach adopted differs from that of \textit{all} other spectrum determinations above $\simeq 5{\times}10^{14}~$eV where the air-shower phenomenon is used to obtain information.
The measurement reported above is the most precise made hitherto and is dominated by systematic uncertainties except at energies above $\simeq 5{\times}10^{19}~$eV. The systematic uncertainties have been discussed in detail and it is shown that the dominant one ($\simeq 14$\%) comes from the energy scale assigned using measurements of the energy loss by ionisation in the atmosphere inferred using the fluorescence technique.
In summary, the principal conclusions that can be drawn from the measurement are:
\begin{enumerate}
\item The flattening of the spectrum near $5{\times}10^{18}~$eV, the so-called ``ankle'', is confirmed.
\item The steepening of the spectrum at around $\simeq 5{\times}10^{19}~$eV is substantiated.
\item A new feature has been identified in the spectrum: in the region above the ankle the spectral index changes from $2.51 \pm 0.03~{\rm (stat.)} \pm 0.05~{\rm (sys.)}$ to $3.05 \pm 0.05~{\rm (stat.)} \pm 0.10~{\rm (sys.)}$ before increasing sharply to $5.1 \pm 0.3~{\rm (stat.)} \pm 0.1~{\rm (sys.)}$ above $5 {\times} 10^{19}~$eV.
\item No evidence for any dependence of the energy spectrum on declination has been found other than a mild excess from the Southern Hemisphere that is consistent with the anisotropy observed above $8{\times}10^{18}~$eV.
\end{enumerate}
A discussion of the significance of these measurements from astrophysical perspectives can be found in~\cite{AugerPRL2019}.
| 23,929 |
\section{Introduction}\label{sec:introduction}}
\else
\section{Introduction}
\label{sec:introduction}
\fi
\IEEEPARstart{T}{he} COVID-19 pandemic has caused significant disruption to daily life around the world. As of August 10, 2020, there have been $19.8$ million confirmed cases worldwide with more than $730$ thousand fatalities. Furthermore, this pandemic has caused significant economic and social impacts.
At the moment, one of the best ways to prevent contracting COVID-19 is to avoid being exposed to the coronavirus. Organizations such as the Centers for Disease Control and Prevention (CDC) have recommended many guidelines including maintaining social distancing, wearing masks or other facial coverings, and frequent hand washing to reduce the chances of contracting or spreading the virus. Broadly, social distancing refers to the measures taken to reduce the frequency of people coming into contact with others and to maintain at least 6 feet of distance between individuals who are not from the same household. Several groups have simulated the spread of the virus and shown that social distancing can significantly reduce the total number of infected cases~\cite{Mao2011}, \cite{pmid23763426}, \cite{pmid26847017}, \cite{pmid19104659}, \cite{pmid18401408}.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth,height=6cm]{Images/Cover_Image2.png}
\caption {\small{Our robot detecting non-compliance to social distancing norms, classifying non-compliant pedestrians into groups and autonomously navigating to the group with currently the most people in it (a group with 3 people in this scenario). The robot encourages the non-compliant pedestrians to move apart and maintain at least 6 feet of social distance by displaying a message on the mounted screen. Our COVID-robot also captures thermal images of the scene and transmits them to appropriate security/healthcare personnel.}}
\label{fig:cover-image}
\vspace{-10pt}
\end{figure}
A key issue is developing guidelines and methods to enforce these social distance constraints in public or private gatherings at indoor or outdoor locations. This gives rise to many challenges
including, framing reasonable rules that people can follow when they use public places such as supermarkets, pharmacies, railway and bus stations, spaces for recreation and essential work, and how people can be encouraged to follow the new rules. In addition, it is also crucial to detect when such rules are breached so that appropriate counter-measures can be employed. Detecting social distancing breaches could also help in contact tracing \cite{social-distance}.
Many technologies have been proposed for detecting excessive crowding or conducting contact tracing, and most of them use some form of communication. Examples of this communication include WiFi, Bluetooth, tracking based on cellular connectivity, RFID, Ultra Wide Band (UWB) etc. Most of these technologies work well only in indoor scenes, though cellular have been used outdoors for tracking pedestrians. In addition, many of these technologies such as RFID, UWB, etc. require additional infrastructure or devices to track people indoors. In other cases, technologies such as WiFi and Bluetooth are useful in tracking only those people connected to the technologies using wearable devices or smartphones. This limits their usage for tracking crowds and social distancing norms in general environments or public places, and may hinder the use of any kind of counter-measures.
\noindent {\bf Main Results:} We present a vision-guided mobile robot (COVID-robot) to monitor scenarios with low or high-density crowds and prolonged contact between individuals. We use a state-of-the-art algorithm for autonomous collision-free navigation of the robot in arbitrary scenarios that uses a hybrid combination of a Deep Reinforcement Learning (DRL) method and traditional model-based method. We use pedestrian detection and tracking algorithms to detect groups of people in the camera's Field Of View (FOV) that are closer than 6 feet from each other. Once social distance breaches are detected, the robot prioritizes groups based on their size, navigates to the largest group and encourages following of social distancing norms by displaying an alert message on a mounted screen. For mobile pedestrians who are non-compliant, the robot tracks and pursues them with warnings.
Our COVID-robot uses inexpensive visual sensors such as an RGB-D camera and a 2-D lidar to navigate and to classify pedestrians that violate social distance constraints as \textit{non-compliant} pedestrians. In indoor scenarios, our COVID-robot uses the CCTV camera setup (if available) to further improve the detection accuracy and check a larger group of pedestrians for social distance constraint violations. We also use a thermal camera, mounted on the robot to wirelessly transmit thermal images. This could help detect persons who may have a high temperature without revealing their identities and protecting their private health information.
\noindent Our main contributions in this work are: \\
\noindent \textbf{1.} A mobile robot system that detects breaches in social distancing norms, autonomously navigates towards groups of \textit{non-compliant} people, and encourages them to maintain at least 6 feet of distance. We demonstrate that our mobile robot monitoring system is effective in terms of detecting social distancing breaches in static indoor scenes and can enforce social distancing in all of the detected breaches. Furthermore, our method does not require the humans to wear any tracking or wearable devices.
\noindent \textbf{2.} We also integrate a CCTV setup in indoor scenes (if available) with the COVID-robot to further increase the area being monitored and improve the accuracy of tracking and pursuing dynamic non-compliant pedestrians. This hybrid combination of static mounted cameras and a mobile robot can further improve the number of breaches detected and enforcements by up to 100\%.
\noindent \textbf{3.} We present a novel real-time method to estimate distances between people in images captured using an RGB-D camera on the robot and CCTV camera using a homography transformation. The distance estimate has an average error of 0.3 feet in indoor environments.
\noindent \textbf{4.} We also present a novel algorithm for classifying non-compliant people into different groups and selecting a goal that makes the robot move to the vicinity of the largest group and enforce social distancing norms.
\noindent \textbf{5.} We integrate a thermal camera with the robot and wirelessly transmit the thermal images to appropriate security/healthcare personnel. The robot does not record temperatures or perform any form of person recognition to protect people's privacy.
We have evaluated our method quantitatively in terms of accuracy of localizing a pedestrian, the number of social distancing breaches detected in static and mobile pedestrians, and our CCTV-robot hybrid system. We also measure the time duration for which the robot can track a dynamic pedestrian. Qualitatively, we highlight the trajectories of the robot pursuing dynamic pedestrians when using only its RGB-D sensor as compared to when both the CCTV and RGB-D cameras are used.
The rest of the paper is organized as follows. In Section 2, we present a brief review of related works on the importance of and emerging technologies for social distancing and robot navigation. In Section 3, we provide a background on robot navigation, collision avoidance, and pedestrian tracking methods used in our system. We describe new algorithms used in our robot system related to grouping and goal-selection, CCTV setup, thermal camera integration, etc in Section 4. In Section 5, we evaluate our COVID-robot in different scenarios and demonstrate the effectiveness of our hybrid system (robot + CCTV) and compare it with cases where only the robot or a standard static CCTV camera system is used.
\section{Related Works}
In this section, we review the relevant works that discuss the effectiveness of social distancing and the different technologies used to detect breaches of social distancing norms. We also give a brief overview of prior work on collision avoidance and pedestrian tracking.
\subsection{Effectiveness of Social Distancing}
Works that have simulated the spread of a virus \cite{Mao2011}, \cite{pmid23763426}, \cite{pmid26847017}, \cite{pmid19104659}, \cite{pmid18401408} demonstrate different levels of effectiveness of different kinds of social distancing measures. Effectiveness of a social distancing measure is evaluated based on two factors: (1). the basic reproduction number $R_o$, and (2) the attack rate. $R_o$ is the average number of people to whom an infected person could spread the virus during the course of an outbreak. The attack rate is the ratio between the total number of infected cases over the entire course of the outbreak~\cite{social-distance}.
For instance, in \cite{Mao2011}, in a workplace setting, the attack rate can be reduced by up to $82\%$ if three consecutive days are removed from the workdays for $R_o = 1.4$ \cite{Mao2011}. Similarly, in an $R_o = 1.4$ setting, maintaining 6 feet or more between persons in the workplace could reduce the attack rate by up to $39.22\%$ \cite{pmid23763426} or reduce the rate by $11\%$ to $20\%$ depending on the frequency of contact with other employees \cite{pmid26847017}. Other works that have studied the effects of self-isolation \cite{pmid19104659}, \cite{pmid16642006}, show that it could reduce the peak attack rate by up to 89\% when $R_o < 1.9$.
\subsection{Emerging Technologies for Social Distancing}
Recently, many techniques have been proposed to monitor whether
people are maintaining the 6-feet social distance. For instance, workers in Amazon warehouses are monitored for social distancing breaches using CCTV cameras \footnote{ Link:
\href{https://www.cnbc.com/2020/06/16/amazon-using-cameras-to-enforce-social-distancing-rules-at-warehouses.html}{Amazon CCTV}}. Other methods include using wearable alert devices \footnote{Link: \href{https://www.safespacer.net}{Wearable devices 1}, \href{https://spectrum.ieee.org/the-human-os/biomedical/devices/wearables-track-social-distancing-sick-employees-workplace}{2}, and \href{https://www.safeteams.co}{3}}. Such devices work using Bluetooth or UWB technologies. Companies such as Apple and Google are developing contact tracing applications that can alert users if they come in contact with a person who could be infected \footnote{\href{https://www.google.com/covid19/exposurenotifications/}{Google and Apple contact tracing}}.
A comprehensive survey of all the technologies that can be used to track people to detect if social distancing norms are followed properly is given in \cite{social-distance}. This includes a discussion of pros and cons of technologies such as WiFi, Zigbee, RFID, Cellular, Bluetooth, Computer Vision, AI, etc. However, almost all of these technologies require new static, indoor infrastructure such as WiFi routers, Bluetooth modules, central RFID hubs, etc. Technologies such as RFID and Zigbee also require pedestrians to use wearable tags to localize them.
Most of these technologies are also mostly limited to indoor scenes, with the exception of cellular-based tracking and do not help in \textit{reacting} to cases where people do not follow social distancing guidelines. In \cite{dog-robot}, a quadruped robot with multiple on-board cameras and a 3-D lidar is used to enforce social distancing in outdoor crowds using voice instructions. Our work is complimentary to these methods and also helps react to social distancing violations. Although we evaluate our system indoors, it was trivially be extended to outdoor scenes in the future.
\subsection{Collision-Free Navigation in Crowded Scenarios}
The problem of collision-free navigation has been extensively studied in robotics and related areas. Recently, some promising methods for navigation with noisy sensor data have been based on Deep Reinforcement Learning (DRL) methods~\cite{JHow1,JHow2}. These methods work well in the presence of sensor uncertainty and produce better empirical results when compared to traditional methods such as Velocity Obstacle-based methods \cite{RVO,ORCA}.
These methods include training a decentralized collision avoidance policy by using only raw data from a 2-D lidar, the robot's odometry, and the relative goal location~\cite{JiaPan1}. The policy is extended by combining it with control strategies~\cite{JiaPan2}. Other works have developed learning-based policies that implicitly fuse data from multiple perception sensors to handle occluded spaces~\cite{crowdsteer} and to better handle the Freezing Robot Problem (FRP)~\cite{densecavoid}. Other hybrid learning and model-based methods include \cite{of-vo}, which predicts the pedestrian movement through optical flow estimation. \cite{frozone} constructs a potential freezing zone that is used by the robot to prevent freezing and improve the pedestrian-friendliness of the robot's navigation.
Our navigation approach is also based on DRL and can be combined with any of these methods.
\section{Background And Overview}
In this section, we provide a brief overview of the collision avoidance scheme used in our system, our pedestrian detection and tracking method, and our criteria for social distancing.
\subsection{DRL-Based Collision Avoidance}
We use an end-to-end Deep Reinforcement Learning-based (DRL) policy \cite{JiaPan1} to generate collision-free velocities for the robot. We chose a DRL-based method because it performs well in the presence of sensor uncertainty, and have better empirical results than traditional collision avoidance methods. The collision avoidance policy is trained in a 2.5-D simulator with a reward function that (i) minimizes the robot's time to reach its goal, (ii) reduces oscillatory motions in the robot, (iii) heads towards the robot's goal, and most importantly, (iv) avoids collisions. At each time instance, the trained DRL policy $\pi_{\theta}$ takes 2-D lidar data observations ($\textbf{o}^t_{lidar}$), the relative goal location ($\textbf{o}^t_{goal}$), and the robot's current velocity ($\textbf{o}^t_{vel}$) as inputs to generate collision-free velocities $\textbf{v}^{DRL}$. Formally,
\begin{equation}
\textbf{v}^{DRL} \sim \pi_{\theta}(\textbf{\textbf{v}}^t | \textbf{o}^t_{lidar}, \textbf{o}^t_{goal}, \textbf{o}^t_{vel}).
\end{equation}
This velocity is then post-processed using Frozone \cite{frozone} to eliminate velocities that lead to the Freezing Robot Problem (FRP).
\subsection{Frozone}
Frozone \cite{frozone} is a state-of-the-art collision avoidance method for navigation in moderate to dense crowds ($\ge 1 $ person$/m^2$) that uses an RGB-D camera to track and predict the future positions and orientations of pedestrians relative to the robot. Its primary focus is to simultaneously minimize the occurrence of FRP \cite{freezing1} and any obtrusion caused by the robot's navigation to nearby pedestrians. FRP is defined as any scenario where the robot's planner is unable to compute velocities that move the robot towards its goal. When navigating among humans, the robot must ensure that it does not freeze, as it severely affects its navigation and causes inconvenience to the humans around it.
Frozone's two core ideas are as follows. The robot first classifies pedestrians into \textit{potentially freezing} (more probable of causing freezing) and \textit{non-freezing} pedestrians based on their walking speeds and directions by predicting their future positions over a time horizon. The robot then constructs and avoids a spatial region called the \textit{Potential Freezing Zone} (PFZ). The PFZ corresponds to the set of locations where the robot has the maximum probability of freezing and being obstructive to the pedestrians around it. Formally, the PFZ is constructed as follows:
\begin{equation}
PFZ = Convex Hull (\hat{\textbf{p}}^{ped}_i), \quad i \in {1,2,...,K.},
\end{equation}
\noindent where $\hat{\textbf{p}}^{ped}_i$ is the predicted future position of the $i^{th}$ pedestrian, and K is the total number of potentially freezing pedestrians. $\hat{\textbf{p}}^{ped}_i$ is calculated as, $\hat{\textbf{p}}^{ped}_i = \textbf{p}^{ped}_i + \textbf{v}^{ped}_i\Delta t, where \quad i \in {1,2,...,K}$.
The symbols $\textbf{p}^{ped}_i$ and $\textbf{v}^{ped}_i$ denote the $i^{th}$ pedestrian's current position and velocity vectors relative to the robot, and $\Delta t$ is the time horizon over which prediction is done. If the distance between the robot and the closest potentially freezing pedestrian is less than a threshold distance, the robot deviates its current velocity direction (computed by the DRL method) away from the PFZ.
\subsection{Pedestrian Detection and Tracking} \label{ped-detect}
A lot of work has been done on object detection and tracking in recent years, especially on methods based on deep learning. For detecting and tracking pedestrians, we use the work done in \cite{ped-detect-track} based on Yolov3 \cite{YOLOv3}, a scheme that achieves a good balance between speed and tracking accuracy. The input to the tracking scheme is an RGB image and the output is a set of bounding box coordinates for all the pedestrians detected in the image.
The bounding boxes are denoted as $\mc{B}= \{ \bb{B}_{k} \ | \ \bb{B} = [\textrm{top left}, m_{\bb{B}}, n_{\bb{B}}], \in \mc{H} \}$, where $\mc{H}$ is the set of all pedestrian detections, $\textrm{top left}, m_{\bb{B}},$ and $n_{\bb{B}}$ denote the top left corner coordinates, width, and height of the $k^{th}$ bounding box $\bb{B}_k$, respectively. Apart from these values, Yolov3 also outputs a unique ID for every person in the RGB image, which remains constant as long as the person remains in the camera's FOV. Since Yolov3 requires RGB images, the images from both the RGB-D and the CCTV cameras can be used for detecting pedestrians.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth,height=4.5cm]{Images/Breach-detect-grouping.png}
\caption {\small{\textbf{a.} The criteria used to detect whether two pedestrians violate the social distance constraint. This figure shows two pedestrians represented as circles in two different scenarios. The increasing size of the circles denotes the passage of time. The green circles represent time instances where the pedestrians maintained $> 6$ feet distance, and the red circles represent instances where they were closer than 6 feet. \textbf{Top:} Two pedestrians passing each other. This scenario is not reported as a breach since the duration of the breach is short. \textbf{Bottom:} Two pedestrians meeting and walking together. This scenario is reported as a breach of social distancing norms. \textbf{b.} A top-down view of how non-compliant pedestrians (denoted as red circles) are classified into groups. The numbers beside the circles represent the IDs of the pedestrians outputted by Yolov3. The compliant pedestrians (green circles) are not classified into groups as the robot does not have to encourage them to maintain the appropriate social distance. In the scenario shown, the robot would first attend to Group 1.}}
\label{fig:breach-criteria}
\vspace{-15pt}
\end{figure}
\subsection{Criteria for Social Distancing Breach} \label{breach-criteria}
We mainly focus on detecting scenarios where individuals do not maintain a distance of at least 6 feet from others \textit{for a given period of time} (we choose a 5-second threshold). We choose to detect this scenario because it is a fundamental social distancing norm during all stages of a pandemic, even as people begin to use public spaces and restrictions are lifted.
An important challenge in detecting when individuals are not maintaining appropriate distances amongst themselves is avoiding false negatives. For example, two or more people passing each other should not be considered a breach, even if the distance between them was less than 6 feet for a few moments (see Figure \ref{fig:breach-criteria}a). Another challenge is detecting pedestrians and estimating the distances between them in the presence of occlusions. This can be addressed in indoor scenarios by using available static mounted CCTV cameras.
\section{Our Method}
In this section, we first discuss how our method effectively detects a breach in social distancing norms. We refer to people who violate social distancing norms as \textit{non-compliant} pedestrians. We then describe how we classify non-compliant pedestrians into groups and compute the goal for the robot's navigation based on the size of each group. Our overall system architecture is shown in figure \ref{fig:system-arch}.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth,height=5.25cm]{Images/Sys_Architecture3.png}
\caption {\small{Overall architecture of COVID-Robot and social distance monitoring: Our method's main components are (i) Pedestrian tracking and localization, (ii) Pairwise distance estimation between pedestrians, (iii) Classifying pedestrians into groups, (iv) Selecting a locked pedestrian in the largest group, (v) Using a hybrid collision avoidance method to navigate towards the locked pedestrian, and (vi) Display an alert message to the non-compliant pedestrians encouraging them to move apart. }}
\label{fig:system-arch}
\end{figure}
\subsection{Breach Detection} \label{breach-detection}
As mentioned in Section \ref{breach-criteria}, if certain individuals do not maintain a distance of at least 6 feet from each other, the system must report a breach. The robot's on-board RGB-D camera and the CCTV camera setup (whenever available) continuously monitor the states of individuals within their sensing range. At any instant, breaches could be detected by the robot's RGB-D camera and/or the CCTV camera.
\subsubsection{Social Distance Estimation Using RGB-D Camera} \label{ped-localization-rgbd}
We first describe how we localize a person detected in the RGB image (Section \ref{ped-detect}) with respect to the robot by using its corresponding depth image from the RGB-D camera. The depth and RGB images from the RGB-D camera have the same widths and heights and are aligned by default to be looking at the same subjects (see figure \ref{fig:social-dist-estimation}). We denote the depth image at any time instant \textit{t} as $I^t$, and the value contained in a pixel at coordinates $(i, j)$ is the proximity of an object at that part of the image.
Formally, $I^t = \{C \in \mathbb{R}^{h \times w} : f < C_{ij} < R,$ and $1 \le i \le w \qquad \text{and} \qquad 1 \le j \le h.$ Here, f is an offset distance from the RGB-D camera from where depth can be accurately measured, and R is the maximum range in which depth can be measured. Symbols \textit{w}, \textit{h}, \textit{i}, and \textit{j} represent the image's width, height, and the indices along the width and height, respectively. Using this data, we localize a detected pedestrian \textit{P} as follows.
First, the detection bounding boxes from the RGB image are superimposed over the depth image. Next, the minimum 10\% of the pixel values inside the bounding box $\bb{B}_P$ are averaged to obtain the mean distance ($d_{avg}$) of pedestrian \textit{P} from the camera. Denoting the centroid of the bounding box $\bb{B}_P$ as $[x^{\bb{B}_P}_{cen}, y^{\bb{B}_P}_{cen}]$, the angular displacement $\psi_P$ of the pedestrian relative to the robot can be computed as:
\begin{equation}
\psi_P = \left(\frac{\frac{w}{2} - x^{\bb{B}_P}_{cen}}{w}\right) * FOV_{cam},
\label{eqn:psi-angle-calculation}
\end{equation}
\noindent where $FOV_{cam}$ is the field of view angle of the camera. This calculates the angle in a coordinate system attached to the robot such that its X-axis is along the robot's forward direction and Y-axis is towards the robot's left. $\psi_P$ can range between $[-\frac{FOV_{cam}}{2}, \frac{FOV_{cam}}{2}]$. The pedestrian's position with respect to the robot is then calculated as $[p^{P}_x, p^{P}_y]$ = $d_{avg}$ * [$\cos{\psi_P}, \sin{\psi_P}$].
To estimate the distances between a pair of pedestrians, say $P_a$ and $P_b$, we use the Euclidean distance function given by,
\begin{equation}
dist(P_a, P_b) = \sqrt{(p^{P_a}_x - p^{P_b}_x) + (p^{P_a}_y - p^{P_b}_y).}
\label{dist-measure}
\end{equation}
\noindent If $dist(P_a, P_b) < 6$ feet for a period of time T (we choose 5 seconds), then the robot reports a breach for that pair of individuals. This process is repeated in a pairwise manner for all the detected individuals or pedestrian, and a list of pairs of non-compliant pedestrian IDs is obtained from the sensor data.
\begin{figure}[t]
\centering
\includegraphics[width = \columnwidth, height = 1.8in]{Images/Ped_Localization_RS.png}
\caption {\small{\textbf{Left:} Two pedestrians detected in the RGB image of the robot's RGB-D camera with the bounding box centroids marked in pink and green. \textbf{Right:} The same bounding boxes superimposed over the depth image from the RGB-D camera. The pedestrians are localized and the distance between them is estimated by the method detailed in Section \ref{ped-localization-rgbd}.
}}
\label{fig:social-dist-estimation}
\end{figure}
\subsubsection{Social Distance Estimation Using a CCTV Camera} \label{ped-localization-cctv}
While the robot's RGB-D camera has the advantage of being mobile and being able to detect breaches anywhere, it is limited by a small FOV and sensing range. If a breach of social distancing occurs outside this sensing range, it will not be reported. To mitigate this limitation, we utilize an existing CCTV camera setup in indoor settings to widen the scope for detecting breaches. Pedestrian detection and tracking are done as described in Section \ref{ped-detect}. We estimate distances between individuals as follows.
\textbf{Homography:} All CCTV cameras are mounted such that they provide an \textit{angled} view of the ground plane. However, to accurately calculate the distance between any two pedestrians on the ground, a top view of the ground plane is preferable. To obtain the top view, we transform the CCTV camera's angled view of the ground plane by applying a homography transformation to four points on the ground plane in the angled view. The four points are selected such that they form the corners of the maximum area of a rectangle that can fit within the FOV of the CCTV camera (see Figure \ref{fig:3-transformations}a and b). Let us call this rectangle the homography rectangle. The four points are transformed as,
\begin{equation}
\begin{bmatrix}
x_{corn, top} \\
y_{corn, top} \\
1
\end{bmatrix}
= M * \begin{bmatrix}
x_{corn, ang} \\
y_{corn, ang} \\
1
\end{bmatrix}
\label{homography}
\end{equation}
\noindent where $x_{corn, ang}$ and $y_{corn, ang}$ denote the pixel coordinates of one of the four points in the angled CCTV view image. $x_{corn, top}$ and $y_{corn, top}$ denote the same point after being transformed to the top view, and M is the scaled homography matrix. The homography matrix is computed using standard OpenCV functions.
\textbf{Distance Estimation Between Pedestrians:} After obtaining the homography matrix, we localize each detected pedestrian within the homography rectangle as follows. We first obtain a point corresponding to the feet of a pedestrian P ([$x^P_{feet, ang}, y^P_{feet, ang}$]) by averaging the coordinates of the bottom left and the bottom right corners of the bounding box of the pedestrian (see Figure \ref{fig:3-transformations}a) in the angled CCTV view. This point is then transformed to the top view using Equation \ref{homography} as $[x^P_{feet, top}, y^P_{feet, top}]^T = M*[x^P_{feet, ang}, y^P_{feet, ang}]^T$ (see Figure \ref{fig:3-transformations}b).
The distance between any two pedestrians $P_a$ and $P_b$ is first calculated by using Equation \ref{dist-measure} with the coordinates [$x^{P_a}_{feet, top}, y^{P_a}_{feet, top}$] and [$x^{P_b}_{feet, top}, y^{P_b}_{feet, top}$]. This distance is then scaled by an appropriate factor S to obtain the real-world distance between the pedestrians. The scaling factor is found by measuring the number of pixels in the image that constitute 1 meter in the real-world.
If the real-world distance between a pair of pedestrians is less than 6 feet for a period of time T, a breach is reported for that pair. A list of all the pairs of non-compliant pedestrian IDs is then obtained.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth,height=5.25cm]{Images/Coordinate_Frames2.png}
\caption {\textbf{a.} The angled view of the homography rectangle marked in red and corners numbered from the CCTV camera. The green dots mark the points corresponding to a person's feet in this view. \textbf{b.} The top view of the homography rectangle after transformation and the origin of the top view coordinate system is marked as $o
_{top}$. The coordinates of the feet points are also transformed using the homography matrix. \textbf{c.} A map of the robot's environment with free space denoted in gray and obstacles denoted in black with a coordinate frame at origin $o_{map}$. The homography rectangle is marked in red and the ground plane coordinate system is shown with the origin $o_{gnd}$. }
\label{fig:3-transformations}
\end{figure*}
\subsection{Enforcing Social Distancing}
Once a breach is detected through the robot's RGB-D camera and/or the CCTV camera, the robot must navigate towards the location of the breach and encourages the non-compliant pedestrians to move away from each other through an alert message. If the non-compliant pedestrians are walking, the robot pursues them until they observe social distancing. Prior to this, the robot must compute the location of the breach relative to itself. We detail this process in the following sections.
\subsubsection{Classifying People into Groups}
In social scenarios, people naturally tend to walk or stand in groups. We define a group as a set of people who are closer than 6 feet from each other (see Figure \ref{fig:breach-criteria}b). Therefore, if the robot attends to a group, it can convey the alert message to observe social distancing to all the individuals in that group. In addition, when there are multiple groups of people breaching the social distancing norms, the robot can prioritize attending to each group based on the number of people in it. We classify non-compliant people into groups based on Algorithm \ref{algo:group-classification}.
\begin{algorithm}[h]
\DontPrintSemicolon
\KwInput{A list nonCompPairs of length $S_{input}$}
\KwOutput{A list grpList}
nonCompPairs $\gets $ List of pedestrian ID pairs breaching social distancing \;
grpList $\gets$ nonCompPairs[0] \;
\For{i \text{from} 1 to $S_{input}$}
{
counter $\gets 0$ \;
\For{j \text{from} 0 to len(grpList)}
{ \tcp*{len() returns the length of the list}
intersection $\gets$ grpList[j] $\cap$ nonCompPairs[i] \;
\If{intersection $\ne \emptyset$}
{ grpList[j] $\gets$ grpList[j] $\cup$ grpList[i] \;
}
\Else
{ counter $\gets$ counter + 1 \;
}
}
\If{counter $==$ len(grpList)}
{ grpList.append(nonCompPairs[i])
}
}
\caption{Group Classification Algorithm.}
\label{algo:group-classification}
\end{algorithm}
In Algorithm \ref{algo:group-classification}, nonCompPairs is a list that contains the IDs of all the pairs of non-compliant pedestrians obtained in Section \ref{breach-detection}. grpList is a list of groups where each group contains the IDs of people who have been assigned to it. For example, if nonCompPairs containing pedestrian IDs 1 to 5 looks like [(1, 2), (1,3), (2,3), (4,5)], then grpList would contain two groups and look like [\{1, 2, 3\}, \{4, 5\}]. Once the number of groups and the number of people in each group are known, the robot \textit{locks} a pedestrian in the group with the most people and navigates towards him/her.
\subsubsection{Locked Pedestrian}
Consider a dynamic group of non-compliant pedestrians. The robot's RGB-D camera or the CCTV camera must be able to track at least one member of that group to efficiently guide the robot towards that group. Our method chooses a person who has the least probability of moving out of the FOV of either the robot's RGB-D camera or the CCTV camera (depending on which camera detected the group), and \textit{locks} on to him/her. This person is called the \textit{locked pedestrian}. The identity of the locked pedestrian is updated as people's positions change. To find the locked pedestrian, we consider the centroid of the bounding box of each person in the largest group. The person whose centroid has the least lateral distance from the center of the image is chosen as the locked pedestrian. That is, the condition for locking a pedestrian is,
\begin{equation}
x^{P_{lp}}_{cent} - \frac{w}{2} = \min_{k \in \mathcal{I}_{\mathcal{G}}} x^{P_k}_{cent} - \frac{w}{2},
\end{equation}
where $\mathcal{I}_{\mathcal{G}}$ is the set of IDs for the detected pedestrians in the current largest group and $P_{lp}$ denotes the locked pedestrian.
\subsubsection{Computing Goal Position Using an RGB-D Camera}
Once a pedestrian is locked, the robot localizes him/her relative to itself using the $d_{avg}$ and equation \ref{eqn:psi-angle-calculation} in Section \ref{ped-localization-rgbd}). That is,
\begin{equation}
o^t_{goal} = d^{lp}_{avg} * [\cos{\psi_{P_{lp}}} \sin{\psi_{P_{lp}}}]^T,
\end{equation}
Where, $o^t_{goal}$ is the location of the goal relative to the robot, $d^{lp}_{avg}$ is the average distance and $\psi_{P_{lp}}$ is the angular displacement of the locked pedestrian from the robot respectively. The DRL method and Frozone use $o^t_{goal}$ to navigate the robot towards the locked pedestrian in a pedestrian friendly way without freezing.
\subsubsection{Computing Goal Position Using a CCTV Camera}
If the CCTV camera detects a breach and a locked pedestrian, the goal computation for the robot requires homogeneous transformations between three coordinate frames: 1. the top-view image obtained after homography, 2. the ground plane, and 3. a map of the environment with which the robot is localized. These three coordinates with origins $o_{top}$, $o_{gnd}$ and $o_{map}$ respectively are shown in figure \ref{fig:3-transformations}b and c. First, the locked pedestrian's location in the ground plane coordinate frame is obtained. This is done as follows.
Let us consider the point in the top-view image corresponding to the feet of the locked pedestrian $[x^{P_{lp}}_{feet, top}, y^{P_{lp}}_{feet, top}]$ and corner point 1 (see figure \ref{fig:3-transformations}) for the homography rectangle $[x_{corn, top}, y_{corn, top}] = o_{top}$. The angle between the two points in the image is calculated as,
\begin{equation}
\theta_{lp-corn, top} = \tan^{-1}(\frac{(y_{corn, top} - y^{P_{lp}}_{feet, top})}{(x_{corn, top} - x^{P_{lp}}_{feet, top})}). \\
\label{angle-measure}
\end{equation}
We consider corner point 1 of the homography rectangle in the real world to be the origin of the coordinate system fixed to the ground plane ($o_{gnd}$) with its X and Y axes aligned with the X and Y axes of the top view image (see Figures \ref{fig:3-transformations}a and c. Therefore, the angle $\theta_{lp-corn, top}$ also corresponds to the angle between the two points on the ground plane $\theta_{lp-corn, gnd}$. The Euclidean distance ($r_{lp-corn, top}$) between the points on the top view image is calculated using Equation \ref{dist-measure}. The real world distance between the points, denoted as $r_{lp-corn, gnd}$ is then obtained by scaling $r_{lp-corn, top}$ using the factor S.
The location of the locked pedestrian in the ground coordinate frame is calculated as $[x^{P_{lp}}_{feet, gnd} y^{P_{lp}}_{feet, gnd}]^T = o_{gnd}^T + r_{lp-corn, gnd}*[\cos\theta_{lp-corn, gnd} \,\,\, \sin\theta_{lp-corn, gnd}]^T$.
The locked pedestrian's location is then converted to the map coordinate frame using a homogeneous transformation matrix as, $[x^{P_{lp}}_{feet, map} y^{P_{lp}}_{feet, map}]^T = H^{map}_{gnd} * [x^{P_{lp}}_{feet, gnd} y^{P_{lp}}_{feet, gnd}]^T$. Now, since the robot and the locked pedestrian are localized with respect to the map coordinate frame, the relative goal location for the robot can be obtained as follows,
\begin{equation}
o^t_{goal} = [x^{P_{lp}}_{feet, map} y^{P_{lp}}_{feet, map}]^T - [x_{robot, map} y_{robot, map}]^T.
\end{equation}
Where $x_{robot, map}$ and $y_{robot_map}$ are the X and Y coordinates of the robot in the map coordinate frame. Using this $o^t_{goal}$, the trained DRL policy computes the collision-free velocity towards the locked pedestrian.
\subsubsection{Multiple Groups and Lawnmower Inspection}
So far, we have discussed how a breach of social distancing norms can be detected using either the robot's RGB-D camera or an existing, independent CCTV camera setup. In the case where both cameras detect several groups of non-compliant pedestrians, the robot attends to the group with the most number of individuals. The robot attends to a group until everyone in the group observes the appropriate distancing measures. Once the robot is done attending to a group, the next largest group is selected and attended to. If the same group is detected in both cameras, the goal data computed using the CCTV camera will be used to guide the robot.
To improve the effectiveness of the integrated robot and CCTV system in detecting new non-compliant groups of pedestrians, the robot inspects the blind spots of the CCTV camera continuously by following the well-known lawnmower strategy. This expands the total area that the system is monitoring at any time instant. In addition, the lawnmower strategy guarantees that 100\% of an environment can be covered by navigating to a few fixed waypoints, although it does not guarantee an increase in the number of breaches detected.
\subsection{Alerting and Encouraging Pedestrians}
Once the robot reaches the vicinity of the locked pedestrian, the robot first displays the reason why they were approached on its mounted screen; the estimated distance between the people in the group. The robot then displays a message encouraging the people to stay apart from each other.
While this is a simplistic approach, this setup can be easily improved with a number of extensions in the future. For instance, the robot can also \textit{talk} to the people in a group by either playing a recorded message, or a message from the security authorities. It can also be extended to include virtual AI applications that can assist people by understanding the context of the scenario.
\subsection{Thermal Camera}
As mentioned previously, the robot is also equipped with a thermal camera that generates images based on the differences in temperatures of different regions that it observes (see Figure \ref{fig:thermal-cam}). Our pedestrian detection scheme detects and tracks people on these images and the results are then sent to appropriate security or healthcare personnel who detects if an individual's temperature signature is higher than normal. Measures can then be initiated to trace the person for future contact tracing. We intentionally choose to have a human in the loop instead of performing any form of facial recognition to protect people's privacy.
Such a system would be useful in places where people's temperatures are already measured by security/healthcare personnel such as airports, hospitals etc. Monitoring people's temperatures remotely reduces exposure for security/healthcare personnel, thus reducing their risk of contracting the coronavirus.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth,height=5.50cm]{Images/Thermal_Image_Combined.png}
\caption {\small{Thermal images generated by the thermal camera that is wirelessly transmitted to appropriate security/healthcare personnel. The temperature signatures of the people irrespective of their orientations. We intentionally have a human in the loop to monitor people's temperature signatures, and we do not perform any form of facial recognition on people to protect their privacy. Pedestrians are detected on the thermal image to aid the personnel responsible for monitoring the area.}}
\label{fig:thermal-cam}
\end{figure}
\section{Results and Evaluations}
In this section, we elaborate on how our system was implemented on a robot, explain the metrics we use to evaluate our system and analyze the effectiveness and the limitations of our method.
\subsection{Implementation}
We implement our method on a Turtlebot 2 robot customized with additional aluminium rods to attach a 15-inch screen to display messages to the non-compliant pedestrians. We specifically chose the Turtlebot 2 due to its ease of customization and its light-weight and tall structure. The pedestrian detection and tracking algorithm is executed on a laptop with an Intel i9 8th generation CPU and an Nvidia RTX2080 GPU mounted on the robot. We use an Intel Realsense (with $70^o$ FOV) RGB-D camera to sense pedestrians and a Hokuyo 2-D lidar ($240^o FOV$) to sense other environmental obstacles.
hTo emulate a CCTV camera setup, we used a simple RGB webcam with a 1080p resolution mounted at an elevation. To process the images from the CCTV camera, we use a laptop with an Intel i7 7th generation CPU and an Nvidia GTX1060 GPU. We use a FLIR C3 thermal camera to generate the temperature signatures of the robot's surroundings. The ROS package for adaptive Monte-Carlo localization is used for locating the robot relative to the map coordinate frame.
\subsection{Metrics}
We use the following metrics to evaluate our method.
\begin{itemize}
\item \textbf{Accuracy of pedestrian localization:} We compare the ground truth location of a pedestrian with the location estimated using our method as detailed in Sections \ref{ped-localization-rgbd} and \ref{ped-localization-cctv}. Higher localization accuracy translates to more accurate distance estimation and goal selection for the robot's navigation.
\item \textbf{Number of breaches detected:} This is the total number of locations in an environment at which a social distancing breach can be detected, given a total number of locations uniformly sampled from the environment. We measure this metric both in the presence and absence of occlusions in the environment. This metric provides a sense of the area in the environment that can be monitored by our system at any time instant. Higher values are better.
\item \textbf{Number of enforcements:} The number of times the robot attended to a breach once it was detected. We again measure this in the presence and absence of occlusions in the environment. Ideally should be equal to the number of breaches detected.
\item \textbf{Tracking Duration for a mobile pedestrian:} We measure the time for which the robot is able to track a walking pedestrian. Since the robot's RGB-D camera has a limited FOV, the robot must rotate itself to track a pedestrian for a longer time. This metric is a measure of the robot's effectiveness in pursuing a mobile locked pedestrian (in a group of people who are walking together).
\end{itemize}
\subsection{Experiments and Analysis}
\subsubsection{Accuracy of Pedestrian Localization}
We perform two sets of comparisons of the ground truth locations versus the estimated pedestrian location using 1. the robot's RGB-D camera, and 2. the CCTV setup. The plots are shown in Figure \ref{fig:localization-accuracy}, with the ground truth locations plotted as green circles and the estimated locations plotted as blue circles. The plot in Figure \ref{fig:localization-accuracy}a shows the pedestrian being localized with respect to a coordinate axis fixed to the robot, with its positive X-axis pointing in the robot's forward direction and the positive Y-axis pointing towards the robot's left. Figure \ref{fig:localization-accuracy}b shows a pedestrian being localized in the ground coordinate frame.
We observe in Figure \ref{fig:localization-accuracy}a that when a pedestrian is closer to the robot and closer to the X-axis of the robot, the localization estimates closely match the ground truth. If a pedestrian is farther away from the robot or near the exterior limits of the RGB-D camera's FOV, the errors between the estimates and the ground truth values increase. This is mainly because the robot localizes a pedestrian based on the centroid of the bounding box of the pedestrian, which is located on the person's torso, whereas the ground truth is measured as a point on the ground.
In addition, the orientation of the pedestrian relative to the RGB-D camera also affects the centroid of the bounding box and the localization estimate. However, since the maximum error between the ground truth and estimated values is within 0.3 meters, its effect on the social distance calculation and goal selection for the robot is within an acceptable limit. The accuracy can also be improved with higher FOV depth cameras in the future.
From Figure \ref{fig:localization-accuracy}b we see a trend similar to the plot in \ref{fig:localization-accuracy}a. The farther away a person is from the origin ($o_{gnd}$), the greater the error between the ground truth and the pedestrian's estimated location. This is due to the approximations in the homography in obtaining the top view from the angled CCTV view, which carries forward to computing [$x^{P_a}_{feet, top}, y^{P_a}_{feet, top}$] (Section \ref{ped-localization-cctv}). However, the maximum error between the estimates and ground truths is again within 0.25 meters. Also, since a pedestrian's location is estimated by the point corresponding to his/her feet, errors due to the pedestrian's orientation are less frequent. The average error in the distance estimation between pedestrians is $\sim$ 0.3 feet.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth,height=5.5cm]{Images/GTvsEstimate4.png}
\caption {\small{Plots of ground truth (blue dots) versus pedestrian localization (red dots) when using the robot's Realsense camera and the static CCTV camera with more FOV. \textbf{a.} The estimates from the Realsense camera tend to have slightly higher errors because we localize pedestrians using averaged proximity values within their detection bounding boxes, which is affected by the size of the bounding boxes. \textbf{b.} Localization using the data from the CCTV camera is more accurate as it tracks a person's feet. This method is not affected by a person's orientation. We observe that in both cases, the localization errors are within the acceptable range of 0.3 meters. }}
\label{fig:localization-accuracy}
\end{figure*}
\subsubsection{Breach Detection and Enforcement}
\begin{table}[]
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|c|c|}
\hline
\rowcolor{lightgray} \multicolumn{4}{|c|}{Case 1: Static Robot No Occlusions} \\
\hline
\textbf{Metric} & \textbf{CCTV-only} & \textbf{Robot-only} & \textbf{Robot-CCTV Hybrid} \\
\hline
Number of breaches detected & 20 & 10 & 30 \\
Number of enforcements & NA & 10 & 30 \\
\hline
\rowcolor{lightgray} \multicolumn{4}{|c|}{Case 2: Static Robot With 50 \% Occlusion} \\
\hline
Number of breaches detected & 20 & 7 & 27 \\
Number of enforcements & NA & 7 & 27 \\
\hline
\rowcolor{lightgray} \multicolumn{4}{|c|}{Case 3: Lawnmower exploration With 50\% Occlusions} \\
\hline
Number of breaches detected & 20 & 20 & 40 \\
Number of enforcements & NA & 20 & 40 \\
\hline
\end{tabular}
}
\caption{\small{Comparison of three configurations in terms of detecting breaches in social distancing norms when two pedestrians are static in any one of 40 points in a laboratory setting. We observe that CCTV + robot configuration has the most number of breaches detected even when the robot is static and outside the CCTV's sensing range. When the robot is mobile, following lawnmower waypoints outside of the CCTV's FOV, it can detect a breach in any of the 20 locations that could not be detected by the CCTV camera.}}
\label{tab:detections-enforcements}
\vspace{-10pt}
\end{table}
In this experiment we compare the performance differences in detecting a social distancing breach and enforcing social distancing guidelines for three configurations: 1. CCTV only, 2. Robot only, and 3. Robot-CCTV hybrid system.
The detection and enforcement capabilities of these systems in dynamic scenes vary extensively depending on the initial orientation of the robot and the walking speed and walking directions of pedestrians. Therefore, we standardize the experiment by comparing the best performances of the three configurations in terms of their ability to detect crowding and social distancing breaches in static scenes and the number of times the robot attended to those breaches in a laboratory setting. We demonstrate the robot's ability to track mobile pedestrians in the next section.
For this experiment, we uniformly sample 40 points in our lab, with 20 points within the FOV of the CCTV camera, and 20 points outside it. Each one of those points could be a location for a social distancing breach at any time instant. We evaluate how many of these points are visible to both cameras and the effect of the robot's mobility. The robot is placed in a fixed location outside the sensing region of the CCTV camera for the static case, and in the mobile case, the robot moves along a lawnmower trajectory outside the CCTV's FOV. The social distancing breaches can also be partially occluded. When a breach is 50\% occluded, we mean a scenario where a person blocks another person such that the half of the human body divided by the sagittal plane is visible to the camera.
The results are shown in Table \ref{tab:detections-enforcements}. As can be seen, the CCTV-only configuration is capable of detecting the standard 20 breaches within its sensing region. It can also handle occlusions between pedestrians better and detect breaches due to the CCTV camera's global view of the environment. It should be noted that this system is an improvement over current CCTV systems where a human manually detects excessive crowding and initiates countermeasures. However, there is no scope for enforcing social distancing at the location of the breaches.
The robot-only configuration detects fewer breaches (10 breaches) than the CCTV setup within the RGB-D camera's sensing region when the robot is static (due to its low FOV). Objects occluding the social distancing breaches also adversely affect the number of detections made by the robot. However, when the robot is moving along a lawnmower trajectory outside the CCTV's FOV, the robot detects the social distancing breaches that could be at any of the 20 locations regardless of whether they are occluded or not.
The robot-CCTV hybrid configuration provides the best performance of the three configurations in terms of detecting novel breaches at the most locations when the robot is static. This is because, when the robot is outside the sensing region of the CCTV camera, the hybrid configuration monitors the largest area in the environment. This configuration also provides better tracking capabilities when a pedestrian is walking (see Section \ref{pursue-walking-ped}). We also note that, in static scenarios, the robot attends to 100\% of the breaches that are detected.
\subsubsection{RGB-D Pedestrian Tracking Duration}
\begin{table}[]
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|}
\hline
\rowcolor{lightgray} \multicolumn{2}{|c|}{Case 1: Maximum Robot Angular Velocity = 0.5 rad/sec} \\
\hline
\footnotesize{\textbf{Pedestrian Velocity (m/sec)}} & \footnotesize{\textbf{Tracking time (sec)}} \\
\hline
0.25 & 20 (\gray{20}) \\
\hline
0.5 & 6.59 (\gray{10})\\
\hline
0.75 & 3.15 (\gray{6.67})\\
\hline
1 & 2.95 (\gray{5})\\
\rowcolor{lightgray} \multicolumn{2}{|c|}{Case 2: Maximum Robot Angular Velocity = 0.75 rad/sec} \\
\hline
0.25 & 20 (\gray{20})\\
\hline
0.5 & 10 (\gray{10})\\
\hline
0.75 & 3.91 (\gray{6.67})\\
\hline
1 & 2.93 (\gray{5})\\
\rowcolor{lightgray} \multicolumn{2}{|c|}{Case 3: Maximum Robot Angular Velocity = 1.0 rad/sec} \\
\hline
0.25 & 20 (\gray{20})\\
\hline
0.5 & 10 (\gray{10})\\
\hline
0.75 & 6.58 (\gray{6.67})\\
\hline
1 & 2.77 (\gray{5})\\
\hline
\end{tabular}
}
\caption{\small{The duration for which the robot tracks a walking pedestrian for different pedestrian walking speeds and maximum angular velocities of the robot. The pedestrian walks 5 meters in a direction perpendicular to the robot's orientation and it has to rotate and track the walking pedestrian. The ideal time for which a pedestrian should be tracked is given in the bracket beside the actual time. The robot can effectively track a pedestrian walking at up to 0.75 m/sec when its angular velocity is 1 rad/sec.}}
\label{tab:ped-tracking-vel}
\vspace{-10pt}
\end{table}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth,height=5.25cm]{Images/Ped_Pursue_Trajectories.png}
\caption {\small{Trajectories of two non-compliant pedestrians (in red) and the robot pursuing them (in green) in the mapped environment shown in figure \ref{fig:3-transformations}c. The pink and blue colors denote the static obstacles in the environment. \textbf{a.} The robot only uses its RGB-D sensors to track the pedestrian. The robot pursues the pedestrians successfully when they move in a smooth trajectory. \textbf{b.} The robot's RGB-D camera is unable to track the pedestrians when they make a sudden sharp turn. \textbf{c.} When the CCTV camera is used to track the pedestrians, the robot follows their trajectories more closely. \textbf{d.} Pedestrians making sharp and sudden turns can also be tracked. The black line denotes the point where the pedestrians leave the CCTV camera's FOV, from where the RGB-D camera tracks the pedestrians. Sharp turns in \textbf{d} again become a challenge.}}
\label{fig:walking-ped-traj}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth,height=4.10cm]{Images/Ped_Pursuit_lab2.png}
\caption {\small{Two mobile non-compliant pedestrians detected by the CCTV camera, pursued by our COVID-Robot in a laboratory setting. The locked pedestrian is marked with a green dot at his feet. Note that the locked pedestrian is changed based on the positions of the two pedestrians in the CCTV footage. The robot pursues them until they maintain the appropriate distance.}}
\label{fig:ped-pursuit}
\end{figure*}
In this experiment, we measure the duration for which the robot-only configuration can track walking pedestrians using only its onboard RGB-D sensor. Since it is limited by its FOV, continuously tracking a pedestrian who is walking out of the RGB-D camera's FOV is challenging. To counteract this limitation and track a pedestrian for a longer time, the robot has to rotate/move towards the pedestrian along the pedestrian's walking direction. We vary the walking speed of a pedestrian moving in a direction that is perpendicular to the orientation of the robot. We also vary the maximum angular velocity of the robot to measure the differences in tracking performance (Table \ref{tab:ped-tracking-vel}).
We observe that the greater the maximum angular velocity of the robot, the better it can track a fast-moving pedestrian. However, since the robot is navigating among humans, we limit the maximum linear and angular velocities to 0.75 m/sec and 0.75 rad/sec, respectively, to minimize the disturbance it causes them. We observe that capping the angular velocity makes it challenging for the robot to track pedestrians walking at $>0.75$ m/sec. Even when the robot is used at its maximum 1 rad/sec angular velocity, pedestrians walking at 1 m/sec are difficult to track. This can only be alleviated in the future when depth cameras improve their range and FOV.
\subsubsection{CCTV-Guided Walking Locked Pedestrian Pursuit} \label{pursue-walking-ped}
We qualitatively demonstrate how a robot pursues two walking non-compliant pedestrians by plotting their trajectories in the cases where the RGB-D (see figure.\ref{fig:walking-ped-traj}a and b) or the CCTV camera (see figure.\ref{fig:walking-ped-traj}c and d) detects him/her. Figure \ref{fig:walking-ped-traj} a shows that when the pedestrians walk in a smooth trajectory without sharp turns, the robot is able to successfully track them throughout their walk.
In figure \ref{fig:walking-ped-traj}b, we observe that when the pedestrians make a sharp turn and manage to go outside the limited FOV of the RGB-D camera, the robot is unable to pursue him/her. The pedestrians were walking at speeds $\sim$0.75 m/sec. This issue is alleviated when the CCTV camera tracks both the pedestrians instead of the RGB-D camera.
Figure \ref{fig:walking-ped-traj}c and d show that the robot is able to track the pedestrians more closely and accurately with the goal data computed using the CCTV's localization. In addition, sudden and sharp turns by the pedestrians are handled with ease, and pedestrians moving at speeds $\sim 0.75$ m/sec can be tracked and pursued, which was not possible with the robot-only configuration. When the pedestrians move out of the CCTV camera's FOV (black line in figures \ref{fig:walking-ped-traj}c and d), the data from the robot's RGB-D camera helps pursue the two pedestrians immediately. However, the pedestrians' sharp turns again becomes a challenge to track.
The robot pursuing two non-compliant pedestrians in our lab setting is shown in figure \ref{fig:ped-pursuit}.
\section{Conclusions, Limitations and Future Work}
We present a novel method to detect breaches in social distancing norms in indoor scenes using visual sensors such as RGB-D and CCTV cameras. We use a mobile robot to attend to the individuals who are non-compliant with the social distancing norm and to encourage them to move apart by displaying a message on a screen mounted on the robot. We demonstrate our method's effectiveness in localizing pedestrians, detecting breaches, and pursuing walking pedestrians. We conclude that the CCTV+robot hybrid configuration outperforms configurations in which only one of the two components is used for tracking and pursuing non-compliant pedestrians.
Our method has a few limitations. For instance, our method does not distinguish between strangers and people from the same household. Therefore, all individuals in an indoor environment are encouraged to maintain a 6-foot distance from each other. Our current approach for issuing a warning to violating pedestrians using a monitor has limitations, and we need to develop better human-robot approaches. As more such monitoring robots are used to check for social distances or collecting related data, this could also affect the behavior of pedestrians in different settings.
We need to perform more studies on the social impact of such robots. Due to COVID restrictions, we have only been able to evaluate the performance of COVID-robot in our low to medium density laboratory settings. Eventually, we want to evaluate the robot's performance in crowded public settings and outdoor scenarios. We also need to design better techniques to improve the enforcement of social distancing by using better human-robot interaction methods.
\section{Introduction}
\section*{Acknowledgment}
This work is supported in part by ARO grant W911NF1910315 and NSF grant 2031901.
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\section{Introduction}
The existence of (generalized) magnetic helicity constraints introduces a fundamental distinction between the Navier-Stokes fluid turbulence and the low-frequency Alfv\'enic turbulence realized in magnetized plasma systems. The resultant Alfv\'enic turbulence has been widely investigated within the context of the simplest model - namely, ideal magnetohydrodynamics (MHD) - but also for models that are often collectively known as ``beyond MHD'' or extended MHD \citep{GP04,Frei14}.
The scientific literature is replete with examples wherein helicity invariants have been exploited to find new relaxed states. The most famous among them are the so-called Woltjer-Taylor states of ideal MHD (${\bf\nabla\times B}= \mu {\bf B}$) that are obtained by minimizing the magnetic energy $\langle|{\bf B}|^2\rangle$ while holding the magnetic helicity $h_m=\langle{\bf A}\cdot{\bf B}\rangle$ fixed \citep{Wol1,Tay1,Berg99,MM12}; henceforth, we shall make use of the notation $\langle \dots \rangle = \int d^3x$ for the sake of simplicity.
A very crucial role played by the constancy of $h_m$ in the evolution of MHD turbulence was identified in early MHD simulations as well as in analytical models: it permitted the inverse cascading of magnetic helicity in 3D models \citep{FPLM75,PFL76},\footnote{However, at scales smaller than the electron skin depth, the inverse cascade of helicity is transformed into a direct cascade as per theory and simulations \citep{MLM17,MMT18}.} whereas in standard fluid turbulence, the transfer of energy and helicity is typically from larger to smaller scales \citep{Moff78,KR80,Bis03,ZMD04,BS05,Gal18}. The latter feature was inherent in the famous conjecture of Andrey Nikolaevich Kolmogorov \citep{Kol41} that led to the equally famous scaling law $E_k\sim k^{-5/3}$ for the kinetic energy spectrum $E_k$ \citep{Fri,Bis03,Dav}. Another seminal result in the realm of MHD turbulence is the Iroshnikov-Kraichnan theory \citep{Iro63,Kra65}, which modelled turbulent fluctuations as weakly interacting Alfv\'enic wave packets and yielded the magnetic energy spectrum $E_k\sim k^{-3/2}$ \citep{Bis03}.
This paper, although motivated by Kolmogorov's legacy, will dwell on precisely those features of Alfv\'enic turbulence that are absent in Navier-Stokes systems. The goal, in the spirit of \citet{Kol41}, is to obtain results of maximal simplicity and, hopefully, of considerable generality that are potentially valid for all Alfv\'enic turbulence irrespective of its origination and evolution. More precisely, we will delve into constraints on Alfv\'enic turbulence imposed by the helicity and energy invariants of extended MHD, and thereby extend prior analyses along similar lines \citep{OSYM,OSM,MNSY}; see also \citet{Hel17}.
As we shall show henceforth, this line of enquiry yields several results of broad scope and interest: (1) Total turbulent energy in each channel - namely, magnetic ($E_m$), kinetic ($E_\mathrm{kin}$) and thermal ($E_\mathrm{th}$) - is determined by a single attribute of turbulence, namely, a characteristic length scale ($L_T=K_T^{-1}$), (2) Complete expressions in terms of a single unknown parameter for all these energies in terms of the invariants and $L_T$, thus enabling us to predict, for instance, the relative energy distribution.
\section{Invariants of extended MHD}
For the sake of simplicity, we will concentrate on a two component (electron-ion) quasineutral plasma. Under the assumption of isotropic pressure with an adiabatic equation of state $(p \propto n^{\gamma})$, each component obeys the following equation of motion \citep{SI,SM1,ML1}:
\begin{equation}\label{eqm1}
{\frac{\partial}{\partial t}}\, {\bf P}_\beta = {\bf v}_\beta\times{\bf\Omega}_\beta-\nabla{\psi_\beta},
\end{equation}
where ${\bf P}_\beta={\bf A} + (m_\beta c/q_\beta){\bf v}_\beta$ is proportional to the canonical momentum,
${\bf\Omega}_\beta = {\bf\nabla\times\bf P}_\beta = {\bf B}+(m_\beta c/q_\beta)\nabla\times{\bf v}_\beta$ represents the generalized vorticity for the species $\beta$ with mass and charge of $m_\beta $ and $q_\beta$, and ${\psi_\beta}= c/q_\beta(h_\beta + 1/2 m_\beta v_\beta^2+ q_\beta \phi)$ encompasses all of the gradient forces; note that $h_\beta$ is the specific enthalpy, and $\phi$ is the electrostatic potential.
Taking the curl of Eq.~(\ref{eqm1}) yields the canonical vortical dynamics \citep{MY98,SM1,AGMD} that is epitomized by
\begin{equation}
\frac{\partial{\bf\Omega}_\beta}{\partial t}=\nabla\times\left({\bf v}_\beta\times
{\bf\Omega}_\beta\right), \label{eqmotion2}
\end{equation}
The low frequency behavior of this system of ideal fluid equations, which is closed via the Amp\`ere's law,
\begin{equation} \label{Amplaw}
{\bf\nabla\times\bf B}=(4\pi/c){\bf J}, \quad\quad {\bf J}=\sum q_\beta n_\beta{\bf v}_\beta,
\end{equation}
is the object of this investigation. Straightforward manipulation of (\ref{eqm1})-(\ref{Amplaw}) yields the following three constants of motion: the total energy
\begin{equation}
E=\Bigg\langle{\frac{B^2}{2}}+{\frac{1}{2}}\sum_\beta n_\beta m_\beta v^2_\beta+{\frac{p}{(\gamma-1)}}\Bigg\rangle,\label{EnCons}
\end{equation}
where $p$ is the total pressure, and two generalized helicities (GH),
\begin{equation}
H_\beta=\frac{1}{2}\langle {\bf P}_\beta\cdot{\bf\Omega}_\beta\rangle,
\label{HelicCons}
\end{equation}
associated with each species. For a perfectly conducting system of $n$ dynamical species, there exist a total of $(n+1)$ bilinear invariants \citep{ML1}. Although it is self-evident, it must nevertheless be emphasized that in any magneto-fluid system, unless the fluid inertia is neglected, it is the generalized helicity $H_\beta$, and \emph{not} the magnetic helicity $H_m= \langle{{\bf A}\cdot{\bf\ B}}\rangle$ that is conserved; for instance, the conservation of $H_m$ in MHD and in Hall MHD holds true because electron inertia is ignored \citep{Turn86}.
To study the constrained Alfv\'en dynamics (including turbulence), it is convenient to work in the equivalent one fluid variables, viz.,
the center-of-mass velocity ${\bf V}$, and the current ${\bf J}$ defined below:
\begin{equation}
{\bf V}=\frac{m{\bf v_e}+ M{\bf v_i}}{m+M}= \mu_e {\bf v_e}+ \mu_i {\bf v_i}, \quad {\bf J}= ne({\bf v_i}-{\bf v_e})
\label{1fluvariables}
\end{equation}
where the two species are identified as electrons (mass $m$ and charge $-e$) and protons (mass $M$ and charge $e$). However, we shall not rewrite
(\ref{eqm1})-(\ref{eqmotion2}) explicitly in terms of ${\bf V}$ and ${\bf J}$ because, in what follows, we will focus only on the invariants
(\ref{EnCons})-(\ref{HelicCons}). The electron and ion helicity invariants
translate into the new variables as
\begin{eqnarray}\label{elec-ion helicities}
2H_e = \left<{\bf \hat A}\cdot {\bf \hat B} + (m/e)^2 {\bf V}\cdot{\bf\nabla\times\bf V} -2 (m/e){\bf V}\cdot{\bf \hat B}\right> \\
2H_i = \left<{\bf \hat A}\cdot {\bf \hat B} + (M/e)^2 {\bf V}\cdot{\bf\nabla\times\bf V} +2 (M/e){\bf V}\cdot{\bf \hat B}\right>
\end{eqnarray}
where ${\bf \hat A}= {\bf A}+\lambda_e^2 {\bf\nabla\times\bf B}$ is the vector potential modified by the contribution stemming from a finite electron skin depth $(\lambda_e^2=c^2/{\omega_{pe}}^2)$; in other words, the second term in ${\bf \hat A}$ is obtained after using the Amp\`ere's law given by (\ref{Amplaw}); it is also derivable by means of the Hamiltonian or Lagrangian formulations from the parent two-fluid model \citep{KC14,AKY15,LMM15,LMM16,DML16}.
Evidently, both helicities comprise of their purely magnetic (${\bf \hat A}\cdot {\bf \hat B}$), purely kinematic (${\bf V}\cdot{\bf\nabla\times\bf V}$), and the mixed (i.e., cross) (${\bf V}\cdot{\bf \hat B}$) components - the chief difference is that the contribution of the kinematic and mixed parts can be far more dominant for the protons (due to $M\gg m$). For further analysis, it is much more transparent to construct the invariant combinations:
\begin{equation}\label{plus- helicity}
H_+= 2\mu_{i}H_i+2\mu_{e}H_e= \left<{\bf \hat A}\cdot {\bf \hat B}\right> + \frac{m}{M} \lambda_i^2 \left<{\bf V}\cdot{\bf\nabla\times\bf V}\right>
\end{equation}
\begin{equation}\label{minus- helicity}
H_-= 2H_i - 2H_e= \lambda_i^2 \left<{\bf V}\cdot{\bf\nabla\times\bf V}\right> +2 \lambda_i \left<{\bf V}\cdot{\bf \hat B}\right>
\end{equation}
where the magnetic field has been normalized to some ambient field strength $B_0$ and and the velocity field is measured in terms of the corresponding Alfv\'en speed $V_A$ (where $V_A^2= B_0^2/(4\pi n M)$), i.e., we have used Alfv\'enic units \citep{ML1}. Notice that, aside from the normalized fields, the only basic parameter is the ion skin depth $\lambda_i$ (where $\lambda_i^2 = c^2/{\omega_{pi}}^2)$ that defines the intrinsic length scale of the system. Of course, the existence of the term proportional to $(m/M)$ serves as a reminder that the electron inertia is not (yet) neglected and the electron length scale ($\lambda_e= {\sqrt {(m/M)}} \lambda_i$) appears in $H_+$ and in the variables ${\bf \hat A}$ and ${\bf \hat B}$. With the above choice of normalization, the helicities acquire the dimensions of length. In what follows, we shall utilize the dimensionless helicities defined to be $h_\pm=H_\pm/\lambda_i$.
These invariants $h_+$ and $h_-$ act in concordance to constrain the total magnetic and kinetic energies of the system. We suppose that the system is embedded in an ambient magnetic field such that ${\bf \hat B} = {\bf \hat B_0} + {\bf \hat b}$, and a similar expression can be constructed for the vector potential ${\bf \hat A}$.\footnote{We implicitly presume that the functions ${\bf \hat B}$ and ${\bf \hat A}$ are well-behaved and that the term $\left<{\bf \hat A_0}\cdot {\bf \hat B_0}\right>$ is finite.} For the time being, we analyze the case where there exists no ambient flow, implying that ${\bf v}$ fully represents the velocity field (i.e., we have ${\bf V}$=${\bf v}$). Note that $\left<{\bf \hat A}\cdot {\bf \hat B}\right>$ will acquire a contribution of the form $\left<{\bf \hat A_0}\cdot {\bf \hat B_0}\right>=H_0$, while the linear terms will vanish on integration \citep{KR80}. Written fully in terms of the (normalized) fluctuating fields denoted by lowercase boldface letters, our normalized invariant equations become
\begin{equation} \label{plus-turb helicity}
h = h_+ - h_0= \frac{\left<{\bf \hat a}\cdot {\bf \hat b}\right>}{\lambda_i} + \frac{m}{M} \lambda_i \left<{\bf v}\cdot{\bf\nabla\times\bf v}\right>,
\end{equation}
\begin{equation}\label{minus-turb helicity}
h_-= \lambda_i \left<{\bf v}\cdot{\bf\nabla\times\bf v}\right> + 2\left<{\bf v}\cdot{\bf \hat b}\right>
\end{equation}
\section{Helicity constraints on turbulence}
From this point onward, our analysis will be purely algebraic and qualitative, as it relies essentially on heuristic considerations. In this paper, we will neglect the electron scale length ($\lambda_e=0$) for the sake of simplicity, although electron inertia can be readily reintroduced; in other words, we investigate the Hall MHD regime \citep{GP04}. In this scenario, the second term on the RHS of (\ref{plus-turb helicity}) becomes vanishingly small, and we end up with ${\bf \hat a}= {\bf a}$ and ${\bf \hat b}= {\bf b}$. We wish to figure out the constraints imposed on Alfv\'enic turbulence by the invariance of $h$ and $h_-$.
\begin{figure}
\includegraphics[width=7.5cm]{EqScale.pdf} \\
\caption{The equipartition length scale ($k_c$) as a function of the helicity ratio $h_-/h$. There are two different solutions for $k_c$, namely, $k_+$ and $k_-$ depending on whether $s_+$ or $s_-$ is adopted. We have plotted $k_+$ and $k_-$ for two choices of $\alpha$, viz. $\alpha = 1$ and $\alpha = 0.1$.}
\label{FigEqScale}
\end{figure}
Now, we introduce a characteristic length scale $L_T$ for the turbulent magnetic field ${\bf b}$; the equivalent wave number is $K_T=1/L_T$. More specifically, because ${\bf b}=\nabla\times{\bf a}$ is valid, we will invoke a phenomenological scaling of the form ${\bf b} \sim K_T {\bf a}$ or ${\bf a} \sim K_T^{-1}{\bf b}$. In other words, one may interpret $K_T$ as the measure of the gradient associated with ${\bf b}$; a similar approach was introduced for the turbulent velocity in \citet[pg. 348]{PFL76}. Note, however, that this mathematical expression is valid \emph{sensu stricto} if ${\bf b}$ is specified to be a Beltrami field, with $K_T$ serving as the corresponding Beltrami parameter; this ansatz is not unreasonable because a number of publications model the turbulent fields as Arn'old-Beltrami-Childress fields \citep{CD95,BS05}. By utilizing the relationship ${\bf a}= K_T^{-1}{\bf b}$ introduced above, (\ref{plus-turb helicity}) reduces to
\begin{equation}\label{H}
h\approx \frac{\langle{\bf b}\cdot{\bf b}\rangle}{k_T}
\end{equation}
where $k_T= \lambda_i K_T$ is the inverse of turbulent scale length measured in units of the ion skin depth. Rewriting (\ref{H})
yields an estimate for the magnetic energy
\begin{equation}\label{Em}
E_m=\frac{\langle{\bf b}\cdot{\bf b}\rangle}{2}\approx \frac{h k_T}{2}
\end{equation}
Following the same procedure we obtain
\begin{equation}\label{Hminus}
h_-\approx 2k_{T} E_\mathrm{kin} +4\alpha \sqrt{E_\mathrm{kin}}\sqrt {E_m}
\end{equation}
where $E_\mathrm{kin}=\left<{\bf v}\cdot{\bf v}\right>/2$. The second term on the RHS represents an alignment condition of sorts, because we suppose that the dimensionless factor $\alpha$ captures the ``projection'' of one turbulent field on the other. This approach is inspired by the fact that, in a special class of exact solutions of nonlinear Alfv\'en waves, ${\bf b}$ is linearly proportional to ${\bf v}$; see, for instance, \citet{Wal44,MK05,MM09,ALM16}. A more general and rigorous strategy for obtaining this term relies upon invoking the well-known Cauchy-Bunyakovsky-Schwarz inequality \citep{JMS}, which yields
\begin{equation}
|\left<{\bf b}\cdot{\bf v}\right>|^2 \leq \left<{\bf b}\cdot{\bf b}\right> \left<{\bf v}\cdot{\bf v}\right>
\end{equation}
and subsequently replacing the inequality in this expression with an equality involving the phenomenological dimensionless factor $\alpha$ that implicitly obeys $0 \leq \alpha \leq 1$ as follows:
\begin{equation}
|\left<{\bf b}\cdot{\bf v}\right>|^2 = \alpha^2 \left<{\bf b}\cdot{\bf b}\right> \left<{\bf v}\cdot{\bf v}\right>
\end{equation}
Lastly, we make use of the definitions of $E_\mathrm{kin}$ and $E_m$ introduced earlier, and take the square root of the above equation to obtain the second term on the RHS of (\ref{Hminus}).
By utilizing (\ref{Em}), (\ref{Hminus}) is readily solved for
\begin{equation}\label{Ekin}
E_\mathrm{kin} \approx \frac{s_\pm^{2}}{k_T}, \quad s_\pm=\frac{-\alpha \sqrt{h} \pm \sqrt{\alpha^2 h + h_-}}{\sqrt{2}}
\end{equation}
where $s$ depends on the constants of motion and the parameter $\alpha$; it will be regulated by the detailed nature of turbulence. The estimates for the global turbulent magnetic and kinetic energies, i.e., (\ref{Em}) and (\ref{Ekin}), are rather robust for all Alfv\'enic turbulence accessible within the two-fluid equations and constitutes one of the salient results in the paper. It can be readily verified that the dominant behavior, contained in the scaling,
\begin{equation}\label{Emag}
E_m \propto {k_T}, \quad\quad\quad E_\mathrm{kin} \propto \frac{1}{k_T}
\end{equation}
holds true (with some corrections on the order of $k_T \lambda_e$) even when the electron dynamics is retained.
Independent of details, Alfv\'enic turbulence is strongly constrained by the ideal invariants of the system. For instance, these systems must obey a definitive, verifiable proportionality emerging from (\ref{Emag}):
\begin{equation}\label{Eratio}
\frac{E_m}{E_\mathrm{kin}} \propto k_T^2
\end{equation}
Therefore, the short-scale turbulence ought to be much richer in magnetic energy while the portion of kinetic energy increases (in relative terms) as one moves toward longer scales (see \citealt[Fig. 3]{SS09}). To put it differently, from (\ref{Eratio}) we see that the two energies may be displaced from equipartition. This lack of equipartition is commonly observed in studies of Hall MHD turbulence, dynamos, and ``reverse'' dynamos \citep{MGM,KM04a,MSMS,MAP07,LM15,LB,LB16}. In particular, the above behavior is consistent with numerical simulations of magnetic and kinetic energy spectra, as seen from \citet[Fig. 2]{MA14} and \citet[Fig. 4]{SP15}; note, however, that the plots in these publications investigate energy spectra and not the global energy budgets. Furthermore, a number of MHD turbulence simulations \citep{WBP11,OMWO} as well as observations of the (turbulent) solar wind \citep{MG82,GVM91,SMBV,CBSM} have revealed an ``excess'' of magnetic energy at small scales as well as differences in the slopes of magnetic and kinetic spectra \citep{BPBP}; the theoretical calculations by \citet{ALM16} suggest that this feature is a generic characteristic of extended MHD. As the invariants are the defining ``labels'' for a given system, once they are specified, we can determine explicit estimates for both ${E_m}$ and $E_\mathrm{kin}$.
\begin{figure}
\includegraphics[width=7.5cm]{TurbScale.pdf} \\
\caption{The turbulent length scale is shown for different solutions and values of $h_-/h$ and $E/h$. The red, black and blue curves correspond to $E/h = 100$, $E/h =1$ and $E/h = 0.01$, respectively. The unbroken and dotted curves correspond to selecting the positive branch of (\ref{ScLeSimp}) with $k_c = k_+$ and $k_c = k_-$ respectively, whereas the dot-dashed and dashed curves constitute the positive branch of (\ref{ScLeSimp}) with $k_c = k_+$ and $k_c = k_-$ respectively. In all cases depicted herein, we have adopted $\alpha = 1$ for simplicity.}
\label{FiTurbScale}
\end{figure}
Notice that, although the estimate for ${E_m}$ is rather simply related to $h$ (which is essentially the magnetic helicity), $E_\mathrm{kin}$ has two solutions $s_\pm$. It is straightforward to verify that it is $s_+$ that must correspond to conventional MHD turbulence - in the limit $h_-\ll h$,
\begin{equation}\label{Splus}
s_+ \simeq \frac{h_-}{2\sqrt{2}\alpha \sqrt{h}}
\end{equation}
The complete expression for $s_+$ is the relevant expression for conventional Hall MHD. The larger root $s_-$ in terms of magnitude will consequently yield a higher kinetic energy. Thus, it must be emphasized that, for a given set of helicities, there are two distinct turbulent energy states: $[E_m, \,E_\mathrm{kin}(s_+)]$ and $[E_m, \,E_\mathrm{kin}(s_-)]$. The ratio $E_m/E_\mathrm{kin}$ is physically relevant since it represents the ratio of the magnetic and kinetic energies. It is possible for this ratio to attain values both greater and smaller than unity. The critical turbulent length scale ($k_c$) at which equipartition is obtained is found by solving for $E_m/E_\mathrm{kin} = 1$, thus leading us to
\begin{equation}\label{kcfin}
k_c = \frac{\sqrt{2} |s_\pm|}{|\sqrt{h}|} = |\alpha| \left(\sqrt{1 + \frac{h_-}{\alpha^2 h}} \pm 1\right).
\end{equation}
As expected, there are two different critical length scales at which equipartition of kinetic and magnetic energies is achieved. An interesting point that emerges from the above formula is that $k_c$ depends only on the ratio $\Gamma = h_-/h$ and not the individual helicities; aside from this ratio, it also depends on $\alpha$. For the case with $\Gamma \gg 1$, we determine that both roots converge to $k_c \approx \sqrt{h_-/h}$. Note, however, that this solution is physically problematic because it corresponds to $L_T \ll \lambda_i$ - in this regime, Hall MHD is not accurate because electron inertia effects (neglected herein) come into play.
On the other hand, when we consider $\Gamma \ll 1$, we find that two divergent values for $k_c$ follow - we obtain $k_{c1} \approx 2$ for one branch and $k_{c2} \approx h_-/(2|\alpha| h)$ in the other. For the first branch, we arrive at $L_{c1} \approx \lambda_i/2$, whereas the second branch yields $L_{c2} \gg \lambda_i$. Thus, for $\Gamma \ll 1$, there is a manifest bifurcation of the equipartition length scales: one of them is comparable to the ion skin depth, while the other is much larger than $\lambda_i$, and presumably comparable to the characteristic system length scale \citep{YMO}. The combination of $\Gamma \ll 1$ and $L_{c2} \gg \lambda_i$ essentially means that the system is dominated by the magnetic helicity (as opposed to the cross helicity and fluid helicity) and that equipartition is being achieved at macroscopic scales \emph{sensu lato}. Hence, this regime is consistent with an ideal MHD-like picture, wherein large-scale behavior and magnetic helicity are dominant. The different values of $k_c$ as a function of $h_-/h$ and $\alpha$ are depicted in Fig. \ref{FigEqScale}.
\begin{figure}
\includegraphics[width=7.5cm]{MinEn.pdf} \\
\caption{The minimum value of $E/h$ that is sufficient to yield real values of $k_T$ is plotted as a function of the helicity ratio $h_-/h$ for different choices of $\alpha$. }
\label{FigMinEn}
\end{figure}
Until now, our analysis has concentrated only on the constraints on the magnetic and kinetic energy imposed by the helicity invariants. Let us now examine these results in conjunction with the conservation of energy, which in the language of preceding considerations, becomes (after having subtracted ambient field energy)
\begin{equation}\label{Energy}
E = \left<\frac{ {\bf b}\cdot{\bf b} +{\bf v}\cdot{\bf v}}{2}+\frac{p}{\gamma-1}\right> = E_m+E_\mathrm{kin}+E_\mathrm{th},
\end{equation}
where $E$ is a constant that denotes the difference between the total energy and ambient magnetic energy. The obvious inference is that having already estimated ${E_m}$ and $E_\mathrm{kin}$ in (\ref{Em}) and (\ref{Ekin}), we find that (\ref{Energy}) allows us to calculate the turbulent thermal energy in terms of the three invariants ($h$, $h_-$ and $E$) of the system.
If we specialize to the special case where the turbulent kinetic energy is negligible for an incompressible plasma, we see that (\ref{Energy}) reduces to
\begin{equation}\label{Energydef}
E= E_m+E_\mathrm{kin},
\end{equation}
and thereby imposes an additional constraint on the system. In fact, we end up constraining the characteristic scale length of turbulence as follows:
\begin{equation}\label{Scalelength}
k_T = \lambda_i K_T=\frac{\lambda_i}{L_T}= \frac{E}{h}\pm\sqrt{ \frac{E^2}{h^2}-\frac{s_\pm^2}{2h}}.
\end{equation}
It is far more transparent to rewrite (\ref{Scalelength}) in terms of $k_c$ because we end up with
\begin{equation}\label{ScLeSimp}
k_T = \frac{E}{h} \left[1 \pm \sqrt{1 - \left(\frac{k_c h}{2 E}\right)^2}\right]
\end{equation}
Although this expression looks deceptively simple, it is quite complex. It has a dependence on $\alpha$, $E/h$ and $h_-/h$ via $k_c$. Moreover, there are four solutions in total: $2$ arising from the $\pm$ in the right-hand-side of (\ref{ScLeSimp}) and $2$ more from the fact that $k_c$ has two different branches as seen from (\ref{kcfin}). After fixing $\alpha$, we have plotted $k_T$ in Fig. \ref{FiTurbScale}. Note that not all of the $4$ solutions are guaranteed to be real, as seen from inspecting this figure.
In order for $k_T$ to be real-valued, the following inequality must hold true:
\begin{equation}\label{InEq}
\Big|\frac{k_c h}{2 E}\Big| \leq 1,
\end{equation}
which imposes a constraint on the parameter space of $\{E,\,h,\,h_-\}$. Hence, depending on the parameters adopted, it is possible for $k_T$ to have either $4$ real roots, $2$ real roots and $1$ complex-conjugate pair, or $0$ real roots and $2$ complex-conjugate pairs (see Fig. \ref{FiTurbScale} for an example). These results, as embodied by the equations (\ref{Scalelength}) and (\ref{ScLeSimp}), are qualitatively similar, albeit derived from a more generic standpoint, to the generalized magneto-Bernoulli mechanism elucidated in \citet{OSYM,MNSY,SMB19}; it has been proposed that this mechanism may constitute a viable explanation for solar flares \citep{KM10}, as opposed to classic paradigms such as fast magnetic reconnection \citep{Bis00,SM11,CLH16}.
Note that (\ref{InEq}) provides us with another means of envisioning $k_c$. It is not only the length scale at which the ratio of magnetic and kinetic energies equals unity, but also the length scale that fulfills the criterion $L_{c1} \geq \lambda_i |h/E|/2$; here, recall that $L_{c1}$ is constructed from the larger root of $k_c$, which we had dubbed $k_{c1}$. By utilizing (\ref{InEq}), we have plotted the minimum value of $E/h$ that suffices to ensure that $k_T$ is real-valued; this lower bound depends on both the helicity ratio $h_-/h$ and $\alpha$.
Upon inspecting (\ref{Scalelength}), we find that $k_T$ is determined almost wholly in terms of the three constants of motion. We notice that there are two different solutions for $k_T$ for a given choice of $s_\pm$, thereby giving rise to multiple length scales that can differ considerably in magnitude. In the above setting, it would seem then that the turbulence in each incompressible Alfv\'enic system, which is defined by its three invariants, should give rise to characteristic length scales that are fully determined or severely constrained by the invariants.
\section{Discussion}
It makes intuitive sense that integral invariants (helicities and energy) would consequently set constraints on the global (i.e., integral) magnetic, kinetic and thermal energies; in fact, they may even formally determine them. Thus, we have obtained explicit relationships between $E_m, E_\mathrm{kin}$ and $E_\mathrm{th}$ on the one hand and $L_T$ on the other, but the essentially heuristic arguments developed in this paper cannot give any specifications for the parameter $\alpha$, which measures the degree of alignment of the two turbulent fields.
Likewise, it is natural to contend that our analysis would not directly yield the $k$-spectrum of $E_m, E_\mathrm{kin}$ and $E_\mathrm{th}$. This information can seemingly emerge only via detailed studies of the Alfv\'enic dynamics, which has been a most active field of investigation in the physics of turbulent plasmas. Thus, to reiterate, our work does not examine the consequences for energy spectra, as it focuses on the \emph{global} energy budgets. The details of the energy spectrum for the various regimes of extended MHD have been explored by \citet{ALM16} in the context of the solar wind. It was shown therein that the spectrum in the MHD regime obeys a Kolmogorov scaling as opposed to the Iroshnikov-Kraichnan scaling, in agreement with prior theoretical and empirical results \citep{GRM,BC16,SHH20}.
By simply harnessing the fundamental plasma invariants (helicities and energy), we were able to formulate certain interesting and possibly generic results for Alfv\'enic turbulence. For a specific set of invariants that remain invariant during whatever dynamics the system undergoes (thus serving as a ``label''), all three components of the total turbulent energy are potentially dictated by a single feature of turbulence that embodies the length scale $L_T$ associated with the small-scale magnetic field; conversely, one may interpret this length scale as being fully determined if the trio of invariants are specified, as seen from (\ref{ScLeSimp}).
Expressing the turbulent energies in terms of $L_T$ also enabled us to deduce some basic constraints on their magnitudes. Although our analysis was expressly concerned with global quantities, we found that our results are compatible with spectral relationships that have been identified in the Hall regime via numerical simulations; while this fact does not validate our predictions, it bolsters their credibility. Apart from establishing some fundamental intrinsic features of Alfv\'enic turbulence as described hitherto, our results can motivate as well as provide a check on detailed simulations.
Lastly, our analysis is valid over a broad range of turbulent scale lengths, namely, ${L_{eq}}^{-1}\ll k_T= \lambda_i K_T \ll M/m$, where $L_{eq}$ is some equilibrium scale length that typically encapsulates the system size. It can be readily extended to and beyond the electron skin depth ($\lambda_e$), but it seems relatively unlikely that the characteristic scale for Alfv\'enic turbulence would enter this regime.
\section*{Acknowledgements}
We thank our reviewer, Mitchell Berger, for the positive and insightful report. This work was partially supported by the US-DOE grant DE-FG02-04ER54742.
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\section{Introduction}
By the classical result of Mazur and Ulam \cite{MU}, every bijective isometry between Banach spaces is affine. This result essentially asserts that the metric structure of a Banach space determines its linear structure. In \cite{Man} Mankiewicz proved that every bijective isometry $f:B_X\to B_Y$ between the unit balls of two Banach spaces $X,Y$ extends to a linear isometry of the Banach spaces. In \cite{Tingley} Tingley asked if the unit balls in this result of Mankiewicz can be replaced by the unit spheres. More precisely, he posed the following problem (that remains unsolved more than thirty years).
\begin{problem}[Tingley, 1987]\label{prob:Tingley} Let $f:S_X\to S_Y$ be a bijective isometry of the unit spheres of two Banach spaces $X,Y$. Can $f$ be extended to a linear isometry between the Banach spaces $X,Y$?
\end{problem}
Here for a Banach space $(X,\|\cdot\|)$ by
$$B_X=\{x\in X:\|x\|\le 1\}\quad\mbox{and}\quad S_X=\{x\in X:\|x\|=1\}$$we denote the unit ball and unit sphere of $X$, respectively.
Tingley's Problem~\ref{prob:Tingley} can be equivalently reformulated in terms of the Mazur--Ulam property, introduced by Cheng and Dong \cite{CD} and widely used in the the literature, see e.g. \cite{BG}, \cite{CAP}, \cite{CAP2}, \cite{JVMCPR}, \cite{Li}, \cite{MO}, \cite{WX}.
\begin{definition} A Banach space $X$ is defined to have the {\em Mazur--Ulam property} if every isometry $f:S_X\to S_Y$ of $S_X$ onto the unit sphere $S_Y$ of an arbitrary Banach space $Y$ extends to a linear isometry of the Banach spaces $X,Y$.
\end{definition}
In fact, Tingley's Problem~\ref{prob:Tingley} asks whether every Banach space has the Mazur--Ulam property. Many classical Banach spaces (including $C(K)$, $c_0(\Gamma)$, $\ell_p(\Gamma)$, $L_p(\mu)$) have the Mazur--Ulam property, see \cite{Ding}, \cite{DL}. By the result of Kadets and Mart\'\i n \cite{KM}, every polyhedral finite-dimensional Banach space has the Mazur--Ulam property.
The main result of this paper is the following theorem that answers Tingley's problem in the class of $2$-dimensional Banach spaces.
\begin{theorem}\label{t:main} Every $2$-dimensional Banach space has the Mazur--Ulam property.
\end{theorem}
Theorem~\ref{t:main} is a corollary of four partial answers to Tingley's problem. The first of them was proved by Cabello S\'anchez in \cite{San}.
\begin{theorem}[Cabello S\'anchez]\label{t:CS} A $2$-dimensional Banach space has the Mazur--Ulam property if it is not strictly convex.
\end{theorem}
A Banach space is called
\begin{itemize}
\item {\em strictly convex} if any convex subset of its unit sphere contains at most one point;
\item {\em smooth} if its unit ball has a unique supporting hyperplane at each point of the unit sphere.
\end{itemize}
The second ingredient of the proof of Theorem~\ref{t:main} was proved by Banakh and Cabello Sanchez in \cite{BCS}.
\begin{theorem}[Banakh, Cabello S\'anchez]\label{t:BCS} A $2$-dimensional Banach space has the Mazur--Ulam property if it is not smooth.
\end{theorem}
The third crucial ingredient of the proof of Theorem~\ref{t:main} concerns absolutely smooth Banach spaces. A $2$-dimensional Banach space $X$ is called {\em absolutely smooth} if there exists a differentiable map $\br:\IR\to S_X$ such that $\|\br'(s)\|=1$ for all $s\in\IR$ and the derivative $\br':\IR\to S_X$ is locally absolutely continuous. For absolutely smooth $2$-dimensional Banach spaces, Tingley's problem was answered in \cite{Ban} as follows.
\begin{theorem}[Banakh] Any isometry between the unit spheres of two absolutely smooth $2$-dimensional Banach spaces extends to a linear isometry of the Banach spaces.
\end{theorem}
Therefore, to derive Theorem~\ref{t:main} it remains to prove
\begin{theorem}\label{t:main2} A $2$-dimensional Banach space has the Mazur--Ulam property if it is strictly convex and smooth but not absolutely smooth.
\end{theorem}
This theorem will be proved in Section~\ref{s:main} after some preparatory work, made in Sections~\ref{s:prep} and \ref{s:dif}.
\section{Preliminaries}\label{s:prep}
In this section we collect some definitions and known results that will be used in the proof our main result.
\subsection{Smoothness properties of real functions}
Let $U$ be an open subset of the real line and $s\in U$. A function $f:U\to X$ to a Banach space $X$ is defined to be
\begin{itemize}
\item {\em Lipschitz} at $s$ if there exists a constant $C$ such that\newline $\|f(s+\e)-f(s)\|\le C\cdot|\e|+o(\e)$ for a small $\e$;
\item {\em differentiable} at $s$ if there is a vector $f'(s)\in X$ such that\newline $f(s+\e)=f(s)+f'(s)\cdot\e+o(\e)$ for a small $\e$;
\item {\em twice differentiable} at $s$ if there are vectors $f'(s),f''(s)\in X$ such that\newline $f(s+\e)=f(s)+f'(s)\cdot\e+\frac12f''(s)\cdot\e^2+o(\e^2)$ for a small $\e$;
\item {\em $C^1$-smooth} if $f$ is differentiable at each point of $U$ and the function $f':U\to X$, $f':u\mapsto f'(u)$, is continuous;
\item {\em absolutely continuous} if for any $\e>0$ there exists $\delta>0$ such that for any points $x_1<y_1<x_2<y_2<\dots<x_n<y_n$ in $U$ with $\sum_{i=1}^n(y_i-x_i)<\delta$ we have $\sum_{i=1}^n\|f(y_n)-f(x_n)\|<\e$;
\item {\em locally absolutely continuous} if for any $s\in U$ there exists a neighborhood $O_s\subseteq U$ of $s$ such that the restriction $f{\restriction}_{O_x}$ is absolutely continuous.
\end{itemize}
For a function $f:U\to X$ we denote by $\dot \Omega_f$ and $\ddot\Omega_f$ the set of points $s\in U$ at which $f$ is differentiable and twice differentiable, respectively.
A subset $A\subseteq \IR$ is called
\begin{itemize}
\item {\em Lebesgue null} if its Lebesgue measure is zero;
\item {\em Lebesgue co-null} if the complement $\IR\setminus A$ is Lebesgue null.
\end{itemize}
By a classical result of Lebesgue \cite[1.2.8]{KK}, the set $\dot\Omega_f$ of differentiability points of any monotone function $f:\IR\to\IR$ is Lebesgue co-null in the real line.
\begin{lemma}\label{l:mac} If a monotone continuous function $f:\IR\to\IR$ is Lipschitz at all but countably many points, then $f$ is locally absolutely continuous.
\end{lemma}
\begin{proof} Let $C$ be the set of points $s\in\IR$ at which $f$ is not Lipschitz. By Theorem 7.1.38 of \cite{KK}, the local absolute continuity of $f$ will follow as soon as we show that for every bounded Lebesgue null set $E\subseteq \IR$ the image $f(E)$ is Lebesgue null. Let $a=\inf E$ and $b=\sup E$. For every $n\in\IN$ consider the closed subset $$X_n=\bigcap_{y\in[a,b]}\{x\in[a,b]:|f(x)-f(y)|\le n\cdot |x-y|\}$$ of $[a,b]$.
It follows that $[a,b]\setminus C=\bigcup_{n\in\IN}X_n$. For every $n\in\IN$ the restriction $f{\restriction}_{X_n}$ is a Lipschitz function with Lipschitz constant $n$. Consequently the set $f(X_n\cap E)$ is Lebesgue null. Then the image
$$f(E)=\bigcup_{x\in E\cap C}\{x\}\cup\bigcup_{n\in\IN}f(E\cap X_n)$$is Lebesgue null being the union of countably many Lebesgue null sets.
\end{proof}
\subsection{Special directions}
\begin{definition} A point $s\in S_X$ on the unit sphere of a Banach space $X$ is called a {\em special direction} if for any bijective isometry $f:S_X\to S_Y$ onto the unit sphere of an arbitrary Banach space $Y$ and any points $a,b\in S_X$ with $b-a=\|b-a\|\cdot s$ we have $f(b)-f(a)=\|b-a\|\cdot f(s)$.
\end{definition}
In the proof of Theorem~\ref{t:main2} we shall use the following theorem, proved in \cite{BCS}.
\begin{theorem}\label{t:key} A $2$-dimensional Banach space $X$ has the Mazur--Ulam property if its sphere contains two linearly independent special directions.
\end{theorem}
\subsection{Natural parameterizations of spheres}
\begin{definition} Let $X$ be a $2$-dimensional Banach space. A map $\br:\IR\to S_X$ is called a {\em natural parameterization} of the sphere $S_X$ is $\br$ is $C^1$-smooth and $\|\br'(s)\|=1$ for every $s\in\IR$.
\end{definition}
The following existence and uniqueness theorems for natural parametrizations were proved in \cite{Ban}.
\begin{theorem}\label{t:e} Every smooth $2$-dimensional Banach space $X$ has a natural parameterization $\br:\IR\to S_X$.
\end{theorem}
\begin{theorem}\label{t:u} Let $X,Y$ be two smooth $2$-dimensional Banach spaces and $\br_X:\IR\to S_X$ and $\br_Y:\IR\to S_Y$ be natural parameterizations of their unit spheres. For any isometry $f:S_X\to S_Y$ there exists an isometry $\Phi:\IR\to\IR$ such that $f\circ\br_X=\br_Y\circ\Phi$.
\end{theorem}
Since each isometry $\Phi:\IR\to\IR$ is of the form $\Phi(x)=ax+b$ for some $a,b\in\IR$ with $|a|=1$, Theorems~\ref{t:e}, \ref{t:u} and \ref{t:BCS} imply the following corollary that will be used in the proof of Theorem~\ref{t:main2}.
\begin{corollary}\label{c:nat} If $\br:\IR\to S_X$ is a natural parameterization of the unit sphere of some $2$-dimensional Banach space, then for any bijective isometry $f:S_X\to S_Y$ between $S_X$ and the unit sphere of an arbitrary Banach space $Y$, the map $f\circ\br_X$ is a natural parameterization of $S_Y$.
\end{corollary}
The following lemma proved in \cite[5.4]{Ban} describes a periodicity property of natural parameterizations.
\begin{lemma}\label{l:r} Let $X$ be a smooth $2$-dimensional Banach space, $\br:\IR\to S_X$ be a natural parameterization of its sphere, and $L=\min\{s\in[0,\infty):\br(L)=-\br(0)\}$. Then $\br(s)=-\br(s+L)=\br(s+2L)$ for every $s\in\IR$.
\end{lemma}
\subsection{Phase shift}
Let $X$ be a smooth $2$-dimensional Banach space and $\br:\IR\to S_X$ be a natural parameterization of its unit sphere. Let $\varphi:\IR\to\IR$ be the function assigning to every $s\in \IR$ the smallest real number such that $\varphi(s)>s$ and $\br'(s)=\br(\varphi(s))$. The function $\varphi$ is called the {\em phase shift} for the parameterization $\br$. Its properties are described in the following lemma taken from \cite[7.1]{Ban}.
\begin{lemma}\label{l:phi} The phase shift $\varphi$ is a continuous non-decreasing function. The Banach space $X$ is strictly convex if and only if the phase shift $\varphi$ is strictly increasing.
\end{lemma}
\begin{lemma}\label{l:difff} If the phase shift $\varphi$ is differentiable at $s\in\IR$, then $\br'$ is differentiable at $s$ and $\br$ is twice differentiable at $s$.
\end{lemma}
\begin{proof} If $\varphi$ is differentiable at $s\in\IR$, then for a small real number $\e$ we have
$$
\begin{aligned}
\br'(s+\e)&=\br(\varphi(s+\e))=\br(\varphi(s)+\varphi'(s)\cdot\e+o(\e))=\\
&=\br(\varphi(s))+\br'(\varphi(s))\cdot(\varphi'(s)\cdot\e+o(\e))+o(\varphi'(s)\cdot\e+o(\e))=\\
&=\br'(s)+\br'(\varphi(s))\cdot\varphi'(s)\cdot\e+o(\e),
\end{aligned}
$$
which means that $\br'$ is differentiable at $s$.
To see that $\br$ is twice differentiable at $s$, observe that for a small real number $\e$ we have
$$
\begin{aligned}
\br(s+\e)-\br(s)&=\int_0^\e\br'(s+t)\,dt=\int_0^\e\br(\varphi(s+t))\,dt=\\
&=\int_0^\e\big(\br(\varphi(s))+\br'(\varphi(s))(\varphi(s+t)-\varphi(s))+o(\varphi(s+t)-\varphi(s))\big)\,dt=\\
&=\br(\varphi(s))\cdot \e+\br'(\varphi(s))\int_0^\e(\varphi'(s)t+o(t))dt+\int_0^\e o(\varphi'(s)t+o(t))dt=\\
&=\br'(s)\cdot \e+\tfrac12\br'(\varphi(s))\cdot\varphi'(s)\cdot\e^2+o(\e^2)
\end{aligned}
$$which means that $\br$ is twice differentiable at $s$.
\end{proof}
By a classical result of Lebesgue \cite[1.2.8]{KK}, the set $\dot\Omega_f$ of differentiability points of any monotone function $f:\IR\to\IR$ is Lebesgue co-null in the real line. This fact and Lemmas~\ref{l:phi}, \ref{l:difff} imply the following lemma.
\begin{lemma}\label{l:full} The sets $\dot\Omega_\varphi\subseteq \dot\Omega_{\br'}\cap\ddot\Omega_\br$ are Lebesgue co-null in the real line.
\end{lemma}
\begin{lemma}\label{l:Las} If $\varphi$ is Lipschitz at all but countably many points, then the Banach space $X$ is absolutely smooth.
\end{lemma}
\begin{proof} By Lemma~\ref{l:mac}, $\varphi$ is locally absolutely continuous and then the function $\br'=\br\circ \varphi$ is locally absolutely continuous being the composition of a $C^1$-smooth function $\br$ and locally absolutely continuous non-decreasing function $\varphi$.
\end{proof}
\subsection{Tingley's Lemma}
We shall need the following lemma proved by Tingley in \cite{Tingley}.
\begin{lemma}[Tingley]\label{l:Tingley} Let $f:S_X\to S_Y$ be an isometry of the unit spheres of finite-dimensional Banach spaces $X,Y$. Then $f(-x)=-f(x)$ for every $x\in S_X$.
\end{lemma}
\section{Some smoothness properties of distances on the sphere}\label{s:dif}
In this section we assume that $X$ is a strictly convex smooth $2$-dimensional Banach space and $\br:\IR\to S_X$ is a natural parameterization of its sphere.
Let $$L=\min\{s\in[0,\infty):\br(s)=-\br(0)\}$$be the half-length of the sphere $S_X$. By Lemma~\ref{l:r}, $\br(s+L)=-\br(s)$ for all $x\in\IR$.
\begin{lemma}\label{l:first} Let $a,b,s\in\IR$ be numbers such that $0\ne\br(b)-\br(a)=\|\br(b)-\br(a)\|\cdot \br(s)$. If the function $\br$ is twice differentiable at $b$ and $s$, then the function $$\nu:\IR\to\IR,\quad\nu:\e\mapsto\|\br(b+\e)-\br(a)\|,$$is twice differentiable at zero.
\end{lemma}
\begin{proof} Let $\br'(b)=x\cdot\br(s)+y\cdot\br'(s)$, $\br''(b)=u\cdot\br(s)+v\cdot\br'(s)$ and $\br''(s)=-\rho\cdot\br(s)+\tau\cdot\br'(s)$ for some real numbers $x,y,u,v,\rho,\tau$.
Given a small $\e$, find a small $\delta$ such that
\begin{equation}\label{eq2}
\br(b+\e)-\br(a)=\|\br(b+\e)-\br(a)\|\cdot\br(s+\delta).
\end{equation}
Taking into account that $\br$ is twice differentiable at $s$ and $b$, we can write
$$\br(s+\delta)=\br(s)+\br'(s)\delta+\tfrac12\br''(s)\delta^2+o(\delta^2)=
(1-\tfrac12\rho\delta^2+o(\delta^2))\br(s)+(1+\tfrac12\tau\delta+o(\delta))\delta\br'(s)$$
and
$$\begin{aligned}
&\br(b+\e)-\br(a)=\br(b)-\br(a)+\br'(b)\e+\tfrac12\br''(b)\e^2+o(\e^2)=\\
&=(\|\br(b)-\br(a)\|+x\e+\tfrac12u\e^2+o(\e^2))\br(s)+(y\e+\tfrac12u\e^2+o(\e^2))\br'(s).
\end{aligned}
$$
Writing the equation (\ref{eq2}) in coordinates, we obtain two equations:
\begin{equation}\label{first}
\|\br(b)-\br(a)\|+x\e+\tfrac12u\e^2+o(\e^2)=\|\br(b+\e)-\br(a)\|\cdot \big(1-\tfrac12\rho\delta^2+o(\delta^2)\big)
\end{equation}
and
\begin{equation}\label{second}
(y+\tfrac12u\e+o(\e))\e=\|\br(b+\e)-\br(a)\|\cdot (1+\tfrac12\tau\delta+o(\delta))\delta.
\end{equation}
The equation (\ref{second}) implies
$$\delta=\frac{y+o(1)}{\|\br(b)-\br(a)\|}\e.$$After substitution of this $\delta$ into the equation (\ref{first}), we obtain
$$
\begin{aligned}
\nu(\e)&=\|\br(b+\e)-\br(a)\|=\frac{\|\br(b)-\br(a)\|+x\e+\tfrac12u\e^2+o(\e)^2}{1-\frac{\rho\cdot(y\e)^2}{2\|\br(b)-\br(a)\|^2}+o(\e^2)}=\\
&=\big(\|\br(b)-\br(a)\|+x\e+\tfrac12u\e^2+o(\e)^2\big)\cdot\big(1+\tfrac{\rho\cdot(y\e)^2}{2\|\br(b)-\br(a)\|^2}+o(\e^2)\big)=\\
&=\|\br(b)-\br(a)\|+x\e+\tfrac12(u+\tfrac{\rho\cdot y^2}{\|\br(b)-\br(a)\|})\e^2+o(\e^2),
\end{aligned}
$$
which means that $\nu$ is twice differentiable at zero.
\end{proof}
\begin{lemma}\label{l:noLip} Let $a,b,s\in\IR$ be numbers such that $0\ne\br(b)-\br(a)=\|\br(b)-\br(a)\|\cdot \br(s)$. If the function $\br$ is twice differentiable at $b$ and the function $$\nu:\IR\to\IR,\quad\nu:\e\mapsto\|\br(b+\e)-\br(a)\|,$$is twice differentiable at zero, then the phase shift $\varphi$ is Lipschitz at $s$.
\end{lemma}
\begin{proof} Assume that the function $\br$ is twice differentiable at $b$ and the function $\nu$ is twice differentiable at zero. Let $\br'(b)=x\br(s)+y\br'(s)$ and $\br''(b)=u\br(s)+v\br'(s)$ for some real numbers $x,y,u,v$.
Since $0\ne\br(b)-\br(a)=\|\br(b)-\br(a)\|\cdot \br(s)$, the strict convexity of $X$ implies that $y\ne0$.
For every $\e\in\IR$ find unique real numbers $\mu(\e),\eta(\e)$ such that $$\br(s+\e)=(1+\mu(\e))\br(s)+(\e+\eta(\e))\br'(s).$$ The differentiability of the function $\br$ at $s$ implies that $\mu(\e)$ and $\eta(\e)$ are of order $o(\e)$ for small $\e$.
Given a small $\e$, find a small $\delta$ such that
\begin{equation}\label{eq:nu}
\br(b+\delta)-\br(a)=\|\br(b+\delta)-\br(a)\|\cdot\br(s+\e)=\nu(\delta)\cdot\br(s+\e).
\end{equation}
Since the function $\br$ is twice differentiable at $b$, we can write
$$
\begin{aligned}
&\br(b+\delta)-\br(a)=\br(b)-\br(a)+\br'(b)\delta+\tfrac12\br''(b)\delta^2+o(\delta^2)=\\
&=(\|\br(b)-\br(a)\|+x\delta+\tfrac12u\delta^2+o(\delta^2))\br(s)+(y\delta+\tfrac12v\delta^2+o(\delta^2))\br'(s)
\end{aligned}
$$
Since the function $\nu$ is twice differentiable at zero, we obtain
$$\nu(\delta)=\|\br(b)-\br(a)\|+\nu'(0)\delta+\tfrac12\nu''(0)\delta^2+o(\delta^2).$$
Writing the equation (\ref{eq:nu}) in coordinates, we obtain the equations
\begin{equation}\label{eq:nu-one}
\|\br(b)-\br(a)\|+x\delta+\tfrac12u\delta^2+o(\delta^2)=(\|\br(b)-\br(a)\|+\nu'(0)\delta+\tfrac12\nu''(0)\delta^2+o(\delta^2))(1+\mu(\e))
\end{equation}
and
\begin{equation}\label{eq:nu-two}
(y\delta+\tfrac12v\delta^2+o(\delta^2))=(\|\br(b)-\br(a)\|+\nu'(0)\delta+\tfrac12\nu''(0)\delta^2+o(\delta^2))(\e+\eta(\e))
\end{equation}
The equation (\ref{eq:nu-two}) implies
$$\delta=\frac{\|\br(b)-\br(a)\|}{y}\e+o(\e).$$
After substitution of $\delta$ into the equation (\ref{eq:nu-one}), we obtain
$$
(x-\nu'(0))\delta+\tfrac12(u-\nu''(0))\delta^2+o(\delta^2)=\mu(\e)(\|\br(b)-\br(a)\|+o(1))
$$Taking into account that $\mu(\e)=o(\e)$, we conclude that $x-\nu'(0)=0$ and hence
$$
\begin{aligned}
\mu(\e)&=\frac{(u-\nu''(0))\delta^2+o(\delta^2)}{2\|\br(b)-\br(a)\|+o(1)}=
\frac{(u-\nu''(0))\delta^2}{2\|\br(b)-\br(a)\|}+o(\delta^2)=\\
&=\frac{(u-\nu''(0))\cdot\|\br(b)-\br(a)\|}{2y^2}\e^2+o(\e^2),\end{aligned}
$$
which means that $\mu$ is twice differentiable at zero and $$\mu''(0)=\frac{(u-\nu''(0))\cdot\|\br(b)-\br(a)\|}{y^2}.$$
Write the vector $\br'(\varphi(s))$ in the basis $\br(s),\br'(s)$ as
$$\br'(\varphi(s))=-P(s)\cdot\br(s)+T(s)\cdot\br'(s)$$for some real numbers $P(s),T(s)$ (called the {\em radial} and {\em tangential supercuravtures} at $s$, see \cite[\S7]{Ban}). Since the vector $\br'(\varphi(s))$ is not collinear to $\br(\varphi(s))=\br'(s)$, the radial supercurvature $P(s)$ is not equal to zero.
Since the function $\br$ is $C^1$-smooth, for small $\e$ we have the equality
$$
\begin{aligned}
&\mu(\e)\br(s)+(\e+\eta(\e)\br'(s)=\br(s+\e)-\br(s)=\int_0^\e\br'(s+u)\,du=\int_0^\e\br(\varphi(s+u))\,du=\\
&=\br(\varphi(s))\cdot\e+\int_0^\e(\br(\varphi(s+u))-\br(\varphi(s))\,du=\br'(s)\cdot\e+\int_0^\e\int_{\varphi(s)}^{\varphi(s+u)}\br'(t)\,dt\,du=\\
&=\br'(s)\cdot\e+\int_0^\e\int_{\varphi(s)}^{\varphi(s+u)}\big(\br'(\varphi(s))+o(1)\big)\,dt\,du=\\
&=\br'(s)\cdot\e+\big(\br'(\varphi(s))+o(1)\big)\int_0^\e(\varphi(s+u)-\varphi(s))\,du=\\
&=\br'(s)\cdot\e+(-P(s)\br(s)+T(s)\br'(s)+o(1))\int_0^\e(\varphi(s+u)-\varphi(s))du
\end{aligned}
$$
and hence
$$\tfrac12\mu''(0)\e^2+o(\e^2)=\mu(\e)=(-P(s)+o(1))\int_0^\e(\varphi(s+u)-\varphi(s))du.$$
Thaking into account that the function $\varphi$ is monotone, we obtain
$$\big|\varphi(s+\e)-\varphi(s)\big|\cdot|\e|\le\Big|\int_\e^{2\e}(\varphi(s+u)-\varphi(s))du\Big|\le\Big|\int_0^{2\e}(\varphi(s+u)-\varphi(s))du\Big|=\Big|\frac{\mu''(0)+o(1)}{2\cdot P(s)}\Big|(2\e)^2,$$
which means that $\varphi$ is Lipschitz at $s$.
\end{proof}
\begin{lemma}\label{l:spec2} If the function $\varphi$ is not Lipschitz at a point $s\in\IR$, then the direction $\br(s)\in S_X$ is special.
\end{lemma}
\begin{proof} Let $f:S_X\to S_Y$ be a bijective isometry of $S_X$ onto the unit sphere $S_Y$ of an arbitrary Banach space $Y$. It is clear that the Banach space $Y$ is $2$-dimensional. Theorems~\ref{t:CS} and \ref{t:BCS} imply that the Banach space $Y$ is strictly convex and smooth. By Corollary~\ref{c:nat}, the composition $\br_Y=f\circ\br:\IR\to S_Y$ is a natural parameterization of the sphere $S_Y$.
Let $\theta:S_X\to S_X$ be the map assigning to each $x\in S_X$ the unique point $\theta(x)\in S_X$ such that $\{x,\theta(x)\}=S_X\cap(x+\IR\cdot\br(s))$. The strict convexity of $X$ implies that the map $\theta$ is well-defined and continuous. Since $\theta\circ\theta$ is the identity map of $S_X$, the map $\theta$ is a homeomorphism of the sphere $S_X$. The definition of $\theta$ ensures that $x-\theta(x)\in\{\|x-\theta(x)\|\cdot \br(s),-\|x-\theta(x)\|\cdot\br(s)\}$ for every $x\in S_X$.
Let $$P=\{x\in S_X:0\ne x-\theta(x)=\|x-\theta(x)\|\cdot \br(s)\}.$$ Then $S_X=P\cup\{x\in S_X:\theta(x)=x\}\cup(-P)$. Observe that $\theta(\br(s))=-\br(s)$ and hence $\br(s)\in P$.
Find a real number $a\in \IR$ such that $\{\br(a),\br(a+L)\}=\{x\in S_X:\theta(x)=x\}$, $\br\big((a,a+L)\big)=P$, and $a<s<a+L$. Then $\br{\restriction}_{(a-L,a)}$ is a homeomorphism of the open interval $(a-L,a)$ onto the open half-sphere $-P$.
The $C^1$-smoothness of the function $\br$ implies the $C^1$-smoothness of the function
$$\Theta=(\br{\restriction}_{(a-L,a)})^{-1}\circ \theta\circ \br{\restriction}_{(a,a+L)}:(a,a+L)\to(a,a-L).$$
Since $\theta(\br(s))=-\br(s)=\br(s-L)$, we have $\Theta(s)=s-L$.
Now consider the continuous map
$$\phi:(a,a+L)\to Y,\;\phi:t\mapsto {f\circ \br(t)-f\circ \theta\circ \br(t)}.$$
\begin{claim} The map $\phi$ is $C^1$-smooth.
\end{claim}
\begin{proof} The function $\br_Y=f\circ \br$ is $C^1$-smooth, being a natural parameterization of the sphere $S_Y$ of the smooth Banach space $Y$.
Now take any $t\in(a,a+L)$ and observe that $\br(t)\in P$, $\theta(\br(t))\in -P$ and hence $\Theta(t)=(\br{\restriction}_{(a-L,a)})^{-1}\circ \theta\circ\br(t)$ is well-defined.
Since
$$f\circ \theta\circ \br(t)=f\circ \br\circ(\br{\restriction}_{(a-L,a)})^{-1}\circ \theta\circ\br(t)=\br_Y\circ\Theta(t),$$the function $f\circ\theta\circ\br$ is is continuously differentiable at $t$ (by the $C^1$-smoothness of the functions $\Theta$ and $\br_Y$).
\end{proof}
The smoothness of the sphere $S_Y$ implies the continuous differentiability of the norm $\|\cdot\|:Y\to\IR$ on the set $Y\setminus \{0\}$. Then the function
$$\Phi:(a,a+L)\to S_Y,\quad \Phi:t\mapsto\frac{\phi(t)}{\|\phi(t)\|},$$
is $C^1$-smooth. By Lemma~\ref{l:Tingley}, $$\phi(s)=f(\br(s))-f(\theta(\br(s)))=f(\br(s))-f(-\br(s))=f(\br(s))-(-f(\br(s)))=2\cdot f(\br(s))$$ and hence $\Phi(s)=f(\br(s))$. Assuming that the function $\Phi$ is not constant, we can find a point $c\in (a,a+L)$ such that $\Phi$ is continuously differentiable at $c$ and the derivative $\Phi'(c)$ is not zero. Then $\Phi$ is a diffeomorphism of some neighborhood $U_c\subseteq (a,a+L)$ onto its image $\Phi(U_c)\subseteq S_Y$.
By Lemma~\ref{l:full}, the sets $\ddot\Omega_\br$ and $\ddot\Omega_{\br_Y}$ are Lebesgue co-null. Since diffeomorphisms preserve Lebesgue co-null sets, the set $U_c\cap \ddot\Omega_{\br}\cap\ddot\Omega_{\br_Y}\cap\Phi^{-1}(\br_Y(\ddot\Omega_{\br_Y}))$ is Lebesgue co-null in $U_c$ and hence contains some point $b$.
Let $a=\Theta(b)$ and observe that $\br(b)-\br(a)=\br(b)-\theta(\br(b))=\|\br(b)-\br(a)\|\cdot\br(s)$. Since $\Phi(b)\in \br_Y(\ddot\Omega_{\br_Y})$, there exists a real number $y\in \ddot\Omega_{\br_Y}$ such that $\Phi(b)=\br_Y(y)$.
Since $b\in\ddot\Omega_\br$ and $\varphi$ is not Lipschitz at $s$, the function $$\nu:\IR\to\IR,\quad \nu:\e\mapsto \|\br(b+\e)-\br(a)\|$$ is not twice differentiable at zero according to Lemma~\ref{l:noLip}.
On the other hand, we have
\begin{multline*}
\br_Y(b)-\br_Y(a)=f(\br(b))-f(\br(a))=f(\br(b))-f(\theta(\br(b)))=\\
=\Phi(b)\cdot\|f(\br(b))-f(\theta(\br(b))\|=\br_Y(y)\cdot \|\br_Y(b)-\br_Y(a)\|
\end{multline*}
and for every $\e$
$$\nu(\e)=\|\br(b+\e)-\br(a)\|=\|f\circ\br(b+\e)-f\circ\br(a)\|=\|\br_Y(b+\e)-\br_Y(a)\|.$$
Since $b,y\in\ddot\Omega_{\br_Y}$, the function
$$\nu:\IR\to\IR,\quad\nu_Y:\e\mapsto\|\br_Y(b+s)-\br_Y(a)\|=\|\br(b+\e)-\br(a)\|,$$
is twice differentiable at zero by Lemma~\ref{l:first}.
This contradiction shows that the function $\Phi:(a,a+L)\to S_Y$ is constant with $f(\br(s))\in \Phi\big((a,a+L)\big)=\{f(\br(s))\}$ and hence the direction $\br(s)\in S_X$ is special.
\end{proof}
\section{Proof of Theorem~\ref{t:main2}}\label{s:main}
Assume that a $2$-dimensional Banach space $X$ is strictly convex, smooth, and not absolutely smooth. Let $\br:\IR\to S_X$ be a natural parameterization of the unit sphere $S_X$ and $\varphi:\IR\to\IR$ is the phase shift. Let $C$ be the set of points $s\in\IR$ at which the function $\varphi$ is not Lipschitz. Since $X$ is not absolutely smooth, the set $C$ is uncountable according to Lemma~\ref{l:Las}. Then we can choose two points $a,b\in C$ such that the vectors $\br(a), \br(b)$ are linearly independent. By Lemma~\ref{l:spec2}, the directions $\br(a),\br(b)$ are special and by Theorem~\ref{t:key}, the Banach space $X$ has the Mazur--Ulam property.
\section{Acknowledgements}
The author expresses his sincere thanks to Javier Cabello S\'anchez for his inspiring paper \cite{CS} that contained a crucial idea of special directions (appearing explicitly in \cite{BCS}), which allowed to handle non-(absolutely)-smooth cases in the proof of Theorem~\ref{t:main}.
| 10,908 |
\section{Personal Memories with Professor Keiji Kikkawa and the
Kikkawa-Type Physics at Ochanomizu}
It is my great pleasure to contribute to the proceedings of the workshop
held at Osaka to celebrate the 60th birthday of Professor Keiji Kikkawa.
I am very much influenced by his physics, especially by his papers on 1)
the light-cone field theory of string (Its Japanese version included in
Soryushiron Kenkyu was my favorite.), 2) his lecture note on the
superstrings given just before the string fever started (I think everybody
should begin with this lecture note when he or she wants to do something
in strings.), 3) hadronic strings with quarks at the ends, and 4) the path
integral formulation of the Nambu-Jona-Lasinio model. Personally,
Professor Kikkawa cited my paper on the dual transformation in gauge
theories at the Tokyo conference in 1978, without which I could not have
survived in our particle physics community and would definitely be
engaged in another job now. Therefore I am greatly indebted to him for his
guidance in physics. It was probably 1979 summer when I went abroad for
the first time with my late friend Dr. Osamu Sawada, and we stayed at
Professor Hirotaka Sugawara's residence in Honolulu. The topical
conference was held at the moment, and Kikkawa-san came to Honolulu to
attend it. Kikkawa-san, Sugawara-san, Sandip Pakvasa-san, Sawada-san,
and myself would always sit on the beautiful seashore and the younger
ones listened to the physics discussion exchanged between Kikkawa-san
and Sugawara-san. The one month stay in Honolulu was one of my great and
most stimulating experiences.
At Kikkawa-san's 60th birthday Conference, everybody was talking about
"p-branes and duality transformations". I really thought we were
timeslipping to 15$\sim$20 years ago. At that time "dual transformation,
membrane and n-dimensionally extended objects (now called p-branes)"
were my favorite themes.~\cite{dual transform} ~\cite{membrane} If my
paper on the membranes (which was the theory of n-branes) gave a little
influence on the famous membrane paper by Kikkawa-san and
Yamasaki-san~\cite{K-Y}, I would be very happy.
After moving to Ochanomizu University in 1987 from KEK, I have been
working with my students mainly on the phenomenological problems of the
non-Kikkawa type physics, including beyond the standard model effects in
the $e^{+} e^{-}\rightarrow W^{+} W^{-}$ process, the effect of the top
condensation in B-physics, the neutrino physics, the CP violating models
and the baryogenesis of our universe. Postdocs, Yasuhiko Katsuki, Kiusau
Teshima, Hirofumi Yamada, Isamu Watanabe, Mohammad Ahmady and
Noriyuki Oshimo did their own physics on the beyond the standard model,
multiple production in perturbative QCD, non-perturbative QCD, linear
collider physics and the two photon process, rare decays and the heavy
quark symmetry in B-physics and CP violation in SUSY and SUSY breaking,
respectively, with the help of the then students, Miho Marui, Kumiko
Kimura, Atsuko Nitta, Azusa Yamaguchi, Fumiko Kanakubo, Tomomi Saito,
Tomoko Uesugi, Tomoko Kadoyoshi, Minako Kitahara and Rika Endo.
I have, however, sometimes come back to the Kikkawa-type physics on
the string, membrane and gravity theories with my postdocs and my
students: For example,\\
(1) Orbifold models were firstly studied with Ikuo
Senda.~\cite{orbifold}\\
(2) Using the light-cone gauge field theory of strings invented by Kaku
and Kikkawa, we with Miho Marui and Ichiro Oda have derived the
Altarelli-Parisi like evolution equation, since the decay function of
strings works naturally in this light-cone frame as has happened similarly
in QCD.~\cite{evolution equation} \\
(3) Knotting of the membrane was studied.~\cite{knot}\\
(4) With Ichiro Oda, Akika Nakamichi and Fujie Nagamori, we studied
four dimensional topological gravities, mainly on their
quantization.~\cite{topological gravity} \\
(5) Relating to this topological nature at high energies, estimation of the
membrane scattering amplitudes is performed with Sachiko Kokubo, giving
an indication of the structural phase transition among the intermediate
shapes of the membranes, when the scattering angle is
changed.~\cite{membrane scattering}\\
Other Kikkawa-type physics performed by our postdocs at Ochanomizu
were;\\
(6) Kiyoshi Shiraishi studied some 5 years ago BPS soliton and
Born-Infeld theories as well as the finite temperature field theories, \\
(7) The dilatonic gravity and black holes were investigated by Ichiro Oda
and Shin'ichi Nojiri, and\\
(8) Hybrid model of continuous and discrete theories are examined by
Toshiyuki Kuruma.
Recently I am very much interested, as for the Kikkawa-type physics,
in\\
(9) the generation of the Einstein gravity from the topological
theory~\cite{string condensation},\\
(10) the swimming of microorganisms viewed from string and membrane
theories,~\cite{swimming} and\\
(11) phase transition dynamics viewed from the field theoretical
membrane theories.\\
The issue (9) is being investigated with Miyuki Katsuki, Hiroto Kubotani
and Shin'ichi Nojiri, the issue (10) is with Masako Kawamura and Shin'ichi
Nojiri and is helped by the Barcelona friends, Sergei Odintsov and Emil
Elizalde, but the last issue (11) is still at the stage of promoting a vague
idea.
In the next section I will mainly explain the issue (9), and will comment
on my vague idea of the issue (11).
\section{Generation of the Einstein Gravity from the Topological 2-Form
Gravity }
The topological 2-form gravity is given by the following chiral action for
the self-dual part:
\begin{equation}
S = \int \frac{1}{2} \epsilon^{\mu\nu\lambda\rho}
\left( B^a_{\mu\nu}(x)R^a_{\lambda\rho}(x) +
\underbrace{ \phi^{ab}(x)B^a_{\mu\nu}(x)B^b_{\lambda\rho}(x)}_
{constraint\: term} \right),\label{chiral action}
\end{equation}
where $B^a_{\mu\nu}(x)$ is the anti-symmetric tensor field or the
Kalb-Ramond field and $R^a_{\lambda\rho}(x)$ is the $SU(2)$ field strengh
for the $SU(2)$ spin connection ${\omega}_{\mu}^{a}$. The constraint
condition expressed by the Lagrange multiplier field ${\phi}^{ab}(x)$ can be
solved naturally by introducing the vierbein and the t' Hooft symbol as
$B_{\mu\nu}^{a} = \frac{1}{2} \eta_{BC}^{a} e_{\mu}^{B}e_{\nu}^{C}$. Then
we have the Einstein action. In the process of solving the constraint the
extra Kalb-Ramond symmetry possessed by the topological "BF" theory is
broken in an ad hoc way. Instead we wish to start with the Kalb-Ramond
invariant action and derive the constraint spontaneously. The Kalb-Ramond
symmetry, $B^a_{\mu\nu} \rightarrow
B^a_{\mu\nu}+\nabla^{ab}_{\mu}\Lambda^b_{\nu} -\nabla^{ab}_{\nu}
\Lambda^b_{\mu}$, was originally the gauge symmetry of strings.
Therefore, by introducing the string field, we write down the following
Kalb-Ramond invariant action:
\begin{eqnarray}
S&=& \int d^4 x\frac{1}{2} \epsilon^{\mu\nu\lambda\rho}
B^a_{\mu\nu}(x)R^a_{\lambda\rho}(x) \nonumber \\
& & + \sum_C \sum_{x_0 (\in C)} \sum_{x( \in C )}
\epsilon^{\mu\nu\lambda\rho}
\left[ \left( \frac{\delta}{\delta C^{\mu\nu}(x)} +
T^a B^a_{\mu\nu}[C; x, x_0]\right) \Psi[C; x_0] \right]
^{\dagger}\nonumber \\
& &~~~\times \left[ \left( \frac{\delta}{\delta C^{\lambda\rho}(x)} +
T^a B^a_{\lambda\rho}[C; x, x_0]\right) \Psi[C; x_0] \right] \nonumber \\
& &+ \sum_{C}\sum_{ x_0} V [ \Psi[C; x_0] ^{\dagger} \Psi[C;
x_0]] .
\label{ non-Abellian K-R action }
\end{eqnarray}
In this expression we need to modify the Kalb-Ramond field
$B^a_{\mu\nu}$ and its transformation to the non-local ones, relfecting
the difficulty of their non-Abelian versions. Now the condensation of the
string fields
\begin{equation}
\frac{1}{2}\phi^{ab}[C; x_0] \equiv \langle\Psi[C; x_0] ^{\dagger} T^a
T^b\Psi[C; x_0] \rangle,
\label{non-Abelian condensation}
\end{equation}
plays the role of the Lagrange multiplier. If the condensation becomes
large for the symmetric (isospin 2 ) part of $(a, b)$, then its coefficient
gives the constraint, leading to the Einstein gravity. For the details refer
to Ref.~\cite{string condensation}.
\section{Phase Transition Dynamics and Field Theory of Membranes}
During the temporal development of the 1st order phase trandition, like
the cooling down of the vapor (unbroken phase), liquid droplets of water
(bubbles of the broken phase) are nucleated, they fuse with themselves,
and finally the whole vessel (the whole space) is filled up with the water
(broken phase). It is really amazing to know that for such a difficult
problem there exists a solvable theory called the Kolmogorov-Avrami
theory~\cite{KA}, if the critical radius of the bubble is vanishing and the
wall velocity is constant. "Solvable" means that we can exactly know the
probability of the arbitrarily chosen N spacetime points to belong to the
broken or the unbroken phase. This may suggest the existence of a solvable
non-relativistic membranic (interfacial) field theory. It is another
Kikkawa-type physics to persue.
| 2,887 |
\section{Introduction}
In the recent years the interest in the Calogero--Moser type of models
\cite{OP}--\cite{R} is considerably revitalized. One of the directions of
this recent development was connected with the notion of the dynamical
$r$--matrices and their interpretation in terms of Hamiltonian reduction
\cite{AT}--\cite{N}. Very recently \cite{BB},\cite{AR}, \cite{S}, \cite{N},
this line of research included also the so--called Ruijsenaars--Schneider
models \cite{RS},\cite{R} which may be seen as relativistic generalizations
of the Calogero--Moser ones \cite{OP}, \cite{KKS}.
In the paper \cite{BB} a {\it quadratic} (dynamical) $r$--matrix Poisson
structure was found for the dynamical system describing the motion of the
solitons of the sine--Gordon model. This system turns out to be a particular
case of the hyperbolic Ruijsenaars--Schneider model corresponding to a
particular value of the parameter $\gamma$ of the model (cf. (\ref{L rel})
below), namely $\gamma=i\pi/2$, when the Lax matrix becomes symmetric in some
gauge,
The case of general hyperbolic Ruijsenaars--Schneider model,
admitting also the rational model as a limiting
case, was considered in \cite{AR}. There was found a {\it linear} $r$--matrix
structure for this model, with the linear dependence of the $r$--matrix on the
elements of the Lax matrix. However, as it stands, the structure found in
\cite{AR} cannot be cast into a quadratic form.
This drawback was overcome in \cite{S}, where the quadratic $r$--matrix Poisson
bracket was found for the general rational and hyperbolic models. Moreover, this
bracket turned out to posess several remarkable properties.
\begin{itemize}
\item
First, the $r$--matrix objects turned out to be independent on the
relativistic parameter $\gamma$ of the model.
\item
Second, and more important, the $r$--matrix
object governing the whole hierarchy of the Lax equations attached to
the Ruijsenaars--Schneider model turned out to be
{\it identical} with the corresponding object governing the
non--relativistic Calogero--Moser hierarchy.
\end{itemize}
A geometric interpretation of this
intriguing property was also provided in \cite{S}. Several open problems were
formulated in \cite{S}, the first of them being the generalization of these
findings to the case of elliptic Ruijsenaars--Schneider model.
Soon after \cite{S} there appeared the paper \cite{N} where an $r$--matrix
quadratic Poisson bracket for the elliptic Ruijsenaars--Schneider model was
found, thus partly solving the mentioned problem. However, despite
the fact that this bracket has the same general structure as the one found in
\cite{S}, it turns out not to generalize the latter. It fails to
have the two remarkable properties pointed out above, and moreover it does not
reduce to the bracket found in \cite{S} in the corresponding (rational or
hyperbolic) limit.
In the present paper we give a proper generalization of the results in
\cite{S} for the elliptic case. Namely, we present a quadratic $r$--matrix
structure for this model enjoying the two properties listed above. An existence
of two different $r$--matrix Poisson brackets for one and the same model is
not contradictory, because of the well--known non--uniqueness of an $r$--matrix.
So in principle both can coexist on their own rights. We hope, however,
that the two remarkable properties pointed out above indicate on some deeper
geometric meaning to be clarified in the future, so
that the result reported here will be accepted as {\it the} $r$--matrix for the
elliptic Ruijsenaars--Schneider model.
\setcounter{equation}{0}
\section{Elliptic models of the Calogero--Moser type.}
The elliptic non--relativistic Calogero--Moser hierarchy is described in
terms of the {\it Lax matrix}
\begin{equation}\label{L nr}
L=L(x,p,\lambda)=\sum_{k=1}^Np_kE_{kk}+\gamma\sum_{k\neq j}\Phi(x_k-x_j,\lambda)E_{kj}.
\end{equation}
Here the function $\Phi(x,\lambda)$ is defined as
\begin{equation}\label{Phi}
\Phi(x,\lambda)=\frac{\sigma(x+\lambda)}{\sigma(x)\sigma(\lambda)},
\end{equation}
where $\sigma(x)$ is the Weierstrass $\sigma$--function. Further,
$\lambda$ is an auxiliary (so called spectral) parameter which
does not enter the equations of motion of the model, but rather serves as a
useful tool for its solution. On the contrary, $\gamma$ is an internal
parameter of the model, usually supposed to be pure imaginary.
The dynamical variables $x=(x_1,\ldots,x_N)^T$ and $p=(p_1,\ldots,p_N)^T$
are supposed to be canonically conjugated, i.e. to have canonical Poisson
brackets:
\begin{equation}\label{can PB}
\{x_k,x_j\}=\{p_k,p_j\}=0,\quad \{x_k,p_j\}=\delta_{kj}.
\end{equation}
The Hamiltonian function of the Calogero--Moser model proper (i.e. of the
simplest representative of the Calogero--Moser hierarchy) is given by
\[
H(x,p)=\frac{1}{2}\sum_{k=1}^N p_k^2-
\frac{1}{2}\gamma^2\sum_{k\neq j}\wp(x_k-x_j)=\frac{1}{2}{\rm tr}L^2(x,p,\lambda)
+{\rm const},
\]
where ${\rm const}=-N(N-1)\gamma^2\wp(\lambda)/2$, and $\wp(x)$ is the
Weierstrass elliptic function.
The elliptic relativistic Ruijsenaars--Schneider hierarchy is also described
in terms of the {\it Lax matrix}
\begin{equation}\label{L rel}
L(x,p,\lambda)=\sum_{k,j=1}^N \frac{\Phi(x_k-x_j+\gamma,\lambda)}
{\Phi(\gamma,\lambda)}b_jE_{kj}.
\end{equation}
The notations are the same as above, and we use an additional abbreviation:
\begin{equation}\label{def b}
b_k=\exp(p_k)\prod_{j\neq k}\left(
\frac{\sigma(x_k-x_j+\gamma)\sigma(x_k-x_j-\gamma)}{\sigma^2(x_k-x_j)}
\right)^{1/2},
\end{equation}
so that in the variables $(x,b)$ the canonical Poisson brackets (\ref{can PB})
take the form
\[
\{x_k,x_j\}=0,\quad \{x_k,b_j\}=b_k\delta_{kj},
\]
\begin{equation}\label{rel PB}
\{b_k,b_j\}=b_kb_j\Big(\zeta(x_j-x_k+\gamma)-
\zeta(x_k-x_j+\gamma)+2(1-\delta_{kj})\zeta(x_k-x_j)\Big).
\end{equation}
Here $\zeta(x)$ is, of course, the Weierstrass $\zeta$--function, i.e.
\[
\zeta(x)=\frac{\sigma'(x)}{\sigma(x)}.
\]
The Hamiltonian function of the Ruijsenaars--Schneider model proper (i.e. of
the simplest member of this hierarchy) is simply
\[
H(x,p)=\sum_{k=1}^N b_k={\rm tr}L(x,p,\lambda).
\]
Let us note that the non--relativistic limit, leading from the
Ruijsenaars--Schneider model to the Calogero--Moser one, is achieved
by rescaling $p\mapsto\beta p$, $\gamma\mapsto\beta\gamma$ and
subsequent sending $\beta\to 0$ (in this limit $L_{{\rm rel}}=
I+\beta L_{{\rm nonrel}}+O(\beta^2)$).
Let us also note that the evolution of either of the non--relativistic or the
relativistic model is governed by the {\it Lax equation} of the form
\begin{equation}\label{Lax eq}
\dot{L}=[M,L],
\end{equation}
where, for example, for the Ruijsenaars--Schneider model one has:
\begin{equation}\label{M kj}
M_{kj}=\Phi(x_k-x_j,\lambda)b_j,\;\;k\neq j,
\end{equation}
\begin{equation}\label{M kk}
M_{kk}=\Big(\zeta(\lambda)+\zeta(\gamma)\Big)b_k+
\sum_{j\neq k}\Big(\zeta(x_k-x_j+\gamma)-\zeta(x_k-x_j)\Big)b_j.
\end{equation}
An $r$--matrix found below enables one to give a general formula for the matrix
$M$ for an arbitrary flow of the corresponding hierarchy (cf. \cite{S} for such
formulas in the rational and hyperbolic cases).
\setcounter{equation}{0}
\section{Dynamical $r$-matrix formulation}
An $r$--matrix formulation of the elliptic Calogero--Moser model was given in
\cite{Skl}, \cite{BS} as a generalization of the previous result obtained in
\cite{AT} for the rational and hyperbolic cases. The result of \cite{Skl} may
be presented in the following form: for the non--relativistic case the
corresponding Lax matrices satisfy a linear $r$--matrix ansatz
\begin{equation}\label{r Anz}
\{L(\lambda)\stackrel{\otimes}{,}L(\mu)\}=
\left[I\otimes L(\mu),r(\lambda,\mu)\right]-\left[L(\lambda)\otimes I,
r^*(\lambda,\mu)\right],
\end{equation}
where the $N^2\times N^2$ matrix $r(\lambda,\mu)$ may be decomposed into the sum
\begin{equation}\label{ras}
r(\lambda,\mu)=a(\lambda,\mu)+s(\lambda).
\end{equation}
Here $a$ is a skew--symmetric matrix
\begin{equation}\label{a}
a(\lambda,\mu)=-\zeta(\lambda-\mu)\sum_{k=1}^N E_{kk}\otimes E_{kk}-
\sum_{k\neq j}\Phi(x_j-x_k,\lambda-\mu)E_{jk}\otimes E_{kj},
\end{equation}
and $s$ is a non--skew--symmetric one:
\begin{equation}\label{s}
s(\lambda)=\zeta(\lambda)\sum_{k=1}^N E_{kk}\otimes E_{kk}+
\sum_{k\neq j}\Phi(x_j-x_k,\lambda)E_{jk}\otimes E_{kk}.
\end{equation}
Here the ''skew--symmetry'' is understood with respect to the operation
\[
r^*(\lambda,\mu)=\Pi r(\mu,\lambda)\Pi\;\;{\rm with}\;\;
\Pi=\sum_{k,j=1}^NE_{kj}\otimes E_{jk}.
\]
So we have
\[
a^*(\lambda,\mu)=-a(\lambda,\mu),
\]
and
\begin{equation}\label{s*}
s^*(\mu)=\zeta(\mu)\sum_{k=1}^N E_{kk}\otimes E_{kk}+
\sum_{k\neq j}\Phi(x_j-x_k,\mu)E_{kk}\otimes E_{jk}.
\end{equation}
(Note that our $r$ is related to the objects $r_{12}$, $r_{21}$ in \cite{Skl}
by means of $r=-r_{21}$ and $r^*=-r_{12}$).
We shall prove that in the relativistic case the corresponding
Lax matrices satisfy the quadratic $r$--matrix ansatz:
\begin{eqnarray}
\{L(\lambda)\stackrel{\otimes}{,}L(\mu)\} & = &
(L(\lambda)\otimes L(\mu))a_1(\lambda,\mu)-
a_2(\lambda,\mu)(L(\lambda)\otimes L(\mu))\nonumber\\
& + & (I\otimes L(\mu))s_1(\lambda,\mu)(L(\mu)\otimes I)-
(L(\lambda)\otimes I)s_2(\lambda,\mu)(I\otimes L(\mu))\nonumber\\
& &\label{as Anz}
\end{eqnarray}
where the matrices $a_1,a_2,s_1,s_2$ satisfy the conditions
\begin{equation}\label{sym}
a_1^*(\lambda,\mu)=-a_1(\lambda,\mu),\quad a_2^*(\lambda,\mu)=-a_2(\lambda,\mu),
\quad s_2^*(\lambda,\mu)=s_1(\lambda,\mu),
\end{equation}
and
\begin{equation}\label{sum}
a_1(\lambda,\mu)+s_1(\lambda,\mu)=a_2(\lambda,\mu)+s_2(\lambda,\mu)
=r(\lambda,\mu).
\end{equation}
The first of these conditions assures the skew--symmetry of the Poisson bracket
(\ref{as Anz}), and the second one garantees that the Hamiltonian flows with
invariant Hamiltonian functions $\varphi(L)$ have the Lax form (\ref{Lax eq})
with the form of the $M$--matrix being governed by the same $r$--matrix as in
the non--relativistic case.
Such general quadratic $r$-matrix structures were discovered several times
independently \cite{FM}, \cite{P}, \cite{S2}. See \cite{S2} for an application
to closely related, but much more simple systems of the Toda lattice type.
{\bf Theorem.} {\it For the Lax matrices of the relativistic model
{\rm (\ref{L rel})} there holds a quadratic $r$--matrix ansatz
{\rm (\ref{as Anz})} with the matrices
\[
a_1=a+w,\quad s_1=s-w,
\]
\[
a_2=a+s-s^*-w,\quad s_2=s^*+w,
\]
and $w$ is an auxiliary matrix}
\begin{equation}\label{w}
w=\sum_{k\neq j}\zeta(x_k-x_j)E_{kk}\otimes E_{jj}.
\end{equation}
Note that all the objects $a$, $s$, $w$ entering these formula do not depend on
$\gamma$, and that (\ref{sum}) is fulfilled, which justifies the title of the
present paper.
The {\bf proof} of this Theorem is based on direct computations, presented in
the Appendix.
\setcounter{equation}{0}
\section{Conclusions}
Now that the formal part of the results in \cite{S} is generalized, it is
tempting to find a geometrical explanation of the phenomena behind it. To
this end one should develop further the theory of Calogero--Moser type models as
Hamiltonian reduced systems.
Certainly, this problem should be supplied with the whole list of open problems
formulated in \cite{S}, \cite{N}.
\setcounter{equation}{0}
\section{Acknowledgements}
The research of the author is financially supported by the DFG (Deutsche
Forschungsgemeinschaft). My pleasant duty is to thank warmly Professor
Orlando Ragnisco (University of Rome) for organizing my visit to Rome, where
this work was done, for useful discussions and collaboration, and his
institution -- for financial support during this visit.
\setcounter{equation}{0}
\section{Appendix: proof of the Theorem}
Let us denote
\[
\{L_{ij}(\lambda),L_{km}(\mu)\}=\pi_{ijkm}L_{ij}(\lambda)L_{km}(\mu),
\]
\[
[L(\lambda)\otimes L(\mu),\,a(\lambda,\mu)]=
\sum_{i,j,k,m=1}^N \alpha_{ijkm}L_{ij}(\lambda)L_{km}(\mu)E_{ij}\otimes E_{km},
\]
and analogously
\[
s(\lambda)(L(\lambda)\otimes L(\mu))=
\sum_{i,j,k,m=1}^N \sigma^{(1)}_{ijkm}L_{ij}(\lambda)L_{km}(\mu)E_{ij}\otimes
E_{km},
\]
\[
s^*(\mu)(L(\lambda)\otimes L(\mu))=
\sum_{i,j,k,m=1}^N \sigma^{(2)}_{ijkm}L_{ij}(\lambda)L_{km}(\mu)E_{ij}\otimes
E_{km},
\]
\[
(I\otimes L(\mu))s(\lambda)(L(\lambda)\otimes I)=
\sum_{i,j,k,m=1}^N \sigma^{(3)}_{ijkm}L_{ij}(\lambda)L_{km}(\mu)E_{ij}\otimes
E_{km},
\]
\[
(L(\lambda)\otimes I)s^*(\mu)(I\otimes L(\mu))=
\sum_{i,j,k,m=1}^N \sigma^{(4)}_{ijkm}L_{ij}(\lambda)L_{km}(\mu)E_{ij}\otimes
E_{km}.
\]
The statement of the Theorem is equivalent to
\begin{equation}\label{form}
\pi_{ijkm}=\alpha_{ijkm}-\sigma^{(1)}_{ijkm}
+\sigma^{(2)}_{ijkm}+\sigma^{(3)}_{ijkm}-\sigma^{(4)}_{ijkm}+w_{ik}
+w_{jm}-w_{im}-w_{jk},
\end{equation}
where $w_{jk}=(1-\delta_{jk})\zeta(x_j-x_k)$ are the coefficients of the auxiliary
matrix $w=\sum_{j\neq k}w_{jk}E_{jj}\otimes E_{kk}$.
According to the Poisson brackets (\ref{rel PB}) we have:
\[
\pi_{ijkm}=\zeta(x_m-x_j+\gamma)-
\zeta(x_j-x_m+\gamma)+2(1-\delta_{jm})\zeta(x_j-x_m)
\]
\[
+(\delta_{jk}-\delta_{jm})
\Big(\zeta(x_k-x_m+\gamma)-\zeta(x_k-x_m+\gamma+\mu)\Big)
\]
\begin{equation}\label{pi}
-(\delta_{im}-\delta_{jm})
\Big(\zeta(x_i-x_j+\gamma)-\zeta(x_i-x_j+\gamma+\lambda)\Big).
\end{equation}
From the definitions of the matrices $a$ and $s$ we have:
\begin{eqnarray*}
\alpha_{ijkm}
& = & (\delta_{ik}-\delta_{jm})\zeta(\lambda-\mu)\\
& & + (1-\delta_{ik})\frac{L_{kj}(\lambda)L_{im}(\mu)}
{L_{ij}(\lambda)L_{km}(\mu)}\Phi(x_i-x_k,\lambda-\mu)\\
& & - (1-\delta_{jm})\frac{L_{im}(\lambda)L_{kj}(\mu)}
{L_{ij}(\lambda)L_{km}(\mu)}\Phi(x_m-x_j,\lambda-\mu);\\ \\
\sigma^{(1)}_{ijkm}
& = & \delta_{ik}\zeta(\lambda)+
(1-\delta_{ik})\frac{L_{kj}(\lambda)}{L_{ij}(\lambda)}
\Phi(x_i-x_k,\lambda);\\ \\
\sigma^{(2)}_{ijkm}
& = & \delta_{ik}\zeta(\mu)+
(1-\delta_{ik})\frac{L_{im}(\mu)}{L_{km}(\mu)}
\Phi(x_k-x_i,\mu);\\ \\
\sigma^{(3)}_{ijkm}
& = & \delta_{im}\zeta(\lambda)+
(1-\delta_{im})\frac{L_{mj}(\lambda)}{L_{ij}(\lambda)}
\Phi(x_i-x_m,\lambda);\\ \\
\sigma^{(4)}_{ijkm}
& = & \delta_{jk}\zeta(\mu)+
(1-\delta_{jk})\frac{L_{jm}(\mu)}{L_{km}(\mu)}
\Phi(x_k-x_j,\mu).\\ \\
\end{eqnarray*}
Using the expressions for the elements of the matrix $L$, we get:
\begin{eqnarray*}
\alpha_{ijkm}
& = & (\delta_{ik}-\delta_{jm})\zeta(\lambda-\mu)\\
& & + (1-\delta_{ik})\frac{\Phi(x_k-x_j+\gamma,\lambda)
\Phi(x_i-x_m+\gamma,\mu)\Phi(x_i-x_k,\lambda-\mu)}
{\Phi(x_i-x_j+\gamma,\lambda)\Phi(x_k-x_m+\gamma,\mu)};\\
& & -(1-\delta_{jm})\frac{\Phi(x_i-x_m+\gamma,\lambda)
\Phi(x_k-x_j+\gamma,\mu)\Phi(x_m-x_j,\lambda-\mu)}
{\Phi(x_i-x_j+\gamma,\lambda)\Phi(x_k-x_m+\gamma,\mu)};\\ \\
\sigma^{(1)}_{ijkm}
& = & \delta_{ik}\zeta(\lambda)+
(1-\delta_{ik})\frac{\Phi(x_k-x_j+\gamma,\lambda)\Phi(x_i-x_k,\lambda)}
{\Phi(x_i-x_j+\gamma,\lambda)};\\ \\
\sigma^{(2)}_{ijkm}
& = & \delta_{ik}\zeta(\mu)+
(1-\delta_{ik})\frac{\Phi(x_i-x_m+\gamma,\mu)\Phi(x_k-x_i,\mu)}
{\Phi(x_k-x_m+\gamma,\mu)};\\ \\
\sigma^{(3)}_{ijkm}
& = & \delta_{im}\zeta(\lambda)+
(1-\delta_{im})\frac{\Phi(x_m-x_j+\gamma,\lambda)\Phi(x_i-x_m,\lambda)}
{\Phi(x_i-x_j+\gamma,\lambda)};\\ \\
\sigma^{(4)}_{ijkm}
& = & \delta_{jk}\zeta(\mu)+
(1-\delta_{jk})\frac{\Phi(x_j-x_m+\gamma,\mu)\Phi(x_k-x_j,\mu)}
{\Phi(x_k-x_m+\gamma,\mu)}.
\end{eqnarray*}
The most laburous part of the further manipulations is the simplification of
the expression for $\alpha_{ijkm}$. This was performed already in \cite{N}, we
give here slightly more details. Following two elliptic identities were used
in \cite{N} to this aim:
\[
\frac{\Phi(X-A,\lambda)\Phi(Y+A,\mu)\Phi(A,\lambda-\mu)-
\Phi(Y+A,\lambda)\Phi(X-A,\mu)\Phi(X-Y+A,\lambda-\mu)}
{\Phi(X,\lambda)\Phi(Y,\mu)}=
\]
\[
\zeta(A)-\zeta(X-Y+A)+\zeta(X-A)-\zeta(Y+A),
\]
and
\[
\frac{\Phi(Y,\lambda)\Phi(X,\mu)\Phi(X-Y,\lambda-\mu)}
{\Phi(X,\lambda)\Phi(Y,\mu)}=
\zeta(\lambda-\mu)+\zeta(X-Y)-\zeta(X+\lambda)+\zeta(Y+\mu).
\]
One gets:
\[
\alpha_{ijkm}=(\delta_{ik}-\delta_{jm})\zeta(\lambda-\mu)
\]
\[
+(1-\delta_{ik})(1-\delta_{jm})\Big(\zeta(x_i-x_k)-\zeta(x_m-x_j)
+\zeta(x_k-x_j+\gamma)-\zeta(x_i-x_m+\gamma)\Big)
\]
\[
+(1-\delta_{ik})\delta_{jm}\Big(\zeta(\lambda-\mu)+\zeta(x_i-x_k)
-\zeta(x_i-x_j+\gamma+\lambda)+\zeta(x_k-x_m+\gamma+\mu)\Big)
\]
\[
-(1-\delta_{jm})\delta_{ik}\Big(\zeta(\lambda-\mu)+\zeta(x_m-x_j)
-\zeta(x_i-x_j+\gamma+\lambda)+\zeta(x_k-x_m+\gamma+\mu)\Big).
\]
Further straightforward manipulations give:
\[
\alpha_{ijkm}=(1-\delta_{ik})\zeta(x_i-x_k)-(1-\delta_{jm})\zeta(x_m-x_j)
\]
\[
+\zeta(x_k-x_j+\gamma)-\zeta(x_i-x_m+\gamma)
\]
\[
+(\delta_{ik}-\delta_{jm})\Big(\zeta(x_k-x_m+\gamma)-
\zeta(x_k-x_m+\gamma+\mu)\Big)
\]
\begin{equation}\label{alfa}
-(\delta_{ik}-\delta_{jm})\Big(\zeta(x_i-x_j+\gamma)-
\zeta(x_i-x_j+\gamma+\lambda)\Big).
\end{equation}
By simplifying the expressions for $\sigma_{ijkm}^{(1-4)}$ one uses
systematically the identity
\[
\frac{\Phi(X,\lambda)\Phi(Y,\lambda)}{\Phi(X+Y,\lambda)}=
\zeta(\lambda)+\zeta(X)+\zeta(Y)-\zeta(X+Y+\lambda).
\]
One gets following expressions:
\begin{eqnarray*}
\sigma^{(1)}_{ijkm}
& = & \zeta(\lambda)+(1-\delta_{ik})\Big(\zeta(x_i-x_k)+\zeta(x_k-x_j+\gamma)-
\zeta(x_i-x_j+\gamma+\lambda)\Big);\\ \\
\sigma^{(2)}_{ijkm}
& = & \zeta(\mu)+(1-\delta_{ik})\Big(\zeta(x_k-x_i)+\zeta(x_i-x_m+\gamma)-
\zeta(x_k-x_m+\gamma+\mu)\Big);\\ \\
\sigma^{(3)}_{ijkm}
& = & \zeta(\lambda)+(1-\delta_{im})\Big(\zeta(x_i-x_m)+\zeta(x_m-x_j+\gamma)-
\zeta(x_i-x_j+\gamma+\lambda)\Big);\\ \\
\sigma^{(4)}_{ijkm}
& = & \zeta(\mu)+(1-\delta_{jk})\Big(\zeta(x_k-x_j)+\zeta(x_j-x_m+\gamma)-
\zeta(x_k-x_m+\gamma+\mu)\Big).
\end{eqnarray*}
It follows after straightforward manipulations:
\[
-\sigma_{ijkm}^{(1)}+\sigma_{ijkm}^{(2)}+\sigma_{ijkm}^{(3)}
-\sigma_{ijkm}^{(4)}=
\]
\[
2(1-\delta_{ik})\zeta(x_k-x_i)+(1-\delta_{im})\zeta(x_i-x_m)-
(1-\delta_{jk})\zeta(x_k-x_j)
\]
\[
+\zeta(x_i-x_m+\gamma)-\zeta(x_k-x_j+\gamma)+\zeta(x_m-x_j+\gamma)
-\zeta(x_j-x_m=\gamma)
\]
\[
+(\delta_{ik}-\delta_{im})\Big(\zeta(x_i-x_j+\gamma)-\zeta(x_i-x_j+\gamma
+\lambda)\Big)
\]
\begin{equation}\label{sigmas}
-(\delta_{ik}-\delta_{jk})\Big(\zeta(x_k-x_m+\gamma)-\zeta(x_k-x_m+\gamma
+\mu)\Big).
\end{equation}
Now it is easy to see that combining (\ref{alfa}), (\ref{sigmas}), and
(\ref{pi}), one gets (\ref{form}), which proves the Theorem.
| 8,336 |
\section{Introduction}
Quarkonium production has traditionally been calculated in the
color singlet model (CSM) \cite{SCH94}. Although the model successfully
describes the production rates for some quarkonium states,
it has become clear that it fails to provide a theoretically
and phenomenologically consistent picture of all production
processes. In hadroproduction of charmonia at fixed target
energies ($\sqrt{s} < 50\,$ GeV), the ratio of the number of $J/\psi$
produced directly
to those arising from decays of higher
charmonium states is under-predicted by at least a factor five
\cite{VAE95}. The $\chi_{c1}$ to $\chi_{c2}$ production ratio
is far too low, and the observation of essentially unpolarized
$J/\psi$ and $\psi'$ can not be reproduced. At Tevatron
collider energies, when fragmentation production dominates,
the deficit of direct $J/\psi$ and $\psi'$ in the color singlet
model is even larger. This deficit has been referred to as the
`$\psi'$-anomaly'
\cite{BRA94,ROY95}.
These discrepancies suggest that the color singlet model is
too restrictive and that other production mechanisms
are necessitated. Indeed, the CSM requires that the quark-antiquark
pair that binds into a quarkonium state be produced
on the time scale $\tau\simeq 1/m_Q$ with the same
color and angular momentum quantum numbers as the eventually formed
quarkonium.
Consequently, a hard
gluon has to be emitted to produce a ${}^3 S_1$ state in the
CSM and costs one power of $\alpha_s/\pi$.
Since the time scale for quarkonium formation is of order
$1/(m_Q v^2)$, where $v$ is the relative quark-antiquark
velocity in the quarkonium bound state, this suppression
can be overcome if one allows for the possibility that the
quark-antiquark pair is in any angular momentum or color
state when produced on time scales $\tau\simeq 1/m_Q$.
Subsequent evolution into the physical
quarkonium state is mediated by emission of soft gluons with momenta
of order $m_Q v^2$. Since
the quark-antiquark pair is small in size, the emission of these
gluons can be analyzed within a multipole expansion.
A rigorous formulation \cite{BOD95} of this picture can be given in
terms of non-relativistic QCD (NRQCD). Accordingly, the production
cross section for a quarkonium state $H$ in the process
\begin{equation}
\label{proc}
A + B \longrightarrow H + X,
\end{equation}
\noindent can be written as
\begin{equation}
\label{fact}
\sigma_H = \sum_{i,j}\int\limits_0^1 d x_1 d x_2\,
f_{i/A}(x_1) f_{j/B}(x_2)\,\hat{\sigma}(ij\rightarrow H)\,,
\end{equation}
\begin{equation}
\label{factformula}
\hat{\sigma}(ij\rightarrow H) = \sum_n C^{ij}_{\bar{Q} Q[n]}
\langle {\cal O}^H_n\rangle\,.
\end{equation}
\noindent Here the first sum extends over all partons in the colliding
hadrons and $f_{i/A}$ etc. denote the corresponding distribution
functions. The short-distance ($x\sim 1/m_Q \gg 1/(m_Q v)$)
coefficients $C^{ij}_{\bar{Q} Q[n]}$
describe the production of a quark-antiquark pair
in a state $n$ and have expansions in $\alpha_s(2 m_Q)$. The
parameters\footnote{Their precise definition is given in
Sect.~VI of \cite{BOD95}.}
$\langle {\cal O}^H_n\rangle$ describe the subsequent
hadronization of the $Q\bar{Q}$ pair into a jet containing
the quarkonium $H$ and light hadrons. These matrix elements
can not be computed
perturbatively, but their relative importance in powers
of $v$ can be estimated from the selection rules for multipole
transitions.
The color octet picture has led to the most plausible explanation of
the `$\psi'$-anomaly' and the direct $J/\psi$ production deficit.
In this picture
gluons fragment into quark-antiquark pairs in a color-octet
${}^3 S_1^{(8)}$ state which then hadronizes into a $\psi'$ (or $J/\psi$)
\cite{BRA95,CAC95,CHO95}. Aside from this striking prediction, the
color octet mechanism remains largely untested. Its verification
now requires considering quarkonium production in other processes
in order to demonstrate the process-independence (universality) of
the production matrix elements $\langle {\cal O}_n^H\rangle$, which is
an essential prediction of the factorization formula (\ref{factformula}).
Direct $J/\psi$ and $\psi'$ production at large $p_t\gg 2 m_Q$ (where
$m_Q$ denotes the heavy {\em quark} mass) is rather unique in that a
single term, proportional to $\langle {\cal O}_8^H ({}^3 S_1)\rangle$,
overwhelmingly dominates the sum (\ref{factformula}). On the other hand,
in quarkonium
formation at moderate $p_t\sim 2 m_Q$ at colliders and in photo-production
or fixed target experiments ($p_t\sim 1\,$GeV),
the signatures of color octet production
are less dramatic, because they are not as enhanced
by powers of $\pi/\alpha_s$
or $p_t^2/m_Q^2$ over the singlet mechanisms. Furthermore,
theoretical predictions are parameterized by
more unknown octet matrix elements and are
afflicted by larger uncertainties. In particular, there are large
uncertainties
due to the increased
sensitivity to the heavy quark mass close to threshold. (The production
of a quark-antiquark pair close to threshold is favored by the rise of
parton densities at small $x$.)
These facts complicate establishing color
octet mechanisms precisely in those processes where experimental data is
most abundant.
Cho and Leibovich \cite{CHO95II} studied direct quarkonium
production at moderate $p_t$ at the Tevatron collider and were able
to extract a value for a certain combination of
unknown parameters $\langle {\cal O}_8^H({}^1 S_0)\rangle$ and
$\langle {\cal O}_8^H({}^3 P_0)\rangle$ ($H=J/\psi,\psi',\Upsilon(1S),
\Upsilon(2S)$). A first test of universality comes from
photo-production \cite{CAC96,AMU96,KO96}, where a different combination
of these two matrix elements becomes important near the elastic
peak at $z\approx 1$, where $z=p\cdot k_\psi / p \cdot k_\gamma$, and
$p$ is the proton momentum. A fit to photo-production data requires
much smaller matrix elements than those found in \cite{CHO95II}.
Taken at face value, this comparison would imply failure of the
universality assumption underlying the non-relativistic QCD approach.
However, the extraction from photo-production should be regarded
with caution since the NRQCD formalism describes
inclusive quarkonium production only after sufficient smearing in $z$
and is not applicable in the exclusive region close to $z=1$, where
diffractive quarkonium production is important.
In this paper we investigate the universality of the color octet
quarkonium production matrix elements
in fixed target hadron collisions and
re-evaluate the failures of the CSM in fixed target production
\cite{VAE95} after inclusion of color octet
mechanisms. Some of the issues involved have already been
addressed by Tang and V\"anttinen \cite{TAN95} and by Gupta and
Sridhar \cite{GUP96}, but a complete survey is still missing. We
also differ from \cite{TAN95} in the treatment of polarized
quarkonium production and the assessment of the importance of color
octet contributions and from \cite{GUP96} in the color
octet short-distance coefficients.
The paper is organized as follows: In Sect.~2 we compile the leading
order color singlet and color octet contributions to the production
rates for $\psi^\prime,~\chi_J,~J/\psi$ as well as bottomonium.
In Sect.~3 we present our numerical results for proton and
pion induced collisions. Sect.~4 is
devoted to the treatment of polarized quarkonium production. As
polarization remains one of the cleanest tests of octet quarkonium
production at large $p_t$ \cite{WIS95,BEN95}, we clarify in detail
the conflicting treatments of polarized production in \cite{BEN95} and
\cite{CHO95II}. Sect.~5 is dedicated to a comparison of the extracted
color-octet matrix elements from fixed target experiments with those
from photo-production and the Tevatron. We argue that
kinematical effects and small-$x$ effects can bias the extraction
of NRQCD matrix elements so that a fit to Tevatron data at large $p_t$
requires larger matrix elements than the fit to fixed
target and photo-production data. The
final section summarizes our conclusions.
\section{Quarkonium production cross sections at fixed
target energies}
\subsection{Cross sections}
We begin with the production cross section for $\psi'$ which does
not receive contributions from radiative decays of higher charmonium
states. The $2\to 2$ parton diagrams produce a quark-antiquark pair
in a color-octet state or $P$-wave singlet state (not relevant
to $\psi'$) and therefore contribute to $\psi'$
production at order $\alpha_s^2 v^7$. (For charmonium $v^2\approx
0.25 - 0.3$, for bottomonium $v^2\approx 0.08 - 0.1$.) The $2\to3$
parton processes contribute to the color singlet processes at order
$\alpha_s^3 v^3$. Using the notation in (\ref{fact}):
\begin{eqnarray}
\label{psiprimecross}
\hat{\sigma}(gg\to\psi') &=& \frac{5\pi^3\alpha_s^2}{12 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\left[\langle {\cal O}_8^{\psi'} ({}^1 S_0)
\rangle+\frac{3}{m_c^2} \langle {\cal O}_8^{\psi'} ({}^3 P_0)\rangle
+\frac{4}{5 m_c^2} \langle {\cal O}_8^{\psi'} ({}^3 P_2)\rangle
\right]\nonumber\\[0.0cm]
&&\hspace*{-1.5cm}
+\,\frac{20\pi^2\alpha_s^3}{81 (2 m_c)^5}\,
\Theta(x_1 x_2-4 m_c^2/s)\,\langle
{\cal O}_1^{\psi'} ({}^3 S_1)\rangle\,z^2\left[\frac{1-z^2+2 z
\ln z}{(1-z)^2}+\frac{1-z^2-2z \ln z}{(1+z)^3}\right]\\[0.2cm]
\hat{\sigma}(gq\to\psi') &=& 0\\[0.2cm]
\hat{\sigma}(q\bar{q}\to \psi') &=& \frac{16\pi^3\alpha_s^2}
{27 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\,\langle {\cal O}_8^{\psi'} ({}^3 S_1)
\rangle
\end{eqnarray}
\noindent Here $z\equiv (2 m_c)^2/(s x_1 x_2)$, $\sqrt{s}$ is the
center-of-mass energy and $\alpha_s$ is normalized at the scale $2 m_c$.
Corrections to these cross sections are suppressed
by either $\alpha_s/\pi$ or $v^2$. Note that the
relativistic corrections
to the color singlet cross section are substantial in specific
kinematic regions $z\to 0,1$ \cite{JUN93}. For $\sqrt{s}>15\,$GeV
these corrections affect the total cross section by less than $50\%$
and decrease
as the energy is raised \cite{SCH94}. Furthermore, notice that we
have expressed the short-distance coefficients in terms of the
charm quark mass, $M_{\psi'}\approx 2 m_c$, rather than the true
$\psi'$
mass.
Although the difference is formally of higher order in $v^2$, this
choice is conceptually favored since the short-distance coefficients
depend only on the physics prior to quarkonium formation. All
quarkonium specific properties which can affect the cross section,
such as quarkonium mass differences,
are hidden in the matrix elements.
The production of $P$-wave quarkonia differs from $S$-waves since
color singlet and color octet processes enter at the same order in $v^2$
as well as $\alpha_s$ in general. An exception is $\chi_{c1}$, which
can not be produced in $2\to 2$ parton reactions through
gluon-gluon fusion in a color singlet state. Since at order $\alpha_s^2$,
the $\chi_{c1}$ would be produced only in a $q\bar{q}$ collision, we
also include the gluon fusion diagrams at order $\alpha_s^3$, which
are enhanced by the gluon distribution. We have for $\chi_{c0}$,
\begin{eqnarray}
\label{chi0cross}
\hat{\sigma}(gg\to\chi_{c0}) &=& \frac{2\pi^3\alpha_s^2}{3 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\frac{1}{m_c^2}
\langle {\cal O}_1^{\chi_{c0}} ({}^3 P_0)
\rangle\\[0.2cm]
\hat{\sigma}(gq\to\chi_{c0}) &=& 0\\[0.2cm]
\hat{\sigma}(q\bar{q}\to \chi_{c0}) &=& \frac{16\pi^3\alpha_s^2}
{27 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\,\langle {\cal O}_8^{\chi_{c0}} ({}^3 S_1)
\rangle\,,
\end{eqnarray}
\noindent for $\chi_{c1}$,
\begin{eqnarray}
\label{chi1cross}
\hat{\sigma}(gg\to\chi_{c1}) &=& \frac{2\pi^2\alpha_s^3}{9 (2 m_c)^5}\,
\Theta(x_1 x_2-4 m_c^2/s)\frac{1}{m_c^2}
\langle {\cal O}_1^{\chi_{c1}} ({}^3 P_1)
\rangle\nonumber\\
&&\hspace*{-1.5cm}
\times\Bigg[\frac{4 z^2\ln z \, (z^8+9 z^7+26 z^6+28 z^5+17 z^4+7 z^3-
40 z^2-4 z-4}{(1+z)^5 (1-z)^4}\nonumber\\
&&\hspace*{-1.5cm}
\,+\frac{z^9+39 z^8+145 z^7+251 z^6+119 z^5-153 z^4-17 z^3-147 z^2-8 z
+10}{3 (1-z)^3 (1+z)^4}\Bigg]
\\[0.2cm]
\hat{\sigma}(gq\to\chi_{c1}) &=& \frac{8\pi^2\alpha_s^3}{81 (2 m_c)^5}\,
\Theta(x_1 x_2-4 m_c^2/s)\frac{1}{m_c^2}
\langle {\cal O}_1^{\chi_{c1}} ({}^3 P_1)
\rangle\left[-z^2\ln z + \frac{4 z^3-9 z+5}{3}\right]\nonumber\\[0.2cm]
\hat{\sigma}(q\bar{q}\to \chi_{c1}) &=& \frac{16\pi^3\alpha_s^2}
{27 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\,\langle {\cal O}_8^{\chi_{c1}} ({}^3 S_1)
\rangle\,,
\end{eqnarray}
\noindent and for $\chi_{c2}$
\begin{eqnarray}
\label{chi2cross}
\hat{\sigma}(gg\to\chi_{c2}) &=& \frac{8\pi^3\alpha_s^2}{45 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\frac{1}{m_c^2}
\langle {\cal O}_1^{\chi_{c2}} ({}^3 P_2)
\rangle\\[0.2cm]
\hat{\sigma}(gq\to\chi_{c2}) &=& 0\\[0.2cm]
\hat{\sigma}(q\bar{q}\to \chi_{c2}) &=& \frac{16\pi^3\alpha_s^2}
{27 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\,\langle {\cal O}_8^{\chi_{c2}} ({}^3 S_1)
\rangle\,.
\end{eqnarray}
\noindent Note that in the NRQCD formalism
the infrared sensitive contributions to
the $q\bar{q}$-induced color-singlet process at order $\alpha_s^3$
are factorized into the color octet matrix elements
$\langle {\cal O}_8^{\chi_{cJ}} ({}^3 S_1)\rangle$, so that
the $q\bar{q}$ reactions at order $\alpha_s^3$ are truly suppressed
by $\alpha_s$. The production of $P$-wave states through
octet quark-antiquark
pairs in a state other than ${}^3 S_1$ is higher order in $v^2$.
Taking into account indirect production of $J/\psi$ from decays of
$\psi'$ and $\chi_{cJ}$ states, the $J/\psi$ cross section is given by
\begin{equation}
\label{jpsicross}
\sigma_{J/\psi} = \sigma(J/\psi)_{dir} + \sum_{J=0,1,2}
\mbox{Br}(\chi_{cJ}\to J/\psi X)\,\sigma_{\chi_{cJ}} +
\mbox{Br}(\psi'\to J/\psi X)\,\sigma_{\psi'}\,,
\end{equation}
\noindent where `Br' denotes the corresponding branching fraction and
the direct $J/\psi$ production cross section
$\sigma(J/\psi)_{dir}$ differs from $\sigma_{\psi'}$ (see
(\ref{psiprimecross})) only by the replacement of $\psi'$ matrix elements
with $J/\psi$ matrix elements. Finally, we note that charmonium production
through $B$ decays is comparatively negligible at fixed target energies.
The $2\to 2$ parton processes contribute only to quarkonium
production at zero transverse momentum with respect to the beam axis.
The transverse momentum distribution of $H$ in reaction (\ref{proc})
is not calculable
in the $p_t<\Lambda_{QCD}$ region, but the total cross section (which averages
over
all $p_t$) is predicted even if the underlying parton process
is strongly peaked at zero $p_t$.
The transcription of the above formulae to bottomonium production is
straightforward. Since more bottomonium states exist below the
open bottom threshold than for the charmonium system,
a larger chain of cascade decays in the
bottomonium system must be included. In particular, there is indirect
evidence from $\Upsilon(3 S)$ production both at the Tevatron
\cite{PAP95} as well as in fixed target experiments (to be discussed
below) that there exist yet unobserved $\chi_b(3P)$ states below
threshold whose decay into lower bottomonium states should also be
included. Our numerical results do not include indirect contributions
from potential $D$-wave states below threshold.
All color singlet cross sections compiled in this section have been
taken from the review \cite{SCH94}. We have checked that the color
octet short-distance coefficients agree with those given in
\cite{CHO95II}, but disagree with those that enter the numerical
analysis of fixed target data in \cite{GUP96}.
\subsection{Matrix elements}
The number of independent matrix elements can be reduced by using the
spin symmetry relations
\begin{eqnarray}
&&\langle {\cal O}^{\chi_{cJ}}_1 ({}^3 P_J) \rangle =
(2 J+1)\,\langle {\cal O}^{\chi_{c0}}_1 ({}^3 P_0) \rangle
\nonumber\\
&&\langle {\cal O}^{\psi}_8 ({}^3 P_J) \rangle =
(2 J+1)\,\langle {\cal O}^{\psi}_8 ({}^3 P_0) \rangle
\\
&& \langle {\cal O}^{\chi_{cJ}}_8 ({}^3 S_1) \rangle =
(2 J+1)\,\langle {\cal O}^{\chi_{c0}}_1 ({}^3 S_1) \rangle
\nonumber
\end{eqnarray}
\noindent and are accurate up to corrections of order $v^2$ ($\psi=J/\psi,
\psi'$ -- identical relations hold for bottomonium). This implies
that at lowest order in $\alpha_s$, the matrix elements
$\langle {\cal O}^H_8 ({}^1 S_0) \rangle$ and
$\langle {\cal O}^H_8 ({}^3 P_0) \rangle$ enter fixed target
production of $J/\psi$ and $\psi'$ only in the
combination
\begin{equation}
\label{delta}
\Delta_8(H)\equiv \langle {\cal O}^H_8 ({}^1 S_0) \rangle +
\frac{7}{m_Q^2}\langle {\cal O}^H_8 ({}^3 P_0) \rangle\,.
\end{equation}
\noindent
Up to corrections in $v^2$, all relevant color singlet production
matrix elements are related to radial quarkonium wave functions at
the origin and their derivatives by
\begin{equation}
\label{wave}
\langle {\cal O}^H_1 ({}^3 S_1) \rangle =
\frac{9}{2\pi} |R(0)|^2
\qquad
\langle {\cal O}^H_1 ({}^3 P_0) \rangle =
\frac{9}{2\pi} |R'(0)|^2.
\end{equation}
\noindent We are then left with three non-perturbative parameters for
the direct production of each $S$-wave quarkonium and two parameters
for $P$-states.
The values for these parameters, which we will use below, are summarized
in tables~\ref{tab1} and \ref{tab2}. Many of the octet matrix elements,
especially for bottomonia, are not established and should
be viewed as guesses. The numbers given in the tables are motivated as
follows: All color singlet matrix elements are computed from the
wavefunctions in the Buchm\"uller-Tye potential tabulated in
\cite{EQ} and using (\ref{wave}). Similar results within $\pm30\%$
could be obtained from leptonic and hadronic decays of quarkonia for
some of the states listed in the tables. The matrix elements
$\langle {\cal O}_8^H ({}^3 S_1)\rangle$ are taken from the fits to
Tevatron data in \cite{CHO95II} with the exception of the $3S$ and $3P$
bottomonium states. In this case, we have chosen the numbers by (rather
ad hoc) extrapolation from the $1S$, $2S$ and $1P$, $2P$ states.
\begin{table}[t]
\addtolength{\arraycolsep}{0.2cm}
\renewcommand{\arraystretch}{1.2}
$$
\begin{array}{|c||c|c|c|c|c|}
\hline
\mbox{ME} & J/\psi & \psi' & \Upsilon(1S) & \Upsilon(2S) &
\Upsilon(3S) \\ \hline
\langle {\cal O}^H_1 ({}^3 S_1) \rangle &
1.16 & 0.76 & 9.28 & 4.63 & 3.54 \\
\langle {\cal O}^H_8 ({}^3 S_1) \rangle &
6.6\cdot 10^{-3} & 4.6\cdot 10^{-3} & 5.9\cdot 10^{-3} &
4.1\cdot 10^{-3} & 3.5\cdot 10^{-3} \\
\Delta_8(H)& \mbox{fitted} & \mbox{fitted} & 5.0\cdot 10^{-2} &
3.0\cdot 10^{-2} & 2.3\cdot 10^{-2} \\
\hline
\end{array}
$$
\caption{\label{tab1} Matrix elements (ME) for the direct
production of a $S$-wave
quarkonium $H$. All values in GeV${}^3$.}
\end{table}
\begin{table}[t]
\addtolength{\arraycolsep}{0.2cm}
\renewcommand{\arraystretch}{1.2}
$$
\begin{array}{|c||c|c|c|c|}
\hline
\mbox{ME} & \chi_{c0} & \chi_{b0}(1P) & \chi_{b0}(2P) &
\chi_{b0}(3P) \\ \hline
\langle {\cal O}^H_1 ({}^3 P_0)\rangle/m_Q^2 &
4.4\cdot 10^{-2} & 8.5\cdot 10^{-2} & 9.9\cdot 10^{-2} &
0.11 \\
\langle {\cal O}^H_8 ({}^3 S_1) \rangle &
3.2\cdot 10^{-3} & 0.42 & 0.32 & 0.25 \\
\hline
\end{array}
$$
\caption{\label{tab2} Matrix elements (ME) for the direct
production of a $P$-wave
quarkonium $H$. All values in GeV${}^3$.}
\end{table}
The combination of matrix elements $\Delta_8(H)$ turns out to be the
single most important parameter for direct production of $J/\psi$ and
$\psi'$. For this reason, we leave it as a parameter to be fitted and later
compared
with constraints available from Tevatron data. For bottomonia we
adopt a different strategy since $\Delta_8(H)$ is of no importance
for the total (direct plus indirect) bottomonium cross section. We therefore
fixed its value using the results of \cite{CHO95II} together with some
assumption on the relative size of $\langle {\cal O}_8^H ({}^1 S_0)
\rangle$ and $\langle {\cal O}_8^H ({}^3 P_0)\rangle$ and an ad hoc
extrapolation for the $3S$ state. Setting $\Delta_8(H)$ to zero for
bottomonia would change the cross section by a negligible amount.
\section{Results}
Figs.~\ref{prifig} to \ref{upsfig} and table \ref{tab3} summarize our
results for the charmonium and bottomonium production cross sections. We
use the CTEQ3 LO \cite{cteq} parameterization for the
parton distributions of the
protons and the GRV LO \cite{grv}
parameterization for pions. The quark masses
are fixed to be $m_c=1.5\,$GeV and $m_b=4.9\,$GeV, as was done in
\cite{CHO95II}. The strong coupling is evaluated at the scale
$\mu=2 m_Q$ ($Q=b,c$) and chosen to coincide with the value
implied by the parameterization of the parton distributions
(e.g., $\alpha_s(2 m_c)\approx 0.23$ for CTEQ3 LO). We comment on these
parameter choices in the discussion below. The experimental data
have been taken from the compilation in \cite{SCH94} with the addition
of results from \cite{AKE93} and the $800\,$GeV proton beam at Fermilab
\cite{SCH95,ALE95}. All data have been
rescaled to the nuclear dependence $A^{0.92}$ for proton-nucleon
collisions and $A^{0.87}$ for pion-nucleon collisions. All cross sections
are given for $x_F>0$ only (i.e. integrated over the forward direction
in the cms frame where most of the data has been collected).
\subsection{$\psi'$}
\begin{figure}[t]
\vspace{0cm}
\epsfysize=8cm
\epsfxsize=8cm
\centerline{\epsffile{pri.eps}}
\vspace*{0cm}
\caption{\label{prifig} Total (solid) and singlet only (dotted)
$\psi^\prime$ production cross section in proton-nucleon collisions
($x_F>0$ only). The solid line is obtained
with $\Delta_8(\psi')=5.2\cdot 10^{-3}\,$GeV${}^3$.}
\end{figure}
The total $\psi'$ production cross section in proton-nucleus
collisions is shown in
Fig.~\ref{prifig}. The color-singlet cross section is seen to be
about a factor of two below the data and the fit, including color octet
processes, is obtained with
\begin{equation}
\Delta_8(\psi') = 5.2\cdot 10^{-3}\,\mbox{GeV}^3\,.
\end{equation}
\noindent The contribution from $\langle {\cal O}^{\psi'}_8({}^3 S_1)
\rangle$ is numerically irrelevant because gluon fusion dominates
at all cms energies considered here.
The relative magnitude of singlet and octet contributions
is consistent with the naive scaling estimate $\pi/\alpha_s\cdot v^4
\approx 1$ (The color singlet cross section acquires an additional
suppression, because it vanishes close to threshold when
$4 m_c^2/(x_1 x_2 s)\to 1$).
It is important to mention that the color singlet prediction has been
expressed in terms of $2 m_c=3\,$GeV and not the physical quarkonium
mass. Choosing the quarkonium mass reduces the color singlet cross
section by a factor of three compared to Fig.~\ref{prifig}, leading to
an apparent substantial $\psi'$ deficit\footnote{This together with
a smaller value for the color singlet radial wavefunction could at
least partially explain the huge discrepancy between the CSM and the
data that was
reported in \cite{SCH95}.}. As explained in Sect.~2, choosing
quark masses is preferred but leads to large normalization uncertainties
due to the poorly known charm quark mass, which could only be partially
eliminated if the color singlet wave function were extracted
from $\psi'$ decays. If, as in open charm production, a small
charm mass were preferred, the data could be reproduced even without
a color octet contribution. Although this appears unlikely (see below),
we conclude that the total $\psi'$ cross section alone does not
provide convincing evidence for the color octet mechanism.
If we neglect the color singlet contribution altogether, we obtain
$\Delta_8(\psi') < 1.0\cdot 10^{-2}\,$GeV${}^3$. This bound is strongly
dependent on the value of $m_c$. Varying $m_c$ between $1.3\,$GeV and
$1.7\,$GeV changes the total cross section by roughly a factor of
eight at $\sqrt{s}=30\,$GeV and even more at smaller $\sqrt{s}$.
Compared to this normalization uncertainty, the variation with the
choice of parton distribution and $\alpha_s(\mu)$ is negligible. This
remark applies to all other charmonium cross sections considered in
this section.
\subsection{$J/\psi$}
\begin{figure}[t]
\vspace{0cm}
\epsfysize=8cm
\epsfxsize=8cm
\centerline{\epsffile{psi.eps}}
\vspace*{0cm}
\caption{\label{psifig} $J/\psi$ production cross sections in
proton-nucleon collisions for $x_F>0$. The dotted line is the
direct $J/\psi$ production rate
in the CSM and the dashed line includes the contribution from the
color-octet processes. The total cross section (solid line) includes radiative
feed-down from the $\chi_{cJ}$ and $\psi'$ states.
The solid line is obtained
with $\Delta_8(J/\psi)=3.0\cdot 10^{-2}\,$GeV${}^3$.}
\end{figure}
The $J/\psi$ production cross section in proton-nucleon collisions is
displayed in Fig.~\ref{psifig}. A reasonable fit is obtained for
\begin{equation}
\Delta_8(J/\psi)=3.0\cdot 10^{-2}\,\mbox{GeV}^3\,.
\end{equation}
\noindent We see that
the color octet mechanism substantially enhances the direct $J/\psi$
production cross section compared to the CSM, as shown by the dashed and
dotted lines in Fig.~\ref{psifig}. The total cross section includes
feed-down from $\chi_{cJ}$ states which is dominated by the color-singlet
gluon fusion process. As expected from the cross section in Sect.~2,
the largest indirect contribution originates from $\chi_{c2}$ states,
because $\chi_{c1}$ production is suppressed by one power of $\alpha_s$
in the gluon fusion channel. The direct $J/\psi$ production fraction
at $\sqrt{s}=23.7\,$GeV ($E=300\,$GeV) is $63\%$, in excellent agreement
with the experimental value of $62\%$ \cite{ANT93}. Note that this
agreement is not a trivial consequence of fitting the color octet
matrix element $\Delta_8(J/\psi)$ to reproduce the observed total
cross section since the indirect contribution is dominated by color
singlet mechanisms and the singlet matrix elements are fixed in terms
of the wavefunctions of \cite{EQ}.
\begin{figure}[t]
\vspace{0cm}
\epsfysize=8cm
\epsfxsize=8cm
\centerline{\epsffile{mc.eps}}
\vspace*{0cm}
\caption{\label{ratfig} Ratio of direct to total $J/\psi$ production
in proton-nucleon collisions as a function of the charm quark mass in
the CSM and after inclusion of color octet processes at $E=300\,$GeV.
The experimental value is $0.62\pm 0.04$.}
\end{figure}
One could ask whether the large sensitivity
to the charm quark mass could be exploited to raise the direct
production fraction in the CSM, thus obviating the need for octet
contributions altogether? As shown in Fig.~\ref{ratfig} this is not the
case, since the charm mass dependence cancels in the direct-to-total
production ratio. It should be mentioned, that expressing
all cross sections in terms of the respective quarkonium masses
increases $\sigma(J/\psi)_{dir}/\sigma_{J/\psi}$, because
$M_{\chi_{cJ}}>M_{J/\psi}$. However, the total color singlet
cross section then decreases further and falls short of the data
by about a factor five. We therefore consider the the
combination of total $J/\psi$ production cross section and
direct production ratio as convincing evidence for an
essential role of color octet mechanisms for direct $J/\psi$
production also at fixed target energies.
\begin{table}[b]
\addtolength{\arraycolsep}{0.05cm}
\renewcommand{\arraystretch}{1.1}
$$
\begin{array}{|c||c|c|c||c|c|c|}
\hline
& pN\,\mbox{th.} & pN\,\mbox{CSM} & pN\,\mbox{exp.} &
\pi^- N\,\mbox{th.} & \pi^- N\,\mbox{CSM} & \pi^- N\,\mbox{exp.}\\
\hline
\sigma_{J/\psi} &
90\,\mbox{nb} & 33\,\mbox{nb} & 143\pm 21\,\mbox{nb} &
98\,\mbox{nb} & 38\,\mbox{nb} & 178\pm 21\,\mbox{nb}\\
\sigma(J/\psi)_{dir}/\sigma_{J/\psi} &
0.63 & 0.21 & 0.62\pm 0.04 &
0.64 & 0.24 & 0.56\pm 0.03 \\
\sigma_{\psi'}/\sigma(J/\psi)_{dir} &
0.25 & 0.67 & 0.21\pm 0.05 &
0.25 & 0.66 & 0.23\pm 0.05 \\
\chi\mbox{-fraction} &
0.27 & 0.69 & 0.31\pm 0.04 &
0.28 & 0.66 & 0.37\pm 0.03 \\
\chi_{c1}/\chi_{c2}\,\mbox{ratio} &
0.15 & 0.08 & - &
0.13 & 0.11 & 1.4\pm 0.4 \\
\hline
\end{array}
$$
\caption{\label{tab3}
Comparison of quarkonium production
cross sections in the
color singlet model (CSM) and the NRQCD prediction
(th.) with experiment at $E=300\,$ GeV and $E=185\,$ GeV (last line
only). The `$\chi$-fraction' is defined by
$\sum_{J=1,2} \mbox{Br}(\chi_{cJ}\to J/\psi X)\,\sigma_{\chi_{cJ}}/
\sigma_{J/\psi}$. The `$\chi_{c1}/\chi_{c2}$'-ratio is defined by
$\mbox{Br}(\chi_{c1}\to J/\psi X)\,\sigma_{\chi_{c1}}/
(\mbox{Br}(\chi_{c2}\to J/\psi X)\,\sigma_{\chi_{c2}})$.}
\end{table}
The comparison of theoretical predictions with the E705 experiment
\cite{ANT93} is summarized in Tab.~\ref{tab3}. Including color octet
production yields good agreement for direct $J/\Psi$ production,
as well as the relative contributions from all $\chi_{cJ}$ states
and $\psi'$. Note that the total cross section from \cite{ANT93}
is rather large in comparison with other data (see Fig.~\ref{psifig}).
In the CSM, the direct production cross section of $7\,$nb should
be compared with the measured $89\,$nb, clearly demonstrating
the presence of an additional numerically large production mechanism.
Note also that our $\psi'$ cross section in the CSM is rather
large in comparison with the direct $J/\Psi$ cross section in
the CSM. A smaller value which compares more favorably with
the data could be obtained if one expressed the cross section
in terms of quarkonium masses \cite{VAE95}. From the point of view
presented here, this agreement appears coincidental since the cross
sections are dominated by octet production.
Perhaps the worst failure of
the theory is the $\chi_{c1}$ to $\chi_{c2}$ ratio
in the feed-down contribution that has been measured in the WA11 experiment
at $E=185\,$GeV \cite{LEM82}. We see that the prediction is far too small
even after inclusion
of color octet contributions. The low rate of $\chi_1$ production is
due to the fact, as already mentioned,
that the gluon-gluon fusion channel is
suppressed by $\alpha_s/\pi$ compared to $\chi_{c2}$
due to angular momentum constraints. Together with $J/\psi$
(and $\psi'$) polarization, discussed in Sect.~4,
the failure to reproduce this ratio
emphasizes the importance of yet other production mechanisms, presumably
of higher twist, which are naively suppressed by $\Lambda_{QCD}/m_c$
\cite{VAE95}.
\subsection{Pion-induced collisions}
\begin{figure}[t]
\vspace{0cm}
\epsfysize=8cm
\epsfxsize=8cm
\centerline{\epsffile{pripi.eps}}
\vspace*{0cm}
\caption{\label{pripifig} Total (solid) and singlet only (dotted)
$\psi^\prime$ production cross section in pion-nucleon collisions
($x_F>0$ only). The solid line is obtained
with $\Delta_8(\psi')=5.2\cdot 10^{-3}\,$GeV${}^3$.}
\end{figure}
\begin{figure}[t]
\vspace{0cm}
\epsfysize=8cm
\epsfxsize=8cm
\centerline{\epsffile{psipi.eps}}
\vspace*{0cm}
\caption{\label{psipifig} $J/\psi$ production cross sections in
pion-nucleon collisions for $x_F>0$. Direct $J/\psi$ production
in the CSM (dashed line) and after inclusion of color-octet processes
(dotted line). The total cross section (solid line) includes radiative
feed-down from the $\chi_{cJ}$ and $\psi'$ states.
The solid line is obtained
with $\Delta_8(J/\psi)=3.0\cdot 10^{-2}\,$GeV${}^3$.}
\end{figure}
The $\psi'$ and $J/\psi$ production cross section in
pion-nucleon collisions are shown in
Figs.~\ref{pripifig} and \ref{psipifig}. The discussion for
proton-induced collisions applies with little modification to the pion
case. A breakdown of contributions to the $J/\psi$ cross section
at $E=300\,$GeV is given in table~\ref{tab3}. The theoretical
prediction is based on the values of $\Delta_8(H)$ extracted from the
proton data. Including color octet contributions can add
little insight into the question
of why the pion-induced cross sections appear to be
systematically larger than expected. This issue has been
extensively discussed in \cite{SCH94}.
The discrepancy may be an indication that, either the
gluon distribution in the pion is not really understood (although
using parameterizations different from GRV LO tends to yield rather
lower theoretical predictions), or that a genuine difference in
higher twist effects for the proton and the pion exists.
\subsection{$\Upsilon(nS)$}
\begin{figure}[t]
\vspace{0cm}
\epsfysize=8cm
\epsfxsize=8cm
\centerline{\epsffile{ups.eps}}
\vspace*{0cm}
\caption{\label{upsfig} Total (direct plus indirect) $\Upsilon(nS)$
production cross sections (for $x_F>0$),
consecutively summed over $n$. The data
point refers to the sum of $n=1,2,3$. }
\end{figure}
If higher twist effects are important for fixed target charmonium
production, their importance should decrease for bottomonium production
and facilitate a test of color octet production. Unfortunately,
data for bottomonium production at fixed target energies
is sparse and does not allow us to
complete this test.
{} Due to the
increase of the quark mass, bottomonium production differs
in several ways from charmonium production, from a theoretical
standpoint. The relative
quark-antiquark velocity squared decreases by a factor of three,
thus, the color octet contributions to direct production of
$\Upsilon(nS)$ are less important
since they are suppressed by $v^4$ (at the same time $\alpha_s(2 m_Q)$
decreases much less). The situation is exactly the opposite for
the production of $P$-wave bottomonia. In this case the color singlet
and octet contributions scale equally in $v^2$. The increased quark mass,
together with an increased relative importance of the octet
matrix element $\langle {\cal O}_8^{\chi_{b0}}
({}^3 S_1)\rangle$ (extracted from Tevatron data in \cite{CHO95II})
as compared to the singlet wavefunction (compare $\chi_{c0}$ with
$\chi_{b0}$ in Tab.~\ref{tab2}),
leads to domination of quark-antiquark pair initiated processes.
Consequently, the direct
$\Upsilon(nS)$ production cross section is at least a factor ten
below the indirect contributions from $\chi_b$-decays.
This observation leads to the
conclusion that the number of $\Upsilon(3S)$ observed by the
E772 experiment \cite{ALD91} can only be explained if $\chi_{bJ}
(3P)$ states that have not yet been observed directly exist below
the open bottom threshold. Such indirect evidence has also been
obtained from bottomonium production at the Tevatron collider
\cite{PAP95}.
To obtain our numerical results shown in Fig.~\ref{upsfig}, we
assumed that these $\chi_{bJ}(3P)$ states decay into $\Upsilon(3S)$
with the same branching fractions as the corresponding $n=2$ states.
The total cross sections are compared with the experimental value
$195\pm 67\,$pb/nucleon obtained from \cite{ALE95} at $E=800\,$GeV
for the sum of $\Upsilon(nS)$, $n=1,2,3$ and show very good agreement.
The color-singlet processes alone would have led to a nine times
smaller prediction at this energy. We should note, however, that
integration of the $x_F$-distribution for $\Upsilon(1S)$ production
given in \cite{ALD91} indicates a cross section about two to three times
smaller than the central value quoted by \cite{ALE95}. The theoretical
prediction for the relative production rates of $\Upsilon(1S):
\Upsilon(2S):\Upsilon(3S)$ is $1:0.42:0.30$ to be compared with
the experimental ratio \cite{ALD91} $1:0.29:0.15$\footnote{
These numbers were taken from the raw data with no concern
regarding the differing efficiencies for the individual states.}.
This comparison
should not be over interpreted since it depends largely on the
rather uncertain octet matrix elements for $P$-wave bottomonia. Due
to lack of more data we also hesitate to use this comparison for
a new determination of these matrix elements.
\section{$\psi'$ and $J/\psi$ Polarization}
In this section, we deal with $\psi'$ and $J/\psi$ polarization at
fixed target energies and at colliders at large transverse momentum.
Before returning to fixed target production in Sect.~4.2, we
digress on large-$p_t$ production. We recall that, at large
$p_t^2\gg 4 m_Q^2$, $\psi'$ and direct $J/\psi$ production is
dominated by gluon fragmentation into color octet quark-antiquark pairs
and expected to yield transversely polarized
quarkonia \cite{WIS95}. The reason for this is that a fragmenting gluon
can be considered as on-shell and therefore transverse. Due to spin
symmetry of NRQCD, the quarkonium inherits the transverse polarization
up to corrections of order $4 m_c^2/p_T^2$ and $v^4$. Furthermore,
it has been shown \cite{BEN95} that including radiative corrections to gluon
fragmentation still leads to more than $90\%$ transversely polarized
$\psi'$ (direct $J/\psi$). Thus, polarization provides one of the
most significant tests for the color octet production
mechanism at large transverse momenta. At moderate $p_t^2\sim 4 m_c^2$,
non-fragmentation contributions proportional to
$\langle {\cal O}_8^H ({}^1 S_0)\rangle$ and
$\langle {\cal O}_8^H ({}^3 P_J)\rangle$ are sizeable \cite{CHO95II}.
Understanding their polarization yield quantitatively is very
important since most of the $p_t$-integrated data comes from the
lower $p_t$-region. The calculation of the polarization yield
has also been attempted in \cite{CHO95II}. However, the
method used is at variance with \cite{BEN95} and leads to an
incorrect result for $S$-wave quarkonia produced through
intermediate quark-antiquark pairs in a color octet $P$-wave state.
In the following subsection we expound on the method discussed in
\cite{BEN95} and hope to clarify this difference.
\subsection{Polarized production}
For arguments sake, let us consider the production of a $\psi'$ in
a polarization state $\lambda$. This state can be reached through
quark-antiquark pairs in various spin and orbital angular momentum states,
and we are led to consider the intermediate quark-antiquark pair
as a coherent superposition of these states. Because of parity
and charge conjugation symmetry, intermediate states with
different spin $S$ and angular momentum $L$ can not
interfere\footnote{Technically, this means that NRQCD matrix elements
with an odd number of derivatives or spin matrices vanish if the
quarkonium is a $C$ or $P$ eigenstate.},
so that the only non-trivial situation occurs for ${}^3 P_J$-states,
i.e. $S=1$, $L=1$.
In \cite{CHO95II} it is assumed that intermediate states with
different $J J_z$, where $J$ is total angular momentum do not
interfere, so that the production cross section can be expressed
as the sum over $J J_z$ of the amplitude squared for production
of a color octet quark-antiquark pair in a ${}^3 P_{J J_z}$ state
times the amplitude squared for its transition into the $\psi'$.
The second factor can be inferred from spin symmetry to be
a simple Clebsch-Gordon coefficient so that
\begin{equation}
\label{cho}
\sigma^{(\lambda)}_{\psi'} \sim \sum_{J J_z}
\sigma(\bar{c} c[{}^3 P^8_{J J_z}])\,|\langle J J_z|1
(J_z-\lambda);1\lambda\rangle|^2\,.
\end{equation}
\noindent We will show that this equation is incompatible with spin
symmetry which requires interference of intermediate
states with different $J$.
A simple check can be obtained by applying (\ref{cho}) to
the calculation of the gluon fragmentation function into
longitudinally polarized $\psi'$. Since the fragmentation functions
into quark-antiquark pairs in a ${}^3 P^8_{J J_z}$ state
follow from \cite{TRI95} by a change of color factor, the
sum in (\ref{cho}) can be computed. The result not only
differs from the fragmentation function obtained in \cite{BEN95}
but contains an infrared divergence which
can not be absorbed into another NRQCD matrix element.
To see the failure of (\ref{cho}) more clearly we return to the
NRQCD factorization formalism. After Fierz rearrangement of color
and spin indices as explained in \cite{BOD95}, the cross
section can be written as
\begin{equation}
\label{factor}
\sigma^{(\lambda)} \sim H_{ai;bj}\cdot S_{ai;bj}^{(\lambda)}\,.
\end{equation}
\noindent In this equation $H_{ai;bj}$ is the hard scattering cross
section, and $S_{ai;bj}$ is the soft (non-per\-tur\-ba\-tive) part that
describes the `hadronization' of the color octet quark pair into
a $\psi'$ plus light hadrons. Note that the
statement of factorization entailed in this equation occurs only on
the cross section and not on the amplitude level. The indices $ij$ and
$ab$ refer to spin and angular momentum in a Cartesian basis $L_a S_i$
($a,i=1,2,3=x,y,z$). Since spin-orbit coupling is
suppressed by $v^2$ in the NRQCD Lagrangian, $L_z$ and $S_z$ are good
quantum numbers. In the specific situation we are considering,
the soft part is simply given by (the notation follows
\cite{BOD95,BEN95})
\begin{equation}
S_{ai;bj}^{(\lambda)}=
\langle 0|\chi^\dagger\sigma_i T^A\left(-\frac{i}{2}
\stackrel{\leftrightarrow}{D}_a\right)
\psi\,{a_{\psi'}^{(\lambda)}}^\dagger
a_{\psi'}^{(\lambda)}\,\psi^\dagger\sigma_j T^A\left(-\frac{i}{2}
\stackrel{\leftrightarrow}{D}_b\right)\chi|0\rangle\,,
\end{equation}
\noindent where $a_{\psi'}^{(\lambda)}$ destroys a $\psi'$
in an out-state with
polarization $\lambda$. To evaluate this matrix element at leading order
in $v^2$, we may use spin symmetry. Spin symmetry tells us that
the spin of the $\psi'$ is aligned with the spin of the $\bar{c} c$ pair,
so $S_{ai;bj}^{(\lambda)}\propto
{\epsilon^i}^*(\lambda)\epsilon^j(\lambda)$. Now all
vectors $S^{(\lambda)}_{ai;bj}$
can depend on have been utilized, and thus by rotational invariance, only the Kronecker
symbol is left to tie up $a$ and $b$. The overall normalization is
determined by taking appropriate contractions, and we obtain
\begin{equation}
\label{decomp}
S_{ai;bj}^{(\lambda)}=
\langle {\cal O}^{\psi^\prime}_8(^3\!P_0)\rangle\,
\delta_{ab}\,{\epsilon^i}^*(\lambda)\epsilon^j(\lambda)\,.
\end{equation}
\noindent This decomposition tells us that to calculate
the polarized production
rate we should project the hard scattering amplitude onto states
with definite $S_z=\lambda$ and $L_z$, square the
amplitude, and then sum over
$L_z$ ($\sum_{L_z}\epsilon_a(L_z)
\epsilon_b(L_z)=\delta_{ab}$ in the rest frame). In other words, the
soft part is diagonal in the $L_z S_z$ basis.
It is straightforward to transform to the $J J_z$ basis. Since
$J_z=L_z+S_z$, there is no interference between intermediate states
with different $J_z$. To see this we write, in obvious
notation,
\begin{equation}
\label{factornew}
\sigma^{(\lambda)} \sim \sum_{J J_z;J' J_z^\prime}
H_{J J_z;J' J_z^\prime}\cdot S_{J J_z;J' J_z^\prime}^{(\lambda)}\,,
\end{equation}
\noindent and using (\ref{decomp}) obtain,
\begin{equation}
S_{J J_z;J' J_z^\prime}^{(\lambda)} =
\langle {\cal O}_8^{\psi'} ({}^3 P_0)\rangle
\sum_M \langle 1M;1\lambda|J J_z\rangle\langle J' J_z^\prime|1 M;1 \lambda
\rangle\,,
\end{equation}
\noindent which is diagonal in $(J J_z)(J' J_z^\prime)$ only after
summation over $\lambda$ (unpolarized production). In general, the
off-diagonal matrix elements cause interference of the following
$J J_z$ states: $00$ with $20$, $11$ with $21$ and $1(-1)$ with $2 (-1)$.
While the diagonal elements agree with (\ref{cho}), the off-diagonal
ones are missed in (\ref{cho}).
To assess the degree of transverse $\psi'$ (direct $J/\psi$)
polarization at moderate $p_t$,
the calculation of \cite{CHO95II} should be redone with the correct
angular momentum projections.
\subsection{Polarization in fixed target experiments}
Polarization measurements have been performed for both $\psi$
\cite{AKE93} and
$\psi^\prime$ \cite{HEI91} production
in pion scattering fixed target experiments.
Both experiments observe an essentially flat angular distribution in
the decay $\psi\to \mu^+ \mu^-$ ($\psi= J/\psi,\psi'$),
\begin{equation}
\frac{d\sigma}{d\cos\theta }\propto 1+ \alpha \cos^2 \theta\,,
\end{equation}
\noindent where the angle $\theta$ is defined as the angle between
the three-momentum vector of the positively charged muon and
the beam axis in the rest frame of the quarkonium. The observed values
for $\alpha$ are $0.02\pm 0.14$ for $\psi'$, measured at
$\sqrt{s}=21.8\,$GeV in the region $x_F>0.25$ and
$0.028\pm 0.004$ for $J/\psi$ measured at $\sqrt{s}=15.3\,$GeV
in the region $x_F>0$. In the CSM, the $J/\psi$'s are predicted to be
significantly transversely polarized \cite{VAE95}, in conflict with
experiment.
The polarization yield of color octet processes can be calculated
along the lines of the previous subsection. We first concentrate on
$\psi'$ production and define $\xi$ as the fraction of longitudinally
polarized $\psi'$. It is related to $\alpha$ by
\begin{equation}
\alpha=\frac{1-3\xi}{1+\xi}\,.
\end{equation}
\noindent For the different intermediate quark-antiquark
states we find the following ratios of longitudinal to transverse
quarkonia:
\begin{equation}
\addtolength{\arraycolsep}{0.3cm}
\begin{array}{ccc}
{}^3 S_1^{(1)} & 1:3.35 & \xi=0.23\\
{}^1 S_0^{(8)} & 1:2 & \xi=1/3\\
{}^3 P_J^{(8)} & 1:6 & \xi=1/7\\
{}^3 S_1^{(8)} & 0:1 & \xi=0
\end{array}
\end{equation}
\noindent where the number for the singlet process (first line) has been
taken from \cite{VAE95}\footnote{This number is $x_F$-dependent and we
have approximated it
by a constant at low $x_F$, where the bulk data is obtained from.
The polarization fractions for the octet $2\to 2$ parton processes
are $x_F$-independent.}. Let us add the following remarks:
(i) The ${}^3 S_1^{(8)}$-subprocess yields pure transverse
polarization. Its contribution to the total polarization is not
large, because gluon-gluon fusion dominates the total rate.
(ii) For the ${}^3 P_J^{(8)}$-subprocess $J$ is not specified,
because interference between intermediate states with different
$J$ could occur as discussed in the previous subsection. As it
turns out, interference does in fact not occur at leading order
in $\alpha_s$, because the only non-vanishing short-distance
amplitudes in the $J J_z$ basis are $00$, $22$ and
$2(-2)$, which do not
interfere.
(iii) The ${}^1 S_0^{(8)}$-subprocess yields unpolarized quarkonia.
This follows from the fact that the NRQCD matrix element is
\begin{equation}
\label{above}
\langle 0|\chi^\dagger T^A{a_{\psi'}^{(\lambda)}}^\dagger
a_{\psi'}^{(\lambda)}\,\psi^\dagger T^A\chi|0\rangle =\frac{1}{3}\,
\langle {\cal O}_8^{\psi'}({}^1 S_0)\rangle\,,
\end{equation}
\noindent independent of the helicity state $\lambda$. At this point,
we differ from \cite{TAN95}, who assume that this channel results
in pure transverse polarization, because the gluon in the
chromomagnetic dipole transition ${}^1 S_0^{(8)}\to {}^3 S_1^{(8)}+g$
is assumed to be transverse. However, one should keep in mind that
the soft gluon is off-shell and interacts with other partons with unit
probability prior to hadronization. The NRQCD formalism
applies only to inclusive quarkonium production. Eq.~(\ref{above})
then follows from rotational invariance.
(iv) Since the ${}^3 P_J^{(8)}$ and ${}^1 S_0^{(8)}$-subprocesses
give different longitudinal polarization fractions, the $\psi'$
polarization depends on a combination of the matrix elements
$\langle {\cal O}_8^{\psi'} ({}^1 S_0)\rangle$ and
$\langle {\cal O}_8^{\psi'} ({}^3 P_0)\rangle$ which is different
from $\Delta_8(\psi')$.
To obtain the total polarization the various subprocesses have to be
weighted by their partial cross sections. We define
\begin{equation}
\delta_8(H)=\frac{\langle {\cal O}_8^{H} ({}^1 S_0)\rangle}
{\Delta_8(H)}
\end{equation}
\noindent and obtain
\begin{eqnarray}
\xi &=& 0.23\,\frac{\sigma_{\psi'}({}^3 S_1^{(1)})}{\sigma_{\psi'}} +
\left[\frac{1}{3}\delta_8(\psi')+\frac{1}{7} (1-\delta_8(\psi'))\right]
\frac{\sigma_{\psi'}({}^1 S_0^{(8)}+{}^3 P_J^{(8)})}{\sigma_{\psi'}}
\nonumber\\
&=& 0.16+0.11\,\delta_8(\psi')\,,
\end{eqnarray}
\noindent where the last line holds at $\sqrt{s}=21.8\,$GeV (The
energy dependence is mild and the above formula can be used with
little error even at $\sqrt{s}=40\,$GeV). Since $0<\delta_8(H)<1$, we
have $0.16<\xi<0.27$ and therefore
\begin{equation}
0.15 < \alpha < 0.44\,.
\end{equation}
\noindent In quoting this range we do not attempt an estimate of
$\delta_8(\psi')$. Note that taking the Tevatron and fixed target
extractions of certain (and different) combinations of
$\langle {\cal O}_8^{\psi'} ({}^1 S_0)\rangle$ and
$\langle {\cal O}_8^{\psi'} ({}^3 P_0)\rangle$ seriously
(see Sect.~5.1), a large value of $\delta_8(\psi')$
and therefore low $\alpha$ would be favored. Within large errors,
such a scenario could be considered consistent with the
measurement quoted earlier. From a theoretical point of view, however,
the numerical violation of velocity counting rules implied by
this scenario would be rather disturbing.
In contrast, the more accurate measurement of polarization for
$J/\psi$ leads to a clear discrepancy with theory. In this case, we
have to incorporate the polarization inherited from decays of
the higher charmonium states $\chi_{cJ}$ and $\psi'$. This task
is simplified by observing that the
contribution from $\chi_{c0}$ and $\chi_{c1}$
feed-down is (theoretically) small as is the octet contribution
to the $\chi_{c2}$ production cross section. On the other hand, the
gluon-gluon fusion process produces $\chi_{c2}$ states only in
a helicity $\pm 2$ level, so that the $J/\psi$ in the subsequent
radiative decay is completely transversely polarized.
Weighting all subprocesses by their partial cross section
and neglecting the small $\psi'$ feed-down, we arrive at
\begin{equation}
\xi = 0.10 + 0.11\,\delta_8(J/\psi)
\end{equation}
\noindent at $\sqrt{s}=15.3\,$GeV, again with mild energy dependence.
This translates into sizeable transverse polarization
\begin{equation}
0.31 < \alpha < 0.63\,.
\end{equation}
\noindent The discrepancy with data could be ameliorated if the observed
number of $\chi_{c1}$ from
feed-down were used instead of the theoretical value. However,
we do not know the polarization yield of whatever mechanism is
responsible for copious $\chi_{c1}$ production.
Thus, color octet mechanisms do not help to solve the
polarization problem and
one has to invoke a significant higher-twist contribution as
discussed in \cite{VAE95}. To our knowledge, no specific mechanism
has yet been proposed that would yield predominantly longitudinally
polarized $\psi'$ and $J/\psi$ in the low $x_F$ region which dominates
the total production cross section. One might speculate that both
the low $\chi_{c1}/\chi_{c2}$ ratio and the large transverse polarization
follow from the assumption of transverse gluons in the gluon-gluon fusion
process, as inherent to the leading-twist approximation. If gluons
in the proton and pion have
large intrinsic transverse momentum, as suggested by the
$p_t$-spectrum in open charm
production, one would be naturally led to higher-twist effects that
obviate the helicity constraint on on-shell gluons.
\section{Other processes}
Direct $J/\psi$ and $\psi'$ production is sensitive to the color octet
matrix element $\Delta_8(H)$ defined in (\ref{delta}). In this section we
compare our extraction of $\Delta_8(H)$ with constraints from
quarkonium production at the Tevatron and in photo-production at fixed
target experiments and HERA.
\subsection{Quarkonium production at large $p_t$}
An extensive analysis of charmonium production data at $p_t>5\,$GeV
has been carried out by Cho and Leibovich \cite{CHO95,CHO95II}, who
relaxed the fragmentation approximation employed earlier
\cite{BRA95,CAC95}. At the lower $p_t$ boundary, the theoretical
prediction is dominated by the ${}^1 S_0^{(8)}$ and ${}^3 P_J^{(8)}$
subprocesses and the fit yields
\begin{eqnarray}
\label{tevme}
\langle {\cal O}^{J/\psi}_8 ({}^1 S_0) \rangle +
\frac{3}{m_c^2}\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle\,
= 6.6\cdot 10^{-2}\nonumber\\
\langle {\cal O}^{\psi'}_8 ({}^1 S_0) \rangle +
\frac{3}{m_c^2}\langle {\cal O}^{\psi'}_8 ({}^3 P_0) \rangle\,
= 1.8\cdot 10^{-2}\,,
\end{eqnarray}
\noindent to be compared with the fixed target values\footnote{
Since there is a strong correlation between the charm quark mass
and the extracted NRQCD matrix elements, we emphasize that both
(\ref{tevme}) and (\ref{fixme}) as well as (\ref{photome}) below
have been obtained with the same $m_c=1.5\,$GeV (or $m_c=1.48\,$ GeV,
to be precise). On the other hand, the apparent agreement of
predictions for fixed target experiments with data claimed in
\cite{GUP96} is obtained from (\ref{photome}) in conjunction with
$m_c=1.7\,$GeV.}
\begin{eqnarray}\label{fixme}
\langle {\cal O}^{J/\psi}_8 ({}^1 S_0) \rangle +
\frac{7}{m_c^2}\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle\,
= 3.0\cdot 10^{-2}\nonumber\\
\langle {\cal O}^{\psi'}_8 ({}^1 S_0) \rangle +
\frac{7}{m_c^2}\langle {\cal O}^{\psi'}_8 ({}^3 P_0) \rangle\,
= 0.5\cdot 10^{-2}\,.
\end{eqnarray}
\noindent If we assume $\langle {\cal O}^{J/\psi}_8 ({}^1 S_0) \rangle =
\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle/m_c^2$, the
fixed target values are a factor seven (four) smaller than the
Tevatron values for $J/\psi$ ($\psi'$). The discrepancy would be lower
for the radical choice $\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle=0$.
While this comparison looks like a flagrant violation of the
supposed process-independence of NRQCD production matrix
elements, there are at least two possibilities that could lead to
systematic differences:
(i) The $2\to 2$ color octet parton processes are
schematically of the form
\begin{equation}
\frac{\langle {\cal O}\rangle}{2 m_c}\,\frac{1}{M_f^2}\,
\delta(x_1 x_2 s-M_f^2)\,,
\end{equation}
\noindent where $M_f$ denotes the final state invariant mass. To leading
order in $v^2$, we have $M_f=2 m_c$. Note, however, that this is
physically unrealistic. Since color must be emitted from the quark pair
in the octet state and neutralized by final-state interactions, the
final state is a quarkonium accompanied by light hadrons with
invariant mass squared of order $M_f^2\approx (M_H+M_H v^2)^2$
since the soft gluon emission carries an energy of order $M_H v^2$,
where $M_H$ is the quarkonium mass. The kinematic effect of this
difference in invariant mass is very large since the gluon
distribution rises steeply at small $x$ and reduces the
cross section by at least a factor two.
The `true' matrix elements would therefore be larger
than those extracted from fixed target experiments at leading order
in NRQCD. Since the $\psi'$ is heavier than the $J/\psi$, the effect
is more pronounced for $\psi'$, consistent with the larger
disagreement with the Tevatron extraction for $\psi'$. Note that
the effect is absent for large-$p_t$ production, since in this
case, $x_1 x_2 s > 4 p_t^2 \gg M_f^2$. If we write $M_f=2 m_c+{\cal O}
(v^2)$, then the difference between fixed target and large-$p_t$ production
stems from different behaviors of the velocity expansion in the two cases.
(ii) It is known that small-$x$ effects increase the
open bottom production cross section at the Tevatron as compared
to collisions at lower $\sqrt{s}$. Since even at large $p_t$,
the typical $x$ is smaller
at the Tevatron than in fixed target experiments, this effect would
enhance the Tevatron prediction more than the fixed target prediction.
The `true' matrix elements would therefore be smaller than
those extracted from the Tevatron in \cite{CHO95II}.
While a combination of both effects could well account for the
apparently different NRQCD matrix elements, one must keep in mind
that we have reason to suspect important higher twist effects
for charmonium production at fixed target energies. Theoretical
predictions for fixed target production are intrinsically less
accurate than at large $p_t$, where higher-twist contributions
due to the initial hadrons are expected to be suppressed by
$\Lambda_{QCD}/p_t$ (if not $\Lambda_{QCD}^2/p_t^2$)
rather than $\Lambda_{QCD}/m_c$.
\subsection{Photo-production}
A comparison of photo-production with fixed target production is
more direct since the same combination of NRQCD matrix elements
is probed and the kinematics is similar. All analyses
\cite{CAC96,AMU96,KO96} find a substantial overestimate of the
cross section if the octet matrix elements of (\ref{tevme}) are used.
The authors of \cite{AMU96} fit
\begin{equation}\label{photome}
\langle {\cal O}^{J/\psi}_8 ({}^1 S_0) \rangle +
\frac{7}{m_c^2}\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle\,
= 2.0\cdot 10^{-2}\,,
\end{equation}
\noindent consistent with (\ref{fixme}) within errors, which we have not
specified. While this agreement is
reassuring, it might also be partly accidental since the extraction
of \cite{AMU96} is performed on the elastic peak, which is not
described by NRQCD. Color octet mechanisms do not
leave a clear signature in the total inelastic photo-production cross
section. The authors of \cite{CAC96} argue that the color-octet
contributions to the energy spectrum of $J/\psi$ are in conflict
with the observed energy dependence in the endpoint region
$z>0.7$, where $z=E_{J/\psi}/E_\gamma$ in the proton rest frame.
This discrepancy would largely disappear if the smaller matrix
element of (\ref{fixme}) or (\ref{photome}) were used rather than
(\ref{tevme}). Furthermore, since in a color octet process soft
gluons with energy $M_H v^2$ must be emitted, but are kinematically
not accounted for, the NRQCD-prediction for the energy distribution
should be smeared over an interval of size $\delta z\sim v^2\sim 0.3$,
making the steep rise of the energy distribution close to $z=1$ is
not necessarily physical.
\section{Conclusion}
We have reanalyzed charmonium production data from fixed target
experiments, including color octet production mechanisms.
Our conclusion is twofold: On one hand, the inclusion of
color octet processes allows us to reproduce the overall normalization
of the total production cross section with color octet matrix
elements of the expected size (if not somewhat smaller)
without having to invoke small values of the charm quark mass.
This was found to be true for bottomonium as well as for charmonium.
Comparing the theoretical
predictions within this framework with the data
implies the existence
of additional bottomonium states below threshold which have not yet
been seen directly.
On the other hand, the present picture of charmonium production at
fixed target energies is far from perfect. The $\chi_{c1}/
\chi_{c2}$ production ratio remains almost an order of magnitude
too low, and the transverse polarization fraction
of the $J/\psi$ and $\psi'$ is too large. We thus confirm the expectation of
\cite{VAE95}
that higher twist effects must be substantial even after including
the octet mechanism.
The uncertainties in the theoretical prediction at fixed target
energies are substantial and preclude a straightforward test
of universality of color octet matrix elements by comparison with
quarkonium production at large transverse momentum. We have
argued that small-$x$, as well as kinematic effects, could bias
the extraction of these matrix elements in different directions
at fixed target and collider energies. The large uncertainties
involved, especially due to the charm quark mass, could hardly
be eliminated by
a laborious calculation of $\alpha_s$-corrections to the
production processes considered here.
To more firmly establish existence of
the octet mechanism there are several experimental
measurements which need to be performed. Data on polarization is presently
only available for charmonium production in pion-induced collisions.
A measurement of polarization at large transverse momentum or
for bottomonium is of crucial importance,
because higher twist effects should be suppressed.
Furthermore, a measurement of direct
and indirect production fractions in the bottom system would provide further
confirmation of the color octet picture and constrain the color octet
matrix elements for bottomonium.
\vspace*{1cm}
\noindent {\bf Acknowledgments.} We thank S.J.~Brodsky,
E.~Quack and V.~Sharma for discussions. IZR acknowledges support from the
DOE grant DE-FG03-90ER40546 and the NSF grant PHY-8958081.
\newpage
| 21,306 |
\section{Introduction}
Hydrodynamics is thought to play a key role in the formation of the visible
structures in the universe, such as bright galaxies and hot intracluster
gas. For this reason there is a great deal of interest in incorporating
hydrodynamical effects into cosmological structure formation simulations in
order to make direct, quantitative comparisons of such simulations to
observed data. In addition to gravitation, a cosmological hydrodynamical
simulation must minimally account for pressure support, shock physics, and
radiative cooling, as these are the fundamental physical processes thought
to play a dominant role in the formation of large, bright galaxies (White
\& Rees 1978). There is already a bewildering array of such studies
published, including Cen \& Ostriker (1992a,b), Katz, Hernquist, \&
Weinberg (1992), Evrard, Summers, \& Davis (1994), Navarro \& White (1994),
and Steinmetz \& M\"{u}ller (1994), to name merely a few. In order to
appreciate the implications of such ambitious studies, it is important that
we fully understand both the physical effects of hydrodynamics under a
cosmological framework and the numerical aspects of the tools used
for such investigations. Basic questions such as how the baryon to dark
matter ratio varies in differing structures (galaxies, clusters, and
filaments) and exactly how this is affected by physical processes such as
shock heating, pressure support, or radiative cooling remain unclear. It
is also difficult to separate real physical effects from numerical
artifacts, particularly given the current limitations on the resolution
which can be achieved. For example, in a recent study of X-ray clusters
Anninos \& Norman (1996) find the observable characteristics of a simulated
cluster to be quite resolution dependent, with the integrated X-ray
luminosity varying as $L_x \propto (\Delta x)^{-1.17}$, core radius $r_c
\propto (\Delta x)^{0.6}$, and emission weighted temperature $T_X \propto
(\Delta x)^{0.35}$ (where $\Delta x$ is the gridcell size of the
simulation). In a study of the effects of photoionization on galaxy
formation, Weinberg, Hernquist, \& Katz (1996) find that the complex
interaction of numerical effects (such as resolution) with microphysical
effects (such as radiative cooling and photoionization heating) strongly
influences their resulting model galaxy population. In this paper we focus
on separating physical from numerical effects in a series of idealized
cosmological hydrodynamical simulations. This study is intended to be an
exploratory survey of hydrodynamical cosmology, similar in spirit to the
purely gravitational studies of Melott \& Shandarin (1990), Beacom {\frenchspacing et al.}\
(1991), and Little, Weinberg, \& Park (1991).
We will examine the effects of pressure support and shock heating in a
mixed baryonic/dark matter fluid undergoing gravitationally driven
hierarchical collapse. This problem is approached with two broad questions
in mind: how stable and reliable is the numerical representation of the
system, and what can we learn about the physics of such collapses? These
questions have been investigated for purely gravitational systems in
studies such as those mentioned above. In those studies numerically it is
found that the distribution of collisionless matter converges to consistent
states so long as the nonlinear collapse scale is resolved. Such
convergence has not been demonstrated for collisional systems, however. It
is not clear that hydrodynamical simulations will demonstrate such
convergence in general, nor if they do that the nonlinear scale is the
crucial scale which must be resolved. Hydrodynamical processes are
dominated by localized interactions on small scales, allowing the smallest
scales to substantially affect the state of the baryonic gas. As an
example, consider a collisional fluid undergoing collapse. Presumably such
a system will undergo shocking near the point of maximal collapse, allowing
a large fraction of the kinetic energy of the gas to be converted to
thermal energy. In a simple case such as a single plane-wave perturbation
(the Zel'dovich pancake collapse), the obvious scale which must be resolved
is the scale of the shock which forms around the caustic. However, in a
hierarchical structure formation scenario there is a hierarchy of collapse
scales, and for any given resolution limit there is always a smaller scale
which will undergo nonlinear collapse. The subsequent evolution of the gas
could well depend upon how well such small scale interactions are resolved,
and changes in the density and temperature of gas on small scales could in
turn influence how it behaves on larger scales (especially if cooling is
important).
In this paper we examine a series of idealized experiments, evolving a
mixed fluid of baryons and collisionless dark matter (dark matter dominated
by mass), coupled gravitationally in a flat, Einstein-de Sitter cosmology.
The mass is seeded with Gaussian distributed initial density perturbations
with a power-law initial power spectrum. We perform a number of
simulations, varying the resolution, the initial cutoff in the density
perturbation spectrum, and the minimum allowed temperature for the baryons.
Enforcing a minimum temperature for the baryons implies there will be a
minimal level of pressure support, and therefore a minimum collapse scale
(the Jeans mass), below which the baryons are pressure supported against
collapse. From the numerical point of view, performing a number of
simulations with identical initial physical conditions but varying
resolution allows us to unambiguously identify resolution effects. By
enforcing a Jeans mass for the baryons we introduce an intrinsic mass scale
to the problem, which may or may not be resolved in any individual
experiment. The hope is that even if the gas dynamical results do not
converge with increasing resolution in the most general case, the system
will converge if the fundamental Jeans mass is resolved.
The effects of the presence (or absence) of a baryonic Jeans mass also
raises interesting physical questions. Though we simply impose arbitrary
minima for the baryon temperatures here, processes such as photoionization
enforce minimum temperatures in the real universe by injecting thermal
energy into intergalactic gas. The Gunn-Peterson test indicates that the
intergalactic medium is highly ionized (and therefore at temperatures $T
\gtrsim 10^4$K) out to at least $z \lesssim 5$. Shapiro, Giroux, \& Babul
(1994) discuss these issues for the intergalactic medium. The dark matter,
however, is not directly influenced by this minimal pressure support in the
baryons, and therefore is capable of collapsing on arbitrarily small
scales. Pressure support provides a mechanism to separate the two species,
and since the dark matter dominates the mass density it can create
substantial gravitational perturbations on scales below the Jeans mass.
While there are many studies of specific cosmological models with detailed
microphysical assumptions, the general problem of the evolution of pressure
supported baryons in the presence of nonlinear dark matter starting from
Gaussian initial conditions has not been investigated in a systematic
fashion.
This paper is organized as follows. In \S \ref{Sim.sec} we discuss the
particulars of how the simulations are constructed and performed. In \S
\ref{Numresults.sec} we characterize the numerical effects we find in these
simulations, and in \S \ref{Physresults.sec} we discuss our findings about
the physics of this problem. Finally, \S \ref{disc.sec} summarizes the
major results of this investigation.
\section{The Simulations}
\label{Sim.sec}
A survey such as this optimally requires a variety of simulations in order
to adequately explore the range of possible resolutions and input physics.
Unfortunately, hydrodynamical cosmological simulations are generally quite
computationally expensive, and therefore in order to run a sufficiently
broad number of experiments we restrict this study to 2-D simulations.
There are two primary advantages to working in 2-D rather than 3-D. First,
parameter space can be more thoroughly explored, since the computational
cost per simulation is greatly reduced and a larger number of simulations
can be performed. Second, working in 2-D enables us to perform much higher
resolution simulations than are feasible in 3-D. While the real universe
is 3-D and we must therefore be cautious about making specific quantitative
predictions based on this work, 2-D experiments can be used to yield
valuable qualitative insights into the behaviour of these systems. For
similar reasons Melott \& Shandarin (1990) and Beacom {\frenchspacing et al.}\ (1991) also
utilize 2-D simulations in their studies of purely gravitational dynamics.
The 2-D simulations presented here can be interpreted as a slice through an
infinite 3-D simulation (periodic in $(x,y)$ and infinite in $z$). The
particles interact as parallel rods of infinite length, obeying a
gravitational force law of the form $\Sub{F}{grav} \propto 1/r$. The
numerical technique used for all simulations is SPH (Smoothed Particle
Hydrodynamics) for the hydrodynamics and PM (Particle-Mesh) for the
gravitation. The code and technique are described and tested in Owen
{\frenchspacing et al.}\ (1996), so we will not go into much detail here. We do note,
however, that while our code implements ASPH (Adaptive Smoothed Particle
Hydrodynamics) as described in our initial methods paper, we are not using
the tensor smoothing kernel of ASPH for this investigation, but rather
simple SPH. The results should be insensitive to such subtle technique
choices since the goal is to compare simulation to simulation, so we employ
simple SPH in order to separate our findings from questions of technique.
All simulations are performed under a flat, Einstein-de Sitter cosmology,
with 10\% baryons by mass ($\Sub{\Omega}{bary}=0.1, \Sub{\Omega}{dm}=0.9,
\Lambda=0$). Thus the mass density is dominated by the collisionless dark
matter, which is linked gravitationally to the collisional baryons. The
baryon and dark matter particles are initialized on the same perturbed
grid, with equal numbers of both species. Therefore, initially all baryons
exactly overlie the dark matter particles, and only hydrodynamical effects
can separate the two species. The baryon/dark matter mass ratio is set by
varying the particle mass associated with each species. The initial density
perturbation spectrum is taken to be a power-law $P(k) = \langle |\delta
\rho(k)/ \bar{\rho}|^2 \rangle \propto k^n$ up to a cutoff frequency $k_c$.
Note that since these are 2-D simulations, for integrals over the power
spectrum this is equivalent in the 3-D to a power spectrum of index $n -
1$. In this paper we adopt a ``flat'' ($n = 0$) 2-D spectrum
\begin{equation}
\Sub{P}{2-D}(k) = \Sub{A}{norm} \left\{ \begin{array}{l@{\quad : \quad}l}
k^0 & k \le k_c \quad \Rightarrow \quad
\Sub{P}{3-D}(k) = k^{-1} \\
0 & k > k_c, \end{array} \right.
\end{equation}
where $\Sub{A}{norm}$ normalizes the power-spectrum. Note that using a
flat cosmology and power-law initial conditions implies these simulations
are scale-free, and should evolve self-similarly in time. We can choose to
assign specific scales to the simulations in order to convert the
scale-free quantities to physical units. All simulations are halted after
60 expansion factors, at which point the nonlinear scale (the scale on
which $\delta \rho/\rho \sim 1$) is roughly 1/8 of the box size.
The Jeans length is the scale at which pressure support makes the gas
stable against the growth of linear fluctuations due to self-gravitation --
the Jeans mass is the amount of mass contained within a sphere of diameter
the Jeans length. The Jeans length $\lambda_J$ and mass $M_J$ are defined
by the well known formula (Binney \& Tremaine 1987)
\begin{equation}
\label{LJ.eq}
\lambda_J = \left( \frac{\pi c_s^2}{G \rho} \right)^{1/2},
\end{equation}
\begin{equation}
\label{MJ.eq}
M_J = \frac{4 \pi}{3} \rho \left( \frac{1}{2} \lambda_J \right)^3
= \frac{\pi \rho}{6} \left( \frac{\pi c_s^2}{G \rho} \right)^{3/2},
\end{equation}
where $\rho$ is the mass density and $c_s$ the sound speed. The baryons
are treated as an ideal gas obeying an equation of state of the form $P =
(\gamma - 1) u \rho$, where $P$ is the pressure and $u$ is the specific
thermal energy. Enforcing a minimum specific thermal energy (and therefore
temperature) in the gas forces a minimum in the sound speed $c_s^2 = \gamma
P/\rho = \gamma (\gamma - 1) u$, which therefore implies we have a minimum
Jeans mass through equation (\ref{MJ.eq}). Note that $\rho$ is the total
mass density (baryons and dark matter), since it is the total gravitating
mass which counts, and therefore $M_J$ as expressed in equation
(\ref{MJ.eq}) represents the total mass contained within $r \le
\lambda_J/2$. If we want the total baryon mass contained within this
radius, we must multiply $M_J$ by $\Sub{\Omega}{bary}/\Omega$.
It is also important to understand how the mass resolution is set for the
baryons by the SPH technique. This is not simply given by the baryon
particle mass, since SPH interpolation is a smoothing process typically
extending over spatial scales of several interparticle spacings. In
general the mass resolution for the hydrodynamic calculations can be
estimated as the amount of mass enclosed by a typical SPH interpolation
volume. If the SPH smoothing scale is given by $h$ and the SPH sampling
extends for $\eta$ smoothing scales, then the mass resolution $M_R$ is
given by
\begin{equation}
\label{MR.eq}
M_R = \frac{4}{3} \pi (\eta h)^3 \rho.
\end{equation}
This is probably something of an overestimate, since the weight for each
radial shell in this interpolation volume (given by the SPH sampling kernel
$W$) falls off smoothly towards $r = \eta h$, but given the other
uncertainties in this quantity equation (\ref{MR.eq}) seems a reasonable
estimate. Note that the resolution limit for the SPH formalism is best
expressed in terms of a mass limit, appropriate for SPH's Lagrangian
nature. For this reason we choose to express the Jeans limit in terms of
the Jeans mass (eq. [\ref{MJ.eq}]) throughout this work, as the Jeans limit
can be equally expressed in terms of a spatial or a mass scale. In N-body
work it is common to express the mass resolution of an experiment in units
of numbers of particles. In our simulations we use a bi-cubic spline
kernel which extends to $\eta = 2$ smoothing lengths, and initialize the
smoothing scales such that the smoothing scale $h$ extends for two particle
spacings. We therefore have a mass resolution in 2-D of roughly 50
particles, or equivalently in 3-D roughly 260 particles.
We perform simulations both with and without a minimum temperature (giving
Jeans masses $M_J=0$, $M_J>0$), at three different resolutions ($N =
\Sub{N}{bary} = \Sub{N}{dm} = 64^2$, $128^2$, and $256^2$), and for three
different cutoffs in the initial perturbation spectrum ($k_c = 32$, 64, and
128). The initial density perturbations are initialized as Gaussian
distributed with random phases and amplitudes, but in such a manner that
all simulations have identical phases and amplitudes up to the imposed
cutoff frequency $k_c$. The cutoff frequencies are the subset of $k_c \in
(32,64,128)$ up to the Nyquist frequency for each resolution $\Sub{k}{Nyq}
= N^{1/2}/2$, so for each resolution we have $k_c(N=64^2) = 32$,
$k_c(N=128^2) \in (32,64)$, and $k_c(N=256^2) \in (32,64,128)$. For each
value of the minimum temperature we therefore have a grid of simulations
which either have the same input physics at differing resolutions ({\em \frenchspacing i.e.},
$k_c=32$ for $N \in [64^2, 128^2, 256^2]$), or varying input physics at
fixed resolution ({\em \frenchspacing i.e.}, $N=256^2$ for $k_c \in [32, 64, 128]$). This allows
us to isolate and study both numerical and physical effects during the
evolution of these simulations. In total we discuss twelve simulations.
For the simulations with a minimum temperature, there is an ambiguity in
assigning a global Jeans mass with that temperature. The density in
equation (\ref{MJ.eq}) is formally the {\em local} mass density, and
therefore the Jeans mass is in fact position dependent through
$\rho(\vec{r})$. Throughout this work we will refer to the Jeans mass at
any given expansion factor as the Jeans mass defined using a fixed minimum
temperature and the average background density, making this mass scale a
function of time only. This is equivalent to taking the zeroth order
estimate of $M_J$, giving us a well defined characteristic mass scale. In
terms of this background density, Figure \ref{MJ.fig} shows the baryon
Jeans mass (in units of the resolved mass via equation [\ref{MR.eq}]) as a
function of expansion. Note that for a given simulation $M_R$ remains
fixed, and it is the Jeans mass which grows as $M_J \propto \rho^{-1/2}
\propto a^{3/2}$. It is apparent that the $N=256^2$ simulations resolve
the Jeans mass throughout most of the evolution, the $N=128^2$ simulations
resolve $M_J$ by $a/a_i \sim 15$, and the $N=64^2$ simulation does not
approach $M_J/M_R \sim 1$ until the end of our simulations at $a/a_i \sim
60$. The specific value of $T_{min}$ used in this investigation is chosen
to yield this behaviour. We discuss physically motivated values for this
minimum temperature in \S \ref{disc.sec}.
\section{Numerical Resolution and the Jeans Mass}
\label{Numresults.sec}
\subsection{Dark Matter}
We will begin by examining the dark matter distribution, as this is a
problem which has been examined previously. Figures \ref{DMRhoMaps.fig}a
and b show images of the dark matter overdensity
$\Sub{\rho}{dm}/\Sub{\bar{\rho}}{dm}$ for the $M_J=0$ simulations. In
order to fairly compare with equivalent images of the SPH baryon densities,
the dark matter information is generated by assigning a pseudo-SPH
smoothing scale to each dark matter particle, such that it samples roughly
the same number of neighboring dark matter particles as the SPH smoothing
scale samples in the baryons. We then use the normal SPH summation method
to assign dark matter densities, which are used to generate these images.
The panels in the figure are arranged with increasing simulation resolution
$N$ along rows, and increasing cutoff frequency $k_c$ down columns. The
diagonal panels represent each resolution initialized at its Nyquist
frequency for $P(k)$. Note that the physics of the problem is constant
along rows, and numerics is constant along columns. If resolution were
unimportant, the results along rows should be identical. Likewise, since
the numerics is held constant along columns, only physical effects can
alter the results in this direction.
Comparing the dark matter densities along the rows of Figure
\ref{DMRhoMaps.fig}a, it is clear that the structure becomes progressively
more clearly defined as the resolution increases. This is to be expected,
since the higher resolution simulations can resolve progressively more
collapsed/higher density structures. The question is whether or not the
underlying particle distribution is systematically changing with
resolution. In other words, do the simulations converge to the same
particle distribution on the scales which are resolved? Figure
\ref{DMRhoMaps.fig}b shows this same set of dark matter overdensities for
the $M_J=0$ simulations, only this time each simulation is degraded to an
equivalent $N=64^2$ resolution and resampled. This is accomplished by
selecting every $n$th node from the higher resolution simulations, throwing
away the rest and suitably modifying the masses and smoothing scales of the
selected particles. Note that now the dark matter distributions look
indistinguishable for the different resolution experiments, at least
qualitatively. This similarity implies that the high frequency small scale
structure has minimal effect on the larger scales resolved in this figure.
Looking down the columns of Figure \ref{DMRhoMaps.fig}a it is clear that
increasing $k_c$ does in fact alter the dark matter particle distribution,
such that the large scale, smooth filaments are progressively broken up
into smaller clumps aligned with the overall filamentary structure. These
differences are lost in the low-res results of Figure \ref{DMRhoMaps.fig}b,
implying that these subtle changes do not significantly affect the large
scale distribution of the dark matter.
In Figures \ref{DMRhoDist.fig}a and b we show the mass distribution
functions for the dark matter overdensity
$f(\Sub{\rho}{dm}/\Sub{\bar{\rho}}{dm})$. Figure
\ref{DMRhoDist.fig}a includes all particles from each simulation (as in
Figure \ref{DMRhoMaps.fig}a), while Figure \ref{DMRhoDist.fig}b is
calculated for each simulation degraded to equivalent $N=64^2$ resolutions
(comparable to Figure \ref{DMRhoMaps.fig}b). The panels are arranged as in
Figure \ref{DMRhoMaps.fig}, with $M_J=0$ and $M_J>0$ overplotted as
different line types. It is clear that the varying Jeans mass in the
baryons has negligible effect on the dark matter, a point we will return to
in \S \ref{Physresults.sec}. The full resolution results of Figure
\ref{DMRhoDist.fig}a show a clear trend for a larger fraction of the mass
to lie at higher densities with increasing resolution. There is also a
similar though weaker trend with increasing $k_c$. However, examining the
resampled results of Figure \ref{DMRhoDist.fig}b it appears that the
results of all simulations converge, bearing out the visual impressions of
Figures \ref{DMRhoMaps.fig}a and b. For the dark matter, with increasing
resolution more information is gained about the highest density/most
collapsed fraction of the mass, but so long as the pertinent nonlinear
scales are resolved the results converge. The underlying particle
distribution does not depend upon the numerical resolution, similarly to
the results discussed in Little {\frenchspacing et al.}\ (1991).
\subsection{Baryons}
We now turn our attention to the baryon distribution. Figures
\ref{BaryRhoMaps.fig}a, b, c, and d show images of the baryon overdensity for
$M_J=0$ and $M_J>0$ at expansions $a/a_i=30$ and $a/a_i=60$. There is a
pronounced trend for the collapsed filaments and clumps to become
progressively more strongly defined as the simulation resolution improves
-- even more so than we see in the dark matter. The tendency to break up
filaments into small scale clumps with increasing $k_c$ is also clearly
evident for the $M_J=0$ case. Additionally, the presence of a nonzero
Jeans mass visibly influences the baryon density distribution in Figures
\ref{BaryRhoMaps.fig}c and d. This is particularly evident in the high
resolution $N=256^2$ column, where the increased pressure support creates a
``puffier'' distribution, wiping out the smallest scale structures in the
baryons. Recall from Figure \ref{MJ.fig} that we naively expect the
presence of the pressure support for $M_J>0$ to affect both $N=128^2$ and
$N=256^2$ at $a/a_i=30$, but not $N=64^2$. Comparing the results of Figures
\ref{BaryRhoMaps.fig}a and c, we indeed see this trend. By $a/a_i=60$, the
effects of the Jeans mass are clearly evident (comparing Figures
\ref{BaryRhoMaps.fig}b and d) for $N=128^2$ and $N=256^2$, though $N=64^2$
still appears relatively unaffected.
Figures \ref{BaryRhoMaps_R64.fig}a and b show images of the baryon densities
for the $k_c=32$ simulations, but in this case resampled to $N=64^2$
resolutions analogous to Figure \ref{DMRhoMaps.fig}b. At expansion $a/a_i
= 30$ (Figure \ref{BaryRhoMaps_R64.fig}a), we see that for $M_J=0$ the
baryons appear to be systematically more tightly collapsed with increasing
simulation resolution, even though they have all been resampled to the same
sampling resolution to produce this image. This supports the view that the
baryon distribution is fundamentally changing with increasing simulation
resolution, in contrast with the dark matter. The $N=128^2$ and $N=256^2$
$M_J>0$ simulations, however, demonstrate very similar baryon density
images, though $N=64^2$ still appears different at $a/a_i=30$. At
$a/a_i=60$ (Figure \ref{BaryRhoMaps_R64.fig}b) we again see for $M_J=0$ a
clear trend with simulation resolution, while the $M_J>0$ runs look
remarkably similar to one another.
Figures \ref{BaryRhoDist.fig}a and b show the full resolution mass
distribution functions of the baryon overdensities
$f(\Sub{\rho}{bary}/\Sub{\bar{\rho}}{bary})$ for all simulations at
$a/a_i=30$ and $a/a_i=60$, respectively. The $M_J=0$ functions show a
strong trend to transfer mass from low to high densities with increasing
resolution, and a similar though weaker trend with $k_c$. However, even at
full resolution the $M_J>0$ simulations show very similar density
distributions once $M_J$ is resolved. The $M_J>0$ simulations also appear
to be relatively insensitive to $k_c$, suggesting that the increased small
scale power is being wiped out by the pressure support. Figures
\ref{BaryRhoDist.fig}c and d show these same baryon density distribution
functions, only for all simulations degraded to $N=64^2$ resolutions.
These bear out our previous observations. In the case with no Jeans mass,
there is no sign of convergence in the baryon distribution as the
resolution is increased. However, when a Jeans mass is present, then the
baryon distributions do converge {\em once the Jeans mass is resolved}.
Figure \ref{KS.fig} presents a more quantitative way to measure this
convergence problem. In this figure we calculate the Kolmogorov-Smirnov
statistic $D(\Sub{\rho}{bary})$, comparing the baryon density distribution
for each simulation to the others at the same expansion and Jeans mass. We
do not expect these simulations to exactly reproduce one another, and
therefore there is little point in assigning significance to the exact
quantitative value of $D$. However, the K-S statistic does provide
objective measures of how similar or dissimilar these distributions are, and
therefore we might expect to learn something by comparing their relative
values. Comparing the upper panels of Figure \ref{KS.fig} we can see that
at $a/a_i=30$ the $N=128^2$ and $N=256^2$ simulations are more similar for
$M_J>0$ than for $M_J=0$, while the $N=64^2$ simulation remains relatively
distinct in both cases. At $a/a_i=60$, however, we can see that for
$M_J>0$ all the simulations appear comparable, while for $M_J=0$ they
remain distinct for the different resolutions.
We therefore have a subtly different picture for the numerical behaviour of
the dark matter and baryons. The critical resolution scale for the dark
matter is the scale of nonlinearity. So long as this scale is resolved,
the dark matter distribution can be expected to converge to a consistent
state on resolved scales. Unfortunately, the distribution and state of the
baryonic particles appears in general to be sensitive to the numerical
resolution. However, it is possible and physically plausible to define a
fundamental collapse scale in the form of the Jeans mass for the baryons,
below which baryonic structure formation is suppressed. This scale can now
be treated as the critical baryonic resolution scale, and we do find that
once this threshold is reached the baryon distribution will reliably
converge as well.
\section{Hydrodynamics and the Baryon Distribution}
\label{Physresults.sec}
\subsection{Shocks and Temperatures}
The results of the previous section indicate that hydrodynamical
interactions on small scales can significantly alter the the final state of
the baryons in ways which propagate upward and affect larger scales. The
tendency for a simulation with a given finite resolution is to
underestimate the ``true'' fraction of high density, collapsed baryons. A
likely cause for this trend is the presence of small scale, unresolved
shocks in the baryon gas. Because shocks provide a mechanism for
transferring the gas's kinetic energy to thermal energy, it is reasonable
to expect that the fashion and degree to which the baryons collapse will be
dependent upon when and how strongly they undergo shocks. In this section
we investigate the thermal state of the baryons, with the goal of
understanding the pattern and importance of shocking in the gas.
In the top row of Figure \ref{RhoTDist.fig} we show the 2-D mass
distribution function of the baryons in terms of their overdensity and
temperature $f(\Sub{\rho}{bary}/\Sub{\bar{\rho}}{bary},T)$ for the $M_J=0$
simulations at $a/a_i=60$. The various resolutions share some gross
properties in the $\rho-T$ plane. The low density gas tends for the most
part to be cool, though there is a tail of low density material with
temperatures up to $T \lesssim 10^4$K. The high density gas is at
relatively high temperatures, with most of the material near $T \sim
10^6$K. However, there is a notable trend for the highest density material
to be somewhat cooler with increasing simulation resolution. This effect
is similar to the behaviour seen in simple 1-D collapse such as the
Zel'dovich pancake (Shapiro \& Struck-Marcell 1985). The highest density
gas is the fraction which collapses earliest, when the background density
is highest. Such gas is placed on a lower adiabat than gas which falls in
at later times, and thus remains cooler. In our case this means that since
higher resolution simulations can resolve higher density clumps (which
therefore form at earlier times), we should tend to see the temperature of
the highest density material fall with increasing resolution.
The high temperature gas is heated by shocks as it falls into the dark
matter dominated potential wells. In order to isolate shock heating from
simple adiabatic compression heating, we calculate the distribution of the
temperature in units of the adiabatic temperature $\Sub{T}{ad}$, given by
\begin{equation}
\Sub{T}{ad} = T_0 \left( \frac{\Sub{\rho}{bary}}{\rho_0} \right)^{\gamma - 1}.
\end{equation}
\Sub{T}{ad} represents the temperature the gas would be at if it were only
heated through simple $P dV$ work. Since the only non-adiabatic process we
allow is shock heating, only gas which has undergone shocking should be at
$T/\Sub{T}{ad} > 1$. In the bottom row of Figure \ref{RhoTDist.fig} we
calculate the distribution
$f(\Sub{\rho}{bary}/\Sub{\bar{\rho}}{bary},T/\Sub{T}{ad})$ for the $M_J=0$
simulations at $a/a_i=60$. The high density fraction of the gas is clearly
strongly shocked in all cases, with $T/\Sub{T}{ad} \sim 10^7-10^9$. There
is a clear trend for $T/\Sub{T}{ad}$ in the high density gas to fall with
resolution, indicating that the highest density fraction of the gas is less
strongly shocked as the resolution increases. Though we do not show the
results at fixed resolution and increasing $k_c$ here, there are also
subtle trends evident with $k_c$ in both the $\rho-T$ and
$\rho-T/\Sub{T}{ad}$ planes. Generally the temperature/shocking
distribution of moderately overdense material grows wider with increasing
$k_c$.
It appears that shocks are indeed the key physical mechanism distinguishing
the different resolution experiments. We find that in general most of the
baryonic material is processed through shocks at some point. We note a
general pattern in which the highest density gas in low resolution
experiments is characteristically more strongly shocked than the highest
density gas in higher resolution experiments. The physical inference of
these trends is that the larger the region which collapses, the stronger
the resulting shock. The underlying physical mechanism for this property
is easily understood. Since the highest density material represents the
gas which collapses earliest, this is also the gas which falls into the
shallowest potential wells. As the structure continues to grow, these
potential wells deepen. Gas which infalls at later times therefore picks
up more kinetic energy, which in turn leads to stronger shocking and higher
temperatures. Once shocking occurs, the state of the baryon gas is
discontinuously and irreversibly altered. In order to properly represent
the physical state of the gas, a simulation must resolve the smallest
scales on which shocks are occuring. This is why enforcing a Jeans mass
allows convergence, since establishing a minimum Jeans mass implies there
is a minimum scale on which baryonic structures can form, forcing a minimum
scale for shocking.
\subsection{Comparing the Baryon \& Dark Matter Distributions}
One of the most fundamental questions we can address is how the
distributions of dark matter and baryons compare to one another.
Comparing the dark matter and baryon density fields for the $M_J=0$ case in
Figures \ref{DMRhoMaps.fig}a and \ref{BaryRhoMaps.fig}b, there is a
distinct impression that the baryons tend to be more tightly clustered than
the dark matter on all collapsed scales. The situation is a bit more
complex for the $M_J>0$ case in Figure \ref{BaryRhoMaps.fig}d. Comparing
the $(N=256^2,k_c=128)$ distributions, it is evident that the $M_J>0$
baryons show a more diffuse structure than that of the $M_J=0$ case, to the
point that some of the smallest scale structures are entirely suppressed.
Bear in mind that the dark matter evolves essentially independently of the
baryons in this dark matter dominated case, so the small scale structures
still form in the overall mass distribution -- the baryons are simply
excluded from them. The large scale structures such as the filaments and
the largest knots are still quite prominent in the $M_J>0$ baryon
distribution, just as for the $M_J=0$ case. These patterns suggest that
the baryons are generically more clustered than the dark matter, down to
the scale set by the Jeans mass. At this scale and lower, the dark matter
continues to form collapsed structures, whereas the baryons are held out of
these structures by the pressure support enforced by the minimum
temperature.
In Figure \ref{B2D.fig} we calculate the baryon to dark matter number
density ratio as a function of baryonic overdensity. The baryon to dark
matter ratio is defined as $\Sub{n}{bary}/\Sub{n}{dm} =
\Sub{\Omega}{dm}\Sub{\rho}{bary}/\Sub{\Omega}{bary}\Sub{\rho}{dm}$, so that
$\Sub{n}{bary}/\Sub{n}{dm} > 1$ corresponds to baryon enrichment, while
$\Sub{n}{bary}/\Sub{n}{dm} < 1$ implies baryon depletion. There is a clear
trend for the highest density material to be baryon enriched, implying that
the cores of the most collapsed structures are relatively enriched in
baryons compared with the universal average. This trend persists even in
the $M_J>0$ simulations, though it is not as pronounced as in the $M_J=0$
case. There is no evidence that a significant fraction of the baryons
exist in regions which are dark matter enhanced. In all simulations
underdense material appears to lie near the universal average mixture
$\Sub{n}{bary}/\Sub{n}{dm} \sim 1$. We also note a trend with resolution,
such that the higher the resolution of the simulation, the greater the
baryon enrichment found in overdense regions.
A simple physical picture can account for these trends. So long as the
density evolution is in the linear regime ($\delta \rho/\rho \ll 1$), the
dark matter and baryons evolve together, remaining at the universal mix of
$\Sub{n}{bary}/\Sub{n}{dm} \sim 1$. During this linear phase the pressure
support (barring any imposed minimum pressure) is orders of magnitude less
important than the gravitational term, so the baryon/dark matter fluid
evolves as a pressureless gas. Once nonlinear collapse begins ($\delta
\rho/\rho \gtrsim 1$), the baryons rapidly fall inward with the dark matter
until they collide near the potential minimum. At this point the baryon
gas shocks, converting the majority of its kinetic energy into thermal
energy, and it stops, forming a hot pressure supported gas at the bottom of
the potential well. In the case with a minimum pressure support, the
collapse proceeds until the pressure term (due to the increase in density)
builds sufficiently to impede the baryons infall, at which point the
baryons slow, separate from the infall, and shock. In either case the dark
matter forms a more diffuse structure supported by velocity dispersion.
This process leads to the generic patterns noted above: on scales below
which the collapse has become nonlinear, the baryons tend to be
characteristically more clustered than the dark matter, at least down to
the minimal point set by the Jeans scale. In either case the critical
factor determining exactly when the baryons separate from the general
inflow is the point at which shocking sets in. We also know from the
numerical observations that this process is resolution dependent, and in
fact the baryon enrichments we see for the $M_J=0$ case in Figure
\ref{B2D.fig} must represent lower limits to the ``true'' baryon
enrichment. The enrichments noted for the $M_J>0$ simulations should be
reliable, to the extent that the specific minimum temperature chosen is
reasonable.
It is somewhat puzzling to note that our measured positive biasing of the
baryons in collapsed structures is at odds with previously published
results. In a study of the cluster formation under the standard $\Omega=1$
Cold Dark Matter (CDM) model, Evrard (1990) finds that while outside of the
cluster environment the baryons and dark matter simply track the universal
average mix, the baryon fraction within the cluster is in fact somewhat
lowered. Kang {\frenchspacing et al.}\ 1994 examine a larger volume of an $\Omega=1$ CDM
cosmology, and find that not only are the overdense regions baryon
depleted, but that their underdense, void structures are baryon enriched.
There are several possible explanations for this disagreement. One
possibility is that this represents a geometric effect, in that our
experiments are 2-D, while these other studies employ fully 3-D
simulations. In our simulations, the ``filaments'' actually represent
walls, and the most collapsed knots are best interpreted as cross-sections
through tubular filaments. The processes of collapsing to a plane, a line,
or a point are certainly different processes, and the isotropy of pressure
support makes these structures progressively more difficult to form. In a
1-D planar collapse, for instance, it is well known that the central
collapse plane will be baryon enriched, while the question of whether or
not a cluster is baryon enriched or depleted is still hotly debated. We
see some evidence for this effect in Figure \ref{B2D.fig}. Looking
particularly at the upper dashed lines in this figure (representing the
baryon enrichment at which 90\% of the mass at that overdensity lies below)
we note our most extreme enrichments occur at moderate overdensities,
roughly in the range $\Sub{\rho}{bary}/\Sub{\bar{\rho}}{bary} \sim
10^1-10^2$. This extremely baryon enriched material represents the
``filaments'' in our simulations (walls in 3-D). It is also possible that
resolution effects play a role here. As pointed out previously, we find a
strong resolution dependence, such that finite resolution tends to
underestimate the fraction of high density, collapsed baryonic material.
Evrard (1990) uses $16^3$ SPH nodes to represent his baryon component,
which for the scale of his box is equivalent to our lowest resolution
simulations. Kang {\frenchspacing et al.}\ (1994) use an entirely different technique to
simulate the hydrodynamics, which relies upon a fixed grid to represent the
baryons. This limits their spatial resolution so that typical clusters are
only a handful of cells across. It is also important to compare these
quantities in the same manner. In Figure \ref{B2D.fig} we calculate the
baryon to dark matter mixture in a manner which follows the baryon mass,
since we sample at the positions of the baryon particles. This naturally
gives the greatest weight to the most prominent baryonic structures. Kang
{\frenchspacing et al.}\ (1994) calculate this distribution in a manner which is volume
weighted, which will tend to give the greatest weight to underdense, void
like regions. Since the baryon fraction appears to be a function of
environment, these differences can be significant. Without further study,
it is difficult to know the true reason for this discrepancy, or how the
actual baryon/dark matter ratio should evolve.
\section{Discussion}
\label{disc.sec}
The results of this investigation can be broken into two broad categories:
what is revealed about the physics of hierarchical collapse in a mixed
baryonic/dark matter fluid, and what is learned about the numerics of
simulations of this process. We find that the dark matter converges to a
consistent state on resolved scales, so long as the nonlinear collapse
scales are well resolved. Increasing the resolution of the experiment does
not fundamentally alter the dark matter distribution, but simply yields
more detailed information about the small scale collapsed structures. This
is in agreement with previous, purely collisionless studies, though we
demonstrate this here including a collisional component.
The numerical story is quite different for the collisional baryonic gas.
We find that in the case where we do not impose a fundamental physical
resolution scale in the baryons, the simulation results do not converge
with increasing resolution. Rather, as the numerical resolution of the
experiment is increased, the collapsed fraction of the baryons is
systematically altered toward a higher density, more tightly bound, and
less strongly shocked state. The physical reason for this behaviour is the
presence of shocks, which allows the evolution on small scales to affect
the overall state of the baryonic mass. With improving resolution the
simulation is able to resolve the collapse of smaller structures at earlier
times. The smaller scale (and therefore earlier) the resolved collapse,
the weaker the resulting shock is found to be. This effect is most obvious
in Figure \ref{RhoTDist.fig}, where there is a systematic trend of higher
density/more weakly shocked material with increasing resolution.
The fact that the dark matter converges in general with resolution, while
the baryons do not, highlights a fundamental difference in the physics of
these two species. While both dark matter and baryon fluids react to the
global and local gravitational potential, the baryons are additionally
subject to purely local hydrodynamical phenomena -- most prominently
shocking in this case. Once strong shocking sets in these hydrodynamical
effects can rise to rival the gravitational force on the baryonic fluid,
allowing the baryons to be strongly influenced by interactions on small
scales in ways which the dark matter is not. This implies that such
small scale interactions can be just as important as the large scale forces
in determining the final state of the baryons. In other words, for the
dark matter there is no back reaction from small to large scales, whereas
the baryons are strongly influenced by interactions on small scales. In
the coupling of these physical processes, gravitation dominates the large
scale structure, but hydrodynamics affects the local arrangements and
characteristics of the baryonic gas. If we want the quantitative results
of such studies to be reliable, we must have reason to believe that the
localized hydrodynamical processes are adequately resolved.
This gloomy picture is alleviated by an important physical effect: the
Jeans mass. Introducing a minimum temperature (and therefore pressure
support) into the baryons creates a fundamental length/mass scale, below
which the baryons are supported by pressure against any further collapse or
structure formation. We find that once we introduce such a minimal scale
into the baryonic component, the simulation results converge as this scale
is resolved. This convergence holds even though the dark matter component
continues to form structures below the baryon Jeans scale. Although the
Jeans scale is dependent upon the local density, we find that the global
Jeans scale defined using the background density is adequate to define the
critical resolution necessary for the hydrodynamics to converge. This
therefore describes an additional resolution scale necessary for
hydrodynamical simulations to meet, much as the nonlinear mass scale
represents the crucial resolution necessary for purely gravitational
systems. Furthermore, our experiments indicate that equation (\ref{MR.eq})
is a reasonable estimate of an SPH simulation's true mass resolution, since
we find that the threshold $M_R \lesssim M_J$ marks the point at which
convergence is achieved.
In these experiments we have tested the effects of the Jeans mass in an
idealized framework by simply imposing an arbitrary minimum temperature
into our system, but there is reason to believe that such minimum
temperatures should exist in the real universe. Based upon observations
such as the Gunn-Peterson test (Gunn \& Peterson 1965), it is known that
the IGM is highly ionized out to redshifts $z \lesssim 5$, which implies a
minimum temperature for the IGM of at least $T \gtrsim 10^4$. Assuming an
Einstein-de Sitter cosmology, a minimum temperature of $T \sim 10^4$
requires a minimum spatial resolution (via eq. [\ref{LJ.eq}])
\begin{equation}
\lambda_J \sim 0.777 \; (1 + z)^{-3/2} \;
\left( \frac{\mu}{0.6} \right)^{-1/2} \; \left( \frac{T}{10^4 \mbox{K}} \right)^{1/2}
\; h^{-1} \mbox{\ Mpc},
\end{equation}
which equates to a baryon mass resolution of (eq. [\ref{MJ.eq}])
\begin{equation}
M_R \lesssim \Sub{\Omega}{bary} \; M_J \sim
6.82 \times 10^{10} \; \Sub{\Omega}{bary} \;
(1 + z)^{-3/2} \; \left( \frac{\mu}{0.6} \right)^{-3/2} \;
\left( \frac{T}{10^4 \mbox{K}} \right)^{3/2} \; h^{-1} \ M_{\sun}.
\end{equation}
This limit can also be expressed in terms of a minimum circular velocity,
which has the advantage of being independent of redshift. The minimum
circular velocity can found as a function of the minimum temperature by
relating the kinetic energy necessary for dynamical support to the internal
energy for equivalent pressure support (Thoul \& Weinberg 1996), yielding
\begin{equation}
\Sub{v}{circ} = \left( \frac{2 k T}{\mu m_p} \right)^{1/2}
\sim 16.6 \left( \frac{\mu}{0.6} \right)^{-1/2} \left( \frac{T}{10^4 \mbox{K}}
\right)^{1/2} \mbox{km/sec}.
\end{equation}
In our $M_J>0$ simulations if we choose to call the scale at which RMS mass
fluctuation is $\Delta M/M \sim 0.5$ to be 8 $h^{-1}$ Mpc at the final
expansion, then our box scale is $L=64 h^{-1}$Mpc and the minimum
temperature corresponds to $\Sub{T}{min} \sim 10^6$K. While there are some
suggestions that the intergalactic medium could be heated to temperatures
as hot as $10^6$K (through mechanisms such as large scale shocks of the
IGM), clearly these simulations do not meet our criteria if we wish to
consider photoionization as setting the minimum temperature. It is also
not clear that the current generation of large-scale hydrodynamical
cosmological simulations meet this criterion, but it should be achievable.
It is still unclear whether or not in the case with no minimum temperature
imposed the baryon distribution will eventually converge. It is well known
that in a purely gravitational system, as structure builds and smaller dark
matter groups merge into larger structures, the dark matter ``forgets''
about the earlier small scale collapses as such small structures are
incorporated into larger halos and disrupted. This is why the dark matter
results converge once the nonlinear mass scale is resolved. While it is
evident from studies such as this that the baryons maintain a longer memory
of their previous encounters, it seems likely that as the baryon gas is
progressively processed through larger scale and stronger shocks, at some
point the previous evolution should become unimportant. At exactly what
level this transition is reached remains uncertain, however, as we see no
evidence for such convergence here.
Radiative cooling must be accounted for in order to model processes such as
galaxy formation, and the inclusion of radiative cooling can only
exacerbate the non-convergence problems we find here. The amount of energy
per unit mass dissipated by radiative cooling is proportional to the
density, and we have already noted that the trend with finite resolution is to
underestimate the local gas density and overestimate the temperature.
Given these tendencies, it is not difficult to envision problems for finite
resolution simulations which will tend to underestimate the effectiveness
of radiative cooling in lowering the temperature (and therefore pressure
support) of the shocked gas. This could lead to perhaps drastic
underestimates of the fraction of cold, collapsed baryons for a given
system, and therefore strongly influence the inferred galaxy formation.
Evrard {\frenchspacing et al.}\ (1994) note this effect when comparing their high and low
resolution 3-D simulations. They find that altering their linear
resolution by a factor of two (and therefore the mass resolution by a
factor of eight) changes the measured total amount of cold collapsed
baryons by a factor of $\sim 3$. They attribute this change to just the
sort of problems we discuss here. Weinberg, Hernquist, \& Katz (1996)
report similar findings and interpretation for simulations with a
photoionizing background. It therefore seems likely it is all the more
important to resolve the minimum mass scale set by the minimum temperature
in systems with radiative cooling.
We find that the majority of the baryonic mass undergoes strong shocking so
long as the nonlinear mass scale exceeds the Jeans mass. At infinite
resolution in the $M_J=0$ case, it is possible that all of the baryonic
material undergoes shocking. As anticipated from previous investigations,
the highest density collapsed fraction is characteristically less shocked
as compared with later infalling material from larger regions. The
underlying cause for this behavior is the fact that potential wells deepen
as structure grows. The highest density material is that which collapses
earliest due to the smallest scale perturbations. This material falls into
relatively shallow potential wells, and is only weakly shocked. As the
structures continue to grow, progressively larger scales go nonlinear and
collapse. The potential wells deepen and infalling material gains more
energy, resulting in stronger shocking and higher temperatures.
Hydrodynamics can also play an important role in determining the
distribution of the baryon mass, particularly in collapsed structures. In
the absence of external mechanisms to heat the baryons (such as energy
input from photoionization), during the linear phase of structure growth
the baryons evolve as a pressureless fluid and simply follow the dominant
dark matter. Once nonlinear collapse sets in, the baryons fall to the
potential minimum, shock, convert their kinetic energy to thermal energy,
and settle. In contrast, the dark matter simply passes though the
potential minimum and creates a more diffuse structure supported by the
anisotropic pressure of random velocities. This difference gives rise to a
characteristic pattern in the baryon/dark matter ratio. Wherever the
evolution is still linear, the baryons and dark matter simply remain at the
universal mix. With the onset of nonlinear collapse, the baryons fall to
the minimum of the potential well where they form a baryon enriched core,
surrounded by a dark matter rich halo. We find that even in the absence of
radiative cooling the cores of collapsed structures can become baryon
enriched by factors of $\Sub{n}{bary}/\Sub{n}{dm} \sim 2$ or more, though
this value is likely resolution and dimension dependent. If the thermal
energy of the baryons is raised to the point that it rivals the potential
energy during the collapse, the baryons will become pressure supported and
stop collapsing at that point. In all cases we find that the dark matter
is relatively unaffected by the baryon distribution. This is due to the
fact that the dark matter dominates the mass density, and therefore the
gravitational potential. In general it appears that under a dark matter
dominated scenario hydrodynamics can substantially alter the
characteristics of the baryonic material (and therefore the visible
universe), such that it does not directly follow the true mass distribution
which is dominated by the the dark matter.
\acknowledgements
We would like to thank David Weinberg for inspiring this project, and for
many useful discussions during its course as well. We would also like to
thank the members of Ohio State's Astronomy Department for the use of their
workstations both for performance and analysis of many of the simulations.
JMO acknowledges support from NASA grant NAG5-2882 during this project.
Some of these simulations were performed on the Cray Y/MP at the Ohio
Supercomputer Center.
\clearpage
\section{References}
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\item Anninos, P., \& Norman, M. L. 1996, \apj, 459, 12
\item Binney, J. \& Tremaine, S. 1987, {\em Galactic Dynamics},
(Princeton: Princeton University Press)
\item Beacom, J. F., Dominik, K. G., Melott, A. L., Perkins, S. P., \&
Shandarin, S. F. 1991, \apj, 372, 351
\item Cen, R., \& Ostriker, J. 1992a, \apj, 393, 22
\item Cen, R., \& Ostriker, J. 1992b, \apj, 399, L113
\item Evrard, A. E.\ 1990, \apj, 363, 349
\item Evrard, A. E., Summers, F. J., \& Davis, M. 1994, \apj, 422, 11
\item Gunn, J. E., \& Peterson, B. A. 1965, \apj, 142, 1633
\item Kang, H., Cen, R., Ostriker, J. P., \& Ryu, D. 1994, \apj, 428, 1
\item Katz, N., Hernquist, L., \& Weinberg, D. H. 1992, \apj, 399, L109
\item Little, B., Weinberg, D. H., \& Park, C. 1991, \mnras, 253, 295
\item Melott, A. L., \& Shandarin, S. F. 1990, \nat, 346, 633
\item Navarro, J. F., \& White, S. D. M. 1994, \mnras, 267, 401
\item Owen, J. M, Villumsen, J. V., Shapiro, P. R., \& Martel, H. 1996,
submitted \apjs\ December 1995
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\item Shapiro, P. R., \& Struck-Marcell, C. 1985, \apjs, 57, 205
\item Steinmetz, M., \& M\"{u}ller, E. 1994, \aap, 281, L97
\item Thoul, A. A., \& Weinberg, D. H. 1996, preprint
\item Weinberg, D. H., Hernquist, L., \& Katz, N. 1996, submitted to \apj,
astro-ph/9604175
\item White, S. D. M., \& Rees, M. J. 1978, \mnras, 183, 341
\end{description}
\clearpage
\figcaption{The ratio of the Jeans mass to the resolved mass ($M_J/M_R$) as
a function of expansion for each of the three resolutions used in this
paper ($N = 64^2, 128^2, 256^2$) for the $M_J > 0$ case. The Jeans mass is
calculated using the average background density of the universe at each
expansion.
\label{MJ.fig}}
\figcaption{Dark matter overdensities
($\protect\Sub{\rho}{dm}/\bar{\rho}_{dm}$) for $M_J=0$ simulations. Panels
arranged with increasing resolution along rows ($N = 64^2, 128^2, 256^2$),
and increasing cutoff in initial input perturbation spectrum down columns
($k_c = 32, 64, 128$). Part a) shows results using the full resolution of
each simulation, while b) is calculated after resampling the simulations
down to equivalent $N=64^2$ resolution. All simulations are shown at the
final time slice (expansion factor $a/a_i = 60$), with grey scale intensity
scaled logarithmically with dark matter density.
\label{DMRhoMaps.fig}}
\figcaption{Normalized dark matter overdensity distribution functions
$f(\protect\Sub{\rho}{dm}/\bar{\rho}_{dm})$ for $M_J=0$ (solid lines) and
$M_J>0$ (dotted lines) simulations. Part a) shows results for full
simulations, while part b) shows all simulations degraded to equivalent
$N=64^2$ resolution.
\label{DMRhoDist.fig}}
\figcaption{Baryon overdensities
$\protect\Sub{\rho}{bary}/\protect\Sub{\bar{\rho}}{bary}$ for a) $M_J=0$ at
$a/a_i=30$, b) $M_J=0$ at $a/a_i=60$, c) $M_J>0$ at $a/a_i=30$, and d)
$M_J>0$ at $a/a_i=60$. Panels arranged as in Figure
\protect\ref{DMRhoMaps.fig}.
\label{BaryRhoMaps.fig}}
\figcaption{Baryon overdensities for $k_c=32$ simulations, resampled to
$N=64^2$ resolution as in Figure \protect\ref{DMRhoMaps.fig}b. Note these
panels represent the same simulations as the top rows of the previous
figures. Shown are expansion factors a) $a/a_i=30$ and b) $a/a_i=60$.
Panels are arranged with increasing simulation resolution ($N = 64^2,
128^2, 256^2$) along rows, and increasing baryon Jeans mass ($M_J = 0$,
$M_J > 0$) down columns.
\label{BaryRhoMaps_R64.fig}}
\figcaption{Normalized baryon overdensity distribution functions
$f(\protect\Sub{\rho}{bary}/\bar{\rho}_{bary})$ for $M_J=0$ (solid lines)
and $M_J>0$ (dotted lines). We show the full resolution results for a)
$a/a_i=30$ and b) $a/a_i=60$, as well as results when each simulation is
degraded to $N=64^2$ resolution at c) $a/a_i=30$ and d) $a/a_i=60$. Panels
arranged as in Figure \protect\ref{DMRhoDist.fig}.
\label{BaryRhoDist.fig}}
\figcaption{Kolmogorov-Smirnov statistic $D$ comparing
$f(\protect\Sub{\rho}{bary})$ between simulations. Each line type
corresponds to one simulation which is compared to each simulation listed
on the ordinate axis, where the simulations are denoted as $N:k_c$. Note
that the K-S statistic for comparing an individual simulation to itself is
formally $D=0$, but for the sake of clarity we have interpolated over these
points in this plot. The panels are arranged with Jeans mass $M_J$
increasing along rows, and expansion $a/a_i$ increasing down columns.
\label{KS.fig}}
\figcaption{Baryon mass distribution for the $M_J=0$ simulations as a
function of overdensity and temperature
$f(\protect\Sub{\rho}{bary}/\protect\Sub{\bar{\rho}}{bary}, T)$ (upper row)
and $f(\protect\Sub{\rho}{bary}/\protect\Sub{\bar{\rho}}{bary}, T/T_{ad})$
(lower row). $T_{ad} = T_0 (\rho/\rho_0)^{\gamma - 1}$ is defined as the
temperature the gas would have due solely to adiabatic processes. Panels
arranged as in Figure \protect\ref{BaryRhoDist.fig}.
\label{RhoTDist.fig}}
\figcaption{Average baryon to dark matter mixture as a function of baryon
overdensity at $a/a_i=60$. The baryon to dark matter mixture is defined as
$\protect\Sub{n}{bary}/\protect\Sub{n}{dm} =
\protect\Sub{\Omega}{dm}\protect\Sub{\rho}{bary}/
\protect\Sub{\Omega}{bary}\protect\Sub{\rho}{dm}$, so that
$\protect\Sub{n}{bary}/\protect\Sub{n}{dm} > 1$ represents baryon enriched
material, while $\protect\Sub{n}{bary}/\protect\Sub{n}{dm} < 1$ is baryon
depleted. In each panel the solid line shows the measured average baryon
to dark matter mixture, while the dashed lines represent the mixtures such
that 10\% of the mass at each overdensity is above and below the enclosed
region. The dotted line shows the universal average
$\protect\Sub{n}{bary}/\protect\Sub{n}{dm} = 1$. The top and bottom rows
represent the $M_J=0$ and $M_J>0$ simulations, respectively.
\label{B2D.fig}}
\end{document}
| 15,978 |
\section{Introduction}
\null\indent
We consider vertex models in D dimensions. As is standard, the
bonds of the lattice
carry variables taking $q$ values (colors).
The model is determined by attributing Boltzmann weights to
the various possible bond configurations around a
vertex~\cite{LiWu72}.
These homogeneous weights are arranged in a matrix, which we denote by
$R$.
The size and form of the matrix $R$ vary according to the number of
colors, and the coordination
number of the lattice. Typical examples we will consider are the 2D
square lattice, the 2D
triangular
lattice, and the 3D cubic lattice:
\setlength{\unitlength}{0.0075in}
\begin{eqnarray*}
\begin{picture}(120,120)(20,700)
\thinlines
\put( -40,755){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $
R^{ij}_{uv} = $}}}
\put( 80,820){\line( 0,-1){120}}
\put( 20,760){\line( 1, 0){120}}
\put(125,765){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $u$}}}
\put( 85,710){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $j$}}}
\put( 85,810){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $v$}}}
\put( 20,765){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $i$}}}
\end{picture}
\qquad & \qquad
\begin{picture}(120,120)(20,700)
\label{fig1}
\thinlines
\put( 50,730){\line( 1, 1){ 60}}
\put( 110,730){\line( -1,1){60}}
\put( 20,760){\line( 1, 0){120}}
\put(110,795){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $v$}}}
\put( 45,710){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $j$}}}
\put(125,765){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $u$}}}
\put( 50,795){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $w$}}}
\put( 110,710){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $k$}}}
\put( 20,765){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $i$}}}
\put( 15,755){\makebox(0,0)[rb]{\raisebox{0pt}[0pt][0pt]{
$R^{ijk}_{uvw}=$}}}
\end{picture}
\qquad & \qquad
\begin{picture}(120,120)(20,700)
\thinlines
\put( 35,715){\line( 1, 1){ 80}}
\put( 80,820){\line( 0,-1){120}}
\put( 20,760){\line( 1, 0){120}}
\put(115,795){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $v$}}}
\put( 25,705){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $j$}}}
\put(125,765){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $u$}}}
\put( 85,710){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $k$}}}
\put( 85,810){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $w$}}}
\put( 20,765){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{ $i$}}}
\put( 15,755){\makebox(0,0)[rb]{\raisebox{0pt}[0pt][0pt]{
$R^{ijk}_{uvw}=$}}}
\end{picture}
\\
\mbox{2D square}\qquad & \qquad \mbox{2D triangular}\qquad & \qquad
\mbox{3D cubic}
\end{eqnarray*}
If the number of colors $q$ is $2$, and we will restrict ourselves to
this case, then the $R$-matrices are of sizes $4\times 4$,
$8\times 8$, and $8 \times 8$ respectively. The difference between {2D triangular}
and {3D cubic} for
example does
not show in the size nor the form of the matrix.
It will, however, appear in the operations we define on
the matrices.
We shall use a number of elementary transformations acting on the
matrices.
These transformations come from the inversion relations and the
geometrical
symmetries of the lattice, in the framework of
integrability~\cite{St79,Ba82,BeMaVi91c,BeMaVi91d},
and beyond integrability~\cite{BeMaVi92}.
They generically form an infinite group
$\Gamma_{lattice}$~\cite{BeMaVi91c,BeMaVi91d}.
The groups $\Gamma_{lattice}$ have a finite number of involutive
generators.
The first one, denoted $I$, is non-linear and does not depend
on the lattice: it is the matrix inversion up to a factor.
The other generators act linearly on $R$, actually by permutations of
the entries, and
represent the geometrical symmetries of the lattice.
For the square lattice, we have two linear transformations, the partial
transpositions $t_l$
and $t_r$~\cite{BeMaVi91d}:
\begin{equation} \nonumber
(t_lR)^{ij}_{uv}= R^{uj}_{iv}, \qquad (t_rR)^{ij}_{uv}= R^{iv}_{uj}
\qquad i,j,u,
v=1..\, q
\end{equation}
The product $t_l t_r$ is the matrix transposition ($l$ stand for `left'
and $r$ stands for `right'
in the standard tensor product structure of $R$).
For the triangular lattice, we have three linear transformations
$\tau_l$, $\tau_m$, $\tau_r$:
\begin{equation}\nonumber
(\tau_lR)^{ijk}_{uvw}= R^{ukj}_{iwv} ,\quad
(\tau_mR)^{ijk}_{uvw}= R^{wvu}_{kji}, \quad
(\tau_rR)^{ijk}_{uvw}= R^{jiw}_{vuk}. \quad
\end{equation}
Finally, for the cubic lattice, we have three linear transformations
$t_l$, $t_m$,
$t_r$~\cite{BeMaVi91d}:
\begin{equation}\nonumber
(t_lR)^{ijk}_{uvw}= R^{ujk}_{ivw}, \quad
(t_mR)^{ijk}_{uvw}= R^{ivk}_{ujw}, \quad
(t_rR)^{ijk}_{uvw}= R^{ijw}_{uvk}, \quad
\end{equation}
and the product $t_l \; t_m\; t_r$ of the three partial transpositions
is the matrix transposition.
All the generators are involutions.
All products $t_l I$, $t_m I$ and $t_r I$ are of infinite order
when acting on a generic matrix, as are $\tau_l I$
and $\tau_r I$.
On the contrary $\tau_m $ and $I$ commute and $(\tau_m I)^2=1$.
It is straightforward to check that:
\begin{equation}
\label{tegaltau}
t_l \; I \; t_l \; = \; \tau_l \; I \; \tau_l
\qquad \mbox{and} \qquad
t_r \; I \; t_r \; = \; \tau_r \; I \; \tau_r
\end{equation}
so that essentially $\Gamma_{triang}$ appears as
a subgroup of $\Gamma_{cubic}$, up to finite factors.
It is important to keep in mind what the ``size'' of the groups $\Gamma$
are. All three $\Gamma_{square}$,
$\Gamma_{triang}$, and $\Gamma_{cubic}$ are infinite, but $\Gamma_{square}$ has
one infinite order generator,
$\Gamma_{triang}$ has two, and $\Gamma_{cubic}$ has three.
The last two groups are thus hyperbolic groups~\cite{MaRo94}, and
studying the triangular
lattice can be a good test-case for the more involved tridimensional
cubic lattice.
The groups $\Gamma$ are the building pieces of the group of automorphisms
of
the Yang-Baxter equations and their higher dimensional
generalizations, and solve the so-called
``baxterization problem''~\cite{BeMaVi91c,BeMaVi91d}.
These equations form overdetermined systems of multilinear equations,
of which the possible solutions are parametrized by algebraic
varieties~\cite{Ma86}.
The overdetermination increases very rapidly with the dimension of the
lattice.
At the same time, the size of $\Gamma$ also explodes.
When looking at solutions of the Yang-Baxter equations and their
generalizations
to higher dimensional lattices,
one faces a conflict between having a more and more overdetermined
system
and a larger and larger group of automorphisms for the set of
solutions.
We will show how this conflict is resolved in some 2D and known 3D
solutions
by a degeneration of the effective realization of the group $\Gamma$, which
becomes finite.
The content of this letter is the description of
a mechanism for such a degeneration, obtained by the linearization
of specific elements of $\Gamma$.
We first show how the free-fermion condition on the asymmetric eight
vertex model~\cite{FaWu70} falls into this scheme.
We then describe the group $\Gamma_{triang}$ for the 32-vertex model on
the triangular lattice. We show that the free-fermion conditions
given in~\cite{Hu66,SaWu75} amount to linearizing
the inverse $I$ and make the realization of $\Gamma_{triang}$ finite.
We finally write and discuss similar conditions for the 32-vertex model
on the cubic (3D) lattice, by analyzing solutions of the tetrahedron
equations~\cite{Za81,Ba86,SeMaSt95}.
One of the results we obtain is that free-fermion
conditions should always appear as quadratic conditions, whatever the size
and form of the matrix $R$ is, and in particular whatever the dimension
and geometry of the lattice are.
There already exists an important literature about free-fermion models.
We may refer to~\cite{Sa80,Sa81,Sa81b}, where an
exploration of the use of grassmannian variables, both for
the construction and the resolution of the models, can be
found.
This work also motivated the interesting 3D construction of~\cite{BaSt84}.
Our approach is based on a direct study of the matrix of Boltzmann weights,
concentrating on the action of the symmetry group $\Gamma$, and provides
another view on this class of models.
\section{Some notations}
At this point it is useful to introduce some notations we will use in
the sequel.
We will denote the equality of two matrices $R$ and $R'$ up to an
overall factor by
$R \simeq R'$.
We always denote by $t$ the full matrix transposition.
We will use various gauge transformations (weak graph
dualities)~\cite{GaHi75},
that is to say the conjugation
by invertible matrices which are tensor products, also defined up to
overall factors,
i.e. transformations of the type
\begin{eqnarray*}
R \longrightarrow g_1^{-1}\otimes g_2^{-1} \otimes \dots \otimes
g_k^{-1}
\cdot R \cdot \; g_1\otimes g_2 \otimes \dots \otimes g_k \; .
\end{eqnarray*}
Define the matrices
\begin{eqnarray}
\sigma_0=\pmatrix{ 1 & 0 \cr 0 & 1 }, \quad
\sigma_1=\pmatrix{ 0 & 1 \cr 1 & 0 }, \quad
\sigma_2=\pmatrix{ 0 & -1 \cr 1 & 0 }, \quad
\sigma_3=\pmatrix{ 1 & 0 \cr 0 & -1 }
\end{eqnarray}
and the matrices $ \sigma_{a_1 a_2 \dots a_k} $
of size $2^k \times 2^k$ ($k$ will be $2$ for the square
lattice, $3$ for the triangular and cubic lattice, and so on) by:
\begin{eqnarray*}
\sigma_{a_1 a_2 \dots a_k} = \sigma_{a_1} \otimes \sigma_{a_2} \otimes
\dots \sigma_{a_k} \; .
\end{eqnarray*}
We denote by $\Sigma_{a_1 a_2 \dots a_k}$ the conjugation by
$\sigma_{a_1 a_2 \dots a_k}$.
Clearly both $t$ and $I$ commute with all $\Sigma_{a_1 a_2 \dots a_k}$,
up to an irrelevant sign.
Moreover, from the fact that
$
\sigma_a \; \sigma_b \; = \; \pm \; \sigma_b \; \sigma_a, \quad \forall
a,b = 0,1,2,3
$,
the gauge transformations $\Sigma$ satisfy
\begin{eqnarray*}
\Sigma_{a_1 a_2 \dots a_k} \; \Sigma_{b_1 b_2 \dots b_k} \; = \; \pm
\; \Sigma_{b_1 b_2 \dots b_k} \; \Sigma_{a_1 a_2 \dots a_k},
\end{eqnarray*}
meaning that they commute up to a factor.
Particular gauge transformations of interest are
\begin{eqnarray*}
\pi \; = \; \Sigma_{3 3 \dots 3},
\end{eqnarray*}
and some transformations acting just by changes of sign of some of the
entries, and denoted
$\epsilon_\alpha$ ($\alpha = l, \; m, \; r,$ ...).
If $k=2$ (square lattice), then $\alpha=l$ or $r$, and
$$ \epsilon_l = \Sigma_{30} , \qquad \epsilon_r = \Sigma_{03}.$$
If $ k=3$ (triangular and cubic lattice):
$$\epsilon_l = \Sigma_{300}, \qquad \epsilon_m = \Sigma_{030}, \qquad
\epsilon_r = \Sigma_{003}. $$
\section{Free-fermion asymmetric eight-vertex model}
The matrix $R$ of the asymmetric eight-vertex model~\cite{Ka74} is of
the form
\begin{equation}
\label{8v}
R = \pmatrix { a & 0 & 0 & d' \cr
0 & b & c' & 0 \cr
0 & c & b' & 0 \cr
d & 0 & 0 & a' }
\end{equation}
Notice that this form is the most general matrix satisfying $\pi \; R =
R$.
The free-fermion condition~\cite{FaWu70} (see also~\cite{BaSt85abc}) is
\begin{equation}
\label{ff8}
a a' - d d' + b b' - c c' =0
\end{equation}
A matrix of the form (\ref{8v}) may be brought, by similarity
transformations, to a block-diagonal
form
\begin{eqnarray*}
R = \pmatrix{ R_1 & 0 \cr 0 & R_2 }, \qquad\mbox{ with }
\qquad R_1 = \pmatrix{ a & d' \cr d & a' } \quad \mbox{and}
\quad R_2 = \pmatrix{ b & c' \cr c & b' }.
\end{eqnarray*}
If one denotes by $\delta_1=a a' - d d'$ and by $\delta_2 = b b' - c c'$ the
determinants of the two blocks then the matrix inverse $I$ written
polynomially
(namely $R \rightarrow det(R) \cdot R^{-1}$) just reads
\begin{eqnarray*}
&
a \rightarrow a' \cdot \delta_2, \quad
a' \rightarrow a \cdot \delta_2, \quad
d \rightarrow -d \cdot \delta_2, \quad
d' \rightarrow -d' \cdot \delta_2,& \\
&
b \rightarrow b' \cdot \delta_1, \quad
b' \rightarrow b \cdot \delta_1, \quad
c \rightarrow -c \cdot \delta_1, \quad
c' \rightarrow -c' \cdot \delta_1. \quad &
\end{eqnarray*}
The condition (\ref{ff8}) may be written as $p_9(R)=0$ with the notations
of~\cite{BeMaVi92}, and is consequently left invariant by $\Gamma_{square}$.
It is straightforward to see that condition (\ref{ff8}) is
$\; \delta_1 = - \delta_2 \;$ and has the effect of linearizing $I$
into
\begin{eqnarray*}
&
a \rightarrow a' , \quad
a' \rightarrow a , \quad
d \rightarrow -d , \quad
d' \rightarrow -d' ,& \\
&
b \rightarrow -b', \quad
b' \rightarrow - b , \quad
c \rightarrow c, \quad
c' \rightarrow c'. \quad &
\end{eqnarray*}
The group $\Gamma$ is then realized by permutations of the entries, mixed
with
changes of signs, and its orbits are thus {\em finite}.
The commutators of partial transpositions and inversion, in the sense
of group theory,
i.e: $ t_l I t_l^{-1} I^{-1} = ( t_l \; I)^2 $ and
$ t_r I t_r^{-1} I^{-1} = ( t_r \; I)^2 $
reduce to a change of sign of the non-diagonal entries of $R$.
These commutators are typical infinite order elements of $\Gamma$, when
acting
on a generic matrix, and their degeneration is a key to the finiteness
of the
realization of $\Gamma$.
If we introduce the grading $gr$
\begin{eqnarray*}
gr(R) = \pmatrix { a & 0 & 0 & d' \cr 0 & -b & -c' & 0 \cr
0 & -c & -b' & 0 \cr d & 0 & 0 & a' }
\end{eqnarray*}
which operates by changing the sign of the entries of only one of the
two blocks,
say $R_2$, then, for any $R$ satisfying (\ref{ff8}), the action of the
inverse reduces to
\begin{eqnarray}
\label{lsq}
I(R) \; \simeq \; t \; \Sigma_{12} \; gr (R)
\end{eqnarray}
where $t$ is matrix transposition.
In other words we have defined, on all matrices satisfying $\pi (R)
=R$, a linear operator
\begin{eqnarray*}
l_{sq} = \; t \; \Sigma_{12} \; gr
\end{eqnarray*}
such that the free-fermion condition (\ref{ff8}) reads
\begin{equation}
I(R) \; \simeq \; l_{sq} (R) \label{lsq1}
\end{equation}
or equivalently
\begin{equation}
R \cdot l_{sq} (R) \; \simeq \; \mbox{ unit matrix }
\end{equation}
The linear transformation $l_{sq}$ satisfies a number of relations:
\begin{eqnarray}
l_{sq}^2 = id, \qquad l_{sq} \; t = t \; l_{sq},
\qquad l_{sq} \; t_\alpha \; l_{sq} \; t_\alpha = \;
\epsilon_\alpha, \quad \alpha=l,\; r
\end{eqnarray}
Such relations ensure that the orbit of $R$ under $\Gamma$ is finite,
as is readily checked,
and specify the changes of signs to which $(t_l \; I)^2$ and $(t_r \;
I)^2$ reduce.
Notice that the definition of $l_{sq}$ is not unique.
\section{Free-fermion conditions for the 32-vertex model on the triangular
lattice}
We consider the free-fermion conditions for the 32-vertex model on a
triangular lattice, and
use the notations of~\cite{SaWu75}:
\begin{equation}
\label{rsawu}
R = \left [\begin {array}{cccccccc} {\it f_{0}}&0&0&{\it f_{23}}&0&{\it
f_{13}}&{
\it f_{12}}&0\\0&{\it f_{36}}&{\it f_{26}}&0&{\it f_{16}}&0&0&{\it
{\bar f}_{45}}\\0&{\it
f_{35}}&{\it f_{25}}&0&{\it f_{15}}&0&0&{\it {\bar f}_{46}}\\{\it
f_{56}}&0&0&
{\it {\bar f}_{14}}&0&{
\it {\bar f}_{24}}&{\it {\bar f}_{34}}&0\\0&{\it f_{34}}&{\it
f_{24}}&0&
{\it f_{14}}&0&0&{\it {\bar f}_{56}}
\\{\it f_{46}}&0&0&{\it {\bar f}_{15}}&0&{\it {\bar f}_{25}}&{\it {\bar
f}_{35}}&
0\\{\it f_{45}}&0&0&{\it
{\bar f}_{16}}&0&{\it {\bar f}_{26}}&{\it {\bar f}_{36}}&0\\0&{\it
{\bar f}_{12}}&
{\it {\bar f}_{13}}&0&{\it {\bar f}_{23}}&0&0&{
\it {\bar f}_{0}}\end {array}\right ]
\end{equation}
This matrix may be brought, by a permutations of lines and columns, into
a block diagonal form:
\begin{equation}
\label{blocks} \nonumber
R= \pmatrix{ R_1 & 0 \cr 0 & R_2 }, \qquad \mbox{ with }
\end{equation}
\begin{equation} \nonumber
R_1 = \left [\begin {array}{cccc} {\it f_{0}}&{\it f_{13}}&{\it
f_{12}}&{\it f_{23}}\\{
\it f_{46}}&{\it {\bar f}_{25}}&{\it {\bar f}_{35}}&{\it {\bar
f}_{15}}\\{\it f_{45}}&
{\it {\bar f}_{26}}&{\it {\bar f}_{36}}&
{\it {\bar f}_{16}}\\{\it f_{56}}&{\it {\bar f}_{24}}&{\it {\bar
f}_{34}}&
{\it {\bar f}_{14}}\end {array}\right ]\, ,
\qquad
R_2 = \left [\begin {array}{cccc} {\it f_{14}}&{\it f_{34}}&{\it
f_{24}}&{\it {\bar f}_{56}}\\{
\it f_{16}}&{\it f_{36}}&{\it f_{26}}&{\it {\bar f}_{45}}\\{\it
f_{15}}&{\it f_{35}}&{\it f_{25}}&
{\it {\bar f}_{46}}\\{\it {\bar f}_{23}}&{\it {\bar f}_{12}}&{\it {\bar
f}_{13}}&{\it {\bar f}_{0}}\end {array}\right ]
\end{equation}
The inverse $I$ written polynomially is now a transformation of
degree $7$.
If one introduces the two determinants $\Delta_1 = det(R_1) $
and $\Delta_2 = det(R_2) $, then each term in the expression of $I(R)$
is a product of a degree three minor, taken within a block, times the
determinant of the other block.
Denoting $\bar{f}_{12}=f_{3456}$ and so on, the free-fermion conditions
of~\cite{Hu66,SaWu75} are:
\begin{eqnarray}
\label{ff32a}
& f_0 f_{ijkl}& = f_{ij} f_{kl} - f_{ik} f_{jl} + f_{il} f_{jk},
\qquad \forall \; i,j,k,l \, = \, 1,\dots, 6 \\
\label{ff32b}
& f_0 \bar{f}_0 & = f_{12} \bar{f}_{12} - f_{13} \bar{f}_{13} + f_{14}
\bar{f}_{14} -
f_{15} \bar{f}_{15} + f_{16} \bar{f}_{16}
\end{eqnarray}
What is remarkable is that, not only the rational variety ${\cal V}$
defined by (\ref{ff32a},\ref{ff32b}), is globally invariant by
$\Gamma_{triang}$, but {\em again} the realization of $\Gamma$ on this
variety is finite.
This comes from
the degeneration of $I$ into a mixture of changes of signs and
permutations of the
entries, as was the case in the previous section.
When relations (\ref{ff32a}, \ref{ff32b}) are satisfied, the action of
$I$ simplifies to
\begin{eqnarray}
\label{ltri}
I(R) \simeq \; l_{tr} (R)
\end{eqnarray}
with
\begin{eqnarray}
l_{tr} \; = \; t \; \Sigma_{121} \; \simeq \; t \; \Sigma_{111} \;
\epsilon_m
\end{eqnarray}
Since we have the prejudice that all free-fermion conditions should
be invariant under $\Gamma_{triang}$, as (\ref{ff32a}, \ref{ff32b}) are,
one should complement (\ref{ltri}) with
\begin{eqnarray}
I \; \tau_l (R) & \simeq & l_{tr}\; \tau_l \; (R) \label{ltril}
\\
I \; \tau_r (R) & \simeq & l_{tr} \; \tau_r \; (R) \label{ltrir}
\end{eqnarray}
We may list some useful relations:
\begin{eqnarray}
\label{sioux00}
& \tau_l \; \Sigma_{abc} = \Sigma_{acb} \; \tau_l , \qquad
\tau_m \; \Sigma_{abc} = \Sigma_{cba} \; \tau_m , \qquad
\tau_r \; \Sigma_{abc} = \Sigma_{bac} \; \tau_r \qquad &\\
\label{sioux01}
& t \; \tau_\alpha = \tau_\alpha \; t \qquad \qquad
\forall \; a,b,c = 0,1,2,3, \; \forall \; \alpha=l,m,r & \\
\label{sioux1}
& ( \tau_\alpha \; \tau_\beta \; \tau_\gamma)^2 = id \qquad\forall
\alpha, \beta, \gamma = l,m,r & \\
\label{sioux2}
& \tau_l \tau_m \tau_r \; I = I \; \tau_l \tau_m \tau_r,
\qquad\qquad
\tau_l \tau_r \tau_l \; I = I \; \tau_l \tau_r \tau_l &
\end{eqnarray}
The linear transformation $l_{tr}$ satisfies in addition:
\begin{eqnarray}
& l_{tr}^2=id, \qquad l_{tr} \; t = t \; l_{tr} & \label{rell1} \\
& l_{tr} \; t_l \; l_{tr} t_l = \epsilon_l, \qquad \label{rell2}
l_{tr} \; t_m \; l_{tr} t_m = id , \qquad
l_{tr} \; t_r \; l_{tr} t_r = \epsilon_r &
\end{eqnarray}
Using relations (\ref{sioux00}) to (\ref{rell2}), it is possible to
show that
the completed system (\ref{ltri},~\ref{ltril},~\ref{ltrir})
is left invariant by the action of the group $\Gamma_{triang}$.
The system is also invariant under the gauge transformations
leaving the form (\ref{rsawu}) stable.
(Hint: The gauge transformations
leaving the form (\ref{rsawu}) stable satisfy
$ \; {}^t\gamma \; \Sigma_{abc} \; \gamma \; \simeq \; \Sigma_{abc} \;
$
when $a,b=1,2$).
Moreover the generic 32-vertex invertible solutions of the
completed system (\ref{ltri},~\ref{ltril},~\ref{ltrir}) satisfy the
free-fermion conditions (\ref{ff32a},~\ref{ff32b}).
It is clear from (\ref{ltri},~\ref{ltril},~\ref{ltrir})
and (\ref{sioux00}) to (\ref{rell2}) that the realization of the
group $\Gamma_{
triang}$
is finite, when conditions (\ref{ff32a}, \ref{ff32b}) are fulfilled.
One should also notice that any linearization condition of the type of
(\ref{ltri})
is a set of {\it quadratic conditions, whatever the size of the matrix
is}.
Indeed they mean that the matrix product of $R$ with some linear
transformed
${\cal L}(R)$ of $R$
is proportional to the unit matrix, i.e:
\begin{eqnarray}
R \cdot {\cal L}(R) \simeq \; \mbox{unit matrix}
\end{eqnarray}
and this is a set of quadratic conditions.
{\bf Remark}: The invertible solutions of (\ref{ltri}) form a group for
the ordinary
matrix product, since $I \cdot l_{tr}$ is an automorphism of the group of
invertible
matrices of the form (\ref{rsawu}), i.e.
$I(l_{tr} (R_1 \cdot R_2)) =I( l_{tr}(R_1)) \cdot I( l_{tr}(R_2))$.
The extra conditions added when completing
the system break this in such a way that the ordinary matrix product of
three solutions is
another solution.
In other words, if $R_1, \; R_2, \; R_3 \; \in {\cal V}$, then
$ R_1 \cdot R_2 \cdot R_3 \; \in {\cal V} $, while $R_1 \cdot R_2 \notin {\cal V}$.
This was actually already the case for solutions of (\ref{ff8}), but
the mechanism is more subtle here
as conditions (\ref{ff32a}, \ref{ff32b}) imply $\Delta_1=+ \Delta_2$.
\section{32-vertex model on the cubic lattice}
We now turn to a solution of the tetrahedron
equations~\cite{SeMaSt95,Ko94,Hi94}.
Let $R$ be of the form
\begin{eqnarray}
\label{serg}
R = \left (\begin {array}{cccccccc}
d&0&0&-a&0&-b&c&0\\0&w&x&0&y&0&0&z\\0&
x&w&0&z&0&0&y\\-a&0&0&d&0&c&-b&0\\0&-y&z&0&w&0&0&-x\\b&0&0&c&0&d&a&0\\
c&0&0&b&0&a&d&0\\0&z&-y&0&-x&0&0&w\end {array}\right )
\end{eqnarray}
The form of (\ref{serg}) is stable \footnote {Notice that the form
(\ref{serg}) is not stable by the circular permutation of the three spaces
$\{l,m,r\}$.} under the group $\Gamma_{cubic}$, and it is natural to look
for invariants of $\Gamma$ in the space of parameters
$\{a,b,c,d,x,y,z,w\}$~\cite{FaVi93}.
There exist five algebraically independent quadratic polynomials in the
entries, transforming
covariantly, and with the same covariance factors under all generators
of $\Gamma_{cubic}$.
They are:
$$ ax, \quad by, \quad cz, \quad dw ,\quad \mbox{ and } \quad
Q = a^{2}+c^{2}-d^{2}-y^{2}-b^{2}+x^{2}-w^{2}+z^{2}\; . $$
We thus have four algebraically independent invariants of
$\Gamma_{cubic}$, say for example
$$ \chi_1 = {{ax}\over{dw}}, \quad
\chi_2 = {{by}\over{dw}}, \quad
\chi_3 = {{cz}\over{dw}}, \quad \mbox{and} \quad
\chi_0 = {{Q}\over{dw}}. $$
A complete analysis shows that there is no other algebraically
independent invariant of
$\Gamma_{cubic}$. A numerical and graphical
study~\cite{BeMaVi91e}, shows
how ``big'' the realization of $\Gamma$ is for generic values of the
above invariants.
These invariants are completely specified in the
solution~\cite{SeMaSt95}, for which
\begin{eqnarray}
& \chi_1 \; = \; \chi_2 \; = \; \chi_3 \; = \; 1 & \label{chi1} \\
& \chi_0 =0 & \label{chi0}
\end{eqnarray}
Out of the four invariants, $\chi_0$ plays a special role.
{\em If $\chi_0=0$, then the
action of $I$ linearizes quite in the same way as in the previous
cases}.
Notice that, strictly speaking, condition (\ref{chi0}) is not so much
an assignment of value to the invariant $\chi_0$ but rather a
vanishing condition for the covariant quantity $Q$.
Recall that assigning a definite value to an
invariant object is meaningful whatever this value is.
On the contrary covariant objects cannot
be assigned a value unless this value is zero.
When $Q=0$, one gets
\begin{eqnarray}
\label{lcub}
I(R) \; \simeq \; l_{c}(R)
\end{eqnarray}
The linear transformation $l_{c}$ may be written
\begin{eqnarray*}
l_{c} = \; t \; \Sigma_{030} \; gr,
\end{eqnarray*}
where $t$ is transposition and $gr$ is
a grading changing the sign of the entries of $R$ belonging to the same
block, say
$\{x,y,z,w\}$.
Notice that the definition of $l_{c}$ is not unique, due to the very
specific form of
(\ref{serg}).
Notice also that $Q =0$ is one of two quadratic conditions
ensuring the equality of the
determinants of the two blocks of $R$ (see~(\ref{blocks})). The other one
is not stable under $\Gamma$.
The linear transformation $l_{c}$ satisfies
\begin{eqnarray}
& l_{c}^2=id, \qquad l_c \; t = t \; l_{c}, & \label{relc1} \\
& l_{c} \; t_\alpha \; l_{c} t_\alpha = \epsilon_\alpha, \qquad
\forall \alpha=l,\; m, \; r &
\label{relc2}
\end{eqnarray}
Any matrix of the form (\ref{serg}) with $Q=0$ obeys
\begin{eqnarray}
& I(R) \; \simeq \; l_{c} (R) & \label{lintet1} \\
& I \; t_\alpha \; (R) \; \simeq \; l_{c}\; t_\alpha \; (R) \qquad
\forall \alpha = l,m,r &
\label{lintet2}
\end{eqnarray}
Using (\ref{relc1},\ref{relc2}), it is straightforward to show that
the complete
system (\ref{lintet1},\ref{lintet2}) is invariant under $\Gamma_{cubic}$
and
that the orbit of $R$ is finite.\footnote{This is also the case for
the bidiagonal solution of the ``constant'' tetrahedron equations
of~\cite{Hi93b}.}
The study of the additional conditions (\ref{chi1}) would take us
beyond the scope of this letter,
but we may make a few remarks.
The first remark is that since among conditions (\ref{chi1},\ref{chi0}),
only (\ref{chi0}) has to do with the finiteness of the realization
of $\Gamma$, (\ref{chi1}) may have nothing to do with
free-fermion conditions.
They are {\em additional constraints} making the resolution
of the tetrahedron equations possible, and this may be understood as follows.
The tetrahedron equations are in essence a compatibility condition for
the existence of
non-trivial solutions of the ``propagation properties''~\cite{Ba73}
(alias ``Zamolodchikov
algebra''~\cite{ZaZa79}, alias ``vacuum curves''~\cite{Kr81,Ko94},
alias ``pre-Bethe Ansatz''
equations~\cite{BeMaVi92,BeBoMaVi93}):
\begin{equation}
\label{pba}
R \; \pmatrix{1 \cr p} \otimes \pmatrix{1 \cr q} \otimes \pmatrix{1 \cr
r} \; \simeq \;
\pmatrix{1 \cr p'} \otimes \pmatrix{1 \cr q'} \otimes \pmatrix{1 \cr
r'}
\end{equation}
What conditions (\ref{chi1},\ref{chi0}) ensure is the existence, for
fixed $R$,
of a {\em one-parameter family of solutions} of~(\ref{pba}).
In the case we consider here, the family happens to be parametrized by a
{\em curve of genus larger than one}.
By eliminating $\{q,q',r,r'\}$ (resp. $\{p,p',r,r'\}$ or
$\{p,p',q,q'\}$)
from (\ref{pba}), one gets conditions relating $p,p'$, (resp. $q,q'$
and $r,r'$).
Such relations are generically of degree 8 (biquartics).
One effect of (\ref{chi1},\ref{chi0}) is that they all reduce to
{\em asymmetric} biquadratic relations, defining three genus one curves
of the form
\begin{eqnarray} \label{asym}
& x^2 y^2 -1 + ( y^2 - x^2) \; \kappa_{xy} =0 &
\end{eqnarray}
\begin{eqnarray*}
& \mbox{with} \quad \kappa_{pp'}= {\displaystyle {b c \; ( d^2-a^2)}
\over{\displaystyle a d \; (b^2-c^2)}} , \quad
\kappa_{qq'} = - {\displaystyle {a c \; (b^2+d^2)}
\over{\displaystyle b d \; (a^2 + c^2)}}, \quad
\kappa_{rr'} = {\displaystyle { a b\; (c^2-d^2)}
\over{\displaystyle c d \; (a^2-b^2) }}. &
\end{eqnarray*}
These three elliptic curves have different (algebraically independent)
moduli.
Their asymmetric character may be an obstacle to the use of (\ref{pba})
in the construction of the Bethe Ansatz states~\footnote{R.J. Baxter,
private communication.}, since the composition of relations of type
(\ref{asym})
reproduces the same type of relations, but alters the value of $\kappa$ by:
\begin{eqnarray} \label{land}
\kappa \longrightarrow {1\over 2} \left( \kappa + {1\over{\kappa}} \right)
\end{eqnarray}
Exceptional values of $\kappa \; (\pm 1, \infty)$, yielding a rationalization
of (\ref{asym}), are fixed points of (\ref{land}).
For these exceptional values, in particular $\kappa=\infty$,
obtained with $d=0$,
the construction of a 3D Bethe Ansatz may be envisaged.
\section{Conclusion}
We have shown, through specific examples, how free--fermion conditions
turn into degeneration conditions of our groups $\Gamma$:
the generically non-linear (rational) infinite realization of $\Gamma$
becomes a linear finite group.
We believe this is a characteristic feature of 2D free-fermion models.
We have shown that the known vertex solution of the tetrahedron equations
does have such a feature.
An appealing issue is to decide whether or not such a statement can be
made about other 3D and higher dimensional models.
Of course the full answer will come from linking directly the phenomenon
we describe with explicit calculus using grassmannian variables.
The particularly simple form of the conditions (combinations of
products of entries with plus and minus signs),
and the linearization process of the inverse
should stem from elementary properties of exponentials of quadratic forms
in anticommuting variables.
Producing new solutions of the tetrahedron equations is another
challenging problem.
What could be done is to look for forms of the matrices $R$ enjoying the
linearization property we have described. This is a rather simple
way to produce ``reasonable'' Ans\"atze for $R$.
The next step would then be to study the so-called propagation properties
(see above) rather than confronting directly the tetrahedron equations
themselves.
Indeed these simpler equations, because they govern the construction of
Bethe Ansatz states --a basic in the field--, underpin 2D, 3D, and higher
dimensional integrability.
\bigskip
{\bf Acknowledgments}: {JMM would like to thank Pr. R.J. Baxter for an
invitation at the Mathematics Dept. A.N.U., Canberra, and for many fruitful
discussions. JMM would also like to thank
Pr. F.Y.Wu for interesting discussions on the 32-vertex model.
CMV would like to thank the members of the High Energy Physics group of SISSA,
Trieste, where this work was completed, for their warm hospitality, and
acknowledges support from the EEC program ``Human Capital and Mobility''.
Both authors benefited from stimulating discussions with M. Bellon and M.
Talon.}
| 11,694 |
\section{Introduction}
The structure of parton multiplicity correlations within QCD jets
and in particular their self-similar properties
have been recently studied\cite{DD93,BMP,*ISMD92,OW93}. The main result is twofold~:
\begin{itemize}
\item The structure of a QCD jet is {\it multifractal}, in the sense that
the multiplicity density of gluons does not occupy uniformally the available
phase-space when it is computed in the Double-Leading-Log
approximation (DLA) of the perturbative QCD expansion. More precisely,
if one measures the inhomogeneities of the multiplicity distribution using the scaled factorial moments of order $q$, one writes~:
\begin{equation}
\label{inter}
{\cal F}_q(\Delta)\equiv {\moy{n(n-1)..(n-q+1)}_\Delta \over \moy{n}_\Delta^q} \propto \Delta^{(q-1)(1-{\cal D}_q/d)}
\end{equation}
where $n$ is the multiplicity of partons registered in a small phase-space interval
$\Delta.$ ${\cal D}_q $ is called the {\it fractal} dimension of rank $q$
of the density of gluons while $d$ is the overall dimension of the phase space under
consideration (in practice, $d=2$ for the solid
angle, and $d=1$ if one has integrated over, say the
azimutal angle, by keeping fixed the opening angle with respect to the jet axis). The multiplicity distribution is uniform if $
{\cal D}_q \equiv d,$ while it is fractal if it is smaller. Then {\it
multifractality} is for a $q-$dependent dimension.
In the DLA, assuming a fixed QCD coupling constant $\alpha_s$, the result is the following:
\begin{equation}
\label{diminter}
{\cal D}_q\eq\gamma_0\ \frac{q+1}{q}
\end{equation}
where $\gamma_0$, considered to be small, is given by~:
\begin{equation}
\label{go}
\gamma_0^2 \eq 4 N_c\ \frac{\alpha_s}{2\pi} \ ;\ \ \ N_c=3
\end{equation}
\item In the running coupling constant scheme, the jet structure is modified
by scaling violation effects. It leads to a multifractal dimension slowly varying with
the angular variables of observation. Furthermore, one observes\cite{BMP}, at small angles ${\cal D}_q \to d,$ i.e. a dynamical saturation of multifractality.
The calculation gives~:
\begin{equation}
\label{inter5}
{\cal D}_q(z)
\simeq \gamma_0\left(\alpha_s(E\theta_0)\right) \frac{1+q }{q}\ \frac{2}{1+\sqrt{1-z}}
\end{equation}
where the scaling variable $z$ is defined as:
\begin{equation}
\label{Scal}
z=\frac{\log{\theta_0/\theta}}{\log{E\theta_0/\Lambda}}.
\end{equation}
Note that the fractal dimension is no more constant and depends
on both the angular direction($\theta_0$) and angular aperture ($\theta$) of the observation window. In practice, the {\it fractal} dimensions have a tendency to increase with $z$ and $\gamma_0$ and to reach the {\it saturation} point where
they become equal to the full dimension $d.$ This saturation effect comes
from the increase of the coupling constant at larger distances and
signals the onset of the non-perturbative regime of
hadronization.
\end{itemize}
We want to reconsider these results by going beyond the approximations
made in Refs.\cite{DD93,BMP,OW93}. Some results exist which include
Next-Leading-Log corrections to the anomalous dimensions\cite{DD93}, but
they do not include the energy-conservation constraints. Indeed, it is well known that pertubative QCD resummation in the leading log approximation (DLA) predict too strong global multiplicity moments, at least when compared to the hadronic final state observed in experimental data on jets\cite{Tran}. In fact, as pointed out in Ref.\cite{D93}, the energy conservation (EC) constraint at the triple parton vertices may
explain the damping of multiparticle moments observed in the data. While the EC correction is perturbatively of higher order ($\alpha_s^{3/2}$ as compared to $\sqrt{\alpha_s})$ and gives a rather moderate effect on the mean multiplicity, the calculation of multiplicity moments of higher rank $q$ gives a contribution of order $q^2\alpha_s^{3/2}$ which become important already for $q=2.$ As we shall demonstrate, this effect is even more important for local correlations in phase-space.
The main goal of our paper is the analytical calculation of
the EC effects on the fractal dimensions ${\cal D}_q$
in the one and two-dimensional angular phase-space. We show that the self-similar structure of correlations is preserved beyond DLA and we give their analytic expression.
As an application we calculate the EC effects on the fractal dimensions of QCD jets produced in $e^+\!-e^-$ collisions at LEP energy\cite{OPAL92,ALEPH92,DELPHI92}.
We find that these effects cannot be neglected either in the phenomenological analysis
or in the theoretical considerations based on the perturbative QCD expansion.
Under the EC effect, the fractal dimensions increase and, in the case of a running $\alpha_s,$ strengthen the saturation phenomenon.
In particular, the fractal dimension ${\cal D}_2$ becomes now of order unity, in agreement with both numerical QCD simulations and experimental data. Note, from the theoretical point of view, the interesting interplay which appears with the property of KNO scaling of multiplicity
distributions\cite{KNO}.
The plan of the paper is the following. In the nextcoming section 2, we recall the QCD formalism for the generating function of multiplicity moments
for the {\it global} jet multiplicity, formulate the EC problem and give an explicit solution for the first global moments.
Section 3 is devoted to the {\it local} multiplicity study of QCD jets, i.e. the correlations/fluctuations in small angular windows (angular intermittency\cite{BMP}). We compute
the fractal dimensions and compare our analytical results (including rather significant EC effects) both with the relevant experimental data and with a computer simulation of QCD jets\cite{DJLM,*Cracow}. Our conclusions can be found
in section 4.
\section{The Global Jet multiplicity distribution}
We will use the following notations:
${\cal Z}(u,Q)$ stands for the QCD generating function of the global factorial multiplicity moments $I\!\! F_q$ (for a gluon of virtuality Q decaying into gluons). The total jet
multiplicity (first global moment) is denoted $N(Q).$
\begin{equation}
\label{notations}
I\!\! F_q\ \equiv {\moy{n(n-1)..(n-q+1)}}_{jet} \eq\ \frac{\partial^q \cal Z}{\partial u^q}\big|_{u=1}.
\end{equation}
When one takes into account energy conservation at the fragmentation vertex,
${\cal Z}$ is governed by an evolution equation which can be sketched as a classical fragmentation mechanism:
\bfg{thb}
\fg{10}{master1.ai}
\end{figure}
\noindent where the black points represent the
generating function of the jet and its sub-jets
while the arrows stand for the jet and sub-jet axis directions.
One obtains the well-known QCD evolution equation\cite{Tran} ~:
\begin{eqnarray}
\label{Z}
&\displaystyle{\frac{\partial {\cal Z}(u,Q)}{\partial \log(Q}} &\eq \int_0^1 \gamma_0^2\frac{dx}{x}\Big|_+{\cal Z}(u,Qx)\ {\cal Z}(u,Q(1-x))\nonumber\\
&&\eq \int_0^1 \gamma_0^2\frac{dx}{x}\left[{\cal Z}(u,Qx){\cal Z}(u,Q(1-x))-{\cal Z}(u,Q) \right]
\end{eqnarray}
where $\frac{dx}{x}\big|_+$ is the principal-value distribution coming from the triple gluon vertex and $\theta$ is the angular aperture of the jet which plays the r\^ole of a time variable. The energy of the primordial parton is $E$ and its virtuality is $Q\simeq E\theta$.
Note that this equation can be enlarged to include other non-leading-log
QCD contributions (the so-called Modified Leading-Log Approximation
MLLA\cite{Tran}).
In fact an earlier study\cite{D93} has shown that the EC effects
play the crucial role in correcting the global moments. We shall thus
focus our discussion on the EC effects
but our method can be enlarged without major problems to the full MLLA equation.
\subsection{The mean multiplicity}
The solution for the mean multiplicity, $N(Q)$ is known\cite{Tran,D93}.
let us rederive it in a language appropriate for further generalization to the correlation problem.
Using Eq. (\ref{Z}), one obtains by differentiation:
\begin{equation}
\label{N}
\frac{\partial N(Q)}{\partial \log(Q)} \eq \int_0^1 \gamma_0^2 \frac{dx}{x}\left(N(Qx)+N(Q(1-x))- N(Q)\right).
\end{equation}
Let us now distinguish the frozen-coupling regime and that with running $\alpha_s.$
\begin{itemize}
\item{\it Frozen coupling constant}
In this case, Eq. (\ref{N}) can be solved using a power-like behaviour for $N$:
\begin{equation}
\label{Power}
N(Q)\eq N_0 Q^\gamma
\end{equation}
with the following link between $\gamma$ and the DLA value $\gamma_0$
of Eq.(\ref{go})~:
\begin{equation}
\label{gamma}
\gamma_0^2=\frac{\gamma^2}{1-\chi(\gamma)}
\end{equation}
The function $\chi$ is defined by:
$$
\chi(x)\ \eq x \left({\Psi(1+x)-\Psi(1)}\right)
$$
where ${\displaystyle \Psi(z)={\partial \log{\Gamma(z)}\over \partial z}}.$
When $x \to 0,$ $\chi \simeq x^2$, $\gamma \simeq \gamma_0-\gamma_0^3/2$ , which confirms that
the EC correction to $N$ is a (${\cal O}(\alpha_s^{3/2})$) correction.
However, this correction appears in the exponent of the multiplicity and is thus not negligeable at LEP energies. In practice, it "renormalizes" the multiplicity
exponent, see Fig.\ref{gog}.
At LEP energy ($E = M_{{\cal Z}_0}$), the correction can reach 20\%.
Note that $\chi$
can be numerically replaced by $x^2$ up to $x=1.5$.
\bfg{thb}
\fg{10}{gog.ai}
\caption{
\protect\small{\bf a.}$\gamma_0$ as a function of the "renormalized"
value $\gamma$; The grey band corresponds to the average value of
$\gamma_0$ at LEP.}
\label{gog}
\end{figure}
\item {\it Running coupling constant}
Let us now reconsider equation (\ref{N}) when $\alpha_s$ is running.
One has:
$$
\gamma_0^2 \eq 4 N_c{\alpha_s(Q)\over 2 \pi}\eq {c_0^2\over\log{(Q/\Lambda)} }\ ;\ \ c_0\eq\frac{6}{\sqrt{33-2n_f}},
$$
where $n_f$ is the number of flavors ($n_f$ = 5 at LEP).
The solution of Eq.(\ref{N}) is of the form $N(Q) = N_0 \expp{2c\sqrt{\log (Q)}}$, where $c$ is a constant. Using the following general identity,
\begin{eqnarray}
\label{Gi}
&N(Q f(x))/N(Q)\simeq
\expp{2c \sqrt {\log{Qf(x)/\Lambda}}
-2c \sqrt {\log{Q/\Lambda}}}\nonumber\\
&\simeq
\expp{c\ \log f(x)/\sqrt {\log{Q/\Lambda}}}\eq \left[f(x)\right]^{\ c /\sqrt{\log{Q/\Lambda}}}.
\end{eqnarray}
one obtains from Eq.(\ref{N}):
\begin{eqnarray}
\label{RN}
&N(Q)\simeq N_0\expp{2 c\sqrt{\log{Q/\Lambda}}}\nonumber\\
&c^2\simeq c_0^2\left(1-\chi{(\gamma )}\right),
\end{eqnarray}
where $\gamma \equiv c/\sqrt{\log{Q/\Lambda}}.$ As already pointed out, $\chi(\gamma) \simeq \gamma^2$ when
$\gamma$ is small, that is the EC effect on the exponent of the
mean multiplicity is a non-leading correction. However, as in the frozen coupling case, it "renormalizes" the
behaviour of the multiplicity.
\end{itemize}
\subsection{The global second moment}
If one goes to the second derivative of equation (\ref{Z}), one obtains the evolution equation for the global factorial moment, $I\!\! F_2$~:
\begin{equation}
\label{GF2}
\frac{\partial I\!\! F_2 (Q)}{\partial \log(Q)} \eq \int_0^1 \gamma_0^2\frac{dx}{x}\left(I\!\! F_2(Qx)+2 N(Qx) N(Q(1-x))+I\!\! F_2(Q(1-x))-I\!\! F_2(Q)\right).
\end{equation}
The way of dealing with this equation is to assume
that $ I\!\! F_2(Q)/N^2(Q)$ is a slowly varying function of
$Q.$ This property, which is one of the KNO scaling
relations $ I\!\! F_q(Q)\propto N^q(Q),$ is known to be correct for
the moments predicted by QCD\cite{Tran}. Using this structure together with the power-like behaviour of $N $, (see Eq. (\ref{Power})) gives~:
\begin{equation}
\label{F2}
I\!\! F_2=\left(\frac{2\Gamma^2(\gamma)}{\gamma\Gamma(2\gamma)(3+\chi(2\gamma)-4\chi(\gamma))} \right)\left( N(Q)\right)^2.
\end{equation}
The result is displayed on Fig.\ref{f2b}. The EC effects are
clearly seen as a serious damping of the KNO ratio $I\!\! F_2/N^2$.
\bfg{thb}
\fg{10}{f2b.ai}
\caption{\protect\small $I\!\! F_2$ as a function of $\gamma$. The DLA result (4/3) is obtained when $\gamma\to 0$. The vertical grey band is for $\gamma_0$ at LEP , while
the horizontal one corresponds to the experimental value quoted in ref. \protect\cite{OPAL92} for one jet (in the forward hemisphere).}
\label{f2b}
\end{figure}
The running of the coupling constant does not modify the result which is very close to the experimental data\cite{OPAL92} ($|bigf_2$ in one hemisphere).
One notices that Eq. (\ref{F2}) leads to a weak violation of KNO scaling due to the dependence of $\gamma$ on $Q.$
\section{The Local Jet multiplicity distribution}
Let us come now to the main topics of our paper, namely the computation of the local correlations between partons.
The characteristic feature of the jet multiplicity structure is that the evolution
equation for the local density of partons is linear. It
can be deduced from the branching structure of jet fragmentation in QCD.
\bfg{thb}
\fg{16}{master2.ai}
\end{figure}
Let us denote by $H(u,Q)$ the generating function for the {\sl density} of factorial moments:
\begin{equation}
\label{density}
H_q\equiv\frac{\partial ^q H}{\partial u^q}\Big|_{u=1}\eq\frac{{F}_q(\Delta)}{\Delta}\equiv \frac{{\moy{n(n-1)..(n-q+1)}}_\Delta}{\Delta}.
\end{equation}
The QCD evolution equation of the multiplicity in an
observation window of size $\theta$ pointing in a definite direction $\theta_0$ from the jet axis is schematically described in the figure. there are 2 contributions, one coming from the parton with a fraction $x$ of the available energy, the other coming from the branch of the fragmentation process with $1-x.$ Once substracted the variation of the phase-space from $\theta_0$ to $\theta$ by defining moments of the {\it density} as in Eq.(\ref{density}), the only evolution comes from the energy degradation along the branch. In the DLA approximation, only one branch contributes to the hadron density evolution, since one parton keeps essentially undisturbed by the branching. In the EC case where we do not neglect the recoil effect, the two branches contribute. Note that one has to take into account the corresponding loss of virtuality $Qx$ and $Q(1-x)$ respectively. One obtains~:
\begin{eqnarray}
\label{H}
&\displaystyle{\frac{\partial H(u,Q)}{\partial t}}
&\eq \int_0^1\gamma_0^2\ \frac{dx}{x}\Big|_+\left( H(u,Qx)+ H(u,Q(1-x))\right)\nonumber\\
&&\eq \int_0^1\gamma_0^2\ \frac{dx}{x}\left(\left[ H(u,Qx)-1\right]
+\left[ H(u,Q(1-x))-H(u,Q)\right]\right)
\end{eqnarray}
where $\log{\theta_0/\theta}\equiv t$ plays the r\^ ole of the evolution parameter and the phase-space splitting depends on the dimensionality, namely $\Delta\propto \theta^d$. One branch of the iteration contributes to
$H(u,Qx)$ and the other to $H(u,Q(1-x))$ in the integrand, while the negative contributions
are necessary to ensure the finiteness and unitarity conditions,
namely $H(1,Q) \equiv 1$ and $\sum_\Delta {\partial H(u,Q)}/{\partial u}\big|_{u=1} = \sum_\Delta
F_1(\Delta) \equiv N(Q).$
The solution of Eq.(\ref{H}) is obtained by using the KNO scaling property of the multiplicity distribution inside a cone of fixed angular aperture $\theta$.
In terms of the generating function $H$, it writes:
\begin{equation}
\label{KNO}
H(u,\lambda Q)\eq H(uN(Q\lambda)/N(Q),Q)
\end{equation}
Inserting the property (\ref{KNO}) into Eq.(\ref{H}), and performing the $q^{th}$ derivative, one gets:
$$
\frac{\partial H_q}
{\partial t} \eq H_q \ \int_0^1 \gamma_0^2 \frac{dx}{x}\left(\left(
{\frac {N(Qx)}{N(Q)}}\right)^q
+\ \left({\frac {N(Q(1-x))}{ N(Q)}}\right)^q - 1\right)
$$
\begin{eqnarray}
\label{Hq}
\eq H_q \ \int_0^1 \gamma_0^2 \frac{dx}{x}\left(
x^{q\gamma} +\
(1-x)^{q\gamma} - 1\right),
\end{eqnarray}
where we have used Eq.(\ref{Gi}). Notice that Eq. (\ref{H}), as well as (\ref{Z}), can be understood in terms of classical fragmentation models\cite{BMP}. The connection between the present formalism and classical fragmentation is also a consequence of the KNO relation (\ref{KNO}).
As a check of this procedure, let us recover the known results
of the DLA approximation\cite{BMP}. Neglecting the EC terms in Eq.(\ref{H}), one obtains~:
\begin{equation}
\label{DLAH}
\frac{\partial H(u,Q)}{\partial t}
\eq \int_0^1\gamma_0^2\ \frac{dx}{x}\left( H(u,Qx)-1\right),
\end{equation}
from wich one obtains the fractal dimensions (\ref{diminter}),(\ref{inter5}).
\subsection{ Frozen coupling }
Equation (\ref{Hq}) reads~:
\begin{equation}
\label{Froze}
\frac{1}{H_q}\frac{\partial H_q}{\partial t}
\eq {\gamma_0^2}\ \frac{(1-\chi(q\gamma))}{q\gamma}\eq \frac{\gamma}{q}\ \frac{1-\chi(q\gamma)}{1-\chi(\gamma)}
\end{equation}
where one makes use of relation(\ref{gamma}).
The corresponding corrections to the normalized factorial moments (\ref{inter}) and
their fractal dimensions can be worked out~:
\begin{equation}
\label{Ecdim}
{\cal D}_q
\eq \frac{\gamma }{q}\left[1+q+\frac{\chi(\gamma)-\chi(q\gamma)}{1-q}\right]
,
\end{equation}
\bfg{thb}
\fg{8}{ec_dims.ai}
\caption{\protect\small The multifractal dimensions ${\cal D}_2, {\cal D}_3, {\cal D}_4 $ as functions of the coupling $\gamma$; The straight line is the DLA result for ${\cal D}_4$ as an example(\ref{diminter}). The saturation limit is the dotted line at $d = 1$ .
}
\label{dim}
\end{figure}
Notice that, while the correction to the unnormalized factorial moments,
see Eq. (\ref{Ecdim}), is
a factor of
order $(1-q^2\gamma^2)/(1-\gamma^2)$ in the exponent and can be important,
the
normalized ones are less sensitive to EC corrections. Since $\chi(\gamma)\simeq \gamma^2$ up to $\gamma \simeq 1.5$, the fractal dimension
(\ref{Froze}) reads
\begin{equation}
\label{Apdim}
{\cal D}_q
\simeq \frac{\gamma (1+\gamma^2)(q+1)}{q},\ \ q\gamma<1.5,
\end{equation}
leading to an enhancement factor with respect to the multiplicity exponent at the same order ($\alpha_s^{3/2}$).
However, when $q\gamma > 1.5,$
the modification becomes more important and $q-$dependent.
One observes an important increase of the fractal dimension, see
Fig.\ref{dim}.
\subsection{ Running coupling }
Let us consider again Eqs.(\ref{Hq}), (\ref{Froze}) with $\gamma_0$ defined with the running coupling and $\gamma$ as in formula (\ref{RN}). Neglecting both the variation of the coupling in the EC correction factors (higher order terms) and the $x$-dependence of the coupling (this approximation have been shown to be quite right in the range ($z<.6$) in ref. \cite{BMP}), one obtains~:
\begin{equation}
\label{Rundim}
{\cal D}_q(z)
\simeq \frac{\gamma }{q}\left[1+q+\frac{\chi(\gamma)-\chi(q\gamma)}{1-q}\right]\ \frac{2}
{1+\sqrt{1-z}},
\end{equation}
where $\gamma$ is expressed as in formula (\ref{gamma}). Notice that the saturation effect is stronger due to the running of the coupling (see, e.g., Fig.\ref{ecrun} for the one-dimensional case).
\bfg{thb}
\fg{13}{ecrun.ai}
\caption
{ $\log[{\cal F}_q(z)/{\cal F}_q(0)]$, as a function of the scaling variable, $z=\log[\theta_0/\theta]/\log[E\theta_0/\Lambda]$. {\bf a:} $q=3$, with and without EC corrections; {\bf b:} $q=2$ (lower), $q=3$ and $q=4$ (upper), including EC corrections; the value $\gamma=.44$ is consistent with the LEP value of $\alpha_s$.
}
\label{ecrun}
\end{figure}
As a phenomenological application of our results, we perform a comparison of our predictions with experimental
results, which while yet preliminary, have been presented as a first detailed analysis of correlations in a jet. As stressed in the experimental paper\cite{Vietri}, the comparison of analytical QCD predictions with data on correlations is not easy due to the hadronization effects. The best candidate for a comparison is the study of particle correlations in a ring
$\theta_0 \pm \theta$ around the jet axis. The variation on $\theta$ allows one to obtain the dependence in $z,$ while considering as an observable the ratio $\log[{\cal F}_q(z)/{\cal F}_q(0)]$ minimizes the hadronization corrections\cite{BMP}.
The results are displayed on Fig. \ref{McDat}. The
result of a Monte-Carlo simulation of the QCD parton cascade \cite{DJLM,*Cracow} is also shown on the same plot. The simulation is useful to take into account
the corrections with respect to analytical calculations which occur when the parton cascade is correctly reproduced, e.g. overlap effects between angular domains, subasymptotic energy-momentum effects, hadronization effects (in part) etc.... For this comparison we make use of the following parameters : $\alpha_s(E\theta_0) =.135 $ for $\theta_0=25^\circ $(corresponding to $\alpha_s=.12$ for the whole jet with $Q_0=45$ Gev). This leads to $\gamma=.44$ or, from Eq.(10),
$\gamma_0 =.5$.
\bfg{thb}
\fg{10}{ec_mc_exp.ai}
\caption
{\protect\small Phenomenological predictions for $\log[{\cal F}_q(z)/{\cal F}_q(0)]$ as a function of the scaling variable $z$.
Upper curves,triangles and circles: $q=3$, lower ones $q=2$.
Continuous lines: the analytical calculations from \protect\ref{inter5}; circles: Preliminary Delphi data from Ref.\protect\cite{Vietri}; squares: Numerical data from the Monte-Carlo simulation\protect\cite{DJLM,*Cracow}}
\label{McDat}
\end{figure}
The analytical QCD predictions happen to be in reasonable agreement with both experimental results and numerical Monte-Carlo simulations, taken into account the approximations considered in the calculation. In particular, some bending of the factorial moments at small aperture angle $\theta$ is qualitatively reproduced. This effect is a typical predictions of the running coupling scheme, which is thus favoured by the data with respect to the frozen coupling case.
The full study of hadronization effects remain to be done. However, we may
notice that when the slope of the intermittency indices (Fig. \ref{ecrun}) goes to zero, the effective coupling $\gamma$ is around $1$ and one enters the not well known strong coupling regime of QCD. As a matter of fact, the only free parameter left in the Monte-Carlo simulation is the effective cut-off of the parton cascade i.e. the maximum value $\alpha_s,\ \alpha_0$, one considers in the development of the parton branching process. In the example considered in the figure, we have found $\alpha_0\simeq .54,$ which is a quite reasonable value for switching on hadronization. All in all it is a positive fact that Local Parton-Hadron Duality seem to work reasonably for such refined quantities as correlations, once some caution is taken to minimize non-perturbative QCD corrections.
\section{Conclusions}
The Energy Momentum Conservation effects (or recoil effects) on the correlation properties of QCD jets are thus under some analytical control. The method proposed to handle these effects is the search for scaling solutions of the QCD generating function of {\it local} multiplicity density moments in angular phase-space. It is interesting to note the deep connection between the KNO-scaling\cite{KNO} and the self-similar properties of correlations in the jet. This structure is reminiscent of general properties of fragmentation models and could be more fundamental than the level of approximation used to derive the result from QCD. Indeed, the same qualitative result is expected from the solution of the equation including EC terms plus other dominant next-to-leading effects (e.g. MLLA approximation scheme, see ref.(\cite{Tran})).
The phenomenological outcome of this study is that the analytical predictions come close to the experimental data and confirmed by the numerical simulation. Parton-Hadron duality for correlations seems to be supported provided hadronization effects can be minimized. Thus the self-similar structure of the QCD branching processes may eventually emerge from the background in the experimental analysis. Hadronization becomes dominant when the factorial moments bend and acquire a zero slope ($z\simeq 0.5$ in terms of the scaling variable of formula (\ref{Scal})). If one would insist to increase $z$ by reducing the opening angle of the detection window,
one would find properties closer to soft processes, and thus difficult to predict theoretically.
From the theoretical point of view, the partial cancellation of EC corrections observed in the scaled factorial moments, leading to a moderate correction to the behaviour determined by the leading-log approximation, is a consistency check of the perturbative resummation predictions. Note however that this remark does not hold for moments of high rank $q\ge 4$, where a stronger correction is expected. The question remains to know whether this indicates a limitation of the perturbative calculations or the signal of a dynamical mechanism.
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| 8,645 |
\section{Moduli spaces of stable maps}
\label{sec1}
\setcounter{equation}{0}
It was M. Gromov \cite{Gr} who first
suggested to construct (and constructed some) topological invariants of
a symplectic manifold $X$
as bordism classes of spaces of pseudo-holomorphic curves in $X$. Recently
M. Kontsevich \cite{Kn} suggested the concept of {\em stable maps} which gives
rise to an adequate compactification of these spaces. We recall here some
basic facts from \cite{Kn} about these compactifications.
Let $(C,p)$ be a compact connected complex curve with only double singular
points
and with $n$ ordered non-singular {\em marked points} $(p_1,...,p_n)$.
Two holomorphic maps
$(C, p) \to X, \ (C', p') \to X$ to an almost-Kahler manifold $X$ are called
{\em equivalent} if they can be identified by a holomorphic isomorphism
$(C, p) \to (C', p')$. A holomorphic map $(C,p)\to X$ is called {\em stable}
if it does not have infinitesimal automorphisms. In other words, a map is
unstable if either it is constant on a genus $0$ irreducible component of
$C$ with $< 3$ {\em special} ($=$ marked or singular) points or
if $C$ is a torus, carries no marked points and the map is constant.
A stable map may have a non-trivial finite automorphism group.
According to Gromov's compactness theorem \cite{Gr}, any sequence of
holomorphic maps
$C\to X$ of a nonsingular compact curve $C$ has a subsequence
Hausdorff-convergent to a holomorphic map $\hat C\to X$ of (may be reducible)
curve $\hat C$ of the same genus $g$ and representing the same total homology
class $d\in H_2(X,\Bbb Z)$. A refinement of this theorem from \cite{Kn} says
that equivalence classes of stable maps $C\to X$ with given $g,n,d$ form a
single
compact Hausdorff space --- the moduli space of stable maps --- which we
denote $X_{g,n,d}$. Here $g=\operatorname{dim} H^1(C,{\cal O})=1-
\chi (C \backslash C^{\mbox{sing}})$.
In the case $X=pt$ the moduli spaces coincide with Deligne-Mumford
compactifications ${\cal M} _{g,n}$ of moduli spaces of genus $g$ Riemannian
surfaces with $n$ marked points. They are compact nonsingular orbifolds
(i.e. local
quotients of nonsingular manifolds by finite groups) and thus bear the rational
fundamental cycle which allows one to build up intersection theory.
In general, the moduli spaces $X_{g,n,d}$ are singular and may have ``wrong"
dimension, and the idea of the program started in \cite{KM, Kn} is to provide
$X_{g,n,d}$ with virtual fundamental cycles insensitive to perturbations of the
almost-Kahler structure on $X$.
In some nice cases however the spaces $X_{g,n,d}$ are already nonsingular
orbifolds of the ``right" dimension.
A compact complex manifold is called {\em
ample} if it is a homogeneous space of its Lie algebra of holomorphic
vector fields.
\begin{thm}[\cite{Kn, BM}]
If $X$ is ample then all non-empty moduli spaces
$X_{n,d}$ of genus $0$ stable maps are compact nonsingular complex
orbifolds of ``right" dimension $\langle c_1(T_x),d\rangle +\operatorname{dim}_{\Bbb C}X+n-3$.
\end{thm}
Additionally, there are canonical morphisms $X_{n,d}\to X_{n-1,d}, \
X_{n,d}\to{\cal M} _{0,n},\ X_{n,d}\to X^n$ between the moduli spaces $X_{n,d}$
called {\em forgetful, contraction} and {\em evaluation} (and defined
by forgetting one of the marked points, forgetting the map and evaluating
the map at marked points respectively). We refer to \cite{Kn, BM} for details
of their construction.
In the rest of this paper we will stick to ample manifolds; we
comment however on which results are expected to hold in greater generality.
A number of recent preprints by B. Behrend -- B. Fantechi, J.Li -- G. Tian,
T. Fukaya -- K. Ono shows that Kontsevich's ``virtual fundamental cycle''
program is being realized successfully and leaves no doubts that these
generalizations are correct. Still some verifications are necessary in order
make them precise theorems.
\section{Equivariant correlators}
\label{sec2}
\setcounter{equation}{0}
The Gromov-Witten theory borrows from quantum field theory the name {\em
(quantum) correlators} for numerical topological characteristics of the moduli
spaces $X_{n,d}$ (characteristic numbers) and borrows from bordism theory
the construction of such correlators as integrals
of suitable wedge-products of various universal cohomology classes
(characteristic classes of the GW theory) over the fundamental cycle.
We list here some such characteristic classes.
\begin{enumerate}
\item Pull-backs of cohomology classes from $X^n$ by the evaluation
maps $e_1\times\dots\times e_n:X_{g,n,d}\to X^n$ at the marked points.
\item Any polynomial of the first Chern classes $c^{(1)},\dots ,c^{(n)}$ of the
line bundles over $X_{g,n,d}$ consisting of tangent lines to the mapped curves
at the marked points. One defines these line bundles (by identifying the
Cartesian product of the forgetful and evaluation maps $X_{n+1,d}\to
X_{n,d}\times X$ with the {\em universal stable map} over $X_{n,d}$) as normal
line bundles to the $n$ embeddings $X_{n,d}\to X_{n+1,d}$ defined by the
$n$ marked points of the universal stable map. We will call these line
bundles {\em the universal tangent lines} at the marked points.
\item Pull-backs of cohomology classes of the Deligne - Mumford spaces
by contraction maps $\pi: X_{n,d}\to {\cal M} _{0,n}$. We will make use of the
classes $A_I:= A_{i_1,...,i_k} $ Poincare-dual to fundamental cycles of fibers
of forgetful maps ${\cal M} _{0,n}\to {\cal M} _{0,k}$.
\end{enumerate}
We define the {\em GW-invariant}
\[ A_I\langle \phi _1,...,\phi _n \rangle _{n,d} :=
\int _{X_{n,d}} \ \pi ^*A_I \wedge e_1^*\phi _1 \wedge ... \wedge e_n^*\phi _n .\]
It has the following meaning in enumerative geometry: it counts the number
of pairs ``a degree-$d$ holomorphic map ${\Bbb C} P^1 \to X$
with given $k$ points mapped to given $k$ cycles, a configuration of $n-k$
marked points mapped to the $n-k$ given cycles''.
\bigskip
Suppose now that the ample manifold $X$ is provided with a Killing action
of a compact Lie group $G$. Then $G$ act also on the moduli spaces of stable
maps. The evaluation, forgetful and contraction maps are $G$-equivariant,
and one can define correlators
$A_I\langle\phi_1,\dots \phi_n\rangle_{n,d}$ of {\em equivariant}
cohomology classes of $X$.
The equivariant cohomology $H^*_G(M)$ of a $G$-space $M$ is defined as the
ordinary cohomology $H^*(M_G)$ of the homotopic quotient
$M_G=EG\times_G M$ --- the total space of the $M$-bundle
$p:M_G\to BG$ associated with the universal principal $G$-bundle
$EG\to BG$. The characteristic class algebra $H^*(BG)=H^*_G$(pt) plays the role
of the coefficient ring of the equivariant theory (so that $H^*_G(M)$ is a
$H^*_G$(pt)-module). If $M$ is a compact manifold with smooth $G$-action, the
push-forward $p_*:H^*_G(M)\to H^*_G$(pt) (``fiberwise integration") provides
the equivariant cohomology of $M$ with intersection theory with values in
$H^*_G$(pt).
We introduce the {\em equivariant GW-invariants},
$A_I(\langle\phi_1,\dots ,\phi_n\rangle_{n,d}$, with
values in $H^*(BG)$, where $\phi_1,\dots ,\phi_n\in H^*_G(X)$. Values of such
invariants on fundamental cycles of maps $B\to BG$ are accountable for
enumeration of rational holomorphic curves in families of complex manifolds
with
the fiber $X$ associated with the principal $G$-bundles over a
finite-dimensional
manifold $B$.
\section{The WDVV equation}
\label{sec3}
\setcounter{equation}{0}
One of the main structural results about Gromov-Witten invariants ---
the composition rule \cite{RT},\cite{MS} --- expresses all genus-0
correlators via the 3-fixed point ones, (we denote them $\langle \phi_1, ...,
\phi _n \rangle _{n,d} $ since the corresponding $A_I=1$)
satisfying additionally the
so-called {\em Witten-Dijkgraaf-Verlinde-Verlinde} equation. We will see here
that the same result holds true for equivariant Gromov-Witten invariants
(at least in the ample case).
Following \cite{WD}, introduce the {\em potential}
\begin{equation}
\label{eq3.1}
F =\sum^\infty_{n=0} \ \frac{1}{n!} \ \sum_d q^d \
\langle t,\dots ,t\rangle_{n,d} \ .
\end{equation}
It is a formal function of $t\in H^*_G(X)$ with values in the coefficient ring
$\L=H^*_G(pt,{\Bbb C}[[q]]$). Here ${\Bbb C}[[q]]$ stands for some completion of
the group algebra ${\Bbb C}[H_2(X,{\Bbb Z})]$ so that the symbol
$q^d=q^{d_1}_1\dots q^{d_k}_k$ represents the class $(d_1,\dots ,d_k)$ in the
lattice ${\Bbb Z}^k=H_2(X,{\Bbb Z})$ of 2-cycles. Fundamental classes of
holomorphic curves in $X$ have non-negative coordinates with respect to a basis
of Kahler forms so that the formal power series algebra ${\Bbb C}[[q]]$ can be
taken on the role of the completion. Strictly speaking, the formula \ref{eq3.1}
defines $F$ up to a quadratic polynomial of $t$ since the spaces $X_{n,0}$ are
defined only for $n\geq 3$.
Denote $\nabla$ the gradient operator with respect to the equivariant
intersection pairing $\langle \ , \ \rangle$ on $H^*_G(X)$. It is defined over the
field of fractions of $H^*_G(pt)$. The WDVV equation is an identity between
third directional derivatives of $F$. It says that
\begin{equation}
\label{eq3.2}
\langle\nabla F_{\a,\b},\nabla F_{\gamma,\d}\rangle
\end{equation}
{\em is totally symmetric with respect to permutations of the four directions}
$\a,\b,\gamma,\d\in H^*_G(X)$.
\begin{thm} The WDVV equation holds for ample $X$.
\end{thm}
Notice that
\begin{equation}
\label{eq3.3}
\langle \nabla\int_X a\wedge t,\nabla\int_X b\wedge t\rangle=
\langle a,b\rangle
\end{equation}
has geometrical meaning of integration $\int_{\Delta\subset X\times X}
a\otimes b$ over the diagonal in $X\times X$.
In order to prove the non-equivariant version of the WDVV equation one
interprets the 4-point correlators $A_{1234}\langle\a,\b,\gamma,\d \rangle_{4,d}$
which are totally symmetric in $\a,\b,\gamma,\d$
as integrals over the fibers $\Gamma_\l$ of the contraction map
$\pi: X_{4,d}\to {\cal M} _4 ={\Bbb C} P^1$ and specializes the cross-ratio $\l$ to
$0, 1$ or $\infty$.
Stable maps corresponding to generic points of, say, $\Gamma_0$ are glued from
a pair
of maps $f_1: ({\Bbb C} P^1,p_1,p_2,a_1)\to X$,
$f_2:({\Bbb C} P^1,p_3,p_4,a_2)\to X$ of degrees $d_1+d_2=d$ with
three marked points each, satisfying the diagonal condition $f_1(a_1)=
f_2(a_2)$. One can treat such a pair as a point in $X_{3,d_1}\times
X_{d_2,3}$ situated on the inverse image $\Gamma_{d_1,d_2}$ of the diagonal
$\Delta \subset X\times X$ under the evaluation map $e_3\times e_3$.
The {\em glueing map} $\displaystyle{\sqcup_{d_1+d_2=d}} \Gamma_{d_1,d_2}\to\Gamma$ is
an isomorphism at generic points and therefore it identifies the analytic
fundamental cycles. This means that
\[
A_{1234}\langle\a,\b,\gamma\,d\rangle_{4,d} =\sum_{d_1+d_2=d}
\langle\nabla\langle\a,\b,t \rangle_{3,d_1},
\nabla\langle\gamma,\d,t \rangle_{3,d_2}\rangle \ .
\]
The above argument applies to the correlators
$A_{1234}\langle\a,\b,\gamma\,d, t,\dots ,t\rangle_{n+4,d}$ with additional marked points
and gives rise to
\begin{equation}
\label{eq3.4}
\langle\nabla F_{\a,\b},\nabla F_{\gamma,\d}\rangle =\sum^\infty_{n=0} \
\frac{1}{n!} \ \sum_d \ q^d
A_{1234}\langle\a,\b,\gamma,\d,t\dots t\rangle_{4+k,d}
\end{equation}
which is totally symmetric in $\a,\b,\gamma,\d$.
Ampleness of $X$ is used here only in order to make sure that the moduli
spaces
have fundamental cycles and that the diagonal in $X\times X$ consists of
regular values of the evaluation map $e_3\times e_3$.
In order to justify the above argument in the equivariant situation, it is
convenient to reduce the problem to the case of tori actions (using maximal
torus of $G$) and use the De Rham version of equivariant cohomology theory.
For a torus $G=(S^1)^r$ acting on a manifold $M$ the equivariant De Rham
complex
\cite{AB} consists of $G$-invariant differential forms on $M$ with
coefficients in
${\Bbb C} [u_1,\dots ,u_r]=H^*_G(pt)$, provided with the coboundary operator
$d_G=d+\sum^r_{s=1} u_si_s$ where $i_s$ are the operators of contraction
by the vector fields generating the action. Applying the ordinary Stokes
formula to $G$-invariant forms and $G$-{\em invariant} chains
we obtain well-defined functionals $H^*_G(M)\to{\Bbb C} [u]$ of
{\em integration over invariant cycles}. The identity \ref{eq3.4} follows
now from the obvious $G$-invariance of the analytic varieties $\Gamma_{\l }$,
$\Gamma$, $\Gamma_{d_1,d_2}$.
A similar argument proves a composition rule that reduces computation of all
equivariant correlators $A_I\langle ... \rangle$ to that of $\langle ... \rangle $.
\section{Ample vector bundles}
\label{sec4}
The following construction was designed by M. Kontsevich in order to extend the
domain of applications of WDVV theory to complete intersections in ample
Kahler manifolds.
Let $E\to X$ be an {\em ample} bundle, that is, a holomorphic vector
bundle spanned by its holomorphic sections. For stable
$f: (C,p)\to X$ (of degree $d$, with $n$ marked points), the spaces
$H^0(C,f^*E)$ form a holomorphic vector bundle $E_{n,d}$ over the moduli
space $X_{n,d}$.
If $f$ is glued from $f_1$ and $f_2$ as in the proof of
(\ref{eq3.4}), then $H^0(C,f^*E)= \operatorname{ker} (H^0(C_1,f^*_1E)
\oplus H^0(C_2,f^*_2E)
\stackrel{e_1-e_2}{\longrightarrow} e_1^*E=e_2^*E)$ where
$e_i: H^0(C_i,f^*_iE)\to e^*_iE$ is defined by evaluation of
sections at the marked point $a_i$.
This allows one to construct a solution $F$ to the WDVV equation
starting with an ample $G$-equivariant bundle $E$ and any invertible
$G$-equivariant multiplicative characteristic class $c$
(the total Chern class would be a good example).
Redefine
\[ \langle a,b\rangle \ := \ \int_X a\wedge b\wedge c(E) \ , \]
\[ \langle t,\dots ,t\rangle_{n,d} \ := \ \int_{X_{n,d}} e^*_1t \wedge
\dots e^*_n t \wedge c(E_{n,d}) \ , \]
\[ F(t) \ = \ \sum^\infty_{n=0} \frac{1}{k!} \sum_d q^d
\langle t,\dots ,t\rangle_{n,d} \ .\]
Then $\langle\nabla F_{\a,\b},\nabla F_{\gamma\d}\rangle$
{\em is totally symmetric in} $\a,\b,\gamma,\d$.
This construction bears a limit procedure from the total Chern class to the
(equivariant) Euler class, and the limit of $F$ corresponds to the GW-theory
on the submanifold $X'\subset X$ defined by an (equivariant) holomorphic
section $s$ of the bundle $E$. Namely, the section $s$ induces a
holomorphic section $s_{n,d}$ of $E_{n,d}$, and the (equivariant) Euler class
$Euler \ (E_{n,d})$ becomes represented by some cycle $[X'_{n,d}]$ situated
in the zero locus $X'_{n,d}:=s^{-1}_{n,d}(0)$ of the induced section. The
variety $X'_{n,d}$ consists of stable maps to $X'$, the Euler cycle
$[X'_{n,d}]$
plays the role of the virtual fundamental cycle in $X'_{n,d}$, and the
correlators
\[ \langle t, ... ,t \rangle _{n,d} := \int _{X_{n,d}} \ e_1^*t ... e_n^*t \ Euler \
(E_{n,d}) = \int _{[X'_{n,d}]} \ e_1^*t ... e_n^*t \]
are correlators of GW-theory on $X'$ between the classes $t$ which come from
the ambient space $X$.
\bigskip
Another solution of the $WDVV$-equation can be obtained from the bundles
$E'_{d,k}:= H^1(C,f^*E^*)$: one should put
$\langle a,b\rangle := \int_X a\wedge b\wedge c^{-1}(E^*)$,
$\langle t,\dots ,t\rangle_{n,d} = \int_{X_{n,d}}
e^*_1t\wedge\dots\wedge e^*_n t\wedge c(E'_{n,d})$ for $d\neq 0$
and $\langle t,\dots ,t\rangle_{n,0}=\int_{X_{n,0}} e^*_1t\wedge
\dots\wedge e^*_n t\wedge c(E^*)$.
\section{Quantum cohomology}
\label{sec5}
One interprets the WDVV equation as the associativity identity for the
{\em quantum cup-product} on $H^*_G(X)$ defined by
\[
\langle \a*\b,\gamma\rangle = F_{\a,\b,\gamma} \ .
\]
It is a deformation of the ordinary cup-product
(with $t$ and $q$ in the role of parameters) in the category
of (skew)-commutative algebras {\em with unity}:
\begin{equation}
\label{eq5.1}
\langle \a*1,\gamma\rangle = \langle\a,\gamma\rangle \ .
\end{equation}
Indeed, the push-forward by the forgetful map
$\pi : X_{n,d}\to X_{n-1,d}$ (with $n\geq 3$) sends
$1\in H^*_G(X_{n,d})$ to 0 unless $d=0$ and $k=3$ in which case
$X_{n,d}=X$ and $X_{n-1,d}$ is not defined.
The structure usually referred in the literature as the
{\em quantum cohomology algebra} corresponds to the restriction of the
deformation $*_{t,q}$ to $t=0$. As it is shown in \cite{KM}, in many
cases the function $F$ can be recovered on the basis of WDVV-equation from
the structural constants
$F_{\a,\b,\gamma}|_{t=0}(q)$ of the quantum cohomology algebra due to the following
symmetry of the potential $F$. Let $u \in H^2_G(X)$ and $(u_1,...,u_k)$ be
its coordinates with respect to the basis of the lattice
$({\Bbb Z} ^k)^*=H^2(X)=H^2_G(X)/H^2_G(pt)$ (so that $u_i \in H^*_G(pt)$).
Then
\begin{equation}
\label{eq5.2}
(F_{\a , \b ,\gamma })_u=\sum_{i=1}^k u_iq_i \partial F_{\a , \b ,\gamma }/\partial q_i
\quad \forall \a,\b,\gamma \in H^*_G(X) \ .
\end{equation}
The identity (\ref{eq5.2}) follows from the obvious push-forward formula
$\pi_*u=d_i u_i$.
The symmetry $(6)$ can be interpreted in the way that the quantum deformation
of the cup-product restricted to $t=0$ is equivalent to the deformation with
$q=1$ and $t$ restricted to the $2$-nd cohomology of $X$ (in the equivariant
setting it is better however to keep both parameters in place ---
see Sections $7, 8$).
\bigskip
In this paper, we will use the term {\em quantum cup-product} for the entire
$(q,t)$-deformation and {\em reserve the name {\em quantum cohomology algebra}
for the restriction of the quantum cup-product to $t=0$}.
I have heard some complaints about such terminology because it allows many
authors to compute quantum cohomology algebras without even mentioning the
deformation in $t$-directions. There are some indications however that (despite
the equivalence $(6)$) the $q$-deformation has a somewhat different nature than
the $t$-deformation. The loop space approach \cite{HG1} and our computations
in Sections $9$ -- $11$ seem to emphasize this distinction.
\bigskip
Quantum cohomology algebras of the classical flag manifolds have been
computed in
\cite{GK}, \cite{K} on the basis of several
conjectures about properties of $U_n$-{\em equivariant}
quantum cohomology (see also \cite{AS} where a slightly different formalism
was applied). The answer (in terms of generators and relations)
for complete flag manifolds $U_n/T^n$ is strikingly related to
conservation laws of Toda lattices. The conjectures named
in \cite{GK} the {\em product, induction} and {\em restriction}
properties and describing behavior of equivariant quantum cohomology under
some natural constructions, were motivated by interpretation of the
quantum cohomology in terms of Floer theory on the loop space $LX$.
Although a construction of the equivariant counterpart of the Floer - Morse
theory on $LX$ remains an open problem, the three conjectured axioms can
be justified within the Gromov-Witten theory. This was done by B.Kim \cite{K2}.
The induction and
restriction properties follow directly from definitions given in this paper
and hold for the entire quantum deformation (not only at $t = 0$), while the
``product'' axiom that the $G_1 \times G_2$-equivariant quantum cohomology
algebra of $X_1\times X_2$ is the tensor product of the $G_i$-equivariant
quantum cohomology algebras of the factors $X_i$ has been verified in
\cite{K2} for ample manifolds.
Behavior of the quantum cup-product at
$t\neq 0$ under the Cartesian product operation on the target manifolds is
much more complicated than the operation of the tensor product.
\section{Floer theory and $D$-modules}
\label{sec6}
Structural constants $\langle \a * \b,\gamma\rangle$ of the quantum cup-product
are derivatives $\partial_{\b}F_{\a,\gamma}$ of the same function. This allows to
interpret the WDVV-equation as integrability condition of some connections
$\nabla_\hbar $ on the tangent bundle $T_H$ of the space $H = H^*(X,{\Bbb C})$.
Namely, put $t = \sum t_{\a}p_{\a}$ where $p_1 = 1,p_2,\dots,p_N$ is a basis
in $H$ and define
\[
\nabla_\hbar = \hbar d - \sum(p_{\a}*)dt_{\a} \wedge:
\Omega^0(T_H) \to \Omega^1(T_H)
\]
where $p_{\a}*$ are operators of quantum multiplication by $p_{\a}$. Then
$\nabla_\hbar \circ \nabla_\hbar = 0$ {\em for each value of the parameter}
$\hbar$.
Notice that the integrability condition that reads ``the system of differential
equations $\hbar \partial_{\a}I = p_{\a} * I$ has solutions $I \in
\Omega^0(TH)$'' is actually obtained as a somewhat combinatorial statement
(the WDVV-equation) about coefficients of the series $F$.
In \cite{HG1}, \cite{HG} we attempted to improve this unsatisfactory
explanation of the integrability property by
describing a direct geometrical meaning of the solutions $I$ in terms of
$S^1$-equivariant Floer theory on the loop space $LX$. Briefly, the universal
covering $\widetilde{LX}$ carries the action of the covering transformation
lattice $\pi_2(X)$ with generators $q_1,\dots,q_k$ and the $S^1$-action by
rotation of loops which preserves natural symplectic forms $\o_1,\dots,\o_k$ on
$LX$ and thus defines corresponding Hamiltonians $H_1,\dots,H_k$ on
$\widetilde{LX}$ (the action functionals). The Duistermaat--Heckman forms
$w_i+ \hbar H_i$ (here $\hbar $ is the generator of $H_{S^1}^*(pt)$)
are equivariantly closed,
and operators $p_i$ of exterior multiplication by these forms have the
following Heisenberg commutation relations with the covering transformations:
\[
p_iq_j - q_jp_i = \hbar q_j\d_{ij}.
\]
Conjecturally, this provides $S^1$-equivariant Floer cohomology of
$\widetilde{LX}$ with a ${{\cal D}}$-module structure which is equivalent to the
above system of differential equations (restricted to $t = 0$, $q \ne 0$) and
reduces to the quantum cohomology algebra in the quasi-classical limit
$\hbar = 0$ (see \cite{HG1, GK}).
In this section we describe solutions to $\nabla_\hbar I = 0$ by imitating the
$S^1$-equivariant Floer theory (which is still to be constructed) within the
framework of Gromov--Witten theory. This construction turns out to be crucial
in our proof in Section 11 of the mirror conjecture for Calabi-Yau projective
complete intersections.
\bigskip
One may think of the graph of an algebraic
loop ${\Bbb C}P^1{\backslash}\{0,\infty\} \to X$ of degree $d$ as of a stable
map ${\Bbb C}P^1 \to X \times {\Bbb C}P^1$ of bidegree $(d,1)$. Our starting
point consists in interpretation of the moduli space $L_d(X)$ of such stable
maps as a degree-$d$ approximation to $\widetilde{LX}$ and application of
equivariant Gromov--Witten theory to the action of $S^1$ on the second factor
${\Bbb C}P^1$ with the fixed points $\{0,\infty\}$.
In the theorem below we assume $X$ to be ample. It is natural to expect
however that the theorem holds true whenever the non-equivariant Gromov--Witten
theory works for $X$ since the $S^1$-action is non-trivial only on the factor
${\Bbb C}P^1$ which is ample on its own.
Let $\langle \ ,\ \rangle$ be the Poincare pairing on $H = H^*(X,{\Bbb C})$.
The equivariant cohomology algebra $H_{S^1}^*(X \times {\Bbb C}P^1)$ is
isomorphic to $H \otimes_{\Bbb C} {\Bbb C}[p,\hbar ]/(p(p-\hbar ))$ with the
$S^1$-equivariant pairing
\[
(\varphi,\psi) = \frac {1}{2\pi i} \oint \frac {\langle \varphi,\psi\rangle
dp}{p(p-\hbar )}.
\]
Localization in $\hbar $ allows to introduce coordinates $\varphi = tp/\hbar +
\tau(\hbar -p)/\hbar $, $\tau,t \in H$, diagonalizing the equivariant pairing:
\[
((\tau,t),(\tau ',t')) = \frac {\langle t,t' \rangle - \langle
\tau,\tau '\rangle }{\hbar }.
\]
The potential
${{\cal F}}(t,\tau,h,q,q_0)$ satisfying the
equivariant WDVV-equation for $X \times {\Bbb C}P^1$ expands as
\[
{{\cal F}} = {{\cal F}}^{(0)} + q_0{{\cal F}}^{(1)} + q_0^2{{\cal F}}^{(2)}\dots
\]
according to contributions of stable maps of degree $0,1,2,\dots$ with respect
to the second factor. Denote $F = F(t,q)$ the potential (\ref{eq3.1}) of
the $GW$-theory for $X$.
\begin{thm}
{\em (a)} ${{\cal F}}^{(0)} = (F(t,q)-F(\tau,q))/\hbar $.
{\em (b)} The matrix $(\Phi_{\a\b}) :=
(\partial^2{{\cal F}}^{(1)}/\partial\tau_{\a}\partial t_{\b})$ is a fundamental
solution of $\nabla_{\pm \hbar }I = 0$:
\[
-\hbar \frac {\partial}{\partial \tau_{\gamma}} \Phi = p_{\gamma}(t)\Phi \ ,\]
\[ \hbar \frac {\partial}{\partial t_{\gamma}} \Phi^* = p_{\gamma}(\tau)\Phi^* \ ,\]
where $p_{\gamma} = (p_{\a}^{\b})_{\gamma}$, $\gamma = 1,\dots,N$, are matrices of quantum
multiplication by $p_1 = 1,\dots,p_N$, and $\Phi^*$ is transposed to $\Phi$.
\end{thm}
\noindent{\em Proof.}\ Moduli spaces of bidegree-$(d,0)$ stable maps to $X \times {\Bbb C}P^1$
coincide with $X_{n,d} \times {\Bbb C}P^1$. This implies (a) and shows that
the WDVV-equation for ${\cal F}$ {\em modulo} $q_0$ follows from the WDVV-equation
for $F$. Part (b) follows now directly from the WDVV-equation for ${\cal F}$
{\em modulo} $q_0^2$ and from
\[
\hbar \frac{\partial }{\partial t_1} \Phi _{ \a \b } = \Phi _{\a \b } =
-\hbar \frac{\partial }{\partial \tau _1 } \Phi _{\a \b }
\]
due to (\ref{eq5.1}) and (\ref{eq5.2}). Here $\partial /\partial t_1, \partial/\partial \t _1$ are
derivatives in the direction $1\in H^*(X)$ of the identity components of
$t$ and $\t$ respectively.
\bigskip
The following corollary is obtained by expressing equivariant correlators
$\Phi_{\a\b}$ via localization of equivariant cohomology classes of moduli
spaces $L_d(X)$ to fixed points of the $S^1$ action.
Define
\begin{equation}
\label{eq6.1}
\psi_{\a\b} = \sum_{n=0}^{\infty} \frac {1}{n!} \sum_d q^d\langle \frac
{p_{\b}}{\hbar +c} ,t,\dots,t,p_{\a}\rangle_{n+2,d}
\end{equation}
where $c$ is the first Chern class of the line bundle over $X_{k,d}$ introduced
in Section $1$ as ``the universal tangent line at the first marked
point'', and $\langle \frac {p_{\b}}{\hbar +c}, p_{\a}\rangle_{2,0} :=
\langle p_{\a}, p_{\b}\rangle $.
\begin{cor}
$\hbar \partial \psi /\partial t_{\gamma } = p_{\gamma}(t)\psi$,
i.e., the matrix $\psi$ is (another) fundamental solution of
$\nabla_{\hbar } I = 0$.
\end{cor}
\noindent{\em Proof.}\ A fixed point in $L_d(X)$ is represented by a stable map $C_0 \cup {\Bbb
C}P^1 \cup C_{\infty} \to X \times {\Bbb C}P^1$ where $\varphi_i: C_l \to X
\times \{i\}$ are stable maps of degrees $d_1 + d_2 = d$ connected by a
``constant loop'' ${\Bbb C}P^1 \stackrel{\simeq}{\rightarrow} \{x\} \times
{\Bbb C}P^1$. Thus components of $L_d(X)^{S^1}$ can be identified with
submanifolds in $X_{d_1,k_1+1}^{(0)} \times X_{d_2,k_2+1}^{(\infty)}$ defined
by the diagonal constraint $e_1(\varphi_0) = e_1(\varphi_{\infty})$, with
$\hbar^2(\hbar + c(0))(\hbar - c(\infty))$ to be the
equivariant Euler class of the normal bundle. This gives rise to
\begin{equation}
\label{eq6.2}
\hbar^2\Phi_{\a\b} = \sum_{\varepsilon,\varepsilon '}
\psi_{\a\varepsilon}(\tau ,-\hbar )\ \eta^{\varepsilon\e'} \psi_{\varepsilon '\b}(t,h)
\end{equation}
where $\sum \eta^{\varepsilon\e'}p_{\varepsilon} \otimes p_{\varepsilon'}$ is the coordinate expression
of the diagonal cohomology class of $X\subset X \times X$.
\bigskip
We give here several reformulations which will be convenient for computation
of quantum cohomology algebras in Sections $9$ -- $10$.
Consider the specialization of the connection $\nabla _{\hbar }$ to the
parameter subspace corresponding to the deformation of the
quantum cup-product along the $2$-nd cohomology (this is accomplished by
putting first $t=0$ and then replacing $q^d$ by $\exp (d,t)$ where
$t=(t_1,...,t_k)$
represents coordinates on $H^2(X)$ with respect to the basis
$p^{(1)},..., p^{k)} \in H^2(X)$ . In this new setting put
\[ s_{\a, \b} := \sum _d e^{dt} \langle p_{\b } \frac{e^{pt/\hbar }}{\hbar + c},
p_{\a }\rangle \]
where $pt:= \sum p^{(i)}t_i$.
{\bf Corollary 6.3.} {\em The matrix $(s_{\a ,\b }(t))$ is a fundamental
solution to
\[ \nabla _{\hbar } \ s = 0 : \ \hbar \frac{\partial}{\partial t_i} \ s = p^{(i)} * s .\] }
{\em Proof.} One should combine Corollary $6.2$ with iterative applications
of the following generalized symmetries $(5),(6)$:
\[ \langle f(c), ..., 1 \rangle _{n+1, d} = \langle \frac{f(0)-f(c)}{c},... \rangle _{n,d}
, \]
\[ \langle f(c), ..., p^{(i)} \rangle _{n+1, d} = d_i \langle f(c),... \rangle _{n,d} +
\langle p^{(i)} \frac{f(0)-f(c)}{c} , ... \rangle _{n,d} .\]
Here $f(c)$ is a function of $c$ with values in $H^*(X)$.
The symmetries are easily verified on the basis of the following geometrical
properties of universal tangent lines:
(i) Consider the push-forward along the map $\pi : X_{n+1,d}\to
X_{n,d} $ (forgetting the last marked point). It is easy to see that
the difference $\pi ^*(c)-c$ between the Chern class of the universal
tangent line at the $1$-st marked point and the pull-back of its counterpart
from $X_{n,d}$ is represented by the fundamental cycle of the section
$i: X_{n,d}\to X_{n+1,d}$ defined by the first marked point.
(ii) $i^* (c) = c$.
In particular $\pi _* (1/(\hbar +c)) = 1/[\hbar (\hbar +c)] $.
{\bf Corollary 6.4.} {\em Consider the functions
\[ s_{\b }:= \sum _d e^{dt }
\langle p_{\b } \frac{e^{pt/\hbar }}{\hbar + c}, 1\rangle _{2,d} \ .\]
Let $P(\hbar \partial/\partial t, \exp t, \hbar)$ be a differential operator
annihilating simultaneously all the functions $s_{\a }$. Then
the relation $P(p^{(1)},..., p^{(k)} ,q_1,...,q_k , 0)=0$ holds
in the quantum cohomology algebra of $X$ (we assume here that $P$
depends only on non-negative powers of $\hbar $).}
The functions $s_{\b }$ --- ``the first components of the vector-solutions
$s_{\a, \b}$'' --- generate a left $\cal{D}$-module with the {\em solution}
locally constant sheaf described by the flat connection $\nabla _{\hbar }$ and
with the characteristic variety isomorphic to the spectrum of the quantum
cohomology algebra.
\bigskip
All results of this section extend literally to the equivariant setting and/or
to the generalization to ample vector bundles described in Section $4$. We will
apply them in this extended form in Sections $9$--$11$.
\bigskip
{\em Remarks.}
$ 1$) The universal formula (\ref{eq6.1}) for solutions of $\nabla_\hbar I = 0$
was perhaps discovered independently by several authors. I first learned this
formula from R. Dijkgraaf. It can also be found in \cite{Db} in the
{\em axiomatic} context of conformal topological field theory. One can prove
it directly from a recursion relation (in the spirit of WDVV-equation) for
so-called {\em gravitational descendents} --- correlators involving the
first Chern classes of the universal tangent lines (or, in a slightly
different manner, by describing explicitly the divisor in $X_{n,d}$
representing $c$).
Our approach provides an interpretation of (\ref{eq6.1}) in
terms of fixed point localization in equivariant cohomology.
$ 2$) One can generalize our theorem to bundles over ${\Bbb C}P^1$ with the
fiber $X$. This seems to indicate that a straightforward ``open-string''
approach to $S^1$-equivariant Floer theory on $\widetilde{LX}$ would be more
powerful and flexible than the approximation by Gromov--Witten theory on $X
\times {\Bbb C}P^1$ described above.
$3$) Although the theorem provides a geometrical interpretation of
solutions to $\nabla_\hbar I = 0$,
it does not eliminate the combinatorial
nature of the integrability condition. Indeed, the theorem is deduced from an
equivariant WDVV-equation which in its turn can be interpreted as an
integrability condition. Of course one can explain it using the $S^1 \times
S^1$-equivariant WDVV-equation on $(X \times {\Bbb C}P^1) \times {\Bbb C}P^1$,
etc. It would be interesting to find out whether this process converges.
\section{Frobenius structures}
\label{sec7}
In \cite{Db}, B. Dubrovin studied geometrical structures defined by
solutions of WDVV-equations on the parameter space and reduced classification
of
generic solutions to the classification of trajectories of some Euler-like
non-autonomous Hamiltonian systems on $so_N^*$. We show here how this approach
to equivariant Gromov--Witten theory yields analogous Hamiltonian systems on
the
affine Lie coalgebras $\widehat{so}_N^*$.
The quantum cup-product on $H = H_G^*(X)$ considered as an $N$-dimensional
vector space over the field of fractions $K$ of the algebra $H_G^*(pt)$ defines
a formal {\em Frobenius structure} on $H$. The structure consists of the
following ingredients.
\begin{enumerate}
\item A symmetric $K$-bilinear inner product $\langle\ ,\ \rangle$,
\item a (formal) function $F: H \to K$ whose third directional derivatives
$\langle
a*b,c\rangle := F_{a,b,c}$ provide tangent spaces $T_tH$ with the Frobenius
algebra structure (i.e. associative commutative multiplication $*$
satisfying $\langle a*b, c\rangle = \langle a, b*c \rangle $).
\item The constant vector field $1\!\!1$ of unities of the algebras
$(T_tH, *)$ whose flow
preserves the multiplication $*$ (i.e. $L_{1\!\!1} (*)=0$).
\item Grading: In the non-equivariant case axiomatically studied
by B. Dubrovin it can be
described by the {\em Euler} vector field $E$, such that
the tensor fields $1\!\!1$, $*$ and $\langle \ , \ \rangle$ are homogeneous (i.e.
are eigen-vectors of the Lie derivative $L_E$) of degrees $-1, 1$ and $D$
respectively (where $D = \operatorname{dim} _{\Bbb C} X$ in the models arising from
the GW-theory). In the equivariant GW-theory this grading axiom should be
slightly modified since the grading of the
structural ring $H_G^*(pt)$ is non-trivial and thus the natural Euler
operator $L_E$ is ${\Bbb C}$-linear but not $K$-linear.
\end{enumerate}
\bigskip
The fact that the multiplication $*$ is defined on tangent vectors to $H$
means that the algebra $(\Omega ^0(T_H), *)$ can be naturally considered as
the algebra $K[L]$ of regular functions on some subvariety $L\subset T^*H$
in the cotangent bundle.
A point $t\in H$ is called {\em
semi-simple} if the algebra $(T_{t}H,*)$ is semi-simple, that is if $L \cap
T_{t}^*H$ consists of $N$ linearly independent points.
Flatness of the connection (defined on $T_H$)
\begin{equation}
\label{eq7.1}
\nabla_\hbar = \hbar d - \sum_{\a} p_{\a} * dt_{\a}
\end{equation}
implies \cite{GK} that $L$ is a Lagrangian submanifold in $T^*H$ near a
semisimple $t$. Following \cite{Db}, introduce local {\em canonical
coordinates} $(u_1,\dots,u_N)$ such that the sections $(du_1,\dots,du_N)$ of
$T^*H$ are the $N$ branches of $L$ near $t$, and transform the connections
$\nabla_\hbar$ to these local coordinates and to a (suitably normalized)
basis $f_1,\dots,f_N$ of vector field on $H$ diagonalizing the $*$-product.
The result of this transformation can be described as follows.
(a) The basis $\{f_i\}$ can be normalized in a way that in the transformed
form
\begin{equation}
\label{eq7.2}
\nabla _{\hbar } = \hbar d - \hbar A^1 \wedge - D^1\wedge
\end{equation}
of the connection $\nabla _{\hbar} $ with
$D^1 = \operatorname{diag}(du_1,\dots,du_N)$, and $A_{ij} =
V_{ij}(u) d(u_i-u_j)/(u_i-u_j)$ for all $i \ne j$,
we will have additionally $A_{ii} =0 \ \forall i$.
(b) The vector field $1\!\!1$ in the canonical coordinates assumes the form
$\sum_k \partial_k$ where $\partial_k := \partial/\partial u_k$ are
the canonical idempotents of the $*$-product:
\begin{equation}
\label{eq7.3}
\partial_i * \partial_j = \d_{ij}\partial_j.
\end{equation}
(c) The (remaining part of the) integrability condition $\nabla_{\hbar} ^2 =0$
reads
$d(A^1) = A^1 \wedge A^1$ or
\begin{equation}
\label{eq7.4}
\partial_i\phi_{\a}^j = \phi_{\a}^i V_{ij}/(u_i-u_j),\ i \ne j,
\end{equation}
where $(\phi_{\a}^j)$ is the transition matrix, $\partial/\partial t_{\a} =
\sum_i \phi_{\a}^i f_i$; it can be reformulated as compatibility of the
PDE system (\ref{eq7.4}) for $(\phi_{\a}^j)$ completed by
\begin{equation}
\label{eq7.5}
\sum_k \partial_k\phi_{\a}^j = 0.
\end{equation}
(d) The Frobenius property $\langle a*b,c\rangle = \langle a,b*c\rangle$ of the
$*$-product shows that the diagonalizing basis $\{f_i\}$ is orthogonal, that
its normalization by $\langle f_i,f_j \rangle = \d_{ij}$ obeys $A_{ii} = 0$
and, additionally, implies anti-symmetricity $A_{ij} = -A_{ji}$, or
\begin{equation}
\label{eq7.6}
V_{ij} = -V_{ji}.
\end{equation}
The presence of the grading axiom (4) of Frobenius structures over $K = {\Bbb
C}$ allows B.Dubrovin to describe anti-symmetric matrices
$V = (V_{ij}) \in so_N^*$
satisfying the integrability conditions (\ref{eq7.4}) and (\ref{eq7.5}) in
{\em quasi-homogeneous} canonical coordinates (i.e. $L_E u_i = u_i$ so that
$E = \sum u_k\partial_k$)
as trajectories of $N$ commuting non-autonomous Hamiltonian
systems (see \cite{Db}):
\[
\partial_iV = \{H_i,V\}
\]
where the Poisson-commuting non-autonomous quadratic Hamiltonians $H_i$
on $so_N^*$ are given by
\[
H_i = \frac {1}{2} \sum_{j \ne i} \frac {V_{ij}V_{ji}}{u_i-u_j} .
\]
Consider now the following model modification of the grading axiom:
$K = {\Bbb C}[[\lambda^{\pm 1}]]$, $\operatorname{deg} \lambda = 1$. In
quasi-homogeneous canonical coordinates $(u_1,\dots,u_N,\lambda)$ the Euler
vector field takes then on
\begin{equation}
\label{eq7.7}
L_E = \sum_k u_k\partial_k + \lambda\partial_{\lambda}.
\end{equation}
Introduce the connection operator
\[
{\Bbb V} = \lambda\partial_{\lambda} - V \in \widehat{so}_N^*
\]
and the qudratic Hamiltonians on the Poisson manifold $\widehat{so}_N^*$
\begin{equation}
\label{eq7.8}
{{\cal H}}_i({\Bbb V}) = \oint H_i(V) \frac {d\lambda}{\lambda} .
\end{equation}
\begin{prop}
The Hamiltonians ${{\cal H}}_1,\dots,{{\cal H}}_N$ are in involution. The operator
${\Bbb V}$ of a Frobenius manifold over $K$ satisfies the non-autonomous system
of Hamiltonian equations
\begin{equation}
\label{eq7.9}
\partial_i{\Bbb V} = \{{{\cal H}}_i,{\Bbb V}\},\ i = 1,\dots,N.
\end{equation}
The columns $\phi_{\a} = (\phi_{\a}^i)$ of the transition matrix are
eigen-functions of the connection operator ${\Bbb V}$:
\begin{equation}
\label{eq7.10}
{\Bbb V}\phi_{\a} := (\lambda\partial_{\lambda} - V)\phi_{\a} = \left( \frac
{n}{2} - \operatorname{deg} t_{\a} + 1\right) \phi_{\a}.
\end{equation}
\end{prop}
\noindent{\em Proof.}\ It can be obtained by a straightforward calculation quite analogous
to that in \cite{Db}.
\bigskip
In our real life the model equations (\ref{eq7.7}--\ref{eq7.10}) describe the
structure of Frobenius manifolds {\em over each semi-simple orbit of the
grading Euler field in the ground parameter space. This parameter space is the
spectrum of the coefficient algebra} $H_G^*(pt,{\Bbb C}) \otimes {\Bbb
C}[q^{\pm 1}_1,\dots,q^{\pm 1}_k]$ (its field of fractions can be taken on the
role of the ground field $K$). An orbit of the Euler vector field in this
parameter space is
semi-simple if the corresponding ${\Bbb C}$-Frobenius algebras are
semi-simple.
The equations (\ref{eq7.7}--\ref{eq7.10}) over semi-simple Euler orbits should
be complemented by the additional symmetries (\ref{eq5.2}).
In the next section we will show how the canonical coordinates of the axiomatic
theory of Frobenius structures emerge from localization formulas in equivariant
Gromov--Witten theory.
\section{Fixed point localization}
\label{sec8}
We consider here the case of a circle $T^1$ acting by Killing transformations
on a compact Kahler manifold $X$ with {\em isolated} fixed points only.
The case of tori actions with isolated fixed points requires only slight
modification of notations which we leave to the reader. Our
results are rigorous for ample $X$ (which includes homogeneous Kahher spaces of
compact Lie group and their maximal tori) while applications to general toric
manifolds (which are typically not ample) yet to be justified.
It is the Borel localization theorem that reduces computations in
torus-equivariant
cohomology to computations near fixed points. Let $\{ p_{\a}\} ,
\a=1,...,N$, be the fixed points of the action. We will denote with the same
symbols $p_{\a}$ the equivariant cohomology class of $X$ which restricts
to $1\in H^*_T(p_{\a})$ at $p_{\a}$ and to $0$ at all the other fixed points.
These classes
are well-defined over the field of fractions ${\Bbb C} (\l )$
of the coefficient ring $H^*_T(pt)={\Bbb C} [\l ]$ and form the basis
of canonical idempotents in the semi-simple algebra $H^*_T(X,{\Bbb C} (\l )$.
The equivariant Poincare pairing reduces to $\langle p_{\a},p_{\b}\rangle =
\d _{\a,\b}/e_{\b}$ where $e_{\a}\in {\Bbb C} [\l]$ is the equivariant Euler class
of the normal ``bundle'' $T_{p_{\a}}X\to p_{\a}$ to the fixed point.
The results described below apply to the setting of Section $4$ of a manifold
$X$ provided with an ample vector bundle in which case $e_{\a}$'s should be
modified accordingly.
\bigskip
The same localization theorem reduces computation of GW-invariants to that
near the fixed point set (orbifold) in the moduli spaces $X_{n,d}$.
A fixed point in
the moduli space is represented by a stable map to $X$ of a (typically
reducible) curve $C$ such that each component of $C$ is mapped to (the
closure of) an orbit of the complexified action $T_{{\Bbb C} }:X$. Any such an
orbit is either one of the fixed points $p_{\a}$ or isomorphic to $({\Bbb C}-0)$
connecting two distinct fixed points $p_{\a}$ and $p_{\b}$ corresponding to
$0$ and $\infty $. Respectively, there are two types of components of $C$:
\noindent (i) Each component of $C$ which carries $3$ or more special points
must be mapped to one of the fixed points $p_{\a}$.
\noindent (ii) All other components are multiple covers $z\mapsto z^d$ of the
non-constant orbits, and their special points may correspond only to $z=0$ or
$\infty$.
The {\em combinatorial structure}
of such a stable map can be described by a tree
whose edges correspond to {\em chains} of components of type (ii) and should
be labeled by the total degree of this chain as a curve in $X$, and vertices
correspond to the ends of the chains. The ends may carry $0$ or $1$ marked
point, or correspond to a (tree of) type-(i) components with $1$ or more
marked points and should be labeled by the indices of these marked points
and by the target point $p_{\a}$.
{\em The fixed stable maps with different combinatorial structure belong to
different connected components of the fixed point orbifold in $X_{n,d}$.}
The results below are based on the observation that a stable map with the
first $k\geq 3$ marked points in a {\em given} generic configuration
(i.e. with the given generic value of the contraction map $X_{n.d}\to
{\cal M} _{0,k}$) must have in an irreducible component $C_0$ in the underlying
curve $C$ which contains this given configuration of $k$ {\em special} points,
(so that the corresponding first $k$ marked points are located on the branches
outgoing these special points of $C_0$). The cause is hidden in the definition
of the contraction map (see \cite{Kn, BM}).
We will call the component $C_0$ {\em special}.
The observation applied to a fixed stable map of the circle action allows to
subdivide all fixed point components in $X_{3+n,d}$ into $N$ {\em types}
$p_i$ according
to the fixed points $p_i$ where the special component is mapped to. We
introduce the superscript notation $(...)^i$ for the contribution (via
Borel's localization formulas) of type-$p_i$ components into various
equivariant correlators. For example,
\[ F^i_{\a \b \gamma } =\sum _{n}\frac{1}{n!} \sum_d q^d
\langle p_{\a },p_{\b },p_{\gamma },t,...,t\rangle _{3+n,d}^i \]
where $t=\sum _{\a=1}^N t_{\a}p_{\a}$ is the general class in
$H^*_T(X,{\Bbb C}(\l))$, so that $F_{\a\b\gamma}=\sum_i F_{\a\b\gamma}^i$.
We introduce also the notations
\begin{itemize}
\item $\Psi _{\a\b}^i$ --- for contributions to $e_iF_{\a\b i}^i$ of those
fixed points which have the third marked point situated directly on the
special component $C_0$ (it is convenient here to introduce the normalizing
factor $e_i\in H^*_T(pt)$, the Euler class of the normal ``bundle'' to the
fixed point $p_i$ in $X$);
\item $\Psi _{\a}^i := \Psi _{\a 1\!\!1}^i=\sum_\b \Psi _{\a\b}^i$;
\item $D_{\a}^i$ --- for contributions to $e_iF_{\a i i}$ of those
fixed points which have the second and third marked points situated directly
on the special component;
\item $\Delta ^i$ --- for contributions to $e_iF_{i i i}$ of those fixed
points which have the first three marked points situated directly on the
special component;
\item $u_i=t_i+$ contributions to $e_iF_{i i}$ of all those fixed point
components in $X_{2+n,d}$ for which the first two marked points belong to
the same vertex of the tree describing the combinatorial structure.
\end{itemize}
The correlators $u_i$ can be also interpreted as contributions to the
{\em genus-$1$} equivariant correlators
\[ \sum _n\frac{1}{n!}\sum_d q^d (t,...,t)_{n,d} \]
with {\em given} complex structure of the elliptic curve of those $T$-invariant
classes which map the (only) genus-$1$ component of the curve $C$ to the
fixed point $p_i$.
\bigskip
{\bf Theorem 8.1.} {\em (a) The functions $u_1(t),...,u_N(t)$ are the canonical
coordinates of the Frobenius structure on $H^*_G(X,{\Bbb C} (\l))$.
(b) The functions $D_{\a}^i(t)$ are eigen-values of the quantum multiplication
by $p_{\a}$: $du_i=\sum _{\a} D_{\a }^i dt_{\a} $.
(c) The transition matrix $(\Psi_{\a}^i)$ provides simultaneous diagonalization
of the quantum cup product: $F_{\a\b\gamma}^i=\Psi_{\a}^i D_{\b}^i \Psi_{\gamma}^i$
and obeys the following orthogonality relations:
\[ \sum_i \Psi_{\a}^i\Psi_{\b}^i=\d _{\a\b}/e_{\b},
\sum _{\a} \Psi_{\a}^i\Psi_{\a}^j =\d _{ij} \ .\]
(d) The Euclidean structure on the cotangent bundle of the Frobenius
manifold (defined by the equivariant intersection pairing in $H^*_T(X)$)
in the canonical coordinates $u_i$ takes on $\langle du_i,du_j \rangle =
(\Delta^i)^2\d_{ij}e_j$ and additionally
\[ (\Delta ^i)^{-1}=\sum_{\a}\Psi_{\a}^i, \ \Psi_{\a}^i=\frac{D_{\a}^i}{\Delta^i},
\ \Psi_{\a\b}^i=\frac{D_{\a}^i D_{\b}^i}{\Delta^i} \ .\]
}
\bigskip
{\em Proof.} We first apply the localization
formula
\[ A_{1234}\langle ... \rangle _{n,d}=\sum _i A_{1234}\langle ...\rangle _{n,d}^i \]
to the $4$-point equivariant correlators with the fixed cross-ratio $z$
of the $4$ marked points and only after this specialize the cross-ratio
to $0$,$1$ or $\infty$. This gives rise to the {\em local} WDVV-identities
\[ \Psi_{\a\b}^i\Psi_{\gamma\d}^i \ \text{\em is totally symmetric in} \ \
{\a,\b,\gamma,\d} \]
which is independent of the global WDVV-equation.
When combined with the global identities
\[ A_{1234}\langle 1\!\!1,p_{\a},p_{\b},p_{\gamma} \rangle _{n,d}
=\langle p_{\a},p_{\b},p_{\gamma} \rangle _{n,d} \]
they yield the orthogonality relation $\sum_i \Psi _{\a}^i\Psi_{\b}^i=
\d_{\a\b}/e_{\b}$ and localization formulas
\[ F_{\a\b\gamma}=\sum_i\Psi_{\a\b}^i\Psi_{\gamma}^i \]
for the structural constants of the quantum cup-product.
A similar argument with $>4$-point correlators $A_{12345...}\langle ...\rangle ^i$
proves the diagonalization
\[ \langle p_{\a}*p_{\b},p_{\gamma}\rangle =
\sum_i \Psi_{\a}^iD_{\b}^i\Psi_{\gamma}^i/e_i \ ,\]
\[ \langle p_{\a}*p_{\b}*p_{\gamma},p_{\d}\rangle =
\sum _i \Psi_{\a}^i D_{\b}^i D_{\gamma}^i \Psi_{\d}^i/e_i \ \]
and the identities
\[ \Psi_{\a\b}^i=\Psi_{\a}^iD_{\b}^i, \
(\Delta ^i)^{-1}=\sum_{\a } \Psi_{\a}^i \ .\]
Finally, the identity $du_i=\sum_{\a} D_{\a}^idt_{\a} $ follows
directly from the definition of $u_i$ and implies that $u_1,...,u_N$
are the canonical coordinates of the Frobenius structure.
\section{ Projective complete intersections}
\label{sec9}
We are going to describe explicitly solutions of the
differential equations arising from quantum cohomology of projective complete
intersections. Lex $X$ be such a non-singular complete intersection in
$Y:={\Bbb C} P^n $ given by $r$ equations of the degrees $(l_1,...,l_r)$. If
$l_1+...+l_r=n+1$ then $X$ is a Calabi-Yau manifold and its quantum cohomology
is described by the mirror conjecture. In this and the next sections we study
respectively the cases $l_1+...+l_r < n$ and $l_1+...+l_r=n$
when the $1$-st Chern class of $X$ is still positive. In the case $l_1+...+l_r
> n+1$ (which from the point of view of enumerative geometry can be considered
as ``less interesting'' for rational curves generically occur only in
finitely many degrees) the ``mirror symmetry'' problem of hypergeometric
interpretation of quantum cohomology differential equations remains open.
Let $E_d$ be the Euler class of the vector bundle over the moduli space
$Y_{2,d}$ of genus $0$ degree $d$ stable maps $\phi : (C, x_0,x_1) \to {\Bbb C} P^n$
with two marked points, with the fiber $H^0(C, \phi ^* H^{l_1}\oplus ...
\oplus \phi^* H^{l_r})$ where $H^l$ is the $l$-th tensor power
of the hyperplane line bundle over ${\Bbb C} P^n$.
Consider the class
$$S_d(\hbar ):=\frac{1}{\hbar + c_1^{(0)}}\ E_d \ \in \ H^*(Y_{2,d})$$
where $c_1^{(0)}$ is the $1$-st Chern class of
the ``universal tangent line at the marked point $x_0$ '',
and $e_0, e_1$ are the evaluation maps. Due to the factor $E_d$ this
class represents the push forward along $X_{2,d} \to Y_{2,d}$ of the class
$1 /(\hbar + c_1^{(0)}) \ \in H^*(X_{2,d})$
(by the very construction of $X_{2,d}$ in Section $4$).
In the cohomology algebra ${\Bbb C} [P]/(P^{n+1})$ of ${\Bbb C} P^n$, consider the class
\[ S(t,\hbar):= e^{Pt/\hbar }\sum _{d=0}^{\infty } e^{dt}
(e_0)_* (S_d(\hbar )) \]
where $(e_0)_*$ represents the push-forward along the evaluation map
(and for $d=0$, when $Y_{2,d}$ is not defined, we take
$Euler \ (\oplus _j H^{\otimes l_j}) $ on the role of $(e_0)_* S_0$).
Considered as a function of $t$, $S$ is a curve in $H^*({\Bbb C} P^n)$ whose
components are solutions of the differential equation we are concerned
about. Indeed, according to Section $6$ a similar sum represents
the solutions of the quantum cohomology differential equation for $X$, and
$S$ is just the push-forward of that sum from $H^*(X)$ to $H^*(Y)$.
(Strictly speaking $S$ carries information only about correlators between
those classes which come from the ambient projective space; also if
$X$ is a surface $\operatorname{rk} H_2(X)$ can be greater than $1$ and $S$ mixes information
about the curves of different degrees in $X$ when they have the same degree
in $Y$.)
\bigskip
{\bf Theorem 9.1.} {\em Suppose that $l_1+...+l_r < n$. Then}
\[ S = e^{Pt/\hbar } \sum _{d=0}^{\infty } e^{dt}
\frac{ \Pi _{m=0}^{dl_1} (l_1P + m\hbar) ... \Pi _{m=0}^{dl_r} (l_rP+m\hbar)}
{\Pi _{m=1}^d (P+m\hbar )^{n+1} } .\]
The formula coincides with those in \cite{HG1, HG} (found by analysis
of toric compactifications of spaces of maps ${\Bbb C} P^1 \to {\Bbb C} P^n$) for
solutions of differential equations in $S^1$-equivariant cohomology of the
loop space.
\bigskip
{\bf Corollary 9.2.} (see \cite{HG1, HG})
{\em The components $s:=\langle P^i, S\rangle , i=0,...,n-r, $
of $S$ form a basis of solutions to the linear differential equation}
\[ (\hbar \frac{d}{dt})^{n+1-r}\ s \ = \
e^t \Pi _{j=1}^r \ l_j\ \Pi _{m=1}^{l_j-1}\
\hbar (l_j\frac{d}{dt} + 1) \ s \ .\]
This implies (combine \cite{HG} with \cite{BVS}) that the solutions
have an integral representation of the form
\[ \int _{\gamma ^{n-r}\subset X'_t} \ e^{(u_0+...+u_n)/\hbar } \ \
\frac{du_0\wedge ... \wedge du_n}{dF_0\wedge dF_1\wedge ... \wedge dF_r} \]
where
\[ F_0=u_0...u_n,\ F_1=u_1+...u_{l_1},\ F_2=u_{l_1+1}+...+u_{l_1+l_2},
... , \ F_r=u_{l_1+...+l_{r-1}+1}+...+u_{l_1+...+u_{l_r}} \]
and the ``mirror manifolds'' $X'_t$ are described by the equations
\[ X'_t=\{ (u_o,...,u_n)\ |\ F_0(u)=e^t,\ F_1(u)=1, ... ,\ F_r(u)=1 \} .\]
This proves for $X$ the mirror conjecture in the form suggested in
\cite{HG}.
\bigskip
{\bf Corollary 9.3.} {\em If $\operatorname{dim} _{{\Bbb C} } X \neq 2$ the cohomology class
$p$ of hyperplane section
satisfies in the quantum cohomology of $X$ the relation}
\[ p^{n+1-r}=l_1^{l_1}...l_r^{l_r} q p^{l_1+...+l_r-r} .\]
When $X$ is a surface the same relation holds true in the quotient of the
quantum cohomology algebra which takes in account only degrees of curves in
the ambient ${\Bbb C} P^n$ (we leave to figure out a precise description of this
quotient to the reader; quadrics ${\Bbb C} P^1\times {\Bbb C} P^1$ in ${\Bbb C} P^3$ provide
a good example:
$(p_1+p_2)^3 = 4q(p_1+p_2) \ \mod p_1^2=q=p_2^2 $.)
This corollary is consistent with the result of A. Beauville \cite{Bea}
describing quantum cohomology of complete intersections with
$\sum l_j \leq n+1 -\sum (l_j-1) $ and with results of M. Jinzenji \cite{J} on
quantum cohomology of projective hypersurfaces ($r=1$) with $l_1<n$.
\bigskip
{\bf Corollary 9.4.} {\em The number of degree $d$ holomorphic maps
${\Bbb C} P^1 \to X^{n-r} \subset {\Bbb C} P^n$,
which send $0$ and $\infty $ to two given
cycles and send $n+1-r$ given points in ${\Bbb C} P^1$ to
$n+1-r$ given generic hyperplane sections, is equal to
$l_1^{l_1}...l_r^{l_r}$ times
the number of degree $d-1$ maps which send $0$ and $\infty $ to the same cycles
and $l_1+...+l_r-r$ given points --- to $l_1+...+l_r-r$ given
hyperplane sections.}
This is the enumerative meaning of Corollary $9.3$; of course in this
formulation numerous general position reservations are assumed.
{\em Control examples.} $1.$ $l_1=...=l_r=1$: The above formulas for quantum
cohomology
and for solutions of the differential equations in the case of a hyperplane
section give rise to the same formulas with $n:=n-1$.
$2$. $n=5, r=1, l=2$: $X$ is the Plucker embedding of the grassmanian
$Gr_{4,2}$. Its quantum cohomology algebra is described by the relations
$c_1^3=2c_1 c_2,\ c_2^2-c_2c_1^2+q=0$ between the Chern classes of the
tautological plane bundle. For the $1$-st Chern class $p=-c_1$ of the
determinant line bundle we deduce the relation $p^5=4pq$ prescribed by
Corollary $9.3$.
\bigskip
We will deduce Theorem $9.1$ from its equivariant generalization.
Consider the space ${\Bbb C} ^{n+1}$ provided with the standard action of the
$(n+1)$-dimensional torus $T$. The equivariant cohomology algebra
of ${\Bbb C} ^{n+1}$ coincides with the algebra of characteristic classes
$H^*(BT^{n+1}) = {\Bbb C} [\l _0,...,\l _n]$. The equivariant cohomology algebra
of the projective space $({\Bbb C} ^{n+1} - 0)/{\Bbb C} ^{\times} $ in these notations
is identified with ${\Bbb C} [p,\l ]/ ((p-\l_0)...(p-\l_n))$ and the push-forward
$H_T^*({\Bbb C} P^n)\to H_T^*(pt)$ is given by the residue formula
\[ f(p,\l ) \mapsto \frac{1}{2\pi i} \oint
\ \frac{f(p,\l )dp}{(p-\l_0)...(p-\l_n)} \ .\]
Here $-p$ can be considered as the equivariant $1$-st Chern class of the
Hopf line bundle provided with the natural lifting of the torus action.
We will use $\phi _i:=\Pi _{j\neq i} (p-\l _j), \ i=0,...,n, $ as a basis in
$H_T^*({\Bbb C} P^n)$.
Consider the $T$-equivariant vector bundle $\oplus _{j=1}^r H^{\otimes l_j}$
and provide it with the fiberwise action of the additional $r$-dimensional
torus $T'$. The equivariant Euler class of this bundle is equal to
$ (l_1p-\l '_1)...(l_rp-\l '_r)$ where ${\Bbb C} [\l ']=H^*(BT')$.
Introduce the equivariant counterpart $S'$ of the class $S$ in the
$T\times T'$-equivariant cohomology of ${\Bbb C} P^n$. This means that we use
the equivariant class $p$ instead of $P$ and replace the Euler classes
$E_d$ and $c_1^{(0)}$ by their equivariant partners.
\bigskip
{\bf Theorem 9.5.} {Let $l_1+...+l_r < n$. Then}
\[ S'= e^{pt/\hbar} \sum _{d=0}^{\infty }
e^{dt} \frac{ \Pi _0^{dl_1} (l_1p-\l '_1 +m\hbar) ...\Pi _0^{dl_r}
(l_rp-\l '_r +m\hbar)}{\Pi _1^d (p-\l _0+m\hbar) ... \Pi _1^d (p-\l _n+m\hbar)}
\ .\]
Theorem $9.1$ follows from Theorem $9.5$ by putting $\l =0, \l '=0$ which
corresponds to passing from equivariant to non-equivariant cohomology.
The vector-function $S'$ satisfies the differential equation
\[ \Pi _{i=0}^r (\hbar \frac{d}{dt} -\l _i) \ S' =
e^t \Pi _{m=1}^{l_1} (l_1\hbar \frac{d}{dt} - \l '_1 + m\hbar ) ...
\Pi _{m=1}^{l_r} (l_r\hbar \frac{d}{dt} - \l '_r + m\hbar ) \ S' .\]
\bigskip
We intend to prove Theorem $9.5$ by means of localization of $S'$ to the
fixed point set of the torus $T$ action on the moduli spaces $Y_{2,d}$.
As it is shown in \cite{Kn}, all correlators of the equivariant theory on
${\Bbb C} P^n$ are computable at least in principle, and in practice the
computation reduces to a recursive procedure which can be understood as a
summation over trees and can be also formulated as a non-linear fixed point
(or critical point) problem. We will see below that in the case of correlators
$\langle \phi _i, S'\rangle $ certain reasons of a somewhat geometrical character
cause numerous cancellations between trees so that the recursive procedure
reduces to a ``summation over chains'' and respectively to a {\em linear}
recurrence equation. The formula of Theorem $9.5$ is simply the solution to
this equation.
\bigskip
In the proof of Theorem $9.5$ below we write down all formulas for for $r=1$
(it serves the case when $X$ is a hypersurface in ${\Bbb C} P^n$ of degree $l < n$).
The proof for $r > 1$ differs only by longer product formulas.
\bigskip
Let us abbreviate $c_1^{(0)}$ as $c$,
denote $E'_d$ the equivariant Euler class of the vector bundle over $Y_{2,d}$
whose fiber over the point $\psi : (C, x_0, x_1) \to Y={\Bbb C} P^n$ consists of
holomorphic sections of the bundle $ \psi ^* (H^l)$
{\em vanishing at} $x_0$, and introduce the following equivariant correlator:
\[ Z_i :=\sum _{d=0}^{\infty } q^d
\int _{Y_{2,d}} e_0^* (\phi _i) \frac{1}{\hbar + c} E'_d .\]
We have
\[ \langle \phi _i, S' \rangle = e^{\l _i t/\hbar } (l\l _i - \l ') (Z_i | _{q=e^t})
\]
\medskip
{\bf Proposition 9.6.}
\[ Z_i = 1 + \sum _{d>0} (\frac{q}{\hbar ^{n+1-l}})^d
\int _{Y_{2,d}} \frac{(-c)^{(n+1-l)d-1}}{1+c/\hbar} \ E'_d \
e_0^*(\phi _i) \ . \]
{\em Proof.} We have just dropped first several terms in the geometrical
series $1/(\hbar + c)$ since their degree added with the degrees of other
factors in the integral over $Y_{2,d}$ is still less than the dimension
of $Y_{2,d}$. It is important here that all the equivariant classes involved
including $\phi _i$ are defined in the equivariant cohomology over
${\Bbb C} [\l ,\l ']$ without any localization.
\medskip
It is a half of the geometrical argument mentioned above. The other half
comes from the description of the fixed point set in $Y_{2,d}$ given
in \cite{K}.
Consider a fixed point of the torus $T$ action on $Y_{2,d}$. It is represented
by a holomorphic map of a possibly reducible curve with complicated
combinatorial structure and with two marked points on some components. Each
component carrying $3$ or more special points is mapped to one of the $n+1$
fixed points of $T$ on ${\Bbb C} P^n$, and the other components are mapped
(with some multiplicity) onto the lines joining the fixed points and
connect the point-mapped components in a tree-like manner.
In the Borel localization formula for $\int e_0^*(\phi _i) ...$ the fixed
point will have zero contribution unless the marked point $x_0$ is mapped
to the $i$-th fixed point in ${\Bbb C} P^n$ (since $\phi _i$ has zero localizations
at all other fixed points.
Consider a fixed point curve $C$ whose marked point $x_0$ is indeed mapped to
the $i$-th fixed point in ${\Bbb C} P^n$. There are two options
(i) the marked point $x_0$ is situated on an irreducible component of $C$
mapped with some degree $d'$ onto the line joining the $i$-th
fixed point with the $j$-th fixed point in ${\Bbb C} P^n$ with $i\neq j$;
(ii) the marked point $x_0$ is situated on a component of $C$ mapped to
the $i$-th fixed point and carrying two or more other special points.
Consider first the option (ii) and the contribution of such a connected
component of the fixed point set in $Y_{2,d}$ to the Borel localization
formula for $\int c^{(n+1-l)d-1} ...$. The connected component itself
is the (product of the) Deligne-Mumford configuration space of, say, $s+1$
special points: the marked point $x_0$, may be the marked point $x_1$, and
respectively $s-1$ or $s$ endpoints of other components of $C$ mapped
onto the lines outgoing the $i$-th fixed point in ${\Bbb C} P^n$.
{\bf Lemma 9.7.} {\em The type (ii) fixed point component
in $Y_{2,d}$ has zero contribution to the Borel localization formula for}
$\int _{Y_{2,d}} c^{(n+1-l)d-1} ... $
{\em Proof.} Restriction of the class $c$ from $Y_{2,d}$ to the type (ii)
fixed point component coincides with the $1$-st Chern class of the line
bundle on the Deligne-Mumford factor ${\cal M} _{0, s+1}$ of the component
defined as ``the universal tangent line as the marked point $x_0$''
and is thus
nilpotent in the cohomology of the component. Since the number of straight
lines in a curve of degree $d$ does not exceed $d$ we find that the dimension
$s-2$ of the factor ${\cal M} _{0,s+1}$ is less than $d$ which in its turn does
not exceed $(n+1-l)d-1$ for $d>0$ (because we assumed that $n+1-l\geq 2$).
\medskip
Consider now the option (i). The irreducible component $C'$ of the curve
$C=C'\cup C''$
carrying the marked point $x_0$ is mapped with the multiplicity $d' \leq d$
onto the line joining $i$-th fixed point in ${\Bbb C} P^n$ with the $j$-th one
{\em while the remaining part $C''\to {\Bbb C} P^n$ of the map represents a fixed
point in $Y_{2,d-d'}$}. Moreover, the normal space to the fixed point
component at the type (i) point (the equivariant Euler class of the normal
bundle occurs in the denominator of the Borel localization formula) is the
sum of (a) such a space $N''$ for $C'' \to {\Bbb C} P^n$, (b) the space $N'$
of holomorphic vector fields along the map $C'\to {\Bbb C} P^n$ vanishing at
the fixed point $j$ factorized by infinitesimal reparametrizations of $C'$,
(c) the tensor product $L$ of the tangent lines to
$C'$ and $C''$ at their intersection point. Since the space $V$ of holomorphic
sections of $H^l$ restricted to $C$ (and vanishing at $x_0$) admits a similar
decomposition $V'\oplus V''$, we arrive to the following linear recursion
relation for $Z_i$.
{\bf Proposition 9.8.} {\em Put $z_i(Q,\hbar ):=
Z_i(\hbar ^{(n+1-l)}Q, \hbar )$. Then}
\[ z_i(Q, \hbar) = 1 + \sum _{j\neq i} \sum _{d'>0} Q^{d'} Coeff \ _i^j (d')
\ z_j(Q, (\l _j-\l _i)/d' ) \]
{\em where}
\[ Coeff \ _i^j(d')=
\frac{[(\l_j -\l_i)/d]^{(n+1-l)d'-1}}{1+(\l _i-\l _j)/d' \hbar}
\frac{Euler \ (V')}{Euler \ (N')} \ \phi _i |_{p=\l _i} \ . \]
{\em Proof.} Here $(\l _i-\l_j)/d' $ is the localization of $c$, and the key
point is that the equivariant Chern class of the line bundle $L$ over
$Y_{2,d-d'}$ is what we would denote $\hbar + c$ {\em for the moduli space}
$Y_{2,d-d'}$ but with $\hbar = (\l _j -\l _i)/d'$. This is how the recursion
for the correlators $z_i$ becomes possible. The rest is straightforward.
{\em Remark.} Our reduction to the linear
recursion relation can be interpreted in the following more geometrical way:
contributions
of all non-isolated fixed points cancel out with some explicit part of the
contribution from {\em isolated} fixed points; the latter are represented by
chains of multiple covers of straight lines connecting the two marked points.
\bigskip
Let us write down explicitly the factor $Coeff \ _i^j(d)$
from Proposition $9.8$
(compare with \cite{Kn}). $Coeff \ _i^j (d) =$
\[ \frac{
\Pi _{m=1}^{ld} (l\l _i -\l' + m(\l _j-\l_i)/d) [(\l_j-\l_i)/d]^{(n+1-l)d-1}
}{
d (1+(\l_i -\l_j)/\hbar d)
\Pi _{\a =0}^n \ _{m=1}^d \ _{(\a ,m)\neq (j, d)} (\l_i-\l_{\a}+m(\l_j-\l_i)/d)
} = \]
(here the product in the numerator is $Euler (V')$, the
denominator --- it is essentially $Euler (N')$ where however the cancellation
with $\phi _i |_{p=\l _i}$ is taken care of --- has been computed using the
exact sequence $0 \to {\Bbb C} \to {\Bbb C} ^{n+1} \otimes H \to T_Y \to 0$ of vector
bundles over $Y= {\Bbb C} P^n$, and the ``hard-to-explain'' extra-factor $d$
is due to the orbifold structure of the moduli spaces (the $d$-multiple map
of $C'$ onto the $(ij)$-line in ${\Bbb C} P^n$ has a discrete symmetry of order
$d$)
\[ = \frac{1}{[(\l_i-\l_j)/\hbar + d]}
\frac{\Pi _{m=1}^{ld} (\frac{(\l_i-\l')d}{\l_j-\l_i} +m)}
{\Pi _{\a =0}^n\ _{m=1}^d \ _{(\a ,m)\neq (j,d)}
(\frac{(\l_i-\l_{\a })d}{\l_j-\l_i} + m) } \ . \]
Now it is easy to check
{\bf Proposition 9.9.} {\em The correlators $z_i(Q, 1/\o )$ are power series
$\sum C_i(d) Q^d $ in $Q$ with coefficients $C_i(d)$ which are {\em reduced}
rational functions of $\o $ with poles of the order $\leq 1$ at
$\o = d'/(\l_j-\l_i)$ with $d'=1,...,d$. The correlators $z_i$ are uniquely
determined by these properties, the recursion relations of Proposition $9.8$
and the initial condition $C_i(0)=1$.}
\bigskip
The proof of Theorem $9.5$ is completed by the following
{\bf Proposition 9.10.} {\em The series}
\[ z_i=\sum _{d=0}^{\infty } Q^d
\frac{\Pi _{m=1}^{ld} ((l\l_i -\l')\o +m)}
{d! \Pi _{\a \neq i} \Pi _{m=1}^d ((\l_i -\l_{\a })\o +m)} \]
{\em satisfy all the conditions of Proposition $9.10$.}
{\em Proof.} The recursion relation is deduced by the decomposition of
the rational functions of $\o $ into the sum of simple fractions (or,
equivalently, from the Lagrange interpolation formula for each numerator
through its values at the roots of the corresponding denominator).
\section{Complete intersections with $l_1+...+l_r=n$}
\label{sec10}
Let $X\subset Y={\Bbb C} P^n$ be a non-singular complete intersection given by
equations of degrees $(l_1,...,l_r)$ with $l_1+...+l_r=n$. There are only
two points where our proof of Theorem $9.1$ would fail for such $X$. One of
them is the Lagrange interpolation formula in the proof of Proposition $9.10$.
Namely, the rational functions of $\o $ there are not reduced ---
the degree $dl$ of the numerator {\em is equal} to the degree $dn$ of the
corresponding denominator. The other one is Lemma $9.7$. Namely, we have the
following lemma instead.
{\bf Lemma 10.1.}
{\em The type (ii) fixed point component in $Y_{2,d}$ makes zero
contribution via Borel localization formulas to $\int _{Y_{2,d}} c^{d-1}...$
{\em unless} it consists of maps $(C'\cup C'', x_0,x_1)\to Y$ where $C'$ is
mapped to a fixed point in ${\Bbb C} P^n$ and carries both marked points, and
$C''$ is a disjoint union of $d$ irreducible components (intersecting $C'$ at
$d$ special points) mapped (each with multiplicity $1$) onto straight lines
outgoing the fixed point. All type (ii) components make zero contribution
to $\int _{Y_{2,d}} c^d ...$.}
Let us modify the results of Section $9$ accordingly. As we will see,
the LHS in Theorem $9.5$ is now only {\em proportional} to the RHS, and we will
compute the proportionality coefficient (a series in $q$) directly.
{\bf Proposition 10.2.} {\em Put $z_i(Q,\hbar ):= Z_i(\hbar Q,\hbar )$. Then}
\[ z_i(Q,\hbar) = 1 + \sum _{d>0} Q^d Coeff\ _i(d) +
\sum _{j\neq i} \sum _{d'>0} Q^{d'}
Coeff \ _i^j(d')\ z_j(Q,(\l_j-\l_i)/d') \]
{\em where $Coeff \ _i(d)$ is equal to the contribution of type (ii) fixed
point components to \newline
$\int _{Y_{2,d}} (-c)^{d-1} E_d' \ e_0^*(\phi _i) $, and}
\[ Coeff \ _i^j(d) = \frac{1}{[(\l_i-\l_j)/\hbar +d]}\
\frac{\Pi _{a=1}^r \Pi _{m=1}^{dl_a} (\frac{(l_a\l_i-\l_a')d}{\l_j-\l_i}+m)}
{\Pi _{\a =0}^n \ _{m=1}^d \ _{(\a ,m)\neq (j,d)}
(\frac{(\l_i-\l_{\a })d}{\l_j-\l_i} +m)} \ .\]
{\bf Corollary 10.3.} {\em The correlators $z_i(Q,1/\o )$ are power series
$\sum _d C_i(d) Q^d$ with coefficients
\[ C_i(d)=P_d(\o ,\l ,\l ')/\Pi _{\a }\Pi _{m=1}^d ((\l_i-\l_{\a })\o +m) \]
where $P_d = P_d^0 \o ^{nd} + ... $ is a polynomial in $\o $ of degree $nd$.
The correlators $z_i$ are uniquely determined by these properties, the
recursion relations of Proposition $10.2$ and the initial conditions
\[ \sum _d Coeff \ _i(d) Q^d = \sum _d Q^d \frac{P_d^0}
{d!\Pi _{\a \neq _i} (\l _i-\l_{a })^d} \ .\]}
{\bf Proposition 10.4.} {\em The series
\[ z_i' =\sum _{d=0}^{\infty } Q^d
\frac{\Pi _{a=1}^r \Pi _{m=1}^{l_a d} ((l_a\l_i-\l_a')\o +m)}
{d!\Pi _{\a \neq i}\Pi _{m=1}^d ((\l_i-\l_{\a })\o +m)} \ \]
satisfy the requirements of Corollary $10.3$ with the initial condition
\[ \sum _d Q^d \frac{\Pi _{a=1}^r (l_a\l_i-\l_a')^{l_a d} }
{ d!\Pi _{\a \neq i}
(\l_i-\l_{\a })^d} \ = \
\exp \{ Q \frac{\Pi _a (l_a\l_i -\l_a')^{l_a}}{\Pi _{\a \neq i}
(\l_i-\l_{\a })} \} . \]}
\bigskip
Now let us compute $Coeff \ _i(d)$ using the description of type (ii) fixed
point components given in Lemma $10.1$.
{\bf Proposition 10.5.} {\em Contribution of the type (ii) fixed point
components to \newline
$\sum _d Q^d \int _{Y_{2,d}} (-c)^{d-1} E'_d \ \phi _i $ is}
\[ \exp \{ Q \frac{\Pi _a (l_a\l_i-\l_a')^{l_a}}
{\Pi _{\a \neq i} (\l_i-\l_{\a })} \} \ \exp \{ - Q \ l_1!...l_r! \} \ .\]
{\em Proof.} Each fixed point component described in Lemma $10.1$ is isomorphic
to the Deligne - Mumford configuration space ${\cal M} _{0,d+2}$. Our computation
is based on the following known formula (see for instance \cite{Kn} ) for
correlators between Chern classes of universal tangent lines at the marked
points:
\[ \int _{{\cal M} _{0, k}} \frac{1}{(w_1+c_1^{(1)}) ... (w_k+c_1^{(k)})} =
\frac{(1/w_1+...1/w_k)^{k-3}}{w_1 ... w_k} \ .\]
Consider the type (ii) fixed point component specified by
the following combinatorial structure of stable maps: $d$ degree $1$
irreducible components join the $i$-th fixed point with the fixed points
with indices $j_1,...,j_d$. Using the above formula and describing explicitly
the normal bundle to this component in $Y_{2,d}$ and localization of the Euler
class $E_d'$ we arrive to the following expression for the contribution of this
component to $\int _{Y_{2,d}} (-c)^{d-1} \phi _i E_d'$:
\[ \Pi _{s=1}^d
\frac{\Pi _{a=1}^r \Pi _{m=1}^{l_a}(l_a\l_i-\l_a' +m(\l_{j_s}-\l_i))}
{(\l_i-\l_{j_s}) \Pi _{\a \neq j_s,i} (\l_{j_s}-\l_{\a })} \ . \]
Summation over all type (ii) components in all $Y_{2,d}$ with weights $Q^d$
gives rise to
\[ \exp \{ - Q \sum _{j\neq i}
\frac{\Pi _a \Pi _{m=1}^{l_a} (l_a\l_i-\l_a' +m(p-\l_i))}
{\Pi _{\a \neq j} (p-\l_{\a})} \ |_{p=\l_j} \} \ .\]
The exponent can be understood as a sum of residues at $p\neq \l_i, \infty $
and is thus opposite to the sum
\[ l_1!...l_r! -
\frac{\Pi _a (l_a\l_i-\l_a')^{l_a}}{\Pi _{\a \neq i} (\l_i-\l_{\a })}
\]
of residues at $\infty $ and $\l_i $.
{\bf Corollary 10.6.} $z_i(Q,1/\o )= z_i'(Q,\o )\ \exp (- l_1! ... l_r! Q)$.
{\em Proof.} Multiplication by a function of $Q$ does not destroy the
recursion relation of Proposition $10.2$ but changes the initial condition.
\bigskip
We have proved the following
{\bf Theorem 10.7.} {\em Suppose $l_1+...+l_r=n$. Then}
\[ S' = e^{ (pt - l_1!...l_r! e^t)/\hbar } \
\sum _{d=0}^{\infty } \ e^{dt} \
\frac{\Pi _0^{dl_1} (l_1p-\l_1'+m\hbar) ... \Pi _0^{dl_r} (l_rp-\l_r'+m\hbar)}
{\Pi _1^d (p-\l_0 +m\hbar ) ... \Pi _i^d (p-\l_n +m\hbar )} \ .\]
\[ S = S' |_{\l =0, \l' =0} =
e^{ (Pt - l_1! ... l_r! e^t)/\hbar }
\frac{ \Pi _{j=1}^r \Pi _{m=0}^{dl_j} (l_j P+m\hbar)}
{\Pi _{m=1}^d (P+m\hbar)^{n+1}} \ \ (\text{mod} \ P^{n+1}) \ .\]
{\bf Corollary 10.8.} {\em Let $D=\hbar d/dt + l_1!...l_r! e^t$. Then}
\[ D^{n+1-r} S = l_1...l_r e^t \Pi _{j=1}^r
(l_j D+\hbar)...(l_j D +(l_j-1)\hbar ) \ S .\]
{\bf Corollary 10.9.} {\em In the quantum cohomology algebra of $X$ the class
$p$ of hyperplane sections satisfies the following relation
(with the same reservation in the case $\operatorname{dim} X \leq 2$ as in Corollary $9.3$):}
\[ (p + l_1!...l_r! q)^{n+1-r}=
l_1^{l_1} ... l_r^{l_r} q (p+l_1!...l_r!q)^{n-r} \ .\]
{\em Control examples.} $1$.
$X=pt$ in ${\Bbb C} P^1$ ($n=1, r=1, l=1$). The above relation
takes on $p+q=q$, or $p=0$. Since $P^2=0$, we also find from Theorem 10.7 that
$ S=P \exp (-e^t) \sum _d e^{dt}/d! = P $, or $\langle 1, S\rangle =1 $ as it should
be for the solution of the differential equation $\hbar d/dt \ s = 0$ that
arises from quantum cohomology of the point.
$2$. $X={\Bbb C} P^1$ embedded as a quadric into ${\Bbb C} P^2$ ($n=2, r=1, l=2$).
We get $(p+2q)^2=4q(p+2q)$, or $p^2=4q^2$. Taking into account that $p$ is
twice the generator in $H^2({\Bbb C} P^1)$ and the line in ${\Bbb C} P^1$ has the degree
$2$ in ${\Bbb C} P^2$ we conclude that this is the correct relation in the quantum
cohomology of ${\Bbb C} P^1$. This example was the most confusing for the author:
predictions of the loop space analysis \cite{HG1} appeared totally unreliable
because they gave a wrong answer for the quadric in ${\Bbb C} P^2$. As we see now,
the loop space approach gives correct results if $l_1+...+l_r<n$ and require
``minor'' modification (by the factor $\exp (-l_1!...l_r! q /\hbar )$ )
in the boundary
cases $l_1+...l_r=n$; the quadric on the plane happens to be one of such cases.
$3$. $n=3, r=1, l=3$. We have $(p+6q)^3 = 27q (p+6q)^2 $, or $p^3=9qp^2+6^3q^2p
+ 27\cdot 28 q^3$. In particular, $\langle p * p, p\rangle = 9 q \langle p, p \rangle +
6^3q^2\langle p, 1\rangle + 27\cdot 28 q^3 \langle 1, 1\rangle = 27 q +0+0$ which indicates
that there should exist $27$ discrete lines on a generic cubical surface in
${\Bbb C} P^3$.
\section{Calabi-Yau projective complete intersections}
\label{sec11}
Let $L_d(Y)$ denote, as in Section $6$, the moduli space of stable maps
$\psi: {\Bbb C} P^1 \to {\Bbb C} P^n \times {\Bbb C} P^1$ of bidegree $(d,1)$ with $2$ marked
points mapped to ${\Bbb C} P^n \times \{ 0\} $ and ${\Bbb C} P^n \times \{ \infty \} $
respectively. Let ${\cal E} _d$ denote the equivariant Euler class of the
vector bundle over $L_d(Y)$ with the fiber
$H^0({\Bbb C} P^1, \psi ^*(E))$ where $E$ is the bundle on ${\Bbb C} P^n \times {\Bbb C} P^1$
induced from our ample bundle $\oplus _a H^{l_a}$ by the projection to the
first factor.
Consider the equivariant correlator
\[ \Phi = \int _Y \ Euler^{-1}(E) \ S'(t,\hbar ) \ S'(\t ,-\hbar) \ = \]
\[ = \sum _{d,d'} e^{dt} e^{d'\t} \sum _i \frac{\Pi _a (l_a \l_i -\l'_a)}
{\Pi _{j\neq i} (\l_i-\l_j)}
\int _{Y_{2,d}} E'_d \frac{e^{pt/\hbar } e_0^*(\phi _i)}{\hbar + c} \
\int _{Y_{2,d'}} E'_{d'} \frac{e^{-p\t /\hbar} e_0^*(\phi _i)}{-\hbar +c} .\]
In the case $l_1+...+l_r < n$ it is easy to check using the explicit formula
for $S'$ from Theorem $9.5$ that
\[ \Phi = \frac{1}{2\pi i} \oint e^{p(t-\t )/\hbar} [ \sum _d e^{d\t }
\frac{\Pi _a \Pi _{m=0}^{l_a d} (l_a p-\l'_a -m\hbar)}
{\Pi _{j=0}^n \Pi _{m=0}^d (p-\l_j - m\hbar)}] \ dp. \]
This is an equivariant version of a formula found in \cite{HG1} in the context
of loop spaces and toric compactifications of spaces of rational maps. Namely,
consider the projective space $L'_d$ of $(n+1)$-tuples of polynomials in
one variable of degree $\leq d$ each, up to a scalar factor (notice that
$L'_d$ has the same dimension $d(n+1)+n$ as $L_d$). It inherits the
component-wise action of the torus $T^{n+1}$ and the action of
$S^1$ by the rotation
of the variable (``rotation of loops''). Integration over the equivariant
fundamental cycle in $L'_d$ is given by the residue formula
\[ f(p,\l , \hbar) \mapsto \frac{1}{2\pi i}\oint \frac{ f dp }
{\Pi _{j=0}^n \Pi _{m=0}^d (p -\l_j - m\hbar ) } .\]
Consider the equivariant vector bundle over $L'_d$ such that substitution
of the $(n+1)$ polynomials into $r$ (invariant) homogeneous equations
in ${\Bbb C} P^n$ of degrees $l_1, ..., l_r$ produces a section of this bundle.
The equivariant Euler class of the bundle is
\[ {\cal E} '_d = \Pi _{a=1}^r \Pi _{m=0}^{l_a d} (l_a p -\l'_a - m\hbar ) .\]
The formula for $\Phi $ indicates that there should exist a close relation
between the spaces $L_d$ and $L'_d$. This relation is described in the
following lemma whose proof will be given in the end of this Section.
{\bf The Main Lemma.} {\em There exists a natural
$S^1\times T^{n+1}$-equivariant map
$\mu : L_d\to L'_d$. Denote $-p$ the equivariant $1$-st Chern class of the Hopf
bundle over $L'_d$ induced by $\mu $ to $L_d$. Then}
\[ \Phi (t,\t) =
\sum _d e^{d\t } \int _{L_d} e^{p(t-\t )/\hbar} {\cal E} _d .\]
Define $\Phi ' (q, z, \hbar ) := \Phi | t=\t +z\hbar, q=e^{\t}$. (By the way
the limit of the series $\Phi '$ at $\hbar =0$ has the topological meaning
of what is called in \cite{GK} the {\em generating volume function}, and the
meaning of this limit procedure in terms of differential equations satisfied
by $\Phi $ is the {\em adiabatic approximation}.)
{\bf Corollary 11.1.} $\Phi '(q,z):= \sum_d q^d \int _{L_d} e^{pz} {\cal E} _d
=$
\[ = \frac{1}{2\pi i} \oint e^{pz} \sum _d \frac{ E_d(p,\l, \l',\hbar )}
{\Pi _{j=0}^n \Pi _{m=0}^d (p-\l_j -m\hbar)} dp \]
{\em where $E_d = \mu _* ({\cal E} _d)$ is a {\em polynomial} (of degree
$<(n+1)d$) of all its variables.}
{\em Proof.} The integrals $E^{(k)} = \int _{L_d} p^k {\cal E} _d, \ k=0,...,
\operatorname{dim} L'_d$, which determine the push-forward $\mu _* ({\cal E})$
are polynomials in $(\l, \l', \hbar )$. The matrix
$\int _{L'_d} p^{i+\operatorname{dim} L'_d - j}$ is triangular with all eigenvalues equal
to $1$. This means that there exists a unique polynomial in $p$ with
coefficients {\em polynomial in} $(\l, \l', \hbar)$ which represents
the push-forward with any given polynomials $E^{(k)}(\l ,\l', \hbar)$.
The last argument also proves
{\bf Proposition 11.2.}{ \em Suppose that a series
\[ s= \sum _d q^d \frac{ P_d(p,\l,\l',\hbar)}
{ \Pi _j \Pi _{m =0}^d (p-\l_j -m\hbar)} \]
with coefficients $P_d$ which are polynomials of $p$ of degree $\leq \operatorname{dim} L_d$
has the property that for every $k=0,1,2,...$ the $q$-series $\oint s p^k dp $
has polynomial coefficients in $(\l, \l', \hbar)$. Then the coefficients of
all $P_d$ are polynomials of $(\l,\l',\hbar)$, and vice versa.}
\bigskip
The coefficient $E_d(p,\l,\l',\hbar)$ in the series $\Phi '$ has the total
degree $(l_1+...+l_r)d+r$ according to the dimension of the vector bundle
whose Euler class it represents. Consider the following operations with the
series $\Phi $:
(i) multiplication by a series of $e^t$ and / or $e^{\t }$;
(ii) simultaneous change of variables $t\mapsto t+f(e^t),
\t \mapsto \t +f(e^{\t} )$.
(iii) multiplication by $\exp [C (f(e^t)-f(e^{\t }))/\hbar ]$
(here the factor $C$ should be a linear function of $(\l, \l')$ in order
to obey homogeneity).
{\bf Proposition 11.3.} {\em The property of the series $\Psi $ to generate
polynomial coefficients $E_d(p,\l,\l' \hbar)$ is invariant with respect
to the operations (i),(ii),(iii).}
{\em Proof.} The polynomiality property of coefficients in $\Phi '$ is
equivalent, due to Proposition $11.2$, to the fact that for all $k$ the
$q$-series $(\partial /\partial z)^k |_{z=0} \Phi ' $ has polynomial coefficients.
Multiplication by a series of $q$ does not change this property, which proves
the invariance with respect to multiplication by functions of $e^{\t }$.
The roles of $t$ and $\t $ can be interchanged by the substitutions
$p \mapsto p+\hbar d, \hbar \mapsto -\hbar $ in each summand of $\Phi $.
This proves the invariance with respect to multiplication by functions
of $e^t$.
The operation (ii) transforms $\Phi '$ to
\[ \frac{1}{2\pi i} \sum _d q^d e^{d f(q)} \oint \
\exp \{ p\frac{z\hbar + f(qe^{z\hbar})-f(q)}{\hbar } \} \
\frac{E_d(p)}{\Pi _j \Pi _m (p-\l_j -m\hbar)} \ dp .\]
Since the exponent is in fact divisible by $\hbar $, the derivatives in $z$
at $z=0$ still have polynomial coefficients. This proves the invariance
with respect to (ii). The case of the operation (iii) is analogous.
\bigskip
We are going to use the above polynomiality and invariance
properties of the correlator $\Phi $ in order to describe quantum cohomology
of {\em Calabi-Yau} complete intersections in ${\Bbb C} P^n$ (in which case
$l_1+...+l_r = n+1$). We will use this polynomiality in conjunction
with recursion relations based on
the fixed point analysis of Sections $9, 10$. The result can be roughly
formulated in the following way: the hypergeometric functions of Theorem $9.5$
in the case $l_1+...+l_r =n+1$ can be transformed to the correlators $S'$ by
the operations (i),(ii),(iii). Notice that in the Calabi -- Yau case
all our formulas are homogeneous with the grading $\deg q = 0,
\deg p =\deg \hbar =\deg \l =\deg \l' =1, \deg z =-1$. In particular the
transformations (i)--(iii) also preserve the degree $\operatorname{dim} L_d$ of the
coefficients $E_d$ in $\Phi '$. In the ``positive'' case $l_1+...+l_r\leq n$
where $\deg q = n+1 -\sum l_a > 0$ the transformations (i)--(iii) in fact
increase degrees of the coefficients $E_d$ and are ``not allowed''. The only
exception is the operation (iii) with $f(q)=\text{const} \ q$ in the case
$l_1+...+l_r=n$ when $\deg q =1$. In Section $10$ we found the right constant
to be $-l_1!...l_r!$
\bigskip
Consider now the correlator $\Phi $,
\[ \Phi = \sum _i \frac{\Pi _a (l_a\l_i -\l'_a)}{\Pi _{j\neq i} (\l_i-\l_j)}
\ e^{\l_i (t-\t )/\hbar } \ Z_i(e^t,\hbar ) \ Z_i(e^{\t }, -\hbar ), \]
in the Calabi-Yau case $l_1+...+l_r=n+1$ (see Section $9$ for a definition
of $Z_i$).
{\bf Proposition 11.4.} {\em $(1)$ The coefficients of the power series
$Z_i(q,\hbar) = \sum _d q^d C_i(d) $ are rational functions
\[ C_i(d)=\frac{ P_d^{(i)} }{d! \hbar ^d \Pi _{j\neq i}
\Pi _{m=1}^d (\l_i -\l_j +m\hbar )} \]
where $P_d^{(i)}$ is a polynomial in $(\hbar , \l, \l')$ of degree $(n+1)d$.
$(2)$ The polynomial coefficients $E_D(p)$ in $\Phi '$
are determined by their values
\[ E_D (\l_i +d\hbar )=\Pi _a (l_a\l_i -\l'_a) \ P_d^{(i)}(\hbar )
P_{D-d}^{(i)}(-\hbar ) \]
at $p=\l_i +d\hbar $, $i=0,...,n$, $d=0,...,D$.
$(3)$ The correlators $ Z_i(q, \hbar)$
satisfy the recursion relation
\[ Z_i(q,\hbar ) = 1+ \sum_d \frac{q^d}{\hbar ^d } \frac{R_{i,d}}{d!} +
\sum _d \sum _{j\neq i} \frac{q^d}{\hbar ^d}
\frac{ Coeff \ _i^j(d) }{\l_i-\l_j+d \hbar } \
Z_j (\frac{q}{\hbar }\frac{(\l_j-\l_i)}{d}, \frac{(\l_j-\l_i)}{d}) \]
where $R_{i,d}=R_{i,d}^{(0)}\hbar ^d + R_{i,d}^{(1)}\hbar ^{d-1} + ...$
is a polynomial of $(\hbar ,\l ,\l')$ of degree $d$, and
\[ Coeff\ _i^j(d)= \
\frac{\Pi _a \Pi _{m=1}^{l_a d} (l_a\l_i-\l'_a +m(\l_j-\l_i)/d)}
{d!\Pi _{\a \neq i}\ _{m=1}^d\ _{(\a,m)\neq (j,d)}
(\l_i-\l_{\a } +m(\l_j-\l_i)/d)} \ .\]
For any given $\{ R_{i,d} \}$ these recursion relations have a unique solution
$\{ Z_i \} $.
$(4)$ Consider the class $\cal{P}$ of solutions $\{ Z_i \}$ to these recursion
relations which give rise to polynomial coefficients
$E_d$ in the corresponding $\Phi $. A solution from $\cal{P}$ is
uniquely determined by the first two coefficients
$R_{i,d}^{(0)}, R_{i,d}^{(1)}, i=0,...,n, d=1,...,\infty ,$
of its initial condition (that is by the first two terms in the
expansion of $Z_i= Z_i^{(0)} + Z_i^{(1)}/\hbar + ... $
as power series in $1/\hbar $).
$(5)$ The class $\cal{P}$ is invariant with respect to the following
operations:
(a) simultaneous multiplication $Z_i \mapsto f(q) Z_i $ by a power series
of $q$ with $f(0)=1$;
(b) changes $ Z_i (q,\hbar) \mapsto e^{ \l_i f(q)/\hbar }
Z_i(q e^{f(q)},\hbar ) $ with $f(0)=0$;
(c) multiplication $Z_i \mapsto \exp (C f(q)/\hbar ) Z_i$ where $C$
is a linear function of $(\l, \l')$ and $f(0)=0$.}
\bigskip
{\em Proof.} $(3)$ We have
\[ Z_i =1+ \sum _{d>0} q^d [ \sum _{k=0}^{d-1} \hbar ^{-k-1} \int _{Y_{2,d}}
E'_d e_0^*(\phi _i) (-c)^k ] + \sum _{d>0} q^d \hbar ^{-d} \int _{Y_{2,d}}
E'_d e_0^*(\phi _i) \frac{(-c)^d}{\hbar + c} \]
where the integrals of the last sum have zero contributions from the type (ii)
fixed point components (Lemmas $9.7, 10.1$). Thus these integrals have a
recursive expression identical to those of Sections $9$ and $10$. The terms
of the double sum constitute the initial condition $\{ R_{i,d} \} $. The
recursion relations have the form of the decomposition of rational functions
of $\hbar $ (coefficients at powers of $Q=q/\hbar $) into the sum of simple
fractions in the case when degrees of numerators exceed degrees of
denominators. This proves existence and uniqueness of solutions to the
recursion relations.
$(1)$ follows directly from the form and topological meaning of the recursion
relations.
$(2)$ follows from the definition of $\Phi $ in terms of $Z_i$.
$(4)$ Perturbation theory: Suppose that two solutions from the class $\cal{P}$
have the same initial condition up to the order $(d-1)$ inclusively. Then
$(2)$ shows that corresponding $E_k$ for these solutions coincide for
$k<d$ and
the variation $\d E_d (p) $ vanishes at $p=\l_i + k\hbar $ for $0<k<d$.
This means that the polynomial $\d E_d$ is divisible by
$\Pi _j \Pi _{m=1}^{d-1} (p-\l_j -m\hbar)$. On the other hand $(1)$ and $(2)$
imply that the variation $\d R_{i,d}$ of the initial condition satisfies
\[ \d R_{i,d} (\hbar ) \ \Pi _a (l_a \l_i -\l'_a) \Pi _{j\neq i} \Pi _{m=1}^d
(\l_i -\l_j + m\hbar ) = \d E_d |_{p= \l_i +\hbar d } \]
(since $R_{i,0}=1$) and thus $\d R_{i,d} $ is divisible by $\hbar ^{d-1}$.
Since $\d R_{i,d}$ is a degree $d$ polynomial, it leaves only the possibility
\[ \d R_{i,d} = \d R_{i,d}^{(0)} \hbar ^d + \d R_{i,d}^{(1)} \hbar ^{d-1} .\]
Thus if two class $\cal{P}$ solutions coincide in orders $\hbar ^{0},
\hbar ^{-1}$ then $\d R_{i,d} =0$, and thus the very solutions coincide.
$(5)$ The operations (a),(b),(c) give rise to the operations of type (i)-(iii)
for corresponding polynomials $E_d$. Thus it suffices to show that the
operations (a),(b),(c) transform a solution
$\{ Z_i \} $ of the recursion relations to another solution.
Consider in our recursion relation the coefficient
$\hbar ^{-d}q^d\ Coeff \ _i^j(d)$
responsible for the simple
fraction with the denominator $(\l_i -\l_j +d\hbar)$. The operations (a), (b),
(c)
cause respectively the following modifications in this coefficient:
\[ q^d\mapsto f(q)q^d/f(Q), \]
\[ q^d\mapsto q^d \exp \{ \frac{\l_i f(q)}{\hbar } + d f(q) -
\frac{d\l_j f(Q)}{(\l_j-\l_i)} \} , \]
\[ q^d\mapsto q^d \exp \{ C\frac{f(q)}{\hbar} +
C\frac{df(Q)}{\l_i-\l_j} \} , \]
where $Q=(\l_j-\l_i)q/d\hbar = q - (\l_i -\l_j + d \hbar ) q/ \hbar d $.
In the case of the change (b), additionally, the argument $q$ in $Z_j$
on the RHS of the recursion relation gets an extra-factor
$\exp [ f(q)-f(Q) ]$.
The difference $Q-q$ and the exponents vanish at $\hbar = (\l_j-\l_i)/d$.
This means that
\[ \frac{q^d}{l_i -\l_j + d\hbar } \mapsto \frac{q^d}{\l_i-\l_j +d\hbar }
+ \ \text{terms without the pole} .\]
The latter terms give contributions to a new initial condition, while the
coefficient \newline $\hbar ^{-d} q^d\ Coeff \ _i^j(d)$ does not change.
It is easy to see that the required properties of the initial condition
(that the degree of $R_{i,d}(\hbar ) $ does not exceed $d$ and $R_{i,0} =1$)
are also satisfied under our assumptions about $f$ (for those contributions
involve $\hbar $ only in the combination $q/\hbar $).
\bigskip
Let us consider now the hypergeometric series
\[ Z_i^* = \sum _{d=0}^{\infty } q^d \
\frac{\Pi _{a=1}^r \Pi _{m=1}^{l_a d} (l_a\l_i -\l'_a +m\hbar)}
{\Pi _{\a =0}^n \Pi _{m=1}^d (\l_i -\l_{\a }+m\hbar)} \]
where $l_1+...+l_r=n+1$.
It is straightforward to see that $\{ Z_i^* \} $ satisfy the recursion
relations of Proposition $11.4 (3)$ (see the proof of Proposition $9.10$)
and that the formulas of Proposition $11.4 (2)$ generate corresponding
\[ \Phi ^* = \frac{1}{2\pi i} \oint \ e^{p(t-\t )/\hbar }
\sum _{d=0}^{\infty } e^{d\t }
\frac{\Pi _{a=1}^r \Pi _{m=0}^{l_a d} (l_a p - \l'_a -m\hbar)}
{\Pi _{i=0}^n \Pi _{m=0}^d (p-\l_i -m\hbar)} dp \]
with polynomial numerators. Thus $\{ Z_i \} $ is a solution from the class
$\cal{P}$.
Computation of the first two terms in the initial condition gives
\[ Z_i^*\ ^{(0)} = f(q)=\sum _{d=0}^{\infty }
\frac{(l_1d)!...(l_rd)!}{(d!)^{n+1}} \ q^d \ ,\]
\[ Z_i^*\ ^{(1)} = \l_i \sum _a l_a [g_{l_a}(q)- g_1(q)] +
(\sum _{\a } \l_{\a }) g_1(q) - \sum _a \l'_a g_{l_a}(q) \]
where
\[ g_l=\sum _{d=1}^{\infty } q^d \frac{\Pi _a (l_a d)!}{(d!)^{n+1}}
\ (\sum _{m=1}^{ld} \frac{1}{m} ) \ .\]
\bigskip
Let us compare these initial conditions with those for $\{ Z_i \}$.
{\bf Proposition 11.5.} $Z_i^{(0)}=1, \ Z_i ^{(1)} =0$.
{\em Proof.} The first statement follows from the definition of $Z_i$ while
the second means that $\int _{Y_{2,d}} E'_d e_0^*(\phi _i) =0$ for all $d>0$.
It is due to the fact that the class $E'_d e_0^*(\phi _i)$
is a pull-back from $Y_{1,d}$.
(In fact we have just repeated an argument proving $(5)$ from Section $5$
and thus the proposition can be deduced from general properties of quantum
cohomology.)
\bigskip
Combining the last two propositions we arrive to the following
\bigskip
{\bf Theorem 11.6.} {\em The hypergeometric solution $\{ Z_i^*(q,\hbar) \} $
coincides with the solution $\{ Z_i(Q,\hbar) \} $ up to transformations
(a),(b),(c). More precisely, perform the following operations
with $\{ Z_i \}$
1) put
\[ Q=q\exp \{ \sum _a l_a [g_{l_a}(q) - g_1(q)]/f(q) \} \ ,\]
2) multiply $Z_i (Q(q),\hbar )$ by
\[ \exp \{ \frac{1}{f(q) \hbar } [\sum_{a} (l_a\l_i- \l'_a) g_{l_a} (q)
-(\sum_{\a} (\l_i-\l_{\a })) g_1(q)] \} ,\]
3) multiply all $Z_i$ simultaneously by $f(q)$.
Then the resulting functions coincide with hypergeometric
series $Z_i^*(q,\hbar)$.}
{\em Proof.} The three steps correspond to consecutive applications of
operations of type (b),(c) and (a) to $\{ Z_i \} $ and transform the
initial condition of Proposition $11.5$ to that for $\{ Z_i^*\} $.
According to Proposition $11.4$ this transforms the whole solution
$\{ Z_i \}$ to $\{ Z_i^* \} $.
\bigskip
Consider the solutions
\[ s_i = \ e^{\l_i T/\hbar } \ Z_i (e^T, \hbar ) \]
to the equivariant quantum cohomology differential equations.
{\bf Corollary 11.7.} {\em The operations
1) change $T=t+ \sum _{\a } l_a [g_{l_a} (e^t) -g_1(e^t)]/f(e^t) $,
2) multiplication by
\[ f(e^t)\exp \{ [g_1(e^t) (\sum _{\a } l_{\a }) - \sum _a \l'_a g_{l_a}(e^t)]
/(\hbar f(e^t)) \} \]
transform $\{ s_i \} $ to the hypergeometric solutions
\[ s_i^* = e^{pt/\hbar } \sum _d e^{dt}
\frac{\Pi _a \Pi _{m=1}^{l_a d} (l_a p -\l'_a +m\hbar)}
{\Pi _{\a } \Pi _{m=1}^d (p -\l_{\a } +m\hbar)} \ |_{p=\l_i } \]
of the differential equation
\[ \Pi _{\a} (\hbar \frac{d}{dt} -\l_{\a } ) s^* = e^t \
\Pi_a \Pi_{m=1}^{l_a} (\hbar l_a \frac{d}{dt} -\l'_a +m\hbar ) \ s^* \ .\]
For $\l'=0 ,\ \l_0+...+\l_n=0$ the solutions $s_i^*$ have the following
integral representation:
\[ \int _{\Gamma ^n\subset \{ F_0(u)=e^t \} }
\frac{u_0^{\l_0}...u_n^{\l_n} \ du_0\wedge ... \wedge du_n }
{ \ F_1(u) \ ... \ F_r(u) \ dF_0} \]
where
\[ F_1=(1-u_1-...-u_{l_1}), \ F_2=(1-u_{l_1+1}-...-u_{l_1+l_2}), \ ...,
F_r=(1-u_{l_1+...+l_{r-1}+1}-...-u_{l_1+...+l_r}) \ \]
and $F_0=u_0...u_n$. }
\bigskip
{\bf Corollary 11.8.} {\em The hypergeometric class $S^*(t,\hbar )\in
H^*({\Bbb C} P^n)={\Bbb C} [P]/(P^{n+1})$,
\[ S^*= e^{Pt/\hbar } \sum_d e^{dt} \frac{\Pi_a \Pi_{m=0}^{l_ad}
(l_aP+m\hbar )}{\Pi_{m=1}^d (P+m\hbar)^{n+1} } \]
whose $n+1-r$ non-zero components are solutions to the Picard-Fuchs equation
\[ (\frac{d}{dt})^{n+1-r} s^* = l_1...l_r e^t \Pi_a \Pi_{m=1}^{l_a-1}
(l_a \frac{d}{dt} + m) s^* \]
for the integrals
\[ \int _{\gamma ^{n-r} \subset X_t'} \frac{du_0\wedge ...\wedge du_n}
{dF_0\wedge dF_1\wedge ... \wedge dF_r} \ ,\]
(here $X_t'=\{ (u_0,...,u_n) | F_0(u)=e^t, F_1(u)=0, ..., F_r(u)=0 \} $)
are obtained from the class $S$ (describing the quantum cohomology
$\cal D$-module for the Calabi-Yau complete intersection
$X^{n-r}\subset {\Bbb C} P^n$),
\[ S=e^{PT/\hbar } \sum_d e^{dT} (e_0)_* (\frac{E_d}{\hbar+c_1^{(0)}}) ,\]
by the change
\[ T=t + \sum _a l_a [g_{l_a}(e^t)-g_1(e^t)]/f(e^t) \]
followed by the multiplication by $f(e^t)$.}
{\em Proof.} Corollary $11.7$ shows that for $\l'=0, \sum \l_{\a }=0$
these change and multiplication transform the corresponding equivariant
classes $S'$ and $S'\ ^*$ to one another. The class $-p$ in the formula for
$s_i^*$ in Corollary $11.7$ is the equivariant Chern class of the Hopf line
bundle over ${\Bbb C} P^n$. In the limit $\l =0$ it becomes $-P$ while $S'$ and
$S'\ ^*$ transform to their non-equivariant counterparts $S$ and $S^*$.
{\em Remarks.} 1) Notice that the components $S_0^*$ and $S_1^*$ in
\[ S^*=l_1...l_r[P^r S_0^* (t)+ P^{r+1} S_1^*(t) + ... + P^n S_n^*(t)] \]
are exactly $f(e^t)$ and $tf(e^t)+\sum_a l_a [g_{l_a }(e^t) - g_1(e^t)]$
respectively. Thus the inverse transformation from $S^*$ to $S$ consists in
division by $S_0^*$ followed by the change $T= S_1^*(t)/S_0^*(t)$ in complete
accordance with the recipe \cite{COGP, BVS, HG1} based on the mirror
conjecture.
2) According to \cite{B} the $(n-r)$-dimensional manifolds $X_t'$ admit a
Calabi-Yau compactification to the family $\bar{X}_t'$ of
{\em mirror manifolds } of the Calabi-Yau complete intersection
$X^{n-r}\subset {\Bbb C} P^n$. The Picard-Fuchs differential equation from
Corollary $11.8$ describes variations of complex structures for $\bar{X}'$.
This proves the mirror conjecture (described in detail in \cite{BVS})
for projective Calabi-Yau complete intersections and confirms the enumerative
predictions about rational curves and quantum cohomology algebras made there
(and in some other papers) on the basis of the mirror conjecture.
3) The description \cite{AM} of the
quantum cohomology algebra of a Calabi-Yau $3$-fold in terms of the numbers
$n_d$ of rational curves of all degrees $d$ (see for instance \cite{HG1} for
the description of the corresponding class $S$ in these terms) has been
rigorously justified in \cite{M}. Combining these results with Corollary
$11.8$ we arrive to the theorem formulated in the introduction.
\bigskip
{\em Proof of The Main Lemma.}
In our construction of the map $\mu: L_d\to L'_d$ we will denote $L_d$
the moduli space of stable maps $C\to {\Bbb C} P^n\times {\Bbb C} P^1$ of bidegree
$(d,1)$ with no marked points (it also has dimension $d(n+1)+n$).
The construction works for any given number
of marked points but produces a map which is the composition of $\mu $ with
the forgetful map. In this form it applies to the submanifold of stable maps
with two marked points confined over $0$ and $\infty $ in ${\Bbb C} P^1$ (this
submanifold is what we denoted $L_d$ in the formulation of The Main Lemma).
Let $\psi: C\to {\Bbb C} P^n \times {\Bbb C} P^1 $ be a stable genus $0$ map of bidegree
$(d,1)$. Then $C=C_0 \cup C_1 ... \cup C_r$ where $C_0$ is isomorphic to
${\Bbb C} P^1$
and $\psi | C_0$ maps $C_0$ onto the graph of a degree $d'\leq d$ map ${\Bbb C} P^1
\to {\Bbb C} P^n$, and for $i=1,...,r$ the bidegree $(d_i,0)$ map $\psi | C_i$
sends $C_i$ into the slice ${\Bbb C} P^n \times \{ p_i \} $ where $p_i \neq p_j$
and $d_1+...+d_r=d-d'$.
The map $\mu :L_d \to L'_d$ assigns to $[\psi ]$ the $(n+1)$-tuples
$(f_0 g : f_1 g : ... : f_n g)$ of polynomials ($=$ binary forms) on
${\Bbb C} P^1$ where
$g$ is the polynomial of degree $d-d'$ with roots $(p_1,...,p_r)$ of
multiplicities $(d_1,...,d_r)$ and the tuples $(f_0:...:f_n)$ of degree $d'$
polynomials (with no common roots, including $\infty $)
is the one that describes the map $\psi | C_0$.
In order to prove that the map $\mu $ is regular let us give it another, more
invariant description.
Denote $\hat{L}_d$ the moduli space of bidegree $(d,1)$ stable maps with
an extra-marked point and pull back to $\hat{L}_d$ the line bundle
\[ H:= Hom (\pi _1^* {\cal O} _{{\Bbb C} P^n} (1), \pi _2^* {\cal O} _{{\Bbb C} P^1} (d)) \]
by the evaluation map $e: \hat{L}_d \to {\Bbb C} P^n\times {\Bbb C} P^1$ (where $\pi _i$
are projections to the factors). Consider the push-forward sheaf
$H^0:=R^0\pi _* e^* (H)$
of the locally free sheaf $e^* H$ along the forgetful map $\pi :\hat{L}_d\to
L_d$. To a small neighborhood $U\subset L_d$, it assigns the ${\cal O} _U$
-module $H^0(\pi ^{-1}(U), e^* H)$ of sections of $e^* H$.
{\em Claim.} {\em $1$) $H^0$ is a rank $1$ locally free sheaf on $L_d$.
$2$) The fiber at $[\psi ]$ of the corresponding line bundle can be
identified with
\[ H^0(C_0, (\psi | C_0)^*(H) \otimes {\cal O} (-[p_1])^{\otimes d_1} ...
\otimes {\cal O} (-[p_r])^{\otimes d_r}) .\]
$3$) The kernel of the natural map
\[ h: H^0(C, \psi ^*\pi _1^*({\cal O}_{{\Bbb C} P^n}(1))) \to
H^0(C, \psi ^*\pi _2^*({\cal O}_{{\Bbb C} P^1}(d))) =
H^0({\Bbb C} P^1, {\cal O} (d)) \]
defined by a nonzero vector in this fiber consists of the sections
vanishing identically on $C_0$.}
Using this, we pick $n+1 $ independent sections of ${\cal O}_{{\Bbb C} P^n}(1)$
(that is homogeneous coordinates on ${\Bbb C} P^n$), define corresponding
sections of $e^* \pi _1^* {\cal O}_{{\Bbb C} P^n} (1)$ and apply the map $h$.
By this we obtain a degree $1$ map from the total space of the line bundle
$H^0$ to the linear space ${\Bbb C} ^{n+1} \otimes H^0({\Bbb C} P^1, {\cal O} (d))$. Since
the homogeneous coordinates on ${\Bbb C} P^n$ nowhere vanish simultaneously,
we obtain a natural map
\[ L_d \to L'_d = Proj ({\Bbb C} ^{n+1}\otimes H^0({\Bbb C} P^1, {\cal O} (d))) \]
which sends $[\psi ]$ to $(f_0 g:...:f_n g)$ and conclude that $\mu $ is
regular.
The remaining statements of The Main Lemma are proved by looking at
localizations of the equivariant class $p$ at the $S^1\times T^{n+1}$-fixed
points in $L'_d$ and $L_d$ (in this paragraph we use the notation $L_d$ for
the same space as in the formulation of The Main Lemma).
The fixed points in $L'_d$ are represented by
the vector-monomials $(0:...:0:x^{d'}:0:...:0)$ where $p$ localizes to
$\l_i + d'\hbar $. A fixed point in $L_d$ is represented by $\psi $ with
$\psi (C_0) = (0: ... :0:1: 0:...:0)$, $r=2$, $p_0=0$, $p_1=\infty $ and
the maps $\psi | C_k : C_k \to {\Bbb C} P^n$, $k=1,2$ representing $T^{n+1}$-fixed
points respectively in $Y_{2,d'}$ and $Y_{2,d-d'}$ such that their (say)
second marked points are mapped to the point $\psi (C_0)$.
This implies that the class $\mu^*(p)$ localizes to $\l_i+d'\hbar $ at such a
fixed point and thus the pull back of $p$ to the fixed point set
\[ \{ [\psi ]\in Y_{2,d'}\times Y_{2,d-d'} | (e_2\times e_2) ([\psi ])
\in \Delta \subset Y\times Y \]
of the $S^1$-action on $L_d$ coincides with the pull back through the common
marked point of the $T^{n+1}$-equivariant class $p+d'\hbar$ on the diagonal
$\Delta = {\Bbb C} P^n$. Now localizations of $\int _{L_d} e^{p(t-\t )} {\cal E} _d $
to the fixed points of $S^1$-action identify the form of the correlator
$\Phi $ given in The Main Lemma with the definition of $\Phi $ as
the convolution of $S'(t,\hbar )$ and $S'(\t ,-\hbar )$.
\bigskip
In order to justify the {\em claim} we need to compute the space of global
sections of the sheaf $e^* (H)$ over the formal neighborhood of the fiber
$\pi ^{-1} ([\psi ])$ of the forgetful map $\pi : \hat{L}_d \to L_d$.
The fiber itself is isomorphic to the tree-like genus $0$ curve $C$.
Let $(x_j,y_j), j=1,...,N\geq r$ be some local parameters on irreducible
components of $C$ near the singular points such that $\varepsilon_j = x_jy_j$ are local
coordinates on the {\em orbifold} $L_d$ near $[\psi ]$ (one should add
some local coordinates $\varepsilon'$ on the stratum $\varepsilon_1=...\varepsilon_N=0$ of stable maps
$C\to {\Bbb C} P^n$ in order to construct a complete
local coordinate system on $L_d$). Such a description of local coordinates on
$L_d$ follows from the very construction of the moduli spaces of stable maps
to ample manifolds; we refer the reader to \cite{Kn, BM} for details.
A line bundle over the neighborhood of $C\subset \hat{L}_d$ can be specified
by the set
\[ u_j (x_j^{\pm 1}, \varepsilon ),
v_j (y_j^{\pm 1}, \varepsilon ), \ j=1,...,N, \]
of non-vanishing functions describing transition maps between trivializations
of the bundle inside and outside the neighborhoods
(with local coordinates $(x_j,y_j,\varepsilon_1,...,\hat{\varepsilon _j},...,\varepsilon_N,\varepsilon')$)
of the double points.
Let us consider first the following model case. Suppose that $C$
consists of $r+1$ irreducible components $(C_0, C_1,...,C_r)$ such that each
$C_j$ with $j>0$ intersects $C_0$ at some point $p_j$. Let $x_j$ be
the local parameter on $C_0$ near $p_j$, and the line bundle
(of the degree $-d_j\leq 0$ on $C_j$) be specified by $v_j=y_j^{-d_j}$.
In the neighborhood of $p_j$ a section of such a bundle is given
by a function $s(x_j,y_j,\hat{\varepsilon _j})$ satisfying
\[ s=y_j^{-d_j}s_j(y_j^{-1}, \varepsilon) \]
where the function $s_j$ represents the section in the trivialization
over the neighborhood of $C_j-p_j$. Here $\hat{\varepsilon_j}$ means that $\varepsilon_j$
is excluded from the set of coordinates $\varepsilon $ (remember that $\varepsilon_j=x_jy_j$).
This implies that $s_j = \varepsilon _j^{d_j} f_j(y_j^{-1}\varepsilon_j, \varepsilon)$ where $f_j$
is some regular function. Thus this section in the neighborhood of $p\in C_0$
is given by a function $ s(x_j, \varepsilon ) = x_j^{d_j}f_j(x_j,\varepsilon)$ with zero of
order $d_j$ at $x_j=0$, and the restriction of this section to the
neighborhood of $C_j$ is determined by $s$.
In other words, the ${\Bbb C} [[\varepsilon ]]$-module of global sections in the
formal neighborhood of $C$ identifies with the module of global sections
on $C_0$ for the line bundle given by the loops $x_j^{-d_j} u_j$
instead of $u_j$ (this corresponds to the subtraction of the divisor
$\sum d_j [p_j]$.
The more general situation where $v_j$ is the product of $y_j^{-d_j}$ with
an invertible function $w_j(y_j, x_j ,\hat{\varepsilon_j })$
preserves the above conclusion with
$w^{-1}s=x_j^{d_j} f_j(x_j,\varepsilon )$ instead of $s$.
Obviously, the above computation bears dependence on additional parameters.
Now we apply our model computation to the neighborhood of a general tree-like
curve $C$ {\em inductively} by decomposing the tree into simpler ones
starting from
the root component $C_0$. We conclude that the ${\Bbb C} [[\varepsilon]] $-module of
sections of the bundle $e^*(H)$ is identified with the module of sections
of some line bundle
over the product of $C_0$ with the polydisk with coordinates
$(\varepsilon_1,...,\varepsilon_r, ..., \varepsilon_N, \varepsilon')$,
and that this line bundle is $e^*(H)$ for $C_0$
(given by the loops $u_j$ in our current
notations) twisted by the loops $x_j^{-d_j}$
in the punctured neighborhoods of the points $(p_1,...,p_r)$, where
$(d_1,...,d_r)$ are the degrees of the maps $\psi |C_j: C_j \to {\Bbb C} P^n$
(in the notations of the {\em claim} so that $d_1+...+d_r=d-d'$).
This implies that the ${\Bbb C} [[\varepsilon ]]$-module ${\cal H}^0$
of global sections can be identified with the
module of those global sections of the degree $d-d'$ locally free sheaf
$(\psi |C_0)^* (H) \otimes {\Bbb C} [[\varepsilon ]]$ which have zeroes of order
$d_j$ at $p_j$ for $j=1,...,r$. In particular
1) ${\cal H}^0$ is a free ${\Bbb C} [[\varepsilon ]]$-module of rank $1$,
2) ${\cal H}^0 \otimes _{{\Bbb C} [[\varepsilon ]] } ({\Bbb C} [[\varepsilon ]]/(\varepsilon))$ is the
$1$-dimensional space $H^0|_{[\psi]} $ described in the {\em claim}, and
3) non-zero vectors in $H^0|_[\psi]$ represent sections of $\psi ^*(H)$ over
$C$ non-zero on $C_0$ (and thus their product with a non-zero on $C_0$
section of
$\psi ^*\pi _1^*({\cal O} _{{\Bbb C} P^n} (1))$ can not vanish identically on $C_0$.)
Factorization by the discrete group $Aut (\psi )$ preserves $(1-3)$ with
${\Bbb C} [[\varepsilon ]]$ replaced by ${\Bbb C} [[\varepsilon ]]^{Aut (\psi )}$.
\newpage
| 40,652 |
\section*{Introduction}
In this paper we develop the Kleinian construction of
hyperelliptic Abelian functions, which is a natural
generalization of the Weierstrass approach in the elliptic
functions theory to the case of a hyperelliptic curve of genus
$g>1$. Kleinian $ \zeta$ and $ \wp$--functions are defined as
\[
\zeta_{i} ( \boldsymbol{ u}) = \frac{ \partial}{ \partial
u_i} \mathrm{ln} \; \sigma ( \boldsymbol{ u}) , \quad
\wp_{ij} ( \boldsymbol{ u}) =- \frac{ \partial^2}{ \partial
u_i \partial u_j} \mathrm{ln} \; \sigma ( \boldsymbol{ u}) , \qquad
i, j=1, \ldots, g, \]
where the vector $ \boldsymbol{ u}$ belongs to Jacobian
$ \mathrm{Jac} (V) $ of the
hyperelliptic curve
$V= \{ (y, x) \in \mathbb{ C}^2:y^2- \sum_{i=0}^{2g+2}
\lambda_{i}x^i=0 \}$
and the $ \sigma ( \boldsymbol{ u}) $ is the Kleinian $ \sigma$--function.
The systematical study of the $ \sigma$--functions,
which may be related
to the paper of Klein \cite{kl88}, was an alternative to
the developments of
Weierstrass \cite{w54a, w54} (the hyperelliptic generalization of
the Jacobi elliptic functions
$ \mathrm{sn}, \mathrm{cn}, \mathrm{dn}$) and the purely
$ \theta$--functional theory G{\"o}ppel \cite{go47} and Rosenhain
\cite{ro51} for genus $2$, generalized further by Riemann. The
$ \sigma$ approach was contributed by Burkhardt \cite{bur88},
Wiltheiss \cite{wi88}, Bolza \cite{bo95}, Baker \cite{ba98} and
others; the detailed bibliography may be found in \cite{kw15}. We
would like to cite separately H.F. Baker's monographs
\cite{ba97, ba07}, worth special attention.
The paper is organized as follows. We recall the basic
facts about hyperelliptic curves in the
Section~ \ref{prelims}. In the Section~ \ref{sigma-defs} we
construct the explicit expression for the fundamental
$2$--differential of the second kind and derive the solution of
the Jacobi inversion problem in terms of the hyperelliptic
$ \wp$--functions. We
give in the Section~3 the proof and the analysis of basic
relations for $ \wp$--functions. It is given an explicit
description of the $ \mathrm{Jac} (V) $ in $ \mathbb{
C}^{g+ \frac{g (g+1) }{2}}$ as the intersection of cubics. We
introduce coordinates $h_{ij}$ (see below \eqref{variables}) , in terms
of which these cubics are the determinants of
$3 \times3$--matrices, inheriting in such a way the structure of
Weierstrass elliptic cubic. The Kummer variety
$ \mathrm{Kum} (V) = \mathrm{Jac} (V) / \pm$ appears to be the
intersection of quartics in
$ \mathbb{ C}^{ \frac{g (g+1) }{2}}$ and is described in a whole by
the condition $ \mathrm{rank} \left ( \{h_{ij}
\}_{i, j=1, \ldots, g+2} \right) <4$.
The Section~4 describes some natural applications of the Kleinian
functions theory.
The paper is based on the recent results partially announced
in \cite{le95, bel96, el95, be96b}. The given
results are already used to describe a $2$--dimensional
Schr{\"o}dinger equation \cite{be95}.
\section{Preliminaries} \label{prelims}
We recall some basic definition from the theory of the
hyperelliptic curves and $ \theta$--functions; see e.g.
\cite{ba97, ba07, fa73, mu75, gh78, fk80} for the detailed exposition.
\subsection{Hyperelliptic curves}
The set of points $V (y, x) $ satisfying the
\begin{eqnarray}
y^2= \sum_{i=0}^{2g+2} \lambda_{i}x^{i}
= \lambda_{2g+2} \prod_{k=1}^{2g+2} (x-e_{k}) =f (x)
\label{curve}
\end{eqnarray}
is a model of a plane { \em hyperelliptic curve} of genus $g$,
realized as a $2$--sheeted
covering over Riemann sphere with the { \em branching points}
$e_1, \ldots, e_{2g+2}$. Any pair $ (y, x) $ in $V (y, x) $
is called an { \em analytic point}; an analytic point, which is not
a branching point is called a { \em regular point}.
The { \em hyperelliptic involution} $ \phi ( \;) $ (the swap of the
sheets of covering) acts as $ (y, x) \mapsto (-y, x) $, leaving the
branching points fixed.
To make $y$ the singlevalued function of $x$ it suffices to
draw $g+1$ cuts, connecting pairs of branching points
$e_i$---$e_{i'}$ for some partition of $ \{1, \ldots, 2g+2 \}$ into
the set of $g+1$ disjoint pairs ${i, i'}$. Those of $e_j$, at
which the cuts start we will denote ${a_i}$, ending points of the
cuts we will denote $b_i$, respectively; except for one of the
cuts which is denoted by starting point $a$ and ending point $b$.
In the case $ \lambda_{2g+2} \mapsto 0$ this point $a \mapsto \infty$. The
equation of the curve, in case $ \lambda_{2g+2}=0$ and
$ \lambda_{2g+1}=4$ can be rewritten as \begin{eqnarray} &y^2=4
P (x) Q (x) , \label{alt-curve} \\ &P (x) = \prod_{i=1}^{g} (x-a_i) ,
\quad Q (x) = (x-b) \prod_{i=1}^{g} (x-b_i) . \nonumber
\end{eqnarray}
The local parametrisation of the point $ (y, x) $ in the vicinity of
a point $ (w, z) $: \[ x=z+ \left \{ \begin{array}{ll} \xi, \quad
& \text{near regular point} \, ( \pm w, z) ; \\ \xi^2, & \text{near
branching point} \, (0, e_i) ; \\ \frac{1}{ \xi}, & \text{near regular
point} \, ( \pm \infty, \infty) ; \\ \frac{1}{ \xi^2}, & \text{near
branching point} \, ( \infty, \infty) \end{array} \right. \]
provides the structure of the { \em hyperelliptic Riemann surface}
--- a one-dimensional compact complex manifold. We will employ the
same notation for the plane curve and the Riemann surface ---
$V (y, x) $ or $V$. All curves and Riemann surfaces through the paper
are assumed to be hyperelliptic, if the converse not stated.
A { \em marking} on $V (y, x) $ is given by the base
point $x_0$ and the canonical basis of cycles
$ (A_1, \ldots, A_g;B_1, \ldots, B_g) $ --- the basis in the group of
one-dimensional homologies $H_1 (V (y, x) , \mathbb{ Z}) $ on the
surface $V (y, x) $ with the symplectic intersection matrix
$I= \left ( \begin{array}{cc}0&- \mathbf{ 1}_g \\ \mathbf{
1}_g&0 \end{array} \right) $, where $ \mathbf{ 1}_g$ is the unit
$g \times g$--matrix.
\subsection{Differentials}
Traditionally three kinds of differential $1$--forms are
distingui \-s \-h \-ed on a Riemann surface.
\subsubsection{Holomorphic differentials}
or the differentials of the first kind,
are the differential $1$--forms
$ \mathrm{d}u$, which can be locally given as
$ \mathrm{d}u= ( \sum_{i=0}^ \infty \alpha_i \xi^i) d \xi$ in the vicinity
of any point $ (y, x) $ with some constants $ \alpha_i \in \mathbb{ C}$.
It can be checked directly, that forms satisfying such a condition
are all of the form $ \sum_{i=0}^{g-1} \beta_i
x^i \frac{ \mathrm{d}x}{y}$. Forms $ \{ \mathrm{d}u_i \}_{i=1}^g$,
\[ \mathrm{d}u_i= \frac{x^{i-1} \mathrm{d}x}{ y}, \quad
i \in 1, \ldots, g \] are the set of { \em canonical
holomorphic differentials} in $H^1 (V, \mathbb{ C}) $. The
$g \times g$--matrices of their $A$ and $B$--periods,
\[2 \omega= \left ( \oint_{A_k} \mathrm{d}u_l \right) , \quad
2 \omega'= \left ( \oint_{B_k} \mathrm{d}u_l \right) \]
are nondegenerate. Under the action of the transformation
$ (2 \omega) ^{-1}$ the vector
$ \mathrm{d} \mathbf{ u}= ( \mathrm{d}u_1, \ldots, \mathrm{d}u_g) ^T$ maps
to the vector of normalized holomorphic differentials
$ \mathrm{d} \mathbf{ v}= ( \mathrm{d}v_1, \ldots, \mathrm{d}v_g) ^T$
--- the vector in
$ H^1 (V, \mathbb{ C}) $ to satisfy the conditions
$ \oint_{A_k} \mathrm{d}v_k= \delta_{kl}, k, l=1 \ldots, g$. It is known,
that $g \times g$ matrix,
\[ \tau= \left ( \oint_{B_k} \mathrm{d}v_l \right) = \omega^{-1} \omega' \]
belongs to the { \em upper Siegel halfspace} $ \mathcal{ S}_g$ of degree
$g$, i.e. it is symmetric and has a positively defined imaginary
part.
Let us denote by $ \mathrm{Jac} (V) $ the { \em Jacobian} of the curve
$V$, i.e. the factor $ \mathbb{ C}^g/ \Gamma$, where
$ \Gamma=2 \omega \oplus 2 \omega'$ is the lattice generated by the
periods of canonical holomorphic differentials.
{ \em Divisor} $ \mathcal{D}$ is a formal sum of subvarieties of
codimension $1$ with coefficients from $ \mathbb{ Z}$. Divisors on
Riemann surfaces are given by formal sums of analytic points
$ \mathcal{ D}= \sum_i^n m_i (y_i, x_i) $, and $ \mathrm{deg} \mathcal{
D}= \sum_i^n m_i$. The { \em effective divisor} is such that $m_i>0
\forall i$.
Let $ \mathcal{D}$ be a divisor of degree $0$, $ \mathcal{
D}= \mathcal{ X}- \mathcal{ Z}$, with $ \mathcal{ X}$ and
$ \mathcal{ Z}$ --- the
effective divisors $ \mathrm{ deg } \; \mathcal{ X}= \mathrm{ deg } \;
\mathcal{Z}=n$ presented by $ \mathcal{ X}= \{ (y_1, x_1) ,
\ldots, (y_n, x_n) \}$ and $ \mathcal{ Z}= \{ (w_1, z_1) ,
\ldots, (w_n, z_n) \} \in (V) ^{n}$, where $ (V) ^n$ is the
$n$--th symmetric power of $V$.
The { \em Abel map} $ \mathfrak{ A}: (V) ^n \rightarrow \mathrm{Jac}
(V) $ puts into correspondence the divisor $ \mathcal{ D}$, with
fixed $ \mathcal{ Z}$, and the point $ \boldsymbol{ u}= (u_1 \ldots,
u_g) ^T \in \mathrm{Jac} (V) $ according to the \[ \boldsymbol{
u}= \int_{ \mathcal{ Z}}^ \mathcal{ X} \mathrm{d} \mathbf{ u},
\quad \text{or} \quad u_i= \sum_{k=1}^n \int_{z_k}^{x_k} \mathrm{d}u_i,
\quad i=1, \ldots, g. \]
The { \em Abel's theorem} says that the points of the
divisors $ \mathcal{ Z}$ and $ \mathcal{ X}$ are respectively the poles and
zeros of a meromorphic function on $V (y, x) $ iff
$ \int^ \mathcal{ X}_ \mathcal{ Z} \mathrm{d} \mathbf{ u}=0 \mod
\Gamma$.
The { \em Jacobi inversion problem} is
formulated as the problem of inversion of the map $ \mathfrak{ A}$,
when $n=g$ the $ \mathfrak{ A}$ is $1 \to1$, except for so called
{ \em special divisors}. In our case special divisors of degree
$g$ are such that at least for one pair $j$ and $k \in 1 \ldots
g$ the point $ (y_j, x_j) $ is the image of the hyperelliptic
involution of the point $ (y_k, x_k) $.
\subsubsection{Meromorphic differentials} or the differentials of the
second kind, are the differential $1$-forms $ \mathrm{d}r$ which can
be locally given as
$ \mathrm{d}r= ( \sum_{i=-k}^ \infty \alpha_i \xi^i) d \xi$ in the
vicinity
of any point $ (y, x) $ with some constants $ \alpha_i$, and
$ \alpha_{ (-1) }=0$. It can be also checked directly, that forms
satisfying such a condition are all of the form $
\sum_{i=0}^{g-1} \beta_i x^{i+g} \frac{ \mathrm{d}x}{y}$ ($ \mod$
holomorphic differential) .
Let us introduce the following { \em canonical Abelian
differentials of the second kind} \begin{equation}
\mathrm{d}r_j= \sum_{k=j}^{2g+1-j} (k+1-j) \lambda_{k+1+j}\frac{x^k
\mathrm{d}x}{
4y}, \quad j=1, \ldots, g. \end{equation} We denote their
matrices of $A$ and $B$--periods,
\[2 \eta= \left (- \oint_{A_k} \mathrm{d}r_l \right) , \quad 2 \eta'
= \left (- \oint_{B_k} \mathrm{d}r_l \right) . \]
From { \em Riemann bilinear
identity}, for the period matrices of the differentials of the
first and second kind follows: \begin{lemma} $2g \times
2g$--matrix $ \mathcal{ G}= \left ( \begin{array} {cc} \omega&
\omega' \\
\eta& \eta' \end{array} \right) $ belongs to $PSp_{2g}$:
\[ \mathcal{ G} \left ( \begin{array}{cc}0&- \mathbf{
1}_g \\ \mathbf{ 1}_g&0 \end{array} \right) \mathcal{ G}^T=-
\frac{ \pi
i}{2} \left ( \begin{array}{cc} 0&- \mathbf{ 1}_g \\ \mathbf{
1}_g&0 \end{array} \right) . \] \label{LPSR}
\end{lemma}
\subsubsection{Differentials of the third kind}
are the differential 1-forms $ \mathrm{d} \Omega$ to have only
poles of order $1$ and $0$ total residue, and so are locally
given in the vicinity of any of the poles as
$ \mathrm{d} \Omega= ( \sum_{i=-1}^ \infty \alpha_i \xi^i) d \xi$
with some constants $ \alpha_i$, $ \alpha_{-1}$ being nonzero. Such
forms ($ \mod$ holomorphic differential) may be presented as:
\[
\sum_{i=0}^{n} \beta_i
\left ( \frac{y+y^+_i}{x-x^+_i}- \frac{y+y^-_i}{x-x^-_i} \right)
\frac{ \mathrm{d}x}{y},
\]
where $ (y^ \pm_i, x^ \pm_i) $ are the analytic points of the poles of
positive (respectively, negative) residue.
Let us introduce the canonical differential of the third kind
\begin{equation}
\mathrm{d} \Omega (x_1, x_2) =
\left ( \frac{y+y_1}{x-x_1}- \frac{y+y_2}{x-x_2} \right)
\frac{ \mathrm{d} x}{2y}, \label{third}
\end{equation}
for this differential we have $ \int_{x_3}^{x_4} \mathrm{d}
\Omega (x_1, x_2) = \int_{x_1}^{x_2} \mathrm{d} \Omega (x_3, x_4) $.
\subsubsection{Fundamental $2$--differential of the second kind}
For
$ \{ (y_1, x_1) , (y_2, x_2) \} \in (V) ^2$
we introduce function $F (x_1, x_2) $ defined by the conditions
\begin{eqnarray}
( \text{i}) . &&F (x_1, x_2) =F (x_2, x_1) , \nonumber \\
( \text{ii}) . &&F (x_1, x_1) =2f (x_1) , \nonumber \\
( \text{iii}) . && \frac{ \partial
F (x_1, x_2) }{ \partial x_2} \big|_{x_2=x_1}
= \frac{ \mathrm{d}f (x_1) }{ \mathrm{d}x_1}. \label{propf}
\end{eqnarray}
Such $F (x_1, x_2) $ can be presented in the following
equivalent forms
\begin{eqnarray}
F (x_1, x_2) &=&2y_2^2+2 (x_1-x_2) y_2 \frac{ \mathrm{d} y_2}{
\mathrm{d}x_2} \nonumber \\
&+& (x_1-x_2) ^2 \sum_{j=1}^gx_1^{j-1} \sum_{k=j}^{2g+1-j} (k-j+1)
\lambda_{k+j+1}x_2^k,
\label{Formf-1} \\
F (x_1, x_2) &=&2 \lambda_{2g+2}x_1^{g+1}x_2^{g+1}+
\sum_{i=0}^{g}x_1^ix_2^i (2 \lambda_{2i}+ \lambda_{2i+1} (x_1+x_2) ) .
\label{Formf-2} \end{eqnarray}
Properties \eqref{propf} of $F (x_1, x_2) $ permit to construct
the { \em global Abelian $2$--differential of the second
kind} with the unique pole of order $2$ along $x_1=x_2$ :
\begin{equation}
\omega (x_1, x_2) = \frac{2y_1 y_2+F (x_1, x_2) }{
4 (x_1-x_2) ^2} \frac{ \mathrm{d}x_1}{ y_1} \frac{ \mathrm{d}x_2}{y_2},
\label{omega-1} \end{equation} which expands in the vicinity of
the pole as \[
\omega (x_1, x_2) = \left ( \frac{1}{2 ( \xi- \zeta) ^2}+O (1)
\right) \mathrm{d} \xi
\mathrm{d} \zeta,
\]
where $ \xi$ and $ \zeta$ are the local
coordinates at the points $x_1$ and $x_2$ correspondingly.
Using the \eqref{Formf-1}, rewrite the \eqref{omega-1} in the form
\begin{equation}
\omega (x_1, x_2) = \frac{ \partial}{ \partial x_2}
\left ( \frac{y_1+y_2}{
2y_1 (x_1-x_2) } \right) \mathrm{d}x_1 \mathrm{d}x_2
+ \mathrm{d} \mathbf{
u}^T (x_1) \mathrm{d} \mathbf{ r} (x_2) , \label{omega-2} \end{equation}
where the differentials $ \mathrm{d} \mathbf{ u}, \mathrm{d} \mathbf{ r}$
are as above. So, the periods of this $2$-form (the double
integrals $ \oint \oint \omega (x_1, x_2) $) are expressible in terms of
$ (2 \omega, 2 \omega') $ and $ (-2 \eta, -2 \eta') $, e.g., we
have for $A$-periods:
\[
\left \{ \oint_{A_i} \oint_{A_k} \omega (x_1, x_2) \right \}_{i, k=1, \ldots, g}=
-4 \omega^T \eta.
\]
\subsection{Riemann $ \theta$-function}
The standard $ \theta$--function $ \theta ( \boldsymbol{ v}| \tau) $ on $ \mathbb{
C}^g \times \mathcal{ S}_g$ is defined by its Fourier series,
\[
\theta ( \boldsymbol{ v}| \tau)
= \sum_{ \boldsymbol{ m} \in \mathbb{ Z}^g} \mathrm{
exp} \; \pi i \left \{ \boldsymbol{ m}^T \tau \boldsymbol{ m}
+2 \boldsymbol{ v}^T
\boldsymbol{ m} \right \} \]
The $ \theta$--function
possesses the periodicity properties $ \forall k \in 1, \ldots, g$
\begin{eqnarray*} && \theta (v_1, \ldots, v_k+1, \ldots, v_g| \tau)
= \theta ( \boldsymbol{ v}| \tau) , \\
&& \theta (v_1+ \tau_{1k}, \ldots, v_k+ \tau_{kk}, \ldots, v_g
+ \tau_{gk}| \tau)
= \mathrm{ e}^{i \pi \tau_{kk}-2 \pi i v_k} \theta ( \boldsymbol{ v}| \tau) .
\end{eqnarray*}
$ \theta$--functions with characteristics $[ \varepsilon]=
\left[ \begin{array}{c} \varepsilon' \\
\varepsilon \end{array} \right]
= \left[ \begin{array}{ccc} \varepsilon_1'&
\ldots& \varepsilon_g'
\\ \varepsilon_1& \ldots& \varepsilon_g
\end{array} \right] \in \mathbb{
C}^{2g}$
\[
\theta[ \varepsilon] ( \boldsymbol{ v}| \tau) = \sum_{ \boldsymbol{ m} \in \mathbb{
Z}^g} \mathrm{ exp} \; \pi i \left \{ ( \boldsymbol{ m}+ \varepsilon') ^T \tau
( \boldsymbol{ m}+ \varepsilon') +2 ( \boldsymbol{ v}+ \varepsilon) ^T
( \boldsymbol{ m}+ \varepsilon') \right \},
\] for which
the periodicity properties are
\begin{eqnarray*}
&& \theta[ \varepsilon] (v_1, \ldots, v_k+1, \ldots, v_g| \tau)
= \mathrm{ e}^{2 \pi i \varepsilon_k'} \theta ( \boldsymbol{
v}| \tau) , \\
&& \theta[ \varepsilon] (v_1+ \tau_{1k}, \ldots, v_k+ \tau_{kk},\ldots, v_g
+ \tau_{gk}| \tau)
= \mathrm{ e}^{i \pi \tau_{kk}-2 \pi i v_k-2 \pi i
\varepsilon_k} \theta ( \boldsymbol{ v}| \tau) .
\end{eqnarray*}
Further, consider half-integer characteristics $[ \varepsilon]$;
the $ \theta$--function $ \theta[ \varepsilon] ( \boldsymbol{ v}|\tau) $
is even or odd whenever $4{ \varepsilon'}^T \varepsilon=0$ or 1
modulo 2. There are $ \frac{1}{2} (4^g+2^g) $ even characteristics
and $ \frac{1}{2} (4^g-2^g) $ odd.
Let $ \boldsymbol{ w}^T= (w_1, \ldots, w_g) \in \mathrm{Jac} (V) $ be
some fixed vector, the function,
\[ \mathcal{
R} (x) = \theta \left ( \int_{x_0}^x \mathrm{d} \mathbf{ v} - \boldsymbol{
w}| \tau \right) , \quad x \in V
\] is called { \em Riemann $ \theta$--function}.
The Riemann $ \theta$--function $ \mathcal{ R} (x) $ is either identically
$0$, or it has exactly $g$ zeros $x_1, \ldots, x_g \in V$, for which
the { \em Riemann vanishing theorem} says that
\[
\sum_{k=1}^g
\int_{x_0}^{x_i} \mathrm{d} \mathbf{ v}= \boldsymbol{ w}+ \mathbf{
K}_{x_0}, \] where $ \mathbf{ K}^T_{x_0}= (K_1, \ldots, K_g) $ is the
vector of Riemann constants with respect to the base point
$x_0$ and is defined by the formula
\begin{equation}
K_j= \frac{1+ \tau_{jj}}{ 2}-
\sum_{l \neq j}
\oint_{A_l} \mathrm{d}v_l (x) \int_{x_0}^x \mathrm{d}v_j,
\quad j=1, \ldots, g.
\label{Rconst} \end{equation}
\section{Kleinian functions} \label{sigma-defs}
Let $ \boldsymbol{ m}, \boldsymbol{ m}' \in \mathbb{ Z}^g$ be two
arbitrary vectors; denote periods $ \mathbf{ E} ( \boldsymbol{ m},
\boldsymbol{
m}') =2 \eta \boldsymbol{ m}+2 \eta' \boldsymbol{ m}'$, $ \boldsymbol{ \Omega} (
\boldsymbol{
m}, \boldsymbol{ m}') =2 \omega \boldsymbol{ m}+2 \omega' \boldsymbol{ m}'$.
\subsection{$ \sigma$--function}
In \cite{kl88, ba98} it was shown, that the properties
\eqref{period} and \eqref{0-expan}
define the function, which plays the central role in the
theory of Kleinian functions.
\begin{definition} An integral function
$ \sigma ( \boldsymbol{ u}) $ is the Kleinian { \em fundamental
$ \sigma$--function} iff
\begin{enumerate} \item for any vector
$ \boldsymbol{ u} \in \mathrm{Jac} (V) $ \begin{equation} \sigma (
\boldsymbol{
u}+ \boldsymbol{ \Omega} ( \boldsymbol{ m}, \boldsymbol{ m}') )
= \mathrm{
exp} \left \{ \mathbf{ E}^T ( \boldsymbol{ m}, \boldsymbol{ m}') ( \boldsymbol{
u}+ \tfrac{1}{2} \boldsymbol{ \Omega} ( \boldsymbol{
m}, \boldsymbol{ m}') ) + \pi i \boldsymbol{ m}^T \boldsymbol{
m}' \right \} \sigma ( \boldsymbol{ u}) . \label{period}
\end{equation}
\item $ \sigma ( \boldsymbol{ u}) $ has $0$ of
$ \left[ \frac{g+1}{2} \right]$ order at $ \boldsymbol{ u}=0$ and
\begin{equation} \lim_{ \boldsymbol{ u} \to 0} \frac{ \sigma ( \boldsymbol{
u}) }{ \delta ( \boldsymbol{ u}) }=1, \label{0-expan} \end{equation} where
$ \delta ( \boldsymbol{ u}) =
\det \left (- \{u_{i+j-1} \}_{i, j=1, \ldots, \left[ \frac{g+1}{2} \right]} \right) $.
\end{enumerate}
\end{definition}
For small genera we have,
$ \sigma=u_1+ \ldots$ for $g=1$ and $2$;
$ \sigma=u_1u_3-u_2^2+ \ldots$ for $g=3$ and $4$;
$ \sigma=-u_3^3+2u_2u_3u_4-u_1u_4^2-u_2^2u_5+u_1u_3u_5+ \ldots$ for $g=5$
and $6$ etc.
We introduce the $ \sigma$-{ \em functions with characteristic},
$ \sigma_{ \boldsymbol{ r}, \boldsymbol{ r}'}$ for vectors
$ \boldsymbol{ r}^T, \boldsymbol{ r}' \in \frac12 \mathbb{ Z}^g/ \mathbb{ Z}^g$ defined
by the formula
\[
\sigma_{ \boldsymbol{ r}, \boldsymbol{ r}'} ( \boldsymbol{ u}) = \mathrm{
e}^{- \mathbf{E}^T ( \boldsymbol{ r}, \boldsymbol{ r}') \boldsymbol{
u}} \frac{ \sigma ( \boldsymbol{ u}+ \boldsymbol{ \Omega} ( \boldsymbol{
r}, \boldsymbol{ r}') ) } { \sigma ( \boldsymbol{ \Omega} ( \boldsymbol{
r}, \boldsymbol{ r}') ) }. \] These functions are completely
analogous to the Weierstrass' $ \sigma_ \alpha$ appearing in the
elliptic theory \cite{ba55}.
\subsubsection{ $ \sigma$--function as
$ \theta$--function}
Fundamental hyperelliptic Kleinian $ \sigma$--function belongs to
the class of generalized $ \theta$--functions. We give the
explicit expression of the $ \sigma $ in terms of standard
$ \theta$--function as follows:
\begin{equation} \sigma ( \boldsymbol{
u}) =C \mathrm{ e}^{ \boldsymbol{ u}^T \varkappa \boldsymbol{
u}}
\theta ( (2 \omega) ^{-1}
\boldsymbol{u}-
\mathbf{K}_{a}|\tau) ,
\label{sigma}
\end{equation}
where $ \varkappa = (2 \omega) ^{-1} \eta$,
$ \mathbf{ K}_{a}$ is the vector of Riemann constants with the base
point $a$ and the constant \[
C= \frac{ \epsilon_4}{ \theta (0| \tau) }
\prod_{r=1}^g \frac{ \sqrt{{P}' (a_r) }}{ \sqrt[4]{f' (a_r) }}
\frac{1}{ \prod_{k<l} \sqrt{e_k-e_l}}, \]
where $ ( \epsilon_4) ^4=1$.
Direct calculation shows, that the function defined by
\eqref{sigma} satisfies \eqref{period} and
\eqref{0-expan}, we only note that,
in our case the vector of Riemann constants
\eqref{Rconst} is, as follows from Riemann vanishing theorem,
\begin{equation}
\mathbf{ K}_{a}= \sum_{k=1}^g \int_{a}^{a_i}
\mathrm{d} \mathbf{ v}. \label{rvector}
\end{equation}
Putting $g=1$ and fixing the elliptic curve $y^2=f (x) =4x^3-g_2
x-g_3$ in \eqref{sigma}, we see that the function
\[ \sigma (u) = \frac{1}{ \vartheta_3 (0| \tau)
\sqrt[4]{ (e_1-e_2) (e_2-e_3) }} \mathrm{ e}^ \frac{ \eta u^2}{
2 \omega} \vartheta_1 \left ( \frac{u}{ 2 \omega} \big| \tau \right) \]
is the standard Weierstrass $ \sigma$--function, were we have used
the standard notation for Jacobi $ \vartheta$--functions (see e.g.
\cite{ba55} ) .
\subsection{Functions $ \zeta$ and $ \wp$}
Kleinian $ \zeta$ and $ \wp$-functions are defined as logarithmic
derivatives of the fundamental $ \sigma$
\begin{eqnarray*}&& \zeta_i ( \boldsymbol{ u}) = \frac{ \partial
\mathrm{
ln} \; \sigma ( \boldsymbol{ u}) }{ \partial u_i}, \quad
i \in 1, \ldots, g; \\ && \wp_{ij} ( \boldsymbol{ u}) =- \frac{ \partial^2
\mathrm{ ln} \; \sigma ( \boldsymbol{ u}) }{ \partial u_i \partial
u_j}, \; \wp_{ijk} ( \boldsymbol{ u}) =- \frac{ \partial^3 \mathrm{
ln} \; \sigma ( \boldsymbol{ u}) }{ \partial u_i \partial
u_i \partial u_k} \ldots, i, j, k, \ldots \in 1, \ldots, g.
\end{eqnarray*}
The functions $ \zeta_i ( \boldsymbol{ u}) $ and $ \wp_{ij} ( \boldsymbol{ u}) $ have the
following periodicity properties
\begin{eqnarray*} \zeta_i ( \boldsymbol{
u}+ \boldsymbol{ \Omega} ( \boldsymbol{ m}, \boldsymbol{ m}') ) &=
& \zeta_i ( \boldsymbol{ u})
+E_i ( \boldsymbol{ m}, \boldsymbol{ m}') , \quad i \in 1, \ldots, g,
\\ \wp_{ij} ( \boldsymbol{ u}+
\boldsymbol{ \Omega} ( \boldsymbol{ m}, \boldsymbol{ m}') )
&=& \wp_{ij} ( \boldsymbol{ u}) , \quad
i, j \in 1, \ldots, g. \end{eqnarray*}
\subsubsection{Realization of the fundamental $2$--differential of
the second kind by Kleinian functions}
The construction is based on the following
\begin{theorem}
\label{the-S} Let $ (y (a_0), a_0) $, $ (y, x) $ and $ ( \nu,
\mu) $ be arbitrary distinct points on $V$ and let $ \{ (y_1,
x_1), \ldots, (y_g, x_g) \}$ and $ \{ ( \nu_1, \mu_1) ,
\ldots, ( \nu_g, \mu_g) \}$ be arbitrary sets of distinct
points $ \in (V) ^g$. Then the following relation is valid
\begin{eqnarray} \lefteqn{ \int_{ \mu}^x \sum_{i=1}^g \int_{
\mu_i}^{x_i} \frac{2yy_i+F (x, x_i) }{4 (x-x_i) ^2} \frac{
\mathrm{d}x}{y} \frac{ \mathrm{d}x_i}{y_i}} \nonumber \\ &&=
\mathrm{ln } \; \left \{ \frac{ \sigma \left (
\int_{a_0}^x \mathrm{d} \mathbf{ u} - \sum_{i=1}^g
\int_{a_i}^{x_i} \mathrm{d} \mathbf{ u} \right) }{ \sigma
\left ( \int_{a_0}^x \mathrm{d} \mathbf{ u}- \sum_{i=1}^g
\int_{a_i}^{ \mu_i} \mathrm{d} \mathbf{ u} \right) } \right
\}- \mathrm{ ln } \; \left \{ \frac{ \sigma \left (
\int_{a_0}^{ \mu} \mathrm{d} \mathbf{ u}- \sum_{i=1}^g
\int_{a_i}^{x_{i}} \mathrm{d} \mathbf{ u} \right) }{ \sigma
\left ( \int_{a_0}^{ \mu} \mathrm{d} \mathbf{
u}- \sum_{i=1}^g \int_{a_i}^{ \mu_i} \mathrm{d} \mathbf{
u} \right) } \right \} , \label{r11} \end{eqnarray} where the function
$F (x, z) $ is given by \eqref{Formf-2}. \end{theorem}
\begin{proof} Let us consider the sum
\begin{equation}
\sum_{i=1}^g \int_{ \mu}^x \int_{ \mu_i}^{x_i} \left[
\omega (x, x_i) +
\mathrm{d} \mathbf{ u}^T (x)
\varkappa
\mathrm{d} \mathbf{ u} (x_i) \right],
\label{r10}
\end{equation}
with $ \omega ( \cdot, \cdot) $ given by \eqref{omega-2}.
It is the normalized Abelian
integral of the third kind with the logarithmic residues in the
points $x_i$ and $ \mu_i$. By Riemann vanishing theorem we can
express \eqref{r10} in terms of Riemann $ \theta$--functions as
\begin{equation}
\mathrm{ln}
\left \{
\tfrac{{ \displaystyle{ \theta}} \left ( \int_{a_0}^x \mathrm{d} \mathbf{
v}- ( \sum_{i=1}^g \int_{a_0}^{x_i} \mathrm{d} \mathbf{ v}- \mathbf{
K}_{a_0}) \right) }
{{ \displaystyle{ \theta}} \left ( \int_{a_0}^{x} \mathrm{d} \mathbf{
v}- ( \sum_{i=1}^g \int_{a_0}^{ \mu_i} \mathrm{d} \mathbf{ v}- \mathbf{
K}_{a_0}) \right) }
\right \}
- \mathrm{ln}
\left \{
\tfrac{{ \displaystyle{ \theta}}
\left ( \int_{a_0}^{ \mu} \mathrm{d} \mathbf{
v}- ( \sum_{i=1}^g \int_{a_0}^{x_i} \mathrm{d} \mathbf{ v}- \mathbf{
K}_{a_0}) \right) }{{ \displaystyle{ \theta}}
\left ( \int_{a_0}^{ \mu} \mathrm{d} \mathbf{
v}- ( \sum_{i=1}^g \int_{a_0}^{ \mu_i} \mathrm{d} \mathbf{ v}- \mathbf{
K}_{a_0}) \right) } \right \}, \label{in-depends}
\end{equation}
and to obtain right hand side of \eqref{r11} we have to combine the
\eqref{sigma}, expression of the vector $ \mathbf{ K}_{a_0}$
\eqref{rvector}, matrix $ \varkappa = (2 \omega) ^{-1} \eta$
and Lemma \ref{LPSR}. Left hand side of
\eqref{r11} is obtained using \eqref{omega-1}. \end{proof}
The fact, that right hand side of the
\eqref{r11} is independent on the arbitrary point $a_0$, to be
employed further, has its origin in the properties of the vector
of Riemann constants. Consider the difference $ \mathbf{ K}_{a_0}-
\mathbf{ K}_{a_0'}$ of vectors of Riemann constants with arbitrary
base points $a_0$ and $a_0'$ by \eqref{Rconst} we find
\[
\mathbf{ K}_{a_0}-
\mathbf{ K}_{a_0'} = (g-1) \int_{a_0'}^{a_0} \mathrm{d} \mathbf{ v},
\]
this property provides that
\[
\int_{a_0}^{x_0} \mathrm{d} \mathbf{
v}- ( \sum_{i=1}^g \int_{a_0}^{x_i} \mathrm{d} \mathbf{ v}- \mathbf{
K}_{a_0}) =
\int_{a_0'}^{x_0} \mathrm{d} \mathbf{
v}- ( \sum_{i=1}^g \int_{a_0'}^{x_i} \mathrm{d} \mathbf{ v}- \mathbf{
K}_{a_0'})
\]
for arbitrary $x_i, $ with $i \in 0, \ldots, g$ on $V$, so the
arguments of $ \sigma$'s in \eqref{r11} which are linear
transformations by $2 \omega$ of the arguments of $ \theta$'s in
\eqref{in-depends}, do not depend on $a_0$.
\begin{cor} \label{cor-P} From Theorem \ref{the-S} for arbitrary
distinct $ (y (a_0) , a_0) $ and $ (y, x) $ on $V$ and arbitrary set of
distinct points $ \{ (y_1, x_1) \ldots, (y_g, x_g) \} \in (V) ^g$ follows:
\begin{equation} \sum_{i, j=1}^g \wp_{ij} \left ( \int_{a_0}^x
\mathrm{d} \mathbf{ u}+ \sum_{k=1}^g \int_{a_k}^{x_k} \mathrm{d} \mathbf{
u} \right) x^{i-1}x_r^{j-1}= \frac{F (x, x_r) -2yy_r}{ 4 (x-x_r) ^2}, \;
r=1, \ldots, g. \label{principal}
\end{equation}
\end{cor}
\begin{proof}
Taking the partial derivative $ \partial^2/ \partial x_r \partial x$
from the both sides of \eqref{r11} and using the hyperelliptic
involution $ \phi (y, x) = (-y, x) $ and
$ \phi (y (a_0) , a_0) = (-y (a_0) , a_0) ) $ we obtain \eqref{principal}. \end{proof}
In the case $g=1$ the formula \eqref{principal} is
actually the addition theorem for the Weierstrass elliptic
functions,
\[
\wp (u+v) =- \wp (u) - \wp (v) + \frac{1}{4} \left[ \frac{ \wp' (u) - \wp' (v) }{
\wp (u) - \wp (v) } \right]^2
\]
on the elliptic curve $y^2=f (x) =4x^3-g_2
x-g_3$.
Now we can give the expression for $ \omega (x, x_r) $ in terms of
Kleinian functions. We send the base point $a_0$ to the branch
place $a$, and for $r \in 1, \ldots, g $ the fundamental
$2$--differential of the second kind is given by \[ \omega (x, x_r) =
\sum_{i, j=1}^g \wp_{ij} \left ( \int_a^x \mathrm{d} \mathbf{ u}-
\sum_{k=1}^g \int_{a_k}^{x_k} \mathrm{d} \mathbf{
u} \right) \frac{x^{i-1} \mathrm{d}x}{y} \frac{x_r^{j-1}
\mathrm{d}x_r}{y_r}. \]
\begin{cor}
$ \forall r \neq
s \in 1, \ldots, g$
\begin{eqnarray}
\sum_{i, j=1}^g \wp_{ij}
\left ( \sum_{k=1}^g \int_{a_k}^{x_k} \mathrm{d} \mathbf{ u} \right)
x_s^{i-1}x_r^{j-1} = \frac{F (x_s, x_r) -2y_sy_r}{ 4 (x_s-x_r) ^2}
. \label{principal11}
\end{eqnarray}
\end{cor}
\begin{proof}
In \eqref{principal} we have for $s \neq r$
\begin{eqnarray*}
& \int \limits_{ \phi (a_0) }^x
\mathrm{d} \mathbf{ u}+ \sum \limits_{k=1}^g
\int \limits_{a_k}^{x_k} \mathrm{d} \mathbf{
u}
=-2 \omega ( \int \limits_{a_0}^{ \phi (x) } \mathrm{d} \mathbf{
v}- ( \sum \limits_{i=1}^g \int \limits_{a_0}^{x_i} \mathrm{d}
\mathbf{ v}- \mathbf{
K}_{a_0}) ) = \\
&-2 \omega ( \int \limits_{x_s}^{ \phi (x) } \mathrm{d} \mathbf{
v}- ( \sum \limits_{ \begin{subarray}{l}i=1 \\
i \neq s \end{subarray}}^g \int \limits_{x_s}^{x_i} \mathrm{d} \mathbf{
v}- \mathbf{ K}_{x_s}) ) = \int \limits_{a_s}^{x} \mathrm{d} \mathbf{
u}+ \sum \limits_{ \begin{subarray}{l}i=1 \\
i \neq s \end{subarray}}^g \int \limits_{a_i}^{x_i} \mathrm{d} \mathbf{
u}
\end{eqnarray*}
and the change of notation $x \to x_s$ gives
\eqref{principal11}. \end{proof}
\subsubsection{Solution of the Jacobi inversion problem}
The equations of Abel map in conditions of Jacobi
inversion problem
\begin{equation}
u_i= \sum_{k=1}^g \int_{a_k}^{x_k} \frac{x^{i-1} \mathrm{d}x}{y},
\label{Abel-map}
\end{equation}
are invertible if the points $ (y_k, x_k) $ are distinct and
$ \forall j, k \in 1, \ldots, g \; \phi (y_k, x_k) \neq (y_j, x_j) $. Using
\eqref{principal} we find the solution of Jacobi inversion problem
on the curves with $a= \infty$ in a very effective form.
\begin{theorem} The Abel preimage of the point $ \boldsymbol{
u} \in \mathrm{Jac} (V) $ is given by the set
$ \{ (y_1, x_1) , \ldots, (y_g, x_g) \} \in (V) ^g$, where
$ \{x_1, \ldots, x_g \}$ are the zeros of the polynomial
\begin{equation} \mathcal{ P} (x; \boldsymbol{ u}) =0 \label{x},
\end{equation}
where \begin{equation} \mathcal{ P} (x; \boldsymbol{ u}) =x^g-x^{g-1}
\wp_{g, g}
( \boldsymbol{ u}) -x^{g-2} \wp_{g, g-1} ( \boldsymbol{ u}) - \ldots-
\wp_{g, 1} ( \mathbf{
u}) , \label{p}
\end{equation}
and $ \{y_1, \ldots, y_g \}$ are given by
\begin{equation} y_k=- \frac{ \partial \mathcal{
P} (x; \boldsymbol{ u}) }{ \partial u_g} \Bigl\lvert_{x=x_k},
\; \label{y} \end{equation} \end{theorem} \begin{proof} We tend
in \eqref{principal} $a_0 \to a= \infty$. Then we take
\begin{equation}
\lim_{x \to \infty}
\frac{F (x, x_r) }{4x^{g-1} (x-x_r) ^2}=
\sum_{i=1}^g \wp_{gi} ( \boldsymbol{ u}) x_r^{i-1}. \label{limit}
\end{equation} The limit in the left hand side of \eqref{limit} is
equal to $x_r^g$, and we obtain \eqref{x}.
We find from \eqref{Abel-map},
\[ \sum_{i=1}^g \frac{x^{k-1}_i}{ y_i}
\frac{ \partial x_i}{ \partial u_j}= \delta_{jk}, \qquad
\frac{ \partial x_k}{ \partial u_g}= \frac{y_k
}{ \prod_{i \neq
k} (x_k-x_i) }. \]
On the other hand we have
\[
\frac{ \partial \mathcal{ P}}{ \partial u_g}
\Bigl\lvert_{x=x_k}= - \frac{ \partial x_k}{ \partial u_g}
\prod_{i \neq k} (x_i-x_k) , \] and we obtain \eqref{y}.
\end{proof}
Let us denote by $ \boldsymbol{ \wp}$, $ \boldsymbol{ \wp}'$ the
$g$--dimensional vectors,
\[
\boldsymbol{ \wp}= \left ( \wp_{g1}, \ldots, \wp_{gg}
\right) ^T,
\quad
\boldsymbol{ \wp}'= \frac{ \partial \boldsymbol{ \wp}}{
\partial u_g}
\] and the companion matrix \cite{hj86} of the polynomial
$ \mathcal{ P} (z; \boldsymbol{ u}) $, given by \eqref{p}
\[ \mathcal{ C}= \mathcal{ B}+ \boldsymbol{ \wp} \mathbf{ e}_g^T,
\quad \text{where} \quad \mathcal{ B}= \sum_{k=1}^g \mathbf{ e}_k
\mathbf{
e}^T_{k-1}. \] The companion matrix $ \mathcal{
C}$ has the property \begin{equation} x_k^n= \mathbf{ X}^T_k \mathcal{
C}^{n-g+1} \mathbf{ e}_g= \mathbf{ X}^T_k \mathcal{ C}^{n-g} \boldsymbol{
\wp}, \quad \forall n \in \mathbb{ Z}, \label{prop-c}
\end{equation} with the vector $ \mathbf{ X}_k^T= (1, x_k,
\ldots, x_k^{g-1}) $, where $x_k$ is one of the roots of
\eqref{x}. From \eqref{principal11} we find $-2y_ry_s=4
(x_r-x_s) ^2 \sum_{i=1}^{g} \wp_{ij} ( \boldsymbol{ u})
x_r^{i-1}x_s^{j-1}-F (x_r, x_s) $. Introducing matrices $ \Pi= (
\wp_{ij}) $, $ \Lambda_0= \mathrm{diag} ( \lambda_{2g-2},
\ldots, \lambda_0) $ and $ \Lambda_1= \mathrm{diag} \; (
\lambda_{2g-1}, \ldots, \lambda_1) $, we have, taking into
account \eqref{prop-c}, \begin{eqnarray*} \lefteqn{2 \mathbf{
X}^T_r \boldsymbol{ \wp}'{ \boldsymbol{ \wp}'}^T \mathbf{
X}_s=-4 \mathbf{ X}^T_r ({ \mathcal{ C}}^2 \Pi-2 \mathcal{
C} \Pi \mathcal{ C}^T+ \Pi { \mathcal{ C}^T}^2 ) \mathbf{
X}_s } \nonumber \\&&+ 4 \mathbf{ X}^T_r ( \mathcal{ C}
\boldsymbol{ \wp} \boldsymbol{ \wp}^T+ \boldsymbol{ \wp}
\boldsymbol{ \wp}^T \mathcal{ C}^T) \mathbf{ X}_s +2
\mathbf{ X}^T_r \Lambda_0 \mathbf{ X}_s+ \mathbf{ X}^T_r (
\mathcal{ C} \Lambda_1+ \Lambda_1 \mathcal{ C}^T) \mathbf{
X}_s. \end{eqnarray*} Whence, (see \cite{be96b}) :
\begin{cor}
The relation
\begin{equation}
2 \boldsymbol{
\wp}'{ \boldsymbol{ \wp}'}^T=-4 ({ \mathcal{
C}}^2 \Pi-2 \mathcal{ C} \Pi \mathcal{ C}^T+ \Pi { \mathcal{ C}^T}^2 )
+ 4 ( \mathcal{ C} \boldsymbol{
\wp} \boldsymbol{ \wp}^T+ \boldsymbol{ \wp} \boldsymbol{ \wp}^T
\mathcal{
C}^T) + \mathcal{ C} \Lambda_1+ \Lambda_1 \mathcal{
C}^T+2 \Lambda_0. \label{principium}
\end{equation}
connects odd functions $ \wp_{ggi}$ with poles order $3$ and even
functions $ \wp_{jk}$ with poles of order $2$ in the field of
meromorphic functions on $ \mathrm{Jac} (V) $.
\end{cor}
\begin{definition} \label{umbral_D} The { \em umbral derivative}
\cite{ro84}
$D_s (p (z) ) $ of a polynomial \newline $p (z) = \sum_{k=0}^n p_k z^k$
is given by \[ D_s p (z) =
\left ( \frac{p (z) }{z^s} \right) _+= \sum_{k=s}^{n}p_kx^{k-s},
\]
where $ ( \cdot) _+$ means taking the purely polynomial part.
\end{definition}
Considering polynomials $p= \prod_{k=1}^{n} (z-z_k) $
and ${ \tilde p}= (z-z_0) p$, the elementary properties of $D_s$ are
immediately deduced:
\begin{eqnarray}
&D_s (p) =z D_{s+1} (p) +p_s=z D_{s+1} (p) +
S_{n-s} (z_1, \ldots, z_n) , \nonumber \\
&D_s ({ \tilde
p}) = (z-z_0) D_s (p) +p_{s-1}= (z-z_0) D_s (p) +S_{n+1-s} (z_1,
\ldots, z_n) , \label{umbral_2}
\end{eqnarray}
where $S_l ( \cdots) $ is the $l$--th order elementary symmetric
function of its variables times $ (-1) ^l$ (we assume
$S_0 ( \cdots) =1$) .
From \eqref{umbral_2} we see that $S_{n-s} (z_0, \ldots, \hat
z_l, \ldots, z_n) = \left (D_{s+1} ({ \tilde
p}) |_{z=z_l} \right) $. This is particularly useful to write
down the inversion of \eqref{y}
\begin{equation}
\wp_{ggk} ( \boldsymbol{ u}) = \sum_{l=1}^g y_l
\left ( \frac{D_k (P (z) ) }
{ \frac{ \partial}{ \partial z}P (z) } \Bigg|_{z=x_l} \right) ,
\label{inversiony}
\end{equation}
where $P (z) = \prod_{k=1}^{g} (z-x_k) $.
It is of importance to describe the set of common zeros of the
functions $ \wp_{ggk} ( \boldsymbol{ u}) $.
\begin{cor}
The vector function $ \boldsymbol{ \wp}' ( \boldsymbol{ u}) $
vanishes iff
$ \boldsymbol{ u}$ is a halfperiod. \end{cor}
\begin{proof} The equations $ \wp_{ggk} ( \boldsymbol{ u}) =0,
\;k \in 1, \ldots, g$ yield due to \eqref{inversiony} the equalities
$y_i=0, \forall i \in 1, \ldots, g$. The latter is possible if and
only if the points $x_1, \ldots, x_g$ coincide with any $g$ points
$e_{i_1}, \ldots, e_{i_g}$ from the set branching points
$e_1, \ldots, e_{2g+2}$. So the point
\[
\boldsymbol{ u}= \sum_{l=1}^{g} \int_{a_l}^{e_{i_l}} \mathrm{d} \mathbf{
u} \in \mathrm{Jac} (V) \] is of the second order in Jacobian and
hence is a halfperiod. \end{proof}
\section{Basic relations}
In this section we are going to derive the explicit algebraic
relations between the generating functions in the field of
meromorphic functions on $ \mathrm{Jac} (V) $. After some
preparations just below, we will in the section~ \ref{hyper-Jac}
find the explicit cubic relations between $ \wp_{ggi}$
and $ \wp_{ij}$. These, in turn, lead
to very special corollaries: the variety
$ \mathrm{Kum} (V) = \mathrm{Jac} (V) / \pm$ is mapped into the space of
symmetric matrices of rank not greater than $3$.
We start with, the conditions
$ \lambda_{2g+2}=0, \, \lambda_{2g+1}=4$ being imposed,
the following Theorem \ref{the-Z}, which is the starting point for
derivation of the basic relations.
\begin{theorem} \label{the-Z}
Let $(y_0, x_0)\in V$ be an arbitrary point and
$ \{ (y_1, x_1) , \ldots, (y_g, x_g) \} \in (V)^g$ be the
Abel preimage of the point $ \boldsymbol{ u} \in
\mathrm{Jac} (V) $. Then \begin{equation} - \zeta_j \left (
\int_{a}^{x_0} \mathrm{d} \mathbf{ u} + \boldsymbol{ u} \right)
= \int_{a}^{x_0} \mathrm{d}r_j + \sum_{k=1}^g \int_{a_k}^{x_k}
\mathrm{d}r_j - \frac{1}{2} \sum_{k=0}^g y_k \left (
\frac{D_j (R' (z) ) }{R' (z) } \Bigg|_{z=x_k} \right) ,
\label{principalz}
\end{equation}
where $ R (z) = \prod_0^g (z-x_j) $ and $
R' (z) = \frac{ \partial}{ \partial z} R (z) $.
And
\begin{eqnarray}
- \zeta_j ( \boldsymbol{ u}) = \sum_{k=1}^g \int_{a_k}^{x_k} \mathrm{d}
r_j- \frac{1}{2} \wp_{gg, j+1} ( \boldsymbol{ u}).
\label{principalz1} \end{eqnarray} \end{theorem}
\begin{proof} Putting in \eqref{r11} $ \mu_i=a_i$ we have
\begin{equation} \mathrm{ln}
\left \{ \frac{ \sigma ( \int_{a_0}^{x} \mathrm{d} \mathbf{ u}- \boldsymbol{ u}) }
{ \sigma ( \int_{a_0}^{x} \mathrm{d} \mathbf{ u}) } \right \}
- \left \{ \frac{ \sigma ( \int_{a_0}^{ \mu} \mathrm{d} \mathbf{ u}- \boldsymbol{ u}) }
{ \sigma ( \int_{a_0}^{ \mu} \mathrm{d} \mathbf{ u}) } \right \} =
\int_{ \mu}^{x} \mathrm{d} \mathbf{ r}^T \, \boldsymbol{
u}+ \sum_{k=1}^g \int_{a_k}^{x_k} \mathrm{d} \Omega (x, \mu) ,
\label{r20} \end{equation}
where $ \mathrm{d} \Omega$ is as in \eqref{third}. Taking derivative over
$u_j$ from the both sides of the equality \eqref{r20}, after that
letting $a_0 \to \mu$ and applying $ \phi (y, x) = (-y, x) $ and
$ \phi ( \nu, \mu) = (- \nu, \mu) $, we have
\[
\zeta_j \left (
\int_{ \mu}^x \mathrm{d} \mathbf{ u}+ \boldsymbol{ u}
\right) + \int_{ \mu}^{x} \mathrm{d} r_j -
\frac12 \sum_{k=1}^g \frac{1}{y_k} \frac{ \partial x_k}{ \partial
u_j} \frac{y_k-y}{x_k-x}
=
\zeta_j \left (
\boldsymbol{ u}
\right) - \frac12 \sum_{k=1}^g \frac{1}{y_k} \frac{ \partial x_k}{
\partial
u_j} \frac{y_k- \nu}{x_k- \mu}.
\]
Put $x=x_0$. Denoting $P (z) = \prod_1^g (z-x_j) $ we find
\begin{eqnarray*}
&& \sum_{k=1}^g \frac{1}{y_k} \frac{ \partial x_k}{ \partial
u_j} \frac{y_k-y}{x_k-x}=
\sum_{k=1}^g \left ( \frac{D_{j} (P (z) ) }{P' (z) } \Bigg|_{z=x_k}
\right)
\frac{y_k-y}{x_k-x} \\=
&& \sum_{k=0}^g
y_k \left ( \frac{D_j (R' (z) ) }{R' (z) } \Bigg|_{z=x_k} \right)
- \sum_{k=1}^g
y_k \left ( \frac{D_{j+1} (P (z) ) }{P' (z) } \Bigg|_{z=x_k} \right) .
\end{eqnarray*}
Hence, using \eqref{inversiony} and adding to both sides
$ \sum_{k=1}^g \int_{a_k}^{x_k} \mathrm{d}r_j$, we deduce
\begin{eqnarray}
&& \zeta_j \left ( \int_{ \mu}^{x_0} \mathrm{d} \mathbf{
u}+ \boldsymbol{ u} \right) + \int_{ \mu}^{x_0} \mathrm{d} r_j
+ \sum_{k=1}^g \int_{a_k}^{x_k} \mathrm{d}r_j -
\frac12 \sum_{k=0}^g
y_k \left ( \frac{D_j (R' (z) ) }{R' (z) } \Bigg|_{z=x_k} \right)
\nonumber \\
&&=
\zeta_j \left (
\boldsymbol{ u}
\right) + \sum_{k=1}^g \int_{a_k}^{x_k} \mathrm{d}r_j
- \frac12 \sum_{k=1}^g \frac{1}{y_k} \frac{ \partial x_k}{ \partial
u_j} \frac{y_k- \nu}{x_k- \mu}- \frac12 \wp_{gg, j+1}. \label{deduce}
\end{eqnarray}
Now see, that the left hand side of the \eqref{deduce} is
symmetrical in $x_0, x_1, \ldots, x_g$, while the right hand side
does not depend on $x_0$. So, it does not depend on any of $x_i$.
We conclude, that it is a constant depending only on $ \mu$.
Tending $ \mu \to a$ and applying the hyperelliptic involution to
the whole aggregate, we find this constant to be $0$. \end{proof}
\begin{cor} For $ (y, x) \in V$ and $ \boldsymbol{
\alpha}= \int_a^{x} \mathrm{d} \mathbf{ u}$ :
\begin{equation}
\zeta_{j} ( \boldsymbol{ u}+ \boldsymbol{ \alpha})
- \zeta_{j} ( \boldsymbol{ u}) - \zeta_{j} ( \boldsymbol{
\alpha}) = \frac{ (-y D_j+ \partial_j) \mathcal{ P} (x; \boldsymbol{
u}) } {2 \mathcal{ P} (x; \boldsymbol{ u}) }
, \label{stickelberger} \end{equation}
where $ \partial_j= \partial/ \partial u_j$.
\end{cor}
\begin{proof} To find $ \zeta_{j} ( \boldsymbol{
\alpha}) $ take the limit
$ \{x_1, \ldots, x_g \} \to \{a_1, \ldots, a_g \}$ in
\eqref{principalz}. The right hand side of
\eqref{stickelberger} is obtained by rearranging $
\frac12 \sum_{k=1}^g \frac{1}{y_k} \frac{ \partial x_k}{ \partial
u_j} \frac{y_k-y}{x_k-x}$. \end{proof}
\begin{cor} The functions
$ \wp_{gggk}$, for $k=1, \ldots, g$ are given by \begin{eqnarray}
\wp_{gggi}= (6 \wp_{gg}+ \lambda_{2g}) \wp_{gi}+6 \wp_{g, i-1}-2 \wp_{g-1, i}
+ \frac{1}{2} \delta_{gi} \lambda_{2g-1}. \label{wpgggi}
\end{eqnarray}
\end{cor}
\begin{proof} Consider the relation
\eqref{principalz1}. The differentials $ \mathrm{d} \zeta_i$,
$i=1, \ldots, g$ can be presented in the following forms
\[
- \mathrm{d} \zeta_i= \sum_{k=1}^g \wp_{ik} \mathrm{d}u_k=
\sum_{k=1}^g \mathrm{d}r_i (x_k) - \frac12
\sum_{k=1}^g \wp_{gg, i+1, k} \mathrm{d}u_k.
\]
Put $i=g-1$.
We obtain for each of the $x_k$, $k=1, \ldots, g$
\[
\left (12x_k^{g+1}+2 \lambda_{2g}x_k^g+
\lambda_{2g-1}x_k^{g-1}
-4 \sum_{j=1}^g \wp_{g-1, j}x_k^{j-1} \right) \frac{ \mathrm{d}x_k}{ y_k}
=2 \sum_{j=1}^g \wp_{gggj}x_k^{j-1} \frac{ \mathrm{d}x_k}{
y_k}. \]
Applying the formula \eqref{p} to eliminate the
powers of $x_k$ greater than $g-1$, and taking into account, that
the differentials $ \mathrm{d}x_k$ are independent, we come to
\[ \sum_{i=1}^g \left[ (6 \wp_{gg}+ \lambda_{2g}) \wp_{gi}+6 \wp_{g, i-1}
-2 \wp_{g-1, i}
+ \frac{1}{2} \delta_{gi} \lambda_{2g-1} \right]x_k^{i-1}=
\sum_{j=1}^g \wp_{gggj}x_k^{j-1}. \] \end{proof}
Let us
calculate the difference
$ \frac{ \partial \wp_{gggk}}{ \partial
u_i}- \frac{ \partial \wp_{gggi}}{ \partial u_k}$ according to the
\eqref{wpgggi}. We obtain
\begin{cor}
\begin{equation}
\wp_{ggk} \wp_{gi}- \wp_{ggi} \wp_{gk}+ \wp_{g, i-1, k}- \wp_{gi, k-1}=0.
\label{wp3}
\end{equation}
\end{cor}
This means that the $1$--form
$ \sum_{i=1}^g ( \wp_{gg} \wp_{gi}+ \wp_{g, i-1}) \mathrm{d} u_i$ is
closed. We can rewrite this as $ \mathrm{d} \boldsymbol{ u}^T \mathcal{
C} \boldsymbol{ \wp}$.
Differentiation of \eqref{wp3} by
$u_g$ yields
\begin{equation}
\wp_{gggk} \wp_{gi}- \wp_{gggi} \wp_{gk}+ \wp_{gg, i-1, k}- \wp_{ggi, k-1}=0.
\label{i-1, k-1}
\end{equation}
And the corresponding closed $1$--form is
$ \mathrm{d} \boldsymbol{ u}^T \mathcal{ C} \boldsymbol{ \wp}'$.
\subsection{Fundamental cubic and quartic
relations} \label{hyper-Jac} We are going to find the relations
connecting the odd functions $ \wp_{ggi}$ and even functions
$ \wp_{ij}$. These relations take in hyperelliptic theory the place
of the Weierstrass cubic relation \[ { \wp'}^2=4 \wp^3-g_2 \wp-g_3,
\]
for elliptic functions, which establishes the meromorphic map
between the elliptic Jacobian $ \mathbb{ C}/ (2 \omega, 2 \omega') $ and
the plane cubic.
The theorem below is based
on the property of an Abelian function to be constant if any
gradient of it is identically $0$, or, if for Abelian
functions $G ( \boldsymbol{ u}) $ and $F ( \boldsymbol{ u}) $ there exist such a
nonzero vector $ \boldsymbol{
\alpha} \in \mathbb{ C}^g$, that
$ \sum_{i=1}^g \alpha_i \frac{ \partial}{ \partial u_i} (G ( \mathbf{
u}) -F ( \boldsymbol{ u}) ) $ vanishes, then
$G ( \boldsymbol{ u}) -F ( \boldsymbol{ u}) $ is a constant.
\begin{theorem} The functions
$ \wp_{ggi}$ and $ \wp_{ik}$ are related by
\begin{eqnarray}
\lefteqn{ \wp_{ggi} \wp_{ggk}=4 \wp_{gg} \wp_{gi} \wp_{gk}-
2 ( \wp_{gi} \wp_{g-1, k}+ \wp_{g, k} \wp_{g-1, i}) } \nonumber \\
&&+
4 ( \wp_{gk} \wp_{g, i-1}+ \wp_{gi} \wp_{g, k-1}) +
4 \wp_{k-1, i-1}-2 ( \wp_{k, i-2}+ \wp_{i, k-2}) \nonumber \\
&&+
\lambda_{2g} \wp_{gk} \wp_{gi}+ \frac{ \lambda_{2g-1}}2
( \delta_{ig} \wp_{kg}+ \delta_{kg} \wp_{ig}) +c_{ (i, k) },
\label{product3}
\end{eqnarray}
where
\begin{equation}c_{ (i, k) }= \lambda_{2i-2} \delta_{ik}
+ \frac12 ( \lambda_{2i-1} \delta_{k, i+1}
+ \lambda_{2k-1} \delta_{i, k+1}) .
\label{cij}
\end{equation}
\end{theorem}
\begin{proof}
We are looking for such a function $G ( \boldsymbol{ u}) $ that
$ \frac{ \partial}{ \partial u_g} ( \wp_{ggi} \wp_{ggk}-G) $ is $0$.
Direct check using \eqref{wp3} shows that
\begin{multline*}
\frac{ \partial}{ \partial u_g} ( \wp_{ggi} \wp_{ggk}- (4 \wp_{gg} \wp_{gi} \wp_{gk}-
2 ( \wp_{gi} \wp_{g-1, k}+ \wp_{g, k} \wp_{g-1, i}) \\
+
4 ( \wp_{gk} \wp_{g, i-1}+ \wp_{gi} \wp_{g, k-1}) +
4 \wp_{k-1, i-1}-2 ( \wp_{k, i-2}+ \wp_{i, k-2}) \\
+
\lambda_{2g} \wp_{gk} \wp_{gi}+ \frac{ \lambda_{2g-1}}2
( \delta_{ig} \wp_{kg}+ \delta_{kg} \wp_{ig}) ) ) =0.
\end{multline*}
It remains to determine $c_{ij}$. From \eqref{principium}, we
conclude, that $c_{ (i, k) }$ for $k=i$ is equal to
$ \lambda_{2i-2}$, for $k=i+1$ to $ \frac12{ \lambda_{2i-1}}$,
otherwise $0$. So $c_{ij}$ is given by \eqref{cij}. \end{proof}
\noindent Consider $ \mathbb{ C}^{g+ \frac{g (g+1) }{2}}$ with
coordinates $ (\boldsymbol{ z}, p=\{ p_{i, j} \}_{i, j=1
\ldots g}) $ with $ \boldsymbol{ z}^T= (z_1, \ldots, z_g)$ and
$p_{ij}=p_{ji}$, then we
have
\begin{cor} The map \[ \varphi:
\mathrm{Jac} (V) \backslash ( \sigma) \to \mathbb{
C}^{g+ \frac{g (g+1) }{2}}, \quad \varphi ( \boldsymbol{
u}) = ( \boldsymbol{ \wp}' ( \boldsymbol{ u}) , \Pi ( \boldsymbol{ u}) ) ,
\]
where $ \Pi= \{ \wp_{ij} \}_{i, j=1, \ldots, g}$, is meromorphic
embedding.
The image
$ \varphi ( \mathrm{Jac} (V) \backslash ( \sigma) ) \subset
\mathbb{ C}^{g+ \frac{g (g+1) }{2}}$ is the intersection of
$ \frac{g (g+1) }{2}$ cubics, induced by \eqref{product3}.
\end{cor}
$ ( \sigma) $ denotes the divisor of $0$'s of $ \sigma$.
Consider projection \[
\pi: \mathbb{ C}^{ \frac{g+g (g+1) }{2}} \to \mathbb{
C}^{ \frac{g (g+1) }{2}}, \quad \pi ( \boldsymbol{ z}, p) =p.
\]
\begin{cor}
The restriction $ \pi \circ \varphi$ is the meromorphic embedding of
the Kummer variety
$ \mathrm{Kum} (V) =
( \mathrm{Jac} (V) \backslash ( \sigma) ) / \pm$ into
$ \mathbb{ C}^{ \frac{g (g+1) }{2}}$. The image
$ \pi ( \varphi ( \mathrm{Jac} (V) \backslash ( \sigma) ) )
\linebreak[3] \subset \mathbb{ C}^{ \frac{g (g+1) }{2}}$ is
the intersection of quartics, induced by \begin{equation} (
\wp_{ggi} \wp_{ggj}) ( \wp_{ggk} \wp_{ggl}) - ( \wp_{ggi}
\wp_{ggk}) ( \wp_{ggj} \wp_{ggl}) =0, \label{Kijkl}
\end{equation} where the parentheses mean, that substitutions
by \eqref{product3} are made before expanding. \end{cor} The
quartics \eqref{Kijkl} have no analogue in the elliptic theory.
The first example is given by genus $2$, where the celebrated
Kummer surface \cite{hu94} appears.
\subsection{Analysis of fundamental relations} \label{hyper-Kum}
Let us take a second look at the fundamental cubics
\eqref{product3} and quartics \eqref{Kijkl}.
\subsubsection{Sylvester's identity}
For any matrix $K$ of entries $k_{ij}$ with $i, j=1, \ldots, N$ we
introduce the symbol $K[{}^{i_1}_{j_1} \cdots{}^{i_m}_{j_n}]$ to
denote the $m \times n$ submatrix:
\[
K[{}^{i_1}_{j_1} \cdots{}^{i_m}_{j_n}]
= \{k_{i_k, j_l} \}_{k=1, \ldots, m; \, l=1, \ldots, n}
\]
for subsets of rows $i_k$ and columns $j_l$.
We will need here the { \em Sylvester's
identity} (see, for instance~ \cite{hj86}) . Let us fix a subset of
indices $ \boldsymbol{ \alpha}= \{i_1, \ldots, i_k \}$, and make up the
$N-k \times N-k$ matrix $S (K, \boldsymbol{ \alpha}) $ assuming that
\[
S (K, \boldsymbol{ \alpha}) _{ \mu, \nu}= \det
K[{}^{ \mu, \boldsymbol{ \alpha}}_{ \nu, \boldsymbol{ \alpha}}]
\] and
$ \mu, \nu$ are not in $ \boldsymbol{ \alpha}$, then
\begin{equation}
\det S (K, \boldsymbol{ \alpha}) = \det
K[{}^{ \boldsymbol{ \alpha}}_{ \boldsymbol{ \alpha}}]^{ (N-k-1) }
\det K .
\label{Sylvester} \end{equation}
\subsubsection{Determinantal form}
We introduce (cf. \cite{le95}) new functions $h_{ik}$ defined by
the formula \begin{eqnarray} h_{ik}&=&4 \wp_{i-1, k-1} -2
\wp_{k, i-2} -2 \wp_{i, k-2} \nonumber \\ &+&
\frac{1}{2} \left ( \delta_{ik} ( \lambda_{2i-2}+ \lambda_{2k-2})
+ \delta_{k, i+1} \lambda_{2i-1}
+ \delta_{i, k+1} \lambda_{2k-1} \right) ,
\label{variables}
\end{eqnarray}
where the indices
$i, k \in 1, \ldots, g+2$.
We assume that $ \wp_{nm}=0$ if $n$ or $m$ is $<1$ and
$ \wp_{nm}=0$ if $n$ or $m$ is $>g$. It is evident that
$h_{ij}=h_{ji}$. We shall denote the matrix of $h_{ik}$ by $H$.
The map \eqref{variables} from $ \wp$'s and $ \lambda$'s to $h$'s
respects the grading
\[ \mathrm{deg} \;h_{ij}=i+j, \qquad
\mathrm{deg} \; \wp_{ij}=i+j+2, \qquad
\mathrm{deg} \; \lambda_{i}=i+2, \] and on a fixed level $L$
\eqref{variables} is
linear and invertible. From the definition follows
\[
\sum_{i=1}^{L-1}h_{i, L-i}= \lambda_{L-2} \Rightarrow \boldsymbol{
X}^T H \boldsymbol{ X}= \sum_{i=0}^{2g+2} \lambda_{i}x^{i}
\]
for $ \boldsymbol{ X}^T= (1, x, \ldots, x^{g+1}) $ with
arbitrary $x \in \mathbb{ C}$. Moreover, for any roots $x_r$ and
$x_s$ of the equation $ \sum_{j=1}^{g+2} h_{g+2, j}x^{j-1}=0$ we
have (cf. \eqref{principal11}) $ y_ry_s = \boldsymbol{ X}_r^T
H \boldsymbol{ X}_s$.
From \eqref{variables} we have
\begin{gather}
-2 \wp_{ggi}= \tfrac{ \partial}{ \partial
u_g}h_{g+2, i}= \tfrac{ \partial}{ \partial
u_i}h_{g+2, g}=- \tfrac12 \tfrac{ \partial}{ \partial
u_i}h_{g+1, g+1}, \notag \\
2 ( \wp_{gi, k-1}- \wp_{g, i-1, k}) = \tfrac{ \partial}{
\partial u_k}h_{g+2, i-1}- \tfrac{ \partial}{ \partial
u_i}h_{g+2, k-1}= \tfrac12 \tfrac{ \partial}{ \partial
u_k}h_{g+1, k}
-\tfrac12 \tfrac{ \partial}{ \partial
u_i}h_{g+2, i}, \notag \\
\intertext{$ \ldots$ etc., \, and (see \eqref{wpgggi}) :}
-2 \wp_{gggi}= \tfrac{ \partial^2}{\partial
u_g^2}h_{g+2, i}=
- \det H[{}^{i, }_{g+1 }{}^{g+1 }_{g+2}] -
\det H[{}^{i-1, }_{g+1,}{}^{g+2 }_{g+2}] -
\det H[{}^{i, }_{g,}{}^{g+2}_{g+2}].\label{dg_H}
\end{gather}
Using \eqref{variables}, we write \eqref{product3} in
more effective form:
\begin{eqnarray}
4 \wp_{ggi} \wp_{ggk}= \tfrac{ \partial}{ \partial
u_g}h_{g+2, i} \tfrac{ \partial}{ \partial
u_g}h_{g+2, k}=- \det
H[{}^{i, }_{k, }{}^{g+1, }_{g+1, }{}^{g+2}_{g+2}]
\label{wpggiwpggk}
\end{eqnarray}
Consider, as an example, the case of genus $1$. We define on the
Jacobian of a curve \[ y^2= \lambda_4 x^4+ \lambda_3 x^3 + \lambda_2
x^2+ \lambda_1 x+ \lambda_0 \] the Kleinian functions:
$ \sigma_{K} (u_1) $ with expansion $u_1+ \ldots$, its second and
third logarithmic derivatives $- \wp_{11}$ and $- \wp_{111}$. By
\eqref{wpggiwpggk} and following the definition \eqref{variables}
\[
-4\wp_{111}^2=\det H
\left[{}_{1, 2, 3}^{1, 2, 3} \right]= \det \left (
\begin{array}{ccc} \lambda_0& \tfrac{1}{2} \lambda_1&-2
\wp_{11} \\ \tfrac12 \lambda_1&4 \wp_{11}+ \lambda_2&
\tfrac12 \lambda_3 \\ -2 \wp_{11}& \tfrac12
\lambda_3& \lambda_4 \end{array} \right) ; \] the determinant
expands as: \[ \wp_{111}^2=4 \wp_{11}^3+ \lambda_2
\wp_{11}^2+ \wp_{11} \frac{ \lambda_1 \lambda_3-4 \lambda_4
\lambda_0}{4}+ \frac{ \lambda_0 \lambda_3^2+ \lambda_4 (
\lambda_1^2-4 \lambda_2 \lambda_0) }{16}, \] and the
\eqref{dg_H}, in complete accordance, gives \[ \wp_{1111}=6
\wp_{11}^2 + \lambda_2 \wp_{11}+ \frac{ \lambda_1
\lambda_3-4 \lambda_4 \lambda_0}{8} \] These equations show
that $ \sigma_K$ differs by $ \mathrm{exp} (- \frac1{12}
\lambda_2 u_1^2) $ from standard Weierstrass $ \sigma_W$ built by
the invariants $g_2= \lambda_4 \lambda_0+ \frac1{12}
\lambda_2^2- \frac14 \lambda_3 \lambda_1$ and $g_3= \det
\left ( \begin{smallmatrix} \lambda_0& \frac14 \lambda_1&
\frac16 \lambda_2 \\ \frac14 \lambda_1& \frac16 \lambda_2&
\frac14 \lambda_3 \\ \frac16 \lambda_2& \frac14 \lambda_3&
\lambda_4 \end{smallmatrix} \right) $ (see, e.g.
\cite{ba55, ww73}) .
Further,
we find, that $ \mathrm{rank} \; H=3$ in generic point of Jacobian,
$ \mathrm{rank} \; H=2$ in halfperiods. At $u_1=0$, where
$ \sigma_K$ has is $0$ of order $1$, we have $ \mathrm{rank} \;
\sigma_K^2 H=3$.
Concerning the general case, on the ground of \eqref{wpggiwpggk},
we prove the following:
\begin{theorem}
$ \mathrm{rank} \;H=3$ in generic point $ \in \mathrm{Jac} (V) $ and
$ \mathrm{rank} \;H=2$ in the halfperiods.
$ \mathrm{rank} \; \sigma ( \boldsymbol{ u}) ^2 H=3$ in generic
point $ \in ( \sigma) $ and
$ \mathrm{rank} \; \sigma ( \boldsymbol{ u}) ^2 H$ in the points
of $ ( \sigma) _{ \mathrm{sing}}$.
\end{theorem}
Here $ ( \sigma) \subset \mathrm{Jac} (V) $ denotes the divisor of
$0$'s of $ \sigma ( \boldsymbol{ u}) $. The
$ ( \sigma) _{ \mathrm{sing}} \subset ( \sigma) $ is the so-called
singular set of $ ( \sigma) $. $ ( \sigma) _{ \mathrm{sing}}$ is the set
of points where $ \sigma$ vanishes and all its first partial
derivatives vanish. $ ( \sigma) _{ \mathrm{sing}}$ is known (see
\cite{fa73} and references therein) to be a subset of dimension
$g-3$ in hyperelliptic Jacobians of $g>3$, for genus $2$ it is
empty and consists of single point for $g=3$. Generally, the
points of $ ( \sigma) _{ \mathrm{sing}}$ are presented by
$ \{ (y_1, x_1) , \ldots, (y_{g-3} \, , x_{g-3} \, ) \} \in (V) ^{g-3}$ such that
for all $i \neq j \in 1, \ldots, g-3, $ $ \phi (y_i, x_i) \neq (y_j, x_j) $.
\begin{proof} Consider the Sylvester's
matrix $S=S \left (H
[{}^{i, j, g+1, g+2}_{k, l, g+1, g+2}], \{g+1, g+2 \}
\right) $.
By \eqref{wpggiwpggk} we have
$S=-4 \left ( \begin{array}{ll}
\wp_{ggi} \wp_{ggk}& \wp_{ggi} \wp_{ggl} \\
\wp_{ggj} \wp_{ggk}& \wp_{ggj} \wp_{ggl}
\end{array} \right) $
and $ \det S=0$, so by \eqref{Sylvester} we see, that
$ \det H [{}^{i, j, g+1, g+2}_{k, l, g+1, g+2} ]
\det H [{}^{g+1, g+2}_{g+1, g+2} ]$ vanishes
identically. As $ \det H [{}^{g+1, g+2}_{g+1, g+2} ]=
\lambda_{2g+2} (4 \wp_{gg}+ \lambda_{2g}) - \frac14 \lambda_{2g+1}^2$
is not an identical $0$, we infer that
\begin{equation}
\det H \left[{}^{i, j, g+1, g+2}_{k, l, g+1, g+2} \right]=0.
\label{subHdet}
\end{equation}
Remark, that this equation is actually the \eqref{Kijkl} rewritten
in terms of $h$'s. Now from the \eqref{subHdet}, putting $j=l=g$,
we obtain for any $i, k$, except for such $ \boldsymbol{ u}$, that
$H \left[{}^{g, g+1, g+2}_{g, g+1, g+2} \right]$ becomes degenerate, and
those where the entries become singular i.e.
$ \boldsymbol{u} \in ( \sigma) $, \begin{equation}
h_{ik}= (h_{i, g}, h_{i, g+1}, h_{i, g+2})
\left (H \left[{}^{g, g+1, g+2}_{g, g+1, g+2} \right] \right) ^{-1}
\left ( \begin{array}{l}
h_{k, g} \\
h_{k, g+1} \\
h_{k, g+2}
\end{array}
\right) . \label{hik}
\end{equation}
This leads to the skeleton decomposition of the matrix $H$
\begin{equation}
H=H \left[{}^{1, \, { \displaystyle \ldots} \, , g+2}_{g, g+1, g+2}
\right]
\left (H \left[{}^{g, g+1, g+2}_{g, g+1, g+2} \right] \right) ^{-1}
H \left[{}^{g, g+1, g+2}_{1, \, { \displaystyle \ldots} \, , g+2}
\right],
\label{skeleton}
\end{equation}
which shows, that in generic point of $ \mathrm{Jac} (V) $ rank
of $H$ equals $3$.
Consider the case $ \det
H [{}^{g, g+1, g+2}_{g, g+1, g+2}]=0$. As by
\eqref{wpggiwpggk} we have $ \det
H [{}^{g, g+1, g+2}_{g, g+1, g+2}]= 4 \wp_{ggg}^2$, this may
happen only iff $ \boldsymbol{u}$ is a halfperiod. And therefore
we have instead of \eqref{subHdet} the equalities $H
[{}^{i, g+1, g+2}_{k, g+1, g+2}]=0$ and consequently
in halfperiods matrix $H$ is decomposed as
\[
H=H \left[{}^{1, \, { \displaystyle \ldots} \, , g+2}_{g+1, g+2}
\right]
\left (H \left[{}^{g+1, g+2}_{g+1, g+2} \right] \right) ^{-1}
H \left[{}^{g+1, g+2}_{1, \, { \displaystyle \ldots} \, , g+2}
\right],\]
having the rank $2$.
Next, consider $ \sigma ( \boldsymbol{ u}) ^2 H$ at the
$ \boldsymbol{ u} \in ( \sigma) $. We have
$ \sigma ( \boldsymbol{ u}) ^2
h_{i, k}=4 \sigma_{i-1} \sigma_{k-1}-2 \sigma_{i} \sigma_{k-2}
-2 \sigma_{i-2} \sigma_{k}, $ where
$ \sigma_{i}= \frac{ \partial}{ \partial u_i} \sigma ( \boldsymbol{
u}) $, and, consequently, the
decomposition
\[
\sigma ( \boldsymbol{ u}) ^2 H|_{ \boldsymbol{
u} \in ( \sigma) }=2 ( \boldsymbol{ s}_1, \boldsymbol{ s}_2,
\boldsymbol{ s}_3) \left ( \begin{array}{rrr} 0&0&-1 \\ 0&2&0 \\
-1&0&0 \end{array} \right) \left ( \begin{array}{r} \boldsymbol{
s}_1^T \\ \boldsymbol{ s}_2^T \\ \boldsymbol{ s}_3^T \end{array}
\right) , \] where $ \boldsymbol{
s}_1= ( \sigma_1, \ldots, \sigma_g, 0, 0) ^T$, $ \boldsymbol{
s}_2= (0, \sigma_1, \ldots, \sigma_g, 0) ^T$ and $ \boldsymbol{
s}_3= (0, 0, \sigma_1, \ldots, \sigma_g) ^T$. We infer, that
$ \mathrm{rank} ( \sigma ( \boldsymbol{ u}) ^2 H) $ is $3$ in generic
point of $ ( \sigma) $, and
becomes $0$ only when $ \sigma_1= \ldots= \sigma_g=0$, is in the
points $ \in ( \sigma) _{ \mathrm{sing}}$, while no other values are
possible.
\end{proof}
{ \em Conclusion}. The map \begin{align*}
h: \boldsymbol{ u} \mapsto&
\{4 \sigma_{i-1} \, \sigma_{k-1}-2 \sigma_{i} \, \sigma_{k-2}
-2 \sigma_{i-2} \, \sigma_{k} \\&- \sigma
(4 \sigma_{i-1, k-1} \, -2 \sigma_{i, k-2} \,
-2 \sigma_{i-2, k}) +
\tfrac{1}{2} \sigma^2 ( \delta_{ik} ( \lambda_{2i-2}+ \lambda_{2k-2}) \\
&+ \delta_{k, i+1} \lambda_{2i-1}
+ \delta_{i, k+1} \lambda_{2k-1}) \}_{i, k \in 1, \ldots, g+2},
\end{align*} induced by
\eqref{variables} establish the meromorphic map of the
$ \big ( \mathrm{Jac} (V) \backslash ( \sigma) _{ \mathrm{sing}} \big) / \pm
$ into the space $Q_3$
of complex symmetric $ (g+2) \times (g+2) $ matrices of
$ \mathrm{rank}$ not greater than $3$.
We give the example of genus $2$ with $ \lambda_6=0$ and
$ \lambda_5=4$:
\begin{equation}
H= \left ( \begin{array}{cccc} \lambda_0& \frac{1}{2} \lambda_1&-2
\wp_{11}&-2 \wp_{12} \\
\frac{1}{2} \lambda_1& \lambda_2+4 \wp_{11}& \frac{1}{2} \lambda_3-
2 \wp_{12}&-2 \wp_{22} \\-2 \wp_{11}& \frac{1}{2} \lambda_3-2
\wp_{12}& \lambda_4+4 \wp_{22}&2 \\
-2 \wp_{12}&-2 \wp_{22}&2&0 \end{array} \right) .
\label{kum} \end{equation}
In this case $ ( \sigma) _{ \mathrm{sing}}= \{ \varnothing \}$,
so the Kummer surface in $ \mathbb{ C} \mathbb{ P}^3$ with coordinates
\newline
$ (X_0, X_1, X_2, X _3) = ( \sigma^2, \sigma^2 \wp_{11}, \sigma^2 \wp_{12}, \sigma^2 \wp_{22}) $ is
defined by the equation $ \det \sigma^2 H=0$.
\subsubsection{Extended cubic relation}
The extension \cite{le95} of \eqref{wpggiwpggk} is given by
\begin{theorem}
\begin{eqnarray}
\mathbf{ R}^T \boldsymbol{ \pi}_{jl} \boldsymbol{ \pi}_{ik}^T
\mathbf{
S}= \frac{1}{4} \det \left ( \begin{array}{cc} H
\left[{}^i_j{}^k_l{}^{g+1}_{g+1}{}^{g+2}_{g+2} \right]
& \mathbf{ S} \\ \mathbf{ R}^T&0
\end{array} \right) , \label{bakergen}
\end{eqnarray}
where $ \mathbf{ R}, \, \mathbf{ S} \in \mathbb{ C}^4$ are
arbitrary vectors and
\[ \boldsymbol{ \pi}_{ik}= \left (
\begin{array}{c} - \wp_{ggk} \\
\wp_{ggi} \\
\wp_{g, i, k-1}- \wp_{g, i-1, k} \\
\wp_{g-1, i, k-1}-
\wp_{g-1, k, i-1}+
\wp_{g, k, i-2}-
\wp_{g, i, k-2}
\end{array}
\right) \]
\end{theorem}
\begin{proof}
Vectors $ \boldsymbol{ { \tilde \pi}}= \boldsymbol{ \pi}_{ik}$ and
$ \boldsymbol{ \pi}= \boldsymbol{ \pi}_{jl}$ solve the equations \[
H \left[{}^i_j{}^k_l{}^{g+1}_{g+1}{}^{g+2}_{g+2} \right] \boldsymbol{
\pi}=0; \quad \boldsymbol{
{ \tilde \pi}}^T{H} \left[{}^i_j{}^k_l{}^{g+1}_{g+1}{}^{g+2}_{g+2}
\right]=0.
\]
The theorem follows.
\end{proof}
The case of genus $2$, when $ \boldsymbol{
\pi}_{21}= (- \wp_{222}, \wp_{221}, - \wp_{211}, \wp_{111}) ^T$
exhausts all the possible $ \wp_{ijk}$--functions, the relation
\eqref{bakergen} was thoroughly studied by Baker \cite{ba07}.
\section{Applications}
\subsection{Matrix realization of hyperelliptic Kummer varieties}
Here we present the explicit matrix realization (see \cite{bel96})
of hyperelliptic Jacobians $ \mathrm{Jac} (V) $ and Kummer varieties
$ \mathrm{Kum} (V) $ of the curves $V$ with the fixed branching point
$e_{2g+2}=a= \infty$. Our approach is based on the results of
Section \ref{hyper-Kum}.
Let us consider the space $ \mathcal{ H}$ of complex symmetric
$ (g+2) \times (g+2) $--matrices $ \mathrm{ H}= \{ \mathrm{
h}_{k, s} \}$, with $ \mathrm{ h}_{g+2, g+2}=0$ and $ \mathrm{
h}_{g+1, g+2}=2$. Let us put in correspondence to $ \mathrm{
H} \in \mathcal{ H}$ a symmetric $g \times g$--matrix $ \mathrm{
A} ( \mathrm{ H}) $, with entries $a_{k, s}= \det \mathrm{ H}
\left[{}_{s, g+1, g+2}^{k, g+1, g+2} \right]$.
From the Sylvester's identity \eqref{Sylvester} follows that
rank of the matrix $ \mathrm{ H} \in \mathcal{ H}$
does not exceed $3$ if and only if rank of the matrix
$ \mathrm{ A} ( \mathrm{ H}) $ does not exceed $1$.
Let us put $K \mathcal{ H}= \left \{ \mathrm{ H} \in \mathcal{ H}:
\mathrm{rank} \mathrm{ H} \leq 3 \right \} $. For each complex
symmetric $g \times g$--matrix $ \mathrm{ A}= \{a_{k, s} \}$ of rank not
greater $1$, there exists, defined up to sign,
a $g$--dimensional column vector $
\mathbf{ z}= \mathbf{ z} ( \mathrm{ A}) $, such that
$ \mathrm{ A}=-4 \mathbf{ z} \cdot \mathbf{ z}^T$.
Let us introduce vectors
$ \mathbf{ h}_k= \{ \mathrm{ h}_{k, s}; \;s=1, \ldots, g \} \,
\in \mathbb{
C}^g$.
\begin{lemma} \label{geom-2}
Map
\begin{eqnarray*}
& \gamma: K \mathcal{ H} \to ( \mathbb{ C}^g/ \pm) \times
\mathbb{ C}^g \times \mathbb{
C}^g \times \mathbb{ C}^1 \\
& \gamma ( \mathrm{ H}) =
- \left ( \mathbf{ z} \left ( \mathrm{ A} ( \mathrm{ H}) \right) ,
\mathbf{
h}_{g+1}, \mathbf{ h}_{g+2}, \mathrm{h}_{g+1, g+1} \right)
\end{eqnarray*}
is a homeomorphism.
\end{lemma}
\begin{proof} follows from the relation:
\[
4 \hat{ \mathrm{ H}}=4 \mathbf{ z} \cdot \mathbf{ z}^T+2
\left ( \mathbf{ h}_{g+2} \mathbf{ h}_{g+1}^T+ \mathbf{ h}_{g+1}
\mathbf{ h}_{g+2}^T \right) - \mathrm{h}_{g+1, g+1} \mathbf{ h}_{g+2} \mathbf{ h}_{g+2}^T
\]
where $ \hat{ \mathrm{ H}}$ is the matrix composed of the column
vectors $ \mathbf{ h}_k, \, k=1, \ldots, g$, and $ \mathbf{ z}= ( \mathbf{
z} \left ( \mathrm{ A} ( \mathrm{ H}) \right) $.
\end{proof}
Let us introduce the $2$--sheeted ramified
covering
$ \pi:J \mathcal{ H} \to K \mathcal{ H}$, which the
covering $ \mathbb{ C}^g \to ( \mathbb{ C}^g/ \pm) $ induces by the map
$ \gamma$.
\begin{cor}
$ \hat \gamma: J \mathcal{ H} \cong \mathbb{ C}^{3g+1}$.
\end{cor}
Now let us consider the universal space
$W_g$ of $g$--th
symmetric powers of hyperelliptic curves
\[V=
\left \{ (y, x) \in \mathbb{ C}^2: \, y^2=4 x^{2g+1} +
\sum_{k=0}^{2g} \lambda_{2g-k}x^{2g-k} \right \} \] as an algebraic
subvariety in $ ( \mathbb{ C}^2) ^g \times
\mathbb{ C}^{2g+1}$ with coordinates
\[ \left \{
\left ( (y_1, x_1) , \ldots, (y_g, x_g) \right) , \; \lambda_{2g}, \ldots,
\lambda_0 \right \}, \] where $ ( \mathbb{ C}^2) ^g$ is $g$--th
symmetric power of the space $ \mathbb{ C}^2$.
Let us define the map \[ \lambda: J \mathcal{ H} \cong
\mathbb{ C}^{3g+1} \to
( \mathbb{ C}^2) ^g \times \mathbb{ C}^{2g+1} \] in the following way:
\begin{itemize}
\item for
$ \boldsymbol{ G}= ( \mathbf{ z}, \mathbf{ h}_{g+1}, \mathbf{
h}_{g+2}, \mathrm{h}_{g+1, g+1}) \in \mathbb{ C}^{3g+1}$ construct
by Lemma
\ref{geom-2} the matrix $ \pi ( \boldsymbol{ G}) = \mathrm{
H}= \{ \mathrm{ h}_{k, s} \} \in K \mathcal{ H}$
\item put
\[ \lambda ( \boldsymbol{ G}) = \{ (y_k, x_k) ,
\lambda_r; \;k=1, \ldots, g, \, r=0, \ldots, 2g,
\} \]
where $ \{x_1, \ldots, x_g \}$ is the set of roots of the equation
$2 x^g+ \mathbf{ h}_{g+2}^T \mathbf{ X}=0$, and
$y_k= \mathbf{ z}^T \mathbf{ X}_k$, and
$ \lambda_r= \sum_{i+j=r+2} \mathrm{h}_{i, j}$.
\end{itemize}
Here
$ \mathbf{ X}_k= (1, x_k, \ldots, x_k^{g-1}) ^T$.
\begin{theorem}. Map $ \lambda$ induces map
$J \mathcal{ H} \cong \mathbb{ C}^{3g+1} \to W_g$ . \end{theorem}
\begin{proof} Direct check shows, that the identity is valid
\[
\mathbf{ X}_k^T \mathrm{ A}
\mathbf{ X}_s+4 \sum_{i, j=1}^{g+2}
\mathrm{h}_{i, j}x_{k}^{i-1}x_s^{j-1}=0, \]
where $ \mathrm{
A}= \mathrm{ A} ( \mathrm{ H}) $ and $ \mathrm{ H}= \pi ( \boldsymbol{
G}) $. Putting $k=s$ and using $ \mathrm{ A}=4 \mathbf{ z} \cdot
\mathbf{ z}^T$, we have $y_k^2=4
x_k^{2g+1}+ \sum_{s=0}^{2g} \lambda_{2g-s}x_k^{2g-s}$. \end{proof}
Now it is all ready to give the description of our realization
of varieties $T^g= \mathrm{Jac} (V) $ and
$K^g= \mathrm{Kum} (V) $ of the hyperelliptic curves.
For each nonsingular curve
$V= \left \{
(y, x) , y^2=4
x^{2g+1} + \sum_{s=0}^{2g} \lambda_{2g-s}x^{2g-s} \right \} $
define the map
\[ \gamma: \;T^g \backslash ( \sigma) \to \mathcal{
H}: \gamma (u) = \mathrm{ H}= \{ \mathrm{h}_{k, s} \}, \] where
$ \mathrm{h}_{k, s}=4 \wp_{k-1, s-1}-2 ( \wp_{s, k-2}+ \wp_{s-2, k})
+ \frac12
[ \delta_{ks} ( \lambda_{2s-2}+ \lambda_{2k-2}) + \delta_{k+1, s}
\lambda_{2k-1}+ \delta_{k, s+1} \lambda_{2s-1}]$.
\begin{theorem} The map $ \gamma$ induces map
$T^g \backslash ( \sigma) \to K \mathcal{ H}$, such that
$ \wp_{ggk} \wp_{ggs}= \dfrac14 a_{ks} ( \gamma (u) ) $, i.e
$ \gamma$ is lifted to
\[
{ \tilde \gamma}:T^g \backslash ( \sigma) \to J \mathcal{ H} \cong \mathbb{
C}^{3g+1} \; \text{with } \; \mathbf{ z}= ( \wp_{gg1}, \ldots,
\wp_{ggg} ) ^T. \]
Composition of maps $ \lambda{ \tilde
\gamma}: \;T^g \backslash ( \sigma) \to W_g$ defines
the inversion of the Abel map $ \mathfrak{ A}: \; (V) ^g \to T^g$ and,
therefore, the map $ \tilde \gamma$ is an embedding.
\label{geom-3}
\end{theorem}
So we have obtained the explicit realization of the Kummer variety
$T^g \backslash ( \sigma) / \pm$ of the hyperelliptic
curve $V$ of genus $g$ as a subvariety in the
variety of matrices $K \mathcal{ H}$. As a consequence of the Theorem
\ref{geom-3}, particularly, follows the new proof of the theorem
by B.A. Dubrovin and S.P. Novikov about rationality of the
universal space of the Jacobians of hyperelliptic curves
$V$ of genus $g$ with the fixed branching point $e_{2g+2}= \infty$
\cite{dn74}.
\subsection{Hyperelliptic $ \Phi$--function}
In this section we construct the linear
differential operators, for which the hyperelliptic curve
$V (y, x) $ is the spectral variety.
\begin{definition}
$ \Phi$--function of the curve $V (y, x) $ with fixed
point $a$
\begin{eqnarray*}
& \Phi: \mathbb{ C} \times \mathrm{Jac} (V) \times V
\to \mathbb{ C} \\
& \Phi (u_0, \boldsymbol{
u}; (y, x) ) = \dfrac{ \sigma ( \boldsymbol{
\alpha}- \boldsymbol{ u}) }{ \sigma ( \boldsymbol{
\alpha}) \sigma ( \boldsymbol{ u}) } \exp (- \frac{1}{2}y u_0+
\boldsymbol{
\zeta}^T ( \boldsymbol{
\alpha}) \boldsymbol{ u}) ,
\end{eqnarray*}
where $ \boldsymbol{
\zeta}^T ( \boldsymbol{
\alpha}) = (
\zeta_1 ( \boldsymbol{
\alpha}) , \ldots,
\zeta_g ( \boldsymbol{
\alpha}) ) $ and $ (y, x) \in V$,
$ \boldsymbol{ u}$ and $ \boldsymbol{
\alpha}= \int_a^x \mathrm{d} \mathbf{ u}$ $ \in \mathrm{Jac} (V) $.
\end{definition}
Particularly, $ \Phi (0, \boldsymbol{
u}; (y, x) ) $ is the Baker function (see \cite[page 421]{ba97} and
\cite{kr77}) .
\begin{theorem} The function $ \Phi= \Phi (u_0, \boldsymbol{
u}; (y, x) ) $ solves the Hill's equation
\begin{equation}
( \partial_g^2-2 \wp_{gg}) \Phi= (x+ \frac{ \lambda_{2g}}{4}) \Phi,
\label{Hill} \end{equation}
with respect to $u_g$, for all $ (y, x) \in V$.
\end{theorem}
\begin{proof} From \eqref{stickelberger}
\[
\partial_g \Phi= \frac{y+ \partial_g{ \mathcal{ P}} (x;
\boldsymbol{ u}) } {2
\mathcal{ P} (x; \boldsymbol{ u}) } \Phi,
\]
where
$ \mathcal{ P} (x; \boldsymbol{ u}) $ is given by \eqref{p},
hence:
\[
\frac{ \partial_g^2 \Phi}{ \Phi}= \frac{y^2- ( \partial_g \mathcal{
P} (x; \boldsymbol{ u}) ) ^2 +2 \mathcal{ P} (x; \boldsymbol{
u}) \partial_g^2 \mathcal{ P} (x; \boldsymbol{ u}) }{4 \mathcal{
P}^2 (x; \boldsymbol{ u}) } \] and by \eqref{product3} and
\eqref{wpgggi} we obtain the theorem. \end{proof}
Let us introduce the vector $ \boldsymbol{ \Psi}= ( \Phi, \Phi_g) $,
where $ \Phi_g$ stands for $ \partial_g \Phi$. Then equation
\eqref{Hill} may be written as
\begin{equation}
\partial_g \boldsymbol{ \Psi}= L_g \boldsymbol{ \Psi},
\, \text{where} \quad L_g= \begin{pmatrix}
0&1 \\
x+2 \wp_{gg}+ \frac{ \lambda_{2g}}{4}&0
\end{pmatrix}. \label{matrix-Hill} \end{equation}
In regard of \eqref{matrix-Hill} and \eqref{stickelberger}, it is
natural to introduce the family of $g+1$ operators, presented by
$2 \times 2$ matrices, \[ \{ L_0, L_1, \ldots, L_g \}, \quad L_k=
\left ( \begin{array}{rr} V_k&U_k \\W_k&-V_k \end{array} \right)
\]
and defined by the equalities
\[
L_k \boldsymbol{ \Psi}={ \partial_k} \boldsymbol{
\Psi}, \quad k \in 0, \ldots, g.
\]
The theory developed in previous sections leads to the following
description of this family of operators.
\begin{prop} Entries of
the matrices $L_k$ are polynomials in $x$ and $2g$--periodic in
$ \boldsymbol{ u}$: \begin{gather} L_k=D_k L_0- \frac12
\begin{pmatrix} 0&0 \\h_{g+2, k}&0 \end{pmatrix}, \notag \\
\intertext{with}
U_0= \frac12 \sum_{i=1}^{g+2}x^{i-1}h_{g+2, i}, \quad
V_0=- \frac14 \sum_{i=1}^{g+2}x^{i-1} \partial_g h_{g+2, i}, \\
\quad \text{and} \quad W_0= \frac14 \sum_{i=1}^{g+2}x^{i-1} \det
\begin{pmatrix} h_{g+1, i}&h_{g+2, g} \\h_{g+2, i}&h_{g+2,
g+1} \notag \end{pmatrix}. \label{entries} \end{gather} And
the compatibility conditions \[ [L_k, L_i]= \partial_k L_i -
\partial_i L_k \] are satisfied. \end{prop} Here $D_k$ is
umbral derivative (see page~ \pageref{umbral_D}) . Proof is
straightforward due to \eqref{stickelberger}, \eqref{variables},
\eqref{dg_H} and \eqref{wpggiwpggk}.
\begin{theorem} \label{the-F} The function
$ \Phi= \Phi (u_0, \boldsymbol{ u}; (y, x) ) $ solves the system of
equations \begin{gather} \left ( \partial_k \partial_l
- \gamma_{kl} (x, \boldsymbol{ u}) \partial_g +
\beta_{kl} (x, \boldsymbol{ u})
\right) \Phi= \tfrac{1}{4}D_{k+l} \big (f (x) \big) \Phi
\notag \\ \intertext{with polynomials in $x$} \begin{aligned}
\gamma_{kl} (x, \boldsymbol{ u}) &=
\tfrac{1}{4} \left[
\partial_k D_l +
\partial_l D_k
\right] \sum_{i=1}^{g+2}x^{i-1}h_{g+2, i} \quad \text{and} \\
\beta_{kl} (x, \boldsymbol{ u}) &= \tfrac{1}{8} \left[
( \partial_g \partial_k +h_{g+2, k} ) D_l +
( \partial_g \partial_l +h_{g+2, l} ) D_k
\right] \sum_{i=1}^{g+2}x^{i-1}h_{g+2, i} \\
&- \frac14 \sum \limits_{j=k+l+2}^{2g+2}x^{j- (k+l+2) }
\left[ \left ( \sum \limits_{ \nu=1}^{k+1} h_{ \nu, j- \nu}
\right) + \left ( \sum \limits_{ \mu=1}^{l+1} h_{j- \mu,
\mu} \right) \right] \end{aligned} \notag \end{gather} for
all $k, l \in 0, \ldots, g$ and arbitrary $ (y, x) \in V$.
\end{theorem}
Here $f (x) $ is as given in \eqref{curve} with
$ \lambda_{2g+2}=0$ and $ \lambda_{2g+1}=4$.
\begin{proof}
Construction of operators $L_k$ yields
\[
\Phi_{lk}= \tfrac12 \left (
\partial_l U_k+
\partial_k U_l \right) { \Phi_g}+ \left (V_lV_k + \tfrac12
( \partial_l V_k+ \partial_k
V_l+U_k W_l+W_k U_l) \right) \Phi.
\]
To prove the theorem we use \eqref{entries}, and it only remains
to notice, that (cf. Lemma \ref{geom-2}) :
\begin{eqnarray*}
&&D_k (V_0) D_l (V_0) + \tfrac12 D_k (U_0) D_l (W_0)
+ \tfrac12 D_l (U_0) D_k (W_0) = \\&&
- \tfrac{1}{16} \left ( \det
H \left[{}_{g+1}^{g+1}{}_{g+2}^{g+2} \right]
\right)
(1, x, \ldots, x^{g+1-k})
H \left[{}_{k}^{l}{}_{ \ldots}^{ \ldots}{}_{g+2}^{g+2} \right]
(1, x, \ldots, x^{g+1-l}) ^T,
\end{eqnarray*}
having in mind that $h_{g+2, g+2}=0$ and $h_{g+2, g+1}=2$, we obtain
the theorem due to properties of matrix $H$.
\end{proof}
Consider as an example the case of genus $2$.
\begin{eqnarray*}
&& ( \partial_2^2-2 \wp_{22})
\Phi= \tfrac{1}{4} (4 x+ \lambda_4) \Phi, \\
&& ( \partial_2 \partial_1+ \tfrac{1}{2} \wp_{222} \partial_2
- \wp_{22} (x+ \wp_{22}+
\tfrac{1}{4} \lambda_4) +2 \wp_{12})
\Phi= \tfrac{1}{4} (4 x^2+ \lambda_4 x + \lambda_3) \Phi, \\
&& ( \partial_1^2+ \wp_{122} \partial_2
-2 \wp_{12} (x+
\wp_{22}+ \tfrac{1}{4} \lambda_4) )
\Phi=
\tfrac{1}{4} (4 x^3+ \lambda_4 x^2 + \lambda_3 x+ \lambda_2) \Phi.
\end{eqnarray*}
And the $ \Phi= \Phi (u_0, u_1, u_2; (y, x) ) $ of the curve $y^2=4
x^5+ \lambda_4 x^4 + \lambda_3 x^3+ \lambda_2x^2+ \lambda_1
x+ \lambda_0$ solves these equations for all $x$.
The most remarkable of the equations of Theorem \ref{the-F} is the
balance of powers of the polynomials $ \gamma_{kl}$, $ \beta_{kl}$
and of the ``spectral part'' --- the umbral derivative
$D_{k+l} (f (x) ) $:
\begin{align*}
& \mathrm{deg}_x \gamma_{kl} (x, \boldsymbol{ u}) \leqslant
g-1- \mathrm{min} (k, l) , \\& \mathrm{deg}_x \beta_{kl} (x, \boldsymbol{
u}) \leqslant 2g- (k+l) , \\
& \mathrm{deg}_x D_{k+l} (f (x) ) =2g+1- (k+l) .
\end{align*}
\subsection{Solution of KdV equations
by Kleinian functions}
The KdV system is the infinite hierarchy of differential equations
\[
u_{t_{k}} = \mathcal{ X}_{k}[u],
\]
the first two are
\[u_{t_{1}} =u_x , \quad \text{and} \quad u_{t_{2}} =
- \tfrac12 (u_{xxx}-6u u_{x}) , \]
and the higher ones are defined by the relation
\[
\mathcal{ X}_{k+1}[u]= \mathcal{ R} \mathcal{ X}_{k}[u],
\]
where $ \mathcal{ R}=- \frac12 \partial_x^2+2u+u_x \partial_x^{-1}$ is the Lenard's
recursion operator.
Identifying time variables $ (t_1=x, t_2, \ldots, t_g) \to (u_{g},
u_{g-1}, \ldots, u_1) $ we have
\begin{prop}
The function $u=2 \wp_{gg} ( \boldsymbol{ u}) $ is a $g$--gap
solution of the KdV system.
\end{prop}
\begin{proof}
Really, we have $u_x= \partial_g 2 \wp_{gg}$ and
by \eqref{wpgggi}
\[u_{t_{2}}= \partial_{g-1} 2 \wp_{gg}=
- \wp_{ggggg}+12 \wp_{gg} \wp_{ggg}. \]
The action of $ \mathcal{ R}$
\[ \partial_{g-i-1}2 \wp_{gg}= \left[-
\partial_{g}^2+8 \wp_{gg} \right] \wp_{gg, g-i}
+4 \wp_{g, g-i} \wp_{ggg} \]
is verified by \eqref{wpgggi} and \eqref{wp3}.
On the $g$--th step of recursion the ``times'' $u_i$ are
exhausted and the stationary equation \[ \mathcal{ X}_{g+1}[u]=0
\] appears. A periodic solution of $g+1$ higher stationary
equation is a $g$--gap potential (see \cite{dmn76}) .
\end{proof}
\section*{Concluding remarks}
The Kleinian theory of hyperelliptic
Abelian functions as, the authors hope, this paper shows is an
important approach alternative to the generally adopted formalism
based directly on the multidimensional $\theta$--functions in
various branches of mathematical physics. Still, a number of
remarkable properties of the Kleinian functions were left beyond
the scope of our paper. We give some instructive examples for the
case of genus two ${\boldsymbol u}=\{u_1, u_2\}$).
\begin{itemize} \item the
addition theorem \[ \frac{\sigma({\boldsymbol u}+{\boldsymbol
v})\sigma({\boldsymbol u}-{\boldsymbol v})}{\sigma^2({\boldsymbol
u})\sigma^2({\boldsymbol v})}= \wp_{22}({\boldsymbol
u})\wp_{12}({\boldsymbol v})-\wp_{12}({\boldsymbol u})\wp_{22}({\boldsymbol
v})+\wp_{11}({\boldsymbol v})-\wp_{11}({\boldsymbol u}) ,
\]
\item the equation, capable of being interpreted as the
Hirota bilinear relation:
\[
\left\{\frac13\Delta\Delta^T+\Delta^T\left(\begin{array}{ccc}
0&0&1\\0&-\frac12&0\\1&0&0\end{array}\right)\epsilon_{\eta,\eta}
\epsilon_{\eta,\eta}\cdot\epsilon_{\eta,eta}^T-(\xi-\eta)^4
\epsilon_{\eta,\xi}\epsilon_{\eta,\xi}^T\right\}\sigma({\boldsymbol u})
\sigma({\boldsymbol u}')\Big|_{{\boldsymbol u}'={\boldsymbol u}},
\]
is identically $0$, where
$\Delta^T=(\Delta_1^2,2\Delta_1\Delta_2,\Delta_2^2)$ with
$\Delta_i=\frac{\partial}{\partial u_i}-\frac{\partial}{\partial
u_i'}$ and also $\epsilon^T_{\xi,\eta}=(1,\eta+\xi,\eta\xi)$.
After evaluation the powers of parameters $\eta$ and $\xi$ are
replaced according to rules $\eta^k,\xi^k \to
\lambda_k\frac{k!(6-k)!}{6!}$ by the constants defining the curve.
\item for the Kleinian $\sigma$--functions the operation is
defined
\[ \sigma(u_1,u_2)=\mathrm{exp}
\left\{\frac{u_2}{u_1} \sum_{k=1}^6 k
\lambda_k\frac{\partial}{\partial \lambda_{k-1}}\right\}
\sigma(u_1,0), \]
which resembles the function executed by vertex operators.
\end{itemize}
We give these formulas with reference to \cite{ba07}.
Another interesting problem is the reduction of
hyperelliptic $\wp$--functions to lower genera. In the case of
genus two it, happens according to the Weierstrass theorem when the
period matrix $\tau$ can be transformed to the form (see e.g.
\cite{ba97,hu94})
\[
\tau=\left(\begin{array}{cc}\tau_{11}&\frac{1}{N}\\
\frac{1}{N}&\tau_{22}\end{array}\right),
\]
where so called {\it Picard number} $N>1$ is a positive integer.
The associated Kummer surface turns in this case to {\it
Pl\"ucker surface}. The reductions of the like were studied in
\cite{bbeim94} in order to single out elliptic potentials among
the finite gap ones. The problems of this kind were treated in
\cite{gw95, gw95a,gw95b} by means of the spectral theory. We
remark that the formalism of Kleinian functions extremely
facilitates the related calculations and makes the solution more
descriptive.
These and other problems of hyperelliptic abelian functions will
be discussed in our forthcoming publications.
Concluding we emphasize, that the Kleinian construction of
the hyperelliptic Abelian functions does not
exclude the theta functional realization but complements it, and
to the authors' experience the combination of the both approaches
makes the whole picture more complete and descriptive.
\section*{Acknowledgments}
The authors are grateful to S.P. Novikov for the attention and
stimulating discussions; we are also grateful to I.M. Krichever,
S.M. Natanson and A.P. Veselov for the valuable discussions.
Special thanks to G. Thieme for the help in the collecting
the classical German mathematical literature.
The research described in this publication was supported in
part by grants no. M3Z000 (VMB) and no. U44000 (VZE) from the
International Science Foundation and also the INTAS grant
no. 93-1324 (VZE and DVL),
and grant no. 94-01-01444 from Russian Foundation of Fundamental
Researches.
| 39,913 |
\section{Introduction}
The measurement of the inclusive deep inelastic lepton-proton
scattering cross section has been of great importance for
the understanding of the substructure of the proton~\cite{JOEL}.
Experiments at HERA extend the previously
accessible kinematic range up to very large squared momentum transfers,
$Q^2 > 5 \cdot 10^4$~GeV$^2$, and to very small values of Bjorken
$x < 10^{-4}$.
The first observations showed a rise of the proton
structure function $F_2(x,Q^2)$ at low $x < 10^{-2}$ with decreasing
$x$~\cite{H1F293,ZEUSF293}, based on data collected in 1992.
This rise was confirmed with the more precise data of 1993
\cite{H1F294,ZEUSF294}.
Such a behaviour is qualitatively expected in the double leading log
limit of Quantum Chromodynamics~\cite{ALVARO}. It is, however, not
clarified whether the linear QCD evolution equations, as the
conventional DGLAP evolution~\cite{DGLAP} in $\ln Q^2$ and/or the BFKL
evolution~\cite{BFKL} in $\ln(1/x)$, describe the rise of $F_2$ or
whether there is a significant effect due to
nonlinear parton recombination \cite{GLR}.
Furthermore, it is unclear whether this rise will persist at low $Q^2$, say
of the order of a few GeV$^2$. For example
Regge inspired models expect $F_2$ to be
\end{titlepage}
flatter for small $Q^2$.
The
quantitative investigation of the quark-gluon interaction dynamics at
low $x$ is one of the major goals of HERA. It
requires high precision for the
$F_2$ measurement and a detailed study of the hadronic final
state behaviour~\cite{H1BFKL}.
The structure functions $F_1(x,Q^2)$, $F_2(x,Q^2)$ and $F_3(x,Q^2)$
are related to the inclusive lepton-photon cross-section
\begin{equation}
\frac{d^2\sigma^{e^+p}}{dxdQ^2} =
\frac{2\pi\alpha^2}{x Q^4}
\left[2xF_1(x,Q^2)+2(1-y)F_2(x,Q^2)
-\left(2y-\frac{y^2}{2}\right)xF_3(x,Q^2)\right]
\end{equation}
and depend on the squared four
momentum transfer $Q^2$ and the scaling variable $x$. These variables
are related to the inelasticity parameter $y$ and to the total squared
centre of mass energy of the collision $s$ as $Q^2= xys$ with $s= 4
E_e E_p$.
However, at low $Q^2$ $xF_3$ can be neglected and the previous
expression can be rewritten as a function of $F_2$ and $R$
\begin{equation}
R \equiv \frac{F_2-2xF_1}{2xF_1} \equiv \frac{F_L}{2xF_1}.
\end{equation}
$R(x,Q^2)$ could not be measured yet at HERA, but can be computed,
supposing that perturbative QCD hold, for a given set of
parton densities. Thus $F_2(x,Q^2)$
can be derived from the double differential cross-section $d^2\sigma/dx dQ^2$
after experimental and QED radiative corrections.
The structure function $xF_3(x,Q^2)$ has not been measured yet, due to
lack of statistics at high $Q^2$. However, simple differential
cross-section $d\sigma/dQ^2$ both on neutral currents (exchange
of a $\gamma$ or $Z^0$) or in charged current (exchange of $W^{\pm}$)
have already been published \cite{H1CC,ZEUSCC}. In the rest of this
paper we will consider only the case of the $\gamma$ exchange.
In 1994
the incident electron energy was $E_e = 27.5$~GeV and the
proton energy was $E_p=820$~GeV. The data were recorded with the H1
\cite{H1DET} and ZEUS \cite{ZEUSDET} detectors.
A salient feature of the HERA collider
experiments is the possibility of measuring not only the scattered
electron but also the complete hadronic final state, apart from losses
near the beam pipe.
This means that the kinematic variables $x,~y$ and $Q^2$
can be determined with complementary methods which
experimentally are sensitive to different systematic effects.
The comparison of the results obtained with different methods
improves the accuracy of the $F_2$ measurement. A convenient
combination of the results ensures maximum coverage of the available
kinematic range.
In this paper after a description of the data samples (section 2) and of
the kinematic reconstruction/event selection used (section 3) we
provide the $F_2$ measurement in section 4. and its interpretation
at low $Q^2$ and in terms of perturbative QCD in section 5, before
giving some prospects in conclusion.
\section{Data Samples}
In 1994 both experiments have reduced the minimum $Q^2$ at which they
could measure $F_2$ using several techniques.
For DIS events at low $Q^2$ the electron is scattered under a large
angle
$\theta_e$ ( the polar angles $\theta$ are defined w.r.t
the proton beam direction, termed "forward" region).
Therefore the acceptance of electrons in the backward region has to be
increased or the incident electron energy to be reduced
to go down in $Q^2$. This was realized as follows.
i) both experiments were able to diminish the region around the backward
beam pipe in which the electron could not be measured reliably in 93,
thus increasing the maximum polar angle of the scattered electron
(cf \cite{H1F295,ZEUSF295} for details).
This large statistic sample, taken with the nominal HERA conditions
is called the "nominal vertex" sample.
Its integrated luminosity is between 2 and 3~pb$^{-1}$, depending
on the analysis/experiment.
ii) Following a pilot exercise performed last year,
58~nb$^{-1}$ of data was collected for which the
interaction point was shifted by +62~cm,
in the forward direction,
resulting in an increase of the electron acceptance.
This sample is refered to as the "shifted vertex" data sample.
In H1 the low $Q^2$ region was also covered by analyzing events which
originated from the
``early'' proton satellite bunch, present during all periods of the
HERA operation, which
collided with an electron bunch at a position
shifted by $+$63~cm. These data, refered
to as the
"satellite" data sample
amount to $\simeq 3\%$ of the total data and correspond to a
total "luminosity" of
68~nb$^{-1}$.
iii) Both experiments used DIS events which underwent initial state
photon radiation detected in an appropriate photon tagger
to measure $F_2$ at lower $Q^2$ (so called "radiative" sample
\cite{H1RAD}). The incident electron energy which participate in the hard
scattering is thus reduced, and so is the $Q^2$.
The luminosity was determined from the measured cross section of
the Bethe Heitler reaction $e^-p \rightarrow e^-p\gamma$, measuring the
hard photon bremsstrahlung data only.
The precision of the luminosity for the nominal vertex position data
amounts to 1.5\%, i.e. an improvement of a factor 3 w.r.t the analysis
of the 1993 data. For the shifted vertex data the luminosity uncertainty
is higher (4.5\% for H1).
The luminosity
of the satellite data sample was obtained from the measured luminosity
for the shifted vertex data multiplied by the efficiency corrected
event ratio in a kinematic region common to both data sets.
The uncertainty of that luminosity determination was estimated to be
5\%.
\section{Kinematics and Event Selection}
The kinematic variables
of the inclusive scattering process $ep \rightarrow eX$
can be reconstructed in different ways using measured quantities from
the hadronic final state and from the scattered electron.
The choice of the reconstruction method for $Q^2$ and $y$
determines the size of systematic errors, acceptance and radiative
corrections. The basic formulae for $Q^2$ and $y$ used in the different
methods are summarized below, $x$ being obtained from $Q^2=xys$.
For the electron method
\begin{equation}
y_e =1-\frac{E'_e}{E_e} \sin^{2}\frac {\theta_e} {2}
\hspace*{2cm}
Q^2_e = \frac{E^{'2}_e \sin^2{\theta_e}}{ 1-y_e}
\end{equation}
The resolution in $Q^2_e$ is $4\%$ while the $y_e$
measurement degrades as $1/y_e$ and cannot be used
for $y_e \leq 0.05$. In the low $y$ region it is, however, possible
to use the hadronic methods for which it is convenient to define the
following variables
\begin{equation}
\Sigma=\sum_h{E_h-p_{z,h}}.
\hspace*{2.cm}
p^{h~2}_T=(\sum_h{p_{x,h}})^2+(\sum_h{p_{y,h}})^2
\end{equation}
Here $E,p_x,p_y,p_z$ are the four-momentum vector components
of each particle and the summation is over all hadronic final
state particles.
The standard definitions for $y_h$ and
$\theta_h$ are
\begin{equation}
y_{h} = \frac{\Sigma}{2E_e}
\hspace*{3.cm}
\tan \frac{\theta_h}{2}=\frac{\Sigma}{p^h_T}
\end{equation}
The combination of $y_h$ and $Q^2_e$ defines the mixed method which
is well suited for medium and low $y$ measurements.
The same is true for the double-angle method which makes use
only of $\theta_e$ and $\theta_h$ and is thus insensitive to
the absolute energy calibration:
\begin{equation}
y_{DA} = \frac{\tan\frac{\theta_h}{2}}
{\tan\frac{\theta_e}{2}+\tan\frac{\theta_h}{2}}
\hspace*{2cm}
Q^2_{DA} = 4 E_e^2 \frac{\cot\frac{\theta_e}{2}}
{\tan\frac{\theta_e}{2}+\tan\frac{\theta_h}{2}}
\end{equation}
The formulae for the $\Sigma$ method
are constructed requiring $Q^2$ and $y$
to be independent of the incident electron energy.
Using the conservation of the total $E-P_z \equiv
\sum_i{E_i-p_{z,i}}$, the sum extending over
all particles of the event, $2E_e$ is replaced by
$ \Sigma + E'_e(1-\cos{\theta_e})$ which gives
\begin{equation}
y_{\Sigma} = \frac{\Sigma}{ \Sigma + E'_e(1-\cos{\theta_e})}
\hspace*{2cm}
Q^2_{\Sigma} = \frac{E^{'2}_e \sin^2{\theta_e}}{ 1-y_{\Sigma}}
\end{equation}
By construction $y_{\Sigma}$ and $Q^2_{\Sigma}$ are independent of
initial state photon radiation.
With respect to $y_h$ the modified quantity $y_\Sigma$ is less
sensitive to the hadronic measurement at high $y$, since the $\Sigma$
term dominates the total $E-P_z$ of the event. At low $y$,
$y_h$ and $y_{\Sigma}$ are equivalent.
H1 measures $F_2$ with the electron and the $\Sigma$ method and after
a complete consistency check uses the electron method for $y>0.15$
and the $\Sigma$ method for $y<0.15$. ZEUS measures $F_2$ with the
electron and the double angle method and for the final results
uses the electron method at low $Q^2$ and the double angle method
for high $Q^2$ ($Q^2 >$ 15~GeV$^2$). The binning in $x$ and $Q^2$
was chosen to match the resolution in these variables.
It was set at 5(8) bins per order in magnitude in x($Q^2$) for the
H1 experiment and about twice as coarse for the ZEUS experiment in
$x$ (similar to H1 in $Q^2$).
The event selection is similar in the two experiments. Events are
filtered on-line using calorimetric triggers which requests an
electromagnetic cluster of at least 5~GeV not vetoed by a trigger
element signing a beam background event. Offline, further
electron identification criteria are applied (track-cluster link,
shower shape and radius) and a minimum energy of 8(11)~GeV is
requested in ZEUS(H1). H1 requests a reconstructed vertex
within 3$\sigma$ of the expected interaction position, while ZEUS
requires that the quantity $\delta = \Sigma +E'_e(1-\cos\theta)$
satisfies 35~GeV $< \delta <$ 65~GeV. If no particle escapes detection,
$\delta = 2 E =$ 55~GeV,
so the $\delta$ cut reduces the photoproduction background and the
size of the radiative corrections. The only significant background
left after the selection comes from photoproduction in which
a hadronic shower fakes an electron. In H1 for instance,
It has been estimated
consistently
both from the data and from Monte Carlo simulation and amounts to
less than 3\% except in a few bins where it can reach values
up to 15\%. It is subtracted statistically bin by bin and an error of
30\% is assigned to it.
The acceptance and the response of the detector has been studied and
understood in great detail by the two experiments: More
than two millions Monte Carlo DIS events were generated using
DJANGO~\cite{DJANGO} and different quark distribution
parametrizations, corresponding to an integrated luminosity of
approximately $20$~pb$^{-1}$. The program is based on
HERACLES~\cite{HERACLES} for the electroweak interaction
and on LEPTO~\cite{LEPTO} to simulate the hadronic
final state. HERACLES includes first order radiative corrections,
the simulation of real bremsstrahlung photons and the longitudinal
structure function. The acceptance corrections were performed using the
GRV~\cite{GRV}
or the MRS parametrization~\cite{MRSH}, which both
describe rather well the HERA
$F_2$ results of 1993 for $Q^2 > 10$~GeV$^2$.
LEPTO uses the colour dipole model (CDM) as implemented in
ARIADNE~\cite{CDM} which is in good agreement with data on the energy
flow and other characteristics of the final
state as measured by H1~\cite{H1FLOW} and ZEUS~\cite{ZEUSFLOW}. For the
estimation of systematic errors connected with the topology
of the hadronic final state, the HERWIG model~\cite{HERWIG} was
used in a dedicated analysis. Based on the GEANT program~\cite{GEANT}
the detector response was simulated in detail.
After this step the Monte Carlo events were subject to the same
reconstruction and analysis chain as the real data.
\section{Structure Function Measurement}
The structure function $F_2(x,Q^2)$
was derived after radiative corrections
from the one-photon exchange cross section
\begin{equation}
\frac{d^2\sigma}{dx dQ^2} =\frac{2\pi\alpha^2}{Q^4x}
(2-2y+\frac{y^2}{1+R}) F_2(x,Q^2)
\label{dsigma}
\end{equation}
Effects due to $Z$ boson exchange are smaller than 2\%.
The structure function ratio $R=F_2/2xF_1 - 1 $
has not been measured yet at HERA and
was calculated using the QCD relation~\cite{ALTMAR} and the
GRV structure function parametrization.
Compared to the 1993 data analyses~\cite{H1F294, ZEUSF294}
the $F_2$ measurement has been extended to lower
$Q^2$ (from $4.5$~GeV$^2$ to $1.5$~GeV$^2$),
and to lower and higher $x$ (from $1.8 \cdot 10^{-4}-0.13$ to
$5 \cdot 10^{-5}-0.32$).
The determination of the structure function requires
the measured event numbers to be converted
to the bin averaged cross section
based on the Monte Carlo acceptance calculation.
All detector efficiencies were determined from the data
using the redundancy of the apparatus. Apart from very small
extra corrections all efficiencies are correctly
reproduced by the Monte Carlo simulation.
The bin averaged cross section was
corrected for higher order QED radiative contributions and a
bin size correction was performed. This determined
the one-photon exchange cross section
which according to eq.\ref{dsigma}
led to the values for $F_2(x,Q^2)$.
Due to the different data sets available: "nominal vertex" data,
"radiative events", "shifted vertex" data and "satellite" data,
which have different acceptances and use
for a given $Q^2, x$ point different parts of the detectors, cross
checks could be made in kinematic regions of overlap.
The results were found to be in very good agreement
with each other for all kinematic reconstruction methods used.
The large available statistics allows to make
very detailed studies on the
detector response: efficiencies and calibration. As a result the
systematic errors on many effects are reduced, compared to
the 1993 data analysis.
Here only a brief summary of these preliminary errors is given,
refering the
reader to the original and forthcoming $F_2$ publications: For the
electron method the main source of error are the energy calibration
(known at 1.5\% level in 1994), the knowledge of the electron
identification efficiency and to a lesser extent the error on the polar
angle of the scattered electron, ($\delta\theta=$1mrad) in
particular at the lowest $Q^2$ and the radiative corrections at
low $x$.
For the $\Sigma$ method, the knowledge of the absolute energy scale
for the hadrons, the fraction of hadrons which stay undetected
in particular at low $x$, due to calorimetric thresholds
and to a lesser extent the electron energy calibration are the
dominating factors. For the double angle method, the major problem
comes from the precision in the resolution of the hadronic angle
at low $x$ and low $Q^2$. Further uncertainties common to all methods
(selection, structure function dependance etc.) were also studied.
The preliminary error on the 1994 data ranges between 10 and 20\%
with expected final values for publication below 10\%.
\begin{figure}[htbp]
\begin{center}
\epsfig{file=fig1.ps,width=17.5cm
bbllx=15pt,bblly=140pt,bburx=600pt,bbury=690pt}
\end{center}
\caption[]{\label{f2x}
\sl Preliminary measurement of the proton structure function $F_2(x,Q^2)$
as function of $x$ in different bins of $Q^2$. The
inner error bar is the statistical error. The full error
represents the statistical and systematic errors added
in quadrature.}
\end{figure}
The preliminary results of H1 and ZEUS are shown in fig.~\ref{f2x}, in bins
of fixed $Q^2$. The rise of $F_2$ at low $x$ is confirmed with the
higher precision, and is now observed down to the lowest $Q^2$
measured (1.5~GeV$^2$). The observed good agreement between H1 and ZEUS
and the smooth transition between the HERA and the fixed target (E665,
NMC) data consolidates this result which can thus be confronted to
theoretical expectations. The steepness of the low $x$ rise increases
visibly with the $Q^2$, a characteristic expected from perturbative
QCD. This rise cannot be attributed to the presence of
"diffractive" events in the DIS sample since their proportion has
been shown to stay essentially constant (~10\%) independently
of $x$ and $Q^2$ \cite{H1DIFF,ZEUSDIFF}.
\section{Low $Q^2$ and Perturbative QCD}
A test of perturbative QCD is displayed in fig.~\ref{f2q} which represents
the results of Next to Leading Order (NLO) QCD fit performed by H1
as explained in \cite{H1QCD} on the data with $Q^2 \le $ 5~GeV$^2$.
In order to constrain the structure function $F_2$ at high $x$,
data from the fixed
target scattering experiment NMC \cite{NMC} and BCDMS \cite{BCDMS}
are used, avoiding
regions where higher twist and target mass effects could become
important.
\begin{figure}[htb]
\begin{center}
\epsfig{file=fig2.ps,height=9.5cm
bbllx=15pt,bblly=150pt,bburx=570pt,bbury=695pt}
\end{center}
\caption[]{\label{f2q}
\sl Preliminary measurement of $F_2(x,Q^2)$.
The H1 data is consistent with the fixed target experiments
BCDMS and NMC. The curve represents the NLO QCD-fit decribed in the text.
The gap visible around 100 GeV$^2$ corresponds to a boundary region
between two calorimeters in the H1 detector, which is not completely
analyzed yet.}
\end{figure}
The $F_2$ behaviour can be well described by the DGLAP
evolution equations within the present preliminary errors. The data
for $Q^2$ values below 5~GeV$^2$ are also
compatible with the extrapolation of the fit in this region,
as can be seen in the figure.
This preliminary result is consistent with the published QCD analysis
of H1 and ZEUS 1993 data, which allowed to determine the gluon
density in the proton, and observe its steep rise at low $x$ as
displayed in fig.~\ref{glu} for $Q^2=$ 20~GeV$^2$.
\begin{figure}[htb]
\begin{center}
\epsfig{file=fig3.ps,height=10cm
bbllx=90pt,bblly=220pt,bburx=500pt,bbury=640pt}
\end{center}
\caption[]{\label{glu}
\sl Preliminary measurement of the gluon density from NLO QCD
fit. The H1 result is shown with the complet systematic
error band, for ZEUS only the value of the gluon density is shown,
the error band is similar to the H1 one.}
\end{figure}
Another test of perturbative QCD lies in observing the asymptotic
behaviour as suggested by early studies \cite{ALVARO}.
Ball and Forte ~\cite{BALL} have recently shown
that evolving a flat input distribution at some $Q_0$, of the order
of 1~GeV$^2$,
with the DGLAP equations
leads to a strong rise of $F_2$ at low $x$
in the region measured by HERA. An interesting feature is that if QCD
evolution is the underlying dynamics of the rise, perturbative QCD predicts
that at large $Q^2$ and small $x$
the structure function
exhibits double scaling in the two variables $\sigma$ and $\rho$ defined
as:
\begin{equation}
\sigma \equiv \sqrt{\log(x_0/x)\cdot \log(t/t_0)}, \ \ \
\rho \equiv \sqrt{\frac{\log(x_0/x)}{\log(t/t_0)}} \ \ \
\mbox{with} \ \ t\equiv \log(Q^2/\Lambda^2)
\end{equation}
In figure~\ref{ball05} the H1 data are presented in the
variables $\sigma$ and $\rho$, taking
the boundary conditions to be $x_0=0.1$ and
$Q^2_0=0.5~{\rm GeV}^2$, and $\Lambda^{(4)}_{\rm LO}=185~{\rm MeV}$.
In a previous analysis \cite{H1QCD} the value $Q^2_0=1~{\rm GeV}^2$ was
chosen,
but the new low $Q^2$ data seems to indicate that $F_2$ is not yet flat
for this $Q^2$ value.
The measured values of $F_2$ are rescaled by
\begin{equation}
R'_F(\sigma,\rho) = 8.1\: \exp \left(\delta \frac{\sigma}{\rho}+\frac{1}{2}
\log(\sigma) +
\log (\frac{\rho}{\gamma})\right),
\end{equation}
to remove the part of the leading subasymptotic behaviour which can be
calculated in a model independent way;
$\log(R'_F F_2)$ is then predicted to rise linearly with $\sigma$.
Scaling in $\rho$ can be shown by multiplying $F_2$
by the factor $R_F\equiv R'_Fe^{- 2\gamma \sigma}$.
Here $\gamma \equiv 2\sqrt{3/b_0}$ with
$b_0$ being the leading order coefficient of the $\beta$ function
of the QCD renormalization group equation for four flavours,
$\delta = 1.36$ for four flavours and three colours.
\begin{figure}[htb]
\begin{center}
\epsfig{file=fig4.ps
height=7cm,bbllx=20pt,bblly=230pt,bburx=530pt,bbury=590pt}
\end{center}
\caption[]{\label{ball05}
\sl The rescaled structure functions
$R_F F_2$ and $R'_FF_2$ (preliminary) plotted versus the variables
$\rho$ and $\sigma$ defined in the text. Only data with $\rho>1.5$ are
shown in b.}
\end{figure}
Fig.~\ref{ball05}a shows $R_FF_2$ versus $\rho$.
Scaling roughly sets in for
$\rho \ge 1.5$.
Fig.~\ref{ball05}b, for $\rho \ge 1.5$, shows
scaling behaviour, namely a linear of
$\log(R'_F F_2)$ with $\sigma$.
\begin{figure}[htbp]
\begin{center}
\epsfig{file=fig5.ps,width=13.5cm
bbllx=40pt,bblly=50pt,bburx=545pt,bbury=780pt}
\end{center}
\caption[]{\label{f2grv}
\sl Preliminary measurement of the proton structure function $F_2(x,Q^2)$
by H1 and ZEUS, compared to the
results of the E665 and NMC experiments and to the ``prediction'' of the
GRV model \cite{GRV} over the full $Q^2$ range.}
\end{figure}
These observations suggest that perturbative QCD could be already
valid at $Q^2=$ 1 or 2~GeV$^2$. Indeed within the present
precision we can observe in fig.~\ref{f2grv} the validity of the
Gl\"uck, Reya, Vogt (GRV) model \cite{GRV} which assumes that all low
$x$ partons are generated "radiatively" starting from a very low
initial $Q^2 =$ 0.34~GeV$^2$ scale, in which both gluon and quark
densities are "valence" like. This result appears surprising since
perturbative QCD does not apply at such a low scale, but the HERA
results and the E665 \cite{E665} preliminary results follow the GRV
expectations as early as 0.8~GeV$^2$. More precise data are needed to
further constrain the model and draw definite conclusions
on the dynamics underlying the low $x$ rise. Nevertheless these results
appear already very promising for the DGLAP evolution equations
which might not need to be supplemented by the BFKL
evolution at low $x$, in the HERA kinematic domain.
Focusing now on low $Q^2$, the persistent rise of $F_2$
at low $x$, when going down in $Q^2$
indicates that the photoproduction regime has not been
reached yet.
This can be seen in fig.~\ref{gp} which display the behaviour of the total
cross-section of the proton-virtual photon system as a function of
$W$, the invariant mass of the $\gamma^* p$ system
(at low $x$, $W \simeq \sqrt{Q^2/x}$). $F_2$ is
related to the total cross-section of the proton-virtual photon
interaction ($\sigma_{tot}(\gamma^* p)) $ via
\begin{equation}
\sigma_{tot}(\gamma^* p) \simeq \frac{4~\pi^2 \alpha}{Q^2} F_2(W,Q^2).
\end{equation}
The $\sigma_{tot}$ growth can be contrasted with the weak rise with
$W$ of the total real photoproduction cross-section
in the same range of $W$ : 20-250~GeV
\cite{H1STOT,ZEUSSTOT}
\begin{figure}[hb]
\begin{center}
\epsfig{file=fig6.ps,height=11cm
bbllx=10pt,bblly=130pt,bburx=550pt,bbury=690pt}
\end{center}
\caption[]{\label{gp}
\sl Preliminary measurement of the cross-section
$\sigma_{tot}(\gamma^{\ast} p)$ as a function of $W$ and $Q^2$.
Results from DIS are compared with the measurements in photoproduction.
For the readability of the plot, not all $Q^2$ bins are shown.}
\end{figure}
The Regge inspired models DOLA~\cite{DOLA}
and CKMT\cite{CTKM} which can describe
the behaviour of $\sigma_{tot}(\gamma p)$ predicts a rather flat
behaviour of $F_2$ at a few GeV$^2$. As shown in fig.~\ref{lowq2},
the DOLA model clearly fails before 1.5~GeV$^2$, while the CKMT model which
assumes that the "bare" pomeron visible at high $Q^2$ has a higher
trajectory intercept ($\sim 0.24$) than the "effective" pomeron involved in
"soft" interactions ($\sim 0.08$) undershoot the data in a less critical
manner. In this same plot we can also notice the similarity above
5~GeV$^2$ between the different parametrizations (GRV, MRS, CTEQ
\cite{GRV, MRSA, CTEQ} )
which use essentially the same data to determine their parton
distributions at the reference scale.
\begin{figure}[htb]
\begin{center}
\epsfig{file=fig7.ps,height=11.cm,angle=270.
bbllx=80pt,bblly=90pt,bburx=535pt,bbury=745pt}
\end{center}
\caption[]{\label{lowq2}
\sl Preliminary measurement of the proton structure function $F_2(x,Q^2)$
in the low $Q^2$ region by H1 and ZEUS,
together with results from the
E665 experiment. Different predictions for $F_2$ are compared to the
data. The DOLA and CKMT curves are only shown for the upper row of $Q^2$
bins; CTEQ3M, MRSG and MRSA' are shown for the lower row;
GRV is shown for the full range.}
\end{figure}
\section{Prospects for Structure function measurements at HERA}
The HERA structure function program is still in its infancy, but has
already provided exciting results at low $x$.
The dynamics underlying the behaviour of the
structure function can be studied in an exclusive way
using jets or particle spectra,
since HERA is a collider equipped with (two) 4$\pi$ detectors.
In the next 2 or 3 years $F_{L}$ will be measured, by taking data at
different beam energies in order to keep $x$ and $Q^2$ constant while
varying $y$, thus improving the knowledge on $F_2$ and on QCD.
The statistics will increase in such a way that a first measurement
of $xF_3$ will be made, and a precise determination of $\alpha_s$
should be possible. In 95, both experiments have upgraded their
detector in the backward area in order to reach lower $Q^2$ ($\simeq$
0.1~GeV$^2$) with good precision. This year will thus be devoted to
understand the questions raised in this paper concerning the low $x$
and low $Q^2$ dynamics and to open up further stringent test of
QCD, in particular about the behaviour of the high parton
density.
\vspace*{1.cm}
{\bf Acknowledgements}
\normalsize
\noindent
I would like to thank the organizers and in particular Vladimir
Petrov to have made such a nice workshop in the quiet town of
Protvino, and to have invited me to discover Russia for the
first time. I would also like to thank my close collaborators,
Ursula Bassler, Beatriz Gonzalez-Pineiro, all the friends of the
H1 structure function group and the ZEUS collaboration
with whom we obtained the results
described above. Special thanks go to Ursula
for her help in the finalization of this
paper.
| 11,946 |
\section{#1}}
\def\baselineskip 24pt \lineskip 10pt{\baselineskip 24pt \lineskip 10pt}
\textheight 8.5in
\textwidth 6in
\oddsidemargin 0pt
\topmargin -30pt
\begin{document}
\begin{titlepage}
\rightline{To Appear: {\em Class. Quantum Grav.}}
\vspace{1in}
\begin{center}
\Large
{\bf Symmetric vacuum scalar--tensor cosmology}
\vspace{1in}
\normalsize
\large{James E. Lidsey$^1$}
\normalsize
\vspace{.7in}
{\em Astronomy Unit, School of Mathematical
Sciences, \\
Queen Mary \& Westfield, Mile End Road, LONDON, E1 4NS, U.K.}
\end{center}
\vspace{1in}
\baselineskip=24pt
\begin{abstract}
\noindent
The existence of point symmetries in the cosmological field equations
of generalized vacuum scalar--tensor theories is
considered within the context of the spatially homogeneous
cosmologies. It is found that such symmetries only occur in the
Brans--Dicke theory when the dilaton field
self--interacts. Moreover, the interaction potential of the dilaton
must take the form of a cosmological constant.
For the spatially flat, isotropic model, it is shown how
this point symmetry may be employed to generate a discrete
scale factor duality in the Brans--Dicke action.
\end{abstract}
\vspace{.7in}
PACS: 11.30.Ly, 98.80.Hw
\vspace{.2in}
$^1$Electronic address: jel@maths.qmw.ac.uk
\end{titlepage}
In this paper we search for point symmetries in the
cosmological field equations of
generalized vacuum scalar--tensor theories of gravity.
Interest in these theories has been widespread in recent years.
They are defined by the action
\begin{equation}
\label{action}
S=\int d^4 x \sqrt{-g} e^{-\Phi}
\left[ R - \omega (\Phi ) \left(
\nabla \Phi \right)^2 -2\Lambda (\Phi) \right] ,
\end{equation}
where $R$ is the Ricci curvature of the space--time and
$g$ is the determinant of the
metric $g_{\mu\nu}$ \cite{ST}. The dilaton field $\Phi$
plays the role of a time--varying gravitational constant
and may self--interact through a potential $\Lambda (\Phi )$.
The function $\omega (\Phi)$ is dimensionless and
determines the precise form of the coupling between the dilaton and graviton.
Each scalar--tensor theory is defined by the functional forms of
$\omega (\Phi)$ and $\Lambda (\Phi)$. A cosmological constant in the
gravitational sector of the theory corresponds to
the special case where $\Lambda (\Phi)$ is a space--time constant.
Action (\ref{action}) provides a
natural background within which deviations from general relativity
may be quantitatively studied. The simplest example is the
Brans--Dicke theory, where $\omega (\Phi)$ is a space--time constant
\cite{BD}.
It is known that inflationary solutions exist in a wide class of
scalar--tensor cosmologies and these theories are therefore
relevant to the study of the very early Universe \cite{INF}.
Indeed, higher--order \cite{W} and higher--dimensional \cite{HD}
theories of gravity may be expressed in a scalar--tensor
form after suitable field redefinitions and the Brans--Dicke theory
with $\omega =-1$ corresponds to a truncated version
of the string effective action \cite{STRING}.
Point symmetries associated with action (\ref{action})
have been discussed previously within the
context of the spatially isotropic Friedmann Universes \cite{PS,PS2}.
It was found that $\omega (\Phi)$ and $\Lambda (\Phi)$
must be related in a certain way if the field equations are to be symmetric.
In this paper we consider whether theory (\ref{action})
admits point symmetries for the more general class of spatially homogeneous
Bianchi Universes. We assume that $\Lambda (\Phi ) \ne 0$ and
that $\omega (\Phi ) >-3/2$ for all physical values of $\Phi$. We find that
such symmetries only exist in these anisotropic cosmologies if strong
restrictions are imposed on the form of Eq. (\ref{action}). In particular,
we show that for the Bianchi type I
model, $\omega (\Phi)$ and $\Lambda (\Phi)$ must both be {\em constant}.
We argue that this conclusion should apply
for the other Bianchi types where a Lagrangian
formulation of the field equations is possible.
The line element for the class of spatially homogeneous space-times is given by
\begin{equation}
\label{metric}
ds^2=-dt^2 +h_{ab} \omega^a \omega^b, \qquad a,b=1,2,3 ,
\end{equation}
where $h_{ab}(t)$ is a function of cosmic time $t$ and
represents the metric on the surfaces of homogeneity and
$\omega^a$ are one--forms. These models have
a topology $R\times G_3$, where $G_3$ represents a Lie group of isometries
that acts transitively on the space--like three--dimensional orbits
\cite{RS}. The Lie algebra of $G_3$ admits
the structure constants ${C^a}_{bc}=m^{ad}\epsilon_{dbc}+
{\delta^a}_{[b} a_{c]}$, where $m^{ab}$ is a symmetric matrix, $a_c
\equiv {C^a}_{ac}$ and $\epsilon_{abc} = \epsilon_{[abc]}$.
The Jacobi identity
${C^a}_{b[c} {C^b}_{de]} =0$ is only satisfied if $m^{ab}a_b =0$, so
$m^{ab}$ must be transverse to $a_b$ \cite{10}. The model belongs to the
Bianchi class A if $a_b=0$ and to the class B if $a_b \ne 0$.
A basis may be found such that $a_b =(a,0,0)$ and
$m^{ab} ={\rm diag} \left[ m_{11},m_{22},m_{33} \right]$,
where $m_{ii}$ take the values $\pm 1$ or $0$.
In the Bianchi class A,
the Lie algebra is uniquely determined up to isomorphisms
by the rank and signature of $m^{ab}$. The six possibilities are
$(0,0,0)$, $(1,0,0)$, $(1,-1,0)$, $(1,1,0)$, $(1,1,-1)$ and $(1,1,1)$ and
these correspond, respectively, to the Bianchi types I, II, ${\rm VI}_0$,
${\rm VII}_0$, VIII and IX. Finally,
the three-metric may be parametrized by $h_{ab}(t)=e^{2\alpha (t)}
\left( e^{2\beta (t)} \right)_{ab}$, where $e^{3\alpha}$ represents the
effective spatial volume of the Universe and
\begin{equation}
\beta_{ab} \equiv
{\rm diag} \left[ \beta_+ +\sqrt{3}\beta_-,\beta_+ -\sqrt{3}\beta_- , -2
\beta_+ \right]
\end{equation}
is a traceless matrix that determines the anisotropy in the models.
The configuration space $Q$ for the Bianchi
models derived from action (\ref{action}) is
therefore four--dimensional and is
spanned by $\{ q_n \equiv \alpha , \Phi ,\beta_{\pm} \}$.
The Lagrangian density $L (q_n ,\dot{q}_n )$ is
defined by $S=\int dt L (q_n , \dot{q}_n )$,
where a dot denotes differentiation
with respect to cosmic time. It may be derived by substituting
the {\em ansatz} (\ref{metric}) into action (\ref{action}) and
integrating over the spatial variables. This procedure is unambiguous
for the class A cosmologies and the action for these models simplifies to
\begin{equation}
\label{actionbianchi}
S = \int dt e^{3\alpha -\Phi} \left[
6\dot{\alpha} \dot{\Phi} -6\dot{\alpha}^2 +6\dot{\beta}^2_+ +
6\dot{\beta}^2_- +\omega (\Phi)
\dot{\Phi}^2 -2\Lambda (\Phi ) +e^{-2\alpha} U(\beta_{\pm})
\right] ,
\end{equation}
where
\begin{equation}
\label{potentialA}
U(\beta_{\pm}) = -e^{-4 \alpha} \left( m_{ab} m^{ab} -\frac{1}{2} m^2
\right)
\end{equation}
is the curvature potential,
$m \equiv {m^a}_a$ and indices are raised and lowered with $h^{ab}$
and $h_{ab}$, respectively \cite{WALD}. In the case of the
type B models, a divergence may arise because the three--curvature
contains a term proportional to $a_b a^b$ \cite{Mac}. In view of this,
we do not consider these models further.
The field equations derived from action (\ref{actionbianchi}) take the
familiar form
\begin{equation}
\label{EL}
\frac{d}{dt} \frac{\partial L}{\partial \dot{q}_n} =
\frac{\partial L}{\partial q_n} .
\end{equation}
Now, a point symmetry of a set of differential
equations such as those given by Eq. (\ref{EL})
may be viewed as a one--parameter
group of transformations acting in the space $TQ$
that is tangent to $Q$ and spanned
by $\{q_n ,\dot{q}_n \}$. One identifies such a symmetry
by introducing a set of arbitrary, real,
differentiable functions $\{ X_n(q)
=X_{\alpha}(q), X_{\Phi}(q), X_{\pm}(q) \}$ and
contracting these with the field equations \cite{PS1}. Thus,
\begin{equation}
\label{equation}
\frac{d}{dt} \left( X_n \frac{\partial L}{\partial \dot{q}_n} \right)
= \left( X_n \frac{\partial}{\partial q_n} +\frac{dX_n}{dt}
\frac{\partial}{\partial \dot{q}_n} \right) L ,
\end{equation}
where summation over $n$ is implied. The right hand side
of this equation is the Lie derivative ${\cal{L}}_{\bf X} L$
of the Lagrangian density with respect to the vector field
\begin{equation}
\label{vector}
{\rm {\bf X}} \equiv
X_n \frac{\partial}{\partial q_n} +\frac{dX_n}{dt}
\frac{\partial}{\partial \dot{q}_n} .
\end{equation}
This vector field belongs to the tangent space and is the infinitesimal
generator of the point transformation. The
Lie derivative (\ref{equation}) determines how the Lagrangian
varies along the flow generated by ${\bf X}$ in $TQ$.
When this derivative
vanishes, the Lagrangian density is {\em constant} along the integral curves
of ${\rm {\bf X}}$. It then follows immediately from
Eq. (\ref{equation}) that the quantity
$i_{\bf {X}} \theta_{L} \equiv X_n {\partial L}/{\partial \dot{q}_n}$
is conserved,
where $i_{\bf {X}}$ denotes the contraction of the vector field ${\bf X}$
with $\theta_L \equiv (\partial L /\partial \dot{q}_n)dq_n$.
Thus, one may uncover a Noether--type
symmetry in the theory by determining the components $\{ X_n (q) \}$ that
satisfy ${\cal{L}}_{\bf X} L=0$ \cite{PS1}.
In general, this equation reduces to
an expression that is quadratic in $\dot{q}_n$ for all values of $n$.
However, the coefficients of these terms are determined
by functions of $q_n$.
Thus, each of the coefficients must vanish identically if
the Lie derivative is to vanish.
This leads to a number of
separate constraints that take the form of first--order, partial
differential equations in $\{ X_n (q) \}$. A further constraint may
arise from terms in the Lagrangian that are independent of $\dot{q}_n$.
A Noether symmetry in the theory
is then identified once a solution to these equations is found.
It can be shown after some algebra that
the Lie derivative of the Lagrangian (\ref{actionbianchi})
with respect to ${\bf X}$ vanishes
if and only if $\{ X_n (q) \}$ satisfy the set of partial differential
equations:
\begin{eqnarray}
\label{1}
6\Lambda X_{\alpha} -2\Lambda X_{\Phi} +2\Lambda 'X_{\Phi} =
e^{-2\alpha} \left[ X_{\alpha} U -X_{\Phi} U +X_+ \frac{\partial U}{\partial
\beta_+} +X_- \frac{\partial U}{\partial \beta_-} \right] \\
\label{3}
9X_{\alpha} -3X_{\Phi} +3 \frac{\partial X_{\alpha}}{\partial \alpha}
-6\frac{\partial X_{\alpha}}{\partial \Phi} +3\frac{\partial
X_{\Phi}}{\partial \Phi} +\omega \frac{\partial X_{\Phi}}{\partial \alpha}
=0 \\
\label{4}
3\omega X_{\alpha} -\omega X_{\Phi} +\omega' X_{\Phi} +6 \frac{\partial
X_{\alpha}}{\partial \Phi}
+2\omega \frac{\partial X_{\Phi}}{\partial \Phi} =0 \\
\label{5}
3X_{\alpha}-X_{\Phi} +2\frac{\partial X_{\alpha}}{\partial \alpha}
-\frac{\partial X_{\Phi}}{\partial \alpha} =0 \\
\label{6}
3X_{\alpha} -X_{\Phi} +2\frac{\partial X_{\pm}}{\partial \beta_{\pm}} =0 \\
\label{7}
\frac{\partial X_+}{\partial \beta_-} +\frac{\partial X_-}{\partial
\beta_+} =0 \\
\label{8}
3\frac{\partial X_{\alpha}}{\partial \beta_{\pm}} +\omega \frac{\partial
X_{\Phi}}{\partial \beta_{\pm}} +
6 \frac{\partial X_{\pm}}{\partial \Phi} =0 \\
\label{9}
-2 \frac{\partial X_{\alpha}}{\partial \beta_{\pm}} +\frac{\partial
X_{\Phi}}{\partial \beta_{\pm}} +2\frac{\partial X_{\pm}}{\partial
\alpha} =0 ,
\end{eqnarray}
where a prime denotes differentiation with respect to $\Phi$.
We will search for non--trivial solutions to these equations
where $\{ X_{\alpha} ,X_{\Phi} , \Lambda (\Phi ) \ne 0 \}$. Moreover,
we shall consider the case where both
sides of Eq. (\ref{1}) are identically zero:
\begin{eqnarray}
\label{2a}
X_{\alpha} = \left( \frac{\Lambda -\Lambda '}{3\Lambda}
\right) X_{\Phi} \\
\label{2b}
X_{\alpha} U -X_{\Phi} U +X_+ \frac{\partial U}{\partial
\beta_+} +X_- \frac{\partial U}{\partial \beta_-} =0 .
\end{eqnarray}
This separation is valid in general for the type I model, since the curvature
potential $U(\beta_{\pm})$ is identically zero in this case. However,
it should also be consistent for the
other Bianchi types. Eq. (\ref{6}) implies that
$\partial X_+/\partial \beta_+ =\partial X_- /\partial \beta_-$.
If we differentiate this constraint
with respect to $\beta_{\pm}$ and compare it with the first
derivative of Eq. (\ref{7}), we find that $X_{\pm}$ satisfy the
one--dimensional Laplace equation:
\begin{equation}
\label{laplace}
\frac{\partial^2 X_{\pm}}{\partial \beta_{\pm}^2} +
\frac{\partial^2 X_{\pm}}{\partial \beta_{\mp}^2} =0 .
\end{equation}
Now, the components of ${\bf X}$ must be real if they
are to correspond to physical solutions.
However, an exponential solution to Eq. (\ref{laplace})
will have the generic form $X_j = \exp \left[ ik \beta_{\pm} \pm
k\beta_{\mp} \right]$, for some arbitrary, real constant $k$.
This suggests that
$X_{\pm}$ can not contain real exponential terms in $\beta_{\pm}$.
Furthermore, Eqs. (\ref{8}) and (\ref{9}) then imply that
the same will be
true for $X_{\alpha}$ and $X_{\Phi}$. This is important because
the curvature
potential (\ref{potentialA}) consists entirely of
exponential terms. We might expect, therefore, that
the components of ${\bf X}$ will be unable to cancel out these
terms in the full expression given by
Eq. (\ref{1}). If so, Eq. (\ref{1}) could only
be satisfied if both sides were identically zero.
When Eq. (\ref{2a}) is valid, Eq. (\ref{5}) simplifies to
\begin{equation}
\label{Xalpha}
\frac{\partial \ln X_{\Phi}}{\partial \alpha} = c(\Phi) = -
\left( \frac{3\Lambda '}{\Lambda + 2\Lambda '} \right) .
\end{equation}
On the other hand,
we may combine Eqs. (\ref{3}) and (\ref{4}) and
eliminate the $\partial
X_{\alpha} / \partial \Phi$ terms. Substituting Eq. (\ref{Xalpha})
into the result then implies that
\begin{equation}
\label{XPhi}
\frac{\partial \ln X_{\Phi}}{\partial \Phi} = f(\Phi) =
\frac{1}{3+2\omega}
\left[ (3+\omega )\frac{\Lambda '}{\Lambda} + \frac{3\Lambda '}{\Lambda
+ 2\Lambda'} \left( 1+\omega -\frac{\Lambda '}{\Lambda} \right) -
\omega' \right] .
\end{equation}
It follows from Eqs. (\ref{Xalpha}) and (\ref{XPhi}) that
$X_{\Phi}$ must be separable in
$\alpha$ and $\Phi$. Inserting a separable ansatz into Eq. (\ref{Xalpha})
then implies that $c(\Phi) \equiv c$ must be {\em independent}
of $\Phi$.
We may also equate Eqs. (\ref{5}) and (\ref{6}) and differentiate
with respect to $\alpha$. The term containing second derivatives in $X_{\pm}$
may then be eliminated by substituting the differential of Eq. (\ref{9}) with
respect to $\beta_{\pm}$. Moreover, substitution of Eq. (\ref{2a}) then
removes any direct dependence on $X_{\alpha}$.
This procedure leads to the very useful constraints
\begin{equation}
\label{second}
\frac{\partial^2 X_{\Phi}}{\partial \alpha^2} = \frac{\partial^2
X_{\Phi}}{\partial \beta^2_+} = \frac{\partial^2 X_{\Phi}}{\partial
\beta_-^2} = c^2 X_{\Phi}
\end{equation}
on the second derivatives of $X_{\Phi}$. These derivatives may be
related to those of $X_{\pm}$ by rewriting Eq. (\ref{6}):
\begin{equation}
\label{6a}
\frac{\partial X_{\pm}}{\partial \beta_{\pm}} = -\frac{c}{4c+6} X_{\Phi} .
\end{equation}
If we differentiate this equation twice with respect to $\beta_{\pm}$, we
find that
\begin{equation}
\label{twiceplus}
\frac{\partial^3 X_{\pm}}{\partial \beta_{\pm}^3} = -\frac{c^3}{4c+6}
X_{\Phi}
\end{equation}
after substitution of Eq. (\ref{second}).
Differentiating Eq. (\ref{6a}) twice with
respect to $\beta_{\mp}$ then implies that
\begin{equation}
\label{twiceminus}
\frac{\partial^3 X_{\pm}}{\partial \beta^2_{\mp} \partial \beta_{\pm}}
=-\frac{c^3}{4c+6} X_{\Phi} .
\end{equation}
However, differentiation of Eq. (\ref{laplace}) with respect to
$\beta_{\pm}$ implies that
\begin{equation}
\label{pm}
\frac{\partial^3 X_{\pm}}{\partial \beta^3_{\pm}} = -
\frac{\partial^3 X_{\pm}}{\partial \beta^2_{\mp} \partial \beta_{\pm}} ,
\end{equation}
so Eqs. (\ref{twiceplus}), (\ref{twiceminus}) and (\ref{pm})
are only consistent if $c=0$. Thus,
$\Lambda (\Phi)$ must be a space--time constant.
Eq. (\ref{2a}) then implies that
$3X_{\alpha}=X_{\Phi}$ if $\Lambda \ne 0$. When this condition is satisfied,
Eq. (\ref{5}) implies that $X_{\alpha}$ and $X_{\Phi}$ must be independent
of $\alpha$, as expected. It then follows from Eq. (\ref{3}) that
these functions must also be independent of $\Phi$. Moreover,
Eq. (\ref{4}) can only be satisfied in this case if $\omega' =0$. Thus,
$\omega (\Phi)$ must also be a space--time constant and this corresponds to
the Brans--Dicke theory.
The solution to Eqs. (\ref{6})--(\ref{9}) for constant $\omega$
is found to be
\begin{eqnarray}
\label{gensol}
X_{\Phi} =h_0 +h_+ \beta_+ +h_-\beta_- ,\qquad X_{\Phi} =3X_{\alpha}
\nonumber \\
X_{\pm} = x_{\pm} +b_{\pm} \beta_{\mp} -\frac{h_{\pm}}{6} (1+\omega )
\Phi -\frac{h_{\pm}}{6} \alpha ,
\end{eqnarray}
where $\{ h_0 , b_+ , h_{\pm} ,x_{\pm} \}$ are arbitrary
constants and $b_-=-b_+$. However, Eq. (\ref{2b}) must also be solved.
This condition is trivial for the type I
model, but it places further restrictions on the
components of ${\bf X}$ in the case of the other Bianchi types.
Since there are no exponential terms in
Eq. (\ref{gensol}), Eq. (\ref{2b})
must reduce to six separate constraints:
\begin{eqnarray}
\label{m}
m_{11}^2 \left[ X_{\alpha}-2X_+ -2\sqrt{3} X_- \right] =0 \nonumber \\
m_{22}^2 \left[ X_{\alpha}-2X_+ +2\sqrt{3} X_- \right] =0 \nonumber \\
m_{33}^2 \left[ X_{\alpha} +4 X_+ \right] =0 \nonumber \\
m_{11} m_{22} \left[ X_{\alpha} -2X_+ \right] =0 \nonumber \\
m_{11}m_{33} \left[ X_{\alpha} +X_+ -\sqrt{3} X_- \right] =0 \nonumber \\
m_{22}m_{33} \left[ X_{\alpha} +X_+ +\sqrt{3} X_- \right] =0 .
\end{eqnarray}
In the case of the type II model, these equations are satisfied when
$h_0=6(x_+ +\sqrt{3} x_-)$ and $h_+=-\sqrt{3}h_- = -6\sqrt{3} b_+$.
For types ${\rm VI}_0$ and ${\rm VII}_0$, the stronger restrictions
$X_-=0$ and $X_{\alpha}=2X_+={\rm constant}$ must apply if a symmetry
is to exist. However, the only solution to Eq. (\ref{m})
for types VIII and IX is $\{ X_n \} =0$, so the field equations
for these two models do not admit non--trivial point symmetries.
Eq. (\ref{gensol}) is
the general solution to Eqs. (\ref{3})--(\ref{9}) when
$\Lambda (\Phi) \ne 0$ and Eqs. (\ref{2a}) and (\ref{2b}) are valid.
When these conditions are satisfied, therefore, we may conclude that
the only vacuum scalar--tensor gravity theory that contains a point symmetry
in anisotropic cosmologies is the Brans--Dicke theory with a
cosmological constant in the gravitational sector of the theory.
The Brans--Dicke theory also exhibits
this symmetry when $\Lambda =0$. This may be verified by
substituting the ansatz $X_{\Phi} (\beta_{\pm}) =3X_{\alpha} (\beta_{\pm})$
into Eqs. (\ref{1})--(\ref{9}). We should emphasize, however, that
other theories may also be symmetric when $\Lambda$ vanishes since the
left hand side of Eq.
(\ref{1}) is trivial in this case. Consequently, Eq. (\ref{2a}) does not
apply, so
Eqs. (\ref{Xalpha}) and (\ref{XPhi}) are not the unique solutions
to Eqs. (\ref{3})--(\ref{5}). This implies that $X_{\Phi}$
could take a more general form to that given in Eq. (\ref{gensol}).
It would be of interest to investigate whether other
theories are indeed symmetric when the dilaton potential vanishes.
Recently, a further symmetry in the Brans--Dicke cosmology was identified
within the context of the spatially
flat, isotropic Friedmann Universe \cite{LIDS}.
It can be shown by direct substitution
that action (\ref{actionbianchi}) is invariant under
a scale factor duality transformation
\begin{eqnarray}
\label{sfd}
\alpha =\frac{2+3\omega}{4+3\omega} z - \frac{2(1+\omega)}{4+3\omega} w
\nonumber \\
\Phi = -\frac{6}{4+3\omega} z-\frac{2+3\omega}{4+3\omega} w
\end{eqnarray}
when $\omega \ne -4/3$ and $U=\beta_{\pm} =\Lambda '=0$.
This symmetry is a generalization of the
scale factor duality exhibited by the string effective action \cite{duality}.
It is a discrete symmetry but
it may be related to the continuous Noether symmetry discussed in this
work.
In the isotropic case the configuration space
is two--dimensional and the Lie derivative of the Lagrangian
vanishes if $X_{\Phi} =3X_{\alpha} ={\rm constant}$ \cite{PS2}.
This Noether symmetry may be employed to generate a new set of variables
$q_n =q_n (Q_k)$ $(n,k =1,2)$. In this case
the vector field (\ref{vector}) transforms to
\begin{equation}
{\bf X} =\left( i_{\rm {\bf X}} dQ_k \right)
\frac{\partial}{\partial Q_k} + \left[ \frac{d}{dt}
\left( i_{\rm {\bf X}} dQ_k \right) \right] \frac{\partial}{\partial
\dot{Q}_k} ,
\end{equation}
where the contraction is over ${\rm {\bf X}}$ and $dQ_k =(\partial
Q_k/\partial q_n) dq_n$ \cite{foliation}.
We may define $\{ Q_k \} $ such that they satisfy
the first--order partial differential equations
\begin{equation}
\label{Qconstraint}
i_{\rm {\bf X}} dw =\epsilon_1(w,z) ,
\qquad i_{\rm {\bf X}} dz =\epsilon_2(w,z) ,
\end{equation}
where $w \equiv Q_1$, $z\equiv Q_2$ and
$\epsilon_l (w,z)$ are particular functions. If we specify
these functions as
$\epsilon_1 =-3 X_{\alpha}$ and $\epsilon_2
=-X_{\alpha}$, respectively, the solution to Eq. (\ref{Qconstraint})
is given by Eq. (\ref{sfd}). Thus, the scale factor
duality of the Brans--Dicke theory may be generated by
the point symmetry associated with the vector field ${\bf X}$.
We will conclude with some general remarks. The symmetry discussed in this
work has a number of applications. Firstly, it
leads to a conserved quantity of the form
$i_{\bf X} \theta_L = X_n \partial L/ \partial
\dot{q}_n $. Since the Lagrangian density is quadratic in
$\dot{q}_n$, the conservation of $i_{\bf X} \theta_L$ results
in an equation that relates the first derivatives of the configuration
space variables $q_n$. This represents a first integral of the
field equations (\ref{EL}). In principle, it should be easier to
solve this constraint, together with the Hamiltonian constraint,
rather than the full system given by Eq. (\ref{EL}).
Thus, the existence of a conservation law implies that
the field equations may be simplified considerably and this may
lead to new solutions. In particular, it would
be interesting to derive new inflationary solutions by this
approach. The search for exact inflationary solutions
in anisotropic cosmologies is well motivated. These solutions would
provide insight into how the anisotropy is effectively washed away
by the accelerated expansion, thereby leading to
the highly isotropic Universe that is observed today.
The question of how the anisotropy may influence the onset of inflation
may also be addressed through exact solutions. Such solutions will exist
since the symmetry is compatible with a cosmological
constant in the gravitational sector of the theory. This term
could also arise, for example,
from the potential energy of a second scalar field that is coupled
to the dilaton field in an appropriate fashion.
We have shown that a Noether symmetry
arises in the Bianchi types I, II, ${\rm VI}_0$ and ${\rm VII}_0$ and have
argued that it is unique to the Brans--Dicke theory
in these cases. However, our conclusions
also apply to other Bianchi models. Although a symmetry of the
form discussed here does not exist for the Bianchi types VIII and IX,
we may consider the high anisotropic limit of all
Bianchi A models where $\beta_{\pm} \gg 1$. In this case, $h_{22}/h_{11}
\ll 1$ and $h_{33}/h_{11} \ll 1$, so
the dominate term in the curvature potential
(\ref{potentialA}) is $m_{11}^2h_{11}^2$.
In effect, this is equivalent to specifying $m_{11}=1$ and $m_{22}
=m_{33}=0$ in Eq. (\ref{potentialA}) and this corresponds to
the type II model. Hence,
the field equations of the Bianchi types VIII and IX will exhibit an
approximate point symmetry if the anisotropy is sufficiently
large. This is interesting because the initial state of the Universe
may well have been very anisotropic due to
quantum effects and the early Universe
is precisely the regime where scalar--tensor gravity is thought
to have been relevant.
We have not considered the Bianchi
class B models directly
in this work because the Lagrangian description of the field
equations is not always consistent \cite{Mac}. However, our conclusions will
apply for those models in this class
that can be expressed in a Lagrangian form,
because the corresponding curvature potential will contain
exponential terms in $\beta_{\pm}$ \cite{WALD}.
Consequently,
the separation of Eq. (\ref{1}) into Eqs. (\ref{2a}) and (\ref{2b})
will apply in these cases also. Thus, the point symmetry associated with
the Brans--Dicke theory arises in a number of different
homogeneous cosmologies.
The question of which scalar-tensor theory may have applied
in the early Universe is currently unresolved.
There are two approaches that one might take in
addressing this question. Firstly, one may
identify the subset of theories that are attracted to the general
relativistic limit at late times \cite{late}. Alternatively, one may
attempt to uncover a deeper principle that strongly favours one
particular theory. Symmetries often provide strong motivation
for selecting a given theory from the space of possible theories. We
have found that the requirement that a point symmetry
exists in the homogeneous, cosmological
field equations of generalized scalar--tensor
gravity is surprisingly restrictive. Indeed, the Brans--Dicke theory
is the only theory to exhibit such a symmetry when
the dilaton field self--interacts. This may be significant because
the Brans--Dicke theory is
consistent with all cosmological observations if $\omega >500$. If
the point symmetry only arose in theories that could not
reproduce Einstein gravity at the present epoch, it would
be uninteresting. However, we have found that the symmetry is associated
with a realistic theory of gravity.
Finally, we considered
the Noether symmetry of the Brans--Dicke theory within the context of the
spatially flat, isotropic Universe. We showed how it is
directly related to a scale factor duality invariance of the theory.
The Noether
symmetry provides new insight into how the duality arises. It would
be of interest to investigate whether the Noether symmetry
associated with the anisotropic cosmologies may be employed in a similar
fashion to uncover more general discrete symmetries in the Brans--Dicke
theory. If such discrete symmetries exist, they could be employed to
map a particular solution of the field equations onto
a new, generally inequivalent, solution. In the isotropic model,
solutions may be generated
in this fashion with and without a cosmological term and, indeed,
inflationary solutions may be found that are driven entirely
by the kinetic energy of the dilaton field \cite{levin}. A similar
approach could be followed in the anisotropic Universes.
To summarize, therefore, we have investigated the existence
of point symmetries in the homogeneous, cosmological
field equations of generalized
vacuum scalar--tensor gravity under the assumption that the dilaton field
self--interacts. In the case of
the spatially flat, anisotropic cosmology, we found that
the Brans--Dicke theory containing a
cosmological constant is the only scalar--tensor theory whose field
equations exhibit a point symmetry. We have argued
that this result also applies for types II, ${\rm VI}_0$ and ${\rm VII}_0$.
We may conclude, therefore, that
the Brans--Dicke theory exhibits
a higher level of symmetry than other scalar--tensor theories.
\vspace{.7in}
The author is supported by the Particle Physics and Astronomy
Research Council (PPARC), UK.
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\end{document}
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\section{Introduction}
The standard model of electroweak interactions predicts the existence of a
phase transition to a high temperature symmetric phase \cite{Lin72}.
The knowledge of fluctuation spectra in the high temperature phase is
essential for understanding its physics.
In the high temperature phase the thermal contribution to the
screening masses of $W$-gauge bosons dominates the contribution (if any)
of the vacuum expectation value of the Higgs field. One has to
distinguish here between the screening scales resulting from
electric and magnetic gauge field fluctuations.
The leading order electric screening mass is ${\cal O}(gT)$. It is
essentially a perturbative quantity, determined by the internal consistency
of the resummed perturbative treatment of the thermodynamics of the system
\cite{Arn93,Karun}.
Beyond leading order it requires a careful non-perturbative definition
\cite{Arn96}.
The non-vanishing magnetic screening mass does play an important role in
controlling the infrared behaviour of the electroweak theory at high
temperature and does influence the nature of the electroweak phase transition
itself. For instance, it is expected that the existence or non-existence of a
first order phase transition in the electroweak theory crucially
depends on the magnitude of the thermal magnetic mass of the $W$-boson
\cite{Buc94,Esp93}. The role of this mass in the symmetric high
temperature phase of the electroweak theory is similar to that of the
magnetic mass generated for the gluons in the high temperature, deconfined
phase of QCD. Also, the thermal magnetic mass is crucial for the infrared
behaviour of QCD at high temperature. In both cases these masses are
expected to be of ${\cal O} (g^2 T)$ \cite{Lin80} up to possible logarithmic
corrections \cite{Kal92}.
The magnetic masses for the gauge bosons in the $SU(2)$ gauge-Higgs
model as well as in QCD are not calculable within the context of high
temperature perturbation theory.
For instance, one might attempt to apply some sort of resummation to the
magnetic sector, represented by a 3-dimensional effective theory in
both cases \cite{Bra95}. A perturbative calculation of the free energy with a magnetic
mass of ${\cal O}(g^2T)$ introduced by adding and subtracting
a corresponding term to the Lagrangian would lead to a series in
$g^2T/m_w(T)$. All evidence gathered so far on this ratio suggest
a large value for it. Any perturbative determination of it is therefore
expected to fail.
Some form of a non-perturbative
approach is needed. In the case of the $SU(2)$ gauge-Higgs model
there have been various Monte Carlo calculations in which the thermal
vector boson as well as the Higgs masses have been calculated with help of gauge
invariant correlators of appropriate quantum numbers
\cite{Fod95,Ilg95,Kaj95,Phi96}.
Another attempt is based on an analytic treatment
of coupled gap equations for the scalar and vector propagators on the
mass shell \cite{Buc95,PhiSint}. Such an analysis leads to the conjecture
that also in the high temperature phase the magnetic
$W$-boson mass is generated essentially by a Higgs-type phenomenon. The
difference being that the expectation value of the order
parameter is much smaller. It therefore is intuitively very appealing to
continue the use of the same gauge invariant operators for the calculation of
the magnetic mass (and also of the Higgs-mass), like in the low temperature
phase.
The numerical calculations of gauge invariant correlators, however, do lead to
a thermal mass for the vector boson which is substantially larger than
the result of the analysis of gap equations in Landau gauge.
This situation is somewhat similar to the case
of $SU(N)$ gauge theories where the analysis of gauge invariant {\it
glueball} operators with the quantum numbers of the {\it gluon} \cite{Gro94}
does lead to much larger screening masses than the direct calculation of the
gluon propagator in Landau gauge \cite{Hel95}. In this case the observed
discrepancy was, however, expected. The
gauge invariant correlation functions correspond to glueball states
at low temperature and {\it melt} into several decoupled gluons with an
effective thermal mass in the high temperature deconfined phase. The gauge
invariant glueball correlation functions thus describe a multiple gluon state
in the high temperature phase. The elementary thermal mass is only visible
in a direct calculation of the gluon propagator.
In the $SU(2)$ gauge-Higgs model, however, on the basis of the above argument
the discrepancy between analytic results
and numerical calculations in the symmetric phase is somewhat
unexpected.
An explanation could be that analytic
calculations are less stable in the symmetric phase.
However, this discrepancy may also hint at a situation
similar to the case of QCD, i.e. the gauge invariant operators may not
project onto single $W$-boson states in the high temperature symmetric phase.
Also the agreement of the bound state model of Dosch {\it et al.}
with the screening masses obtained from the spectroscopy of gauge invariant
operators \cite{Dos95} points to such an interpretation.
In order to get closer to a clarification of this problem we
need a more detailed quantitative understanding of both the behaviour of
correlation functions for gauge fields in fixed gauges as well as that
of gauge independent correlation functions with quantum numbers of the
gauge bosons.
It is the purpose of this paper to study in detail the behaviour of
correlation functions for the gauge potentials in Landau gauge.
This will be done within the context of the 3-dimensional
$SU(2)$ gauge-Higgs model which is obtained as an effective theory for the
finite temperature electroweak model by integrating out heavy static modes
corresponding to the zeroth component of the gauge fields \cite{Kaj96}.
We extract from the exponential fall-off of correlation functions of the
gauge fields the $W$-boson magnetic mass in Landau gauge and compare with
corresponding calculations of gauge invariant operators with quantum numbers
of the $W$-boson as well as the Higgs boson. We have performed these
calculations on a large number of different lattice sizes in order to
control finite size effects. In addition we have performed calculations
for the pure $SU(2)$ gauge theory in order to test the conjecture that the
$W$-boson mass in the symmetric phase is closely related to the magnetic
mass in the $SU(2)$ gauge theory. We do not address the problem of gauge
invariance of the masses extracted from the gauge boson propagators and
the related issue of influence of Gribov copies in this paper.
These problems have been discussed in the context of calculations for
photon and gluon propagators \cite{Nak91}.
The paper is organized as follows. In section 2 we give the basic
definitions for the 3-d gauge-Higgs model and its relation to the
(3+1)-dimensional finite temperature $SU(2)$ Higgs model. In section 3 we
discuss the calculation of the $W$-boson propagator in Landau gauge.
The analysis of gauge invariant scalar and vector correlation functions
is presented in section 4. Finally we give our conclusions in section 5.
\section{Reduced EW-model}
The physics of the longest range fluctuations of the finite temperature
electroweak theory is described by an effective 3-d theory which is matched
to the complete model at the distance scale $a\sim (gT)^{-1}$
\cite{Kaj96,Kaj93,Kaj94,Jak94}.
Though this effective theory coincides formally with the superrenormalisable
3-d gauge--Higgs model, the optimal quantitative description of the original
physics is expected to be obtained by choosing the lattice spacing according to
$\Theta\equiv aT\sim 1$.
The lattice formulation of the effective theory is given by
\begin{eqnarray}
&
S^{3D}_{lat}={\beta \over 2}\sum_P{\rm Tr}U_P(x)+{1\over 2} \sum_{x,i}
{\rm Tr}\Phi^\dagger_xU_{x,i}\Phi_{x+\hat{i}}-{1\over 2\kappa}\sum_x{1\over 2}
{\rm Tr} \Phi^\dagger_x\Phi_x\nonumber\\
&
-{\lambda_3\over 24}\sum_x({1\over 2}{\rm Tr}
\Phi^\dagger_x\Phi_x)^2,
\end{eqnarray}
where $U_P$ is the standard plaquette variable of the $SU(2)$ lattice gauge
theory and $\Phi_x$ is a complex $2\times 2$ matrix which in terms of
the real weak isosinglet-triplet decomposition of the
complex Higgs doublet is given by $\Phi_x \equiv \Phi_0+\tau_i\Phi_i$.
The relationship of the dimensionless lattice couplings $\beta,~\lambda_3,~
\kappa$ to the couplings of the original $T=0$, $SU(2)$ gauge--Higgs system
is given by the following sequence of equations:
\begin{equation}
\beta ={4\over g_3^2\Theta},~~~ g_3^2=g^2(1-{g\over 20\pi}\sqrt{5\over 6}),
\label{beta}
\end{equation}
\begin{equation}
\lambda_3=\biggl(
{3\over 4}{m_H^2\over m_W^2}g^2-{27\over 160\pi}\sqrt{5\over 6}g^3
\biggr)\Theta,~~m_H^2={\lambda\over 3}v^2,~~m_W^2={g^2\over 4}v^2,
\label{lambda}
\end{equation}
\begin{equation}
{1\over \kappa}=m^2a^2+({3\over 16}g^2+{1\over 12}
\lambda -{3g^3\over 16\pi}\sqrt{5\over 6})\Theta^2
-\Theta\Sigma (L)({3\over 2}g^2+\lambda -{15g^3\over 32\pi}\sqrt
{5\over 6})+6.
\label{kappa}
\end{equation}
In these equations $g,\lambda, m^2$ represent the renormalised parameters
of the original theory and $m_H,~m_W$ are the $T=0$ masses of the Higgs and
$W$ bosons. $\Sigma (L)$ is a slightly size dependent geometrical
factor which is known exactly for any lattice size $L$. In particular one
obtains in the infinite volume limit
$\Sigma (\infty )= 0.252731$.
The above relations are obtained when one integrates first over the non-static
Matsubara modes with 1-loop accuracy and in a second reduction step also
over the static $A_0$ mode \cite{Kaj94}.
Then the coupling relations are accurate to ${\cal O}(g^3,\lambda^{3/2})$.
The $A_0$-integration has been realised with help of an
iterated 1-loop calculation \cite{Pol95}. Although it will not be relevant
for the following discussion it is worth to note that
the ${\cal O}(g^3\Theta)$ term in Eq.~(\ref{kappa}) which has been
estimated with this iterative technique deviates from the result given
in \cite{Kaj94}.
Our numerical calculations have been performed with the parameters
\begin{equation}
\beta =9.0 \quad , \quad \lambda_3 = 0.313646~~,
\end{equation}
at the physical $W$-boson mass, $m_W=80.6$ GeV.
Choosing also $\Theta =1$
this does correspond to a Higgs mass of about 80~GeV (Eq.~(\ref{lambda})).
Eq.~(\ref{kappa}) does then relate
$\kappa$ to the ratio $m^2/T^2$. After finding $\kappa_c$,
its value is easily translated into $T_c$ (with one more
input of $v=246$ GeV). This can be performed for each
value of $\kappa$. Our correlation measurements covered the range
$0.170\leq \kappa\leq 0.180$. The $\kappa-$range can be translated with
help of Eq.~(\ref{kappa}) into a temperature interval around the
critical value, $0.6~T_c~\raise0.3ex\hbox{$<$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}~T~\raise0.3ex\hbox{$<$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}~(4-5)~T_c$.
We expect that more accurate
mappings between the couplings will not modify qualitatively the temperature
interval covered.
We have chosen to work at a rather large value of the Higgs mass.
For this choice of parameters the nature of the phase transition has not yet
been clarified. It could be either
continuous or only very weakly first order. The study of the gap equations
in \cite{Buc95} indicates a smooth crossover. We shall give an
argument below in favor of a second order transition. One therefore should be
able to find a critical value of the hopping
parameter at which the symmetry restoring takes place. This has been
determined by us as
\begin{equation}
\kappa_c = 0.17467(2)~~.
\end{equation}
We will report in more detail about the determination of $\kappa_c$ and
an analysis of the order of the phase transition elsewhere.
\section{W-boson propagator in Landau gauge}
\subsection{Landau gauge fixing}
Our analysis of the $W$-boson propagator in Landau gauge closely follows the
approach used in the calculation of the gluon propagator in finite
temperature QCD \cite{Hel95,Kar95}. We define the gauge fields,
$A_{\mu}({\bf x})$, from the $SU(2)$ link variables, $U_{x,\mu}$
\begin{equation}
A_{\mu}({\bf x}) = {i \over 2g}
\left( U^{\dagger}_{x,\mu} - U_{x,\mu} \right) \ .
\label{a_mu}
\end{equation}
The Landau gauge condition $|\partial_{\mu}A^{\mu}({\bf x})|^2 = 0$ is
then realized on each lattice configuration by maximizing
the trace of the link fields, $U_{x,\mu}$,
\begin{equation}
\Sigma = \sum_{x, \mu} \Tr \, \left[ U_{x,\mu} + U^{\dagger}_{x,\mu}
\right]~~.
\label{spur_summe_1b}
\end{equation}
The maximization has been performed using an overrelaxation algorithm
combined with a FFT-algorithm \cite{Dav88,Man88} until the Landau gauge
condition has been satisfied within an accuracy of $10^{-9}$. Typically
this required about 500 iterative maximization steps.
On the gauge fixed configurations we analyze correlation functions of gauge
fields averaged over $(x,y)$-planes,
\begin{equation}
\tilde{A}_\mu (z) = \sum_{x,y} A_\mu (x,y,z)~~,
\label{Aave}
\end{equation}
in order to improve the projection onto the zero momentum excitations.
The correlation functions of these averaged fields are then calculated in
the transverse $z$-direction,
\begin{equation}
G_w (z) = \left\langle \Tr \; \tilde{A}_\mu (0)\tilde{A}_\mu (z)
\right\rangle~~.
\label{corr}
\end{equation}
For large separations $z$ these correlation functions do project onto the
$W$-boson propagator mass.
\subsection{W-boson propagator}
We have analyzed the $W$-boson propagator in Landau gauge in the 3-d $SU(2)$
gauge-Higgs model.
Calculations have been performed for a large number of hopping parameter values.
We have used an overrelaxed heat-bath algorithm and performed typically
50.000 iterations\footnote{We call an iteration a
combination of 4 overrelaxation steps followed by one heat-bath update.
The heat-bath update of the scalar fields was optimized by shifting a
suitably chosen quadratic term of the Higgs field from the accept reject
decision into the generation of the cartesian gaussian components.}
per $\kappa$-value.
After every tenth iteration we have then fixed the Landau gauge and calculated
the gauge fixed correlators as well as a set of gauge invariant operators.
Most calculations have been performed on a lattice of size
$16^2\times 32$. In order to get control over finite lattice size effects
we have
performed additional calculations on lattices of size $L^2 \times 32$
with $L$ ranging from 4 to 24 as well as $16^2\times L_z$ with $L_z$, ranging
from 16 to 128. These calculations have been performed at two values of
the hopping parameter below and above $\kappa_c$.
The statistics accumulated for our detailed finite size analysis at these
$\kappa$-values is summarized in Table~\ref{corrfit.tab}.
The analysis of the volume dependence
allowed us to select a suitable ansatz for the fits of correlation
functions on the $16^2\times 32$ lattice which minimize finite size
effects in the determination of the $W$-boson propagator mass.
\begin{figure}[htb]
\epsfig{file=c11vgl_17450.eps, width=73mm, height=65mm}
\hfill
\epsfig{file=c11vgl_17484.eps, width=73mm, height=65mm}
\caption{Gauge field correlation functions on $16^2 \times L_z$ lattices
with $L_z = 32$ (squares), 40 (circles), 48 (upper triangles), 64 (lower
triangles) and 128 (diamonds). Shown are correlation functions in the
symmetric phase at $\kappa=0.1745$ (a) and the symmetry broken phase at
$\kappa=0.17484$ (b). The curves give fits for $z\ge 8$. The fitting
parameters are listed in Table 1. }
\label{wcorr.fig}
\end{figure}
In Fig.~\ref{wcorr.fig} we show the correlation function $G_w (z)$ defined in
Eq.(\ref{corr}) for various lattice sizes at $\kappa = 0.1745$ and $\kappa =
0.17484$.
The correlation functions show a slower decay at short
distances which also has been observed in the analysis of the gluon
propagator \cite{Hel95,Nak95}. This behaviour is also evident from the
analysis of local masses which approach a plateau at large distances from
below.
We define local masses in two different ways either as solution of the equation
\beqn
{ G_w (z-1) - G_w (z) \over G_w (z) -G_w (z+1) } =
{ G_w^{fit} (z-1) - G_w^{fit} (z) \over G_w^{fit} (z) - G_w^{fit} (z+1) }\quad ,
\label{wlocala}
\eqn
or
\beqn
{ G_w (z-1) \over G_w (z)} = { G_w^{fit} (z-1) \over G_w^{fit} (z) }\quad .
\label{wlocalb}
\eqn
Here $G_w (z)$ denotes the calculated values for the correlation functions
and
\beqn
G_w^{fit} (z) = A~\biggr(\exp{(-m_w z)} + \exp{(-m_w (L_z - z))}\biggl)~+~B~~.
\label{wfit}
\eqn
\begin{table}
\begin{center}
\begin{tabular}{|r|l|l|l|r|}
\hline
$L_z$ & \multicolumn{1}{|c|}{$m_w$} & \multicolumn{1}{|c|}{$A$} &
\multicolumn{1}{|c|}{$B$} & \# iterations \\
\hline
\multicolumn{5}{|c|}{$\kappa = 0.1745$} \\
\hline
32 & 0.166(~7) & 10.3~(2) & -0.80(15) & 190.000 \\
40 & 0.194(14) & 10.9~(6) & -0.18(13) & 40.000 \\
48 & 0.179(11) & 10.1~(9) & -0.14(~8) & 40.000 \\
64 & 0.174(~9) & 9.3~(8) & -0.045(22) & 90.000 \\
128 & 0.163(12) & 8.6~(9) & -0.012(13) & 60.000 \\
\hline
\multicolumn{5}{|c|}{$\kappa = 0.17484$} \\
\hline
32 & 0.308(~6) & 3.4~(1) & -0.012(19) & 80.000 \\
64 & 0.291(11) & 3.2~(2) & -0.009(~4) & 40.000 \\
\hline
\end{tabular}
\end{center}
\caption{Results of fits to the correlation functions shown in
Fig.~1. }
\label{corrfit.tab}
\end{table}
While the ansatz given in Eq.~(\ref{wlocala}) is independent of the constant
$B$ the second version given in Eq.~(\ref{wlocalb}) is not. In the latter
case we use $B \equiv 0$ to define the local masses.
Results for these are shown in Fig.~\ref{wlocal.fig}. We note
that there is no apparent dependence on $L_z$ visible in the analysis based
on Eq.~(\ref{wlocala}) while there is a significant
volume dependence if we use ths second ansatz. We do, however, find
that both forms yield consistent results for $L_z \ge 64$. This suggests
that we can minimize finite lattice size effects by allowing for a constant
in our ansatz for a global fit to the correlation functions. We also see
from Fig.~\ref{wlocal.fig} that the local masses do
develop a plateau for $z \raise0.3ex\hbox{$>$\kern-0.75em\raise-1.1ex\hbox{$\sim$}} 8$.
\begin{figure}[htb]
\begin{center}
\epsfig{file=m11vgl.eps, width=73mm, height=65mm}
\hfill
\epsfig{file=m10vgl.eps, width=73mm, height=65mm}
\end{center}
\caption{Local masses calculated at $\kappa = 0.1745$
from the correlation functions shown in Fig.~1a. In (a) we show local masses
extracted according to Eq.~(3.5) while (b) gives the result according to
Eq.(3.6). The latter does assume $B=0$. The horizontal lines give the error band
resulting from the fit on a $16^2\times 128$ lattice. }
\label{wlocal.fig}
\end{figure}
We therefore have fitted the correlation functions only for distances $z \ge 8$
using the ansatz given in Eq.~(\ref{wfit}).
The fit results for the correlation functions shown in
Fig.~\ref{wcorr.fig} are summarized in Table~\ref{corrfit.tab}. We note that
these fits yield values for the propagator masses $m_w$ which are within
errors independent of $L_z$. Moreover, the constant $B$ rapidly drops to
zero with increasing $L_z$. We find that it is well described by an
exponential decrease, $B \sim \exp{(-0.1 L_z)}$. We also note that the
constant $B$ is consistent with being zero in the symmetry broken phase
already for $L_z = 32$.
A similar analysis has been performed for the dependence of the correlation
functions on the transverse lattice size. In that case simulations have been
performed on lattices of size $L^2 \times 32$ with $L$ ranging from 4 to 24.
From the above analysis of finite size effects on propagator masses below
and above $\kappa_c$ we conclude that masses can reliably be extracted from
correlation functions already on lattices of size $16^2 \times 32$ using
a fit of the form given in Eq.~(\ref{wfit}). The results obtained this way for
a large number of $\kappa$ values are shown in Fig.~\ref{propmasses.fig}. We
note that within our numerical accuracy the propagator mass $m_w$ is
independent of $\kappa$ in the symmetric phase while it rises rapidly above
$\kappa_c$. We also have performed a calculation at $\kappa =0$, $\lambda_3 =
0$, i.e. in the pure $SU(2)$ gauge theory. This yields a value for the
propagator mass which is consistent with those obtained in the symmetric
phase of the $SU(2)$ gauge-Higgs model close to $\kappa_c$. This pure gauge
value also is shown in Fig.~\ref{propmasses.fig} as a filled circle.
A fit to the data for $m_w$ below $\kappa_c$ yields
\begin{figure}[htb]
\begin{center}
\epsfig{file=masse_11.eps, height=80mm}
\end{center}
\caption{Propagator masses obtained from
fits to correlation functions calculated on lattices of size $16^2\times 32$.
The curves show fits as explained in the text. The filled circle at
$\kappa=0$ gives the result for the pure $SU(2)$ gauge theory. }
\label{propmasses.fig}
\end{figure}
\beqn
m_w = 0.158 \pm 0.002 \quad, \quad \kappa \le \kappa_c~.
\label{mwfitb}
\eqn
Above $\kappa_c$ the mass increases rapidly. A good fit to the data in the
entire range $\kappa \ge \kappa_c$ is obtained with the ansatz
\beqn
m_w = 0.158 + a (\kappa - \kappa_c)^\beta \quad \kappa \ge \kappa_c~.
\label{mwfita}
\eqn
In order to be more sensitive to the critical behaviour close to $\kappa_c$
we have restricted the fit to the interval $\kappa_c \le \kappa \le 0.176$.
In this case we find for the two free parameters $a=4.0(4)$
and $\beta = 0.384(15)$.
We note that the exponent $\beta$ turns out to be consistent with that
of the $O(4)$ spin model in 3-dimensions. In a recent
Monte Carlo analysis \cite{Kan95} this exponent has been found to be
$\beta = 0.3836(46)$ which is in agreement with
results obtained from the $(4-\epsilon)$-expansion.
Through the Higgs-mechanism the $W$-boson mass in the $SU(2)$
gauge-Higgs model is linked to the scalar field expectation value.
It thus seems plausible that also the temperature dependence of the
$W$-boson mass close to $\kappa_c$ is controlled
by the exponent $\beta$. Although the agreement is quite striking we
stress that without a more detailed finite size analysis we can, at
present, not rule out a smooth crossover as found in Ref.~\cite{Buc95}
or a critical behaviour controlled by a one-component scalar field as
suggested in Ref.~\cite{Kaj95}.
The results for the propagator mass discussed so far are in good
quantitative agreement with analytic calculations of Ref. \cite{Buc95}.
Expressing our result in terms of $g^2$
we find in the symmetric phase $m_w = 0.35(1) g^2T$. This should be compared
with the approximate value $m_w \simeq 0.28g_3^2$ quoted in Ref.~\cite{Buc95} for
large values of $\lambda /g^2$. Also the functional form of $m_w (\kappa)$
in the symmetry broken phase is in good agreement with results obtained from
the analysis of gap equations when the $\kappa$-dependence is transformed
into temperature dependence with help of Eq.~(\ref{kappa}).
However, the emerging physical picture is rather different from what is
proposed in \cite{Buc95}.
Based on an analysis of coupled gap equations for the scalar
and vector propagators in Landau gauge it has been concluded in \cite{Buc95}
that also above
$T_c$ the propagator mass is determined by the vacuum expectation value of
the Higgs field. This expectation value is actually much smaller than at $T=0$
(mini-Higgs mechanism) and is proportional to $g^2$. For
scalar and gauge couplings corresponding to equal Higgs and $W$-boson masses
at zero temperature they do find a smooth crossover between the low and high
temperature regimes which are distinguished by different magnitudes of
the scalar field expectation values. However, it also is found that the high
temperature magnetic mass is rather insensitive to the actual value
of $\lambda /g^2$.
Our calculation suggests a different physical mechanism. Since the pure gauge
propagator mass is compatible with the result of the Higgs-model in the
symmetric phase, we believe
that the magnetic mass is of {\it fully thermal origin}, without any high T
(low $\kappa$) Higgs effect. Above $\kappa_c$ a non-analytic contribution
adds to the thermal value which can be attributed to the onset
of the Higgs-effect in the low temperature phase. Although we can, at
present, not rule out a smooth crossover behaviour we note that the
temperature dependence in the vicinity of $\kappa_c$ is well described
by a non-analyticity characteristic for continuous phase
transitions in Higgs models.
\section{Gauge invariant vector and scalar correlators}
In the symmetry broken phase the $W$-boson propagator mass increases
rapidly with $\kappa$. This tendency
agrees well with both the low temperature vector mass extracted from
the gap equation (when $g \simeq 2/3$
and $a=1/T$ of our simulation are taken into account) and with the masses
obtained from gauge invariant correlators in lattice units by
\cite{Ilg95,Kaj95,Phi96}. These latter simulations do, however, use a
smaller value for $\lambda$ than we do. It therefore is important to
compare for
our choice of couplings the agreement of the masses extracted from gauge
invariant vector operators with the propagator mass.
We have calculated correlators for the standard gauge
invariant vector and scalar operators
\beqn
O_{v,i} (x) = {\rm Tr} \sigma_3 \Phi^\dagger_xU_{x,i} \Phi_{x+\hat{i}}
\quad i=1,~2\quad ,
\label{cordefv}
\eqn
as well as two different scalar operators,
\beqn
\quad O_s^a = {\rm det} \Phi_x ~~,
\quad O_s^b = \sum_{i=1}^2{\rm Tr} \Phi^\dagger_x U_{x,i} \Phi_{x+\hat{i}} ~~,
\label{cordefs}
\eqn
In analogy to Eq.~(\ref{corr}) we define operators $\tilde{O}$ which project
onto zero momentum states.
The long distance behaviour of the scalar correlator $G_s^\alpha (z) =
\langle \tilde{O}_s^\alpha (0) \tilde{O}_s^\alpha (z) \rangle$,
$\alpha=a,~b$, does then yield the scalar (Higgs) mass while the correlator
$G_v (z) = \sum_{i=1,2}\langle \tilde{O}_{v,i} (0) \tilde{O}_{v,i} (z) \rangle $
defines the mass of a vector particle with the quantum numbers of the $W$-boson.
In both cases we also have performed an analysis of the finite size
dependence of masses extracted from these correlation functions. Some
results are given in Fig.~\ref{localmsw.fig}. This shows that also in this
case the masses may reliably be analyzed on lattices of size $16^2 \times
32$.
\begin{figure}[htb]
\begin{center}
\epsfig{file=ma3.eps, width=73mm, height=65mm}
\hfill
\epsfig{file=ma1.eps, width=73mm, height=65mm}
\end{center}
\caption{Vector ($a$) and scalar ($b$) masses at $\kappa = 0.1745$ (squares)
and 0.17484 (circles) obtained
from fits to gauge invariant correlation functions calculated on lattices
of size $L^2\times 32$ with $L$ ranging from 4 to 24. Curves show
exponential fits for the volume dependence of the masses. }
\label{localmsw.fig}
\end{figure}
The scalar (Higgs) mass has been calculated on the same set of
configurations used for the analysis of the $W$-boson propagator in Landau
gauge. Fits have been performed for distances $z\raise0.3ex\hbox{$>$\kern-0.75em\raise-1.1ex\hbox{$\sim$}} (2-4)$.
We do find quite a different behaviour for the scalar mass in the
symmetric phase while it is similar in magnitude and functional dependence
to the propagator mass in the symmetry broken phase. In all cases we
obtained consistent results from the two scalar operators defined in
Eq.~(\ref{cordefs}). The masses extracted from $G_s^a$ are shown in
Fig.~\ref{scalarmass.fig}.
\begin{figure}[htb]
\begin{center}
\epsfig{file=masse_1.eps, height=80mm}
\end{center}
\caption{Scalar masses obtained from fits to the gauge invariant correlation
function $G_s^a$ calculated on lattices of size $16^2\times 32$. }
\label{scalarmass.fig}
\end{figure}
The scalar mass becomes very small at $\kappa_c$. The behaviour is consistent
with a second order or very weak first order phase transition. The
functional dependence of the scalar mass below and above $\kappa_c$ is
clearly different. We therefore have fitted the masses to the ansatz
\beqn
m_s = c_{\pm} + a_{\pm} |(\kappa - \kappa_c)|^{\nu_{\pm}} \quad,
\label{mwfits}
\eqn
where the subscript +/- refers to the broken/symmetric phases. We find
consistent fits for $c_{\pm} \equiv 0$ as well as $c_{\pm} \ne 0$. The fit
parameters for both cases are summarized in Table~\ref{fits.tab}.
\begin{table}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\multicolumn{3}{|c|}{$\kappa \le \kappa_c$} &
\multicolumn{3}{|c|}{$\kappa \ge \kappa_c$} \\
\hline
\multicolumn{6}{|c|}{fit with $c_\pm \ne 0$} \\
\hline
$c_-$ & $a_-$ & $\nu_-$ &$c_+$ & $a_+$ & $\nu_+$ \\
\hline
0.04~(2) & 26~(9) & 0.58~(6) & 0.073~(20) & 6~(2) & 0.45~(6) \\
\hline
\multicolumn{6}{|c|}{fit with $c_\pm \equiv 0$} \\
\hline
- & 14~(2) & 0.48~(4) & - & 3.0~(5) & 0.31~(3) \\
\hline
\end{tabular}
\end{center}
\caption{Results of the fits to the masses extracted from the scalar correlation
function $G_s^a$.
In Fig.~6. we have shown the fits with $c_\pm \equiv 0$.}
\label{fits.tab}
\end{table}
We note that also this behaviour is quite similar to the results obtained from
gap equations \cite{Buc95}.
From Table~\ref{fits.tab} we conclude that the determination of critical
exponents is quite sensitive to the inclusion or exclusion of a constant
term in the fits. This shows that also here we will need a rather
detailed finite size analysis close to
$\kappa_c$ to draw definite conclusions. Still a few observations may be
appropriate already at this point: When approaching
$\kappa_c$ from below (in the symmetric phase) we find that $\nu$ is consistent
with the mean field
value 1/2. This has also been observed in another simulation of the
3-dimensional model \cite{Kaj95}.
However, when approaching $\kappa_c$ in the broken phase we
find a smaller value for the exponent $\nu$. This is particularly
true when we exclude the constant term in our fits. In any case, it does seem
that a large exponent, i.e. $\nu > 1/2$, like in the 3-d Ising model
($\nu \sim 2/3$) or the $O(4)$-model ($\nu \simeq 0.75$ \cite{Kan95}) is
ruled out by our data. It seems that the temperature dependence of the
scalar mass is very similar to that of the $W$-boson propagator mass,
although in the case of a second order phase transition both should
depend on different critical exponents. As we have argued in the
previous section the temperature dependence of the $W$-boson propagator
mass is expected to be controlled by the exponent
$\beta$ while the scalar mass is controlled by the correlation length
exponent $\nu$.
This may hint at are more complex dynamics close to $\kappa_c$ than
described by the universality class of 3-$d$, scalar spin models and may
even indicate the possible existence of a tricritical point.
A similar detailed analysis of the gauge invariant vector correlation function
on our data sets failed because the signal disappeared already at
rather
short distances ($z\simeq 4$) in the statistical noise. The construction of
improved operators may help in this channel \cite{Phi96}.
We could calculate the vector mass at the two $\kappa$ values close
to $\kappa_c$, where we did perform the finite size analysis (see
Fig.~\ref{localmsw.fig}). Here we extracted
the masses in the vector channel from a fit to the correlation functions for
$z \ge 2$. This yields
\begin{eqnarray}
m_v &=& \cases{ 0.557 \pm 0.087 \quad,\quad \kappa = 0.1745 \cr
0.356 \pm 0.028 \quad,\quad \kappa = 0.17484 }
\label{vectormass}
\end{eqnarray}
We note that in the symmetric phase at $\kappa=0.1745$ the mass in the
vector channel is more than twice as large as the mass extracted from
the $W$-boson propagator ($m_w \simeq 0.158$), whereas these masses are
similar in the symmetry broken phase. For instance, we find
from Table~\ref{corrfit.tab} at $\kappa = 0.17484$ for the $W$-boson
propagator mass $m_w = 0.308~(6)$, which is
compatible with the value given in Eq.~\ref{vectormass} for $m_v$.
\section{Conclusions}
In this paper we have established the existence of the exponential
decay of gauge field (link-link) correlations of the 3-d gauge-Higgs system
in Landau gauge which leads to a non-vanishing magnetic screening mass in
the $W$-boson propagator. In the high temperature (small $\kappa$) regime
this characteristic length scale agrees with the same quantity of
the pure gauge system.
This equality cannot be a coincidence, and gets further support from
the study of the gauge invariant excitation spectrum by Philipsen {\it et al.}
\cite{Phi96}. They report the non-mixing of a $0^{++}$ state composed of
gauge plaquettes with those operators having the same quantum numbers
and also involving Higgs fields. This decoupling phenomenon is actually
expected at high temperature; it corresponds to the separation of the heavy
scalar modes from the dynamics of the weakly screened magnetic fluctuations
which are described by an
effective theory both in the case of QCD and the gauge-Higgs system
\cite{Bra95}.
On the basis of this apparent decoupling we have argued that the magnetic
vector fluctuations do not receive any contribution to their screening mass
from a Higgs-type mechanism in the high temperature phase.
The onset of the additional mass generation through the Higgs-mechanism can be
observed as a well-localized increase of the effective mass above $\kappa_c$.
Our present calculations, which have been performed with a set of couplings
corresponding to $m_H\approx 80GeV$, suggest the existence of a second order
phase transition. While the temperature dependence of the $W$-boson
propagator mass close to $\kappa_c$ is consistent with the critical behaviour
expected from an $O(4)$ symmetric effective theory, we seem to find
deviations from this picture for the scalar mass.
However, only a very careful finite size analysis can substantiate this
observation and should allow to
distinguish from a smooth crossover suggested by \cite{Buc95} or the
critical behaviour of an effective theory possibly controlled by a
one-component scalar field as suggested in \cite{Kaj95}.
The propagator masses and the gauge invariant spectrum agree well in the
broken symmetry phase.
An important issue is to clarify why the two kinds of operators which yield the
same mass in the symmetry broken phase cease to couple to the same state in the
symmetric phase. Further investigations of gauge invariant and gauge
dependent correlation functions should lead to progress on this
question. One possibility would be, for instance, to construct also simple
non-gauge invariant two-particle operators whose correlators in the Landau
gauge could reproduce the results of the gauge invariant spectroscopy.
Another important next step towards the clarification of the nature of the
symmetric phase and its fundamental degrees of freedom is the thorough
investigation of the contribution
of the static sector to the equation of state of the finite temperature
gauge-Higgs system. Our analysis suggests that the thermodynamics
in the symmetric phase may be described in terms of almost free massive
degrees of freedom having the mass explored in the present paper.
\medskip
\noindent
{\bf Acknowledgements:}
The computations have been performed on Connection Machines at the
H\"ochstleistungs\-rechenzentrum (HLRZ) in J\"ulich, the University of
Wuppertal and the Edinburgh Parallel Computing Center (EPCC).
We thank the staff of these institutes for their support.
The work of FK has been supported through the Deutsche
Forschungsgemeinschaft under grant Pe 340/3-3. JR has partly been supported
through the TRACS program at the EPCC. A.P. acknowledges a grant from OTKA.
| 11,068 |
\section{Introduction}
\label{intro}
The first multimessenger observation of a binary neutron star (BNS) merger in 2017, combining gravitational waves (GWs) with a variety of electromagnetic (EM) signals across the entire spectrum, marked a major milestone in the investigation of these extraordinary astrophysical events (\citealt{LVC-BNS,LVC-Hubble,LVC-MMA,LVC-GRB}; see, e.g., \citealt{Ciolfi2020c,Nakar2020} and refs.~therein).
Among the numerous discoveries, the coincident detection of the gamma-ray signal GRB\,170817A and the following observation of a multiwavelength afterglow confirmed the long-standing hypothesis that BNS mergers can produce relativistic jets and power short gamma-ray bursts (SGRBs) (\citealt{LVC-GRB,Goldstein2017,Hallinan2017,Savchenko2017,Troja2017,Lazzati2018,Lyman2018,Mooley2018a,Mooley2018b,Ghirlanda2019}).
Moreover, this SGRB was observed from a viewing angle $\approx\!15^\circ-20^\circ$ away from the main jet propagation axis, offering unprecedented insights into the angular structure of the relativistic outflows associated with SGRBs (\citealt{Mooley2018b,Ghirlanda2019}; see, e.g., \citealt{Ioka2019} and refs.~therein).
Despite this breakthrough discovery, key open questions remain on both the nature of the SGRB central engine (either a massive neutron star or an accreting black hole) and the jet launching mechanism itself.
Observational data from GRB\,170817A and the accompanying afterglow signals directly probed the properties of the relativistic outflow emerging from the baryon-polluted environment surrounding the merger site, but not the physical conditions of the system at the time the incipient jet was initially launched.
In order to connect the ultimate jet structure and its EM signatures with the properties of the incipient jet (initial opening angle and power, total energy, etc.) and the post-merger environment, including the jet launching time with respect to merger, a growing effort is devoted to model the breakout and propagation of collimated relativistic outflows following BNS mergers. Such an effort, strongly boosted by the observation of GRB\,170817A, includes semi-analytical models (e.g., \citealt{Salafia2020,Lazzati2020,Hamidani2021} and refs.~therein) as well as two- or three-dimensional (magneto)hydrodynamic simulations in the framework of special or general relativity (e.g., \citealt{Nagakura2014, Lazzati2018,Xie2018,Kathirgamaraju2019,Geng2019,Nathanail2020,Murguia2021,Urrutia2021,Nathanail2021,Gottlieb2021} and refs.~therein).
While the physical description provided by the above modelling effort is continuously improving, current studies share one important limitation:
the density, pressure, and velocity distributions characterizing the surrounding environment at the jet launching time are the result of hand-made prescriptions that should reproduce a typical post-merger system, but have no direct connection with any specific merging BNS.\footnote{An exception is represented by \citet{Nativi2021}, where the environment is set by importing data from a newtonian simulation of neutrino-driven winds produced by a massive neutron star remnant \citep{Perego2014}.}
As a first step towards a consistent end-to-end description covering merger, jet launching, and jet propagation, we present here the first three-dimensional (3D) special relativistic hydrodynamic simulations of incipient SGRB jets where the initial conditions of the surrounding environment are directly imported from the outcome of a fully general relativistic BNS merger simulation.
We discuss the details of the setup and the adopted prescriptions, along with the results of a number of simulations testing different aspects of our approach.
Our findings on a fiducial model reveal the severe limitations of employing hand-made environment initial data as opposed to the outcome of actual BNS merger simulations. Moreover, they demonstrate the importance of including the effects of the gravitational pull from the central object.
Referring to the same merging BNS, we vary the time (after merger) at which the jet is launched and show how this affects the final structure and properties of the escaping jet.
This work serves mostly as a demonstration of the approach, paving the way for future explorations of the relevant parameter space and the first application to events like GRB\,170817A.
The paper is organized as follows. Section~\ref{setup} presents our setup in terms of numerical methods, initial data, grid structure, boundary conditions, and jet injection properties. Our fiducial model is discussed in Section~\ref{fiducial100}, where we also consider test cases where we remove the contribution of external forces (including gravity) and, in one case, we also substitute the initial surrounding environment with a much simpler matter distribution inspired by prescriptions typically adopted in the literature.
In Section~\ref{fiducial200}, we consider a different jet injection time and discuss the impact on the final outcome. Finally, a summary of the work and concluding remarks are given in Section~\ref{summary}.
\section{Physical and numerical setup}
\label{setup}
We perform our special relativistic hydrodynamic simulations using the publicly available code PLUTO, version 4.4 \citep{Mignone2007-PLUTO1,Mignone2012-PLUTO2}.
The code provides a multi-physics, multi-algorithm modular environment designed to solve conservative problems in different spatial dimensions and systems of coordinates, especially in presence of strong discontinuities.
We carry out our simulations using the HLL Riemann solver, piecewise parabolic reconstruction and $3^{\rm rd}$-order Runge Kutta time stepping in 3D spherical coordinates $(r,\theta,\phi)$.
When setting the computational domain in the radial direction, we ``excise'' the central region up to a radius of $r_\mathrm{exc}\!=\!380$\,km, i.e.~we do not evolve the inner part of the system. A careful choice of the boundary conditions on the corresponding spherical surface allows for angle dependent ingoing and outgoing fluxes according to the combined effects of gravitational pull and radial pressure gradients (see Section~\ref{gravity}). Moreover, the incipient jet is introduced into the computational domain from the same surface and thus our jet prescription (Section~\ref{jet}) refers to its properties at 380\,km from the central engine.
We also note that general relativistic effects, which are not accounted for in PLUTO, can be safely neglected above 380\,km.
The initial setup of our simulations is based on the outcome of a previous general relativistic BNS merger simulation, from which data are imported.
The physical and numerical setup of such BNS merger simulation is identical to the one employed in \citet{Ciolfi2020a}, except that in this case magnetic fields are not present.
In particular, the BNS system at hand has the same chirp mass as the one estimated for GW170817 \citep{LVC-170817properties}, with mass ratio $q\!\simeq\!0.9$, and the equation of state (EOS) adopted for neutron star (NS) matter is a piece-wise polytropic approximation of the APR4 EOS \citep{Akmal:1998:1804} as implemented in \cite{Endrizzi2016}. The above choices lead to a long-lived supramassive NS as the merger remnant, which would survive the collapse to a black hole (BH) for much longer than the evolution time covered by the simulation, i.e.~up to 156\,ms after merger.
The BNS merger simulation employs a 3D Cartesian grid with 7 refinements levels and finest grid spacing of $\approx\,$250$\,$m, extending up to $\approx\,$3400$\,$km along all axes. To save computational resources, we also enforced reflection symmetry across the $z\!=\!0$ equatorial plane. An artificial constant density floor of $\rho^*\simeq6.3\times10^4\,$g/cm$^3$ is also set in the numerical domain, corresponding to a total mass of $\simeq3.5\times10^{-3}\,M_{\odot}$.
We refer the reader to \citet{Ciolfi2017,Ciolfi2019} and \citet{Ciolfi2020a} for further details about numerical codes and methods.
In this work, we adopt the paradigm in which a SGRB jet is launched by the accreting BH system resulting from the eventual collapse of the massive NS remnant. While the SGRB central engine is still a matter of debate, this scenario remains the leading one and also finds support in BNS merger simulations (\citealt{Ruiz2016,Ciolfi2020a}; see, e.g., \citealt{Ciolfi2020b} for a review).
For the time being, we assume that the collapse occurs at a chosen time along the massive NS remnant evolution and we use the physical conditions of the system at that time to start our PLUTO simulations. After a short (order $\sim\!10$\,ms) transition interval to account for the effects of the forming BH-accretion disk system, we introduce the jet with a given set of properties.
This choice allows us to control the parameters of the injection, which is very convenient for a first investigation, and to explore the effects of a different collapse time on an otherwise identical system.\footnote{Importing data from BNS merger simulations directly covering the collapse and the formation of an incipient jet would represent a further crucial step towards a fully consistent end-to-end description and should be the goal of future studies.}
In the following, we discuss in detail our prescriptions, including data import, grid settings, boundary conditions, treatment of external forces, jet injection, and more.
\begin{figure*}
\includegraphics[width=1.6\columnwidth]{fig1.pdf}
\caption{Meridional view of the rest-mass density as obtained by importing data from the reference BNS merger simulation at 101$\,$ms post-merger (see text).
The left panel shows the result of the PostCactus interpolation, while the right panel shows the final setup on the PLUTO grid. The white circle of 380\,km radius in the right panel corresponds to the excised region (that we do not evolve in PLUTO).}
\label{fig0}
\end{figure*}
\subsection{Data import and computational grid}
\label{import}
Our reference BNS merger simulation leads to a remnant NS of gravitational mass $M_0\!\simeq\!2.596\,M_{\odot}$ and follows its evolution up to 156\,ms after merger. Along the evolution, we save 3D outputs of rest-mass density, pressure, 3-velocity, and specific internal energy every $\simeq\!5$\,ms.
For a chosen time at which the remnant NS is assumed to collapse, we import the corresponding data in PLUTO according to the following steps:
\begin{itemize}[leftmargin=+.3cm]
\item First, data are mapped onto a uniform 3D Cartesian grid using the PostCactus Python package\footnote{\url{https://github.com/wokast/PyCactus}} and setting the resolution of the new grid to be the same of the BNS merger simulation at $380\,$km from the origin (i.e.~$\simeq\!8.2$\,km, sixth refinement level).
In this step, we exploit the equatorial symmetry to obtain the data on the full domain (i.e.~for both positive and negative $z$).
\item Second, we apply a $90^{\circ}$ clockwise-rotation around the $x$-axis. In this way, $y\!=\!0$ becomes the new equatorial plane (and reflection symmetry plane) of the BNS merger simulation, while the $y$-axis becomes the new orbital axis.
This avoids dealing with the singularity at $\theta=0$ in our spherical coordinate system.
\item Then, data are imported in PLUTO and interpolated onto a 3D spherical coordinate grid where we remove the region $r\!<\!r_\mathrm{exc}(=380\,\mathrm{km})$.
A logarithmic increase in the grid spacing along the radial direction is adopted, allowing us to retain high resolution close to the inner boundary, where it is required, while significantly lowering the number of grid points at larger and larger distances.
\item The artificial density floor or ``atmosphere'' employed in the BNS merger simulation ($\simeq6.3\times10^4\,$g/cm$^3$) is appropriate within a distance of order $\sim\!1000$\,km, but at larger scales it needs to be replaced with a medium with density and pressure that rapidly decrease with distance.
More specifically, we import density and pressure values in the region $r_\mathrm{exc}\le r < 1477$\,km, while at larger radii we replace the artificial floor contribution with a function decaying as $r^{-a}$, where $a\!=\!5$.
At $r\!>\!2500\,$km, we only retain such decaying artificial atmosphere and do not use anymore data imported from the BNS merger simulation.
In this way, we have the freedom to fill the remaining computational domain up to the outer radial boundary, which is set to $r_{\max} = 2.5\times10^6$\,km.
In Appendix~\ref{atmo}, we analyze the impact of the atmosphere on the final outcome of our simulations by showing the results obtained with different power-law exponents for the decaying density and pressure.
\end{itemize}
As an example of the PostCactus interpolation, we report in the left panel of Figure~\ref{fig0} the results obtained for the rest-mass density at $101\,$ms after merger. On the right panel of the same Figure, we show instead the final result of the procedure discussed above to import data into the PLUTO computational grid.
To better illustrate the corresponding atmosphere replacement, we also show in Figure~\ref{import-1D} the radial profiles of the rest-mass density along the $x$- and $y$-axes.
The physical quantities from the BNS merger simulation that are used for setting up the initial data are rest-mass density, pressure, and 3-velocities, while the specific internal energy is recomputed via the EOS.
In PLUTO, we employ the Taub EOS, which corresponds to an ideal gas EOS with $\Gamma=4/3$ in the highly relativistic limit and $\Gamma=5/3$ in the non-relativistic limit, with a smooth and continuous behaviour at intermediate regimes \citep{Mignone2007}.
Since this EOS does not match exactly the EOS of the BNS merger simulation at the low densities of interest, the specific internal energy in the PLUTO setup does not coincide with the one of the original BNS merger data.
In order to ensure that such a mismatch has no relevant impact on the final conclusions of our study, we performed twice the same simulation where either (i) the pressure is directly imported and the specific internal energy is derived from the Taub EOS or (ii) the opposite. The comparison is discussed in Appendix~\ref{EOS}.
As fiducial resolution, we adopt $756\times252\times504$ points along $r$, $\theta$, and $\phi$, respectively. With a logarithmic radial grid, this yields the smallest grid spacing (at $r_\mathrm{exc}\!=\!380$\,km) of $\Delta r\!\simeq\!4.4$\,km, $r \Delta \theta\!\simeq\!4.4$\,km, and $r \Delta \phi\!\simeq\!4.7$\,km.
We also note that, to avoid the polar axis singularity, $\theta$ varies within the range $[0.1,\pi-0.1]$, while $\phi$ covers the whole $[0,2\pi]$ interval.
A resolution study is presented in Appendix~\ref{res}, where we show results for the same model with four different resolutions (including the fiducial one).
\subsection{Gravity, pressure gradients, and boundary conditions}
\label{gravity}
While general relativistic effects are not important at $r\!\gtrsim\!380$\,km, the (Newtonian) gravitational pull from the central object remains an ingredient that must be taken into account.
In particular, gravity causes the fall-back of the inner part of the slowly expanding material that constitutes the surrounding environment through which the incipient jet has to drill and, as we demonstrate in Section~\ref{forces} (see also Figure~\ref{1DExtrap_NEW}), this has an impact on the final jet energetics and collimation.
For this reason, we introduce the Newtonian gravitational acceleration
\begin{equation}\label{g}
\Vec{g} = -\,G\dfrac{M_0}{r^2}\hat{r}\, ,
\end{equation}
where $M_0$ is the gravitational mass of the merger remnant (specified above) and $G$ is the gravitational constant.\footnote{External forces (including gravity) are introduced in our simulations using the BodyForceVector() function provided by PLUTO. In order to consistently treat them in the special relativistic case, we corrected the relevant equations in the latest PLUTO release (version 4.4). More details can be found in the PLUTO User's Guide at \url{http://plutocode.ph.unito.it/documentation.html}.}
Before introducing the collapse to a BH and the subsequent launching of a jet, we tested our ability to simply reproduce the ongoing evolution of the remnant NS and surrounding environment, based on the information provided by the BNS merger simulation.
As we show in Appendix~\ref{extr}, a simple and reliable prescription can be adopted once the post-merger dynamics, under the combined action of gravitational pull, centrifugal support, and pressure gradients, has settled to a quasi-stationary state (i.e. later than $\sim\!120$\,ms after merger, for the case at hand).
At this stage, the angle-averaged radial velocity at $r\!=\!380\,$km is nearly constant in time, while the rest-mass density and pressure show an approximately linear increase (Figure~\ref{meandata_126}). The prescription consists of imposing, as radial boundary conditions at the excision radius, the initial rest-mass density, pressure, and 3-velocity as imported from the BNS merger simulation (with their original angular distributions) multiplied by a time-dependent factor that reflects the above angle-averaged trends.
For $\theta$ and $\phi$ coordinates, we impose instead zero-gradient (i.e.~``outflow'') and periodic boundary conditions, respectively.
A direct comparison with the original evolution up to 156\,ms after merger (final time of the BNS merger simulation) shows a nice match in all quantities (Appendix~\ref{extr}). In contrast, this is no longer the case when gravity is neglected (Figure~\ref{1DExtrap_NEW}).
The above result is particularly relevant as a basis to extrapolate the evolution beyond what is originally covered by the BNS merger simulation. In Section~\ref{fiducial200}, as an example, we exploit it to study the case in which the remnant NS is assumed to collapse at $201$\,ms after merger (i.e. 45\,ms beyond the reach of the original merger simulation).
\begin{figure}
\includegraphics[width=\columnwidth]{fig2.pdf}
\caption{Radial profile of the rest-mass density along the $x$- and $y$-axes for data imported from the reference BNS merger simulation at 101$\,$ms post-merger.
The blue and black dots refer to the result of the PostCactus interpolation and to the final setup in PLUTO, respectively. The vertical red- and green-dashed lines indicate, respectively, the excision radius and the radial distance (1477$\,$km) above which we replace the uniform artificial density floor with a profile decaying as $r^{-5}$ (shown with a magenta-dashed line). See text for further details.}
\label{import-1D}
\end{figure}
\begin{figure*}
\includegraphics[width=2.0\columnwidth,keepaspectratio]{fig3.pdf}
\caption{Meridional view of the rest-mass density at different times for our fiducial model (see Section~\ref{fiducial}). Left panel refers to the initial jet launching time. As in Figure~\ref{fig0}, the white circle of 380\,km radius corresponds to the excised region.}
\label{rho_small}
\end{figure*}
We now turn to consider the adopted prescription to handle the evolution after the remnant NS is assumed to collapse.
In this case, there is no direct information from the BNS merger simulation on how the evolution should proceed. As discussed above (see also Section~\ref{forces}), gravitational pull is a necessary ingredient. We introduce it along with zero-gradient radial boundary conditions, thus allowing the material to eventually cross our inner boundary $r_\mathrm{exc}\!=\!380\,$km while falling-back towards the center.
Boundary conditions for $\theta$ and $\phi$ coordinates, are again zero-gradient and periodic, respectively.
In addition, there is another aspect that should be taken into account. In a realistic evolution, BH formation does not make the pressure gradient support at 380\,km distance disappear instantaneously, but rather leads to a gradual transition in which such support fades away in time. Moreover, the characteristic timescale for the transition strongly depends on the angle with respect to the spin axis: along this axis, we expect a rather short timescale, no longer than a few tens of ms (see, e.g., \citealt{Ruiz2016}), while orthogonal to it the timescale should be similar to the accretion timescale of the disk surrounding the BH.
In order to introduce such a transition, we add an extra fading-away acceleration term. For simplicity, we consider an isotropic acceleration with the same form of a gravitational force with opposite sign
\begin{equation}\label{push}
\Vec{a} =G\dfrac{M_\mathrm{eff}(r,t)}{r^2}\hat{r} \, ,
\end{equation}
where the ``effective mass'' $M_\mathrm{eff}$ is a function of both the radial coordinate and time.
At the excision radius and at the time of data import ($t_\mathrm{in}$), we set
\begin{equation}\label{MHE}
M_{\mathrm{eff}}(r_\mathrm{exc},t_\mathrm{in})=-\,\left[r^2\dfrac{1}{\Bar{\rho}G}\dfrac{d\Bar{P}}{dr}\right]_{r_\mathrm{exc},t_\mathrm{in}} \, ,
\end{equation}
where $\Bar{P}$ and $\Bar{\rho}$ are the angle-averaged pressure and rest-mass density extracted from the BNS merger simulation.
This corresponds to having at the inner radial boundary the same initial (angle-averaged) acceleration opposing the gravitational pull as in the original system.
To limit the effect in the vicinity of the excision, we set the radial dependence as a linear decrease such that $M_{\mathrm{eff}}$ becomes zero at a characteristic radius $r^*\!=\!700$\,km (roughly twice the excision radius), i.e.
\begin{equation}\label{linear}
M_{\mathrm{eff}}(r,t)=\begin{cases}M_{\mathrm{eff}}(r_\mathrm{exc},t) \frac{r^*-r}{r^*-r_\mathrm{exc}}&\ \mathrm{if}\ r_{\mathrm{exc}}\le r\le r^* \\0&\ \mathrm{if}\ r>r^*\end{cases} \, .
\end{equation}
Finally, we set the time dependence as follows:
\begin{equation}\label{collapse}
M_{\mathrm{eff}}(r,t)=M_{\mathrm{eff}}(r,t_c)\exp{\left(-\dfrac{t-t_c}{\tau}\right)} \, ,
\end{equation}
where $t_c$($=\!t_\mathrm{in}$) is the collapse time and $\tau$ is defined as
\begin{equation}\label{timescale}
\tau \equiv \tau_j-(\tau_d-\tau_j)\sin^2{\alpha} \, ,
\end{equation}
with $\alpha$ the angle with respect to the orbital axis (or the remnant/BH spin axis, i.e.~$\theta,\phi\!=\!\pi/2$).
In the above expression, $\tau_d$ is the accretion timescale of the BH-disk system, while $\tau_j$ is a characteristic timescale connected to the delay between the collapse and the jet launching time.
Along the BH spin axis, matter is rapidly accreted on a timescale $\tau_j$, allowing the jet to emerge, while on the orbital plane, matter accretes on the much longer timescale $\tau_d$ (see discussion above).
In this work, we set $\tau_j=14.5$\,ms and $\tau_d=0.3$\,s, which is consistent in order of magnitude with what found in BNS merger simulations (e.g., \citealt{Ruiz2016}).
\subsection{Jet injection}
\label{jet}
In our PLUTO simulations, an incipient relativistic jet is introduced into the system from the inner radial boundary ($r_{\mathrm{exc}}=380\,$km), starting $11\,$ms after the time chosen for the collapse of the remnant NS.
We assume a time-dependent ``top-hat'' (i.e.~uniform) jet contained within a half-opening angle of $10^\circ$ from the $y$-axis, which corresponds to the direction orthogonal to the orbital plane of the BNS merger.
The injection is two-sided, with identical properties in the $y\!>\!0$ and $y\!<\!0$ regions (we recall that the simulation is in full 3D, without imposed symmetries).
Outside the injection angle, radial boundary conditions at $r_\mathrm{exc}$ are kept as zero-gradient for the rest-mass density, pressure, and angular components of the 3-velocity ($\mathrm{v}_{\theta},\mathrm{v}_{\phi}$).
The radial velocity obeys to the same condition as long as $\mathrm{v}_r\!<\!0$, otherwise we set $\mathrm{v}_r=0$.
Moreover, to ensure numerical stability near the injection region, we change the reconstruction to the more dissipative piecewise linear for $r_{\mathrm{exc}}\le r<385\,\mathrm{km}$ and within an angular distance of $30^{\circ}$ from the $y$-axis.
For the incipient jet properties at the initial injection time, we set a Lorentz factor $\Gamma_{0}\!=\!3$ (with purely radial outgoing motion), a specific enthalpy $h_0\!=\!100$ (corresponding to a terminal Lorentz factor $\Gamma_{\infty} \equiv h_0\Gamma_0 = 300$), and a two-sided luminosity of
\begin{equation}\label{lum0}
L_0= 4\pi r_\mathrm{exc}^2\int_0^{\alpha_\mathrm{j}}(h_0\Gamma_{0}^2\rho_0 c^2-P_0) \mathrm{v}_0\sin{\alpha'}d\alpha'=3\times10^{50}\,\mathrm{erg/s} \, ,
\end{equation}
where $\alpha'$ is the angle with respect to the jet axis, $\alpha_\mathrm{j}$ is the jet half-opening angle in radians, $\mathrm{v}_0$ is the radial velocity (corresponding to $\Gamma_{0}=3$), and $\rho_0$ and $P_0$ are the comoving rest-mass density and pressure, respectively.
In the above expression, $P_0$ is determined from $\rho_0$ and $h_0$ via the Taub EOS. Therefore, $\rho_0$ is the only remaining free parameter and can be adjusted to give the desired $L_0$.
We then impose an exponential time decay in luminosity $L(t) = L_0 \,e^{-t/\tau_d}$, with characteristic timescale $\tau_d=0.3$\,s (the same as the BH-disk accretion timescale; see Section~\ref{gravity}).
Such a decay in luminosity is achieved by means of an exponential decay with double characteristic timescale $2\tau_d$ in both the incipient jet radial velocity $\mathrm{v}_0$ and the term $(h_0\Gamma_{0}^2\rho_0 c^2-P_0)$.
The above incipient jet properties, which are within the expected range for a SGRB jet (e.g., \citealt{Lazzati2020} and refs.~therein), are employed in all the simulations discussed in this work. Investigating the effects of different injection properties is beyond our present scope and will be the subject of future studies.
\begin{figure*}
\includegraphics[width=2\columnwidth,keepaspectratio]{fig4.pdf}
\caption{Radial profiles of the angle-averaged rest-mass density, pressure, and radial velocity at 112$\,$ms after merger in our fiducial simulation (blue dots). Red lines represent the analytical fits described in Section~\ref{ball}. The vertical orange dashed line in the right panel marks the distance up to which the nearly homologous expansion regime holds. In all panels, the vertical green dashed line marks the distance at which the analytic density profile gives the same total mass of the environment as the original data.}
\label{fitbolla}
\end{figure*}
\section{Collapse at 0.1\,seconds after merger}
\label{fiducial100}
In this Section, we discuss the outcome of simulations where we set the collapse time of the remnant NS to $101$\,ms after merger.
The evolution is followed up to slightly more than 1\,s after merger.
We start by discussing our fiducial model, where the prescriptions presented in Section~\ref{setup} are applied in full.
In addition, we consider two more simulations, one where we do not include the external forces (i.e.~the gravitational pull and the extra outward acceleration accounting for the fading-away radial pressure gradients close to the excision; see Section~\ref{gravity}) and another one where we additionally replace the post-merger environment with a spherically symmetric matter distribution in homologous expansion, as often assumed in SGRB jet propagation models.
\subsection{Fiducial model}
\label{fiducial}
For our first jet simulation in PLUTO (hereafter ``fiducial'' case or model), we started from BNS merger data imported at 101\,ms post-merger as described in Section~\ref{import}. The corresponding initial data for the rest-mass density are shown in the right panel of Figure~\ref{fig0}.
The excised region (of radius 380\,km) is surrounded by a slowly expanding (maximum radial velocity $\simeq\!0.07\,c$) cloud of material of mass $\simeq\!0.02\,M_\odot$ and extending up to a radius of $\sim\!2000$\,km, with density declining with distance by a few orders of magnitude. The higher density inner region ($380\,\mathrm{km}\!\leq\!r\!\lesssim\!500$\,km) presents a significant deviation from an isotropic distribution, with a lower density funnel along the orbital axis (or remnant NS spin axis).
This is a common feature observed in BNS merger simulations (e.g., \citealt{Ciolfi2020b} and refs.~therein), resulting from the combination of the gravitational pull and the non-isotropic centrifugal support and pressure gradients.
As we discuss in this Section, the presence of such a funnel can significantly affect the initial propagation of an incipient jet.
Following the prescriptions described in the previous Section~\ref{setup}, we assume that 101\,ms is the time at which the remnant NS collapses to a BH.
After 11\,ms of evolution accounting for the formation of a central BH-disk system, i.e.~at 112\,ms after merger, we inject into the system a relativistic beam with the chosen properties (specified in Section~\ref{jet}).
In the first panel of Figure~\ref{rho_small}, we show the rest-mass density distribution at the initial time of injection.
Due to the further outflow of matter emerging from the excision surface, the total mass of the surrounding environment is about 30\% larger with respect to 101\,ms.
In Figure~\ref{fitbolla}, we report the angle-averaged rest-mass density, pressure and radial velocity at the same time.
From the right panel, we notice that the expansion is nearly homologous up to more than 600\,km.
As the incipient jet starts to propagate though the surrounding environment, a high collimation is maintained up to the breakout time, around 30\,ms later (Figure~\ref{rho_small}, second panel).
As the injection continues, more and more energy is transferred laterally to the material surrounding the jet, leading to a hot and high-pressure interface (or cocoon) and eventually to an emerging jet with a certain angular structure (see below).
On small scales, the ensuing evolution is characterized by a widening of the low density funnel excavated by the jet (Figure~\ref{rho_small}, third and fourth panels), due to a changing pressure balance with the surrounding material.
An important contribution to this effect is given by the continuous accretion of the innermost and most dense material. At 200\,ms after merger, for instance, the mass outside the excised region has already decreased by a factor of $\simeq\!4$.
Along with the above widening effect, the evolution on small scales also reveals the development of Kelvin-Helmholtz instability vortices at the jet-cocoon interface, which lead to episodes of increased baryon loading of the funnel due to portions of material that are occasionally brought in (see, e.g., \citealt{Gottlieb2019} for a discussion of a similar process).
One of this episodes is illustrated in Figure~\ref{KH_NEW} (top panels), where we show a meridional view of rest-mass density and Lorentz factor at 462\,ms after merger.
Such a process perturbs the recollimation shock at the base of the jet, contributing to the development of intermittency and deviations from axisymmetry, and reducing both the collimation and the overall efficiency in converting the injected power into radial kinetic energy.
In the central panels of Figure~\ref{KH_NEW}, we show the analogous case where we set $M_\mathrm{eff}\!=\!0$, i.e.~we remove the extra outward acceleration term mimicking the fading-away support via radial pressure gradients near the excision. In this case, matter falls back towards the center more rapidly and at 462\,ms after merger the mass above 380\,km radius is already 40\% lower than in the fiducial case.
The jet-cocoon interface is less turbulent, with only minor episodes of baryon loading within the funnel.
These differences show that a more realistic description of the post-collapse phase near the excision has a potentially relevant impact.
Figure~\ref{large_scale} (upper panels) shows in full scale (order $\sim\!10^5$\,km) the rest-mass density, internal energy density, and Lorentz factor at the end of our fiducial simulation, i.e.~1012\,ms after merger. At this time, the injection power has significantly declined ($L(t)\propto e^{-t/\tau_d}$ with $\tau_d\!=\!0.3$\,s) and the jet is composed by an ultra-relativistic ``head'' (hereafter referring to the outer high Lorentz factor portion of the outflow) whose front has reached $\simeq\!2.7\times10^5$\,km, followed by a less collimated, slower, hotter, and more turbulent tail.
The maximum Lorentz factor at the jet's head is $\Gamma\!\simeq\!40$.
From the meridional view of the Lorentz factor (Figure~\ref{large_scale}, top right panel), we also notice clear deviations from axisymmetry.
This reflects the fact that the surrounding environment imported from the BNS merger simulation is not perfectly axisymmetric and the following evolution amplifies further such deviations.
\begin{figure}
\includegraphics[width=\columnwidth]{fig5.pdf}
\caption{Meridional view of rest-mass density (left) and Lorentz factor (right) at 462\,ms after merger. Top, central, and bottom rows refer to the fiducial simulation (Sect.~\ref{fiducial}), the one with $M_\mathrm{eff}\!=\!0$ (see Sect.~\ref{gravity}), and the one without external forces (Sect.~\ref{forces}), respectively.}
\label{KH_NEW}
\end{figure}
\begin{figure*}
\includegraphics[height=0.95\textheight]{fig6.pdf}
\caption{Meridional view of rest-mass density, internal energy density, and Lorentz factor at 1012$\,$ms after merger (left to right). Top, central, and bottom rows refer to the fiducial simulation (Sect.~\ref{fiducial}), the one without external forces (Sect.~\ref{forces}), and the one with simplified isotropic environment (Sect.~\ref{ball}), respectively.}
\label{large_scale}
\end{figure*}
\begin{figure}
\includegraphics[width=\columnwidth]{fig7.pdf}
\caption{Radial profiles of maximum Lorentz factor at 1012\,ms after merger (maximum value achieved at each radial distance). North and south profiles are shown for the same three cases of Figure~\ref{large_scale}: our fiducial simulation (top), the one without external forces (center), and the one with simplified isotropic surrounding environment (bottom).}
\label{gamma_max}
\end{figure}
Figure~\ref{gamma_max} (top panel) shows the radial profile of the Lorenz factor at 1012\,ms after merger, where the maximum value is reported for each radius. At the jet's head, north and south profiles are nearly identical to each other.
We also investigate the angular dependence of the Lorentz factor at the jet's head (and at 1012\,ms after merger), by computing the radial-average $\Bar{\Gamma}$ within the interval $r\!\in (24,27)\times10^4$\,km along different directions.
In the top-left panels of Figures~\ref{GammaVSalpha_xy} and \ref{GammaVSalpha_yz}, we consider in particular the resulting angular profiles on the $xy$- and $yz$-planes, respectively, referring to the north side only (south profiles are very similar). The profiles are given in terms of the angle $\alpha$, measuring the angular distance from the injection axis (or $y$-axis) and with positive/negative sign for positive/negative $x$ and $z$, respectively.
On the $xy$-plane, the angular distribution of $\Bar{\Gamma}$ appears very asymmetric between positive and negative $\alpha$ values and the maximum occurs $\simeq\!0.7^{\circ}$ away from the injection axis. On the $yz$-plane, the distribution is only slightly asymmetric and the (higher) maximum is achieved for $\alpha\!\simeq\!0.8^{\circ}$. The rather different profiles in the two planes confirm that the full 3D distribution strongly deviates from axisymmetry.
The direction containing the absolute maximum of $\Bar{\Gamma}$ in 3D is tilted by $\simeq\!0.9^{\circ}$ with respect to the $y$-axis (or injection axis).
It is worth noting that angular profiles like those obtained on the $xy$- and $yz$-planes cannot be reproduced by a simple Gaussian or power-law function. In particular, the central peak ($\Bar{\Gamma}\!\gtrsim\!20$) can be nicely fit with a skewed function
\begin{equation}
\mathrm{S}(\alpha)\propto e^{-\frac{(\alpha-\Bar{\alpha})^2}{2\sigma^2}}\left[1+\mathrm{erf}\left(\beta\dfrac{(\alpha-\Bar{\alpha})^2}{2\sigma^2}\right)\right] \, ,
\end{equation}
with characteristic half-width $\sigma\!=\!1.66^{\circ}$ and $3.34^{\circ}$, respectively (analogous to the Gaussian $\sigma$ parameter).
However, the full profiles present additional lateral wings, particularly prominent on the $xy$-plane, that are hard to fit with any simple function and cannot be neglected when considering the jet energetics (see below).
We also analyze the energy content of the emerging outflow (in the region $r\!>\!3000$\,km) at 1012\,ms after merger. The total kinetic energy is $\simeq\!3.7\!\times\!10^{49}$\,erg, while the internal energy is $\simeq\!5.4\!\times\!10^{48}$\,erg, i.e.~about 15\% of the kinetic one. This ratio confirms a substantial (but not yet complete) conversion of heat into outflowing motion.
The sum of the above kinetic and internal energies accounts for $\simeq\!48.5\%$ of the total injected energy ($\simeq\!8.7\!\times\!10^{49}$\,erg), where about half of the latter is instead lost due to the gravitational pull acting on the environment material (see also the next Section, where the effects of removing the gravitational pull are discussed).
At the jet's head, taking as a reference the shell given by the radial interval $r\!\in (24,27)\times10^4$\,km and defining the ``core'' as the region within an angle of $\sigma\!=\!3.34^{\circ}$ from the maximum Lorentz factor direction (chosen as the largest $\sigma$ among the $\Bar{\Gamma}$ angular profiles on the $xy$- and $yz$-planes), we obtain
\begin{align*}
&\mathrm{E_{kin,core} \simeq 2.947\times10^{48}\,} \mathrm{erg} \, , \ &\mathrm{E_{kin,shell} \simeq 6.806\times10^{48}\,} \mathrm{erg} \, ,\\
&\mathrm{E_{tot,core} \simeq 3.790\times10^{48}\,} \mathrm{erg} \, , \ &\mathrm{E_{tot,shell} \simeq 8.842\times10^{48}\,} \mathrm{erg} \, .
\end{align*}
In the core, kinetic energy contributes to 78\% of the total energy. The contribution of the core total energy compared to the whole shell is $\simeq 43\%$.
The bottom-left panels of Figures~\ref{GammaVSalpha_xy} and \ref{GammaVSalpha_yz} show the angular profiles (on the $xy$- and $yz$-planes, respectively) of the isotropic equivalent energy $E_\mathrm{iso}$ (kinetic plus internal) of the jet's head, i.e.~within the radial interval $r\!\in (24,27)\times10^4$\,km. Also in this case, only the profiles on the north side are reported, since the ones on the south side are nearly coincident.
As for the radial-averaged Lorentz factor, the $E_\mathrm{iso}$ angular profiles are characterized by a slightly offset and asymmetric central peak that is well reproduced by a skewed function (with characteristic half-width $\sigma\!\simeq\!2^{\circ}$) and by additional lateral wings that are highly (moderately) prominent and asymmetric on the $xy$-plane ($yz$-plane).
In 3D, $E_\mathrm{iso}$ reaches a maximum of $\simeq\!2.8\times10^{51}$\,erg, occurring about $0.8^{\circ}$ away from the injection axis.
\subsection{Impact of external forces}
\label{forces}
Our second simulation is analogous to the fiducial one (discussed in the previous Section), except that in this case we switch off the acceleration terms that account for the gravitational pull and the fading-away radial pressure gradient support near the excision (see Section~\ref{gravity}).
Although the inclusion of the above external forces makes the simulations arguably more consistent, the corresponding effects are commonly neglected in SGRB jet propagation studies (excluding those based on general relativistic simulations, e.g.~\citealt{Kathirgamaraju2019,Nathanail2021}).
Here, we aim at assessing whether this choice might have a significant impact on the final jet properties.
A visual comparison at the final simulation time (1012\,ms after merger) is provided in Figure~\ref{large_scale}, where the upper row refers to the fiducial model, while the central row refers to the simulation without external forces.
Looking at the rest-mass density close to the excision surface, we note that in the latter case a more massive environment surrounds the low density funnel, due to the fact that in absence of a gravitational pull material does not fall back towards the center nor gets accreted.
As a consequence, not only the widening effect depicted in Figure~\ref{rho_small} is substantially reduced, maintaining a higher degree of collimation,
but also the generation of Kelvin-Helmholtz instabilities discussed in the previous Section is strongly inhibited. To illustrate this, in the bottom panels of Figure~\ref{KH_NEW} we show the rest-mass density and Lorentz factor on the meridional plane at 462\,ms after merger, to be compared with the fiducial case in the top panels of the same Figure.
The shear at the jet-cocoon interface is much more stable and no episodic baryon loading of the funnel is noticed. This results in an essentially unperturbed recollimation shock, which allows the incipient jet to preserve a higher and more stable Lorentz factor.
At the same time, the internal energy density (see Figure~\ref{large_scale}) reveals a hotter and more uniform outflow up to $\sim\!10^5$\,km.
At the end of the simulation (see Figure~\ref{large_scale}), the jet reaches a maximum Lorentz factor almost a factor of 2 higher than the fiducial case, as also reported in Figure~\ref{gamma_max}. The acceleration is however more gradual and the distance reached by the outflow at the final time is similar (if not even slightly smaller). Moreover, because of the higher and more persistent collimation and Lorentz factor at the base of the outflow, the jet's head is now followed by a rather well-defined tail extending down to the excision surface.
Computing the internal and kinetic energies of the emerging outflow for $r\!>\!3000$\,km and at 1012\,ms after merger, we find that they sum up to $\simeq\!1.1\times10^{50}$\,erg, i.e.~about 127\% of the total injected energy. On the one hand, this indicates that the absence of the gravitational pull, unlike the fiducial case, preserves the injected energy. On the other hand, it shows that additional energy residing in the initial environment is carried along by the incipient jet.
From this example, we conclude that neglecting the external forces (in particular gravity) can have major effects on the final outcome.
\subsection{Realistic versus simplified environment}
\label{ball}
For the simulation discussed in this Section, we further reduce the degree of realism by not only neglecting the external forces (like in the previous Section), but also substituting the matter distribution and velocities of the surrounding environment imported from the BNS merger simulation with simpler analytic prescriptions.
In particular, we adopt the common assumptions of (i) a spherically symmetric matter distribution with density and pressure decreasing with radius as a power-law and (ii) homologous expansion.
In order to produce an initial setup with the above assumptions that is the closest to what we have in the case of imported BNS merger data, we fit the angle-averaged rest-mass density and pressure at 112\,ms after merger with power-law radial functions. Similarly, we fit the angle-averaged radial velocity with a linear function of the radius.
We limit the fits within a distance of $\simeq\!645$\,km, up to which the three angle-averaged profiles are rather well reproduced (Figure~\ref{fitbolla}). The resulting analytical functions are:
\begin{align}
&\Bar{\rho}_{\mathrm{fit}}(r) = 1.056\times10^8\times\left(\dfrac{r}{380\,\mathrm{km}}\right)^{-3.981}\, \mathrm{g/cm^3}\\
&\Bar{P}_{\mathrm{fit}}(r) = 6.408\times10^{25}\times\left(\dfrac{r}{380\,\mathrm{km}}\right)^{-3.320}\, \mathrm{dyne/cm^2}\\
&\Bar{\mathrm{v}}_\mathrm{fit}(r) / c = 0.047\times\left(\dfrac{r}{380\,\mathrm{km}}\right)-0.037 \, .
\end{align}
Then, for our simulation setup, we impose an isotropic environment following the above profiles and extended up to a radial distance of $\simeq\!843\,$km. Such a distance is chosen in order to have a total mass of the environment equal to the one in the fiducial simulation. Finally, we add an artificial atmosphere identical to the one imposed in the fiducial case (see Section~\ref{import}).
\begin{figure}
\includegraphics[width=\columnwidth]{fig8.pdf}
\caption{Angular profiles (north side only) on the $xy$-plane of the radial-averaged Lorentz factor (top) and isotropic-equivalent energy $E_\mathrm{iso}$ (bottom) at the jet's head (see text for details) and at 1012\,ms after merger. Left panels refer to our fiducial simulation, while right panels refer to the one with simplified isotropic surrounding environment (Section~\ref{ball}).
Here, $\alpha$ is the angle with respect to the injection axis (i.e. the $y$-axis), with positive/negative sign for positive/negative $x$.
The green curve in the left panels corresponds to the skewed function that best-fits the central peak of the profile. The vertical dashed blue line marks the angular position of the peak.}
\label{GammaVSalpha_xy}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{fig9.pdf}
\caption{Same as Figure~\ref{GammaVSalpha_xy} for the $yz$-plane. In this case, the sign of $\alpha$ corresponds to the sign of $z$.}
\label{GammaVSalpha_yz}
\end{figure}
\begin{figure*}
\includegraphics[width=1.6\columnwidth]{fig10.pdf}
\caption{Same as Figure~\ref{rho_small} for the model with collapse at 201\,ms (and jet launching at 212\,ms).}
\label{rho_small2}
\end{figure*}
Figures~\ref{large_scale} and \ref{gamma_max} (bottom row) show the simulation results in terms of rest-mass density, internal energy density, and Lorentz factor at the final time of 1012\,ms after merger.
To evaluate the effects of a simplified analytical and isotropic environment, we compare with the results of the previous simulation (without external forces, central row in Figs.~\ref{large_scale} and \ref{gamma_max}). The impact is substantial. The absence of a lower density funnel along the injection axis in the surrounding material makes it much harder for the jet to emerge, resulting in a breakout time delayed by $\simeq\!100\,$ms, a final maximum Lorentz factor around 30 (more than a factor of 2 lower and notably with a non-negligible north/south difference), and a less compact jet's head that has reached only $\simeq\!2.5\times10^5$\,km at its front.
On the other hand, after the initial breakout, the collimation at the base of the jet is more persistent and the jet's tail maintains a conical structure (Figure~\ref{large_scale}, bottom row). Finally, the overall jet structure is nearly axisymmetric and well aligned with the injection axis, differently from the cases with a non-isotropic initial environment (Figure~\ref{large_scale}).
To further illustrate the effects of a simplified isotropic environment on the emerging jet structure, the right panels of Figures~\ref{GammaVSalpha_xy} and \ref{GammaVSalpha_yz} report the angular profiles of radial-averaged Lorentz factor (top) and isotropic equivalent energy (bottom) at the jet's head and at 1012\,ms after merger, on $xy$- and $yz$-planes, respectively.
We refer again to the jet's head, which is defined in this case by the radial range
$r\!\in (18,25)\times10^4$\,km (see Figure~\ref{gamma_max}, bottom panel).
A direct comparison with the fiducial case (left panels of the same Figures) reveals not only much lower peak values, but also significantly smaller deviations from axisymmetry, i.e.~there are smaller differences between positive and negative $\alpha$ values on each plane, as well as between the two planes.
These results indicate that simplified analytical prescriptions for the surrounding environment, corresponding to what is often assumed, may substantially weigh on the jet dynamics and morphology when compared to the more realistic conditions obtained in BNS merger simulations.
\section{Collapse at 0.2\,seconds after merger}
\label{fiducial200}
In this Section, we discuss the results of a simulation with collapse time of the remnant NS set to $201\,$ms after merger. Unlike the fiducial case presented in Section~\ref{fiducial}, here the collapse time is not covered by the BNS merger simulation, which is limited to $156\,$ms. Therefore, in order to obtain the initial data for the following incipient jet evolution, we first need to continue or extrapolate the evolution from 156 to 201\,ms.
We treat such a case with a double purpose: (i) showing the feasibility of this kind of extrapolation and (ii) investigating the effect of a significantly different remnant NS lifetime (by a factor 2 in this case).
The data imported at the latest available time of the BNS merger simulation provide a different environment around the remnant NS. Figure~\ref{rho_small2} (left panel) shows in particular the rest-mass density at that time. Compared to 101\,ms post-merger (Figure~\ref{rho_small}), we observe a larger cloud of slowly expanding material resulting from the nearly isotropic baryon-loaded wind from the remnant NS. Radial motion is nearly homologous and radial velocity reaches a maximum of $\approx\!0.03\,c$ at $\approx\!1.5\times10^3\,$km.
The inner and higher density region up to $\approx\!500\,$km remains rather unchanged and a lower density funnel along the y-axis is still present.
In order to evolve the system in PLUTO from 156 to 201\,ms post-merger, we first perform a 30\,ms test simulation from 126 to 156\,ms after merger following the prescriptions given at the beginning of Section~\ref{gravity}, in particular for the radial boundary conditions at the excision radius.
The result, discussed in Appendix~\ref{extr}, shows a good match with the actual BNS merger simulation, giving us confidence to extrapolate the evolution at later times.
We then import the original data at 156\,ms after merger and use a prolongation of the same time-varying radial boundary conditions to evolve up to 201\,ms (i.e.~for 45\,ms, not much longer than the 30\,ms of the test).
At $201\,$ms post-merger, the remnant NS is assumed to collapse and the following evolution consists, as in the fiducial case and the other cases discussed in Section~\ref{fiducial100}, of 11\,ms of post-collapse rearrangement and the subsequent jet injection.
The only missing ingredient to evolve the system after collapse is the initial value of the effective mass $M_\mathrm{eff}$ (see Section~\ref{gravity}), which cannot be retrieved in this case from the BNS merger simulation.
As shown in Figure~\ref{Meff_decay}, the last part of the original simulation reveals a clear decreasing trend in the radial pressure gradients, which corresponds to a decreasing effective mass at the excision radius. An exponential decay with characteristic damping time of 58.8\,ms reproduces well the decreasing profile.
Adopting the corresponding fitting function, we obtain the value of $M_\mathrm{eff}$ at the desired time (201\,ms after merger).
Also in this case, the simulation covers up to 900\,ms after the jet launching time, i.e.~up to $1112\,$ms after merger.
Figure~\ref{rho_small2} shows the rest-mass density at the time of jet launching and around the time the jet itself breaks out of the surrounding environment (central and right panels, respectively).
We note that the total rest mass outside the excised region at the jet launching time ($\simeq\!9.4\times 10^{-2}\,M_\odot$) is a factor $\simeq\!3.6$ larger than in our fiducial model, due to the longer remnant NS lifetime. As a consequence, the incipent jet takes significantly longer to break out ($\simeq\!90$\,ms instead of $\simeq\!30$\,ms).
\begin{figure}
\includegraphics[width=\columnwidth]{fig11.pdf}
\caption{Evolution of the effective mass at the excision radius (380\,km) at times $\geq\!141\,$ms after merger. Red triangles are the values of $M_{\mathrm{eff}}(r_{\mathrm{exc}})$ extracted from the BNS merger simulation, while the dashed blue curve corresponds to an exponential fit (see text).}
\label{Meff_decay}
\end{figure}
The effects of a longer NS lifetime and a later jet launching on the final jet structure (900\,ms after jet launching) can be appreciated in Figure~\ref{large_scale2}, where we show the rest-mass density and the Lorentz factor at large scales on the $xy$-plane.
Even though the jet injection parameters are the same as in the fiducial model, we observe important differences that are to be attributed to the different initial environment.
The jet needs to drill through a more massive and extended cloud of material, spending more power to break out and reaching, after 900\,ms, a significantly smaller radial distance (jet's head front is located at $\simeq\!2.2\times10^5\,$km, to be compared with $\simeq\!2.7\times10^5\,$km of the fiducial model).
Also the maximum Lorentz factor at the jet's head is smaller ($\simeq\!22$ instead of $\simeq\!40$).
At the same time, the denser environment close to the excision surface maintains a higher degree of collimation and axisymmetry in the outflow.
For a given BNS merger model, the time interval between merger and jet launching is confirmed as a key parameter in determining the ultimate jet structure, which, in turn, shapes the corresponding radiative signatures. This offers good prospects for constraining such a time interval via the comparison with the observations (e.g., \citealt{Zhang2019,Gill2019,Lazzati2020,Beniamini2020}).
\section{Summary and conclusions}
\label{summary}
In this paper, we presented 3D special relativistic hydrodynamic simulations of incipient SGRB jets propagating through the baryon-polluted environment surrounding the remnant of a BNS merger. For the first time, we employ initial data for the environment obtained by directly importing the outcome (i.e.~density, pressure, and velocity distributions) of a general relativistic BNS merger simulation.
This represents a first key step towards a consistent end-to-end description connecting the details of a specific BNS merger with the ultimate EM signatures associated with the breakout and propagation of an emerging SGRB jet.
The simulations are performed with the PLUTO code, using rotated spherical coordinates (with polar axis lying on the BNS orbital plane) and logarithmic spacing along the radial direction.
A central sphere of $380\,$km radius is excised and suitable boundary conditions are adopted on the corresponding spherical surface.
We also include the gravitational pull of the central object (with a mass of $\simeq\!2.596\,M_{\odot}$) and, after collapse, an extra (time and space dependent) radial acceleration term added to mimic a gradual fading away of the radial pressure gradient support close to the excision surface, as expected in a realistic post-collapse evolution.
The computational domain is initially filled with an artificial atmosphere with density and pressure scaling with distance as $\propto\!r^{-5}$ and zero velocity (also tested along with different choices of the power-law exponent).
Finally, we adopt the Taub EOS. Being slightly different from the EOS used in the reference BNS merger simulation (at the low rest-mass densities of interest, i.e.~$<\!10^8$\,g/cm$^3$), small differences arise in the specific internal energy of the initial setup. However, dedicated benchmarks show that the influence on the final outcome is minor.
For the jet injection, we adopt the paradigm in which the central engine powering the relativistic outflow is an accreting BH-disk system, formed after the eventual collapse of a massive NS remnant.\footnote{In principle, our approach is also applicable to the case of a massive NS central engine. In this case, however, the formation of a jet might be difficult to justify unless directly covered (or at least strongly supported) by the BNS merger simulation itself.}
In this work, we assume that the collapse occurs at a chosen time after merger. This is the time at which we import data from the reference BNS merger simulation and start the evolution in PLUTO.
The jet is launched after a short time window (set here to 11\,ms) from the collapse of the remnant NS, compatibly with the expected delay characterizing the formation of an incipient jet from a newly formed BH-disk system.
The incipient jet properties are the same in all our jet simulations.
A top-hat outflow is continuously injected from the excision surface and within a half-opening angle of $10^{\circ}$ around the remnant spin axis (or orbital axis of the BNS).
The initial luminosity, Lorentz factor, and specific enthalpy are $L_0\!=\!3\!\times\!10^{50}$\,erg/s, $\Gamma_0\!=\!3$, and $h_0\!=\!100$, respectively.
An exponential time decay in luminosity is enforced with a characteristic timescale of 0.3\,s, which is consistent with the order of magnitude of typical accretion timescales of BH-disk systems formed in BNS mergers.
Our simulations probe two different collapse times, namely 101 and 201\,ms after merger.
For our fiducial case (collapse at 101\,ms), we also repeat the simulation without the contribution of external forces (i.e.~gravity and the extra acceleration compensating for the missing radial pressure gradients after collapse) and in one case we also substitute the environment with an isotropic and homologously expanding one. \\
The main results of our study can be summarized as follows:
\begin{itemize}
\item \textit{Realistic post-merger environment.}
The density and velocity distributions of the material surrounding the merger remnant at the jet launching time, which depend on the details of the specific BNS system, can deviate significantly from the simplified isotropic and homologously expanding medium often considered in SGRB jet propagation studies. Comparing a reference model (with 112\,ms post-merger jet launching time and no external forces) with an equivalent one where the environment is substituted by the best-fitting isotropic and homologously expanding medium having the same total mass (see Section~\ref{fiducial100}), we find major differences in the outcome. The presence of a lower density funnel along the remnant spin axis allows the incipient jet to breakout more efficiently, retaining a higher energy. This, in turn, results in a much larger maximum Lorentz factor ($\Gamma \gtrsim\!70$ vs.~$\simeq\!30$, at 1012\,ms after merger) and a higher degree of collimation at the jet's head. Another relevant effect is caused by deviations from axisymmetry in the environment (e.g., due to the remnant recoil in unequal mass mergers), which make the final jet slightly misaligned with respect to the orbital axis and significantly less axisymmetric in structure.\footnote{This may also translate in uncertainties in GW-based Hubble constant estimates (e.g., \citealt{LVC-Hubble,Hotokezaka2019}).} \\
This example poses a strong caveat for any model neglecting the anisotropy in matter distribution of the post-merger environment. Moreover, it shows that the final jet properties can be affected by other features that are typically not considered, such as deviations from axisymmetry or velocity distributions that are more complex than a simple homologous expansion.
\begin{figure}
\includegraphics[width=\columnwidth]{fig12.pdf}
\caption{Meridional view of rest-mass density (left) and Lorentz factor (right) at $1112\,$ms after merger for the case with remnant NS collapse at $201$\,ms. The spatial and color scales used here are the same adopted in Figure~\ref{large_scale}.}
\label{large_scale2}
\end{figure}
\item \textit{Impact of gravity.} A proper description of the environment dynamics should take into account the gravitational pull of the central object, as also shown by the direct comparison with the BNS merger simulation results (e.g., Figure~\ref{1DExtrap_NEW}).
When gravity is included, the dynamics of the jet propagation is significantly affected. In particular, the surrounding material is allowed to fall back towards the central engine. The falling material directly encountered by the incipient jet acts as an obstacle, dissipating part of the jet energy into heat and turbulent motions. At the same time, accretion keeps reducing the overall mass of the environment, changing the lateral pressure balance between the jet and the surrounding material in favour of the former and leading to a significant widening of the jet's opening angle shortly above the injection radius.
The decreasing collimation ultimately results in a more compact jet's head followed by a much wider and slower outflow.
Finally, due to the gravitational pull acting on the environment material, the internal and kinetic energies of the emerging outflow only carry about half of the total injected energy.
Compared to the equivalent case with no gravitational pull, the maximum Lorentz factor achieved is much lower (almost a factor of 2 at 1012\,ms after merger), more energy is deposited in the cocoon, and there is no well defined jet's tail.\\
In conclusion, the gradual and continuous accretion of the surrounding material, while being overlooked in most SGRB jet propagation models, can have a strong influence on the emerging jet properties.
\item \textit{Fiducial model.} Our fiducial simulation, where the remnant NS collapses at 101\,ms after merger and the incipient jet is launched 11\,ms later, results in a final jet that has successfully emerged from the BNS merger environment.
At 1012\,ms after merger, the internal-to-kinetic energy ratio of the outflow (for $r\!>\!3000$\,km) is about 15\%, indicating an advanced stage of conversion of heat into motion. The angular profiles of Lorentz factor and isotropic-equivalent energy at the jet's head reveal a central narrow core (of half-opening angle $\simeq\!3^{\circ}$ and $\simeq\!2^{\circ}$, respectively) surrounded by a wider and moderately relativistic outflow carrying a significant fraction of the total energy.
These angular profiles also appear very different on the $xy$- and $yz$-planes, showing strong deviations from axisymmetry. Moreover, Lorentz factor and isotropic-equivalent energy peak along a direction that is slightly tilted with respect to the injection axis (by $0.7^{\circ}-0.9^{\circ}$). Such angular dependences cannot be reproduced via simple Gaussian or power-law functions. A skewed normal function can fit well the central core, but not the very asymmetric lateral wings.
This result suggests that employing simple functions to fit SGRB jet angular structures (as revealed, e.g., by afterglow observations) may require some caution.
\item \textit{Dependence on the jet launching time.} When considering a jet launching time of 212\,ms post-merger (almost double with respect to the fiducial case), the very same incipient jet has to drill through a significantly more massive environment (factor $\simeq\!3.6$). As a consequence, more energy is dissipated into the surrounding material, it takes longer to break out ($\simeq\!90$\,ms vs.~$\simeq\!30$\,ms), and the maximum Lorentz factor reached is lower ($\simeq\!22$ vs.~$\simeq\!40$ at 900\,ms after jet launching). On the other hand, the more expanded and massive environment provides a more efficient collimation at the base of the jet, also resulting in a more axisymmetric final structure. The comparison with the fiducial case confirms that the time interval between merger and jet launching can have a strong influence, thus offering the opportunity to tightly constrain such a key parameter via observations. \\
While we consider here jet launching times of up to $\approx\!200$\,ms after merger, significantly longer delays are possible. In the case of GRB\,170817A, for instance, there is an ample range of delays favoured by different authors, going from a few hundred ms (e.g., \citealt{Zhang2019,Lazzati2020}) to order $\sim\!1$\,s (e.g., \citealt{Gill2019,Murguia2021}).
Within our setup, considering later jet launching times would require longer BNS merger simulations, beyond our current reach, and/or more extended extrapolations. The latter, to be considered reliable, would likely demand a more refined scheme (compared to what is done here) and in-depth testing against BNS merger simulation results.
A possible consequence of a later jet launching time could be that, due to the slower rate of change in the environment mass, the final outcome will depend less on the time delay itself. This represents an interesting issue open for investigation.
\end{itemize}
The main aim of this work is to introduce a new approach to address the problem of SGRB jet propagation in BNS merger environments, showing the potential advantages of employing the outcome of BNS merger simulations as initial data.
The prescriptions and assumption adopted here, while attempting to offer a more realistic description of some aspects of the system dynamics, leave plenty of room for further improvement.
In particular, we do not include magnetic fields, which are a key ingredient in SGRB jet production and evolution.
Furthermore, the incipient jet is introduced by hand and not produced self-consistently in the BNS merger simulation.
Overcoming the above limitations (among others) should represent a priority in future studies.
\section*{Acknowledgements}
We thank the anonymous referee for very helpful remarks and constructive comments on the manuscript.
We also thank Om Sharan Salafia and Stefano Ascenzi for useful discussions.
J.V.K. kindly acknowledges the CARIPARO Foundation for funding his PhD fellowship within the PhD School in Physics at the University of Padova.
All the simulations were performed on GALILEO and MARCONI machines at CINECA. In particular, we acknowledge CINECA for the availability of high performance computing resources and support through awards under the ISCRA and the MoU INAF-CINECA initiatives (Grants \texttt{IsB18\_BlueKN, IsB21\_SPRITZ, INA20\_C6A49, INA20\_C7A58}) and through a CINECA-INFN agreement, providing the allocations \texttt{INF20\_teongrav} and \texttt{INF21\_teongrav}.
\section*{Data Availability}
The data underlying this article will be shared on reasonable request to the corresponding authors.
\bibliographystyle{mnras}
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\section{Introduction}
\label{sec:intro}
Cosmic Rays have been known and studied for more than a century now (see e.g. \cite{Amato14,Blasi13} for recent reviews). They are highly energetic charged particles, mainly protons and He nuclei, with a minor fraction of heavier nuclei (1\%), electrons (2\%) and anti-matter particles (positrons and anti-protons, 1 \textperthousand). Their origin was associated with Supernova (SN) explosions in the Galaxy already in the 1930s and the mechanism by which they would be accelerated up to very high energies in the blast waves emerging from SN explosions was proposed in the late 1970s (see \S~\ref{sec:observations}).
This association, however, has not yet found direct proof, and, as we discuss below, some recent developments, in both theory and high energy astrophysical observations, have actually cast doubts on whether Supernova Remnants (SNRs) should be considered as the primary CR accelerators in the Galaxy throughout the entire energy range, up to PeV ($10^{15}$ eV) energies. At the same time, direct CR measurements have shown that particle transport is more complex than traditionally considered, and that additional sources are likely needed at least for CR positrons (see e.g.~\cite{amatoblasi18} for a recent review, and Sec.~\ref{sec:cranti} for more details).
In recent years it has become increasingly clear that understanding the processes that govern the acceleration and transport of energetic particles is a necessary key to unveil a number of unsettled questions in Astrophysics. Energetic particles - which we will often refer to as Cosmic Rays, in the following - play a role in many astrophysical systems. Just to mention a few, from the smallest to the largest scales, CRs affect the behaviour of planetary magnetospheres \citep{Griessmeier15}, they are believed to be the main ionizing agents penetrating dense gas clouds and regulating star formation \citep{Padovani20}, they are suspected to be the primary drivers of galactic winds and the rulers of galactic feedback \citep{Buck20}, and might even have a role in the generation of intergalactic magnetic fields \citep{Donnert18}. In some of these situations the interaction between CRs and the ambient plasma is simply described in terms of momentum exchange, in some others the nature of CRs as charged particles, carrying an electric current, is instead fundamental. In all cases, the correct description of the interaction between CRs and the ambient medium is an extremely challenging plasma physics problem (see e.g. \cite{Zweibel13}).
All the information we can use to try to understand this problem comes from two kinds of observations: direct detection of CRs at Earth and detection of non-thermal radiation from astrophysical sources. In the last two decades both types of observations have experienced enormous developments.
In terms of indirect detection, most of the news have come from high energy telescopes observing the sky in the X-ray and gamma-ray bands. The launch of {\it Chandra} was quickly followed by two important results in terms of CR physics. First of all, the detection of non-thermal keV photons from SNRs highlighted the presence of multi-TeV electrons in these sources: since in most of the acceleration mechanisms considered in Astrophysics - and certainly in the one proposed for SNRs - the acceleration process only depends on particle rigidity, this was also taken as indirect evidence of the presence of multi-TeV hadrons in SNRs. In addition, the excellent spatial resolution of {\it Chandra} allowed to resolve the region of non-thermal X-ray emission, showing it to be very thin, with a typical size of few \% of pc (see \cite{Vink12} for a review). Such a small size, when interpreted as due to synchrotron energy losses of the emitting electrons, indicates the presence of a magnetic field of several $\times$ 100 $\mu$G downstream of the SNR shock. Such a large value of the magnetic field cannot simply result from shock compression of the typical interstellar field ($B_{\rm ISM}\approx 3 \mu$G), but rather requires an extremely efficient amplification process to be at work.
At the same time of these discoveries, also theory was making a major step forward with the identification of a mechanism through which CRs could provide magnetic field amplification well beyond the quasi-linear theory limit of $\delta B/B\approx 1$. The process invoked is the non-resonant streaming instability \citep{Bell04} which we will discuss more in \S~\ref{sec:cracc}: this is a current induced instability that operates in situations where the CR energy density dominates over that of the background magnetic field. This condition is easily realized in young, fast expanding SNRs, if CRs are accelerated with efficiency of order 10\%, as required by the SNR-CR connection paradigm. The X-ray observations showing evidence for amplified magnetic fields were then taken as a proof of this connection: the amplified magnetic field implied that efficient CR acceleration was ongoing, and provided, at the same time, the ideal conditions for CRs to reach very high energies.
At higher photon energies, the gamma-ray satellites {\it AGILE} and {\it Fermi}, observing the sky in the 20 MeV - 300 GeV energy range, and the Cherenkov detectors H.E.S.S., VERITAS and MAGIC operating between tens of GeV and $\sim 10$ TeV, were providing, in the meantime, interesting but somewhat surprising data. For the first time, direct evidence of relativistic protons was found in two middle-aged ($\sim 20-30$ kyr old) SNRs interacting with molecular clouds, W44 \citep{FermiW44,AgileW44} and IC443 \citep{AgileIC443,FermiIC443}. These sources were not expected to be efficient CR accelerators and indeed the proton spectrum inferred from observations is cut-off at relatively low energies, $\sim 10$ GeV in W44, $\sim 10$ TeV in IC443. Follow-up theoretical studies showed that, in fact, at least in the case of W44, we are likely not witnessing fresh acceleration of CRs, with particles directly extracted from the interstellar plasma crossing the shock, but rather re-acceleration of particles from the galactic CR pool \citep[e.g.][]{,Uchiyama+10,Cardillo+16}. As far as young sources are concerned, gamma-ray observations have been partly disappointing, failing to provide indisputable evidence of hadronic acceleration \citep{FunkRev}, and often showing steeper spectra than theoretically predicted in the case of efficient acceleration \citep{Amato14}. We will discuss gamma-ray observations of SNRs and the implications of these findings on CR acceleration in SNRs in \S~\ref{sec:observations} and \S~\ref{sec:cracc}, respectively.
In terms of direct detection, experiments such as PAMELA, AMS-02 and CREAM have provided us with very precise measurements of the CR spectrum and composition up to TeV energies. Among the many discoveries, the most important ones in terms of implications on our understanding of CR acceleration and transport are: 1) the detection of a hardening in the spectra of protons, He nuclei, and virtually all primary nuclei \citep{Adriani11pHe,AMS02p,AMS02He,CREAM}, at $R\approx 300$ GV, where $R=cp/Ze$ is the particle rigidity, with $p$ the particle momentum, $Z$ its atomic number, $e$ the electron charge and $c$ the speed of light; 2) the hardening found in the spectrum of secondary CRs at $R\approx 200$ GV, with a change of spectral slope that is $\sim$ twice as large as that of primaries \citep{AMS02Sec}; 3) the energy dependence of the ratio between secondary and primary cosmic rays, that has now been measured with excellent statistics up to TV rigidities \citep{AMS02BC} and provides invaluable constraints on the energy dependence of particle transport in the Galaxy; 4) a rise in the fraction of positrons-to-electrons at energies larger than 30 GeV \citep{PamelaPos,AMS02frac}, possibly suggesting the presence of an additional source of positrons in the Galaxy; 5) the spectrum of anti-p which is unexpectedly very close to that of protons and positrons \citep{AMS02Pbar}. We will discuss these discoveries and their implications for the origin of Cosmic Rays in \S~\ref{sec:crtransp}.
\section{Testing the SNR paradigm}
\label{sec:observations}
While already in the 1930s \cite{PhysRev.46.76.2} suggested that CRs might originate from supernovae, the SNR paradigm was later formulated based on energetic arguments: SN explosions in the Galaxy can easily provide the power needed to sustain the CR population \citep{1957RvMP...29..235M,1964ocr..book.....G}. Assuming that the locally measured CR energy density, $w_{cr}$ $\sim$ 1 eV/cm$^3$, is representative of the CR energy density everywhere in the Galaxy, the CR production rate is $ \dot{W}_{CR} = V w_{cr}/t_{conf}$ with $t_{conf}$ and $V$ the CR confinement time and volume respectively. Using $t_{conf}\approx 15$ Myr \citep{Yanasak2001} and describing the Galaxy as a cylinder of radius $R_d\approx 15$kpc and height $H_d\approx 5$ kpc \citep{EvoliBe,MorlinoFe20} we can estimate
$\dot W_{CR}\approx 1-3 \times {10}^{41}$ erg/s.
SN events happen in the Galaxy about every 30 years and typically release 10$^{51}$ erg. Thus the power needed to sustain the galactic CR population turns out to be about 10 $\%$ of the power provided by SN events. The appeal of the SNR hypothesis was then increased by the formulation of the theory of diffusive shock acceleration \citep{Bell78,BlandOstr78}: this acceleration process, expected to be active at most shock waves, predicts the spectrum of accelerated particles to be a power-law, with an index close to $-2$ in the case of a strong shock, such as the blast wave of a SN explosion. Such a spectrum perfectly fits what inferred for the sources of Galactic CRs, lending support to the association. An equally remarkable, but much more recent finding, concerns the acceleration efficiency: kinetic simulations of diffusive shock acceleration (DSA) in a regime that is close to represent a SN blast wave have finally become available in the last decade, and they show that for parallel shock waves (magnetic field perpendicular to the shock surface) the acceleration efficiency is 10-20\% \citep{Caprioli14a}, exactly as required for the SNR-CR paradigm to work.
Testing the SNR paradigm for the origin of CRs has long been one of the top priorities in High Energy Astrophysics. Since the early days, photons with energies in the range between GeVs and TeVs have provided a unique tracer of the CR population far from Earth: the interactions of CRs with the environment are expected, indeed, to produce radiation in this energy range. The process that most directly reveals the presence of hadrons is the decay of neutral pions produced when CR hadrons collide inelastically with ambient gas in the ISM: this process is commonly referred to as the {\it hadronic} production mechanism. As a rule of thumb, an inelastic nuclear collision will produce gamma rays with $\sim$ 10 \% of the energy of the parent cosmic ray, so, for instance, photons of several tens to hundreds of TeV for CRs close to PeVs. Typically, the spectrum of the hadronic radiation at TeV mimics the parent CR spectrum shifted to lower energy by a factor 20-30 \citep{PhysRevD.74.034018,PhysRevD.90.123014}. In the same energy range, radiation can originate from {\it leptonic} production mechanisms, mainly inverse Compton (IC) scattering of CR electrons off ambient radiation fields, and non thermal electron bremsstrahlung \citep{RevModPhys.42.237}, which plays an important role in dense gas regions at sub GeV to GeV energies. A recurring difficulty when trying to infer the CR population from the GeV and TeV $\gamma$-ray emission from SNRs is to break the hadronic-leptonic degeneracy and unveil the dominant emission process: the spectral and morphological features of the emission are crucial to this task.
The gamma-ray spectrum is also affected by the particles' energy losses. For gamma-ray emitting particles, a number of loss mechanisms that are important at lower energies, like ionization and bremsstrahlung, have a negligible impact. The most relevant loss processes affect electrons, whose synchrotron and Inverse Compton emission must be properly taken into account when trying to disentangle the radiative contribution from leptons and hadrons. Electron synchrotron, particularly efficient in the strong magnetic fields believed to be associated with particle accelerators, has a fundamental role in cooling the CR electron population and thus shape the leptonic $\gamma$-ray spectrum. Therefore, observations of non-thermal radio and X-ray synchrotron emission can provide invaluable constraints on the electron population, and clues to solve the hadronic-leptonic degeneracy.
The SNR paradigm implies that young SNRs accelerate CRs up to the knee and that on average each SNR provides roughly 10$^{50}$ erg in accelerated particles. In order to test whether SNRs are the main contributors to the Galactic CR population, it is thus crucial to assess the energetics in electrons and protons and the spectra of accelerated particles in young SNRs.
Studies of the GeV-to-TeV spectra of the remnants, of their morphology and of the spatial correlation between TeV gamma-rays and the distribution of ambient gas and of X-ray radiation are the three most powerful instruments to extract the population of accelerated particles in the SNRs. GeV to TeV gamma-ray studies of both young SNRs \citep{1994A&A...287..959D}, and dense molecular clouds close to middle-aged SNRs \citep{1994A&A...285..645A} were undertaken to assess the role of SNRs in the origin of Galactic CRs, by understanding and disentangling the dynamics of the coexisting and competing processes of acceleration and escape or release of particles in the interstellar medium (ISM). A useful repository of high energy observations of SNRs is the Manitoba Catalogue (\url{http://snrcat.physics.umanitoba.ca/}).
Several young shell-type SNRs have been observed at TeV energies. Among these, the best studied ones are Cas~A \citep{2001A&A...370..112A,2007A&A...474..937A,2017MNRAS.472.2956A}, Tycho \citep{2011ApJ...730L..20A,2017ApJ...836...23A}, and RX~J1713.7-3946 \citep{2004Natur.432...75A,2018A&A...612A...6H}. We will discuss the latter as an example of how the SNR paradigm is tested with radiation from X-rays to gamma-rays.
As an example of multi-wavelength studies concerning middle-aged SNRs showing the so-called pion bump, a clear signature of the presence of relativistic protons, we will discuss the case of the brightest GeV source among these: W44.
\subsection{Young supernova remnants: the case of RX J1713.7-3946}
\label{sec:rxj}
The supernova remnant RX J1713.7-3946, possibly associated with a star explosion seen by Chinese astronomers in the year AD393 \citep{article1996,article1997}, is one of the brightest sources of gamma rays in the TeV energy range. The sky map of RX J1713.7-3946, obtained with the H.E.S.S. telescope, is shown in the top left panel of Fig.~\ref{RXJmap} \citep{2018A&A...612A...6H}. The morphology of the TeV shell resembles closely the XMM X-ray shell \citep{2004A&A...427..199C}, plotted with blue contours in the top left panel of the figure. The other five panels of Fig.~\ref{RXJmap} show the radial profiles at keV and TeV photon energies from different regions of the shell: in four cases, the TeV supernova remnant extends, in radius, beyond the X-ray emitting shell \citep{2004Natur.432...75A,2007Natur.449..576U,2008ApJ...685..988T,2018A&A...612A...6H}.
The main shock position and extent are visible in the X-ray data and the $\gamma$-ray emission extending further is either due to accelerated particles escaping the acceleration (shock) region or particles in the shock precursor region.
VHE (Very High Energy) electrons, up to hundreds of TeV, are accelerated in the shell of RX J1713-3946. These particles emit hard X-rays through synchrotron processes and TeV radiation through inverse Compton scattering off CMB photons very efficiently. A crucial piece of evidence in support of efficient particle acceleration comes from hard X-ray measurements with Suzaku \citep{2007Natur.449..576U}. Suzaku detected X-ray hot-spots brightening and disappearing within one year timescale. If the X-ray variability is associated to the acceleration and immediate synchrotron cooling of the accelerated electrons, then the magnetic field in the shell must be roughly two orders of magnitude higher than the average magnetic field in the Galaxy, about 100 $\mu$Gauss. Broadband X-ray spectrometric measurements of RXJ1713.7-3946 indicate also that electron acceleration proceeds in the most effective {\it Bohm-diffusion} regime. Finally, the presence of strongly amplified magnetic fields lends further support to the idea that not only electrons, but also protons up to 100 TeV are accelerated in the shell, as these amplified magnetic fields would be the result of CR induced instabilities \citep{2007Natur.449..576U}.
The high-energy part of the spectrum of RX~J1713.7-3946, from X-rays to TeV photons, can be used to infer the population of particles producing this emission. The particle spectrum of the young SNR in the {\it Left panel} of Fig.~\ref{RXJparticle} is characterised by a spectral index close to -2, as predicted by DSA, up to a few TeV (=${10}^{12}$ eV). While the hard spectrum shows that particles are efficiently accelerated in the shell of this young SNR, the cut-off at few TeV suggests that RX~J1713.7-3946 is currently accelerating electrons and protons up to 100 TeV but not up to PeV energies \citep{2018A&A...612A...6H}.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{aa29790-16-fig2.jpg}
\caption{Top left panel: H.E.S.S. gamma-ray skymap of RX~J1713.7-3946 \citep{ 2004Natur.432...75A,2018A&A...612A...6H}. In blue the contours of the shell as observed by XMM in X-rays \citep{2004A&A...427..199C}. The
morphology of the TeV shell resembles closely the X-ray shell. Top right panel and following: radial profiles of the emission at selected places along the shell from H.E.S.S. and XMM-Newton. In four out of five regions of the shell, the TeV supernova remnant is more extended than the X-ray one as a result of possible escape of high energy particles from the shell. The coordinate "Radius", used for the one-dimensional profiles, refers to the mean radius "r" of the shell over which average is performed.}
\label{RXJmap}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{aa29790-16-fig6.jpg}
{\caption{{\it Left Panel:} The black data points represent the GeV (Fermi-LAT) and TeV (H.E.S.S.) spectral energy distribution of the radiation from the shell of RX J1713.7-3946. The lines represent the multi-wavelength spectra predicted by NAIMA \cite{naima}, assuming either a leptonic or hadronic origin of the emission. {\it Right Panel:} The spectral energy distributions of the electrons and protons producing the leptonic and hadronic emissions shown on the left panel.}
\label{RXJparticle}}
\end{figure}
As mentioned, protons are likely accelerated in the supernova shell through the same mechanism accelerating electrons, but it is not clear what fraction of the energy input by the supernova explosion goes into the acceleration of electrons and what fraction goes into the acceleration of protons. Both protons or electrons can be, in fact, responsible for the TeV emission from RX~J1713.7-3946 ({\it leptonic-hadronic degeneracy}, see e.g. \cite{MorlinoRXJ}). If electrons are responsible for the emission from the shell, by inverse Compton scattering of the ambient radiation fields, one speaks of {\it leptonic scenario}. On the other hand, if the emission is produced by protons colliding with the ambient gas, one speaks of {\it hadronic scenario}. In Fig.~\ref{RXJparticle} the measurements of the shell emission carried out by Fermi LAT \citep{2011ApJ...734...28A}, HESS \citep{2018A&A...612A...6H} and Suzaku \citep{2008ApJ...685..988T} are compared to leptonic ({\it Left panel}) and hadronic ({\it Right panel}) model predictions by \cite{naima}. The total energetics in accelerated electrons and protons in the relevant leptonic and hadronic models of gamma-rays can be estimated by assuming a distance to the source of about 1~kpc \citep{article1996,article1997}. The required budget in electrons is determined only by the reported gamma-ray fluxes, if the target radiation is the CMB: one finds $W_{\rm e} \simeq 1.2 \times 10^{47}$erg. On the other hand, the total energy budget of protons in hadronic models depends on the highly uncertain ambient gas density: $W_{\rm p} \simeq 5 \times {10}^{49} (n_H/1 \mathrm{ cm^{-3})^{-1}}$erg. The leptonic model in the {\it Left panel} corresponds to the electron IC scattering off CMB photons and an infrared radiation field with energy density 0.415\,eV cm$^{-3}$ and temperature T = 26.5 K.
The leptonic model requires a break in the electron spectrum at 2.5\,TeV with the index of the electron energy distribution changing from $\Gamma=1.7$ to $\Gamma=3$ beyond the break. If due to synchrotron cooling, this spectral break implies a magnetic field of about $140~\mu{\mathrm{ G}}$, which is much larger than what the X-ray flux allows within the same scenario, $B\approx 15 \mu \mathrm{G}$. Similar difficulties arise if one tries to explain the break as a result of Inverse Compton cooling, in which case a photon field of energy density of 140\,eV\,cm$^{-3}$, would be required, about 100 times larger than the average galactic value.
If the emission is hadronic in origin, a spectrum of protons with index $\Gamma=1.5$, steepening to $\Gamma=1.9$ at about 1 TeV, and with an exponential cut-off at 79\,TeV is required. The break in the proton population spectrum can be explained if the hadronic emission is produced mostly in dense clumpy regions. In fact, low energy protons can be efficiently excluded from dense gas region during the timescale of $\sim$ 1000 years since the SNR explosion. The exclusion of low energy cosmic rays would also explain the hard $\gamma$-rays detected by the Fermi LAT telescope \citep{2012ApJ...746...82F,2014MNRAS.445L..70G,2012ApJ...744...71I}, and the lack of thermal X-ray emission from the shell. The latter has traditionally been one of the strongest argument against a possible hadronic origin of the emission from the shell of RXJ1713.7-3946, since it implies an ambient gas density as low as $0.1 \rm cm^{-3}$ \citep{2010ApJ...712..287E,2012ApJ...746...82F,2014MNRAS.445L..70G}. Finally hadronic gamma-ray production in gas condensations results in a narrow angular distribution of the radiation, which is beyond the reach of the current generation of Imaging Air Cherenkov Telescopes (IACTs), but could be tested with the upcoming Cherenkov Telescope Array (CTA) which will have an angular resolution of about 1-2\,arcmin \citep{2010ApJ...708..965Z,2018APh...100...69A}.
The conclusion one reaches, after investigating the spectrum and morphology of RX~J1713.7$-$3946 as currently known, is that neither the hadronic nor the leptonic scenario is fully satisfactory. Each emission mechanism has strengths and weaknesses when compared with observations, suggesting that the ambient conditions might differ in different parts of the remnant, making one or the other process locally dominant. While for some specific SNRs one of the two scenarios might indeed be favored (see e.g. \cite{FunkRev}), the general conclusions is that no known SNR has proved to accelerate particles beyond 100 TeV.
\subsection{Molecular Clouds close to middle aged SNRs: the case of W44}
\label{sec:mc}
Molecular clouds are regions of the Galaxy, typically a few tens of parsecs in radius, where the density of cold molecular gas is often orders of magnitude higher than elsewhere in the diffuse
ISM. Stars are believed to be born in these clouds. Radio observations of the rotational $1\rightarrow 0$ line emission of carbon monoxide are mainly used to trace the distribution of molecular gas \citep{2015ARA&A..53..583H}. The cloud Galactocentric distance is usually estimated using a kinematical distance method, through each the radial velocity of the cloud is related to the rotation velocity of the Galaxy. Cross calibrations with the distance of spiral arms or known objects with precise parallax determination are also carried out \citep{2009ApJ...699.1153R,2014ApJ...783..130R}.
Giant molecular clouds, which are typically 5 to 200 parsecs in diameter and have masses of 10 thousand to 10 million solar masses \citep{Murray_2011}, are excellent laboratories for CR physics. In these clouds, the hadronic channel of gamma-ray production is enhanced by the high target density, and easily dominates over the leptonic production mechanism. Contrary to the warmer atomic gas phase, which is homogeneously distributed in the Galaxy, one or a few giant molecular clouds are essentially the dominant contributions to the gas column density along a given direction in the sky. The emission enhancement associated with the clouds makes it possible to precisely locate along a given line of sight where the CR population produces the emission. Molecular clouds are thus used to perform a sort of {\it tomography}, and obtain a three dimensional view of the CR distribution in the Galaxy \citep{1981Natur.292..430I,1996A&A...309..917A,2001SSRv...99..187A,2010PASJ...62..769C,PhysRevD.101.083018,Baghmanyan_2020}.
The plasma conditions in molecular clouds are generally different from those in the diffuse ISM. In addition to the gas density, also the magnetic field energy density and turbulence level are enhanced (see e.g. \cite{Crutcher12} for a comprehensive review). This will lead to a suppression of the diffusion coefficient and effective exclusion of lower energy CRs from the cloud
\citep{2007ApESS.309..365G}. If CRs can penetrate clouds, the $\gamma$-ray emission from $\pi^0$-decay depends only upon the total mass of the cloud, $M_{cl}$, its distance from the Earth, $d$, and the CR flux within the cloud, $\Phi_{CR}$. The latter is thus determined as $\Phi_{CR} \propto \frac{\Phi_{\gamma} d^2} {M_{cl}}$, where ${\Phi_{\gamma}}$ is the $\gamma$-ray flux from the cloud. Lower energy cosmic rays can be effectively excluded from penetrating clouds, which results in peculiar features in the gamma-ray spectrum from clouds, such as hardenings with respect to the average interstellar spectrum \citep{2007ApESS.309..365G}.
The role of molecular clouds in testing the SNR paradigm is made particularly crucial by the time evolution of particle acceleration in SNRs. Within a DSA scenario, SNRs accelerate the highest energy CRs (in principle up to a few PeVs, but see \S~\ref{sec:cracc}) at the transition between the free expansion and the Sedov phase, which typically happens a few tens to a few hundred years after the supernova explosion, depending on the explosion type and properties of the surrounding medium. During the Sedov phase the SN shock slows down and the magnetic field intensity decreases, so that the most energetic particles cannot be confined any longer and are free to escape. In practice, a SNR can accelerate the highest energy particles only for a short time. This fact, coupled to the low rate of PeV particles accelerating events (see \S~\ref{sec:cracc}) makes the chances of observing a SNR when it still acts as a {\it PeVatron} very low. The runaway CRs can illuminate molecular clouds located close to the SNRs and this enhanced gamma-ray emission can thus provide crucial insights on the parental population of runaway CRs, which would otherwise escape the SNR without leaving a footprint \citep{1979ICRC....1..191M,1981Natur.292..430I,2007ApESS.309..365G, 1996A&A...309..917A,2001SSRv...99..187A}. The emission produced by these runaway CRs is essential to trace back acceleration up to PeV energies.
Gamma rays in association with dense molecular clouds located close to SNRs have been detected both at GeV and TeV energies. Depending on the location of massive clouds, on the acceleration history and on the timescales of the particle escape into the interstellar medium (which depend in turn on the diffusion coefficient), a broad variety of energy distributions of gamma rays is produced, from very hard spectra (much harder than the spectrum of the SNR itself) to very steep ones \citep{1996A&A...309..917A,2009MNRAS.396.1629G,2010PASJ...62.1127C}.
W44 is a 20,000 years old SNR located on the Galactic Plane at a distance of roughly 3 kpc from the Sun. W44, which is the brightest middle-aged SNR at GeV energies, is a perfect laboratory to test the coexisting processes of acceleration and escape of cosmic rays from SNRs. Due to the slow shock speed, high energy particles are expected to have already escaped from this source. A population of protons with spectral index close to -2.3 and a cutoff at about 80 GeV is likely responsible for the gamma-ray emission from the remnant measured by AGILE and Fermi-LAT \citep{FermiW44,AgileW44}. The presence of such a low energy cutoff might be the effect of the escape of the highest energy cosmic rays. Indeed, the remnant is thought to be currently a rather poor accelerator and most of its gamma-ray emission can be interpreted as due to re-acceleration of ambient CRs, rather than to acceleration of fresh particles \citep{Uchiyama+10,Cardillo+16}.
Because of their short lives, SNRs are often found within the giant molecular gas complexes where they were born as massive stars. W44 resides within a giant gas complex of 10$^6$ solar masses, homogeneously distributed over an extended region surrounding the remnant \citep{2001ApJ...547..792D}. North west and south east of the remnant, \citet{2012ApJ...749L..35U} discovered two bright sources, which the authors associated to runaway cosmic rays colliding with the dense gas clouds. \cite{Peron_2020} re-analysed both the Fermi-LAT data and the gas data from the clouds around W44 and noted that, despite the gas is homogeneously distributed, the GeV emissivity is enhanced only in the two regions which can be associated to regions of enhanced CR density, {\it CR clouds}, rather than gas clouds as previously thought. This phenomenon suggests the existence of a preferential path for CR escape, likely linked to the magnetic field structure in the vicinity of this source.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{IC443}
\includegraphics[width=0.49\textwidth]{W44}
\caption{ Gamma-ray spectra of IC443 ({\it Left Panel}) and W44 ({\it Right Panel}) as measured with the Fermi LAT \citep{FermiW44}, in blue, and AGILE \citep{AgileW44}, in magenta. The pion bump feature at about 1 GeV is evident in the spectra of both SNRs. For IC433 the data-points of MAGIC \citep{2007ApJ...664L..87A} and VERITAS \citep{2009ApJ...698L.133A} are also shown. Color-shaded areas denote the best-fit broadband smooth broken power law (60 MeV to 2 GeV); gray-shaded bands show Fermi LAT systematic errors below 2 GeV. The data points at the highest energy (4 $ \times {10}^{10}$ - 1$ \times {10}^{11}$ eV) suggest a hardening of the spectrum, which might be produced by runaway CRs illuminating a molecular cloud (MC) located in front of the shells.}
\label{W44}
\end{figure}
\section{Recent developments on CR acceleration in SNRs}
\label{sec:cracc}
In spite of all the recent observational developments, providing several hints of efficient CR acceleration in SNRs, a number of unsettled questions remain and challenge the association. In particular, one of the strongest arguments to look for alternative particle accelerators has to do with the difficulties SNRs have at reaching PeV energies. This aspect of the CR-SNR connection has long appeared as the most delicate.
As already mentioned, CR acceleration in SNRs is thought to be well described within the framework of diffusive shock acceleration (DSA). Acceleration mechanisms alternative to DSA have also been proposed and studied (see e.g. \cite{Lazarian20}). In this article, however, we focus on DSA, which is still the best studied and the most promising, in this context, to reach the highest energies. The idea at the basis of the theory is that particles gain energy each time they cross the shock thanks to the discontinuity of the fluid velocity field at the shock, that leaves an unscreened electric field. The energy gain of the particle is a constant fraction ($v_{\rm sh}/c$ with $v_{\rm sh}$ the shock velocity) of its energy before the crossing. The particle moves diffusively between crossings, scattering on low frequency magnetic turbulence and continuously changing its pitch-angle. Reaching high energies requires a large number of crossings, which have to occur in a time shorter than the minimum between the duration of the system $t_{\rm life}$ and the time-scale over which energy losses become important $t_{\rm loss}$. In the case of protons losses are negligible and the energy is limited by the time for which the SNR is active as an accelerator, $t_{\rm life}<10^4$ yr. Then $t_{\rm life}$ must be compared with the acceleration time, $t_{\rm acc}$, which depends on how quickly the particle is able to go back to the shock after each crossing: we can estimate $t_{\rm acc}(E)\approx D(E)/v_{\rm sh}^2$, where $D(E)$ is the particle diffusion coefficient, typically an increasing function of the particle energy $E$. For relatively low turbulence levels one can estimate $D(E)$ in quasi-linear theory, writing it as (see e.g. \cite{Amato14}):
\begin{equation}
D(E)\approx \frac{v(E) r_L(E)}{3}\frac{1}{kW(k)}
\label{eq:D(E)}
\end{equation}
where $v(E)$ is the particle velocity, $r_L(E)$ its Larmor radius, and $W(k)$ is the spectrum of magnetic fluctuations causing the diffusion, with
\begin{equation}
\int_{1/L}^\infty W(k')dk'=\frac{\delta B^2}{B_0^2}\ ,
\label{eq:wk}
\end{equation}
and $k$ the wave-number resonant with the particle of energy $E$, namely $k=1/r_L(E)$.
Assuming that the scattering is due to turbulence that is injected by SNRs on a typical scale $L\approx$ 100 pc with $\delta B/B_0\approx 1$, and then develops a Kolmogorov-type spectrum $W(k)\propto k^{-5/3}$, from Eqs.~\ref{eq:D(E)} and \ref{eq:wk} above\footnote{one may wonder about the appropriateness of using quasi-linear theory for turbulence levels as large as $\delta B/B_0\approx 1$: while this is not fully justified on theoretical ground, numerical simulations find it to provide a decent description of wave-particle interactions (see e.g. \cite{Holcomb19})}, one can estimate the diffusion coefficient as $D(E)=7\times 10^{27}$ (E/{GeV})$^{1/3}$ cm$^2$/s, which, as we will discuss below, is not very different from what CR observations indicate ($D(1 \mathrm{GeV})\approx 10^{28}$ cm$^2$/s).
If this estimate of the diffusion coefficient were the appropriate one to describe particle transport in the vicinity of a SNR shock, the maximum achievable energy in these systems would only be $E_{\rm max}\sim$ few GeV, and could only improve to $\sim$ 100 TeV if particle transport were described by Bohm diffusion, namely $\delta B/B_0\approx 1$ at all relevant scales \citep{LagageCesarsky83}. It is then clear that in order to reach PeV energies, not only efficient - Bohm like - scattering is needed, but also largely amplified magnetic fields.
As already mentioned above, this is exactly what X-ray observations of young SNRs were found to show: aside from the short variability time-scales already discussed in Sec.~\ref{sec:rxj}, X-ray synchrotron emission is seen to be confined to rims of thickness $\Delta \sim {\rm few} \times$ 0.01 pc \citep{Vink12}. The extremely small value of $\Delta$ can be interpreted either as a result of radiative losses, that kill the electron population with increasing distance from the shock, or as a result of magnetic field damping \citep{Rettig2012}. If we interpret the thickness of the rims as the distance traveled by the emitting electrons in a synchrotron loss time, we can write $\Delta\approx \sqrt{D(E)\ t_{\rm sync}(E)}=0.04\ {\rm pc}\ B_{-4}^{-3/2}$ with $B_{-4}$ the magnetic field downstream of the shock in units of $10^{-4}$G. It is clear then that a magnetic field amplified by a factor of $\sim$100 with respect to the interstellar value ($B_{\rm ISM}\approx 3 \mu$G) is implied in the rims, if they are to be interpreted as the result of radiative losses. On the other hand, interpreting $\Delta$ as a result of magnetic damping requires a value of the magnetic field which is still in the same range estimated above \citep{Pohl+05}, and hence largely amplified.
There are several mechanisms by which a magnetic field can be amplified at a shock \citep{Bykov+12}: some involve fluid instabilities \citep{GiacaloneJokipii07,Ohira16}, some involve MHD instabilities and CR related effects \citep{Beresnyak+09}, some others CR induced instabilities, either of resonant type (\cite{AmatoBlasi06} and references therein) or of non-resonant type \citep{Bell04,Reville+08,AmatoBlasi09}. Except for purely fluid instabilities \citep{GiacaloneJokipii07}, all other classes require efficient CR acceleration, and in this sense the detection of amplified magnetic fields can be considered as indirect evidence of efficient CR acceleration in SNRs. However the different mechanisms proposed are not equivalent in terms of the spectrum of magnetic fluctuations they produce, and hence in terms of consequences on particle acceleration. Particle scattering is only efficient with resonant waves. Hence magnetic field amplification (MFA, hereafter) will increase scattering efficiency and lead to high maximum energies of the accelerated particles only if there is enough power on scales comparable with the particle Larmor radii. The most promising MFA mechanism in this sense turns out to be the non-resonant CR streaming instability (Bell's instability).
The super-Alfv\'enic streaming of energetic particles has long been known to induce the growth of magnetic fluctuations at wavelengths that are resonant with the gyro-radii of the exciting particles \citep{Skilling75}: this is the so-called resonant streaming instability. While creating fluctuations on the right scales, this instability can only lead to MFA up to a level $\delta B/B_0\lesssim 1$, and hence it is not powerful enough to explain the field strength deduced in SNRs, nor to guarantee particle acceleration up to the knee. In more recent times, however, it has been recognized that a more powerful instability arises if the CR current is large enough to twist the ambient magnetic field on the scale of the Larmor radius ($r_{L,0}$) of the particles that carry the current ($J_{\rm CR}>(c/4 \pi)(B_0/r_{L,0})$), or, equivalently, if the CR energy density, $U_{\rm CR}$, is larger than $c/v_D$ times the magnetic energy density \citep{Bell04}:
\begin{equation}
U_{\rm CR}>\frac{c}{v_D} \frac{B_0^2}{4 \pi}\ .
\label{eq:bell2}
\end{equation}
where $v_D$ is the bulk velocity of CRs. If this condition is satisfied, the magnetic field grows very rapidly. The basic physical process can be described a follows: the CR current induces a compensating return current in the ISM plasma; the force $\vec J_{\rm ret}\wedge \vec B$ induces transverse plasma motion; the current associated to this motion acts as a source of magnetic field; the result is that the magnetic field lines associated to right-hand polarized waves are stretched and for these modes the field is amplified.
In the vicinity of a shock that is accelerating particles, Eq.~\ref{eq:bell2} can be turned into a condition for the acceleration efficiency $\xi_{\rm CR}$:
\begin{equation}
\xi_{\rm CR}>\frac{c B_0^2}{8 \pi \rho_{\rm ISM} v_{\rm sh}^3}\ .
\label{eq:bell3}
\end{equation}
Here $\xi_{\rm CR}$ is the fraction of incoming flow energy ($\rho_{\rm ISM} v_{\rm sh}^2/2$) that is converted into accelerated particles, $\rho_{\rm ISM}$ is the mass density of the ISM plasma and $v_{\rm sh}$ is the velocity of the blast wave, also coincident with the CR bulk velocity. Eq.~\ref{eq:bell3} makes it clear that the possibility for the non-resonant streaming instability to operate strongly depends on the shock velocity: detailed calculations show that it is indeed a viable MFA mechanism in the vicinity of the fast shocks of young SNRs \citep{AmatoBlasi09}, but then stops working early during the Sedov-Taylor phase of expansion of the blast wave, typically after few hundred to few thousand years, depending on the ambient medium density and magnetic field strength.
The growth initially occurs on very small scales ($k\approx (4\pi/c)(J_{\rm CR}/B_0>> r_{\rm L,0}$, but quickly the power moves to larger and larger scales \cite{RiquelmeSpitkovsky09}, possibly due to some mean-field dynamo process \citep{Bykov11, Rogachevskii12}, and numerical simulations show that the final outcome of the instability, when it develops at a shock, is a spectrum of fluctuations with $W(k)\propto k^{-1}$, leading to Bohm diffusion of the particles \citep{CaprioliSpitkovsky14C}. When saturation is reached, the magnetic energy density is a substantial fraction of the CR energy density \citep{CaprioliSpitkovsky14B}.
The level of MFA that the non-resonant streaming instability can provide is in the correct range to explain the magnetic field strength inferred in SNRs \citep{Schure+12}, and as we just mentioned the associated scattering is efficient \citep{CaprioliSpitkovsky14C}, and yet recent studies have cast doubt on the fact that these sources might be able to accelerate particles up to PeV energies \citep{Cardillo+15}. In fact, assuming that MFA is primarily due to the streaming of particles that leave the acceleration region, it is possible to build a description of the shock as a self-regulating system: when the level of turbulence in the upstream is low, a large fraction of particles can escape from the shock; when this happens, however, the large current in the upstream causes the growth of turbulence and escape is reduced; if the fraction of escaping particles becomes too small, then the turbulence level is reduced again favouring escape. The self-regulation mechanism qualitatively illustrated above translates into a quantitative prescription for the maximum energy of shock accelerated particles as a function of the system parameters. The current of escaping particles, which is what determines the level of MFA, is in turn determined by the shock velocity and the spectrum of accelerated particles, including its total energy content (which determines the amount of energy available), its slope and the maximum particle energy (which determine the current, once the energy content is fixed). When writing the equation describing $E_{\rm max}$ as a function of time during the expansion of a SN blast wave, one finds that $E_{\rm max}$ is an ever decreasing function of time during the SNR evolution, so that the relevant value of $E_{\rm max}$ is that reached at the beginning of the Sedov-Taylor phase, namely the highest possible after a sufficient amount of mass has been processed by the blast wave. It is clear then that a higher maximum energy will be achieved in systems that enter the Sedov-Taylor phase earlier in time after the explosion. This occurs for type II explosions expanding in the dense and slow wind of a progenitor red super-giant star. In this case, assuming a $\propto E^{-2}$ spectrum, the maximum achievable energy reads:
\begin{equation}
E_{\rm max}\approx 0.5 \left(\frac{\xi_{\rm CR}}{0.1}\right)
\left(\frac{E_{\rm SN}}{10^{51}\ {\rm erg}}\right)
\left(\frac{M_{\rm ej}}{M_\odot}\right)^{-1}
\left(\frac{\dot M}{10^{-5} M_\odot /{\rm yr}}\right)^{\frac{1}{2}}
\left(\frac{v_w}{10\ {\rm km\ s}^{-1}}\right)^{-\frac{1}{2}}
{\rm PeV}
\label{eq:emax}
\end{equation}
where $\xi_{\rm CR}$ is the CR acceleration efficiency, $E_{\rm SN}$ and $M_{\rm ej}$ are the energy released and the mass of material ejected in the SN explosion, and $\dot M$ and $v_w$ are the mass loss rate and speed of the progenitor's wind. If one takes into account that the mass of the ejecta in a type II SN explosion is more likely around $10\ M_\odot$, it is immediately clear that reaching the {\it knee} is very challenging in this framework. One is forced to invoke extremely energetic events, or extreme acceleration efficiency, or finally extreme properties of the progenitor. In all cases these must be rare events. In fact, the product of $\xi_{\rm CR}$, $E_{\rm SN}$ and the frequency in the Galaxy of the PeV producing SN explosions are constrained by the overall flux of CRs measured flux at Earth. When all of this is taken into account, one determines an event rate that cannot exceed a few in $10^4$ yr \citep{Cristofari20}.
In reality the possibility for SNRs to reach the {\it knee} becomes even more challenging when taking into account the fact that the accelerated CR spectrum is likely $\propto E^{-p}$ with $p>2$. Such a spectrum corresponds to a smaller current for a given $E_{\rm max}$ and total energy in accelerated particles, hence requiring even more extreme parameters to reach $E_{\rm max}\approx$ PeV (e.g., \cite{Cardillo+15}). On the other hand a spectrum steeper than $E^{-2}$ is exactly what gamma-ray observations of the majority of young SNRs require and what recent progress on propagation also seems to require \citep{Evoli+19}: the CR spectrum released in the ISM will be steeper than $E^{-2}$ only if the spectrum in the source is itself steeper than $E^{-2}$ \citep{SchureBell14,Cardillo+15} and a steep injection spectrum of CRs is exactly what the most recent CR data seem to suggest, as we discuss in \S~\ref{sec:crtransp}.
\section{Cosmic ray transport in the Galaxy}
\label{sec:crtransp}
As mentioned in \S~\ref{sec:intro} the last decade has also been rich of observational progress providing interesting constraints on the properties of CR transport throughout the Galaxy. In this section we will review the main findings and try to put them in a coherent theoretical framework.
The first important discovery was that of a hardening in the spectrum of protons and He nuclei \citep{Adriani11pHe, Aguilar+15p,Aguilar+15He} and also, though with somewhat lower statistics, of heavier nuclei \citep{Ahn+10}. Within the framework of diffusive transport, the detection of a break in the spectrum of CRs can be interpreted as a signature of a change either in the injection spectrum or in the diffusion coefficient. During propagation through the Galaxy, CRs primarily loose energy due to adiabatic expansion and ionization. In addition, particles of a given species can be also lost due to spallation or decay. However, if one focuses on stable primary CRs, with energy above a few tens of GeV, energy losses during propagation can be neglected and a very simple expression for the diffuse steady state spectrum in the Galaxy can be found. If we assume that CRs are injected in the Galaxy at a constant rate
\begin{equation}
Q_{\rm p}(E)\propto E^{-\gamma_{\rm inj}}\ ,
\label{eq:crinj}
\end{equation}
and that particles then propagate diffusively with an average diffusion coefficient
\begin{equation}
D(E)\propto E^\delta
\label{eq:crdiff}
\end{equation}
the steady state spectrum of stable CR nuclei in the Galaxy will be given by the product between injection and confinement time, $Q(E)\times \tau_{\rm esc}$. This will read, for primary nuclei:
\begin{equation}
N_{\rm p}(E)\propto Q(E)\frac{H^2}{D(E)}\propto E^{-\gamma_{\rm inj}-\delta}\ ,
\label{eq:nprim}
\end{equation}
where the confinement time has been taken to be $\tau_{\rm esc}=H^2/D(E)$, with $H$ the size of the magnetized halo in which CRs are confined (see e.g. \citep{amatoblasi18} for a more refined description). It is then clear then that the detected hardening implies a change in $\gamma_{\rm inj}$ or $\delta$. Several models have been proposed invoking either of the two \citep[][and references therein]{TomassettiHardening,HorandelHardening}. A possibility is that this feature is signaling the importance of non-linear effects in CR propagation. An early suggestion \citep{Blasi+12} was that the break could point to the transition between scattering in self-generated and external turbulence. It had indeed been suggested since the '70s \citep{Wentzel74,Cesarsky80} that at scales comparable with the Larmor radius of GeV particles, CRs could be an important source of turbulence in the galaxy through the resonant streaming instability. At larger scales, on the other hand, CRs become too few, given their steep spectrum, and the main source of scattering would become the large scale turbulence present in the Galaxy. The latter is usually assumed to be injected by SNRs on a scale of order tens to 100 pc and then cascade to smaller scales developing a Kolmogorov type spectrum $k^{-5/3}$. Such a spectrum translates in a diffusion coefficient with $\delta\approx 1/3$, flatter than the low energy value $\delta\approx 0.7$ that is appropriate to describe CR self-generated turbulence. A back of the envelope calculation places the transition between self-generated and external turbulence at a CR rigidity in the range 200-300 GV \citep{Blasi+12}, tantalizingly close to the value of 336 GV at which AMS-02 detects the hardening in the proton spectrum.
This explanation of the hardening as due to a change in the properties of galactic transport entails a clear prediction for the spectrum of secondary CR nuclei. These are injected in the Galaxy as a result of spallation of primaries. If we approximate the spallation cross-section as independent of energy, their injection will be
\begin{equation}
Q_{\rm sec}(E)\propto N_p(E) \sigma_{\rm sp}c n_{\rm ISM}\propto E^{-\gamma_{\rm inj}-\delta}\ ,
\label{eq:secinj}
\end{equation}
where $n_{\rm ISM}$ is the target gas density. Their equilibrium spectrum in the Galaxy will then read:
\begin{equation}
N_{\rm sec}(E)\propto Q_{\rm sec}(E)\tau_{\rm esc}\propto E^{-\gamma_{\rm inj}-2\delta},
\label{eq:nsec}
\end{equation}
which implies that any hardening $\Delta$ of the spectrum of primaries due to a change in the slope of the diffusion coefficient will reflect in a hardening of the spectrum of secondaries equal to $2\Delta$: this expectation is perfectly consistent with the analysis of secondaries performed by AMS-02 \citep{Aguilar+18sec}.
Within this interpretation of the observed hardenings, the transport of CRs through the Galaxy becomes a complex non-linear problem, where the diffusion coefficient and the spectra of all nuclei need to be determined self-consistently. Once this complex problem is solved, however, and all available AMS-02 data are reproduced, the low energy Voyager data \citep{Stone+13} are automatically reproduced \citep{Aloisio+15}. One important result that comes out of this analysis is directly related to the Boron-over-Carbon (B/C) measurements of AMS-02 at high energies \citep{Aguilar+16BC}. The B/C ratio has traditionally been considered the primary indicator of CR transport: B is the most abundant stable secondary and its mostly produced by the spallation of C, though the contribution by N and O is not negligible. In practice the B/C ratio provides a direct measurement of the grammage (mass per unit surface, or mass density integrated along the path-length) encountered by CRs during their propagation from their sources to Earth. If one compares Eqs.~\ref{eq:nsec} and \ref{eq:nprim}, for the spectrum of secondary and primary nuclei, respectively, one immediately sees that the ratio between the two is expected to scale with energy exactly as the diffusion coefficient. In fact this ratio, as measured by AMS-02 \citep{Aguilar+16BC}, well agrees with a high energy slope of the diffusion coefficient $\delta\approx 0.4$ \citep{Aloisio+15,Evoli+19}, but an additional, energy independent contribution seems to be required to well reproduce not only the highest, but also the lowest energy \citep{Cummings+16} available data. The additional grammage needed, $X_s \approx 0.15\ {\rm g\ cm}^{-2}$, is independent of energy and of the order of what particles can accumulate within a SNR - or any source with an ambient density of order $1\ {\rm cm}^{-3}$ and a duration $\approx 10^4\ {\rm yr}$ - during acceleration \citep{Aloisio+15,Bresci+19}. Within this modeling, namely if hardenings are interpreted as a result of turbulence self-generation at low energies, the contribution $X_s$ turns out to be fundamental also for explaining another recent surprise found in CR data, namely the spectrum of anti-protons, which we will discuss in \S~\ref{sec:cranti}.
Before concluding this section we would like to remark a most important consequence of a scenario in which CR propagation at high energy is described by a diffusion coefficient $\delta\approx 0.4$: CRs must be injected in the galaxy with a spectrum $E^{-\gamma_{\rm inj}}$ with $\gamma_{\rm inj}\approx 2.3$. This combination of injection and propagation parameters, and in particular the relatively weak energy dependence of the diffusion coefficient, helps to explain the low level of anisotropy detected at TeV energies, as shown by \cite{BlasiAmato12a,BlasiAmato12b,Sveshnikova+13}. On the other hand, as discussed in \S~\ref{sec:cracc}, such a steep spectrum makes it very difficult for SNRs to be the primary sources of PeV CRs.
Within a DSA scenario SNRs accelerate the highest energy CRs (up to at least a few PeV) at the transition between the free expansion and the Sedov-Taylor phase, which typically happens a few hundred years after the supernova explosion. During the Sedov-Taylor phase the SN shock slows down and the magnetic field intensity decreases, so that the shock cannot confine any longer the most energetic particles, which escape the SNR. This means that a SNR can act as a PeVatron for a relatively short time. Considering the rate of SN explosions in the entire Galaxy (about 3 per century), the chances to observe a SNR when it is still a PeVatron are thus very low, and any proof of emission from PeV CRs even from young SNRs is challenging to find.
\section{Alternative CR sources}
While gamma-ray observations have proven that SNRs are efficient accelerators of cosmic ray electrons, and possibly protons, up to 100\,TeV, the hypothesis that acceleration of cosmic rays proceeds up to PeV energies has been rejected in all known young SNRs because of the clear cut-offs detected at several TeVs in their spectra.
The GeV-to-TeV radiation from young shell-type SNRs, which had been long expected to provide final evidence to settle the question of the origin of cosmic rays, can be either of hadronic or leptonic origin. Typically, a 1~TeV $\gamma$-ray photon is emitted either by an electron or a proton of about 10~TeV. The cooling time for 10 TeV electrons is $\approx 5 \times 10^4$yr (for IC scattering off the CMB photons) while for protons the cooling time is $5 \times 10^7 (n/1 \rm cm^{-3})^{-1}$yr (see e.g. \cite{2004vhec.book.....A}). Thus the ratio of the luminosity in IC gamma-rays to $\pi^0$-decay gamma-rays is of the order of $10^3 \times (\frac{W_e}{W_p}) \times {(\frac{n}{\mathrm{1cm}^{-3}})}^{-1}$. The leptonic contribution to the emission is thus dominant over the hadronic one unless $\frac{W_e}{W_p} << 10^{-3}$ or alternatively if the gas density inside the shell is $n >> 1 \, \rm {cm^{-3}}$. This latter condition, however, if realized, would have the drawback of slowing down the shock wave very quickly and thus prevent efficient acceleration. On the other hand, a much larger energy density in protons than in electrons could result from rapid cooling of the electron population, especially in the presence of an amplified magnetic field, as inferred for young SNRs. If $B>10\ \mu$G, only a small fraction of their energy $w_{MBR}/w_{\rm B} \approx 0.1 (\rm {B}/10 \mu \rm G)^{-2}$, is released in IC gamma-rays, making the conditions for detection of $\pi^0$ decay more favourable.
To recap, no known SNR has proved to accelerate particles beyond 100 TeV. Additionally, the power in accelerated protons within SNRs derived from $\gamma$-ray observations depends on the highly uncertain local gas density $n$ and on the magnetic and radiation fields within the remnant. As a result, it has so far been impossible to unequivocally prove that the population of SNRs in the Galaxy injects the necessary power (about 10$^{50}$erg per supernova event) to sustain the CR population. In fact simulations of the SNR population show how the choice of acceleration and ISM parameters lead to remarkably different SNR populations as CR sources \citep{2017MNRAS.471..201C}.
Some other classes of astrophysical sources, such as super-bubbles or star forming regions (see \S~\ref{sec:CR:SC}) or remnants of GRBs in our Galaxy \citep{2006ApJ...642L.153A}, and, in particular, the Centre of our Galaxy (see \S~\ref{sec:CR:GC}), have long been proposed as alternative accelerators of particles and major contributors to the population of Galactic cosmic rays up to PeV energies \citep{1980ApJ...237..236C,1982ApJ...253..188V,1983SSRv...36..173C,2004ApJ...601L..75T,2006ApJ...644.1118R,2001AstL...27..625B,2004A&A...424..747P,2005ApJ...628..738H,2006ApJ...642L.153A,2019NatAs...3..561A}.
\subsection{Galactic Centre}
\label{sec:CR:GC}
A breakthrough in the quest for the origin of the highest energy CRs in the Galaxy was the discovery of a powerful PeVatron in the Centre of the Milky Way \citep{2016Natur.531..476H}.
The nucleus of the Milky Way is a very active region with numerous sources of non-thermal radiation and constitutes a unique laboratory for the study of very high energy astrophysical processes
within the Galaxy and in external active galactic nuclei. The
Galactic Centre (GC) is thought to host a super-massive
black hole (SMBH) of $2.6 \times 10^6$ solar masses \citep{2008ApJ...689.1044G,Gillessen_2009} located very close to the dynamical centre of the Galaxy and coincident with the compact radio source Sagittarius A* (Sgr A*). Sgr A* emits radiation in X-rays and infrared through accretion of mass onto the BH. The region within 400 pc from the Centre of the Galaxy, called the Central Molecular Zone (CMZ), contains around 5\% of the total Galactic molecular gas \citep{1998ApJS..118..455O,1999ApJS..120....1T}. The molecular CO line towards the GC being somehow optically thick due to the high gas density, the gas distribution there is mapped additionally with other lines, such the CS radio line \citep{1999ApJS..120....1T}. While the presence of this dense gas and dust prevents observations of this region in the optical and ultra-violet wavelengths, the Galactic Centre region is extremely bright at radio, infrared, X-ray and gamma-ray frequencies.
At very high energy \citep{2018A&A...612A...6H}, the GC hosts a bright point-like source, HESS J1745-290, which, within errors, coincides spatially with Sgr A* and presents an energy spectrum with a steepening below 10 TeV. While the association of HESS J1745-290 with Sgr A* is well motivated in terms of spatial coincidence and of required energetics, {the pulsar wind nebula (PWN)} candidate G359.95-0.04 is also a viable counterpart of HESS J1745-290. HESS J1745-290 is surrounded by an extended component of VHE gamma-ray emission, correlated spatially with the CMZ. The spectrum of this radiation is a pure power law, with a spectral index -2.3 and without evidence of a spectral cutoff.
Because of their different energy distributions, the central point-like source and the extended emission cannot have the same origin. The diffuse VHE gamma-ray emission of the CMZ could be in principle produced through interactions of either relativistic protons with the ambient gas or of relativistic electrons with the radiation fields. However, a leptonic origin of the diffuse gamma-rays can be excluded. The power-law acceleration spectra of electrons should extend to about 100 TeV, which is extremely difficult because of the severe Inverse Compton and synchrotron radiative losses in the GC region. For the same reason, leptons hardly can escape the sites of their acceleration and propagate over the tens of parsec extended region of the HESS diffuse emission.
The spatial correlation between the gamma-ray emission from the Galactic Ridge and the ambient gas supports a hadronic origin of the emission. The spectrum of the parental protons -- with a spectral index close to -2.4 -- should extend to energies close to 1\,PeV. Assuming a cutoff in the parent proton spectrum, the corresponding secondary gamma-ray spectrum deviates from the HESS data at 68$\%$, 90$\%$ and 95$\%$ confidence levels for cutoffs at 2.9\,PeV, 0.6\,PeV and 0.4\,PeV, respectively. This makes the discovery of the diffuse emission from the Galactic Centre the first robust claim of detection of a Galactic cosmic ray PeVatron.
The CR energy density in the CMZ, obtained combining the gamma-ray and gas distributions, is an order of magnitude higher than the {\it sea} of cosmic rays. The radial distribution of CRs follows a 1/r dependence, where r is the distance to the GC. This means that the PeVatron should be located within 10 pc from the GC and the total energy output in protons above 10 TeV over the whole region should amount to W$_{CR}$ = 10$^{49}$ erg. Such a modest energy output could be provided by a single supernova remnant event, such as Sgr A East, a Pulsar Wind Nebula. If the super-massive black hole, Sgr~A*, is the GC PeVatron, then the particles producing the HESS extended emission are accelerated either in the vicinity of the SMBH, close to event horizon, or at the termination of a relativistic outflow, either a jet or a wind. This jet or wind should be injected close to the black hole and carry a substantial fraction of energy, extracted from the accretion disk. Indeed, the power required by the inferred CR distribution corresponds to $\gtrsim 1\%$ of the accretion power of the central black hole, and is 2-3 orders of magnitude larger than the bolometric luminosity of Sgr~A*. The source, however, could have been more active in the past \citep{2016Natur.531..476H}.
Alternatively, the extended TeV emission from the Galactic center could be the combined result of acceleration within three powerful star clusters, the Arches, the Quintuplet and the Nuclear cluster \citep{2016Natur.531..476H,2019NatAs...3..561A}. In this case, the continuous injection over millions of years, required by the 1/r distribution of the diffuse emission, could result from the characteristic ages of massive clusters, roughly 10$^6$ years \citep{2019NatAs...3..561A}.
All the proposals discussed above assume that particle transport in the GC vicinity can be described through the same simple model of a spatially uniform diffusion coefficient adopted for the rest of the Galaxy. If this assumption is released, however, the GC excess can be interpreted as a result of a spatially dependent diffusion coefficient, as was proposed by \cite{gaggero2017prl}, based on Fermi-LAT data showing the spatial dependence of the gamma-ray galactic diffuse emission.
\subsection{Star Clusters}
\label{sec:CR:SC}
Collective stellar winds and SNR shocks in clusters and associations of massive stars have long been suggested as possible, alternative or additional contributors to the Galactic cosmic ray flux \citep{1980ApJ...237..236C,1982ApJ...253..188V,1983SSRv...36..173C}. Core-collapse SN progenitor stars and colliding wind binaries evolve in giant molecular clouds and mostly remain close to their birthplaces in groups of loosely bound associations or dense stellar clusters. The winds of multiple massive stars in such systems can collide and form collective cluster winds which drive a giant bubble, a so called superbubble, filled with a hot (T = 10$^6$~K) and tenuous (n $<$ 0.01~cm$^{−3}$) plasma.
At the termination shock of the stellar cluster wind, turbulence can build up, in the form of MHD fluctuations and weak shocks \citep{2004ApJ...601L..75T,2006ApJ...644.1118R,2001AstL...27..625B}. Turbulence in the superbubble interiors can accelerate particles to very high energies, not only through the 1$^{\rm st}$ order Fermi process, but also via the 2$^{\rm nd}$ order mechanism \citep{2001AstL...27..625B}. Supernova explosions of massive stars in thin and hot superbubbles can also produce efficient particle acceleration at the boundary of the superbubbles or at MHD turbulence and further amplify existing MHD turbulence \citep{2010A&A...510A.101F}. It has also been recognized that multiple shocks can result in efficient acceleration beyond PeV energies \citep{2000APh....13..161K}. The interaction of the accelerated particles with the ambient medium - often including dense molecular clouds - or with electromagnetic fields, leads to the efficient production of VHE gamma rays.
Recent observations at TeV energies of massive star-forming regions and stellar
clusters, such as 30~Doradus in the LMC \citep{2015Sci...347..406H}, Westerlund 1 \citep{2012A&A...537A.114A} and the Cygnus region \citep{2002A&A...393L..37A,2007ApJ...664L..91A,2007ApJ...658.1062K,2009ApJ...700L.127A,2012ApJ...745L..22B} in our own Galaxy, support the hypothesis that star forming regions are sites of high energy particle acceleration, and give new impulse to the $\gamma$-ray research in this field.
The primary objectives of these gamma-ray observations are: 1) to constrain the fraction of mechanical energy of the stellar wind transferred to relativistic particles and hence gamma rays; 2) to unveil the physics of particle acceleration and propagation in Galactic stellar clusters and superbubbles.
Furthermore, high-energy phenomena are receiving increasing attention
also from the point of view of their impact on
the life cycle of interstellar matter and star-formation processes.
The rate and efficiency of the star formation process depends, in fact, on the balance between the self-gravity of dense molecular cores and the countervailing forces which act to support the clouds. The most important of these are likely to be thermal pressure, turbulence, and magnetic fields. In order for magnetic support to be effective, a population of ionized particles must be present in the core. Since molecular clouds are opaque to ultraviolet radiation from stars, the main ionizing agent is thought to be low-energy CRs, and the magnetic support of the cloud is critically dependent on their abundance.
The Cygnus region hosts some of the most remarkable star-forming systems in the Milky Way, including Cygnus X, a star forming region at only 1.5 kpc from the Sun, with a total mass in molecular
gas of a few million solar masses - at least 10 times the total mass in all other close-by star-forming regions, such as Carina or Orion - and a total mechanical stellar wind energy input of $10^{39}$ erg s$^{-1}$, which corresponds to several per cent of the kinetic energy input by SNe in the entire Galaxy. Cygnus X hosts many young star clusters and several groups of O- and B-type stars, called OB associations. One of these associations, Cygnus OB2, contains 65 O stars and nearly 500 B stars. These super stars have created cavities filled with hot, thin gas surrounded by ridges of cool, dense gas where stars are now forming, which strongly emit at GeV energies, called the Fermi Cocoon \citep{2011Sci...334.1103A}. At TeV energies the Cygnus region shows two distinctive regions. One is possibly connected to the Cygnus\,X complex, the star association Cygnus\,OB2 and the Fermi cocoon observed at GeV energies.
In the other region detected at TeV energies, the Cyg\,OB1 region, the Milagro collaboration discovered MGRO\,J2019+37 \citep{2007ApJ...664L..91A}, a very hard and extended source, possibly related to the massive star-forming region associated with the HII region Sharpless\,104 (Sh\,2-104) \citep{2004ApJ...601L..75T}. Particle acceleration in shocks driven by the winds from the Wolf-Rayet stars in the young cluster Berk 87 in the Cyg OB1 association have also been proposed as a possible origin of the VHE gamma rays \citep{2007MNRAS.382..367B,2007MNRAS.377..920B}. VERITAS resolved the Milagro source into two sources, one of which, VER\,J2019+378, is a bright, 1 degree extended source, that likely accounts for the bulk of the Milagro emission, coincident with the star formation region Sh\,2-104. Its spectrum in the range 1-30\,TeV is well fitted with a power-law model of photon index 1.75, among the hardest values measured in the VHE band. The TeV counterpart of the Fermi-Cocoon has been studied with the ARGO detector up to about 10 TeV and with the HAWC detector up to 200 TeV \citep{2014ApJ...790..152B,Hona:2019g7}. The spectral energy distribution shows a significant softening at a few TeV: this is revealed by the comparison between the ARGO and HAWC data and the Fermi-LAT data. This break in the $\gamma$-ray spectrum might hint at a cut-off in the injected CR spectrum \citep{Hona:2019g7}, or possibly be explained as due to suppressed diffusion in the high turbulence environment of the Cygnus super-bubble, which would confine low energy particles while higher energy ones escape.
\citet{2019NatAs...3..561A} argued that three ultra-compact clusters located in the Galactic Centre power the HESS diffuse emission from the CMZ. The $1/r$ dependence of the CR density on distance from the star cluster is, in fact, a distinct signature of continuous injection of CRs over the cluster lifetime and following diffusion through ISM. The efficiency of conversion of kinetic energy of powerful stellar winds can be as high as 10 percent. This implies that the population of young massive stars can provide production of CRs at a rate of up to $10^{41}$ erg/s, which is sufficient to support the flux of Galactic CRs without invoking other source populations. This Galactic center PeVatrons together with the other massive star forming regions, such as Westerlund\,1 and Cyg\,OB2, would represent the major factories of Galactic CRs \citep{2019NatAs...3..561A}. A broader review of this subject, including a discussion of the subtleties associated with acceleration and propagation of CRs in these environments, may be found in \cite{Bykov+20}.
\section{Implications of anti-matter data and CR leptons}
\label{sec:cranti}
One final subject we want to address in this paper concerns the implications of recent observational results on the spectrum of CR electrons, positrons and anti-protons. While electrons can be directly accelerated from the ISM plasma, positrons and anti-protons are extremely few in the ordinary ISM and hence they have long been considered to be purely secondaries, namely a byproduct of CR interactions during propagation in the Galaxy. If one assumes that $e^+$ and $\bar p$ are pure secondaries, the straightforward expectation is that the ratio between their fluxes and that of protons (their primary parent particles) should monotonically decrease with energy, as is true for the B/C ratio discussed in \S~\ref{sec:crtransp}. In fact, the ratio between $e^+$ and $p$ should decrease even faster, because positrons are subject to radiation losses which further steepen the spectrum at high energies (see e.g. \cite{amatoblasi18} for extended discussion). Direct observations of the flux of positrons \citep{Adriani+09,Aguilar+13pos} and anti-protons \citep{Aguilar+16pbar} contradict this expectation, showing that the spectra of $e^+$, $\bar p$ and $p$ are all parallel to one another.
Several different scenarios have been proposed in the literature to explain this finding (e.g. \cite{Blum+13,Cowsik+14,Cowsik+16,Lipari17,Eichler17}). An interesting possibility is that CRs accumulate sizeable grammage in the vicinity of their sources, where the diffusion coefficient is reduced with respect to the Galactic average. Here diffusion is energy dependent and this is where most of the Boron is produced. Then, in the rest of the Galaxy, diffusion must be energy independent and faster than usually assumed. Most $e^+$ and $\bar p$ would be produced by particles that accumulate most of their grammage in this second region: at a given rigidity their parent particles are $\sim 10$ times more energetic than those that produce B nuclei, and hence spend a shorter time in the source vicinity.
While it is not fully clear that these assumptions would not violate other constraints, like those provided by anisotropy, a possible physical justification of such a scenario could stem from the effects of self-generated turbulence (see Sec.~\ref{sec:cracc}) which could considerably reduce the diffusion coefficient in the vicinity of CR sources. In terms of theory the viability of this scenario is not fully clear, especially due to the unknowns related to the abundance of neutrals, which can effectively damp the CR induced resonant streaming instability \citep{DAngelo+16,Nava+16}. However, there are observational indications that enhanced confinement indeed occurs, at least in the vicinity of some sources. A striking example are the so-called TeV haloes \citep{Abeysekara+17} surrounding evolved pulsars. Many more cases are expected to be found with next-generation gamma-ray telescopes, such as CTA.
In reality, given the present uncertainties in many of the parameters involved, among which, most notably, the $\bar p$ production cross-section \citep{Korsmeier+18}, the $\bar p/p$ ratio is not in a statistically significant disagreement with the standard description of CR propagation through the Galaxy, nor with the B/C ratio, when all available information is taken into account and the subtle effects of re-acceleration are included in the description \citep{Bresci+19}. Indeed the high precision of the currently available data prompts theory to move forward and include second-order effects: one of these is the finite probability that during propagation particles might encounter an active CR accelerator (e.g. a SNR shock) and be re-accelerated. This has no effect on CR primaries, but flattens the spectrum of CR secondaries. Taking this phenomenon into account, together with the hardening of primaries, the energy dependence of the $\bar p$ production cross section, and, finally, the source grammage mentioned in \S~\ref{sec:crtransp}, allows one to reproduce the $B/C$ ratio, the $\bar p/p$ ratio and all available data on stable nuclei \citep{Bresci+19}.
Differently from $\bar p$, the spectrum of $e^+$ is not easy to reproduce in the standard CR propagation scenario, unless additional sources of primary positrons are invoked.
The leptonic cosmic ray population is a small fraction of the total CR population. For instance at $\sim$1 GeV the ratio of the hadronic versus leptonic fluxes measured at Earth is roughly 100 : 1 (https://lpsc.in2p3.fr/crdb/). Differently from hadrons, leptons are heavily affected by energy losses due to synchrotron and inverse Compton processes in the interstellar magnetic and radiation fields. If positrons were purely secondary products of CR interactions, their spectrum would have to be steeper than that of protons at energies where losses are important. As recently shown by \cite{2020PhRvD.101j3030D}, in order for this not to happen, the escape time from the Galaxy would have to be so short that protons would not be able to produce the observed $\bar p$ flux.
The anomalous positron spectrum has been at the origin of a huge number of papers, proposing either astrophysical or particle physics (Dark Matter annihilation related) solutions \citep{Aguilar+19pos}. Limiting the discussion to astrophysical scenarios, important constraints on the plausibility of the different proposals come from the maximum age and distance of the highest energy lepton sources. As shown in Fig.~\ref{fig:my_label}, electrons and positrons are now detected up 20 TeV. The electron spectrum shows roughly a constant slope up to an energy of about 1 TeV, above which the spectrum becomes steeper. One possibility to interpret the break at 1 TeV is to relate it to the transition between a regime where a large number of lepton accelerators contribute to the spectrum, to a regime where only a few or even only one single electron source very close to the Sun contributes \citep{Recchia19}.
The number of contributing sources depends, in turn, on the average intensity of the magnetic and radiation fields and on the diffusion coefficient in the Galaxy. The latter, in particular, seems to need some revision in light of the recent measurements of secondary nuclei, both stable and unstable ones, provided by AMS-02 \citep{Aguilar+18sec}.
For the standard average values of the magnetic and radiation fields in the Galaxy, the cooling time of leptons with energies above few tens of GeV can be approximately estimated as:
\begin{equation}
t_{\rm cool} \approx 10^6\, \left[\left(\frac{U_{\rm rad}}{0.3\ {\rm eV}{\rm cm}^{-3}}\right)+\left(\frac{B}{3\mu G}\right)^2\right]^{-1}\left(\frac{E}{1 \mathrm{TeV}}\right)^{-1}\ .
\end{equation} Losses limit the distance $\lambda$ from which leptons of a given energy can reach us to $\lambda \approx 2\sqrt{D(E)t_{\rm cool}}(E)$, with $D(E)$ the galactic diffusion coefficient.
Using for $D(E)$ the estimate provided by \cite{EvoliBe}, which allows to reproduce all the available AMS-02 data on both primary and secondary nuclei (both stable and unstable), one can estimate that the highest energy leptons detected at the Earth cannot come from further than $\lambda({\rm 1 TeV})\approx 3.5$ kpc. This is a much larger distance than typically estimated in the past (see e.g. \cite{PhysRevD.52.3265}) and ensures that many sources contribute to the lepton flux even at the highest energies \citep{EvoliPosPRL}.
The most promising sources of primary positrons are likely Pulsar Wind Nebulae (PWNe). These are the nebulae formed by the highly relativistic magnetized wind produced by a fast spinning, highly magnetized neutron star (see e.g. \cite{Amato19} for a recent review). The magnetosphere of such a star is a very efficient anti-matter factory and the pulsar wind contains electrons and positrons in about equal amounts. At the wind termination shock, these particles are accelerated up to PeV energies with a spectrum in the form of a broken power-law, harder than $E^{-2}$ at energies below $\approx 500$ GeV and softer than $E^{-2}$ at higher energies. As discussed by \cite{Bykov+17}, the accelerated particles are released in the ISM after the pulsar leaves its associated Supernova Remnant to become a Bow Shock Pulsar Wind Nebula. Once taken into account, the contribution of CR leptons from these sources allows to very well explain both the overall flux of leptons and the positron fraction with very reasonable values of the few free parameters \citep{EvoliPosPRL}.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{electrons.pdf}
\includegraphics[width=0.49\textwidth]{positronfraction.pdf}
\caption{{\it Left Panel:} Differential spectrum of electrons and positrons multiplied by E$^3$. Cosmic-ray electrons can be measured using space-born instruments such as AMS or Fermi-LAT up to 1 $\sim$TeV and CALET or DAMPE up to $\sim$10 TeV. Ground-based Cherenkov telescopes, such as H.E.S.S., MAGIC and VERITAS, which benefit from very large effective areas, have measured the flux up to 20 TeV. The black line shows the proton spectrum multiplied by 0.01 \citep{2007APh....28..154S}. {\it Right Panel:} Positron fraction. Not all experiments are able to measure the electrons and positrons separately. Here measurements by \citet{PhysRevLett.122.101101,2009Natur.458..607A,2004PhRvL..93x1102B} are reported, together with a number of model predictions: the black curve is for pure secondary production in the standard scenario \citep{1998ApJ...493..694M}; the blue one includes a more sophisticated propagation model \citep{PhysRevLett.111.021102}; the green one invokes dark matter decay \citep{2013IJMPA..2830040I}; the red one includes a contribution from pulsars \citep{PhysRevD.88.023001}. The Figure is from \cite{2020PTEP.2020h3C01P}.}
\label{fig:my_label}
\end{figure}
\section{Conclusions and Outlook}
In this article we have tried to review some recent theoretical and observational developments in the quest for the origin of galactic Cosmic Rays. High energy observations, in the X-rays, gamma-rays and VHE gamma-rays have opened new possibilities to test the long standing paradigm that wants these particles mainly produced in the blast waves of supernova explosions.
X-ray observations have provided undisputable proof that SNRs can accelerate electrons up to tens of TeV, and at the same time have shown the presence of amplified magnetic fields in their environment. These magnetic fields might be taken as a hint of efficient particle acceleration, although alternatives are possible. For sure their strengths are in the range needed to make SNRs act as PeVatrons, accelerating particles up to the {\it knee} energy, where the galactic cosmic ray proton spectrum is expected to end. In spite of this, theory points to a rare occurrence and an extremely short duration of the PeVatron phase for SNRs (Sec.~\ref{sec:cracc}).
Multi-TeV photons (10-100 TeV photons) are the most direct messengers of the presence of protons in the critical knee region. Secondary electrons, produced as decay products of charged pions, may provide complimentary information through their X-ray synchrotron emission, but this channel has a lower radiative efficiency and secondary leptons are often subdominant with respect to primaries. In the end, the spectrum of the highest energy gamma-rays qualifies as the best source of information about the extension of the parent proton spectrum, and hence is the only direct tool to identify the PeVatrons and obtain essential information about the physics of particle acceleration and the formation of the {\it knee}.
Detailed gamma-ray studies of young objects in this class have failed to show clear evidence of acceleration above $\sim$ 100 TeV (Sec.~\ref{sec:rxj}). Clear evidence of accelerated hadrons has only been found in middle-aged SNRs interacting with molecular clouds. The wealth of target for nuclear interactions makes these objects ideal candidates to be bright in gamma-rays of hadronic origin, without contamination from leptonic Inverse Compton emission. However, these sources are not very efficient as accelerators and the maximum energy they achieve is limited in all cases to $\lesssim$ 10 TeV (Sec.~\ref{sec:mc}).
At the same time, observations in the TeV band have highlighted the existence of other kinds of sources showing relatively hard gamma-ray spectra extending above 100 TeV, in association with the galactic center region (Sec.~\ref{sec:CR:GC}) and with young stellar clusters (Sec.~\ref{sec:CR:SC}). How propagation in these complex environments, very likely more turbulent than the average ISM in the Galaxy, affects the spectrum finally released is yet to be thoroughly investigated.
On the other hand, direct measurements of CRs, and in particular high statistics spectra of primary and secondary nuclei, have provided new important insights on the properties of CR transport in the Galaxy, highlighting the presence of a change in propagation properties in the energy range 200-300 GeV. A possible interpretation of this finding is that below such energy CRs are mostly scattered by self-generated turbulence, another bit of information that stresses the importance of non-linearities in CR physics, and the importance of understanding how these particles affect not only the environment of their acceleration sites, but also the properties of the ISM at large (Sec.~\ref{sec:crtransp}).
Finally, a fundamentally important piece of information that we discussed, has to do with the anti-matter component of CRs (Sec.~\ref{sec:cranti}). Direct detection experiments showed that the leptonic anti-matter component of CRs can hardly be interpreted as of pure secondary origin. Additional sources of primary positrons seem to be required. The most natural candidates are pulsars and their nebulae, well known to be excellent positron factories. These will release their positrons only at late stages of their evolution, after leaving the parent SNRs. Crucial information about the particle release by these sources is likely to come from deeper studies of the gamma-ray halos that have been found to surround some of them \cite{TevHalos}.
In a time when a combination of new and surprising observations and theoretical difficulties are pushing the CR physics community to a deep re-evaluation of the SNR-CR connection, crucial tests are expected to come from high energy astrophysical observations. All the recent surprises and issues that we have discussed through this article suggest that different astrophysical sources might contribute at different energies and at different levels to the CR population. This prompts to the need of a census of all possible CR contributors, a purpose that is well served by an unbiased survey of the Galactic Plane in the crucial multi-TeV energy range. Such survey is currently being undertaken with the high altitude water Cherenkov detector, HAWC \citep{2017APS..APR.X4008M}. A PeVatron search conducted with detailed spectroscopy, higher sensitivity above 10 TeV and higher angular resolution is a key scientific goal of the upcoming Cherenkov Telescope Array \citep{2019scta.book.....C}. In the search for PeVatrons in the Galaxy, CTA will benefit from a wider energy range and a much better angular resolution: while the latter will be essential to establish correlations with the gas maps and sources observed in other wavelengths, the former will finally give us access to the hundreds of TeV energy range, which is the realm where the hadronic mechanism is believed to be strongly dominant over the leptonic one.
\section*{Acknowledgments}
We thank the anonymous reviewers for their careful reading of the manuscript and their many useful comments, which in our opinion have substantially improved the paper.
EA acknowledges support by INAF and ASI through Grants SKA-CTA INAF 2016, INAF-MAINSTREAM 2018 and ASI/INAF No. 2017-14-H.O and by the National Science Foundation under Grant No. NSF PHY-1748958.
SC acknowledges the support from Polish Science Centre grant, DEC-2017/27/B/ST9/02272.
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\section{Introduction}
The ability to render 3D scenes from arbitrary viewpoints can be seen as a big step in the evolution of digital multimedia, and has applications such as mixed reality media, graphic effects, design, and simulations. Often such renderings are based on a number of high resolution images of some original scene, and it is clear that to enable many applications, the data will need to be stored and transmitted efficiently over low-bandwidth channels (e.g., to a mobile phone for augmented reality).
Traditionally, the need to compress this data is viewed as a separate need from rendering. For example, light field images (LFI) consist of a set of images taken from multiple viewpoints. To compress the original views, often standard video compression methods such as HEVC \citep{hevc} are repurposed \citep{lfi_depthbased, lfi_eval}. Since the range of views is narrow, light field images can be effectively reconstructed by ``blending'' a smaller set of representative views \citep{lfi_wasp, lfi_depthbased, lfi_cnn, lfi_linear, lfi_gan}. Blending based approaches, however, may not be suitable for the more general case of arbitrary-viewpoint 3D scenes, where a very diverse set of original views may increase the severity of occlusions, and thus would require storage of a prohibitively large number of views to be effective.
A promising avenue for representing more complete 3D scenes is through neural representation functions, which have shown a remarkable improvement in rendering quality \citep{nerf, srn, nsvf, graf}. In such approaches, views from a scene are rendered by evaluating the representation function at sampled spatial coordinates and then applying a differentiable rendering process. Such methods are often referred to as implicit representations, since they do not explicitly specify the surface locations and properties within the scene, which would be required to apply some conventional rendering techniques like rasterization \citep{rasterization}. However, finding the representation function for a given scene requires training a neural network. This makes this class of methods difficult to use as a rendering method in the existing framework, since it is computationally infeasible on a low-powered end device like a mobile phone, which are often on the receiving side. Due to the data processing inequality, it may also be inefficient to compress the original views (the training data) rather than the trained representation itself, because the training process may discard some information that is ultimately not necessary for rendering (such as redundancy in the original views, noise, etc.).
In this work, we propose to apply neural representation functions to the scene compression problem by compressing the representation function itself. We use the NeRF model \citep{nerf}, a method which has demonstrated the ability to produce high-quality renders of novel views, as our representation function. To reduce redundancy of information in the model, we build upon the model compression approach of \citet{smc}, applying an entropy penalty to the set of discrete reparameterized neural network weights. The \emph{compressed NeRF} (cNeRF) describes a radiance field, which is used in conjunction with a differentiable neural renderer to obtain novel views (see Fig. \ref{fig:teaser}). To verify the proposed method, we construct a strong baseline method based on the approaches seen in the field of light field image compression. cNeRF consistently outperforms the baseline method, producing simultaneously superior renders and lower bitrates. We further show that cNeRF can be improved in the low bitrate regime when compressing multiple scenes at once. To achieve this, we introduce a novel parameterization which shares parameters across models and optimize jointly across scenes.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{images/teaser2.pdf}
\caption{Overview of cNeRF. The sender trains an entropy penalized neural representation function on a set of views from a scene, minimizing a joint rate-distortion objective. The receiver can use the compressed model to render novel views.}
\label{fig:teaser}
\end{figure}
\section{Background}
We define a multi-view image dataset as a set of tuples $D = \{(V_n, X_n)\}_{n=1}^N$, where $V_n$ is the camera pose and $X_n$ is the corresponding image from this pose. We refer to the 3D ground truth that the views capture as the scene. In what follows, we first provide a brief review of the neural rendering and the model compression approaches that we build upon while introducing the necessary notation.
\textbf{Neural Radiance Fields (NeRF)}~~ The neural rendering approach of \cite{nerf} uses a neural network to model a radiance field. The radiance field itself is a learned function $g_\theta : \mathbb{R}^5 \rightarrow (\mathbb{R}^3, \mathbb{R}^+)$, mapping a 3D spatial coordinate and a 2D viewing direction to a RGB value and a corresponding density element. To render a view, the RGB values are sampled along the relevant rays and accumulated according to their density elements. The learned radiance field mapping $g_\theta$ is parameterized with two multilayer perceptrons (MLPs), which \cite{nerf} refer to as the ``coarse'' and ``fine'' networks, with parameters $\theta_c$ and $\theta_f$ respectively. The input locations to the coarse network are obtained by sampling regularly along the rays, whereas the input locations to the fine network are sampled conditioned on the radiance field of the coarse network. The networks are trained by minimizing the distance from their renderings to the ground truth image:
\begin{align}\label{eqn:nerf_loss}
L &= \underbrace{\sum_{n=1}^N \big\Vert \hat{X}_n^{c}(\theta_c; V_n) - X_n \big\Vert_2^2}_{\textstyle L_c(\theta_c)} + \underbrace{\sum_{n=1}^N \big\Vert \hat{X}_n^{f}(\theta_f; V_n, \theta_c) - X_n \big\Vert_2^2}_{\textstyle L_f(\theta_f ;\theta_c)}
\end{align}
Where $|| \cdot ||_2$ is the Euclidean norm and the $\hat{X}_n$ are the rendered views. Note that the rendered view from the fine network $\hat{X}_n^{f}$ relies on both the camera pose $V_n$ and the coarse network to determine the spatial locations to query the radiance field. We drop the explicit dependence of $L_f$ on $\theta_c$ in the rest of the paper to avoid cluttering the notation. During training, we render only a minibatch of pixels rather than the full image. We give a more detailed description of the NeRF model and the rendering process in Appendix Sec. \ref{sec:nerf_rendering}.
\textbf{Model Compression through Entropy Penalized Reparameterization}~~ The model compression work of \citet{smc} reparameterizes the model weights $\Theta$ into a latent space as $\Phi$. The latent weights are decoded by a learned function $\mathcal{F}$, i.e. $\Theta = \mathcal{F}(\Phi)$. The latent weights $\Phi$ are modeled as samples from a learned prior $q$, such that they can be entropy coded according to this prior. To minimize the rate, i.e. length of the bit string resulting from entropy coding these latent weights, a differentiable approximation of the self-information $I(\bm{\phi}) = -\log_2(q(\bm{\phi}))$ of the latent weights is penalized. The continuous $\Phi$ are quantized before being applied in the model, with the straight-through estimator \citep{ste} used to obtain surrogate gradients of the loss function. Following \cite{BaLaSi17}, uniform noise is added when learning the continuous prior $q(\bm{\phi} + \bm{u})$ where $u_i \sim U(-\frac{1}{2}, \frac{1}{2}) ~ \forall ~ i$. This uniform noise is a stand-in for the quantization, and results in a good approximation for the self-information through the negative log-likelihood of the noised continuous latent weights. After training, the quantized weights $\tilde{\Phi}$ are obtained by rounding, $\tilde{\Phi} = \lfloor \Phi \rceil$, and transmitted along with discrete probability tables obtained by integrating the density over the quantization intervals. The continuous weights $\Phi$ and any parameters in $q$ itself can then be discarded.
\section{Method}
To achieve a compressed representation of a scene, we propose to compress the neural scene representation function itself. In this paper we use the NeRF model as our representation function. To compress the NeRF model, we build upon the model compression approach of \cite{smc} and jointly train for rendering as well as compression in an end-to-end trainable manner. We subsequently refer to this approach as cNeRF. The full objective that we seek to minimize is:
\begin{equation}
\mathcal{L}(\Phi, \Psi) = \underbrace{L_c(\mathcal{F}_c(\tilde{\Phi}_c)) + L_f(\mathcal{F}_f(\tilde{\Phi}_f))}_{\text{Distortion}} + \lambda \underbrace{\sum\nolimits_{\bm{\phi} \in \Phi} I(\bm{\phi})}_{\text{Rate}}
\end{equation}
where $\Psi$ denotes the parameters of $\mathcal{F}$ as well any parameters in the prior distribution $q$, and we have explicitly split $\Phi$ into the coarse $\Phi_c$ and fine $\Phi_f$ components such that $\Phi= \{\Phi_c, \Phi_f$\}. $\lambda$ is a trade-off parameter that balances between rate and distortion. A rate--distortion (RD) plot can be traced by varying $\lambda$ to explore the performance of the compressed model at different bitrates.
\textbf{Compressing a single scene}~~ When training cNeRF to render a single scene, we have to choose how to parameterize and structure $\mathcal{F}$ and the prior distribution $q$ over the network weights. Since the networks are MLPs, the model parameters for a layer $l$ consist of the kernel weights and biases $\{W_l, b_l\}$. We compress only the kernel weights $W_l$, leaving the bias uncompressed since it is much smaller in size. The quantized kernel weights $\tilde{W}_l$ are mapped to the model weights by $\mathcal{F}_l$, i.e. $W_l = \mathcal{F}_l(\tilde{W}_l)$. $\mathcal{F}_l$ is constructed as an affine scalar transformation, which is applied elementwise to $\tilde{W}_l$:
\begin{equation}\label{eqn:single_scene}
\mathcal{F}_l(\tilde{W}_{l, ij}) = \alpha_l \tilde{W}_{l, ij} + \beta_l
\end{equation}
We take the prior to be factored over the layers, such that we learn a prior per linear kernel $q_l$. Within each kernel, we take the weights in $\tilde{W}_l$ to be i.i.d. from the univariate distribution $q_l$, parameterized by a small MLP, as per the approach of \cite{BaLaSi17}. Note that the parameters of this MLP can be discarded after training (once the probability mass functions have been built).
\begin{figure}
\centering
\includegraphics[width=\textwidth]{images/lego_fern_zoom.pdf}
\caption{Renderings of the synthetic Lego scene and real Fern scene from the uncompressed NeRF model, at 32 bits per parameter (bpp), and from cNeRF with $\lambda \in \{0.0001, 0.01\}$.}
\label{fig:zooms1}
\end{figure}
\textbf{Compressing multiple scenes}~~ While the original NeRF model is trained for a single scene, we hypothesize that better rate--distortion performance can be achieved for multiple scenes, especially if they share information, by training a joint model.
For a dataset of $M$ scenes, we parameterize the kernel weights of model $m$, layer $l$ as:
\begin{align}
W^{m}_l &= \mathcal{F}^{m}_l(\tilde{W}^{m}_l, \tilde{S}_l) \nonumber \\
&= \alpha^{m}_l \tilde{W}^{m}_{l} + \beta^{m}_l + \gamma_l \tilde{S}_l \label{eqn:multiscene_param}
\end{align}
Compared to Eqn. \ref{eqn:single_scene}, we have added a shift, parameterized as a scalar linear transformation of a discrete shift $\tilde{S}_l$ , that is shared across all models $m \in \{1,..., M\}$. $\tilde{S}_l$ has the same dimensions as the kernel $W^{m}_l$, and as with the discrete latent kernels, $\tilde{S}_l$ is coded by a learned probability distribution. The objective for the multi-scene model becomes:
\begin{equation}\label{eqn:multiscene_loss}
\mathcal{L}(\Phi, \Psi) = \sum_{m=1}^M \bigg[ L_c^m(\mathcal{F}^{m}_c(\tilde{\Phi}^{m}_c, \tilde{\Phi}^{s}_c)) + L_f^m(\mathcal{F}^{m}_f(\tilde{\Phi}^{m}_f, \tilde{\Phi}^{s}_f)) + \lambda \sum_{\bm{\phi} \in \Phi^m} I(\bm{\phi}) \bigg] + \lambda \sum_{\bm{\phi} \in \Phi^s} I(\bm{\phi})
\end{equation}
where $\Phi^s$ is the set of all discrete shift $\tilde{S}$ parameters, and the losses, latent weights and affine transforms are indexed by scene and model $m$. Note that this parameterization has \textit{more} parameters than the total of the $M$ single scene models, which at first appears counter-intuitive, since we wish to reduce the overall model size. It is constructed as such so that the multi-scene parameterization contains the $M$ single scene parameterizations - they can be recovered by setting the shared shifts to zero. If the shifts are set to zero then their associated probability distributions can collapse to place all their mass at zero. So we expect that if there is little benefit to using the shared shifts then they can be effectively ignored, but if there is a benefit to using them then they can be utilized. As such, we can interpret this parameterization as inducing a soft form of parameter sharing.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{images/baseline.pdf}
\caption{A comparison of four (zoomed in) renderings from cNeRF with $\lambda=0.0001$ and HEVC + LLFF with QP=30. HEVC + LLFF shows obvious artifacts such as ghosting around edges and an overall less crisp rendering.}
\label{fig:baseline_comparison}
\end{figure}
\section{Experiments}
\textbf{Datasets}~~ To demonstrate the effectiveness of our method, we evaluate on two sets of scenes used by \cite{nerf}:
\begin{itemize}
\item \textit{Synthetic}. Consisting of $800\times800$ pixel$^2$ views taken from either the upper hemisphere or entire sphere around an object rendered using the Blender software package. There are 100 views taken to be in the train set and 200 in the test set.
\item \textit{Real}. Consisting of a set of forward facing $1008\times756$ pixel$^2$ photos of a complex scene. The number of images varies per scene, with $1/8$ of the images taken as the test images.
\end{itemize}
Since we are interested in the ability of the receiver to render novel views, all distortion results (for any choice of perceptual metric) presented are given on the test sets.
\textbf{Architecture and Optimization}~~ We maintain the same architecture for the NeRF model as \cite{nerf}, consisting of 13 linear layers and ReLU activations. For cNeRF we use Adam \citep{adam} to optimize the latent weights $\Phi$ and the weights contained in the decoding functions $\mathcal{F}$. For these parameters we use initial learning rate of $5\times 10^{-4}$ and a learning rate decay over the course of learning, as per \cite{nerf}. For the parameters of the learned probability distributions $q$, we find it beneficial to use a lower learning rate of $5\times 10^{-5}$, such that the distributions do not collapse prematurely. We initialize the latent linear kernels using the scheme of \cite{glorot}, the decoders $\mathcal{F}$ near the identity.
\textbf{Baseline}~~ We follow the general methodology exhibited in light field compression and take the compressed representation of the scene to be a compressed subset of the views. The receiver then decodes these views, and renders novel views conditioned on the reconstructed subset. We use the video codec HEVC to compress the subset of views, as is done by \cite{lfi_depthbased}. To render novel views conditioned on the reconstructed set of views, we choose the Local Light Field Fusion (LLFF) approach of \cite{llff}. LLFF is a state-of-the-art learned approach in which a novel view is rendered by promoting nearby views to multiplane images, which are then blended. We refer to the full baseline subsequently as HEVC + LLFF.
\begin{figure}[t]
\includegraphics[width=.5\textwidth]{images/blender_baseline_rds}\hfill%
\includegraphics[width=.5\textwidth]{images/llff_baseline_rds}
\caption{Rate--distortion curves for both the cNeRF and HEVC + LLFF approaches, on two synthetic (left) and two real (right) scenes. We include uncompressed NeRF, which is the rightmost point on the curve and a fixed size across scenes. We truncate the curves for HEVC + LLFF, since increasing the bitrate further does not improve PSNR. See Fig. \ref{fig:baseline_qp} for the full curves.}
\label{fig:rd_curves}
\end{figure}
\begin{table}[t]
\centering \small
\begin{tabular}{l|r:rrr:rr}
& \textbf{NeRF} & \multicolumn{3}{c:}{\textbf{cNeRF ($\lambda=1e^{-4}$)}} & \multicolumn{2}{c}{\textbf{HEVC + LLFF} (QP=30)} \\
Scene & PSNR & PSNR & Size (KB) & Reduction & PSNR & Size (KB) \\
\hline
\rule{0pt}{2ex}Chair & 33.51 & 32.28 & 621 & 8.32$\times$ & 24.38 & 5,778 \\
Drums & 24.85 & 24.85 & 870 & 5.94$\times$ & 20.25 & 5,985 \\
Ficus & 30.58 & 30.35 & 701 & 7.37$\times$ & 21.49 & 3,678 \\
Hotdog & 35.82 & 34.95 & 916 & 5.65$\times$ & 30.18 & 2,767 \\
Lego & 32.61 & 31.98 & 707 & 7.31$\times$ & 23.92 & 4,665 \\
Materials & 29.71 & 29.17 & 670 & 7.71$\times$ & 22.49 & 3,134 \\
Mic & 33.68 & 32.11 & 560 & 9.23$\times$ & 28.95 & 3,032 \\
Ship & 28.51 & 28.24 & 717 & 7.21$\times$ & 24.95 & 5,881 \\
\hdashline
\rule{0pt}{2ex}Room & 31.06 & 30.65 & 739 & 7.00$\times$ & 26.27 & 886\\
Fern & 25.23 & 25.17 & 990 & 5.22$\times$ & 22.16 & 2,066 \\
Leaves & 21.10 & 20.95 & 1,154 & 4.48$\times$ & 18.15 & 3,162 \\
Fortress & 31.65 & 31.15 & 818 & 6.32$\times$ & 26.57 & 1,149 \\
Orchids & 20.18 & 20.09 & 1,218 & 4.24$\times$ & 17.87 & 2,357 \\
Flower & 27.42 & 27.21 & 938 & 5.51$\times$ & 23.46 & 1,009 \\
T-Rex & 27.24 & 26.72 & 990 & 5.22$\times$ & 22.30 & 1,933 \\
Horns & 27.80 & 27.28 & 995 & 5.20$\times$ & 20.71 & 2,002
\end{tabular}
\caption{Results comparing the uncompressed NeRF model,cNeRF and HEVC + LLFF baseline. We pick $\lambda$ and QP to give a reasonable trade-off between bitrate and PSNR. The reduction column is the reduction in the size of cNeRF as compared to the uncompressed NeRF model, which has a size of 5,169KB. Note that for all scenes, cNeRF achieves both a higher PSNR and a lower bitrate than HEVC + LLFF.}
\label{tab:main}
\end{table}
\subsection{Results}
\textbf{Single scene compression}~~ To explore the frontier of achievable rate--distortion points for cNeRF, we evaluate at a range of entropy weights $\lambda$ for four scenes -- two synthetic (Lego and Ficus) and two real (Fern and Room). To explore the rate--distortion frontier for the HEVC + LLFF baseline we evaluate at a range of QP values for HEVC. We give a more thorough description of the exact specifications of the HEVC + LLFF baseline and the ablations we perform to select the hyperparameter values in Appendix Sec. \ref{sec:baseline_ablations}. We show the results in Fig. \ref{fig:rd_curves}. We also plot the performance of the uncompressed NeRF model -- demonstrating that by using entropy penalization the model size can be reduced substantially with a relatively small increase in distortion. For these scenes we plot renderings at varying levels of compression in Fig. \ref{fig:zooms1} and Fig. \ref{fig:zooms2}. The visual quality of the renderings does not noticeably degrade when compressing the NeRF model down to bitrates of roughly 5-6 bits per parameter (the precise bitrate depends on the scene). At roughly 1 bit per parameter, the visual quality has degraded significantly, although the renderings are still sensible and easily recognisable. We find this to be a surprising positive result, given that assigning a single bit per parameter is extremely restrictive for such a complex regression task as rendering. Indeed, to our knowledge no binary neural networks have been demonstrated to be effective on such tasks.
Although the decoding functions $\mathcal{F}$ (Eqn. \ref{eqn:single_scene}) are just relatively simple scalar affine transformations, we do not find any benefit to using more complex decoding functions. With the parameterization given, most of the total description length of the model is in the coded latent weights, not the parameters of the decoders or entropy models. We give a full breakdown in Tab. \ref{tab:bits_breakdown}.
Fig. \ref{fig:rd_curves} shows that cNeRF clearly outperforms the HEVC + LLFF baseline, always achieving lower distortions at a (roughly) equivalent bitrate. Reconstruction quality is reported as peak signal-to-noise ratios (PSNR). The results are consistent with earlier demonstrations that NeRF produces much better renderings than the LLFF model \citep{nerf}. However, it is still interesting to see that this difference persists even at much lower bitrates. To evaluate on the remaining scenes, we select a single $\lambda$ value for cNeRF and QP value for HEVC + LLFF. We pick the values to demonstrate a reasonable trade-off between rate and distortion. The results are shown in Tab. \ref{tab:main}. For every scene the evaluated approaches verify that cNeRF achieves a lower distortion at a lower bitrate. We can see also that cNeRF is consistently able to reduce the model size significantly without seriously impacting the distortion. Further, we evaluate the performance of cNeRF and HEVC + LLFF for other perceptual quality metrics in Tab. \ref{tab:metrics} and \ref{tab:metrics2}. Although cNeRF is trained to minimize the squared error between renderings and the true images (and therefore maximize PSNR), cNeRF also outperforms HEVC + LLFF in both MS-SSIM \citep{ms-ssim} and LPIPS \citep{lpips}. This is significant, since the results of \cite{nerf} indicated that for SSIM and LPIPS, the LLFF model had a similar performance to NeRF when applied to the real scenes. We display a comparison of renderings from cNeRF and HEVC + LLFF in Fig. \ref{fig:baseline_comparison}.
\textbf{Multi-scene compression}~~For the multi-scene case we compress one pair of synthetic scenes and one pair of real scenes. We train the multi-scene cNeRF using a single shared shift per linear kernel, as per Eqn. \ref{eqn:multiscene_param}. To compare the results to the single scene models, we take the two corresponding single scene cNeRFs, sum the sizes and average the distortions. We plot the resulting rate--distortion frontiers in Fig. \ref{fig:multiscene_rd}. The results demonstrate that the multi-scene cNeRF improves upon the single scene cNeRFs at low bitrates, achieving higher PSNR values with a smaller model. This meets our expectation, since the multi-scene cNeRF can share parameters via the shifts (Eqn. \ref{eqn:multiscene_param}) and so decrease the code length of the scene-specific parameters. At higher bitrates we see no benefit to using the multi-scene parameterization, and in fact see slightly worse performance. This indicates that in the unconstrained rate setting, there is no benefit to using the shared shifts, and that they may slightly harm optimization.
\begin{figure}[t]
\includegraphics[width=.5\textwidth]{images/lego_ficus_rd}\hfill%
\includegraphics[width=.5\textwidth]{images/fern_room_rd}
\caption{Rate--distortion curves for comparing the multi-scene model with a single shared shift to the single scene models. The models are shown for two synthetic (left) and two real scenes (right).}
\label{fig:multiscene_rd}
\end{figure}
\section{Related work}
\textbf{Scene Compression}
A 3D scene is typically represented as a set of images, one for each view. For a large number of views, compressing each image individually using a conventional compression method can require a large amount of space. As a result, there is a body of compression research which aims to exploit the underlying scene structure of the 3D scene to reduce space requirements. A lot of research has been focused on compressing light field image (LFI) data \citep{lfi_wasp, lfi_depthbased, lfi_linear, lfi_gan, lfi_cnn}. LFI data generally consists of multiple views with small angular distances separating them. This set of views can be used to reconstruct a signal on the 4D domain of rays of the light field itself, thus permitting post-processing tasks such as novel view synthesis and refocusing. A majority of works select a representative subset of views to transmit from the scene. These are compressed and transmitted, typically using a video codec, with the receiver decoding these images and then rendering any novel view for an unobserved (during training) camera pose. Reconstruction for novel camera poses can be performed using traditional methods, such as optical flow \citep{lfi_depthbased}, or by using recent learned methods that employ convolutional neural networks \citep{lfi_cnn} and generative adversarial networks \citep{lfi_gan}.
A contrasting approach to multi-view image compression is proposed by \cite{dsic}, in which a pair of images from two viewpoints is compressed by conditioning the coder of the second image on the coder of the first image. It is important to emphasise that we are not studying this kind of approach in this work, since we wish the receiver to have the ability to render novel views.
\textbf{Neural Rendering} is an emerging research area which combines learned components with rendering knowledge from computer graphics. Recent work has shown that neural rendering techniques can generate high quality novel views of a wide range of scenes \citep{nerf, srn, nsvf, graf}. In this work we build upon the method of \cite{nerf}, coined as a Neural Radiance Field (NeRF), for single scene compression and then extend it with a novel reparameterization for jointly compressing multiple scenes. Training neural representation networks jointly across different scenes (without compression) has been explored by \cite{srn} and \cite{nsvf}, who use a hypernetwork \citep{hypernetwork} to map a latent vector associated with each scene to the parameters of the representation network. \cite{nsvf} note that the hypernetwork approach results in significant degradation of performance when applied to the NeRF model (a loss of more than 4 dB PSNR). In contrast, our approach of shared reparameterization is significantly different from these methods.
\textbf{Model Compression}
There is a body of research for reducing the space requirements of deep neural networks. Pruning tries to find a sparse set of weights by successively removing a subset of weights according to some criterion \citep{pruning1, pruning2}. Quantization reduces the precision used to describe the weights themselves \citep{binary, ternary}. In this work we focus instead on weight coding approaches \citep{miracle,smc} that code the model parameters to yield a compressed representation.
\section{Discussion and Conclusion}\label{sec:discussion}
Our results demonstrate that cNeRF produces far better results as a compressed representation than a state-of-the-art baseline, HEVC+LLFF, which follows the paradigm of compressing the original views. In contrast, our method compresses a representation of the radiance field itself. This is important for two reasons:
\vspace{-0.2cm}
\begin{itemize}
\item Practically, compressing the views themselves bars the receiver from using more complex and better-performing rendering methods such as NeRF, because doing this would require training to be performed at the receiving side after decompression, which is computationally infeasible in many applications.
\item Determining the radiance field and compressing it on the sending side may have coding and/or representational benefits, because of the data processing inequality: the cNeRF parameters are a function of the original views, and as such must contain equal to or less information than the original views (the training data). The method is thus relieved of the need to encode information in the original views that is not useful for the rendering task.
\end{itemize}
\vspace{-0.2cm}
It is difficult to gather direct evidence for the latter point, as the actual entropy of both representations is difficult to measure (we can only upper bound it by the compressed size). However, the substantial performance improvement of our method compared to HEVC+LLFF suggests that the radiance field is a more economical representation for the scene.
The encoding time for cNeRF is long, given that a new scene must be trained from scratch. Importantly though, the decoding time is much less, as it is only required to render the views using the decompressed NeRF model. cNeRF enables neural scene rendering methods such as NeRF to be used for scene compression, as it shifts the complexity requirements from the receiver to the sender. In many applications, it is more acceptable to incur high encoding times than high decoding times, as one compressed data point may be decompressed many times, allowing amortization of the encoding time, and since power-constrained devices are often at the receiving side. Thus, our method represents a big step towards enabling neural scene rendering in practical applications.
\newpage
| 7,741 |
\section*{Introduction}
Generative music is a subfield of computational musicology in which the focus lies on the
automatic creation of musical material. This creation is based on algorithms accepting
inputs to influence the result obtained, and having a randomized behavior in the sense that
two executions of the algorithm with the same inputs produce different results. Several
very different approaches exist. For instance, some of them use Markov chains, others
genetic algorithms~\cite{Mat10}, still others neural networks~\cite{BHP20}, or even formal
grammars~\cite{Hol81,HMU06}. The way in which such algorithms represent and manipulate
musical data is crucial. Indeed, the data structures used to represent musical phrases
orient the nature of the operations we can define of them. Considering operations producing
new phrases from old ones is important to specify algorithms to randomly generate music. A
possible way for this purpose consists to give at input some musical phrases and the
algorithm creates a new one by blending them through operations. Therefore, the willingness
to endow the infinite set of all musical phrases with operations in order to obtain suitable
algebraic structures is a promising approach. Such interactions between music and algebra is
a fruitful field of investigation~\cite{Mor18,Jed19}.
In this work, we propose to use tools coming from combinatorics and algebraic combinatorics
to represent musical phrases and operations on them, in order to introduce generative
music algorithms close to the family of those based on formal grammars. More precisely, we
introduce the music box model, a very simple model to represent polyphonic phrases, called
multi-patterns. The infinite set of all these objects admits the structure of an operad.
Such structures originate from algebraic topology and are used nowadays also in algebraic
combinatorics and in computer science~\cite{Men15,Gir18}. Roughly speaking, in these
algebraic structures, the elements are operations with several inputs and the composition
law is the usual composition of operators. Since the set of multi-patterns forms an
operad, one can regard each pattern as an operation. The fallout of this is that each
pattern is, at the same time, a musical phrase and an operation acting on musical phrases.
In this way, our music box model and its associated operad provide an algebraic and
combinatorial framework to perform computations on musical phrases.
All this admits direct applications to design random generation algorithms since, as
introduced by the author in~\cite{Gir19}, given an operad there exist algorithms to generate
some of its elements. These algorithms are based upon bud generating systems, which are
general formal grammars based on colored operads~\cite{Yau16}. In the present work, we
propose three different variations of these algorithms to produce new musical phrases from
old ones. More precisely, our algorithm works as follows. It takes as input a finite set of
multi-patterns and an integer value to influence the size of the output. It works
iteratively by choosing patterns from the initial collection in order to alter the current
one by performing a composition using the operad structure. As we shall explain, the initial
patterns can be colored in order to forbid some compositions and avoid in this way some
musical intervals for instance. These generation algorithms are not intended to write
complete musical pieces; they are for obtaining, from short old patterns, a similar but
longer one, presenting possibly new ideas to the human composer.
This text is organized as follows. Section~\ref{sec:music_box_model} is devoted to setting
our context and notations about music theory and to introduce the music box model. In
Section~\ref{sec:operads}, we begin by presenting a brief overview of operad theory and we
build step by step the music box operad. For this, we introduce first an operad on sequences
of scale degrees, an operad on rhythm patterns, and then an operad on monophonic patterns to
end with the operad of multi-patterns. Three random generation algorithms for multi-patterns
are introduced in Section~\ref{sec:random_generation}. Finally,
Section~\ref{sec:applications} provides some concrete applications of the previous
algorithms. We focus here on random variations of a monophonic musical phrase as input
leading to random changes of rhythm, harmonizations, and arpeggiations.
In this version of this work, most of the proofs of the announced results are omitted due to
lack of space. A computer implementation of all the presented algorithms is, as well as its
source code and {\bf concrete} examples, available at~\cite{Gir20}.
\subsubsection*{General notations and conventions}
For any integer $n$, $[n]$ denotes the set $\{1, \dots, n\}$. If $a$ is a letter and $n$ is
a nonnegative integer, $a^n$ is the word consisting in $n$ occurrences of $a$. In
particular, $a^0$ is the empty word $\epsilon$.
\section{The music box model} \label{sec:music_box_model}
The purpose of this section is to set some definitions and some conventions about music
theory, and introduce multi-patterns that are abstractions of musical phrases.
\subsection{Notes and scales}
We fit into the context of an $\eta$ tone equal temperament, also written as $\eta$-TET,
where $\eta$ is any nonnegative integer. An \Def{$\eta$-note} is a pair $(k, n)$ where $0
\leq k \leq \eta - 1$ and $n \in \mathbb{Z}$. We shall write $\Note{k}{n}$ instead of $(k, n)$. The
integer $n$ is the \Def{octave index} and $k$ is the \Def{step index} of $\Note{k}{n}$. The
set of all $\eta$-notes is denoted by $\mathcal{N}^{(\eta)}$. Despite this level of generality,
and even if all the concepts developed in the sequel work for any~$\eta$, in most
applications and examples we shall consider that $\eta = 12$. Therefore, under this
convention, we simply call \Def{note} any \Def{$12$-note} and write $\mathcal{N}$ for
$\mathcal{N}^{(12)}$. We set in this context of $12$-TET the ``middle $C$'' as the note
$\Note{0}{4}$, which is the first step of the octave of index~$4$. An \Def{$\eta$-scale} is
an integer composition $\bm{\lambda}$ of $\eta$, that is a sequence $\Par{\bm{\lambda}_1, \dots,
\bm{\lambda}_\ell}$ of nonnegative integers satisfying
\begin{math}
\bm{\lambda}_1 + \dots + \bm{\lambda}_\ell = \eta.
\end{math}
The \Def{length} of $\bm{\lambda}$ is the number $\ell(\bm{\lambda}) := \ell$ of its elements. We
simply call \Def{scale} any $12$-scale. For instance, $(2, 2, 1, 2, 2, 2, 1)$ is the major
natural scale, $(2, 1, 2, 2, 1, 3, 1)$ is the harmonic minor scale, and $(2, 1, 4, 1, 4)$ is
the Hirajoshi scale. This encoding of a scale by an integer composition is also known under
the terminology of interval pattern.
A \Def{rooted scale} is a pair $\Par{\bm{\lambda}, r}$ where $\bm{\lambda}$ is a scale and $r$ is a
note. This rooted scale describes a subset $\mathcal{N}_{\Par{\bm{\lambda}, r}}$ of $\mathcal{N}$ consisting
in the notes reachable from $r$ by following the steps prescribed by the values $\bm{\lambda}_1$,
$\bm{\lambda}_2$, \dots, $\bm{\lambda}_{\ell(\bm{\lambda})}$ of $\bm{\lambda}$. For instance, if $\bm{\lambda}$ is the
Hirajoshi scale, then
\begin{equation}
\mathcal{N}_{\Par{\bm{\lambda}, \Note{0}{4}}}
= \Bra{\dots,
\Note{7}{3}, \Note{8}{3},
{\bf \Note{0}{4}},
\Note{2}{4},
\Note{3}{4},
\Note{7}{4},
\Note{8}{4}, \Note{0}{5}, \dots}.
\end{equation}
If $\bm{\lambda}$ is the major natural scale, then
\begin{equation}
\mathcal{N}_{\Par{\bm{\lambda}, \Note{2}{4}}} =
\Bra{\dots,
\Note{1}{4},
{\bf \Note{2}{4}}, \Note{4}{4}, \Note{6}{4}, \Note{7}{4},
\Note{9}{4}, \Note{11}{4}, \Note{1}{5},
\Note{2}{5}, \dots}.
\end{equation}
\subsection{Patterns}
We now introduce degree patterns, rhythm patterns, patterns, and finally multi-patterns.
A \Def{degree} $d$ is any element of $\mathbb{Z}$. Negative degrees are denoted by putting a bar
above their absolute value. For instance, $-3$ is denoted by $\bar{3}$. A \Def{degree
pattern} $\mathbf{d}$ is a finite word $\mathbf{d}_1 \dots \mathbf{d}_\ell$ of degrees. The
\Def{arity} of $\mathbf{d}$, also denoted by $|\mathbf{d}|$, is the number $\ell$ of its elements.
Given a rooted scale $(\bm{\lambda}, r)$, a degree pattern $\mathbf{d}$ specifies a sequence of notes
by assigning to the degree $0$ the note $r$, to the degree $1$ the following higher note in
$\mathcal{N}_{\Par{\bm{\lambda}, r}}$ next to $r$, to the degree $\bar{1}$ the lower note in
$\mathcal{N}_{\Par{\bm{\lambda}, r}}$ next to $r$, and so on. For instance, the degree pattern $1 0
\bar{2} \bar{3} 5 0 7$ specifies, in the context of the rooted scale $\Par{\bm{\lambda},
\Note{0}{4}}$ where $\bm{\lambda}$ is the major natural scale, the sequence of notes
\begin{equation*}
\Note{2}{4}, \Note{0}{4}, \Note{9}{3}, \Note{7}{3}, \Note{9}{4},
\Note{0}{4}, \Note{0}{5}.
\end{equation*}
A \Def{rhythm pattern} $\mathbf{r}$ is a finite word $\mathbf{r}_1 \dots \mathbf{r}_\ell$ on the
alphabet $\Bra{{\square}, {\blacksquare}}$. The symbol ${\square}$ is a \Def{rest} and the symbol ${\blacksquare}$ is
a \Def{beat}. The \Def{length} of $\mathbf{r}$ is $\ell$ and the \Def{arity} $|\mathbf{r}|$ of
$\mathbf{r}$ is its number of occurrences of beats. The \Def{duration sequence} of a rhythm
pattern $\mathbf{r}$ is the unique sequence $\Par{\alpha_0, \alpha_1, \dots,
\alpha_{|\mathbf{r}|}}$ of nonnegative integers such that
\begin{equation}
\mathbf{r}
=
{\square}^{\alpha_0} \enspace {\blacksquare} \; {\square}^{\alpha_1}
\enspace \cdots \enspace
{\blacksquare} \; {\square}^{\alpha_{|\mathbf{r}|}}.
\end{equation}
The rhythm pattern $\mathbf{r}$ specifies a rhythm wherein each beat has a relative duration:
the rhythm begins with a silence of $\alpha_0$ units of time, followed by a first beat
sustained $1 + \alpha_1$ units of time, and so on, and finishing by a last beat sustained $1
+ \alpha_{|\mathbf{r}|}$ units of time. We adopt here the convention that each rest and beat
last each the same amount of time of one eighth of the duration of a whole note. Therefore,
given a tempo specifying how many there are rests and beats by minute, any rhythm pattern
encodes a rhythm.
For instance, let us consider the rhythm pattern
\begin{equation}
\mathbf{r} := {\square} {\blacksquare} {\blacksquare} {\square} {\blacksquare} {\square} {\square} {\square} {\blacksquare} {\blacksquare} {\square} {\blacksquare}.
\end{equation}
The duration sequence of $\mathbf{r}$ is $\Par{1, 0, 1, 3, 0, 1, 0}$ so that $\mathbf{r}$
specifies the rhythm consisting in an eighth rest, an eighth note, a quarter note, a half
note, an eighth note, a quarter note, and finally an eighth note.
A \Def{pattern} is a pair $\mathbf{p} := (\mathbf{d}, \mathbf{r})$ such that $|\mathbf{d}| = |\mathbf{r}|$.
The \Def{arity} $|\mathbf{p}|$ of $\mathbf{p}$ is the arity of both $\mathbf{d}$ and $\mathbf{r}$, and the
\Def{length} $\ell(\mathbf{p})$ of $\mathbf{d}$ is the length $\ell(\mathbf{r})$ of~$\mathbf{r}$.
In order to handle concise notations, we shall write any pattern $(\mathbf{d}, \mathbf{r})$ as a
word $\mathbf{p}$ on the alphabet $\Bra{{\square}} \cup \mathbb{Z}$ where the subword of $\mathbf{p}$ obtained by
removing all occurrences of ${\square}$ is the degree pattern $\mathbf{d}$, and the word obtained
by replacing in $\mathbf{p}$ each integer by ${\blacksquare}$ is the rhythm pattern $\mathbf{r}$. For
instance,
\begin{equation} \label{equ:example_pattern}
1 {\square} {\square} \bar{2} {\square} 1 2
\end{equation}
is the concise notation for the pattern
\begin{equation}
\Par{1 \bar{2} 1 2, {\blacksquare} {\square} {\square} {\blacksquare} {\square} {\blacksquare} {\blacksquare}}.
\end{equation}
For this reason, thereafter, we shall see and treat any pattern $\mathbf{p}$ as a finite word
$\mathbf{p}_1 \dots \mathbf{p}_\ell$ on the alphabet $\Bra{{\square}} \cup \mathbb{Z}$. Remark that the length of
$\mathbf{p}$ is $\ell$ and that its arity is the number of letters of $\mathbb{Z}$ it has.
Given a rooted scale $(\bm{\lambda}, r)$ and a tempo, a pattern $\mathbf{p} := (\mathbf{d}, \mathbf{r})$
specifies a musical phrase, that is a sequence of notes arranged into a rhythm. The notes of
the musical phrase are the ones specified by the degree pattern $\mathbf{d}$ and their relative
durations are specified by the rhythm pattern~$\mathbf{r}$.
For instance, consider the pattern
\begin{equation}
\mathbf{p} :=
0 {\square} 1 2 \bar{1} {\square} 0 1 \bar{2} {\square} \bar{1} \bar{0} 0 {\square} {\square} {\square}.
\end{equation}
By choosing the rooted scale $\Par{\bm{\lambda}, \Note{9}{3}}$ where $\bm{\lambda}$ is the harmonic
minor scale, and by setting $128$ as tempo, one obtains the musical phrase
\begin{abc}[name=PhraseExample1,width=.9\abcwidth]
X:1
T:
K:Am
M:8/8
L:1/8
Q:1/8=128
A,2 B, C ^G,2 A, B, | F,2 ^G, A, A,4 |
\end{abc}
For any positive integer $m$, an \Def{$m$-multi-pattern} is an $m$-tuple $\mathbf{m} :=
\Par{\mathbf{m}^{(1)}, \dots, \mathbf{m}^{(m)}}$ of patterns such that all $\mathbf{m}^{(i)}$ have the same
arity and the same length. The \Def{arity} $|\mathbf{m}|$ of $\mathbf{m}$ is the common arity of all
the $\mathbf{m}^{(i)}$, and the \Def{length} $\ell(\mathbf{m})$ of $\mathbf{m}$ is the common length of
all the $\mathbf{m}^{(i)}$. An $m$-multi-pattern $\mathbf{m}$ is denoted through a matrix of
dimension $m \times \ell(\mathbf{m})$, where the $i$-th row contains the pattern $\mathbf{m}^{(i)}$
for any $i \in [m]$. For instance,
\begin{equation}
\mathbf{m} :=
\begin{MultiPattern}
0 & {\square} & 1 & {\square} & 1 \\
{\square} & \bar{2} & \bar{3} & {\square} & 0
\end{MultiPattern}
\end{equation}
is a $2$-multi-pattern having arity $3$ and length $5$. The fact that all patterns of an
$m$-multi-pattern must have the same length ensures that they last the same amount of units
of time. This is important since an $m$-multi-pattern is used to handle musical sequences
consisting in $m$ stacked voices. The condition about the arities of the patterns, and
hence, about the number of degrees appearing in these, is a particularity of our model and
comes from algebraic reasons. This will be clarified later in the article.
Given a rooted scale $(\bm{\lambda}, r)$ and a tempo, an $m$-multi-pattern $\mathbf{m}$ specifies a
musical phrase obtained by considering the musical phrases specified by each $\mathbf{m}^{(i)}$,
$i \in [m]$, each forming a voice. For instance, consider the $2$-multi-pattern
\begin{equation}
\mathbf{m} :=
\begin{MultiPattern}
0 & 4 & {\square} & 4 & 0 & 0 \\
\bar{7} & \bar{7} & 0 & {\square} & \bar{3} & \bar{3}
\end{MultiPattern}.
\end{equation}
By choosing the rooted scale $\Par{\bm{\lambda}, \Note{9}{3}}$ where $\bm{\lambda}$ is the minor natural
scale and by setting $128$ as tempo, one obtains the musical phrase
\begin{abc}[name=MultiPatternExample1,width=.47\abcwidth]
X:1
T:
K:Am
M:8/8
L:1/8
Q:1/8=128
V:voice1
A,1 E2 E1 A,1 A,1
V:voice2
A,,1 A,,1 A,2 E,1 E,1
\end{abc}
Due to the fact that $m$-multi-patterns evoke paper tapes of a programmable music box, we
call \Def{music box model} the model just described to represent musical phrases by
$m$-multi-patterns within the context of a rooted scale and a tempo.
\section{Operad structures} \label{sec:operads}
The purpose of this section is to introduce an operad structure on multi-patterns, called
music box operad. The main interest of endowing the set of multi-patterns with the structure
of an operad is that this leads to an algebraic framework to perform computations on
patterns.
\subsection{A primer on operads} \label{subsec:primer_operads}
We set here the elementary notions of operad theory used in the sequel. Most of them come
from~\cite{Gir18}.
A \Def{graded set} is a set $\mathcal{O}$ decomposing as a disjoint union
\begin{equation}
\mathcal{O} := \bigsqcup_{n \in \mathbb{N}} \mathcal{O}(n),
\end{equation}
where the $\mathcal{O}(n)$, $n \in \mathbb{N}$, are sets. For any $x \in \mathcal{O}$, there is by
definition a unique $n \in \mathbb{N}$ such that $x \in \mathcal{O}(n)$ called \Def{arity} of $x$ and
denoted by~$|x|$.
A \Def{nonsymmetric operad}, or an \Def{operad} for short, is a triple $\Par{\mathcal{O},
\circ_i, \mathbf{1}}$ such that $\mathcal{O}$ is a graded set, $\circ_i$ is a map
\begin{equation}
\circ_i : \mathcal{O}(n) \times \mathcal{O}(m) \to \mathcal{O}(n + m - 1),
\qquad i \in [n],
\end{equation}
called \Def{partial composition} map, and $\mathbf{1}$ is a distinguished element of
$\mathcal{O}(1)$, called \Def{unit}. This data has to satisfy, for any $x, y, z \in \mathcal{O}$,
the three relations
\begin{equation} \label{equ:operad_axiom_1}
\Par{x \circ_i y} \circ_{i + j - 1} z = x \circ_i \Par{y \circ_j z},
\quad i \in [|x|], \enspace j \in [|y|],
\end{equation}
\begin{equation} \label{equ:operad_axiom_2}
\Par{x \circ_i y} \circ_{j + |y| - 1} z = \Par{x \circ_j z} \circ_i y,
\quad 1 \leq i < j \leq |x|,
\end{equation}
\begin{equation} \label{equ:operad_axiom_3}
\mathbf{1} \circ_1 x = x = x \circ_i \mathbf{1},
\quad i \in [|x|].
\end{equation}
Intuitively, an operad is an algebraic structure wherein each element can be seen as an
operator having $|x|$ inputs and one output. Such an operator is depicted as
\begin{equation}
\begin{tikzpicture}
[Centering,xscale=.2,yscale=.25,font=\scriptsize]
\node[NodeST](x)at(0,0){$x$};
\node(r)at(0,2){};
\node(x1)at(-3,-2){};
\node(xn)at(3,-2){};
\node[below of=x1,node distance=1mm](ex1){$1$};
\node[below of=xn,node distance=1mm](exn){$|x|$};
\draw[Edge](r)--(x);
\draw[Edge](x)--(x1);
\draw[Edge](x)--(xn);
\node[below of=x,node distance=6mm]{$\dots$};
\end{tikzpicture}
\end{equation}
where the inputs are at the bottom and the output at the top. Given two operations $x$ and
$y$ of $\mathcal{O}$, the partial composition $x \circ_i y$ is a new operator obtained by
composing $y$ into $x$ onto its $i$-th input. Pictorially, this partial composition
expresses as
\begin{equation} \label{equ:partial_compostion_on_operators}
\begin{tikzpicture}[Centering,xscale=.24,yscale=.26,font=\scriptsize]
\node[NodeST](x)at(0,0){$x$};
\node(r)at(0,2){};
\node(x1)at(-3,-2){};
\node(xn)at(3,-2){};
\node(xi)at(0,-2){};
\node[below of=x1,node distance=1mm](ex1){$1$};
\node[below of=xn,node distance=1mm](exn){$|x|$};
\node[below of=xi,node distance=1mm](exi){$i$};
\draw[Edge](r)--(x);
\draw[Edge](x)--(x1);
\draw[Edge](x)--(xn);
\draw[Edge](x)--(xi);
\node[right of=ex1,node distance=4mm]{$\dots$};
\node[left of=exn,node distance=4mm]{$\dots$};
\end{tikzpicture}
\circ_i
\begin{tikzpicture}[Centering,xscale=.17,yscale=.25,font=\scriptsize]
\node[NodeST](x)at(0,0){$y$};
\node(r)at(0,2){};
\node(x1)at(-3,-2){};
\node(xn)at(3,-2){};
\node[below of=x1,node distance=1mm](ex1){$1$};
\node[below of=xn,node distance=1mm](exn){$|y|$};
\draw[Edge](r)--(x);
\draw[Edge](x)--(x1);
\draw[Edge](x)--(xn);
\node[below of=x,node distance=6mm]{$\dots$};
\end{tikzpicture}
=
\begin{tikzpicture}[Centering,xscale=.44,yscale=.3,font=\scriptsize]
\node[NodeST](x)at(0,0){$x$};
\node(r)at(0,1.5){};
\node(x1)at(-3,-2){};
\node(xn)at(3,-2){};
\node[below of=x1,node distance=1mm](ex1){$1$};
\node[below of=xn,node distance=1mm](exn){$|x| + |y| - 1$};
\node[right of=ex1,node distance=8mm]{$\dots$};
\node[left of=exn,node distance=9mm]{$\dots$};
\draw[Edge](r)--(x);
\draw[Edge](x)--(x1);
\draw[Edge](x)--(xn);
%
\node[NodeST](y)at(0,-2.5){$y$};
\node(y1)at(-1.6,-4.5){};
\node(yn)at(1.6,-4.5){};
\node[below of=y1,node distance=1mm](ey1){$i$};
\node[below of=yn,node distance=1mm](eyn){\qquad $i + |y| - 1$};
\draw[Edge](y)--(y1);
\draw[Edge](y)--(yn);
\node[below of=y,node distance=7mm]{$\dots$};
%
\draw[Edge](x)--(y);
\end{tikzpicture}.
\end{equation}
Relations~\eqref{equ:operad_axiom_1}, \eqref{equ:operad_axiom_2},
and~\eqref{equ:operad_axiom_3} become clear when they are interpreted into this context of
abstract operators and rooted trees.
Let $\Par{\mathcal{O}, \circ_i, \mathbf{1}}$ be an operad. The \Def{full composition} map of
$\mathcal{O}$ is the map
\begin{equation}
\circ :
\mathcal{O}(n) \times
\mathcal{O}\Par{m_1}
\times \dots \times
\mathcal{O}\Par{m_n}
\to \mathcal{O}\Par{m_1 + \dots + m_n},
\end{equation}
defined, for any $x \in \mathcal{O}(n)$ and $y_1, \dots, y_n \in \mathcal{O}$ by
\begin{equation} \label{equ:full_composition_maps}
x \circ \Han{y_1, \dots, y_n}
:= \Par{\dots \Par{\Par{x \circ_n y_n} \circ_{n - 1}
y_{n - 1}} \dots} \circ_1 y_1,
\end{equation}
Intuitively, $x \circ \Han{y_1, \dots, y_n}$ is obtained by grafting simultaneously the
outputs of all the $y_i$ onto the $i$-th inputs of~$x$.
Let $\Par{\mathcal{O}', \circ'_i, \mathbf{1}'}$ be a second operad. A map $\phi : \mathcal{O} \to
\mathcal{O}'$ is an \Def{operad morphism} if for any $x \in \mathcal{O}(n)$, $\phi(x) \in
\mathcal{O}'(n)$, $\phi(\mathbf{1}) = \mathbf{1}'$, and for any $x, y \in \mathcal{O}$ and $i \in [|x|]$,
\begin{equation} \label{equ:operad_morphisms}
\phi(x \circ_i y) = \phi(x) \circ'_i \phi(y).
\end{equation}
If instead~\eqref{equ:operad_morphisms} holds by replacing the second occurrence of $i$ by
$|x| + 1 - i$, then $\phi$ is an \Def{operad antimorphism}. We say that $\mathcal{O}'$ is a
\Def{suboperad} of $\mathcal{O}$ if for any $n \in \mathbb{N}$, $\mathcal{O}'(n)$ is a subset of
$\mathcal{O}(n)$, $\mathbf{1} = \mathbf{1}'$, and for any $x, y \in \mathcal{O}'$ and $i \in [|x|]$, $x
\circ_i y = x \circ'_i y$. For any subset $\mathfrak{G}$ of $\mathcal{O}$, the \Def{operad
generated} by $\mathfrak{G}$ is the smallest suboperad $\mathcal{O}^{\mathfrak{G}}$ of
$\mathcal{O}$ containing $\mathfrak{G}$. When $\mathcal{O}^{\mathfrak{G}} = \mathcal{O}$ and
$\mathfrak{G}$ is minimal with respect to the inclusion among the subsets of
$\mathfrak{G}$ satisfying this property, $\mathfrak{G}$ is a \Def{minimal generating
set} of $\mathcal{O}$ and its elements are \Def{generators} of~$\mathcal{O}$.
The \Def{Hadamard product} of $\mathcal{O}$ and $\mathcal{O}'$ is the operad $\mathcal{O}
\boxtimes \mathcal{O}'$ defined, for any $n \in \mathbb{N}$, by $\Par{\mathcal{O} \boxtimes
\mathcal{O}'}(n) := \mathcal{O}(n) \times \mathcal{O}'(n)$, endowed with the partial composition map
$\circ''_i$ defined, for any $\Par{x, x'}, \Par{y, y'} \in \mathcal{O} \boxtimes
\mathcal{O}'$ and $i \in \Han{\left|\Par{x, x'}\right|}$, by
\begin{equation}
\Par{x, x'} \circ''_i \Par{y, y'} :=
\Par{x \circ_i y, x' \circ'_i y'},
\end{equation}
and having $\Par{\mathbf{1}, \mathbf{1}'}$ as unit.
\subsection{The music box operad}
We build an operad on multi-patterns step by step by introducing an operad on degree
patterns and an operad on rhythm patterns. The operad of patterns is constructed as the
Hadamard product of the two previous ones. Finally, the operad of multi-patterns if a
suboperad of an iterated Hadamard product of the operad of patterns with itself.
Let $\mathsf{DP}$ be the graded collection of all degree patterns, wherein for any $n \in \mathbb{N}$,
$\mathsf{DP}(n)$ is the set of all degree patterns of arity $n$. Let us define on $\mathsf{DP}$
the partial composition $\circ_i$ wherein, for any degree patterns $\mathbf{d}$ and $\mathbf{d}'$,
and any integer $i \in [|\mathbf{d}|]$,
\begin{footnotesize}
\begin{equation}
\mathbf{d} \circ_i \mathbf{d}' :=
\mathbf{d}_1 \dots \mathbf{d}_{i - 1}
\Par{\mathbf{d}_i + \mathbf{d}'_1} \dots \Par{\mathbf{d}_i + \mathbf{d}'_{|\mathbf{d}'|}}
\mathbf{d}_{i + 1} \dots \mathbf{d}_{|\mathbf{d}|}.
\end{equation}
\end{footnotesize}
For instance,
\begin{equation}
0 {\bf 1} 234 \circ_2 {\bf \bar{1}10} = 0 {\bf 021} 234.
\end{equation}
We denote by $\epsilon$ the empty degree pattern. This element is the only one
of~$\mathsf{DP}(0)$.
\begin{Proposition} \label{prop:operad_degree_patterns}
The triple $\Par{\mathsf{DP}, \circ_i, 0}$ is an operad.
\end{Proposition}
\begin{proof}
This is the consequence of the fact that $\Par{\mathsf{DP}, \circ_i, 0}$ is the image of
the monoid $\Par{\mathbb{Z}, +, 0}$ by the construction $\mathsf{T}$ defined
in~\cite{Gir15}. Since this construction associates an operad with any monoid, the
result follows.
\end{proof}
We call $\mathsf{DP}$ the \Def{degree pattern operad}.
\begin{Proposition} \label{prop:generating_set_operad_degree_patterns}
The operad $\mathsf{DP}$ admits $\Bra{\epsilon, \bar{1}, 1, 00}$ and $\Bra{\epsilon,
\bar{1} 1}$ as minimal generating sets.
\end{Proposition}
Let $\mathsf{RP}$ be the graded collection of all rhythm patterns, wherein for any $n \in \mathbb{N}$,
$\mathsf{RP}(n)$ is the set of all rhythm patterns of arity $n$. Let us define on $\mathsf{RP}$
the partial composition $\circ_i$ wherein, for any rhythm patterns $\mathbf{r}$ and $\mathbf{r}'$,
and any integer $i \in [|\mathbf{r}|]$, $\mathbf{r} \circ_i \mathbf{r}'$ is obtained by replacing the
$i$-th occurrence of ${\blacksquare}$ in $\mathbf{r}$ by $\mathbf{r}'$. For instance,
\begin{equation}
{\blacksquare} {\blacksquare} {\square} {\blacksquare} {\square} {\square} {\blacksquare}
\circ_3
{\square} {\blacksquare} {\square} {\blacksquare}
=
{\blacksquare} {\blacksquare} {\square} \; {\square} {\blacksquare} {\square} {\blacksquare} \; {\square} {\square} {\blacksquare}.
\end{equation}
We denote by $\epsilon$ the empty rhythm pattern. This element is not the only one of
$\mathsf{RP}(0)$ since $\mathsf{RP}(0) = \Bra{{\square}^\alpha : \alpha \in \mathbb{N}}$.
\begin{Proposition} \label{prop:operad_rhythm_patterns}
The triple $\Par{\mathsf{RP}, \circ_i, {\blacksquare}}$ is an operad.
\end{Proposition}
We call $\mathsf{RP}$ the \Def{rhythm pattern operad}.
\begin{Proposition} \label{prop:generating_set_operad_rhythm_pattern}
The operad $\mathsf{RP}$ admits $\Bra{\epsilon, {\square}, {\blacksquare} {\blacksquare}}$ as minimal generating
set.
\end{Proposition}
Let $\mathsf{P}$ be the operad defined as
\begin{equation}
\mathsf{P} := \mathsf{DP} \boxtimes \mathsf{RP}.
\end{equation}
Since a pattern is a pair $(\mathbf{d}, \mathbf{r})$ where $\mathbf{d}$ is a degree pattern and
$\mathbf{r}$ is a rhythm pattern of the same arity, for any $n \in \mathbb{N}$, $\mathsf{P}(n)$ is in
fact the set of all patterns of arity $n$. For this reason, $\mathsf{P}$ is the graded set of
all patterns. We call $\mathsf{P}$ the \Def{pattern operad}. For instance, by using the
concise notation for patterns,
\begin{equation}
{\square} \bar{2} {\bf 1} {\square} 1 \circ_2 {\bf 0 {\square} \Bar{1}}
= {\square} \bar{2} \; {\bf 1 {\square} 0} \; {\square} 1.
\end{equation}
We denote by $\epsilon$ the empty pattern.
\begin{Proposition} \label{prop:generating_set_operad_patterns}
The operad $\mathsf{P}$ admits $\Bra{\epsilon, {\square}, \bar{1}, 1, 00}$ and $\Bra{\epsilon,
{\square}, \bar{1} 1}$ as minimal generating sets.
\end{Proposition}
A consequence of Proposition~\ref{prop:generating_set_operad_patterns} is that any pattern
$\mathbf{p}$ expresses as a tree on the internal nodes in $\Bra{\epsilon, {\square}, \bar{1}, 1, 00}$
or in $\Bra{\epsilon, {\square}, \bar{1} 1}$. For instance, the pattern
$\mathbf{p} := \bar{1} {\square} {\square} 1 {\square} 3$ expresses as the trees
\begin{equation}
\begin{tikzpicture}[Centering,xscale=0.26,yscale=0.23]
\node(1)at(0.00,-9.14){};
\node(10)at(8.00,-6.86){${\square}$};
\node(15)at(10.00,-13.71){};
\node(3)at(2.00,-6.86){${\square}$};
\node(5)at(4.00,-4.57){${\square}$};
\node(8)at(6.00,-6.86){};
\node[NodeST](0)at(0.00,-6.86){$\bar{1}$};
\node[NodeST](11)at(9.00,-4.57){$00$};
\node[NodeST](12)at(10.00,-6.86){$1$};
\node[NodeST](13)at(10.00,-9.14){$1$};
\node[NodeST](14)at(10.00,-11.43){$1$};
\node[NodeST](2)at(1.00,-4.57){$00$};
\node[NodeST](4)at(3.00,-2.29){$00$};
\node[NodeST](6)at(5.00,0.00){$00$};
\node[NodeST](7)at(6.00,-4.57){$1$};
\node[NodeST](9)at(7.00,-2.29){$00$};
\draw[Edge](0)--(2);
\draw[Edge](1)--(0);
\draw[Edge](10)--(11);
\draw[Edge](11)--(9);
\draw[Edge](12)--(11);
\draw[Edge](13)--(12);
\draw[Edge](14)--(13);
\draw[Edge](15)--(14);
\draw[Edge](2)--(4);
\draw[Edge](3)--(2);
\draw[Edge](4)--(6);
\draw[Edge](5)--(4);
\draw[Edge](7)--(9);
\draw[Edge](8)--(7);
\draw[Edge](9)--(6);
\node(r)at(5.00,2){};
\draw[Edge](r)--(6);
\end{tikzpicture}
\quad
\mbox{or}
\quad
\begin{tikzpicture}[Centering,xscale=0.21,yscale=0.20]
\node(0)at(0.00,-14.17){};
\node(10)at(10.00,-8.50){};
\node(12)at(12.00,-5.67){${\square}$};
\node(14)at(14.00,-8.50){$\epsilon$};
\node(16)at(16.00,-8.50){};
\node(2)at(2.00,-14.17){$\epsilon$};
\node(4)at(4.00,-11.33){$\epsilon$};
\node(6)at(6.00,-8.50){${\square}$};
\node(8)at(8.00,-8.50){${\square}$};
\node[NodeST](1)at(1.00,-11.33){$\bar{1} 1$};
\node[NodeST](11)at(11.00,0.00){$\bar{1} 1$};
\node[NodeST](13)at(13.00,-2.83){$\bar{1} 1$};
\node[NodeST](15)at(15.00,-5.67){$\bar{1} 1$};
\node[NodeST](3)at(3.00,-8.50){$\bar{1} 1$};
\node[NodeST](5)at(5.00,-5.67){$\bar{1} 1$};
\node[NodeST](7)at(7.00,-2.83){$\bar{1} 1$};
\node[NodeST](9)at(9.00,-5.67){$\bar{1} 1$};
\draw[Edge](0)--(1);
\draw[Edge](1)--(3);
\draw[Edge](10)--(9);
\draw[Edge](12)--(13);
\draw[Edge](13)--(11);
\draw[Edge](14)--(15);
\draw[Edge](15)--(13);
\draw[Edge](16)--(15);
\draw[Edge](2)--(1);
\draw[Edge](3)--(5);
\draw[Edge](4)--(3);
\draw[Edge](5)--(7);
\draw[Edge](6)--(5);
\draw[Edge](7)--(11);
\draw[Edge](8)--(9);
\draw[Edge](9)--(7);
\node(r)at(11.00,2.25){};
\draw[Edge](r)--(11);
\end{tikzpicture}
\end{equation}
respectively for the two previous generating sets.
For any positive integer $m$, let $\OperadP'_m$ be operad defined through the iterated
Hadamard product
\begin{equation}
\OperadP'_m :=
\underbrace{\mathsf{P} \boxtimes
\cdots \boxtimes \mathsf{P}}_{\footnotesize m \mbox{ terms}}.
\end{equation}
Let also $\OperadP_m$ be the subset of $\OperadP'_m$ restrained on the $m$-tuples
$\Par{\mathbf{m}^{(1)}, \dots, \mathbf{m}^{(m)}}$ such that $\ell\Par{\mathbf{m}^{(1)}} = \dots =
\ell\Par{\mathbf{m}^{(m)}}$.
\begin{Theorem} \label{thm:operad_multi_patterns}
For any positive integer $m$, $\OperadP_m$ is an operad.
\end{Theorem}
Since an $m$-multi-pattern is an $m$-tuple $\Par{\mathbf{m}^{(1)}, \dots, \mathbf{m}^{(m)}}$ where all
$\mathbf{m}^{(i)}$ have the same arity and the same length, for any $m \in \mathbb{N}$, $\OperadP_m$ is
the graded set of all $m$-multi-patterns. By Theorem~\ref{thm:operad_multi_patterns},
$\OperadP_m$ is an operad, called \Def{$m$-music box operad}.
By using the matrix notation for $m$-multi-patterns, we have for instance respectively in
$\OperadP_2$ and in $\OperadP_3$,
\begin{equation}
\begin{MultiPattern}
{\square} & \bar{2} & \bar{1} & {\square} & 0 \\
0 & 1 & {\square} & {\square} & 1
\end{MultiPattern}
\circ_2
\begin{MultiPattern}
{\bf 1} & {\square} & {\bf 0} & {\bf 0} \\
{\bf \bar{3}} & {\square} & {\bf 0} & {\bf 4} \\
\end{MultiPattern}
=
\begin{MultiPattern}
{\square} & \bar{2} & {\bf 0} & {\square} & {\bf \bar{0}} & {\bf \bar{0}} & {\square} & 0 \\
0 & {\bf \bar{2}} & {\square} & {\bf 1} & {\bf 5} & {\square} & {\square} & 1
\end{MultiPattern}.
\end{equation}
This definition of the $m$-music box operad $\OperadP_m$ explains why all the patterns of
an $m$-multi-pattern must have the same arity. This is a consequence of the general
definition of the Hadamard product of operads.
For any sequence $\Par{\alpha_1, \dots, \alpha_m}$ of integers of $\mathbb{Z}$ and $\beta \in \mathbb{N}$,
let
\begin{equation}
\phi_{\Par{\alpha_1, \dots, \alpha_m}, \beta} : \OperadP_m \to \OperadP_m
\end{equation}
be the map such that, for any $\mathbf{m} := \Par{\mathbf{m}^{(1)}, \dots, \mathbf{m}^{(m)}} \in
\OperadP_m$,
\begin{math}
\phi_{\Par{\alpha_1, \dots, \alpha_m}, \beta}\Par{\mathbf{m}}
\end{math}
is the $m$-multi-pattern obtained by multiplying each degree of $\mathbf{m}^{(j)}$ by $\alpha_j$
and by replacing each occurrence of ${\square}$ in $\mathbf{m}$ by $\beta$ occurrences of ${\square}$.
For instance,
\begin{equation}
\phi_{\Par{2, 0, -1}, 2}\Par{
\begin{MultiPattern}
1 & {\square} & {\square} & 2 \\
{\square} & 1 & {\square} & 3 \\
3 & 1 & {\square} & {\square}
\end{MultiPattern}}
=
\begin{MultiPattern}
2 & {\square} & {\square} & {\square} & {\square} & 4 \\
{\square} & {\square} & 0 & {\square} & {\square} & 0 \\
\bar{3} & \bar{1} & {\square} & {\square} & {\square} & {\square}
\end{MultiPattern}.
\end{equation}
\begin{Proposition} \label{prop:morphism_operad_multi_patterns}
For any positive integer $m$, any sequence $\Par{\alpha_1, \dots, \alpha_m}$ of
integers, and any nonnegative integer $\beta$, the map $\phi_{\Par{\alpha_1, \dots,
\alpha_m}, \beta}$ is an operad endomorphism of~$\OperadP_m$.
\end{Proposition}
Let also the map $\mathrm{mir} : \OperadP_m \to \OperadP_m$ be the map such that, for any
$\mathbf{m} \in \OperadP_m$, $\mathrm{mir}(\mathbf{m})$ is the $m$-multi-pattern obtained by reading the
$\mathbf{m}$ from right to left. For instance,
\begin{equation}
\mathrm{mir}\Par{
\begin{MultiPattern}
1 & {\square} & {\square} & 2 \\
{\square} & 1 & {\square} & 3 \\
3 & 1 & {\square} & {\square}
\end{MultiPattern}}
=
\begin{MultiPattern}
2 & {\square} & {\square} & 1 \\
3 & {\square} & 1 & {\square} \\
{\square} & {\square} & 1 & 3
\end{MultiPattern}.
\end{equation}
\begin{Proposition} \label{prop:mirror_operad_multi_patterns}
For any positive integer $m$, the map $\mathrm{mir}$ sending any $m$-multi-pattern to its
mirror is an operad anti-automorphism of $\OperadP_m$.
\end{Proposition}
Due to the $m$-music box operad and more precisely, to the operad structure on
$m$-multi-patterns, we can see any $m$-multi-pattern as an operator. Therefore, we can build
$m$-multi-patterns and then musical sequences by considering some compositions of small
building blocks $m$-multi-patterns. For instance, by considering the small
$2$-multi-patterns
\begin{equation}
\mathbf{m}_1 :=
\begin{MultiPattern}
0 & {\square} \\
{\square} & 0
\end{MultiPattern},
\enspace
\mathbf{m}_2 :=
\begin{MultiPattern}
1 & 0 & 1 \\
\bar{7} & 0 & 0
\end{MultiPattern},
\enspace
\mathbf{m}_3 :=
\begin{MultiPattern}
1 & 2 & {\square} & 3 \\
\bar{1} & 0 & {\square} & 1
\end{MultiPattern},
\end{equation}
one can build a new $2$-multi-pattern by composing them as specified by the tree
\begin{equation} \label{equ:example_composition_tree_multi_patterns}
\begin{tikzpicture}[Centering,xscale=0.35,yscale=0.2]
\node(0)at(0.00,-2.25){};
\node(10)at(6.00,-5){};
\node(3)at(1.00,-7.75){};
\node(5)at(2.00,-7.75){};
\node(6)at(3.00,-7.75){};
\node(7)at(4.00,-5){};
\node(9)at(5.00,-5){};
\node[NodeST](1)at(2.00,0.00){$\mathbf{m}_2$};
\node[NodeST](2)at(2.00,-2.75){$\mathbf{m}_1$};
\node[NodeST](4)at(2.00,-5.50){$\mathbf{m}_2$};
\node[NodeST](8)at(5.00,-2.75){$\mathbf{m}_3$};
\draw[Edge](0)--(1);
\draw[Edge](10)--(8);
\draw[Edge](2)--(1);
\draw[Edge](3)--(4);
\draw[Edge](4)--(2);
\draw[Edge](5)--(4);
\draw[Edge](6)--(4);
\draw[Edge](7)--(8);
\draw[Edge](8)--(1);
\draw[Edge](9)--(8);
\node(r)at(2.00,2){};
\draw[Edge](r)--(1);
\end{tikzpicture}.
\end{equation}
This produces the new $2$-multi-pattern
\begin{equation} \label{equ:example_multi_pattern}
\begin{MultiPattern}
1 & 1 & 0 & 1 & {\square} & 2 & 3 & {\square} & 3 \\
\bar{7} & {\square} & \bar{7} & 0 & 0 & \bar{1} & 0 & {\square} & 1
\end{MultiPattern}.
\end{equation}
Besides, by Proposition~\ref{prop:morphism_operad_multi_patterns}, the image
of~\eqref{equ:example_multi_pattern} through the map, for instance, $\phi_{\Par{-1, 2}, 3}$
is the same as the $2$-multi-pattern obtained
from~\eqref{equ:example_composition_tree_multi_patterns} by replacing each
$2$-multi-patt\-ern appearing in it by its image by~$\phi_{\Par{-1, 2}, 3}$.
\section{Generation and random generation} \label{sec:random_generation}
We exploit now the music box operad to design three random generation algorithms devoted to
generate new musical phrases from a finite set of multi-patterns. This relies on colored
operads and bud generating systems, a sort of formal grammars introduced in~\cite{Gir19}.
\subsection{Colored operads and bud operads}
We provide here the elementary notions about colored operads~\cite{Yau16}. We also explain
how to build colored operads from an operads.
A \Def{set of colors} is any nonempty finite set
\begin{math}
\mathfrak{C} := \Bra{\mathtt{b}_1, \dots, \mathtt{b}_k}
\end{math}
wherein elements are called \Def{colors}. A \Def{$\mathfrak{C}$-colored set} is a set
$\mathcal{C}$ decomposing as a disjoint union
\begin{equation}
\mathcal{C}
:=
\bigsqcup_{\substack{
a \in \mathfrak{C} \\
u \in \mathfrak{C}^*}}
\mathcal{C}(a, u),
\end{equation}
where $\mathfrak{C}^*$ is the set of all finite sequences of elements of $\mathfrak{C}$, and the
$\mathcal{C}(a, u)$ are sets. For any $x \in \mathcal{C}$, there is by definition a
unique pair $(a, u) \in \mathfrak{C} \times \mathfrak{C}^*$ such that $x \in \mathcal{C}(a,
u)$. The \Def{arity} $|x|$ of $x$ is the length $|u|$ of $u$ as a word, the \Def{output
color} $\mathrm{out}(x)$ of $x$ is $a$, and for any $i \in [|x|]$, the \Def{$i$-th input color}
$\mathrm{in}_i(x)$ of $x$ is the $i$-th letter $u_i$ of $u$. We also denote, for any $n \in \mathbb{N}$, by
$\mathcal{C}(n)$ the set of all elements of $\mathcal{C}$ of arity $n$. Therefore, a
colored graded set is in particular a graded set.
A \Def{$\mathfrak{C}$-colored operad} is a triple $\Par{\mathcal{C}, \circ_i, \mathbf{1}}$
such that $\mathcal{C}$ is a $\mathfrak{C}$-colored set, $\circ_i$ is a map
\begin{equation}
\circ_i : \mathcal{C}(a, u) \times \mathcal{C}\Par{u_i, v}
\to \mathcal{C}\Par{a, u \circ_i v},
\qquad i \in [|u|],
\end{equation}
called \Def{partial composition} map, where $u \circ_i v$ is the word on $\mathfrak{C}$
obtained by replacing the $i$-th letter of $u$ by $v$, and $\mathbf{1}$ is a map
\begin{equation}
\mathbf{1} : \mathfrak{C} \to \mathcal{C}(a, a),
\end{equation}
called \Def{colored unit} map. This data has to satisfy
Relations~\eqref{equ:operad_axiom_1} and~\eqref{equ:operad_axiom_2} when their left and
right members are both well-defined, and, for any $x \in \mathcal{C}$, the relation
\begin{equation}
\mathbf{1}(\mathrm{out}(x)) \circ_1 x = x = x \circ_i \mathbf{1}\Par{\mathrm{in}_i(x)},
\qquad i \in[|x|].
\end{equation}
Intuitively, an element $x$ of a colored operad having $a$ as output color and $u_i$ as
$i$-th input color for any $i \in [|x|]$ can be seen as an abstract operator wherein colors
are assigned to its output and to each of its inputs. Such an operator is depicted as
\begin{equation}
\begin{tikzpicture}[Centering,xscale=.25,yscale=.33,font=\scriptsize]
\node[NodeST](x)at(0,0){$x$};
\node(r)at(0,2){};
\node(x1)at(-3,-2){};
\node(xn)at(3,-2){};
\node[below of=x1,node distance=1mm](ex1){$1$};
\node[below of=xn,node distance=1mm](exn){$|x|$};
\draw[Edge](r)edge[]node[EdgeLabel]{$a$}(x);
\draw[Edge](x)edge[]node[EdgeLabel]{$u_1$}(x1);
\draw[Edge](x)edge[]node[EdgeLabel]{$u_{|x|}$}(xn);
\node[below of=x,node distance=8mm]{$\dots$};
\end{tikzpicture},
\end{equation}
where the colors of the output and inputs are put on the corresponding edges. The partial
composition of two elements $x$ and $y$ in a colored operad expresses pictorially as
\begin{equation}
\begin{tikzpicture}[Centering,xscale=.3,yscale=.35,font=\scriptsize]
\node[NodeST](x)at(0,0){$x$};
\node(r)at(0,1.75){};
\node(x1)at(-3,-2){};
\node(xn)at(3,-2){};
\node(xi)at(0,-2){};
\node[below of=x1,node distance=1mm](ex1){$1$};
\node[below of=xn,node distance=1mm](exn){$|x|$};
\node[below of=xi,node distance=1mm](exi){$i$};
\draw[Edge](r)edge[]node[EdgeLabel]{$a$}(x);
\draw[Edge](x)edge[]node[EdgeLabel]{$u_1$}(x1);
\draw[Edge](x)edge[]node[EdgeLabel]{$u_{|x|}$}(xn);
\draw[Edge](x)edge[]node[EdgeLabel]{$u_i$}(xi);
\node[right of=ex1,node distance=5mm]{$\dots$};
\node[left of=exn,node distance=5mm]{$\dots$};
\end{tikzpicture}
\circ_i
\begin{tikzpicture}[Centering,xscale=.2,yscale=.35,font=\scriptsize]
\node[NodeST](x)at(0,0){$y$};
\node(r)at(0,2){};
\node(x1)at(-3,-2){};
\node(xn)at(3,-2){};
\node[below of=x1,node distance=1mm](ex1){$1$};
\node[below of=xn,node distance=1mm](exn){$|y|$};
\draw[Edge](r)edge[]node[EdgeLabel]{$u_i$}(x);
\draw[Edge](x)edge[]node[EdgeLabel]{$v_1$}(x1);
\draw[Edge](x)edge[]node[EdgeLabel]{$v_{|y|}$}(xn);
\node[below of=x,node distance=8mm]{$\dots$};
\end{tikzpicture}
=
\begin{tikzpicture}[Centering,xscale=.41,yscale=.45,font=\scriptsize]
\node[NodeST](x)at(0,0){$x$};
\node(r)at(0,1.5){};
\node(x1)at(-3,-2){};
\node(xn)at(3,-2){};
\node[below of=x1,node distance=1mm](ex1){$1$};
\node[below of=xn,node distance=1mm](exn){$|x| \! + \! |y| \! - \! 1$};
\node[right of=ex1,node distance=8mm]{$\dots$};
\node[left of=exn,node distance=9mm]{$\dots$};
\draw[Edge](r)edge[]node[EdgeLabel]{$a$}(x);
\draw[Edge](x)edge[]node[EdgeLabel]{$u_1$}(x1);
\draw[Edge](x)edge[]node[EdgeLabel]{$u_{|x|}$}(xn);
%
\node[NodeST](y)at(0,-2.5){$y$};
\node(y1)at(-1.6,-4.5){};
\node(yn)at(1.6,-4.5){};
\node[below of=y1,node distance=1mm](ey1){$i$};
\node[below of=yn,node distance=1mm](eyn){\quad $i \! + \! |y| \! - \! 1$};
\draw[Edge](y)edge[]node[EdgeLabel]{$v_1$}(y1);
\draw[Edge](y)edge[]node[EdgeLabel]{$v_{|y|}$}(yn);
\node[below of=y,node distance=10mm]{$\dots$};
%
\draw[Edge](x)edge[]node[EdgeLabel]{$u_i$}(y);
\end{tikzpicture}.
\end{equation}
Besides, most of the definitions about operads recalled in
Section~\ref{subsec:primer_operads} generalize straightforwardly to
colored operads. In particular, one can consider the full composition map of a colored
operad defined by~\eqref{equ:full_composition_maps} when its right member is well-defined.
Let us introduce another operation, specific to colored operads. Let $\Par{\mathcal{C},
\circ_i, \mathbf{1}}$ be a colored operad. The \Def{colored composition} map of
$\mathcal{C}$ is the map
\begin{equation}
\odot :
\mathcal{C}(a, u) \times \mathcal{C}(b, v) \to \mathcal{C},
\quad
a, b \in \mathfrak{C},
\enspace
u, v \in \mathfrak{C}^*,
\end{equation}
defined, for any $x \in \mathcal{C}(a, u)$ and $y \in \mathcal{C}(b, v)$, by using the
full composition map, by
\begin{equation}
x \odot y := x \circ \Han{y^{(1)}, \dots, y^{(|x|)}},
\end{equation}
where for any $i \in [|x|]$,
\begin{equation}
y^{(i)} :=
\begin{cases}
y & \mbox{if } \mathrm{in}_i(x) = \mathrm{out}(y), \\
\mathbf{1}\Par{\mathrm{in}_i(x)} & \mbox{otherwise}.
\end{cases}
\end{equation}
Intuitively, $x \odot y$ is obtained by grafting simultaneously the outputs of
copies of $y$ into all the inputs of $x$ having the same color as the output color of~$y$.
Let us describe a general construction building a colored operad from a noncolored one
introduced in~\cite{Gir19}. Given a noncolored operad $\Par{\mathcal{O}, \circ_i, \mathbf{1}}$ and a
set of colors $\mathfrak{C}$, the \Def{$\mathfrak{C}$-bud operad} of $\mathcal{O}$ is the
$\mathfrak{C}$-colored operad $\mathsf{B}_\mathfrak{C}(\mathcal{O})$ defined in the following way.
First, $\mathsf{B}_\mathfrak{C}(\mathcal{O})$ is the $\mathfrak{C}$-colored set defined, for any $a
\in \mathfrak{C}$ and $u \in \mathfrak{C}^*$, by
\begin{equation}
\mathsf{B}_\mathfrak{C}(\mathcal{O})(a, u) := \Bra{(a, x, u) : x \in \mathcal{O}(|u|)}.
\end{equation}
Second, the partial composition maps $\circ_i$ of $\mathsf{B}_\mathfrak{C}(\mathcal{O})$ are
defined, for any $(a, x, u), \Par{u_i, y, v} \in \mathsf{B}_\mathfrak{C}(\mathcal{O})$ and $i \in
[|u|]$, by
\begin{equation} \label{equ:partial_composition_map_bud_operad}
(a, x, u) \circ_i \Par{u_i, y, v} := \Par{a, x \circ_i y, u \circ_i v}
\end{equation}
where the first occurrence of $\circ_i$ in the right member
of~\eqref{equ:partial_composition_map_bud_operad} is the partial composition map of $\mathcal{O}$ and the
second one is a substitution of words: $u \circ_i v$ is the word obtained by replacing in
$u$ the $i$-th letter of $u$ by $v$. Finally, the colored unit map $\mathbf{1}$ of
$\mathsf{B}_\mathfrak{C}(\mathcal{O})$ is defined by $\mathbf{1}(a) := \Par{a, \mathbf{1}, a}$ for any $a
\in \mathfrak{C}$, where $\mathbf{1}$ is the unit of $\mathcal{O}$. The \Def{pruning} $\mathrm{pr}((a, x, u))$ of an element $(a, x, u)$ of
$\mathsf{B}_\mathfrak{C}(\mathcal{O})$ is the element $x$ of~$\mathcal{O}$.
Intuitively, this construction consists in forming a colored operad
$\mathsf{B}_\mathfrak{C}(\mathcal{O})$ out of $\mathcal{O}$ by surrounding its elements with an output
color and input colors coming from $\mathfrak{C}$ in all possible ways.
We apply this construction to the $m$-music box operad by setting, for any set $\mathfrak{C}$
of colors,
\begin{equation}
\mathsf{B}\OperadMP_m^\mathfrak{C} := \mathsf{B}_\mathfrak{C}\Par{\OperadP_m}.
\end{equation}
We call $\mathsf{B}\OperadMP_m^\mathfrak{C}$ the \Def{$\mathfrak{C}$-bud $m$-music box operad}. The
elements of $\mathsf{B}\OperadMP_m^\mathfrak{C}$ are called \Def{$\mathfrak{C}$-colored
$m$-multi-patterns}. For instance, for $\mathfrak{C} := \Bra{\mathtt{b}_1, \mathtt{b}_2, \mathtt{b}_3}$,
\begin{equation}
\Par{\mathtt{b}_1,
\begin{MultiPattern}
1 & {\square} & 0 & {\square} & 1 \\
\bar{7} & {\square} & 0 & 0 & {\square}
\end{MultiPattern},
\mathtt{b}_2 \mathtt{b}_2 \mathtt{b}_1}
\end{equation}
is a $\mathfrak{C}$-colored $2$-multi-pattern. Moreover, in the colored operad
$\mathsf{B}\OperadMP_2^\mathfrak{C}$, one has
\begin{multline}
\Par{\mathtt{b}_3,
\begin{MultiPattern}
0 & 1 & {\square} \\
\bar{1} & {\square} & 0
\end{MultiPattern},
\mathtt{b}_2 \mathtt{b}_1}
\circ_2
\Par{\mathtt{b}_1,
\begin{MultiPattern}
{\bf 1} & {\bf 1} & {\bf 2} \\
{\bf 2} & {\bf \bar{1}} & {\bf \bar{2}}
\end{MultiPattern},
{\bf \mathtt{b}_3 \mathtt{b}_3 \mathtt{b}_2}}
\\
=
\Par{\mathtt{b}_3,
\begin{MultiPattern}
0 & {\bf 2} & {\bf 2} & {\bf 3} & {\square} \\
\bar{1} & {\square} & {\bf 2} & {\bf \bar{1}} & {\bf \bar{2}}
\end{MultiPattern},
\mathtt{b}_2 {\bf \mathtt{b}_3 \mathtt{b}_3 \mathtt{b}_2}}.
\end{multline}
The intuition that justifies the introduction of these colored versions of patterns and of
the $m$-music box operad is that colors restrict the right to perform the composition of two
given patterns. In this way, one can for instance forbid some intervals in the musical
phrases specified by the patterns of a suboperad of $\mathsf{B}\OperadMP_m^\mathfrak{C}$ generated by
a given set of $\mathfrak{C}$-colored $m$-multi-patterns. Moreover, given a set
$\mathfrak{G}$ of $\mathfrak{C}$-colored $m$-multi-patterns, the elements of the suboperad
${\mathsf{B}\OperadMP_m^\mathfrak{C}}^\mathfrak{G}$ of $\mathsf{B}\OperadMP_m^\mathfrak{C}$ generated by
$\mathfrak{G}$ are obtained by composing elements of $\mathfrak{G}$. Therefore, in some
sense, these elements inherit from properties of the patterns~$\mathfrak{G}$.
The next section uses these ideas to propose random generation algorithms outputting new
patterns from existing ones in a controlled way.
\subsection{Bud generating systems and random generation}
\label{subsec:bud_generating_systems}
We describe here a sort of generating systems using operads introduced in~\cite{Gir19}.
Slight variations are considered in this present work. We also design three random
generation algorithms to produce musical phrases.
A \Def{bud generating system}~\cite{Gir19} is a tuple $\Par{\mathcal{O}, \mathfrak{C}, \mathcal{R},
\mathtt{b}}$ where
\begin{enumerate}[label={\em (\roman*)}]
\item $\Par{\mathcal{O}, \circ_i, \mathbf{1}}$ is a noncolored operad, called \Def{ground operad};
\item $\mathfrak{C}$ is a finite set of colors;
\item $\mathcal{R}$ is a finite subset of $\mathsf{B}_\mathfrak{C}(\mathcal{O})$, called
\Def{set of rules};
\item $\mathtt{b}$ is a color of $\mathfrak{C}$, called \Def{initial color}.
\end{enumerate}
For any color $a \in \mathfrak{C}$, we shall denote by $\mathcal{R}_a$ the set of all rules of
$\mathcal{R}$ having $a$ as output color.
Bud generating systems are devices similar to context-free formal grammars~\cite{HMU06}
wherein colors play the role of nonterminal symbols. These last devices are designed to
generate sets of words. Bud generating systems are designed to generate more general
combinatorial objects (here, $m$-multi-patterns). More precisely, a bud generating system
$\Par{\mathcal{O}, \mathfrak{C}, \mathcal{R}, \mathtt{b}}$ allows us to build elements of $\mathcal{O}$ by
following three different operating modes. We describe in the next sections the three
corresponding random generation algorithms. These algorithms are in particular intended to
work with $\OperadP_m$ as ground operad in order to generate $m$-multi-patterns.
Hereafter, we shall provide some examples based upon the bud generating system
\begin{equation} \label{equ:example_bud_generating_system}
\mathcal{B} :=
\Par{\OperadP_2, \Bra{\mathtt{b}_1, \mathtt{b}_2, \mathtt{b}_3},
\Bra{\mathbf{c}_1, \mathbf{c}_2, \mathbf{c}_3, \mathbf{c}_4, \mathbf{c}_5}, \mathtt{b}_1}
\end{equation}
where
\begin{equation}
\mathbf{c}_1 :=
\Par{\mathtt{b}_1,
\begin{MultiPattern}
0 & 2 & {\square} & 1 & {\square} & 0 & 4 \\
\bar{5} & {\square} & {\square} & 0 & 0 & 0 & 0
\end{MultiPattern},
\mathtt{b}_3 \mathtt{b}_2 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_3},
\end{equation}
\begin{equation}
\mathbf{c}_2 := \Par{\mathtt{b}_1,
\begin{MultiPattern}
1 & {\square} & 0 \\
0 & {\square} & 1
\end{MultiPattern},
\mathtt{b}_1 \mathtt{b}_1},
\quad
\mathbf{c}_3 := \Par{\mathtt{b}_2,
\begin{MultiPattern}
\bar{1} \\
\bar{1}
\end{MultiPattern},
\mathtt{b}_1},
\end{equation}
\begin{equation}
\mathbf{c}_4 := \Par{\mathtt{b}_2,
\begin{MultiPattern}
0 & 0 \\
0 & 0
\end{MultiPattern},
\mathtt{b}_1 \mathtt{b}_1},
\quad
\mathbf{c}_5 := \Par{\mathtt{b}_3,
\begin{MultiPattern}
0 \\
0
\end{MultiPattern},
\mathtt{b}_3}.
\end{equation}
Moreover, to interpret the generated multi-patterns, we choose to consider a tempo of $128$
and the rooted scale $\Par{\bm{\lambda}, \Note{9}{3}}$ where $\bm{\lambda}$ is the Hirajoshi scale.
Let $\mathcal{B} := (\mathcal{O}, \mathfrak{C}, \mathcal{R}, \mathtt{b})$ be a bud generating system.
Let $\xrightarrow{\circ_i}$ be the binary relation on $\mathsf{B}_\mathfrak{C}(\mathcal{O})$ such
that
\begin{equation}
(a, x, u) \xrightarrow{\circ_i} (a, y, v)
\end{equation}
if there is a rule $r\in \mathcal{R}$ and $i \in [|u|]$ such that
\begin{equation}
(a, y, v) = (a, x, u) \circ_i r.
\end{equation}
An element $x$ of $\mathcal{O}$ is \Def{partially generated} by $\mathcal{B}$ if there is an
element $(\mathtt{b}, x, u)$ such that $\Par{\mathtt{b}, \mathbf{1}, \mathtt{b}}$ is in relation with
$(\mathtt{b}, x, u)$ w.r.t. the reflexive and transitive closure of~$\xrightarrow{\circ_i}$.
For instance, by considering the bud generating
system~\eqref{equ:example_bud_generating_system}, since
\begin{multline}
\Par{\mathtt{b}_1,
\begin{MultiPattern}
0 \\
0
\end{MultiPattern},
\mathtt{b}_1}
\xrightarrow{\circ_i}
\Par{\mathtt{b}_1,
\begin{MultiPattern}
1 & {\square} & 0 \\
0 & {\square} & 1
\end{MultiPattern},
\mathtt{b}_1 \mathtt{b}_1}
\\
\xrightarrow{\circ_i}
\Par{\mathtt{b}_1,
\begin{MultiPattern}
1 & {\square} & 0 & 2 & {\square} & 1 & {\square} & 0 & 4 \\
0 & {\square} & \bar{4} & {\square} & {\square} & 1 & 1 & 1 & 1
\end{MultiPattern},
\mathtt{b}_1 \mathtt{b}_3 \mathtt{b}_2 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_3}
\\
\xrightarrow{\circ_i}
\Par{\mathtt{b}_1,
\begin{MultiPattern}
1 & {\square} & 0 & 2 & 2 & {\square} & 1 & {\square} & 0 & 4 \\
0 & {\square} & \bar{4} & {\square} & {\square} & 1 & 1 & 1 & 1 & 1
\end{MultiPattern},
\mathtt{b}_1 \mathtt{b}_3 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_3},
\end{multline}
the $2$-multi-pattern
\begin{equation}
\begin{MultiPattern}
1 & {\square} & 0 & 2 & 2 & {\square} & 1 & {\square} & 0 & 4 \\
0 & {\square} & \bar{4} & {\square} & {\square} & 1 & 1 & 1 & 1 & 1
\end{MultiPattern}
\end{equation}
is partially generated by~$\mathcal{B}$.
The \Def{partial random generation algorithm} is the algorithm defined as follows:
\begin{itemize}
\item Inputs:
\begin{enumerate}
\item A bud generating system
$\mathcal{B} := (\mathcal{O}, \mathfrak{C}, \mathcal{R}, \mathtt{b})$;
\item An integer $k \geq 0$.
\end{enumerate}
\item Output: an element of $\mathcal{O}$.
\end{itemize}
\begin{enumerate}
\item Set $x$ as the element $(\mathtt{b}, \mathbf{1}, \mathtt{b})$;
\item Repeat $k$ times:
\begin{enumerate}
\item Pick a position $i \in [|x|]$ at random;
\item If $\mathcal{R}_{\mathrm{in}_i(x)} \ne \emptyset$:
\begin{enumerate}
\item Pick a rule $r \in \mathcal{R}_{\mathrm{in}_i(x)}$ at random;
\item Set $x := x \circ_i r$;
\end{enumerate}
\end{enumerate}
\item Returns $\mathrm{pr}(x)$.
\end{enumerate}
This algorithm returns an element partially generated by $\mathcal{B}$ obtained by applying
at most $k$ rules to the initial element $(\mathtt{b}, \mathbf{1}, \mathtt{b})$. The execution of the
algorithm builds a composition tree of elements of $\mathcal{R}$ with at most $k$ internal
nodes.
For instance, by considering the bud generating
system~\eqref{equ:example_bud_generating_system}, this algorithm called with $k := 5$ builds
the tree of colored $2$-multipatterns
\begin{equation}
\begin{tikzpicture}[Centering,xscale=0.22,yscale=0.14]
\node(0)at(-1.00,-3.75){};
\node(10)at(7.00,-11.25){};
\node(11)at(8.00,-11.25){};
\node(12)at(8.00,-3.75){};
\node(14)at(11.00,-7.50){};
\node(2)at(2.00,-7.50){};
\node(4)at(4.00,-7.50){};
\node(6)at(4.00,-11.25){};
\node(7)at(5.00,-11.25){};
\node(9)at(6.00,-11.25){};
\node[NodeST](1)at(2.00,-3.75){$\mathbf{c}_3$};
\node[NodeST](13)at(11.00,-3.75){$\mathbf{c}_5$};
\node[NodeST](3)at(5.00,0.00){$\mathbf{c}_1$};
\node[NodeST](5)at(5.00,-3.75){$\mathbf{c}_2$};
\node[NodeST](8)at(6.00,-7.50){$\mathbf{c}_1$};
\draw[Edge](0)--(3);
\draw[Edge](1)--(3);
\draw[Edge](10)--(8);
\draw[Edge](11)--(8);
\draw[Edge](12)--(3);
\draw[Edge](13)--(3);
\draw[Edge](14)--(13);
\draw[Edge](2)--(1);
\draw[Edge](4)--(5);
\draw[Edge](5)--(3);
\draw[Edge](6)--(8);
\draw[Edge](7)--(8);
\draw[Edge](8)--(5);
\draw[Edge](9)--(8);
\node(r)at(5.00,3){};
\draw[Edge](r)--(3);
\end{tikzpicture}
\end{equation}
which produces the $2$-multi-pattern
\begin{equation}
\begin{MultiPattern}
0 & 1 & {\square} & 2 & {\square} & 1 & 3 & {\square} & 2 & {\square} & 1 & 5 & {\square} & 0 & 4 \\
\bar{5} & {\square} & {\square} & \bar{1} & 0 & {\square} & \bar{4} & {\square} & {\square} & 1 & 1 & 1
& 1 & 0 & 0
\end{MultiPattern}.
\end{equation}
Together with the aforementioned interpretation, the generated musical phrase is
\begin{abc}[name=PhraseExample2,width=.9\abcwidth]
X:1
T:
K:Am
M:8/8
L:1/8
Q:1/8=128
V:voice1
A,1 B,2 C2 B,1 E2 C2 B,1 A2 A,1 F1
V:voice2
A,,3 F,1 A,2 B,,3 B,1 B,1 B,1 B,1 A,1 A,1
\end{abc}
Let $\xrightarrow{\circ}$ be the binary relation on $\mathsf{B}_\mathfrak{C}(\mathcal{O})$ such that
\begin{equation}
(a, x, u) \xrightarrow{\circ} (a, y, v)
\end{equation}
if there are rules $r_1, \dots, r_{|x|} \in \mathcal{R}$ such that
\begin{equation}
(a, y, v)
=
(a, x, u) \circ \Han{r_1, \dots, r_{|x|}}.
\end{equation}
An element $x$ of $\mathcal{O}$ is \Def{fully generated} by $\mathcal{B}$ if there is an element
$(\mathtt{b}, x, u)$ such that $\Par{\mathtt{b}, \mathbf{1}, \mathtt{b}}$ is in relation with $(\mathtt{b}, x,
u)$ w.r.t. the reflexive and transitive closure of~$\xrightarrow{\circ}$.
For instance, by considering the bud generating
system~\eqref{equ:example_bud_generating_system}, since
\begin{multline}
\Par{\mathtt{b}_1,
\begin{MultiPattern}
0 \\
0
\end{MultiPattern},
\mathtt{b}_1}
\\
\xrightarrow{\circ}
\Par{\mathtt{b}_1,
\begin{MultiPattern}
0 & 2 & {\square} & 1 & {\square} & 0 & 4 \\
\bar{5} & {\square} & {\square} & 0 & 0 & 0 & 0
\end{MultiPattern},
\mathtt{b}_3 \mathtt{b}_2 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_3}
\\
\xrightarrow{\circ}
\Par{\mathtt{b}_1,
\begin{MultiPattern}
0 & 1 & {\square} & 2 & {\square} & 1 & {\square} & 0 & 2 & {\square} & 1 & {\square} & 0 & 4 & 4 \\
\bar{5} & {\square} & {\square} & \bar{1} & 0 & {\square} & 1 & \bar{5} & {\square} & {\square} & 0 & 0
& 0 & 0 & 0
\end{MultiPattern},
\right.
\\
\left.
\mathtt{b}_3 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_3 \mathtt{b}_2 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_3
\mathtt{b}_3},
\end{multline}
the $2$-multi-pattern
\begin{equation}
\begin{MultiPattern}
0 & 1 & {\square} & 2 & {\square} & 1 & {\square} & 0 & 2 & {\square} & 1 & {\square} & 0 & 4 & 4 \\
\bar{5} & {\square} & {\square} & \bar{1} & 0 & {\square} & 1 & \bar{5} & {\square} & {\square} & 0 & 0
& 0 & 0 & 0
\end{MultiPattern}
\end{equation}
is fully generated by $\mathcal{B}$.
The \Def{full random generation algorithm} is the algorithm defined as follows:
\begin{itemize}
\item Inputs:
\begin{enumerate}
\item A bud generating system
$\mathcal{B} := (\mathcal{O}, \mathfrak{C}, \mathcal{R}, \mathtt{b})$;
\item An integer $k \geq 0$.
\end{enumerate}
\item Output: an element of $\mathcal{O}$.
\end{itemize}
\begin{enumerate}
\item Set $x$ as the element $(\mathtt{b}, \mathbf{1}, \mathtt{b})$;
\item Repeat $k$ times:
\begin{enumerate}
\item If all $\mathcal{R}_{\mathrm{in}_i(x)}$, $i \in [|x|]$, are nonempty:
\begin{enumerate}
\item Let $\Par{r_1, \dots, r_{|x|}}$ be a tuple of rules such that each $r_i$
is picked at random in
$\mathcal{R}_{\mathrm{in}_i(x)}$;
\item Set $x := x \circ \Han{r_1, \dots, r_{|x|}}$;
\end{enumerate}
\end{enumerate}
\item Returns $\mathrm{pr}(x)$.
\end{enumerate}
This algorithm returns an element synchronously generated by $\mathcal{B}$ obtained by
applying at most $k$ rules to the initial element $(\mathtt{b}, \mathbf{1}, \mathtt{b})$. The execution
of the algorithm builds a composition tree of elements of $\mathcal{R}$ of height at most~$k +
1$ wherein the leaves are all at the same distance from the root.
For instance, by considering the bud generating
system~\eqref{equ:example_bud_generating_system}, this algorithm called with $k := 2$ builds
the tree of colored $2$-multipatterns
\begin{equation}
\begin{tikzpicture}[Centering,xscale=0.19,yscale=0.085]
\node(0)at(0.00,-18.00){};
\node(1)at(1.00,-18.00){};
\node(12)at(9.00,-18.00){};
\node(14)at(11.00,-18.00){};
\node(16)at(12.50,-18.00){};
\node(18)at(13.50,-18.00){};
\node(19)at(14.50,-18.00){};
\node(21)at(15.50,-18.00){};
\node(23)at(17.00,-18.00){};
\node(3)at(2.00,-18.00){};
\node(4)at(3.00,-18.00){};
\node(5)at(4.00,-18.00){};
\node(7)at(5.50,-18.00){};
\node(9)at(6.50,-18.00){};
\node[NodeST](10)at(9.00,0.00){$\mathbf{c}_2$};
\node[NodeST](11)at(9.00,-12.00){$\mathbf{c}_5$};
\node[NodeST](13)at(11.00,-12.00){$\mathbf{c}_3$};
\node[NodeST](15)at(13.00,-6.00){$\mathbf{c}_1$};
\node[NodeST](17)at(13.00,-12.00){$\mathbf{c}_2$};
\node[NodeST](2)at(2.00,-12.00){$\mathbf{c}_1$};
\node[NodeST](20)at(15.00,-12.00){$\mathbf{c}_2$};
\node[NodeST](22)at(17.00,-12.00){$\mathbf{c}_5$};
\node[NodeST](6)at(4.00,-6.00){$\mathbf{c}_2$};
\node[NodeST](8)at(6.00,-12.00){$\mathbf{c}_2$};
\draw[Edge](0)--(2);
\draw[Edge](1)--(2);
\draw[Edge](11)--(15);
\draw[Edge](12)--(11);
\draw[Edge](13)--(15);
\draw[Edge](14)--(13);
\draw[Edge](15)--(10);
\draw[Edge](16)--(17);
\draw[Edge](17)--(15);
\draw[Edge](18)--(17);
\draw[Edge](19)--(20);
\draw[Edge](2)--(6);
\draw[Edge](20)--(15);
\draw[Edge](21)--(20);
\draw[Edge](22)--(15);
\draw[Edge](23)--(22);
\draw[Edge](3)--(2);
\draw[Edge](4)--(2);
\draw[Edge](5)--(2);
\draw[Edge](6)--(10);
\draw[Edge](7)--(8);
\draw[Edge](8)--(6);
\draw[Edge](9)--(8);
\node(r)at(9.00,5.50){};
\draw[Edge](r)--(10);
\end{tikzpicture}
\end{equation}
which produces the $2$-multi-pattern
\begin{small}
\begin{equation}
\begin{MultiPattern}
2 & 4 & {\square} & 3 & {\square} & 2 & 6 & {\square} & 2 & {\square} & 1 & {\square} & 0 & 1 & {\square}
& 2 & {\square} & 1 & {\square} & 1 & {\square} & 0 & 4 \\
\bar{5} & {\square} & {\square} & 0 & 0 & 0 & 0 & {\square} & 1 & {\square} & 2 & {\square} & \bar{4}
& {\square} & {\square} & 0 & 1 & {\square} & 2 & 1 & {\square} & 2 & 1
\end{MultiPattern}.
\end{equation}
\end{small}
Together with the aforementioned interpretation, the generated musical phrase is
\begin{abc}[name=PhraseExample3,width=1.0\abcwidth]
X:1
T:
K:Am
M:8/8
L:1/8
Q:1/8=128
V:voice1
C1 F2 E2 C1 B2 C2 B,2 A,1 B,2 C2 B,2 B,2 A,1 F1
V:voice2
A,,3 A,1 A,1 A,1 A,2 B,2 C2 B,,3 A,1 B,2 C1 B,2 C1 B,1
\end{abc}
Let $\xrightarrow{\ColoredComposition}$ be the binary relation on $\mathsf{B}_\mathfrak{C}(\mathcal{O})$ such
that
\begin{equation}
(a, x, u) \xrightarrow{\ColoredComposition} (a, y, v)
\end{equation}
if there is a rule $r \in \mathcal{R}$ such that
\begin{equation}
(a, y, v) = (a, x, u) \odot r.
\end{equation}
An element $x$ of $\mathcal{O}$ is \Def{colorfully generated} by $\mathcal{B}$ if there is an
element $(\mathtt{b}, x, u)$ such that $\Par{\mathtt{b}, \mathbf{1}, \mathtt{b}}$ is in relation with
$(\mathtt{b}, x, u)$ w.r.t. the reflexive and transitive closure of~$\xrightarrow{\ColoredComposition}$.
For instance, by considering the bud generating
system~\eqref{equ:example_bud_generating_system}, since
\begin{multline}
\Par{\mathtt{b}_1,
\begin{MultiPattern}
0 \\
0
\end{MultiPattern},
\mathtt{b}_1}
\xrightarrow{\ColoredComposition}
\Par{\mathtt{b}_1,
\begin{MultiPattern}
0 & 2 & {\square} & 1 & {\square} & 0 & 4 \\
\bar{5} & {\square} & {\square} & 0 & 0 & 0 & 0
\end{MultiPattern},
\mathtt{b}_3 \mathtt{b}_2 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_3}
\\
\xrightarrow{\ColoredComposition}
\Par{\mathtt{b}_1,
\begin{MultiPattern}
0 & 2 & {\square} & 2 & {\square} & 1 & {\square} & 1 & {\square} & 0 & 4 \\
\bar{5} & {\square} & {\square} & 0 & 0 & {\square} & 1 & 0 & {\square} & 1 & 0
\end{MultiPattern},
\mathtt{b}_3 \mathtt{b}_2 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_3}
\\
\xrightarrow{\ColoredComposition}
\Par{\mathtt{b}_1,
\begin{MultiPattern}
0 & 1 & {\square} & 2 & {\square} & 1 & {\square} & 1 & {\square} & 0 & 4 \\
\bar{5} & {\square} & {\square} & \bar{1} & 0 & {\square} & 1 & 0 & {\square} & 1 & 0
\end{MultiPattern},
\mathtt{b}_3 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_1 \mathtt{b}_3},
\end{multline}
the $2$-multi-pattern
\begin{equation}
\begin{MultiPattern}
0 & 1 & {\square} & 2 & {\square} & 1 & {\square} & 1 & {\square} & 0 & 4 \\
\bar{5} & {\square} & {\square} & \bar{1} & 0 & {\square} & 1 & 0 & {\square} & 1 & 0
\end{MultiPattern}
\end{equation}
is colorfully generated by $\mathcal{B}$.
The \Def{colored random generation algorithm} is the algorithm defined as follows:
\begin{itemize}
\item Inputs:
\begin{enumerate}
\item A bud generating system
$\mathcal{B} := (\mathcal{O}, \mathfrak{C}, \mathcal{R}, \mathtt{b})$;
\item An integer $k \geq 0$.
\end{enumerate}
\item Output: an element of $\mathcal{O}$.
\end{itemize}
\begin{enumerate}
\item Set $x$ as the element $(\mathtt{b}, \mathbf{1}, \mathtt{b})$;
\item Repeat $k$ times:
\begin{enumerate}
\item Pick a rule $r \in \mathcal{R}$ at random;
\item Set $x := x \odot r$;
\end{enumerate}
\item Returns $\mathrm{pr}(x)$.
\end{enumerate}
This algorithm returns an element colorfully generated by $\mathcal{B}$ obtained by applying
at most $k$ rules to the initial element $(\mathtt{b}, \mathbf{1}, \mathtt{b})$. The execution of the
algorithm builds a composition tree of elements of height at most~$k + 1$.
For instance, by considering the bud generating
system~\eqref{equ:example_bud_generating_system}, this algorithm called with $k := 3$ builds
the tree of colored $2$-multipatterns
\begin{equation}
\begin{tikzpicture}[Centering,xscale=0.35,yscale=0.085]
\node(1)at(0.00,-12.50){};
\node(10)at(4.00,-12.50){};
\node(11)at(4.50,-12.50){};
\node(13)at(5.50,-18.75){};
\node(15)at(6.50,-18.75){};
\node(17)at(7.50,-18.75){};
\node(19)at(8.00,-12.50){};
\node(20)at(8.50,-12.50){};
\node(22)at(9.50,-18.75){};
\node(24)at(10.50,-12.50){};
\node(3)at(1.50,-12.50){};
\node(6)at(2.50,-18.75){};
\node(8)at(3.50,-18.75){};
\node[NodeST](0)at(0.00,-6.25){$\mathbf{c}_5$};
\node[NodeST](12)at(5.50,-12.50){$\mathbf{c}_5$};
\node[NodeST](14)at(6.50,-12.50){$\mathbf{c}_5$};
\node[NodeST](16)at(7.50,-12.50){$\mathbf{c}_3$};
\node[NodeST](18)at(8.00,-6.25){$\mathbf{c}_1$};
\node[NodeST](2)at(1.50,-6.25){$\mathbf{c}_3$};
\node[NodeST](21)at(9.50,-12.50){$\mathbf{c}_5$};
\node[NodeST](23)at(10.50,-6.25){$\mathbf{c}_5$};
\node[NodeST](4)at(4.00,0.00){$\mathbf{c}_1$};
\node[NodeST](5)at(2.50,-12.50){$\mathbf{c}_5$};
\node[NodeST](7)at(3.50,-12.50){$\mathbf{c}_3$};
\node[NodeST](9)at(4.00,-6.25){$\mathbf{c}_1$};
\draw[Edge](0)--(4);
\draw[Edge](1)--(0);
\draw[Edge](10)--(9);
\draw[Edge](11)--(9);
\draw[Edge](12)--(9);
\draw[Edge](13)--(12);
\draw[Edge](14)--(18);
\draw[Edge](15)--(14);
\draw[Edge](16)--(18);
\draw[Edge](17)--(16);
\draw[Edge](18)--(4);
\draw[Edge](19)--(18);
\draw[Edge](2)--(4);
\draw[Edge](20)--(18);
\draw[Edge](21)--(18);
\draw[Edge](22)--(21);
\draw[Edge](23)--(4);
\draw[Edge](24)--(23);
\draw[Edge](3)--(2);
\draw[Edge](5)--(9);
\draw[Edge](6)--(5);
\draw[Edge](7)--(9);
\draw[Edge](8)--(7);
\draw[Edge](9)--(4);
\node(r)at(4.00,4.69){};
\draw[Edge](r)--(4);
\end{tikzpicture}
\end{equation}
which produces the $2$-multi-pattern
\begin{equation}
\begin{MultiPattern}
0 & 1 & {\square} & 1 & 2 & {\square} & 2 & {\square} & 1 & 5 & {\square} & 0 & 1 & {\square} & 1
& {\square} & 0 & 4 & 4 \\
\bar{5} & {\square} & {\square} & \bar{1} & \bar{5} & {\square} & {\square} & \bar{1} & 0 & 0 & 0
& \bar{5} & {\square} & {\square} & \bar{1} & 0 & 0 & 0 & 0
\end{MultiPattern}.
\end{equation}
Together with the aforementioned interpretation, the generated musical phrase is
\begin{abc}[name=PhraseExample4,width=1.0\abcwidth]
X:1
T:
K:Am
M:8/8
L:1/8
Q:1/8=128
V:voice1
A,1 B,2 B,1 C2 C2 B,1 A2 A,1 B,2 B,2 A,1 F1 F1
V:voice2
A,,3 F,1 A,,3 F,1 A,1 A,1 A,1 A,,3 F,1 A,1 A,1 A,1 A,1
\end{abc}
\section{Applications: exploring variations of patterns} \label{sec:applications}
We construct here some particular bud generated systems devoted to work with the algorithms
introduced in Section~\ref{subsec:bud_generating_systems}. They generate variations of a
single $1$-multi-pattern $\mathbf{p}$ given at input, with possibly some auxiliary data. Each
performs a precise musical transformation of~$\mathbf{p}$.
\subsection{Random temporizations}
Given a pattern $\mathbf{p}$ and an integer $t \geq 1$, we define the \Def{temporizator bud
generating system} $\BudSystem^\mathrm{tem}_{\mathbf{p}, t}$ of $\mathbf{p}$ and $t$ by
\begin{equation}
\BudSystem^\mathrm{tem}_{\mathbf{p}, t}
:=
\Par{\OperadP_1, \mathfrak{C}, \Bra{\mathbf{c}_1, \mathbf{c}_2, \mathbf{c}'_1, \dots, \mathbf{c}'_t},
\mathtt{b}_1}
\end{equation}
where $\mathfrak{C}$ is the set of colors $\Bra{\mathtt{b}_1, \mathtt{b}_2, \mathtt{b}_3}$ and $\mathbf{c}_1$,
$\mathbf{c}_2$, $\mathbf{c}'_1$, \dots, $\mathbf{c}'_t$ are the $\mathfrak{C}$-colored $1$-multi-patterns
\begin{subequations}
\begin{equation}
\mathbf{c}_1 := \Par{\mathtt{b}_1, \mathbf{p}, \mathtt{b}_2^{|\mathbf{p}|}},
\quad
\mathbf{c}_2 := \Par{\mathtt{b}_2, \mathbf{p}, \mathtt{b}_2^{|\mathbf{p}|}},
\end{equation}
\begin{equation}
\mathbf{c}'_j :=
\Par{\mathtt{b}_2,
\begin{MultiPattern}
0 & {\square}^j
\end{MultiPattern},
\mathtt{b}_3},
\quad j \in [t].
\end{equation}
\end{subequations}
The temporizator bud generating system of $\mathbf{p}$ and $t$ generates a version of the pattern
$\mathbf{p}$ composed with itself where the durations of some beats have been increased by at most
$t$. The colors, and in particular the color $\mathtt{b}_3$, prevent multiple compositions of
the colored patterns~$\mathbf{c}'_j$, $j \in [t]$, in order to not overly increase the duration
of some beats.
For instance, by considering the pattern
\begin{math}
\mathbf{p} := 0 2 {\square} 1 {\square} 0 4
\end{math}
and the parameter $t := 2$, the partial random generation algorithm ran with the bud
generating system $\BudSystem^\mathrm{tem}_{\mathbf{p}, t}$ and $k := 16$ as inputs produces the
pattern
\begin{equation}
0 2 {\square} {\square} {\square} 1 {\square} 3 {\square} {\square} {\square} 2 {\square} 1 5 {\square} 0 {\square} {\square} 4
{\square}.
\end{equation}
Together with the interpretation consisting in a tempo of $128$ and the rooted scale
$\Par{\bm{\lambda}, \Note{9}{3}}$ where $\bm{\lambda}$ is the Hirajoshi scale, the generated musical
phrase is
\begin{abc}[name=PhraseExample5,width=.85\abcwidth]
X:1
T:
K:Am
M:8/8
L:1/8
Q:1/8=128
A,1 C4 B,2 E4 C2 B,1 A2 A,3 F2
\end{abc}
\subsection{Random rhythmic variations}
Given a pattern $\mathbf{p}$ and a rhythm pattern $\mathbf{r}$, we define the \Def{rhythmic bud
generating system} $\BudSystem^\mathrm{rhy}_{\mathbf{p}, \mathbf{r}}$ of $\mathbf{p}$ and $\mathbf{r}$ by
\begin{equation}
\BudSystem^\mathrm{rhy}_{\mathbf{p}, \mathbf{r}}
:=
\Par{\OperadP_m, \mathfrak{C}, \Bra{\mathbf{c}_1, \mathbf{c}_2, \mathbf{c}_3}, \mathtt{b}_1}
\end{equation}
where $\mathfrak{C}$ is the set of colors $\Bra{\mathtt{b}_1, \mathtt{b}_2, \mathtt{b}_3}$ and $\mathbf{c}_1$,
$\mathbf{c}_2$, and $\mathbf{c}_3$ are the three $\mathfrak{C}$-colored $1$-multi-patterns
\begin{subequations}
\begin{footnotesize}
\begin{equation}
\mathbf{c}_1 := \Par{\mathtt{b}_1, \mathbf{p}, \mathtt{b}_2^{|\mathbf{p}|}},
\enspace
\mathbf{c}_2 := \Par{\mathtt{b}_2, \mathbf{p}, \mathtt{b}_2^{|\mathbf{p}|}},
\enspace
\mathbf{c}_3 := \Par{\mathtt{b}_2, \mathbf{r}', \mathtt{b}_3^{|\mathbf{r}|}},
\end{equation}
\end{footnotesize}
\end{subequations}
where $\mathbf{r}'$ is the pattern $\Par{0^{|\mathbf{r}|}, \mathbf{r}}$. The rhythmic bud generating
system of $\mathbf{p}$ and $\mathbf{r}$ generates a version of the pattern $\mathbf{p}$ composed with
itself where some beats are repeated accordingly to the rhythm pattern $\mathbf{r}$. The
colors, and in particular the color $\mathtt{b}_3$, prevent multiple compositions of the colored
pattern~$\mathbf{c}_3$. Observe that when $\mathbf{r} = \epsilon$, each composition involving
$\mathbf{c}_3$ deletes a beat in the generated pattern.
For instance, by considering the pattern
\begin{math}
\mathbf{p} := 1 {\square} 0 1 1 {\square} 2
\end{math}
and the rhythm pattern
\begin{math}
\mathbf{r} := {\blacksquare} {\blacksquare} {\square} {\blacksquare} {\square},
\end{math}
the partial random generation algorithm ran with the bud generating system
$\BudSystem^\mathrm{rhy}_{\mathbf{p}, \mathbf{r}}$ and $k := 8$ as inputs produces the pattern
\begin{footnotesize}
\begin{multline}
2 2 {\square} {\square} 2 {\square} {\square} 1 2 2 {\square} 3 {\square} 1 {\square} 0 1 1 {\square} 2 2 {\square} 1 2 2
{\square} {\square} 2 {\square} 2 {\square}
\\
3 1 {\square} 2 2 {\square} {\square} 2 {\square}.
\end{multline}
\end{footnotesize}
Together with the interpretation consisting in a tempo of $128$ and the rooted scale
$\Par{\bm{\lambda}, \Note{9}{3}}$ where $\bm{\lambda}$ is the minor natural scale, the generated musical
phrase is
\begin{abc}[name=PhraseExample6,width=1.0\abcwidth]
X:1
T:
K:Am
M:8/8
L:1/8
Q:1/8=128
V:voice1
C1 C3 C3 B,1 C1 C2 D2 B,2 A,1 B,1 B,2 C1 C2 B,1 C1 C3 C2 C2 D1 B,2 C1 C3 C2
\end{abc}
\subsection{Random harmonizations}
For any pattern $\mathbf{p}$ and an integer $m \geq 1$, we denote by $[\mathbf{p}]_m$ the
$m$-multi-pattern $\Par{[\mathbf{p}]_m^{(1)}, \dots, [\mathbf{p}]_m^{(m)}}$ satisfying $[\mathbf{p}]_m^{(i)} =
\mathbf{p}$ for all $i \in [m]$.
Given a pattern $\mathbf{p}$ and a degree pattern $\mathbf{d}$ of arity $m \geq 1$, we define the
\Def{harmonizator bud generating system} $\BudSystem^\mathrm{har}_{\mathbf{p}, \mathbf{d}}$ of $\mathbf{p}$
and $\mathbf{d}$ by
\begin{equation}
\BudSystem^\mathrm{arp}_{\mathbf{p}, \mathbf{d}}
:=
\Par{\OperadP_m, \mathfrak{C}, \Bra{\mathbf{c}_1, \mathbf{c}_2, \mathbf{c}_3}, \mathtt{b}_1}
\end{equation}
where $\mathfrak{C}$ is the set of colors $\Bra{\mathtt{b}_1, \mathtt{b}_2, \mathtt{b}_3}$ and $\mathbf{c}_1$,
$\mathbf{c}_2$, and $\mathbf{c}_3$ are the three $\mathfrak{C}$-colored $m$-multi-patterns
\begin{subequations}
\begin{equation}
\mathbf{c}_1 := \Par{\mathtt{b}_1, [\mathbf{p}]_m, \mathtt{b}_2^m},
\quad
\mathbf{c}_2 := \Par{\mathtt{b}_2, [\mathbf{p}]_m, \mathtt{b}_2^m},
\end{equation}
\begin{equation}
\mathbf{c}_3 :=
\Par{\mathtt{b}_2,
\begin{MultiPattern}
\mathbf{d}_1 \\
\mathbf{d}_2 \\
\vdots \\
\mathbf{d}_m
\end{MultiPattern},
\mathtt{b}_3}.
\end{equation}
\end{subequations}
The harmonizator bud generating system of $\mathbf{p}$ and $\mathbf{d}$ generates an harmonized
version of the pattern $\mathbf{p}$ composed with itself, with chords controlled by~$\mathbf{d}$. The
colors, and in particular the color $\mathtt{b}_3$, prevent multiple compositions of the colored
pattern~$\mathbf{c}_3$.
For instance, by considering the pattern
\begin{math}
\mathbf{p} := 2 1 0 2 {\square} 1 {\square} 0 {\square}
\end{math}
and the degree pattern
\begin{math}
\mathbf{d} := 0 5 \bar{7},
\end{math}
the partial random generation algorithm ran with the bud generating system
$\BudSystem^\mathrm{har}_{\mathbf{p}, \mathbf{d}}$ and $k := 3$ as inputs produces the
$3$-multi-pattern
\begin{equation}
\begin{MultiPattern}
2 & 1 & 0 & 2 & {\square} & 1 & {\square} & 0 & {\square} \\
2 & 6 & 5 & 2 & {\square} & 1 & {\square} & 0 & {\square} \\
2 & \bar{6} & \bar{7} & 2 & {\square} & 1 & {\square} & 0 & {\square}
\end{MultiPattern}.
\end{equation}
Together with the interpretation consisting in a tempo of $128$ and the rooted scale
$\Par{\bm{\lambda}, \Note{9}{3}}$ where $\bm{\lambda}$ is the minor natural scale, the generated musical
phrase is
\begin{abc}[name=PhraseExample7,width=.45\abcwidth]
X:1
T:
K:Am
M:8/8
L:1/8
Q:1/8=128
V:voice1
C1 B,1 A,1 C2 B,2 A,2
V:voice2
C1 G1 F1 C2 B,2 A,2
V:voice3
C1 B,,1 A,,1 C2 B,2 A,2
\end{abc}
\subsection{Random arpeggiations}
Given a pattern $\mathbf{p}$ and a degree pattern $\mathbf{d}$ of arity $m \geq 1$, we define the
\Def{arpeggiator bud generating system} $\BudSystem^\mathrm{arp}_{\mathbf{p}, \mathbf{d}}$ of $\mathbf{p}$
and $\mathbf{d}$ by
\begin{equation}
\BudSystem^\mathrm{arp}_{\mathbf{p}, \mathbf{d}}
:=
\Par{\OperadP_m, \mathfrak{C}, \Bra{\mathbf{c}_1, \mathbf{c}_2, \mathbf{c}_3}, \mathtt{b}_1}
\end{equation}
where $\mathfrak{C}$ is the set of colors $\Bra{\mathtt{b}_1, \mathtt{b}_2, \mathtt{b}_3}$ and $\mathbf{c}_1$,
$\mathbf{c}_2$, and $\mathbf{c}_3$ are the three $\mathfrak{C}$-colored $m$-multipatterns
\begin{subequations}
\begin{equation}
\mathbf{c}_1 := \Par{\mathtt{b}_1, [\mathbf{p}]_m, \mathtt{b}_2^m},
\quad
\mathbf{c}_2 := \Par{\mathtt{b}_2, [\mathbf{p}]_m, \mathtt{b}_2^m},
\end{equation}
\begin{equation}
\mathbf{c}_3 :=
\Par{\mathtt{b}_2,
\begin{MultiPattern}
\mathbf{d}_1 & {\square} & {\square} & \dots & {\square} \\
{\square} & \mathbf{d}_2 & {\square} & \dots & {\square} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
{\square} & {\square} & \dots & {\square} & \mathbf{d}_m
\end{MultiPattern},
\mathtt{b}_3}.
\end{equation}
\end{subequations}
The arpeggiator bud generating system of $\mathbf{p}$ and $\mathbf{d}$ generates an arpeggiated
version of the pattern $\mathbf{p}$ composed with itself, where the arpeggio is controlled
by~$\mathbf{d}$. The colors, and in particular the color $\mathtt{b}_3$, prevent multiple
compositions of the colored pattern~$\mathbf{c}_3$. Observe in particular that when $\mathbf{d} =
0^m$, each composition involving $\mathbf{c}_3$ creates a repetition of a same beat over the $m$
stacked voices.
For instance, by considering the pattern
\begin{math}
\mathbf{p} := 0 {\square} 2 1 3 {\square} 1
\end{math}
and the degree pattern
\begin{math}
\mathbf{d} := 0 2 4,
\end{math}
the partial random generation algorithm ran with the bud generating system
$\BudSystem^\mathrm{arp}_{\mathbf{p}, \mathbf{d}}$ and $k := 8$ as inputs produces the
$3$-multi-pattern
\begin{scriptsize}
\begin{equation}
\begin{MultiPattern}
0 & {\square} & 2 & 1 & {\square} & {\square} & 3 & {\square} & 1 & {\square} & 2 & {\square} & 4 & 3 & 5
& {\square} & 3 & {\square} & {\square} & 1 & {\square} & {\square} & 3 & {\square} & 1 \\
0 & {\square} & 2 & {\square} & 3 & {\square} & 3 & {\square} & 1 & {\square} & 2 & {\square} & 4 & 3 & 5
& {\square} & {\square} & 5 & {\square} & {\square} & 3 & {\square} & 3 & {\square} & 1 \\
0 & {\square} & 2 & {\square} & {\square} & 5 & 3 & {\square} & 1 & {\square} & 2 & {\square} & 4 & 3 & 5
& {\square} & {\square} & {\square} & 7 & {\square} & {\square} & 5 & 3 & {\square} & 1
\end{MultiPattern}.
\end{equation}
\end{scriptsize}
Together with the interpretation consisting in a tempo of $128$ and the rooted scale
$\Par{\bm{\lambda}, \Note{0}{3}}$ where $\bm{\lambda}$ is the major natural scale, the generated musical
phrase is
\begin{abc}[name=PhraseExample8,width=1.0\abcwidth]
X:1
T:
K:Am
M:8/8
L:1/8
Q:1/8=128
V:voice1
C,2 E,1 D,3 F,2 D,2 E,2 G,1 F,1 A,2 F,3 D,3 F,2 D,1
V:voice2
C,2 E,2 F,2 F,2 D,2 E,2 G,1 F,1 A,3 A,3 F,2 F,2 D,1
V:voice3
C,2 E,3 A,1 F,2 D,2 E,2 G,1 F,1 A,4 C3 A,1 F,2 D,1
\end{abc}
\bibliographystyle{plain}
| 35,837 |
\section{Introduction}
\noindent The idea that the Higgs may arise as a composite pseudo Nambu-Goldstone boson (pNGB) from a new strongly-interacting sector provides one of the best-motivated solutions to the naturalness problem of the weak scale~\cite{Panico:2015jxa,Bellazzini:2014yua}. The minimal realistic model \cite{Agashe:2004rs} is based on a strong sector with global symmetry $\mathcal{G} = SO(5)$, whose spontaneous breaking to $\mathcal{H} = SO(4)$ at scale $f \sim \mathrm{TeV}$ yields four real pNGBs, identified with the components of the Standard Model (SM)-like Higgs doublet $H$.
Non-minimal models offer, in addition, an appealing link to the dark matter (DM) puzzle~\cite{Frigerio:2012uc}. If one of the extra pNGBs contained in $\mathcal{G}/\mathcal{H}$, assumed to be a SM singlet and labeled $\chi$, is stable (owing, for example, to a discrete $Z_2$ symmetry~\cite{Frigerio:2012uc}, or because it is the lightest particle charged under a continuous $U(1)$ symmetry~\cite{Balkin:2017aep}), it constitutes a compelling weakly interacting massive particle (WIMP) DM candidate. Not only is $\chi$ naturally light and weakly coupled in the same way the Higgs is, but also its leading coupling to the SM is the {\it derivative Higgs portal},
\begin{equation}
\frac{1}{f^2}\partial_\mu |H|^2 \partial^\mu |\chi|^2 \,,
\end{equation}
which is extremely suppressed at the small momentum transfers that characterize DM scattering with heavy nuclei, $|t| / f^2 \lesssim (100\;\mathrm{MeV})^2/ (1 \;\mathrm{TeV})^2 \sim 10^{-8}$. This type of WIMP is therefore naturally compatible with the existing strong exclusions from direct detection experiments. At the same time, the interaction strength of DM annihilation is $s / f^2 \simeq 4 m_\chi^2 / f^2$, which if the DM acquires a radiative mass $m_\chi \sim 100\;\mathrm{GeV}$ is in the right range to obtain the observed relic density through thermal freeze-out.
This simple and attractive picture can, however, be significantly altered by explicit symmetry breaking effects. Some sources of explicit breaking of the Goldstone shift symmetries are in fact necessary, in order to provide a potential and Yukawa couplings for $H$ and at least the mass for $\chi$. Generically, these sources also introduce non-derivative couplings between the DM and the SM, in particular the {\it marginal Higgs portal},
\begin{equation}
\lambda |H|^2 |\chi|^2 \,,
\end{equation}
which is strongly constrained by direct detection. The main purpose of this paper is to construct and analyze realistic models where $\lambda$ is very suppressed, either because it is proportional to the Yukawas of the light SM fermions or because it arises at higher-loop order, while at the same time $\chi$ obtains a mass of $O(100)$ GeV at one loop. These models then retain the most appealing features of pNGB DM discussed above, and should in our view be considered as very motivated targets for experiments that will search for WIMPs in the near future.
The irreducible sources of explicit $\mathcal{G}$ breaking are the gauging of $SU(2)_L\times U(1)_Y \subset \mathcal{H}$, and the couplings of the SM fermions. If $\chi$ is a SM-singlet, as assumed in this paper, the SM gauging does not give it a potential at one loop. The fermions are linearly coupled to operators of the strong sector, thus realizing the partial compositeness mechanism \cite{Kaplan:1991dc}. Hence we must specify the incomplete $\mathcal{G}$ representations (spurions) that $q_L = (t_L, b_L)^T, t_R$ and $b_R$ are embedded into, where we focus on the third-generation quarks since the first two generations of quarks and the leptons have much smaller Yukawa couplings. The choice of the spurions fixes the explicit breaking of the DM shift symmetry and therefore the strength of the non-derivative couplings of $\chi$. Three qualitatively different scenarios can be identified:
\vspace{1mm}
\noindent {\bf Leading breaking by top quark couplings.} This case was first discussed in Ref.~\cite{Frigerio:2012uc} and later analyzed extensively in the $SO(6)/SO(5)$ model, where the DM is a real scalar stabilized by a $Z_2$ symmetry~\cite{Marzocca:2014msa}, and by the authors of this paper in the $SO(7)/SO(6)$ model, where the DM is a complex pNGB stabilized by a $U(1)$~\cite{Balkin:2017aep}. Top loops make the DM heavier than the Higgs, and the global symmetry causes the marginal portal coupling to be generated with size comparable to that of the Higgs quartic $\lambda_h\,$,
\begin{equation}
\lambda \lesssim \frac{\lambda_h}{2} \qquad\; \text{and} \qquad m_\chi \gg m_h\,.
\end{equation}
As a consequence, once the observed $\lambda_h \simeq 0.13$ is reproduced, $\lambda$ is automatically of few percent, corresponding to DM-nucleon cross sections $\sigma_{\rm SI}^{\chi N} \sim 10^{-46}\,\mathrm{cm}^2$ that are currently being probed by XENON1T. This setup does not fully realize the pNGB DM picture in the sense described above and has been well covered in previous work, so it will not be studied further in this paper. Nevertheless, for completeness we provide a very short summary of its features in Sec.~\ref{eq:tRbreaking}.
\vspace{1mm}
\noindent {\bf Leading breaking by bottom quark couplings.} Although this case was also first discussed in Ref.~\cite{Frigerio:2012uc}, here we focus on a different parametric regime. The marginal portal coupling and DM mass are both generated at one loop, but scale very differently with the bottom Yukawa,
\begin{equation} \label{eq:bottomScalings}
\lambda \propto y_b^2 \qquad\; \text{and} \qquad m_\chi \propto (y_b g_{\ast})^{1/2} f\,,
\end{equation}
where $g_\ast$ is the strong sector coupling. Hence $\lambda$ is so small ($\lambda \ll 10^{-3}$) that it is irrelevant for direct detection, but $\chi$ can be sufficiently heavy ($m_\chi \sim 100\;\mathrm{GeV}$) that its annihilation via the derivative Higgs portal yields the correct DM relic density. The explicit breaking of the $\chi$ shift symmetry, however, also generates the operator $y_b \bar{q}_L H b_R\, |\chi|^2/f^2$, which yields small DM-nucleon cross sections $\sigma_{\rm SI}^{\chi N} \sim 10^{-47}\,\mathrm{cm}^2$. Part of the parameter space will therefore be tested in next-generation direct detection experiments such as LZ. This setup is discussed in Sec.~\ref{eq:bRbreaking}, with further important details provided in Appendix~\ref{sec:appA}.
\vspace{1mm}
\noindent {\bf DM shift symmetry unbroken by SM fermion couplings.} It is possible to embed all SM quarks into spurions that preserve the DM shift symmetry \cite{Frigerio:2012uc,Balkin:2017aep}. In this case some beyond-the-SM source of explicit breaking is required, in order to generate a mass for the DM. In Sec.~\ref{sec:gaugeBreaking}, which contains the main results of this paper, we focus on the case where $\chi$ is a complex scalar stabilized by a $U(1)_{\rm DM}\subset \mathcal{H}$ symmetry, and show that gauging $U(1)_{\rm DM}$ with coupling $g_D$ can naturally produce a one-loop mass
\begin{equation}
m_\chi \propto g_D f
\end{equation}
of $O(100)$ GeV for $\chi$, while $\lambda$ is strongly suppressed as it is only generated at higher loop order. This setup realizes the crucial feature of the previously-advertised pNGB DM picture, namely the DM scattering on nuclei is too suppressed to be within the foreseeable reach of direct detection experiments. Incidentally, let us mention that gauging $U(1)_{\rm DM}$ may also increase the theoretical robustness of the DM stability, an aspect that will be briefly addressed in our final remarks of Sec.~\ref{sec:outlook}.
The presence of a dark photon $\gamma_D$ yields a rich phenomenology, which we study in detail. Importantly, we take zero kinetic mixing between the $U(1)_{\rm DM}$ and the SM hypercharge. This is motivated by our explicit analysis of the $SO(7)/SO(6)$ model, where the kinetic mixing is forbidden because $C_D$, the charge conjugation associated to $U(1)_{\rm DM}$, is an accidental symmetry of the theory, at least in the limit where subleading spurions for the SM fermions are neglected. This point is thoroughly discussed in Appendix~\ref{app:KinMixing}. First we examine, in Sec.~\ref{subsec:masslessDP}, the case where the dark photon is massless and therefore constitutes dark radiation. The dark sector phenomenology shares several aspects with those considered in Refs.~\cite{Feng:2008mu,Ackerman:mha,Feng:2009mn,Agrawal:2016quu},\footnote{See also the scenario of Refs.~\cite{Fan:2013yva,Fan:2013tia}, where only a subdominant component of the DM is assumed to be charged under a hidden $U(1)$, and in addition features dissipative dynamics.} predicting an array of signals in cosmology, astroparticle, and collider experiments. These signatures place constraints on the parameter space and will allow this scenario to be further probed in the near future. Subsequently, we consider in Sec.~\ref{subsec:massiveDP} the possibility that $\gamma_D$ acquires a mass through the St\"uckelberg mechanism. Here again $C_D$ invariance plays an important role, making $\gamma_D$ stable for $m_{\gamma_D} < 2m_\chi$, when the decay to $\chi \chi^\ast$ is kinematically forbidden. We identify a region of parameters where both $\chi$ and $\gamma_D$ behave as cold DM, and discuss the novel features of this two-component-DM regime.
\vspace{1mm}
Our discussion is phrased within a low-energy effective field theory (EFT), but as already stated we support our parameter choices with concrete examples that arise in the composite Higgs model based on the $SO(7)/SO(6)$ symmetry breaking pattern \cite{Balkin:2017aep}. This construction can yield each of the three above scenarios depending on the region of parameters one focuses on, and is therefore well suited as theory backdrop. Other previous work on composite Higgs models with pNGB DM includes Refs.~\cite{Chala:2012af,Barnard:2014tla,Kim:2016jbz,Chala:2016ykx,Ma:2017vzm,Ballesteros:2017xeg,Balkin:2017yns,Alanne:2018xli}, whereas Refs.~\cite{Fonseca:2015gva,Brivio:2015kia,Bruggisser:2016ixa,Bruggisser:2016nzw} performed studies employing EFTs.
The remainder of this paper is organized as follows. In Sec.~\ref{sec:effLagr} we introduce the EFT we use to describe the pNGB Higgs and DM, as well as its essential phenomenological implications. Section~\ref{sec:fermionbreak} presents concisely the scenarios where the DM shift symmetry is explicitly broken by the couplings of the SM fermions. In Sec.~\ref{sec:gaugeBreaking} we analyze in depth the case where the DM shift symmetry is preserved by the couplings of the SM fermions, but broken by the gauging of the $U(1)$ symmetry that stabilizes the DM. Finally, Sec.~\ref{sec:outlook} provides some closing remarks. Appendices~\ref{sec:appA}, \ref{sec:appB} and \ref{app:KinMixing} contain details on important aspects of the $SO(7)/SO(6)$ composite Higgs model, while Appendix~\ref{app:PhenoResults} collects formulas relevant to our phenomenological analysis.
\section{Effective Lagrangian for the Higgs and DM pNGBs} \label{sec:effLagr}
\noindent The low-energy effective Lagrangian for the pNGBs, namely the Higgs doublet $H$ and the SM-singlet DM, taken to be a complex scalar $\chi$ stabilized by a $U(1)_{\rm DM}$ symmetry,\footnote{For real DM $\eta$ that is stable due to a $Z_2$ symmetry, we simply replace $\chi \to \eta /\sqrt{2}$ in $\mathcal{L}_{\rm eff}$.} has the form
\begin{equation}
\mathcal{L}_{\rm eff} = \mathcal{L}_{\rm GB} + \mathcal{L}_f - V_{\rm eff}\,,
\end{equation}
where $\mathcal{L}_{\rm GB}$ contains only derivative interactions, whose structure is determined by the non-linearly realized global symmetry. $\mathcal{L}_f$ contains the couplings to the SM fermions, which originate from elementary-composite mixing couplings that break $\mathcal{G}$ explicitly. These elementary-composite mixings, together with the gauging of a subgroup of $\mathcal{G}$ that includes the SM electroweak symmetry, generate the radiative potential $V_{\rm eff}$. We discuss first the leading order Lagrangian $\mathcal{L}_{\rm GB}$, and then turn to the effects of the explicit symmetry breaking, contained in $\mathcal{L}_{f} - V_{\rm eff}$.
\subsection{Two-derivative Lagrangian}\label{subsec:LGB}
The most general two-derivative, $SU(2)_L\times SU(2)_R \times U(1)_{\rm DM}\subset \mathcal{H}$ invariant Lagrangian\footnote{More precisely, this is the most general $SU(2)_L\times SU(2)_R \times U(1)_{\rm DM}$ invariant Lagrangian where $SU(2)_R$ is only broken by the gauging of hypercharge.\vspace{1.5mm}} that arises from the nonlinear sigma model kinetic term is\footnote{We do not include in $\mathcal{L}_{\rm GB}$ operators containing $\chi^*\overset\leftrightarrow{\partial_\mu}\chi \equiv \chi^\ast \partial_\mu \chi - \partial_\mu \chi^\ast \chi$, which vanish trivially in the $SO(6)/SO(5)$ model where $\chi \to \eta/\sqrt{2}$ with real $\eta$, and are forbidden in the $SO(7)/SO(6)$ model by custodial $SO(4) \simeq SU(2)_L \times SU(2)_R$ invariance, since $H$ and $\chi$ belong to the same irreducible representation of $\mathcal{H} = SO(6)$. Notice also that $\chi^*\overset\leftrightarrow{\partial_\mu}\chi$ is odd under the charge conjugation associated to $U(1)_{\rm DM}$.}
\begin{equation}\label{eq:LGB}
\mathcal{L}_{\rm GB} \,=\, |D^\mu H|^2 + |\partial^\mu \chi|^2 + \frac{c_H}{2f^2} \partial_\mu |H|^2 \partial^\mu |H|^2 + \frac{c_d}{f^2} \partial_\mu |H|^2 \partial^\mu |\chi |^2 + \frac{c_\chi}{2f^2} \partial_\mu |\chi |^2 \partial^\mu |\chi |^2 .
\end{equation}
We could have written four additional operators,
\begin{equation}
\frac{c_1}{f^2} |D_\mu H|^2 |H|^2 \,,\qquad \frac{c_2}{f^2} |D_\mu H |^2 |\chi |^2 \,, \qquad \frac{c_3}{f^2} |\partial_\mu \chi |^2 |H|^2 \,,\qquad \frac{c_4}{f^2} |\partial_\mu \chi |^2 |\chi |^2 \,,
\end{equation}
but these can be removed through the $O(1/f^2)$ field redefinition
\begin{equation}
\label{eq:FieldRedef}
H\rightarrow \Big(1 - \frac{c_1}{2f^2} |H|^2 - \frac{c_2}{2 f^2} |\chi |^2 \Big) H \,,\qquad \chi\rightarrow \Big( 1 - \frac{c_3}{2 f^2} |H|^2 -\frac{c_4}{2f^2} |\chi|^2 \Big) \chi \, .
\end{equation}
Notice that for $c_1=c_2=c_3=c_4=-2/3$ these are the leading terms of
\begin{equation}
\frac{\sin (\pi /f )}{\pi}\, \pi^a \rightarrow \frac{\pi^a}{f}\,,\qquad \pi = \sqrt{\vec{\pi}^{\,T}\vec{\pi}}\,,
\end{equation}
where $\vec{\pi}$ is the GB vector~\cite{Gripaios:2009pe}. This redefinition has customarily been adopted in studies of the $SO(6)/SO(5)$ and $SO(7)/SO(6)$ models because in the basis of Eq.~(\ref{eq:LGB}), which also coincides with the SILH basis \cite{Giudice:2007fh} when restricted to Higgs interactions, the scalar potential is a simple polynomial and the vacuum expectation value (VEV) of the Higgs is equal to $v \simeq 246\;\mathrm{GeV}$. In those models the coefficients take the values $c_H = c_d = c_\chi = 1$, which we often adopt as reference in the following.
The ``derivative Higgs portal'' operator parametrized by $c_d$, which constitutes the only interaction between the DM and the SM contained in $\mathcal{L}_{\rm GB}$, allows the DM to annihilate to SM particles via $s$-channel Higgs exchange, and the observed DM relic density to be produced via the freeze-out mechanism. This fixes the interaction strength $c_d /f^2$ as a function of the DM mass, as shown by the blue curve in Fig.~\ref{fig:fvsmchi}, which was obtained by solving the Boltzmann equation for the $\chi$ number density using micrOMEGAs~\cite{Belanger:2018mqt}. For $m_\chi > m_h$ the relation is very simple, being approximately determined by
\begin{equation} \label{eq:RAapproxAnalytical}
1 = \frac{\Omega_{\chi + \chi^\ast}}{\Omega_{\rm DM}} \simeq \frac{\langle \sigma v_{\rm rel} \rangle_{\rm can}}{\tfrac{1}{2} \langle \sigma v_{\rm rel} \rangle}\,, \qquad \langle \sigma v_{\rm rel} \rangle \simeq \frac{c_d^2 m_\chi^2}{\pi f^4}\,
\end{equation}
hence
\begin{equation} \label{eq:RAapprox}
\frac{f}{c_d^{1/2}} \approx 1.1\;\mathrm{TeV} \left( \frac{m_\chi}{130\;\mathrm{GeV}} \right)^{1/2},
\end{equation}
where $\langle\, \cdot \, \rangle$ denotes thermal average, $\Omega_{\rm DM} = 0.1198\, h^{-2}$ \cite{Aghanim:2018eyx}, $\langle \sigma v_{\rm rel} \rangle_{\rm can} \approx 2 \times 10^{-26} \,\mathrm{cm}^3\, \mathrm{s}^{-1}$ is the canonical value of the thermal cross section~\cite{Steigman:2012nb}, and the dominant $\chi \chi^\ast \to WW, ZZ, hh$ channels were included in the annihilation.\footnote{The cross section for annihilation to $t\bar{t}$ scales as $\sigma_{t\bar{t}} \,v_{\rm rel} \sim N_c m_t^2 / (\pi f^4)$, as opposed to $\sigma_{WW,ZZ,hh} \,v_{\rm rel} \sim m_\chi^2 / (\pi f^4)$, therefore $t\bar{t}$ is important only for $m_\chi$ not much larger than $m_t$. See the right panel of Fig.~\ref{fig:fvsmchi}.}
Crucially, the derivative Higgs portal also leads to negligibly small cross sections for the scattering of DM with heavy nuclei: the amplitude for $q\chi \to q\chi$ scattering mediated by Higgs exchange is proportional to $|t|/f^2 \lesssim (100\;\mathrm{MeV})^2/ (1 \;\mathrm{TeV})^2 \sim 10^{-8}$, where we took $100\;\mathrm{MeV}$ as a rough estimate of the maximum momentum transfer. The expected strength of the direct detection signal is then set by the interactions contained in $\mathcal{L}_f - V_{\rm eff}$, which depend on the explicit breaking of the global symmetry.
\begin{figure}[t]
\centering
\includegraphics[width=.495\textwidth]{f_vs_mchi_RA.pdf}
\includegraphics[width=.495\textwidth]{AnnBR.pdf}
\caption{{\it Left panel:} value of the global symmetry breaking scale $f$ that allows to reproduce the observed DM relic density via the derivative Higgs portal, as function of the DM mass. In solid blue the full Boltzmann solution, in dashed orange the approximate relation given in Eq.~\eqref{eq:RAapprox}. The gray lines show the $95\%$ CL lower bounds achievable from the measurement of the $hVV$ couplings at current and future colliders, assuming $c_H = c_d$. {\it Right panel:} fractions for annihilation to the different SM final states. $\bar{f}f$ denotes the sum over all light quarks and leptons.}
\label{fig:fvsmchi}
\end{figure}
The other important effect encapsulated in $\mathcal{L}_{\rm GB}$ is that $h$, due to its pNGB nature, has all its couplings rescaled by a universal factor with respect to their SM values: writing in unitary gauge $H = (0\,, \, \tilde{h}/\sqrt{2})^T$ we have
\begin{equation}
\tilde{h} = v + \Big( 1 - \frac{c_H}{2} \frac{v^2}{f^2} \Big)\, h\, .
\end{equation}
A robust and model-independent probe of this effect is the measurement of the $hVV$ couplings ($V=W,Z$). In Fig.~\ref{fig:fvsmchi} we compare the projected sensitivity on this observable of current and future colliders \cite{Thamm:2015zwa} with the pNGB DM parameter space, under the assumption that $c_H = c_d$.
\subsection{Explicit symmetry breaking effects}\label{subsec:Lf-Veff}
The most general effective Lagrangian coupling the pNGBs to the third generation quarks is
\begin{equation}\label{eq:Lfermions}
\mathcal{L}_f = - y_t \bar{q}_L \widetilde{H} t_R \left( 1 - \frac{c_t}{f^2} |H|^2 - \frac{c_t^\chi}{f^2} |\chi|^2 \right) -y_b \bar{q}_L H b_R \left( 1 - \frac{c_b}{f^2} |H|^2 - \frac{c_b^\chi}{f^2} |\chi |^2\right) + \text{h.c.}.
\end{equation}
The general form of the one-loop scalar potential generated by the explicit symmetry breaking is, up to quartic order in the fields,
\begin{equation}\label{eq:Veff}
V_{\rm eff} = \mu_h^2 |H|^2 + \lambda_h |H|^4 + \mu_{\rm DM}^2 |\chi |^2 +\lambda_{\rm DM} |\chi |^4 + 2 \lambda |H|^2 |\chi |^2\,.
\end{equation}
The parameters $\mu_h^2$ and $\lambda_h^2$ are fixed by requiring the observed mass and VEV for the SM-like Higgs. We only consider regions of parameters where $\langle \chi \rangle = 0$, so that $U(1)_{\rm DM}$ is not spontaneously broken and $\chi$ is stable. This imposes a mild constraint on the parameter space of the fermionic sector~(see Appendix~\ref{sec:appA} for a concrete example), whereas the gauging of $U(1)_{\rm DM}$ automatically yields $\mu^2_{\rm DM} > 0\,$.
In addition to providing the DM with a mass $m_\chi^2 = \mu_{\rm DM}^2 + \lambda v^2$, the explicit symmetry breaking can affect its phenomenology in important ways. The annihilation to SM particles is still dominated by $s$-channel Higgs exchange, but now the $\chi^\ast \chi h$ coupling has both a derivative and a non-derivative component,
\begin{equation} \label{eq:annihilationGEN}
\mathcal{M} (\chi \chi^\ast \to \mathrm{SM}) \propto \Big(c_d \frac{s}{f^2} - 2 \lambda\Big)v \simeq \Big(c_d \frac{4m_\chi^2}{f^2} - 2 \lambda \Big)v\,.
\end{equation}
A priori, for $m_\chi > m_t$ the $\chi^\ast \chi \bar{t}t$ interaction proportional to $c_t^\chi$ can also give an important contribution to $\chi^\ast \chi \to t\bar{t}$. As we will discuss momentarily, however, in the models we consider $c_t^\chi$ is suppressed or altogether absent, hence Eq.~\eqref{eq:annihilationGEN} is a good approximation of the strength for annihilation to SM particles.
DM scattering with nuclei proceeds via $t$-channel Higgs exchange and through the contact interactions parametrized by $c_q^\chi$. The effective interactions with the SM quarks $q$ have the form
\begin{equation} \label{eq:ddGEN}
2m_q a_q \, \bar{q}q \chi^\ast \chi\,, \qquad a_q \approx \frac{\lambda}{m_h^2} + \frac{c_q^\chi}{2f^2}\,.
\end{equation}
As already emphasized, the contribution of the derivative Higgs portal is negligible.
Note that, for any relevant values of the parameters, the DM self-interactions mediated by $c_\chi$ and $\lambda_{\rm DM}$ are far too small to have any effects on cosmological scales.
\subsection{Origins of explicit breaking and DM scenarios}\label{subsec:scenarios}
Two irreducible sources of explicit symmetry breaking, which generate at least some of the interactions contained in Eqs.~\eqref{eq:Lfermions} and \eqref{eq:Veff}, are the gauging of the SM electroweak subgroup $SU(2)_L \times U(1)_Y \subset \mathcal{H}$ and the Yukawa couplings for the SM fermions. The SM gauging only contributes to the scalar potential and, under our assumption that the DM is a SM singlet, at one-loop level generates only $\mu_h^2$ and $\lambda_h$. In the fermion sector, Yukawas are assumed to arise via the partial compositeness mechanism~\cite{Kaplan:1991dc}: the elementary fermions couple linearly to operators of the strong sector,
\begin{equation}\label{eq:mixLagr}
\mathcal{L}^{\rm UV}_{\rm mix} \sim \lambda_q f\, \bar{q}_L \mathcal{O}_q + \lambda_t f\, \bar{t}_R \mathcal{O}_t + \lambda_{q^\prime} f\, \bar{q}_L \mathcal{O}_{q^{\,\prime}} + \lambda_b f\, \bar{b}_R \mathcal{O}_b + \text{h.c.},
\end{equation}
where we have ignored the flavor structure and put our focus on the masses of the third generation of quarks~\cite{Csaki:2008zd}. We have included mixings of the left-handed quark doublet with two distinct operators, as it is in general required to generate both the top and bottom Yukawa couplings. For example, in the $SO(6)/SO(5)$ and $SO(7)/SO(6)$ models the global symmetry is extended by an unbroken $U(1)_X$, hence if $t_R$ and $b_R$ are coupled to operators with different $X$ charge, two distinct embeddings of $q_L$ are needed in order to generate both $y_t$ and $y_b$. At low energies Eq.~\eqref{eq:mixLagr} leads to mass mixing between the elementary fermions and the composite resonances, and as a result the physical SM fields are linear combinations of elementary and composite degrees of freedom. Their compositeness fractions are defined schematically as $\epsilon_{L, R}^{t}\sim \lambda_{q,\,t} f / \sqrt{ m_{\ast q,\, t}^2 + \lambda^2_{q,\,t} f^2 }$ and $\epsilon_{L, R}^{b}\sim \lambda_{q^{\prime},\,b} f / \sqrt{m_{\ast q^{\prime},\,b}^2 + \lambda^2_{q^{\prime},\,b} f^2}$, where $m_{ \ast q,\, t, \, q^\prime, \,b}$ are the relevant masses of the resonances in the top and bottom sectors. The Yukawas have the form
\begin{equation}
\label{eq:yukawa}
y_\psi \simeq \frac{M_{\ast \psi}}{f} \,\epsilon_L^\psi \, \epsilon_R^\psi \,,\qquad (\psi = t, b)
\end{equation}
where $M_{\ast \psi}$ is a combination of the resonance mass parameters.
Since the elementary fermions do not fill complete $\mathcal{G}$ representations, Eq.~\eqref{eq:mixLagr} breaks explicitly at least part of the global symmetry. The Higgs shift symmetry must be broken by the couplings of both the top and bottom, in order to generate the observed values of $y_{t,b\,}$, $v$ and $m_h$. However, whether each of these couplings breaks or preserves the $\chi$ shift symmetry is a priori unknown, and all possibilities deserve close scrutiny. The three scenarios discussed in this paper are listed in Fig.~\ref{fig:GrandSummary}, along with the Feynman diagrams that dominate the annihilation and direct detection of DM in each case. In Sec.~\ref{sec:fermionbreak} we consider the scenarios where the leading breaking of the DM shift symmetry originates from the SM quarks, focusing in particular on the bottom. Then, in Sec.~\ref{sec:gaugeBreaking} we study the scenario where the fermion sector is fully symmetric, and the leading explicit breaking arises from the gauging of the $U(1)_{\rm DM}$ symmetry that stabilizes the DM.
\begin{figure}[t]
\centering
\includegraphics[scale=.6]{summary_table_EXT_2.png}
\caption{Schematic summary of the three scenarios discussed in this paper. The EFT coefficients $c_d, c_b^\chi$ and $\lambda$ were defined in Eqs.~\eqref{eq:LGB}, \eqref{eq:Lfermions} and \eqref{eq:Veff}, respectively. In the third scenario we denote with $\gamma_D$ the dark photon associated to the gauging of $U(1)_{\rm DM}$ with coupling $g_D$, and mark the gauge interactions in green.}
\label{fig:GrandSummary}
\end{figure}
\section{Dark Matter shift symmetry broken by fermions}\label{sec:fermionbreak}
\noindent In this section we briefly discuss the possibility that the leading breaking of the DM shift symmetry originates from the couplings of the top quark or the bottom quark.
\subsection{Breaking of the DM shift symmetry by top quark couplings} \label{eq:tRbreaking}
This scenario has been discussed extensively in Refs.~\cite{Frigerio:2012uc,Marzocca:2014msa,Balkin:2017aep}, and is realized e.g. for $\mathcal{O}_{q, \,t} \sim \mathbf{7}_{2/3}$ under $SO(7)\times U(1)_X$. In this case $t_R$ interactions break the shift symmetries of $\chi$ and make the DM rather heavy, typically $m_\chi \sim 200$-$400$ GeV for $f \gtrsim \mathrm{TeV}$. At the same time the marginal Higgs portal coupling is generated with size closely related to that of the Higgs quartic, $\lambda \lesssim \lambda_h/2 \simeq 0.065$. These rough estimates imply that $ \lambda f^2 \lesssim m_\chi^2$, hence from Eq.~\eqref{eq:annihilationGEN} we read that $\lambda$ plays a subleading but non-negligible role in DM annihilation. In addition, $\lambda$ determines the DM-nucleon scattering cross section as~\cite{Balkin:2017aep}
\begin{equation} \label{eq:DDxsectionTop}
\sigma_{\rm SI}^{\chi N} \simeq \frac{f_N^2}{\pi} \frac{\lambda^2 m_N^4 }{ m_h^4 m_\chi^2} \;\approx\; 1.6 \times 10^{-46} \,\mathrm{cm}^2\, \left(\frac{\lambda}{0.02}\right)^2 \left(\frac{300\;\mathrm{GeV}}{m_\chi}\right)^2 , \qquad (t_R \;\mathrm{breaking})
\end{equation}
where $f_N \simeq 0.30$ contains the dependence on the nucleon matrix elements (since Higgs exchange dominates, all SM quarks contribute to the signal strength). The XENON1T experiment is currently probing cross sections of the size of Eq.~\eqref{eq:DDxsectionTop}, and part of the parameter space has recently been excluded by its latest results \cite{Aprile:2018dbl}. Notice that we have consistently neglected the effects of $c_t^\chi$: this is because the viable parameter space features large mixing of $t_R$ with the fermionic resonances, which strongly suppresses this coefficient~\cite{Balkin:2017aep}.
\subsection{Breaking of the DM shift symmetry by bottom quark couplings} \label{eq:bRbreaking}
A different scenario is obtained if the DM shift symmetry is fully preserved by the interactions of the top quark, but it is broken by those of the bottom. As a concrete example we take $\mathcal{O}_q \sim \mathbf{7}_{2/3}, \mathcal{O}_t \sim \mathbf{21}_{2/3}$ and $\mathcal{O}_{q^\prime, \, b} \sim \mathbf{7}_{-1/3}$ under $SO(7)\times U(1)_X$, in which case only the couplings of $b_R$ to the strong sector break the $\chi$ shift symmetries. Only the essential features of the setup are presented here, while a detailed discussion is provided in Appendix~\ref{sec:appA}. We focus on the region of parameter space where $\epsilon_L^b \sim \epsilon_R^b \sim \sqrt{y_b f / M_{\ast b}}\,$, which in turn lead to the scalings in Eq.~\eqref{eq:bottomScalings} with $g_\ast \sim M_{\ast b}/f$. As a result, the $\chi$ mass can be of $O(100)$~GeV while the portal coupling remains very suppressed. Quantitatively, we estimate
\begin{subequations}\label{eq:parametersbBreaking}
\begin{alignat}{2}
m_\chi \simeq \sqrt{\mu^2_{\rm DM}} &\,\approx\, 120\;\mathrm{GeV} \left( \frac{M_{\ast b}}{8\;\mathrm{TeV}} \right)^{3/2} \left( \frac{1\;\mathrm{TeV}}{f} \right)^{1/2},\\ \lambda &\,\approx\, 3\,\times 10^{-4} \left( \frac{M_{\ast b}}{8\;\mathrm{TeV}} \right)^2 \left( \frac{1\;\mathrm{TeV}}{f} \right)^{2}.
\end{alignat}
\end{subequations}
The above parametrics have been confirmed by a numerical scan of the $SO(7)/SO(6)$ model whose results are reported in Appendix~\ref{sec:appA}. The important message contained in Eq.~\eqref{eq:parametersbBreaking} is that since $\lambda f^2 \ll m_\chi^2$, $\chi$ annihilation proceeds dominantly via the derivative portal, and the DM is heavy enough that the correct relic density can be reproduced for $f \sim \mathrm{TeV}$, see Fig.~\ref{fig:fvsmchi}. In addition, we have $c_b^\chi \simeq 1$ and $\lambda f^2 \ll m_h^2$ in Eq.~\eqref{eq:ddGEN}, so the scattering with nuclei is dominated by the $\chi^\ast \chi b\bar{b}$ contact interaction. The DM-nucleon scattering cross section is
\begin{align} \label{eq:DDxsectionBottom} \nonumber
\sigma_{\rm SI}^{\chi N} &\simeq \frac{\tilde{f}_N^2}{\pi} \frac{m_N^4}{ 4 f^4 m_\chi^2} \;\\&\approx\; 1.0\,\mbox{-}\, 5.6 \times 10^{-47} \,\mathrm{cm}^2\, \left(\frac{1\;\mathrm{TeV}}{f}\right)^4 \left(\frac{100\;\mathrm{GeV}}{m_\chi}\right)^2 , \quad (b_R \;\mathrm{breaking})
\end{align}
where the range of values accounts for the theory uncertainty on the couplings of the first and second generation quarks. The lower estimate corresponds to breaking of the DM shift symmetry only by the bottom quark ($c_b^\chi = 1$ and $c_q^\chi = 0$ for all $q\neq b$, case I), yielding a nucleon form factor $\tilde{f}_N \simeq 0.066$. The higher estimate corresponds to breaking by all down-type quarks ($c_{d,s,b}^\chi = 1$ and $c_{u,c,t}^\chi = 0$, case II),\footnote{This is the pattern obtained by extending the embeddings $\mathcal{O}_q \sim \mathbf{7}_{2/3}$, $\mathcal{O}_t \sim \mathbf{21}_{2/3}$ and $\mathcal{O}_{q^\prime, \, b} \sim \mathbf{7}_{-1/3}$ to all three generations.} yielding $\tilde{f}_N \simeq 0.15$. The extremely suppressed cross sections in Eq.~\eqref{eq:DDxsectionBottom} will be probed by next-generation experiments such as LZ \cite{Mount:2017qzi}, for which they constitute a very motivated target.
A summary of the current constraints and future reach on the $(m_\chi, f)$ parameter space is shown in Fig.~\ref{fig:BottomSummary}, where we have set $c_d = c_b^\chi = 1$, $c_t^\chi = \lambda = 0$.
\begin{figure}[t]
\centering
\includegraphics[scale=.45]{mf_RA.pdf}
\caption{Parameter space of the model where the bottom quark gives the leading breaking of the DM shift symmetry. The coefficients of the effective Lagrangian are set to $c_d = c_b^\chi = 1$, $c_t^\chi = \lambda = 0$. To draw the exclusions from direct and indirect detection we have assumed that all of the observed DM is composed of $\chi$ particles, irrespective of the thermal value of the $\chi$ density predicted at each $(m_\chi, f)$ point.}
\label{fig:BottomSummary}
\end{figure}
Points lying on the blue curve reproduce the observed DM relic density. The red-shaded region is ruled out by current XENON1T results~\cite{Aprile:2018dbl} assuming case I for the DM-nucleon cross section, whereas the dashed red line corresponds to the exclusion for case II. The solid gray (dashed gray) lines show the expected sensitivity achieved by LZ~\cite{Mount:2017qzi} for case I (case II). The region $m_\chi < m_h/2$ is also constrained by LHC searches for invisibly-decaying Higgses. The current $95\%$ CL bound $\mathrm{BR}(h\rightarrow \chi^\ast \chi) < 0.24$ \cite{CMS:2018awd} rules out the region shaded in orange, which extends up to $f \simeq 1.2\;\mathrm{TeV}$ for very light $\chi$. The projected HL-LHC limit $\mathrm{BR}(h\to \chi^\ast \chi) < 0.08$ \cite{ATLAShl-lhc}, corresponding to the dotted orange curve, will extend the reach to $f \simeq 1.6\;\mathrm{TeV}$. Finally, the region shaded in purple is excluded by searches for present-day DM annihilations from dwarf spheroidal galaxies (dSphs) performed at Fermi-LAT~\cite{Ackermann:2015zua}. This bound was derived by comparing the total cross section for DM annihilation in our model to the limit reported by Fermi for the $b\bar{b}$ final state, and should therefore be taken as approximate. Additional indirect detection constraints~\cite{Cuoco:2016eej,Cui:2016ppb} arise from the measurement of the antiproton spectrum by AMS-02~\cite{Aguilar:2016kjl}. These are, however, affected by systematic uncertainties whose sizes are under active debate. We have therefore chosen to quote only the more conservative bounds from dSphs. Figure~\ref{fig:BottomSummary} shows that most of the best-motivated parameter space, with $80\;\mathrm{GeV} \lesssim m_\chi \lesssim 200\;\mathrm{GeV}$ and $0.8 \;\mathrm{TeV} \lesssim f \lesssim 1.4\;\mathrm{TeV}$, is currently untested but within reach of LZ.
\section{Dark Matter shift symmetry broken by $U(1)_{\rm DM}$ gauging} \label{sec:gaugeBreaking}
\noindent It is possible to couple all the elementary quarks to the strong sector in a way that preserves the DM shift symmetry \cite{Frigerio:2012uc,Balkin:2017aep}. For example, in the $SO(7)/SO(6)$ model this is achieved with $\mathcal{O}_q \sim \mathbf{7}_{2/3}$, $\mathcal{O}_{u,d} \sim \mathbf{21}_{2/3}$ for all three generations. This setup gives $c_q^\chi = 0$ in Eq.~\eqref{eq:Lfermions} and no contribution to $\mu^2_{\rm DM}, \lambda_{\rm DM}$ and $\lambda$ in Eq.~\eqref{eq:Veff} from the fermion sector, while at the same time top loops easily produce a realistic Higgs potential. In this case, some additional explicit breaking should be responsible for generating the DM mass. If $\chi$ is a complex scalar, a natural possibility is that the explicit breaking originates from the gauging of $U(1)_{\rm DM}$. In the $SO(7)/SO(6)$ coset the generators associated with the real and imaginary parts of $\chi$ together with the $U(1)_{\rm DM}$ generator form an $SU(2)^\prime \sim \{X^{\rm Re}, X^{\rm Im}, T^{\rm DM}\}$, hence gauging $U(1)_{\rm DM}$ generates a radiative mass for $\chi$ in very similar fashion to the contribution of photon loops to the charged pion mass in the SM.
From the effective theory point of view, the effects of gauging $U(1)_{\rm DM}$ with coupling $g_D$ can be taken into account by replacing in $\mathcal{L}_{\rm GB}$ in Eq.~\eqref{eq:LGB},
\begin{equation} \label{eq:gaugingEFT}
| \partial^\mu \chi |^2 \quad \to \quad |(\partial^\mu - i g_D A_{D}^{\mu}) \chi|^2 - \frac{1}{4} F_D^{\mu \nu} F_{D\mu \nu} + \frac{1}{2}\, m_{\gamma_D}^2 A_{D\mu} A_{D}^{\mu}\,,
\end{equation}
where we took $\chi$ to have unit charge. Note that to be general we have included a mass term for the dark photon $\gamma_D$, which can arise via the St\"uckelberg mechanism without spontaneous breaking of $U(1)_{\rm DM}$. The one-loop DM mass and marginal portal coupling are
\begin{equation} \label{eq:potentialGauging}
m_\chi = \sqrt{ \mu_{\rm DM}^2} \simeq \sqrt{ \frac{3 \alpha_D}{2 \pi}} \,m_\rho \approx 100\;\mathrm{GeV} \left( \frac{\alpha_D}{10^{-3}} \right)^{1/2} \left( \frac{m_\rho}{5\;\mathrm{TeV}} \right), \qquad\quad \lambda = 0\,,
\end{equation}
where $\alpha_D \equiv g_D^2 / (4\pi)$ and the loop that generates $m_\chi$ was cut off at $m_\rho$, the mass of vector resonances (in the $SO(7)/SO(6)$ model, this is the mass of the $\mathbf{15}$ multiplet of $SO(6)$). The estimate for the DM mass in Eq.~\eqref{eq:potentialGauging} is valid as long as $m_{\gamma_D} \ll m_\rho$, which we assume. Importantly, since the Higgs is uncharged under $U(1)_{\rm DM}$ the marginal portal coupling is not generated at one loop, leading from Eq.~\eqref{eq:ddGEN} to an extremely suppressed DM-nucleon cross section. We find it remarkable that such a simple model is effectively inaccessible to direct detection experiments.
The introduction of the dark photon has significant impact on the phenomenology. It is important to stress that in Eq.~\eqref{eq:gaugingEFT} we have not included the operator $\varepsilon B_{\mu \nu} F_{D}^{\mu \nu}/2$ that mixes kinetically $U(1)_{\rm DM}$ and the SM hypercharge. The choice to set $\varepsilon = 0$ in the EFT is motivated by the $SO(7)/SO(6)$ model, where the kinetic mixing is forbidden by $C_D$, the charge conjugation of $U(1)_{\rm DM}$, which is an accidental symmetry (provided it is respected by subleading spurionic embeddings of the SM fermions, see Appendix~\ref{app:KinMixing}). In particular, in the low-energy theory $C_D$ transforms $A_{D}^\mu \to - A_{D}^\mu$ and $\chi \to - \chi^\ast$, whereas all SM fields are left unchanged. An additional, important consequence of this discrete symmetry is that the dark photon is stable if $m_{\gamma_D} < 2m_\chi$, when the $\gamma_D \to \chi \chi^\ast$ decay is kinematically forbidden. The complete discussion of kinetic mixing, as well as the details on the implementation of $C_D$ as an $O(6)$ transformation that we call $P_6$, are contained in Appendix~\ref{app:KinMixing}.
The dark sector, composed of the DM and the dark photon, is thus characterized by the four parameters $\{m_\chi, f, \alpha_D, m_{\gamma_D}\}$. In the remainder of this section we analyze its phenomenology in detail, beginning in Sec.~\ref{subsec:masslessDP} with the simplest setup where the dark photon is massless, and later moving to the massive case in Sec.~\ref{subsec:massiveDP}.
\subsection{Phenomenology for massless dark photon} \label{subsec:masslessDP}
Setting $m_{\gamma_D} = 0$ leaves the three-dimensional parameter space $\{m_\chi, f, \alpha_D\}$. We begin the discussion with a summary of the thermal history of the model. At early times the dark sector, composed of $\chi$ and $\gamma_D$, and the visible sector are kept in kinetic equilibrium by elastic $\chi f \to \chi f$ scatterings mediated by Higgs exchange, where $f$ denotes the still-relativistic SM fermions. These processes are effective down to temperatures $T \ll m_\chi$, but eventually they become slower than the Hubble expansion rate and the dark and visible sectors decouple. The corresponding decoupling temperature $T_{\rm dec}$ is defined through \cite{Gondolo:2012vh} $H(T_{\rm dec}) = \gamma (T_{\rm dec})/2$, where $H(T) = \pi \sqrt{g_\ast (T)} \,T^2 / (3 \sqrt{10} M_{\rm Pl})$ is the Hubble parameter for a radiation-dominated Universe ($g_\ast (T)$ is the total number of relativistic degrees of freedom including both the visible and dark sectors, and $M_{\rm Pl}$ is the reduced Planck mass), whereas $\gamma (T)$ is the momentum relaxation rate, which scales as $\gamma \sim (T/ m_\chi) n_f \langle \sigma_{\chi f} v_{\rm rel} \rangle$. Using the exact expression of $\gamma (T)$ given in Ref.~\cite{Gondolo:2012vh} we calculate\footnote{For simplicity, in deriving $T_{\rm dec}$ the total number of relativistic degrees of freedom was set to the approximate constant value $g_\ast = g_{\ast, \rm vis} + g_{\rm dark} = 75.75 + 2 = 77.75$, which corresponds to $m_\tau < T < m_b$.} $T_{\rm dec}$ as a function of $m_\chi$ and $f$, finding that it is typically between $1$ and $3\;\mathrm{GeV}$ as shown in the left panel of Fig.~\ref{fig:DarkPhoton}.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{kin_decoupling2.pdf}\hspace{1mm}
\includegraphics[width=0.49\textwidth]{DeltaNeff3.pdf}
\caption{{\it Left panel:} temperature of kinetic decoupling between the dark and visible sectors. {\it Right panel:} contribution of the dark photon to $\Delta N_{\rm eff}$ at photon decoupling, calculated from Eq.~\eqref{eq:deltaNeff}. In the evaluation of $g_{\ast s, \rm vis}(T_{\rm dec})$ we assumed $150\;\mathrm{MeV}$ as temperature of the QCD phase transition. The region shaded in red corresponds to the current CMB constraint $\Delta N_{\rm eff} \lesssim 0.6$, while the dashed red line shows the projected Stage-IV CMB bound $\Delta N_{\rm eff} \lesssim 0.04$.}
\label{fig:DarkPhoton}
\end{figure}
The massless dark photon behaves as radiation at all temperatures. The strongest constraint on new relativistic degrees of freedom arises from Cosmic Microwave Background (CMB) measurements of the Hubble parameter, usually formulated in terms of the effective number of light neutrino species $N_{\rm eff}$. In our model the dark photon gives a contribution \cite{Ackerman:mha}
\begin{equation} \label{eq:deltaNeff}
\Delta N_{\rm eff} = N_{\rm eff} - 3.046 = \frac{8}{7}\, \frac{g_{\rm dark}(T)}{2} \left( \frac{T}{T_\nu} \right)^{4} \left( \frac{g_{\rm dark} (T_{\rm dec})}{ g_{\rm dark} (T)} \frac{g_{\ast s, \rm vis} (T)}{g_{\ast s, \rm vis} (T_{\rm dec})} \right)^{4/3} ,
\end{equation}
where $T \sim 0.3\;\mathrm{eV}$ is the photon temperature at decoupling, $N_{\rm eff} = 3.046$ is the SM prediction, $T/T_\nu = (11/4)^{1/3}$ and $g_{\ast s, \rm vis} (T) = 3.91$. To obtain Eq.~\eqref{eq:deltaNeff} we have used the fact that below $T_{\rm dec}$ the entropies of the dark and visible sectors are separately conserved. Since $\chi$ is already non-relativistic at kinetic decoupling, we have $g_{\rm dark} (T_{\rm dec}) = g_{\rm dark} (T) = 2$ and $\Delta N_{\rm eff}$ is determined by the number of SM relativistic degrees of freedom at $T_{\rm dec}$. As shown in the right panel of Fig.~\ref{fig:DarkPhoton}, as long as $T_{\rm dec} \gg 100\;\mathrm{MeV}$ the current bound $\Delta N_{\rm eff} \lesssim 0.6$ \cite{Aghanim:2018eyx} ($95\%$ CL) is easily satisfied. As we have seen, the typical decoupling temperature is $1\,$-$\,3$ GeV, corresponding to $\Delta N_{\rm eff} \approx 0.07\,$-$\,0.09$. Such values could be probed in future Stage-IV CMB measurements, which are expected to constrain $\Delta N_{\rm eff} \lesssim 0.04$ at $95\%$ CL \cite{Abazajian:2013oma}. A similar, but slightly weaker, current bound is obtained from Big-Bang nucleosynthesis \cite{Cooke:2013cba}.
In addition, the Compton scattering process $\chi \gamma_D\to \chi \gamma_D$ delays kinetic decoupling of the DM compared to the standard WIMP scenario \cite{Ackerman:mha,Feng:2009mn}, suppressing the matter power spectrum on small scales and leading to a minimum expected DM halo mass. For weak-scale DM and typical coupling $\alpha_D\sim 10^{-3}$, though, $\chi\,$-$\,\gamma_D$ kinetic decoupling takes place at temperature of $O(\mathrm{MeV})$ and the minimum halo mass is too small to be testable with current observations~\cite{Feng:2009mn}.
Having established that the massless dark photon does not conflict with cosmological observations, we turn to the DM phenomenology. The $\chi \chi^\ast$ pairs undergo $s$-wave annihilation both to SM particles via the derivative Higgs portal, and to $\gamma_D \gamma_D$ with amplitude mediated by the scalar QED interactions in Eq.~\eqref{eq:gaugingEFT}. The cross section for the latter is
\begin{equation} \label{eq:annihilationDarkPhoton}
\langle \sigma_{\gamma_D \gamma_D} v_{\rm rel} \rangle = \frac{2\pi \alpha_D^2}{m_\chi^2}\,,
\end{equation}
where we took the leading term in the velocity expansion. Notice that the ``mixed'' dark-visible annihilation $\chi \chi^\ast \to \gamma_D h$ is instead $p$-wave suppressed: the amplitude vanishes at threshold, because spin cannot be conserved for $m_{\gamma_D} = 0$.\footnote{The $p$-wave suppression applies also for $m_{\gamma_D} \neq 0$, since the longitudinal polarization does not contribute to the amplitude due to $U(1)_{\rm DM}$ invariance.\vspace{1.5mm}} Therefore this process has only a very small impact on the freeze-out. The requirement to obtain the observed relic density yields a two-dimensional manifold in the parameter space, whose features are best understood by considering slices with fixed $f$.
As discussed in Sec.~\ref{subsec:LGB}, there exists then only one value of the DM mass which gives the correct relic density by annihilation only through the derivative Higgs portal: for example, for $f = 1\,(1.4)\,\mathrm{TeV}$ this is $m_\chi^{(f)} \approx 122\,(194)\,\mathrm{GeV} $. For $m_\chi > m_\chi^{(f)}$ the derivative portal coupling strength $\sim m_\chi^2 / f^2$ is too large, yielding DM underdensity for any value of $\alpha_D$. Conversely, for $m_\chi <m_\chi^{(f)}$ the $\chi \chi^\ast \to \gamma_D \gamma_D$ annihilation compensates for the reduced derivative portal for an appropriate value of $\alpha_D$. Comparing Eqs.~\eqref{eq:RAapproxAnalytical} and \eqref{eq:annihilationDarkPhoton}, the two annihilation channels have equal strength when $\alpha_D^2 \sim m_\chi^4 / (2\pi^2 f^4)$, which since $m_\chi /f \sim 1/10$ corresponds to $\alpha_D \sim 2 \times 10^{-3}$. For very light DM, $m_\chi \ll m_h/2$, only annihilation to dark photons is relevant and the coupling is fixed to $\alpha_D \approx 7 \times 10^{-4}\, (m_\chi / 30\;\mathrm{GeV})$ by the analog of Eq.~\eqref{eq:RAapproxAnalytical}.
\begin{figure}
\begin{minipage}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{RA_fraction.pdf}
\label{fig:1}
\end{minipage}
\begin{minipage}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{indDet_SE_dSphs.pdf}
\vspace{+0.0cm}
\label{fig:2}
\end{minipage}
\vspace{-1cm}
\caption{{\it Left panel:} contours of observed DM relic density for representative values of $f$. The inset shows the fraction of annihilations to dark photons. Contours of constant vector resonance mass $m_\rho$ are also shown, as dashed grey lines. {\it Right panel:} the colored curves show $\langle \sigma_{\rm SM} v_{\rm rel} \rangle_{\rm SE}$, the present-day annihilation cross section to SM particles including Sommerfeld enhancement, calculated along the relic density contours shown in the left panel. The black line is the observed $95\%$ CL upper limit from the dSphs analysis in Ref.~\cite{Fermi-LAT:2016uux}. The yellow band corresponds to $95\%$ uncertainty on the expected limit in the same analysis. We also show, as dashed black line, the observed limit from the analysis of a smaller dSphs sample \cite{Ackermann:2015zua}. The quoted experimental limits were obtained assuming DM annihilation to $b\bar{b}$.}
\label{fig:RD+ID}
\end{figure}
These features are illustrated by the left panel of Fig.~\ref{fig:RD+ID}, where contours of the observed relic abundance in the $(m_\chi, \alpha_D)$ plane are shown. Notice that in the window $55\;\mathrm{GeV} \lesssim m_\chi \lesssim 62.5\;\mathrm{GeV}$ the DM is always underdense, because the annihilation to SM particles is too strongly enhanced by the Higgs resonance. To help identify the most plausible parameter space we also show contours of constant vector resonance mass $m_\rho$, as obtained from the one-loop expression of the $\chi$ mass in Eq.~\eqref{eq:potentialGauging}.\footnote{Precisely, we employed Eq.~\eqref{eq:gaugeDMmass} with $f_\rho = f$.} We expect $1 \lesssim m_\rho /f \lesssim 4\pi$, although stronger lower bounds can arise from electroweak precision tests and from direct searches for the $\rho$ particles at colliders.
The massless dark photon mediates a long-range force between DM particles, which leads to the non-perturbative Sommerfeld enhancement (SE)~\cite{Sommerfeld:1931} of the annihilation cross section. For $s$-wave annihilation the cross section times relative velocity including SE is
\begin{equation} \label{eq:SECoulomb}
(\sigma v_{\rm rel})_{\rm SE} = (\sigma v_{\rm rel})_0 \,S(\alpha_D / v_{\rm rel}), \qquad\quad S(\zeta) = \frac{2\pi \zeta }{1 - e^{-2 \pi \zeta}} \,,
\end{equation}
where $ (\sigma v_{\rm rel})_0$ is the perturbative result, e.g. $ (\sigma v_{\rm rel})_0 = 2\pi \alpha_D^2 / m_\chi^2$ for $\chi \chi^\ast \to \gamma_D \gamma_D$. The SE is important when the ratio $\alpha_D /v_{\rm rel}$ is not too small, and scales as $S \simeq 2\pi \alpha_D/ v_{\rm rel}$ for $\alpha_D/v_{\rm rel} \gtrsim 1/2$. Assuming a Maxwell-Boltzmann distribution for the DM velocity, the thermally averaged cross section times relative velocity including SE can be written in the approximate form~\cite{Feng:2010zp}
\begin{equation} \label{eq:SEthermalAverage}
\langle \sigma v_{\rm rel} \rangle_{\rm SE} = (\sigma v_{\rm rel})_0\, \overline{S}_{\rm ann}\,, \qquad \overline{S}_{\rm ann} = \sqrt{\frac{2}{\pi}} \frac{1}{v_0^3 N} \int_{0}^{v_{\rm max}} dv_{\rm rel} \, S(\alpha_D / v_{\rm rel}) v_{\rm rel}^2 \,e^{- \frac{v_{\rm rel}^2}{2 v_0^2}}
\end{equation}
where $v_0$ is the most probable velocity. The maximal relative velocity $v_{\rm max}$ and the normalization constant $N$ depend on whether we consider early-Universe annihilation around the time of freeze-out, in which case $v_{\rm max} = \infty$ and $N = 1$, or present-day annihilation in a galaxy halo, where $v_{\rm max} = 2 \,v_{\rm esc}$ with $v_{\rm esc}$ the escape velocity, and $N = \mathrm{erf} ( z/ \sqrt{2} ) - \sqrt{ 2/ \pi }\,z\, e^{- z^2/2}\,$, $z \equiv v_{\rm max} / v_0$. We have checked that Eq.~\eqref{eq:SEthermalAverage} agrees within a few percent with the full numerical treatment. At DM freeze-out the typical DM speed is $v_0 = \sqrt{2 / x_{\rm fo}} \sim 0.3$ since $x_{\rm fo} \equiv m_\chi / T_{\rm fo} \sim 25$, so for the typical coupling $\alpha_D \sim 10^{-3}$ the SE enhancement is negligible. Today, however, DM particles are much slower, with typical relative velocities of $10^{-3}$ in the Milky Way (MW), and $\lesssim 10^{-4}$ in dwarf galaxies. For the MW we take $v_0 = 220\,\mathrm{km/s}$ and $v_{\rm esc} = 533\,\mathrm{km/s}$ \cite{Piffl:2013mla}, obtaining a typical SE of $\overline{S}_{\rm ann} \approx 6.9$ for $\alpha_D = 10^{-3}$. For a dwarf galaxy with representative parameters $v_0 = 10\,\mathrm{km/s}$ and $v_{\rm esc} = 15\,\mathrm{km/s}$ \cite{Cirelli:2016rnw} we find $\overline{S}_{\rm ann} \approx 150$, again for $\alpha_D = 10^{-3}$. If the DM has a sizeable annihilation to SM particles, these large enhancements lead to conflict with bounds from indirect detection of DM.
The strongest constraint comes from the non-observation by the Fermi-LAT \cite{Ackermann:2015zua,Fermi-LAT:2016uux} of excess gamma ray emission from dSphs, which are the most DM-dominated galaxies known. For $m_\chi \sim 100\;\mathrm{GeV}$ the current exclusion on $\langle \sigma v_{\rm rel} \rangle$ is about the thermal relic value. In the right panel of Fig.~\ref{fig:RD+ID} we show the total cross section for $\chi$ annihilation to SM particles, including the SE, calculated along contours in the $\{m_\chi, f, \alpha_D\}$ parameter space where the observed relic density is reproduced. Due to the large SE the region $m_\chi > m_h/2$, where an $O(1)$ fraction of DM annihilations produce SM particles, is ruled out by dSphs analyses. Notice that the experimental limits shown in Fig.~\ref{fig:RD+ID} were obtained assuming DM annihilates to $b\bar{b}$ only, whereas our $\chi$ annihilates to a combination of SM final states (see the right panel of Fig.~\ref{fig:fvsmchi}), but the uncertainty due to this approximation is mild and cannot change the conclusion that the region $m_\chi > m_h/2$ is excluded. Furthermore, in our analysis we have neglected the effects of bound state formation, which has the same parametric dependence on $\alpha_D / v_{\rm rel}$ as the SE and is expected to further enhance the signal from dSphs by an $O(1)$ factor (see Ref.~\cite{Cirelli:2016rnw} for a comprehensive analysis). On the other hand, bound state formation has negligible impact on freeze-out for the relatively light DM we consider in this work, $m_\chi \sim 100\;\mathrm{GeV}$ \cite{vonHarling:2014kha}.
Additional, important constraints on the DM self-interaction mediated by the dark photon arise from observations of DM halos. The strongest such bounds come from the triaxial structure of galaxy halos, in particular from the well-measured nonzero ellipticity of the halo of NGC720 \cite{Buote:2002wd}. This disfavors strong self interactions, which would have reduced the anisotropy in the DM velocity distribution via the cumulative effect of many soft scatterings \cite{Feng:2009mn}. In the nonrelativistic limit the scattering of two DM particles is dominated by dark photon exchange. The differential cross section in the center of mass frame is
\begin{equation} \label{eq:SIDMxsec}
\frac{d \sigma}{d \Omega} \simeq \frac{\alpha_D^2}{4m_\chi^2 v_{\rm cm}^4 (1 - \cos \theta_{\rm cm})^2}
\end{equation}
where we only retained the leading singular behavior at small $\theta_{\rm cm}$, which is the same for same-charge $\chi \chi \to \chi \chi$ and opposite-charge $\chi \chi^\ast \to \chi \chi^\ast$ scattering. Notice the very strong velocity dependence $\propto v_{\rm cm}^{-4}$, which implies that constraints from galaxies are much stronger than those from clusters. The authors of Ref.~\cite{Feng:2009mn} obtained a constraint by requiring that the relaxation time to obtain an isotropic DM velocity distribution be longer than the age of the Universe,
\begin{equation}
\tau_{\rm iso} \equiv \langle E_k \rangle / \langle \dot{E}_k \rangle = \mathcal{N} m_{\chi}^3 v_0^3 (\log \Lambda)^{-1} / ( \sqrt{\pi} \alpha_D^2 \rho_\chi ) > 10^{10}\,\mathrm{years}
\end{equation}
where $E_k = m_\chi v^2/2$, $\dot{E}_k$ is the rate of energy transfer proportional to $d\sigma/d\Omega$, $\mathcal{N}$ is an $O(1)$ numerical factor, $v_0$ is the velocity dispersion (very roughly $250\,\mathrm{km/s}$ in NGC720), $\rho_\chi = m_\chi n_\chi$ is the $\chi$ energy density and the ``Coulomb logarithm'' $\log \Lambda$ originates from cutting off the infrared divergence arising from Eq.~\eqref{eq:SIDMxsec}. The ellipticity bound was recently reconsidered by the authors of Ref.~\cite{Agrawal:2016quu}, who found it to be significantly relaxed compared to the original calculation of Ref.~\cite{Feng:2009mn}. We do not review their thorough analysis here, but simply quote the result
\begin{equation}\label{eq:ellipticity}
\alpha_D < 2.4 \times 10^{-3} \left( \frac{m_\chi}{100\;\mathrm{GeV}} \right)^{3/2}. \qquad (\mathrm{ellipticity})
\end{equation}
Although Ref.~\cite{Agrawal:2016quu} considered Dirac fermion DM, their ellipticity bound directly applies to our model, because the leading term of the self-scattering cross section in Eq.~\eqref{eq:SIDMxsec} is the same for fermions and scalars.\footnote{Notice that Fig.~4 in Ref.~\cite{Agrawal:2016quu} was drawn requiring $\Omega_{X} = 0.265$ for the DM density, instead of the correct $2\,\Omega_X = 0.265$. As a result, for $m_X < 200\;\mathrm{GeV}$ (where the SE is negligible) their relic density contour should be multiplied by $\sqrt{2}$. We thank P.~Agrawal for clarifications about this point.} Furthermore, there exist several reasons \cite{Agrawal:2016quu} to take even the bound in Eq.~\eqref{eq:ellipticity} with some caution, including the fact that it relies on a single galaxy, and that the measured ellipticity is sensitive to unobservable initial conditions (for example, a galaxy that recently experienced a merger may show a sizeable ellipticity even in the presence of strong DM self-interactions). Therefore we also quote the next most stringent constraint, obtained by requiring that the MW satellite dSphs have not evaporated until the present day as they traveled through the Galactic DM halo \cite{Kahlhoefer:2013dca}. This yields
\begin{equation} \label{eq:dSphsEvap}
\alpha_D < 5 \times 10^{-3} \left( \frac{m_\chi}{100\;\mathrm{GeV}} \right)^{3/2}, \qquad (\mathrm{dwarf\;survival})
\end{equation}
which stands on a somewhat more robust footing than ellipticity, but is not free from caveats either~\cite{Agrawal:2016quu}.
\begin{figure}[t]
\centering
\includegraphics[width=0.495\textwidth]{1_2TeV_RAplot_lessEllipt.pdf}
\includegraphics[width=0.495\textwidth]{1_4TeV_RAplot_lessEllipt.pdf}
\caption{Parameter space of the model where the gauging of $U(1)_{\rm DM}$ gives the leading breaking of the DM shift symmetry, for $f = 1.2\;\mathrm{TeV}$ (left panel) and $f = 1.4\;\mathrm{TeV}$ (right panel). The coefficients of the effective Lagrangian are set to $c_d = 1$, $c_t^\chi = c_b^\chi = \lambda = 0$, $m_{\gamma_D} = 0$. The exclusions from Fermi dwarfs were drawn assuming that all of the observed DM is composed of $\chi$ particles, irrespective of the thermal value of the $\chi$ density predicted at each point in parameter space.}
\label{fig:masslessDPsummary}
\end{figure}
A summary of all constraints on our parameter space is shown in Fig.~\ref{fig:masslessDPsummary}, for the choices $f = 1.2$ and $1.4\;\mathrm{TeV}$. While the region $m_\chi > m_h/2$ is ruled out by gamma ray observations from dSphs, for $m_\chi < m_h/2$ the strongest bounds arise from ellipticity and dwarf evaporation. In light of the previous discussion, however, we do not interpret these as strict exclusions, but rather note that they constitute an important class of probes of our setup, which may in the near future provide important evidence in favor of, or against, DM self-interactions mediated by a massless dark photon. Such self-interactions could also have interesting implications~\cite{Agrawal:2016quu} for the small-scale issues of the collisionless cold DM paradigm~\cite{Tulin:2017ara}. A complementary test of the light DM mass region is the search for invisible $h\to \chi^\ast \chi$ decays at the LHC,\footnote{The Higgs can also decay to $\gamma_D \gamma_D$ via a $\chi$ loop. The decay width for $m_{\gamma_D} = 0$ is $\Gamma(h\to \gamma_D \gamma_D) = m_h^3 \alpha_D^2 c_d^2 v^2 | F\big(\tfrac{m_h^2}{4m_\chi^2}\big) |^2 / (64\pi^3 f^4)$, where $F(\tau)$ is given in Eq.~\eqref{eq:hgaDgaD}. Numerically, for $m_\chi < m_h/2$ this is negligible compared to $\Gamma(h \to \chi^\ast \chi)$, while for $m_\chi > m_h/2$ it is too small to be observable: e.g. for $m_\chi = 100\;\mathrm{GeV}$ and $f = 1\;\mathrm{TeV}$ we have $\Gamma(h\to \gamma_D \gamma_D) \sim 10^{-12} \;\mathrm{GeV}$.} which will be sensitive to $f\lesssim 1.6\;\mathrm{TeV}$ by the end of the high-luminosity phase (see Fig.~\ref{fig:BottomSummary}).
\subsection{Phenomenology for massive dark photon} \label{subsec:massiveDP}
\enlargethispage{-5mm}We regard the mass of the dark photon as a free parameter of our model. Having extensively discussed the simplest possibility $m_{\gamma_D} = 0$ in Sec.~\ref{subsec:masslessDP}, we turn here to the study of the massive case. The physics is qualitatively different if $m_{\gamma_D} < m_\chi$ or $m_{\chi} < m_{\gamma_D}$, so we analyze these two regions separately. Our main findings are that (1) the region $m_{\gamma_D} < m_\chi$ is ruled out, unless $\gamma_D$ is so light that it still behaves as radiation today, and (2) for $m_\chi \lesssim m_{\gamma_D} < 2m_\chi$ we obtain a two-component DM setup with novel properties. Table~\ref{Tab:summaryDarkPhoton} summarizes the mileposts in the $m_{\gamma_D}$ parameter space.
\begin{table}
\begin{center}
\setlength{\doublerulesep}{0.04pt}
\begin{tabular}{ccc}
\hline
&& \\[-0.7cm]
\multirow{2}{*}{$m_{\gamma_D} < 6 \times 10^{-4} \;\mathrm{eV}$} & \multirow{2}{*}{$\;\; \checkmark /\, \text{X} \;\;$} & $\gamma_D$ is dark radiation today, \\
& &$\;$ strong constraints from SE of $\chi \chi^\ast \to \mathrm{SM} \;$ \\
&& \\[-0.7cm]
\hline
\multirow{2}{*}{$\;6 \times 10^{-4} \;\mathrm{eV} < m_{\gamma_D} \lesssim 3m_\chi / 25\;$} & \multirow{2}{*}{$\;\; \text{X} \;\;$} & $\gamma_D$ is relativistic at freeze-out, \\
& & ruled out by warm DM bounds/overabundant \\
&& \\[-0.7cm]
\hline
$ 3m_\chi / 25 < m_{\gamma_D} < m_\chi$ & $\;\; \text{X} \;\;$ & $\;\gamma_D$ is non-relativistic at freeze-out, overabundant $\,$ \\
&& \\[-0.7cm]
\hline\hline
$ m_\chi \lesssim m_{\gamma_D} < 2 m_\chi$ & $\;\; \checkmark \;\;$ & both $\gamma_D$ and $\chi$ are cold DM \\
&& \\[-0.7cm]
\hline
$2 m_\chi < m_{\gamma_D} $ & $\;\; \checkmark \;\;$ & $\gamma_D$ is unstable \\
\hline
\end{tabular}
\end{center}
\caption{Overview of the different regions in the dark photon mass space. The second column indicates whether each region satisfies ($\checkmark$) or conflicts with ($\text{X}$) experimental constraints, while the third column summarizes the key features.}
\label{Tab:summaryDarkPhoton}
\end{table}
\subsubsection{Light dark photon: $m_{\gamma_D} < m_\chi$}
If $m_{\gamma_D} < m_\chi$, the dark photon abundance freezes out almost simultaneously with the $\chi$ abundance. Assuming $\gamma_D$ is still relativistic at freeze-out, i.e. $m_{\gamma_D} \lesssim 3 T_{\rm fo}^\chi \approx 3 m_\chi / 25$, the ratio of its number density to the SM entropy density $s_{\rm SM} = (2 \pi^2 /45) g_{\ast s, \rm vis} T^3$ is $r_{\gamma_D} = n_{\gamma_D} / s_{\rm SM} = 45\, \zeta (3) g_{\gamma_D} / ( 2\pi^4 g_{\ast s, \rm vis}) \approx 0.01$, where we assumed that the dark and visible sectors are still in kinetic equilibrium at freeze-out, and took $g_{\gamma_D} = 3$, $g_{\ast s, \rm vis} \sim 80$. Since after freeze-out there are no $\gamma_D$-number-changing interactions in equilibrium (the scattering $\gamma_D \chi \to (h^\ast \to f\bar{f}\,) \chi$ is extremely suppressed), $r_{\gamma_D}$ is conserved.\footnote{Before kinetic decoupling of the dark and visible sectors only $n_{\gamma_D}/s_{\rm tot}$ is conserved, where $s_{\rm tot}$ is the total entropy, but $s_{\rm tot} \approx s_{\rm SM}$ since $g_{\gamma_D} \ll g_{\ast s, \rm vis}$.} As the Universe cools the dark photon becomes non relativistic, its energy density being $\Omega_{\gamma_D} = m_{\gamma_D} r_{\gamma_D} s_{\rm SM}$. Requiring that today this does not exceed the observed DM density yields
\begin{equation} \label{eq:overclosureBound}
\Omega_{\gamma_D} < \Omega_{\rm DM} \qquad \to \qquad m_{\gamma_D} < 40\;\mathrm{eV} \qquad (\mathrm{dark\; photon \;over\text{-}abundance})
\end{equation}
where we used $g_{\ast s, \rm vis} (T_0) = 3.91$.
Stronger constraints are derived from studies of ``mixed DM'' models, where the DM consists of an admixture of cold and non-cold particles. Recently, Ref.~\cite{Diamanti:2017xfo} obtained bounds on the fraction $f_{\rm ncdm}$ of the non-cold DM component, assumed to be a thermal relic, for a wide range of masses, by combining observations of the CMB, baryon acoustic oscillations (BAO) and the number of dwarf satellite galaxies of the MW. In our model, if the dark photon freezes out when relativistic it constitutes a hot DM component. Its temperature at late times is obtained from entropy conservation, $T_{\gamma_D} / T = [g_{\ast s, \rm vis} (T) / g_{\ast s, \rm vis} (T_{\rm dec})]^{1/3} \approx 0.37$, where $T$ is the SM photon temperature and we took $ g_{\ast s, \rm vis} (T_{\rm dec}) = 75.75$. The fraction of non-cold DM is
\begin{equation} \label{eq:fncdm_theory}
f_{\rm ncdm} \simeq \frac{\Omega_{\rm ncdm}}{ \Omega_{\rm DM}} = \frac{\rho_{\gamma_D, 0}}{\rho^c_0 \Omega_{\rm DM}} = \frac{r_{\gamma_D} s_{\rm SM, 0}}{\rho^c_0 \Omega_{\rm DM}} \begin{cases}
\frac{\pi^4 T_{\gamma_D, 0}}{30\, \zeta (3)} \\
m_{\gamma_D}
\end{cases} \approx
\begin{cases}
5.8 \times 10^{-6} & m_{\gamma_D} \lesssim 3 \,T_{\gamma_D,0} \\
0.024\left( \frac{m_{\gamma_D}}{1\;\mathrm{eV}} \right) & m_{\gamma_D} \gtrsim 3 \,T_{\gamma_D,0}
\end{cases}
\end{equation}
where the first (second) expression applies to the case where the dark photon is still relativistic (non-relativistic) today, with $3 \,T_{\gamma_D,0} \approx 2.6 \times 10^{-4}\;\mathrm{eV}$. In the first equality we assumed $\Omega_{\rm ncdm} \ll \Omega_{\rm DM}$ since the non-cold component is in practice constrained to be small, while $ \rho^c = 3 H^2 M_{\rm Pl}^2$ is the critical density. The prediction in Eq.~\eqref{eq:fncdm_theory} can be compared with the bounds given in Ref.~\cite{Diamanti:2017xfo}, after correcting for the fact that there the non-cold relic was assumed to have temperature equal to that of the SM neutrinos, hence the mass needs to be rescaled by a factor $T_{\gamma_D}/T_\nu \approx 0.52$. The result is shown in Fig.~\ref{fig:fncdm}, from which we read a $95\%$ CL bound
\begin{equation} \label{eq:ncdm_Bound}
m_{\gamma_D} < 6 \times 10^{-4}\;\mathrm{eV}, \qquad (\mathrm{CMB} + \mathrm{BAO} + \mathrm{MW}\;\mathrm{satellites})
\end{equation}
roughly equivalent to the requirement that $\gamma_D$ be still relativistic today. For dark photon masses that satisfy the overclosure bound of Eq.~\eqref{eq:overclosureBound} the relevant observables are CMB and BAO measurements, while the MW satellite count becomes important at higher masses, of order $\mathrm{keV}$ \cite{Diamanti:2017xfo}. In the region $m_{\gamma_D} \lesssim 1\;\mathrm{eV}$, where the dark photon behaved as radiation at photon decoupling, the constraints shown in Fig.~\ref{fig:fncdm} are stronger than those derived purely from $\Delta N_{\rm eff}$. This is due to the inclusion of BAO, which are sensitive to the suppression of the matter power spectrum on small scales caused by the free-streaming of the hot DM component.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{fncdm_vs_mncdm2.pdf}
\caption{The fraction of non-cold DM embodied by the dark photon as predicted by our model (dashed blue), compared to the $2\sigma$ (thick red) and $3\sigma$ (thin red) upper bounds from Ref.~\cite{Diamanti:2017xfo}.}
\label{fig:fncdm}
\end{figure}
For dark photon masses satisfying Eq.~\eqref{eq:ncdm_Bound}, the phenomenology for $m_{\gamma_D} = 0$ discussed in Sec.~\ref{subsec:masslessDP} still applies. The $\chi$ annihilation is unaffected, including the SE, as the dark photon mediates an effectively long-range force: its wavelength is much larger than the Bohr radius of the $(\chi^\ast \chi)$ bound state, $m_{\gamma_D} \ll \alpha_D m_\chi / 2$. In addition, the Coulomb limit of Eq.~\eqref{eq:SECoulomb} is still appropriate, since the average momentum transfer is much larger than the mediator mass, $m_{\gamma_D} \ll m_{\chi} v_{\rm rel} / 2$ \cite{Petraki:2016cnz}. In the calculation of the ellipticity bound for massless dark photon \cite{Agrawal:2016quu} the infrared divergence that arises from integrating Eq.~\eqref{eq:SIDMxsec} over angles was cut off at the inter-particle distance, $ \lambda_P = ( m_\chi / \rho_\chi )^{1/3} \sim 5\;\mathrm{cm} $, where the numerical value was estimated for a representative DM mass $m_\chi = 100\;\mathrm{GeV}$ and density $\rho_\chi \sim 1 \;\mathrm{GeV}/\mathrm{cm}^3$ in the DM-dominated outer region ($\mathrm{r}\geq 6\;\mathrm{kpc}$) of NGC720 \cite{Humphrey:2010hd}. When $m_{\gamma_D} > 1/\lambda_P \sim 4 \times 10^{-6}\, \mathrm{eV}$, it is $1/m_{\gamma_D}$ that must be taken as IR cutoff. However, since the cutoff only enters logarithmically in the expression of the timescale for velocity isotropization, the ellipticity bound discussed for $m_{\gamma_D} = 0$ applies essentially unchanged to the whole region defined by Eq.~\eqref{eq:ncdm_Bound}. The same holds for the bound from dwarf galaxy survival.\footnote{The small dark photon masses in Eq.~\eqref{eq:ncdm_Bound} are legitimate from an EFT standpoint. Still, it has recently been conjectured~\cite{Reece:2018zvv} that quantum gravity forbids arbitrarily small St\"uckelberg masses: local quantum field theory would break down at $\Lambda_{\rm UV} \sim (m_{\gamma_D} M_{\rm Pl}/g_D)^{1/2}$. Taking $g_D \sim 0.1$ as needed to obtain the observed relic density for $\chi$, Eq.~\eqref{eq:ncdm_Bound} corresponds then to a troublesome $\Lambda_{\rm UV} \lesssim 4\;\mathrm{TeV}$. The conjecture does not apply, however, if $m_{\gamma_D}$ arises from a dynamical symmetry breaking~\cite{Reece:2018zvv}. This topic is currently under debate~\cite{Craig:2018yld}.}
For $ 3 T_{\rm fo}^\chi \approx 3 m_\chi / 25 \lesssim m_{\gamma_D} < m_\chi $ the dark photon freezes out non-relativistically, but is nevertheless over-abundant.
\subsubsection{Heavy dark photon: $m_{\chi} < m_{\gamma_D}$}
In the region $m_\chi \lesssim m_{\gamma_D} < 2m_\chi$ both $\gamma_D$ and $\chi$ are stable and freeze out when non-relativistic, naturally giving rise to a two-component cold DM model.
The features of this region are best explained by fixing $f$ and $m_\chi > m_\chi^{(f)}$, so that $\chi$ would be under-abundant in isolation, owing to its too strong annihilation to SM particles via the derivative Higgs portal. Requiring that the heavier dark photon provides the remaining DM fraction then gives a contour in the $(m_{\gamma_D}/m_\chi, \alpha_D)$ plane, shown in the left panel of Fig.~\ref{fig:2compMain} for $f = 1\;\mathrm{TeV}$ and some representative choices of $m_\chi$. The relic densities of $\chi$ and $\gamma_D$ were computed solving the coupled Boltzmann equations with micrOMEGAs \cite{Belanger:2018mqt}. To understand the basic features of Fig.~\ref{fig:2compMain}$\,$-left, a useful first approximation is to treat the freeze-outs of $\chi$ and $\gamma_D$ as decoupled processes, since in this limit the relic density of $\chi$ is simply fixed by the freeze-out of $\chi \chi^\ast \to \mathrm{SM}$ and therefore completely determined by $f$ and $m_{\chi}$. This simplified picture does receive important corrections in some regions of parameter space, as we discuss below. Focusing first on the $m_\chi = 300\;\mathrm{GeV}$ case, four qualitatively different regions arise in our analysis:\enlargethispage{-12mm}
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth]{ratioVSalpha_fixedMchi.pdf} \hspace{1mm}
\includegraphics[width=0.4925\textwidth]{indirectDetection.pdf}
\caption{{\it Left panel:} contours in the $(m_{\gamma_D}/m_\chi, \alpha_D)$ plane where the sum of the $\chi$ and $\gamma_D$ densities matches the observed total DM density, $\Omega_{\chi + \chi^\ast} + \Omega_{\gamma_D} = \Omega_{\rm DM}$, assuming $f = 1\;\mathrm{TeV}$ and for representative values of $m_\chi > m_\chi^{(f)} \approx 122\;\mathrm{GeV}$. The solid portions highlight the range of $\alpha_D$ where $m_\chi$ can be obtained from dark photon loops cut off at $2.5\;\mathrm{TeV} < m_\rho < 4\pi f$ (see Eq.~\eqref{eq:gaugeDMmass}), where the lower bound comes from the $S$ parameter, $\widehat{S} \sim m_W^2/ m_\rho^2 \lesssim 10^{-3}$ (see e.g. Ref.~\cite{Giudice:2007fh}). {\it Right panel:} effective cross section for present-day DM annihilation to SM particles, calculated along the relic density contours in the left panel. Also shown are the observed $95\%$ CL limits from dSphs in the $WW$ channel \cite{Ackermann:2015zua} (dashed lines), together with the $95\%$ CL uncertainties on the expected limits (colored regions). For reference, the black solid line shows $\langle \sigma v_{\rm rel} \rangle_{\rm can}$, the cross section expected for a single thermal relic that annihilates entirely to SM particles.}
\label{fig:2compMain}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item The non-degenerate region, $ 2 m_\chi - m_h \approx 1.6\, m_\chi < m_{\gamma_D} < 2 m_\chi $. The dark photon freeze-out is determined by the semi-annihilation process $\gamma_D h \to \chi \chi^\ast$, which is kinematically allowed at zero temperature. Hence the relic density contour is approximately given by $n_h^{\rm eq} \langle \sigma_{\gamma_D h \to \chi \chi^\ast} v_{\rm rel} \rangle = \mathrm{constant} $, where the LHS is evaluated at the $\gamma_D$ freeze-out temperature, $T_{\gamma_D}^{\rm fo} \approx m_{\gamma_D} / 25$, and the thermally averaged cross section is given in Eq.~\eqref{eq:semiannihilationGammah}. As $m_{\gamma_D}/m_\chi$ decreases, the dark fine structure constant increases exponentially to compensate for the suppression of the Higgs number density, $\alpha_D \propto \exp \big(\frac{m_h}{m_\chi} \frac{25}{m_{\gamma_{D}}/m_\chi} \big)$, where we dropped subleading power corrections. The importance of semi-annihilation processes, which change the total DM number by one unit (rather than two units as for ordinary annihilation), was discussed for the first time in Ref.~\cite{DEramo:2010keq}.
\item The intermediate region, $ 1.3\,m_\chi \lesssim m_{\gamma_D} \lesssim 1.6\,m_\chi \approx 2 m_\chi - m_h$. The $\gamma_D$ freeze-out is still determined by $\gamma_D h \to \chi \chi^\ast$, which however is now forbidden at zero temperature. Using detailed balance, the relic density contour is given by $n_h^{\rm eq} \langle \sigma_{\gamma_D h \to \chi \chi^\ast} v_{\rm rel} \rangle = (n_{\chi}^{\mathrm{eq}\,2} / n_{\gamma_D}^{\rm eq}) \langle \sigma_{\chi \chi^\ast \to \gamma_D h} v_{\rm rel} \rangle = \mathrm{constant} $, where the LHS is evaluated at $T_{\gamma_D}^{\rm fo} \approx m_{\gamma_D} / 25$ and the cross section can be found in Eq.~\eqref{eq:semiannihilationChiChi}. The dependence of $\alpha_D$ on $m_{\gamma_D}/m_\chi$ is exponential and faster than in the non-degenerate region, $\alpha_D \propto \exp \big[\big(2 -\tfrac{m_{\gamma_D}}{m_\chi}\big) \frac{25}{m_{\gamma_D}/m_\chi}\big]$, where power corrections were neglected.
\item The degenerate region, $ m_\chi \lesssim m_{\gamma_D} \lesssim 1.3\,m_\chi $. As $m_{\gamma_D}/m_\chi$ decreases the semi-annihilation is increasingly Boltzmann suppressed, while the rate of the annihilation $\gamma_D \gamma_D \to \chi \chi^\ast$ increases as $\alpha_D^2\,$. Therefore the dark photon freezes out when its annihilation to $\chi \chi^\ast$ goes out of equilibrium. The relic density contour is approximately described by $\langle \sigma_{\gamma_D \gamma_D \to \chi \chi^\ast} v_{\rm rel} \rangle = \mathrm{constant}$, where the cross section is given in Eq.~\eqref{eq:annihilationChiChi}. The resulting variation of $\alpha_D$ is slow in comparison to the regions dominated by semi-annihilation, thus explaining the nearly flat behavior of the contours. Importantly, in this region the evolutions of the $\chi$ and $\gamma_D$ densities are tightly coupled, and the injection of $\chi$ particles due to the $\gamma_D \gamma_D \to \chi \chi^\ast$ process gives a larger $\chi$ abundance than the one expected based on the simplified decoupled picture. This interesting type of system was first studied numerically in Ref.~\cite{Belanger:2011ww}, and we provide here analytical insight into its dynamics. After the yields $Y_{\chi, \gamma_D}$ become much larger than their equilibrium values, they obey the simplified Boltzmann equations ($x \equiv m_\chi / T$)
\begin{subequations}
\begin{alignat}{2}
\widehat{\lambda}^{-1} x^2\, \frac{dY_\chi}{dx} &\,=\, - \langle \sigma v_{\rm rel} \rangle_{\mathrm{SM}} Y_\chi^2 + \tfrac{1}{2} \langle \sigma v_{\rm rel} \rangle_{\gamma_D \gamma_D} Y_{\gamma_D}^2 \\
\widehat{\lambda}^{-1} x^2\, \frac{dY_{\gamma_D}}{dx} &\,=\, - \langle \sigma v_{\rm rel} \rangle_{\gamma_D \gamma_D} Y_{\gamma_D}^2
\end{alignat}
\end{subequations}
where $\widehat{\lambda} \equiv (2\sqrt{10}\, \pi / 15) (g_{\ast s} m_\chi M_{\rm Pl} / \sqrt{g_\ast})$, while $\langle \sigma v_{\rm rel} \rangle_{\mathrm{SM}}$ refers to $\chi \chi^\ast \to \mathrm{SM}$ and $\langle \sigma v_{\rm rel} \rangle_{\gamma_D \gamma_D}$ to $\gamma_D \gamma_D \to \chi \chi^\ast$. The analytical solution of this system gives at $x \gg 1$
\begin{equation} \label{eq:2CDMsolution}
\frac{1}{a_\sigma}\left(\frac{2Y_{\chi}}{Y_{\gamma_D}}\right)^2 \simeq 1 + \tfrac{1}{2}\left( a_\sigma + \sqrt{a_\sigma (a_\sigma + 4)}\, \right), \qquad a_\sigma \equiv \frac{\langle \sigma v_{\rm rel} \rangle_{\gamma_D \gamma_D}}{\langle \sigma v_{\rm rel} \rangle_{\mathrm{SM}}/2}\;,
\end{equation}
where $a_\sigma$ goes to a constant since both processes are $s$-wave. This result is obtained by solving a quadratic equation, whose other root yields $dY_\chi / dx > 0$ and is therefore unphysical. For $a_\sigma \ll 1$, as verified in the $m_\chi = 300, 600$ GeV examples, the RHS of Eq.~\eqref{eq:2CDMsolution} goes to $1$ and the formula expresses the equality of the fluxes that enter and leave the $\chi$ population, $Y_{\gamma_D}^2 \langle \sigma v_{\rm rel} \rangle_{\gamma_D \gamma_D} = (2Y_\chi)^2 \langle \sigma v_{\rm rel} \rangle_{\mathrm{SM}}/2\,$. Correspondingly, the relative $\chi$ density is suppressed (albeit still larger than in the simplified decoupled picture), $2n_\chi / n_{\gamma_D} \simeq a_\sigma^{1/2}$. In the $m_\chi = 150\;\mathrm{GeV}$ example we have $a_\sigma = O(1)$ instead: in this regime the annihilation to the SM is not as efficient, leading to an accumulation of the $\chi$ particles injected by $\gamma_D\gamma_D$ annihilation and therefore to a large relative $\chi$ abundance, $2n_\chi / n_{\gamma_D} \simeq \mathrm{few}$.
\item The very degenerate and forbidden~\cite{Griest:1990kh} region, $m_{\gamma_D} \lesssim m_\chi $. The dark photon freeze-out is determined by $\gamma_D \gamma_D \to \chi \chi^\ast$, but $\alpha_D$ increases very rapidly as $m_{\gamma_D}/m_\chi$ is decreased toward and eventually slightly below $1$, in order to compensate for the kinematic suppression.
\end{enumerate}
The previous discussion focused on the $m_{\chi} = 300\;\mathrm{GeV}$ benchmark. The features of the relic density contour for $m_{\chi} = 600\;\mathrm{GeV}$ are very similar. On the contrary, in the case $m_\chi = 150\;\mathrm{GeV}$ we have $2m_\chi - m_{h} \approx 1.2\, m_\chi$ and as a consequence we observe a direct transition from the non-degenerate to the degenerate region, while the intermediate region is absent.
The right panel of Fig.~\ref{fig:2compMain} shows the effective cross section for DM annihilation to SM particles today, computed along the relic density contours. All processes that yield SM particles were included in the numerical evaluation, but we have checked that $\chi \chi^\ast \to \mathrm{SM}$ is always dominant and the subleading channels (such as $\gamma_D \chi \to h \chi$ and $\gamma_D \gamma_D \to \mathrm{SM}$, the latter of which proceeds at one loop) contribute at the sub-percent level.\footnote{Note that due to the large mass of the dark photon, in this case the Sommerfeld enhancement of the $\chi \chi^\ast \to \mathrm{SM}$ annihilation is negligible.} Two different regimes can be observed. In the non-degenerate region the freeze-outs of $\chi$ and $\gamma_D$ can be treated as independent to a good approximation, hence from Eq.~\eqref{eq:RAapproxAnalytical} the effective cross section is reduced compared to the standard thermal value $ \langle \sigma v_{\rm rel} \rangle_{\rm can} \approx 2 \times 10^{-26} \,\mathrm{cm}^3\, \mathrm{s}^{-1}$ by a factor $\langle \sigma v_{\rm rel} \rangle_{\rm can} \,/ (\tfrac{1}{2}\langle \sigma v_{\rm rel} \rangle_{\chi \chi^\ast \to\, \mathrm{SM}} ) < 1$. For $m_\chi = 600 \;\mathrm{GeV}$ the suppression amounts to more than one order of magnitude. Conversely, in the degenerate region the already discussed injection of $\chi$ particles from $\gamma_D \gamma_D$ annihilations compensates the increased $\langle \sigma v_{\rm rel} \rangle_{\chi \chi^\ast \to\, \mathrm{SM}}\,$, resulting in effective cross sections that are numerically close to $\langle \sigma v_{\rm rel} \rangle_{\rm can}$.
Finally, if $ 2 m_\chi < m_{\gamma_D}$ the dark photon is unstable, with decay width $\Gamma(\gamma_D \to \chi^\ast \chi) = (\alpha_D m_{\gamma_D} / 12) (1 - 4m_\chi^2/m_{\gamma_D}^2)^{3/2}$. In the early Universe, the inverse decay process keeps the dark sector in chemical equilibrium until $H \sim \langle \Gamma \rangle \, n_{\gamma_D} / n_\chi$, when the ratio of the number densities is
\begin{equation}
\frac{n_{\gamma_D}}{n_{\chi}} \sim \frac{H}{\Gamma} \sim \frac{10\, T^2}{M_{\rm Pl} \alpha_D m_{\gamma_D}} < 10^{-12}\, \left( \frac{m_{\gamma_D}}{100\;\mathrm{GeV}} \right) \left( \frac{10^{-3}}{\alpha_D} \right),
\end{equation}
where we assumed that $T < m_{\gamma_D}$ at this point, and neglected $O(1)$ factors. Thus, the subsequent decay of the remaining dark photons has negligible impact on the $\chi$ relic density, which can effectively be computed considering only the freeze-out of $\chi\chi^\ast$ annihilations to SM particles, with the results summarized in Fig.~\ref{fig:fvsmchi}. In the region $2 m_\chi < m_{\gamma_D}$ the only phenomenologically relevant imprint of the dark photon is the one-loop mass for $\chi$, estimated in Eq.~\eqref{eq:potentialGauging}.
\vspace{0.5cm}
\section{Closing remarks} \label{sec:outlook}
\noindent We have considered models where the Higgs doublet $H$ and the DM $\chi$ have common origin as pNGBs of a spontaneously broken global symmetry. We have shown that the shift symmetry of $\chi$ can be broken in such ways that a mass of $O(100)\;\mathrm{GeV}$ is generated at one loop, whereas the non-derivative couplings between $\chi$ and the SM are small, naturally leading to suppressed direct detection. In a first realization the DM, taken to be either a real or complex scalar, acquires mass from bottom quark loops. Correspondingly the operator $y_b \bar{q}_L H b_R\, |\chi|^2/f^2$ is generated with $O(1)$ coefficient, leading to very suppressed cross sections for DM-nucleus scattering that will be probed by LZ. In a second realization, which constitutes the central subject of this work, the DM is a complex scalar whose mass arises from the gauging of the $U(1)_{\rm DM}$ stabilizing symmetry. The direct detection signal is out of reach even at future experiments, but the dark sector -- now including $\chi$ and the dark photon $\gamma_D$ as light fields -- can be tested both at colliders and in cosmology and astroparticle experiments. As concerns the latter, especially important observables are $\Delta N_{\rm eff}$ and the effects of long-range DM self-interactions if $m_{\gamma_D} = 0$, and indirect DM detection if $m_{\gamma_D} > m_\chi\,$.
We wish to remark that promoting $U(1)_{\rm DM}$ to a local symmetry may in fact be preferred, based on both model-specific and more general theoretical considerations. Specifically, gauging $U(1)_{\rm DM}$ ensures that any subleading couplings of the SM fermions to the strong sector, which were neglected in our discussion, automatically preserve the DM stability.\footnote{We thank K.~Agashe and M.~Frigerio for enlightening comments on this point.} More generally, several arguments exist that suggest quantum gravity does not conserve continuous global symmetries (see Refs.~\cite{Kallosh:1995hi,ArkaniHamed:2006dz,Banks:2006mm} and further references therein). If that is the case, then Planck-scale suppressed operators can destabilize the DM, potentially leading to conflict with observations~\cite{Mambrini:2015sia}, although this strongly depends on the assumptions made about the coefficients of the higher-dimensional operators. In any case, the issue is absent if $U(1)_{\rm DM}$ is gauged.
We conclude with some further comments about the collider phenomenology, focusing on signals that involve the pNGB DM (overviews of the ``standard'' signatures of composite Higgs models can be found in Refs.~\cite{Panico:2015jxa,Bellazzini:2014yua}). As already discussed, for $m_\chi < m_h/2$ the searches for invisible Higgs decays provide a powerful probe of the derivative Higgs portal operator. In contrast, the experimental prospects are less favorable for $m_\chi > m_h/2$, when the intermediate Higgs is off shell. For the marginal Higgs portal the reach was studied in Ref.~\cite{Craig:2014lda} for the LHC and future hadron colliders, including all relevant production channels (monojet, $t\bar{t}h$ and vector boson fusion), and in Ref.~\cite{Chacko:2013lna} for future lepton colliders. For real DM with $m_\chi = 100\;\mathrm{GeV}$ the sensitivity was found to extend up to $\lambda \sim 0.7$ at the High-Luminosity LHC and $\lambda \sim 0.3$ at a $100\;\mathrm{TeV}$ $pp$ collider with $30$ ab$^{-1}$ (neglecting systematic uncertainties~\cite{Craig:2014lda}), and $\lambda \sim 0.6$ at the $1\;\mathrm{TeV}$ ILC \cite{Chacko:2013lna}. For the derivative Higgs portal relevant to pNGB DM, we need to replace $\lambda \to c_d M^2_{\chi \chi^\ast}/(2f^2)$ where $M^2_{\chi \chi^\ast} \geq 4m_\chi^2$ is the squared invariant mass of the DM pair. Although the momentum-dependent coupling gives harder kinematic distributions and therefore sensitivity to smaller cross sections compared to the momentum-independent case, the LHC reach in the monojet channel is negligible~\cite{Barducci:2016fue}. A detailed assessment of the reach of future colliders on the derivative portal is, to our knowledge, not yet available. Another class of signals arises from composite resonances that are charged under the DM-stabilizing symmetry. For fermionic top partners, the reach at future hadron colliders was shown to exceed that on resonances with only SM quantum numbers~\cite{Chala:2018qdf}. For vector resonances, the pair production (via Drell-Yan and vector boson fusion) and the production in association with DM yield final states with $W$ and/or $Z$ bosons, missing transverse energy, and possibly Higgs bosons. The latter signatures, although difficult to discover due to the suppressed cross sections, constitute a robust feature of models where the Higgs and the DM arise as pNGB, regardless of the details of the specific construction.
\vspace{1cm}
\noindent {\bf Acknowledgments}
\noindent We have benefited from conversations with K.~Agashe, M.~Frigerio, M.~Garny, R.~Harnik, A.~Katz, J.~Serra, Y.~Shadmi, and Y.~Tsai. We also thank P.~Agrawal for clarifications about Ref.~\cite{Agrawal:2016quu}. This work has been partially supported by the DFG Cluster of Excellence 153 ``Origin and Structure of the Universe,'' by the Collaborative Research Center SFB1258, and the COST Action CA15108. RB is supported by the Minerva foundation. MR is supported by the Studienstiftung des deutschen Volkes. The work of ES was initiated at the Aspen Center for Physics, which is supported by NSF grant PHY-1607611. ES thanks the organizers of the CERN-Korea TH Institute ``Physics at the LHC and beyond'' for a stimulating environment and partial support in the final stages of this work. ES is also grateful to the GGI for hospitality and to the INFN for partial support as this paper was being completed.
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\subsection{Methods}
| 9 |
\section{Introduction}
Designing and mounting a sensor setup for a research vehicle requires much time and engineering effort.
Furthermore, the amount of data needed for highly automated driving is soaring, and the acquisition and annotation of sensor data are expensive and time-consuming, yielding a limited collection of publicly available data sets used for HAD research. In addition, most data sets consist only of data acquired with a fixed sensor setup.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{tex_images_or_tables/sensor_data_abstraction.pdf}
\caption{Sensor data abstraction provides an interface between a sensor and further applications.}
\label{fig:sens_dat_abs}
\end{figure}
The properties of the sensor or sensor setup can implicitly bias the data and methods developed on it. This bias might reduce the transferability of methods and algorithms to new sensor setups. This bias might reduce the transferability of methods and algorithms to new sensor setups.
In this work, we want to explore the benefit of using sensor data abstraction in HAD. As displayed in fig. \ref{fig:sens_dat_abs}, we see sensor data abstraction as an interface between sensor data and machine learning applications, improving existing models' performance and utility. We envision a unified and sensor-independent abstraction of sensor data, enabling a broad perception pipeline for different sensor setups.
In the following chapter \ref{sec: SOTA}, we will introduce state-of-the-art sensor setups in HAD-related datasets. In section \ref{sec:towards_sensor_abstraction} we examine the abstraction of data from single sensors, which we refer to as \emph{marginal sensor data abstraction} and the abstraction of data from sensor setups, which we refer to as \emph{joint sensor data abstraction}. Finally, section \ref{sec:outlook} summarizes our findings and points towards challenges and further research directions.
\section{State of the Art} \label{sec: SOTA}
The following section will first introduce several open-access datasets. They provide a basis for the methods presented in this paper. After that, different sensor modalities and their characteristics will be discussed. At the end of this chapter, sec. \ref{sec:degrees_of_abstraction} will introduce the field of sensor data abstraction and standard definitions.
\subsection{Sensor Setups in Datasets} \label{sec:datasets}
For the abstraction of sensor data, it is necessary to understand the implicit properties of the utilized sensors and their setup. These have a strong influence on the type and quality of the resulting data. For this purpose, we examine popular HAD-related datasets as listed in \cite{fen2021DeepMultimodal}.
While camera images and lidar point clouds are typically included, only recently datasets included radar dat
Sensor data is usually decoupled from the physical sensor by a hardware abstraction layer (raw sensor data) and provided as point clouds or images. This hardware-related signal processing, whose resulting representation contains implicit sensors' characteristics, is not examined in this paper. In the following, we will only refer to this resulting representation as \emph{sensor data}, shown in fig. \ref{fig:sens_dat_abs}.
\subsection{Sensor Modalities} \label{sec:sensor_modalities}
There are common characteristics, such as position and orientation, that differ for each data collection process and affect all sensor types. The influence of other common features, such as resolution, the field of view (FOV), and manufacturing tolerances, depending on the sensor configuration. A sensor's characteristics are implied in its data representation, biasing the method used to process the data. This risk exists with both learned and engineered methods. An example of this bias can be found in \cite{DBLP:journals/corr/DoerschGE15}. The authors tried to find the relative position of corresponding image patches with a convolutional neural network (CNN). However, the CNN seems to learn a trivial solution by locating the patches based on the chromatic aberration implied to the image due to lens characteristics. The sensor characteristics lead to a specific profile which is shown in fig. \ref{fig: compare_sensors} for camera, lidar and radar.
\begin{figure}
\includegraphics{tex_images_or_tables/spider_sensorcomp.pdf}
\centering
\caption{Quantitative comparison between the three major sensor modalities. From bad (center) to excellent (rim).}
\label{fig: compare_sensors}
\end{figure}
Based on \cite{rosique2019systematic} we define seven parameters to visualize the strengths and weaknesses inside that profile.
The values are based on ideal sensors. To determine the possible marginal, abstract spaces, one can use the filled areas inside the plot.
As seen in the figure, the sum of all characteristics also results in a shared space. This space could be covered in a sensor fusion and, therefore, in a shared (joint) abstraction. For further research, both spaces for marginal and joint abstraction need to be considered.
\subsection{Sensor Data Abstraction} \label{sec:degrees_of_abstraction}
Kirman et al. first coined the term \emph{abstract sensor} as a module that provides a \say{mapping from the actual sensor space to the observation space} \cite{kir1991SensorAbstractionsa}. As their observation space is well designed, the resulting \say{abstract sensor reading} is human-interpretable. Jha et al. provide a similar understanding of a \say{sensor abstraction layer}, which performs several preprocessing steps of input data before given to the perception layer \cite{jha2019MLBasedFault}.
From a software development perspective, sensor abstraction is often understood as a consistent and unified interface for many sensors, which can have different sensing technologies or are made from different companies \cite{kab2006VirtualSensors}
. This concept neglects the implicit information included in the sensor setup but might provide it as metadata.
In contrast to classical, hand-engineered abstractions, such as mathematical transformations, in Machine Learning, neural networks are frequently used to learn latent spaces, abstractions of input data. These are typically not directly human-interpretable. Latent spaces can be utilized for specific perception tasks, such as object detection or state representation for end-to-end systems \cite{che2020InterpretableEndtoend}.
\section{Towards Sensor Abstraction}
\label{sec:towards_sensor_abstraction}
As shown in the previous sec. \ref{sec:degrees_of_abstraction}, there exist a multitude of meanings for sensor (data) abstraction in the literature. We aim to provide insights into sensor-independent and unified environment representations for modern perception approaches. Thus we define the term as:
\begin{displayquote}
\emph{Sensor data abstraction is a learned mapping of sensor data from $1-n$ sensors to a unified abstraction suitable as input data for neural networks}
\end{displayquote}
Based on this definition, which is also shown in fig. \ref{fig:sens_dat_abs}, we will discuss the abstraction of single sensors, sensor setups and informed abstraction based on meta-data in the following chapter. The latter are highly motivated by the sensor's characteristics shown in fig. \ref{fig: compare_sensors}.
\subsection{Marginal Sensor Data Abstraction}
\label{sec: marginal}
As mentioned in sec. \ref{sec:sensor_modalities} the implicit encoding of sensor parameters to the representation is a problem. For camera data, as an example, this problem can be partially solved by empiric models, optimized on large-scale datasets with a wide variety of used cameras.
In detail, such CNN-based models often consist of a two-stage architecture \cite{Jiao_2019}. Such two-stage architectures include a \emph{backbone stage}, which encodes the sensor data to an abstract latent space, and a \emph{head stage}, which uses the extracted features to perform tasks like classification, object detection, or human pose estimation segmentation.
From vast datasets like COCO \cite{lin2015microsoft}, backbones can learn to extract generalized features such that transferring to new data or tasks can be done by retraining.
A model trained on COCO can avoid sensor bias due to the sheer amount of images taken from various perspectives with multiple consumer grade cameras.
Unlike cameras, lidar and radar sensors represent the environment as point clouds, i. e. a radar generates data with additional values like velocity and magnitude. In literature there are various methods, e.g. PointNet \cite{PointNet}, capable of directly working on 3D data.
In summary, an abstraction of various sensors can be achieved empirically inter alia with state-of-the-art two-stage perception models, given enough and variable data. For HAD datasets, this is currently not the case. Therefore, one core question is how to acquire suitable, sensor-specific datasets regarding scale and variety, given that marginal sensor abstraction architectures are relatively mature at this point.
\subsection{Joint Sensor Data Abstraction}\label{sec: joint}
Besides marginal sensor abstractions as discussed in sec. \ref{sec: marginal}, we want to outline the necessity and feasibility of a joint sensor abstraction and highlight early results in this research field. As the authors in \cite{fay2020DeepLearning} demonstrate, multi-modal sensor setups are needed to provide a safe perception pipeline for HAD. As an example, camera-only sensor setups are vulnerable to harsh weather, which can be compensated with additional sensor(s), as seen in fig. \ref{fig: compare_sensors}.
We can see joint abstraction as a subset of \emph{sensor fusion}. While most fusion approaches are based on a high level of data abstraction, joint sensor abstraction focuses on \emph{low-level} fusion where raw sensor data from multiple sensors is fused. While this approach contains the most information, it is also extremely complex \cite{ott2013FusionData}
. Nobis et al. have also shown that it is possible to learn the optimal level for fusion \cite{nobis2020deep}.
The datasets mentioned in sec. \ref{sec:datasets} act as a basis for these approaches. They were recorded by research vehicles with certain sensor setups regarding the number, types, manufacturers, positions, and orientations of the utilized sensors. Such configurations can be split into three categories: \emph{Complementary} for the combination of independent sensors, \emph{competitive} for reliability and \emph{cooperative} for enhanced quality by integrating higher-level measurements which cannot be measured directly \cite{dur1988SensorModels}. Information about such configurations is implicitly included in the sensor data and also explicitly available as meta-data, which we will examine in sec. \ref{sec:metadata}. An example of a \emph{complementary} is the usage of multiple cameras, each observing disjunctive parts of the world. In \emph{competitive} setups, the output of multiple, redundant sensors can be used as input for a voting model. Examples for \emph{cooperative} would be a mapping of camera data to point clouds or a transformation of point clouds onto an image representation (e. g., RGB-Depth) before a marginal abstraction. This mapping of one sensor into the representation of another gives us a \textit{novel sensor}. This \textit{novel sensor} contains information sensed by its parental sensors but novel characteristics. If the marginal abstraction is done as described in sec. \ref{sec: marginal}, an empiric model has the freedom to decide which components of this novel sensor representation are helpful.
Based on \emph{mid-level} and \emph{high-level} fusion approaches and \emph{competitive} as well as \emph{cooperative} configurations, many abstraction architectures can be engineered. Examples are mapping of multi-modal sensor data onto a common representation, e. g., radar and lidar onto image formats, merging posterior to a marginal abstraction, or pure transformations
\cite{nobis2020deep}. Due to their nature of equalized input data representation, such approaches are currently prevalent for learned abstraction approaches, as Cui et al. demonstrate \cite{cui2021DeepLearning}. Famous examples are end-to-end models, which encode multi-modal sensor data, mostly camera and lidar, into an abstract latent space to perform the complete driving task based on this environment representation. Che et al. present a method that shows the quality of the latent space by constructing a birds-eye view based on it \cite{che2020InterpretableEndtoend}. Jointly trained variational autoencoders follow similar research directions \cite{kor2019JointlyTraineda}.
Due to the high complexity of \emph{low-level} approaches, potentially suitable neural network architectures are not well understood yet based on the different types of input data representation. Also, current approaches suffer under model inflexibility - new sensing modalities typically need full retraining of the network \cite{fen2021DeepMultimodal}. Therefore, further research is necessary.
Regardless of the joint abstraction, the sensors' bias remains, and its impact on the data could be even higher due to coherence's between the sensors in one setup.
\subsection{Utilizing Metadata for Abstraction} \label{sec:metadata}
So far, we have only dealt with abstraction in terms of sensor data. However, an automotive sensor setup usually comes with auxiliary data describing sensors themselves or geometric relations between multiple sensors. We refer to this data as metadata. Following the concept of \emph{informed machine learning}, this metadata might be helpful for sensor data abstraction \cite{von2019InformedMachine}. An example for this is CamConvs \cite{facil2019camconvs}, which explicitly encodes the intrinsic parameters of cameras to its image representation to overcome a sensor bias, yielding considerable improvements in the task of mono depth estimation when train and test images are acquired with different cameras. The research question arises if and how metadata can be utilized to improve both marginal (intrinsic parameters) and joint (intrinsic and extrinsic parameters) abstraction. There lies much potential in metadata regarding joint abstraction. There is still no consensus among the industry leaders in HAD regarding an optimal sensor setup. Eslami et al. have shown that it is possible to learn a scene representation by providing two different camera perspectives as input and predict the scene from a third camera viewpoint. Based on this integration of metadata, it is possible to utilize the prediction error to analyze optimal positions for additional sensors. A significant error suggests that the existing sensors could not capture the content from the new viewpoint \cite{esl2018NeuralScene}. Further research for multi-modal approaches is necessary.
\subsection{General pipeline for multiple sensor configurations} \label{sec:general_pipeline}
Based on the current situation of available sensor setups in HAD, the research question arises how to represent multiple sensor setups, such that utilization to new setups is given. As introduced in the beginning, we envision a unified and sensor-independent abstraction of sensor data, enabling a general perception pipeline for different sensor setups as shown in fig. \ref{fig:data_flow}. While there are many open questions, this would enable the transferability of whole perception pipelines between multiple, with different sensor configurations equipped, autonomous vehicles.
Due to the complementary nature of the installed sensors, one cannot expect that it will be possible to utilize arbitrary setups which will lead to the same abstract environment model as all others. Therefore, a core research question is the space, in which the positions as well as the combinations of sensors can be altered without effects on the abstract representation. For this purpose, the shown concepts and research directions from this section will be key contributors to potential solutions regarding this research field.
\begin{figure}[h!]
\centering
\includegraphics[width=\columnwidth]{tex_images_or_tables/sensor_data_to_abstraction.pdf}
\caption{Data flow from sensors to an environment model.
The abstraction layer allows for a decoupling of observations and the state of a system.
}
\label{fig:data_flow}
\end{figure}
\section{Conclusion and Outlook}
\label{sec:outlook}
This paper defined the theme of sensor data abstraction for the field of automotive applications. Furthermore, we discovered limitations and multiple critical paths towards sensor data abstraction. Further research could study different ways to realize marginal and joint abstraction concerning utilization and transferability.
A possible solution to overcome the limitation of available and unbiased data is to build a vast data set with identical trajectories and variations in the sensor setups used. However, this is not feasible. Further research would therefore be to integrate the real sensor setups into simulation environments such as Carla \cite{Dosovitskiy17}. With such a simulation, it would be less effort to generate data than building multiple test vehicles. The authors will continue their research on this topic. Since such latent spaces have the potential for further applications, we will also focus on topics such as input augmentation, data compression \cite{tol2018GenerativeNeural}, corner case detection \cite{bre2020CornerCases} and prediction \cite{neu2021VariationalAutoencoderBased}.
\section{\large Acknowledgment}
This work results from the KI Data Tooling project supported by the Federal Ministry for Economic Affairs and Energy (BMWi), grant numbers 19A20001I, 19A20001J, 19A20001L and 19A20001O.
| 3,973 |
\section*{Acknowledgements}
\noindent
We express our gratitude to our colleagues in the CERN
accelerator departments for the excellent performance of the LHC. We
thank the technical and administrative staff at the LHCb
institutes.
We acknowledge support from CERN and from the national agencies:
CAPES, CNPq, FAPERJ and FINEP (Brazil);
MOST and NSFC (China);
CNRS/IN2P3 (France);
BMBF, DFG and MPG (Germany);
INFN (Italy);
NWO (Netherlands);
MNiSW and NCN (Poland);
MEN/IFA (Romania);
MSHE (Russia);
MICINN (Spain);
SNSF and SER (Switzerland);
NASU (Ukraine);
STFC (United Kingdom);
DOE NP and NSF (USA).
We acknowledge the computing resources that are provided by CERN, IN2P3
(France), KIT and DESY (Germany), INFN (Italy), SURF (Netherlands),
PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex
LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil),
PL-GRID (Poland) and NERSC (USA).
We are indebted to the communities behind the multiple open-source
software packages on which we depend.
Individual groups or members have received support from
ARC and ARDC (Australia);
AvH Foundation (Germany);
EPLANET, Marie Sk\l{}odowska-Curie Actions and ERC (European Union);
A*MIDEX, ANR, IPhU and Labex P2IO, and R\'{e}gion Auvergne-Rh\^{o}ne-Alpes (France);
Key Research Program of Frontier Sciences of CAS, CAS PIFI, CAS CCEPP,
Fundamental Research Funds for the Central Universities,
and Sci. \& Tech. Program of Guangzhou (China);
RFBR, RSF and Yandex LLC (Russia);
GVA, XuntaGal and GENCAT (Spain);
the Leverhulme Trust, the Royal Society
and UKRI (United Kingdom).
\section{Introduction}%
\label{sec:introduction}
The quark model~\cite{GellMann:1964nj,Zweig:1981pd,Zweig:1964jf}
describes a doubly charmed baryon as a system of two bound charm quarks and a light quark ({\ensuremath{\Pu}}\xspace, {\ensuremath{\Pd}}\xspace or {\ensuremath{\Ps}}\xspace).
There are three doubly charmed, weakly decaying states expected: a {\ensuremath{\Xires_{\cquark\cquark}}}\xspace isodoublet $(ccu,ccd)$ and an
{\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace isosinglet $(ccs)$, each with spin-parity $J^{P} = 1/2^{+}$. Theoretical models~\cite{Fleck:1989mb,Berezhnoy:1998aa, Ebert:2002ig, Chang:2006eu} predict that the light quark moves with a large relative velocity with respect to the bound ($cc$)-diquark inside the baryon and experiences a short-range of QCD potential.
The {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace baryon with mass $3620.6 \pm 1.6$\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace was first observed by the LHCb collaboration in the {\ensuremath{\Lz^+_\cquark}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace\pip decay\footnote{Inclusion of charge-conjugated processes is implied throughout this paper.}~\cite{LHCb-PAPER-2017-018}, and confirmed in the ${\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\pion^+}}\xspace$ decay~\cite{LHCb-PAPER-2018-026}. The search for {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{+}}}\xspace via its decay to {\ensuremath{\Lz^+_\cquark}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace was updated recently by the LHCb collaboration, and no significant signal was found~\cite{LHCb-PAPER-2019-029}. The {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace mass is predicted to be in the range $3.6-3.9\ensuremath{\aunit{Ge\kern -0.1em V\!/}c^2}\xspace$~\cite{Ebert:2002pd,Gershtein:2000pd,Ebert:1997pc,Roncaglia1:1995pd,Roncaglia2:1995pd,Korner:1994np,Narodetskii:2001hep,Zachary:2014pd} and its lifetime is predicted to be $75-180\ensuremath{\aunit{fs}}\xspace$~\cite{Haiyang:2018pd,hsi2008lifetime, Ebert:2002ig, guberina1999inclusive,Karliner:2014gca,Kiselev:1998sy,Kiselev:2001fw}.
Due to destructive Pauli interference~\cite{Haiyang:2018pd}, the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace and {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace baryons have a larger lifetime than that of the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{+}}}\xspace baryon which is shortened by the contribution from
$W$ boson exchange between the charm and down quarks. In proton-proton ($pp$) collisions at a centre-of-mass energy of 13\aunit{Te\kern -0.1em V}\xspace, the production cross-section of the doubly charmed baryons is predicted to be within the range of $60-1800\aunit{nb}\xspace$~\cite{Berezhnoy:1998aa,Kiselev:2001fw,Ma:2003zk,Chang:2006xp,Chang:2006eu,Zhang:2011hi,Chang:2005bf}, which is between $10^{-4}$ and $10^{-3}$ times that of the total charm quark production~\cite{Kiselev:2001fw}.
The production cross-section of the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace baryon is expected to be about 1/3 of those of the
{\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{+}}}\xspace and {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace baryons due to the presence of an ${\ensuremath{\Ps}}\xspace$ quark~\cite{Berezhnoy:2018pd}. A discovery of the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace baryon would validate a prediction of the Standard Model. Besides, the study of doubly charmed baryons will also
deepen our understanding of perturbative and non-perturbative QCD dynamics.
\begin{figure}[!htp]
\centering
\includegraphics[width=0.55\linewidth]{Figs/Fig1.pdf}
\caption{Example Feynman diagram for the ${\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace\ensuremath{\rightarrow}\xspace{\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace$ decay.}
\label{fig:feynman1}
\end{figure}
In this paper, a search for the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace baryon via the ${\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace \ensuremath{\rightarrow}\xspace
{\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace {\ensuremath{\kaon^-}}\xspace {\ensuremath{\pion^+}}\xspace$ decay, which is predicted to have a relatively large branching fraction~\cite{Shi:2017dto,Jiang:2018}, is presented. The data are collected by the LHCb experiment in $pp$ collisions at a centre-of-mass energy of 13\aunit{Te\kern -0.1em V}\xspace in the period from 2016 to 2018. A possible Feynman diagram for this decay
is shown in Fig.~\ref{fig:feynman1}.
In order to avoid experimenter's bias, the results of the analysis were not examined until the full procedure had been finalised.
Two different selections are developed: selection A is optimised to maximise the hypothetical signal sensitivity and selection B is optimised for the production ratio measurement.
The analysis strategy is defined as follows:
\mbox{selection A} is first used to search for ${\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace$ signal
and evaluate its significance as a function of ${\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace$ mass.
If evidence for a signal with a global significance above 3 standard deviations after considering the look-elsewhere effect would be found, the mass would be measured and Selection B would be employed to measure the production cross-section of the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace baryon;
else, upper limits on the production ratio $R$ as a function of the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace mass for different lifetime hypotheses would be set. The production ratio $R$, relative to the ${\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace \ensuremath{\rightarrow}\xspace {\ensuremath{\Lz^+_\cquark}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace\pip$ decay, is defined as
\begin{equation}
\label{eq:RXicc}
R \equiv \frac{\sigma({\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace)\times{\ensuremath{\mathcal{B}}}\xspace({\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace \ensuremath{\rightarrow}\xspace
{\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace)\times{\ensuremath{\mathcal{B}}}\xspace({\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace \ensuremath{\rightarrow}\xspace
{\ensuremath{\Pp}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace)}{\sigma({\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace)\times{\ensuremath{\mathcal{B}}}\xspace({\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace \ensuremath{\rightarrow}\xspace {\ensuremath{\Lz^+_\cquark}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace\pip)\times{\ensuremath{\mathcal{B}}}\xspace({\ensuremath{\Lz^+_\cquark}}\xspace \ensuremath{\rightarrow}\xspace
{\ensuremath{\Pp}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace)},
\end{equation}
where $\sigma$ is the baryon production cross-section and
$\mathcal{B}$ is the branching fraction of the corresponding decays. Both the ${\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace$ and ${\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace$ baryons are required to be in the rapidity range of 2.0 to 4.5 and have transverse momentum between 4 and 15 \ensuremath{\aunit{Ge\kern -0.1em V\!/}c}\xspace.
The production ratio is evaluated as
\begin{equation}
\label{eq:alphaN}
R = \frac{\varepsilon_{\text{norm}}}{\varepsilon_{\text{sig}}}
\frac{N_{\text{sig}}}{N_{\text{norm}}}
\equiv \alpha N_{\text{sig}},
\end{equation}
where $\varepsilon_{\text{sig}}$ and $\varepsilon_{\text{norm}}$
refer to the efficiencies of the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace signal
and the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace normalisation decay mode, respectively,
$N_{\text{sig}}$ and $N_{\text{norm}}$ are the corresponding yields, and $\alpha$
is the single-event sensitivity.
The lifetime of the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace baryon is unknown and strongly affects the selection efficiency,
hence upper limits on $R$
are quoted as a function of the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace baryon mass
for a discrete set of lifetime hypotheses.
\section{Detector and simulation}%
\label{sec:Detector}
The \mbox{LHCb}\xspace detector~\cite{LHCb-DP-2008-001,LHCb-DP-2014-002} is
a single-arm forward
spectrometer covering the \mbox{pseudorapidity} range $2<\eta <5$,
designed for the study of particles containing {\ensuremath{\Pb}}\xspace or {\ensuremath{\Pc}}\xspace
quarks. The detector includes a high-precision tracking system
consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region~\cite{LHCb-DP-2014-001},
a large-area silicon-strip detector located
upstream of a dipole magnet with a bending power of about
$4{\mathrm{\,Tm}}$, and three stations of silicon-strip detectors and straw
drift tubes~\cite{LHCb-DP-2013-003,LHCb-DP-2017-001}
placed downstream of the magnet.
The tracking system provides a measurement of the momentum, \ensuremath{p}\xspace, of charged particles with
a relative uncertainty that varies from 0.5\% at low momentum to 1.0\% at 200\ensuremath{\aunit{Ge\kern -0.1em V\!/}c}\xspace.
The minimum distance of a track to a primary $pp$-collision vertex (PV), the impact parameter (IP),
is measured with a resolution of $(15+29/\ensuremath{p_{\mathrm{T}}}\xspace)\ensuremath{\,\upmu\nospaceunit{m}}\xspace$,
where \ensuremath{p_{\mathrm{T}}}\xspace is the component of the momentum transverse to the beam, in\,\ensuremath{\aunit{Ge\kern -0.1em V\!/}c}\xspace.
Different types of charged hadrons are distinguished using information
from two ring-imaging Cherenkov detectors\cite{LHCb-DP-2012-003}.
The online event selection is performed by a trigger~\cite{LHCb-DP-2012-004},
which consists of a hardware stage, based on information from the calorimeter and muon
systems, followed by a software stage, which applies a full event
reconstruction.
Simulated samples are required to develop the event selection and
to estimate the detector acceptance and the
efficiency of the imposed selection requirements.
Simulated $pp$ collisions are generated using
\mbox{\textsc{Pythia}}\xspace~\cite{Sjostrand:2007gs,*Sjostrand:2006za}
with a specific \mbox{LHCb}\xspace configuration~\cite{LHCb-PROC-2010-056}.
A dedicated generator, \mbox{\textsc{GenXicc2.0}}\xspace~\cite{Chang:2009va},
is used to simulate the doubly charmed baryon production.
Decays of unstable particles
are described by \mbox{\textsc{EvtGen}}\xspace~\cite{Lange:2001uf}, in which final-state
radiation is generated using \mbox{\textsc{Photos}}\xspace~\cite{davidson2015photos}. The
interaction of the generated particles with the detector, and its response,
are implemented using the \mbox{\textsc{Geant4}}\xspace
toolkit~\cite{Allison:2006ve, *Agostinelli:2002hh} as described in
Ref.~\cite{LHCb-PROC-2011-006}.
Unless otherwise stated,
simulated events are generated with an {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace ({\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace) mass of 3738\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace (3621\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace)
and a lifetime of 160\ensuremath{\aunit{fs}}\xspace (256\ensuremath{\aunit{fs}}\xspace).
\section{Reconstruction and selection}%
\label{sec:reconstruction_and_selection}
The {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace signal mode is reconstructed by combining a {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace candidate with kaon and pion candidates coming from the same vertex. The {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace candidates are firstly formed by combining three tracks originating from the same vertex, displaced with respect to the PV; at least one track is required to satisfy an inclusive software trigger based on a multivariate classifier~\cite{pmlr-v14-gulin11a, LHCb-PROC-2015-018}, and the three tracks must satisfy particle identification (PID) requirements to be compatible with a ${\ensuremath{\Pp}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace$ hypothesis. Then the {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace candidates with good vertex quality and invariant mass within the region of 2450 to 2486 \ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace are combined with two extra tracks, identified as {\ensuremath{\kaon^-}}\xspace and {\ensuremath{\pion^+}}\xspace , to reconstruct a {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace candidate. The {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace mass region is defined as $2468\pm18 \ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace$ where the mean value is the known {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace mass~\cite{PDG2020} and
the width is corresponding to three
times the mass resolution.
To improve further the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace signal purity, a multivariate classifier based on a boosted decision tree (BDT)~\cite{Breiman,Hocker:2007ht,*TMVA4} is developed to suppress combinatorial background.
The classifier is trained using simulated {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace events as signal and wrong-sign ${\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^-}}\xspace$ combinations in data with mass
in the interval 3600 to 4000\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace to represent background.
For \mbox{selection A}, no specific trigger requirement is applied. A multivariate selection is trained with two sets of variables which show good discrimination between {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace signal and background. The first set contains variables related to the reconstructed {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace candidates, including the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace decay vertex-fit quality, such as \ensuremath{\chi^2_{\text{IP}}}\xspace, the pointing angle and the flight-distance \ensuremath{\chi^2}\xspace. Here \ensuremath{\chi^2_{\text{IP}}}\xspace is the difference in \ensuremath{\chi^2}\xspace of the PV reconstructed with and without the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace candidate, the pointing angle is the three-dimensional angle between the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace
candidate momentum direction and the vector joining the PV and the reconstructed {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace decay vertex, while the flight-distance \ensuremath{\chi^2}\xspace is defined as the \ensuremath{\chi^2}\xspace of the hypothesis that the decay vertex of the candidate coincides with its associated PV. The second set adds variables related to the decay products ({\ensuremath{\Pp}}\xspace, {\ensuremath{\kaon^-}}\xspace and {\ensuremath{\pion^+}}\xspace from the {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace decay, and {\ensuremath{\kaon^-}}\xspace and {\ensuremath{\pion^+}}\xspace from the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace decay), including momentum, transverse momentum, \ensuremath{\chi^2_{\text{IP}}}\xspace and PID variables.
The threshold of the multivariate output is determined by maximising
the figure of merit $\varepsilon/\left(5/2 + \sqrt{N_B}\right)$~\cite{Punzi:2003bu}, where $\varepsilon$ is the estimated MVA selection efficiency,
$5/2$ corresponds to 5 standard deviations in a Gaussian significance test,
and $N_B$ is the expected number of background candidates
in the signal region, estimated with the wrong-sign ${\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^-}}\xspace$ combinations in the mass region of $\pm12.5\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace$ around the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace mass of 3738\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace used in the simulation, taking into account the difference of the background level for the signal sample and the wrong-sign sample.
After the multivariate selection, events may still contain more than one {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace candidate in the signal region although the probability to produce more than one {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace is small. The reconstructed {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace candidates could suffer from background from candidates reconstructed with clone tracks, i.e. reconstructed tracks sharing a large portion of their detector hits. Clone tracks could be included in a {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace candidate, when one is used for the {\ensuremath{\pion^+}}\xspace candidate from the {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace decay and its clone for the {\ensuremath{\pion^+}}\xspace candidate from the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace decay. To avoid that, candidates with the angle between each pair of identically charged tracks smaller than 0.5~mrad are removed. The {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace candidates could also be formed by the same five final tracks but with two tracks interchanged, \mbox{\itshape e.g.}\xspace the {\ensuremath{\kaon^-}}\xspace ({\ensuremath{\pion^+}}\xspace) candidate from the {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace decay is swapped with the {\ensuremath{\kaon^-}}\xspace ({\ensuremath{\pion^+}}\xspace) candidate from the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace decay. In this case, only one candidate is chosen randomly.
For \mbox{selection B},
the multivariate selection is similar to \mbox{selection A} except that the
PID variables of the {\ensuremath{\kaon^-}}\xspace and {\ensuremath{\pion^+}}\xspace candidates from the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace decay are not used in the training to ease the efficiency determination. Furthermore, an additional hardware trigger requirement is imposed on candidates for both the signal and the normalisation modes to minimise differences between data and simulation.
The data sets are split into two disjoint subsamples. One subsample is triggered on signals associated with one of the reconstructed {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace candidates with high transverse energy deposits in the calorimeters (TOS), and the other is triggered on signals exclusively unrelated to the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace candidate (exTIS).
The reconstruction and selection requirements of the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace normalisation mode are similar to those in the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{+}}}\xspace search~\cite{LHCb-PAPER-2019-029,LHCb-PAPER-2019-035}.
Both {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace and {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace candidates are required to be in the fiducial region of
rapidity $2.0<y<4.5$ and transverse momentum $4<\ensuremath{p_{\mathrm{T}}}\xspace<15\ensuremath{\aunit{Ge\kern -0.1em V\!/}c}\xspace$.
\section{Yield measurements}
\label{sec:yield}
\begin{figure}[tb]
\centering
\includegraphics[width=0.70\linewidth]{Figs/Fig2.pdf}
\caption{Invariant mass $m({\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace)$ distribution of selected {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace candidates from (black points) selection A, with (blue solid line) the fit with the largest local significance at the mass of 3876\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace superimposed.}
\label{fig:fit_shape}
\end{figure}
After applying \mbox{selection A} to the full data sample,
the invariant mass distribution $m({\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace)$ of selected ${\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace$ candidates is shown in Fig.~\ref{fig:fit_shape}.
To improve the mass resolution,
the variable
$m({\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace)$ is defined as the difference of the reconstructed mass of the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace and {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace candidates plus the known {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace mass~\cite{PDG2020}.
The $m({\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace)$ distribution is fitted with a sum of signal and background components, where the signal component is described by the sum of two Crystal Ball functions~\cite{Skwarnicki:1986xj} and the background component by a second-order Chebyshev function. The parameters of the signal shape are fixed from simulation, where the width is found to be around 5.5\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace. The parameters of background shape are obtained from a fit to the wrong-sign ${\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^-}}\xspace$ combinations. An unbinned maximum likelihood fit is performed with the peak position varied in steps of \\
2 \ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace, and the largest signal contribution is found for an {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace mass of 3876\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace.
The local significance of the signal peak is quantified with a $p$-value, which is calculated as the likelihood ratio of the background plus signal hypothesis and the background-only hypothesis~\cite{Wilks:1938dza,Narsky:2000}.
The local $p$-value is plotted in Fig.~\ref{fig:pvalue} as a function of mass, $m({\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace)$, showing a dip around 3876\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace, which has the largest local significance, corresponding to $3.2$
standard deviations. The global significance is evaluated with pseudoexperiments, by taking into account the look-elsewhere effect~\cite{Gross:2010qma} in the mass range from 3600\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace to 4000\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace, and is estimated to be $1.8$ standard deviations.
As no excess above 3 standard deviations is observed,
upper limits on the production ratios are set by using \mbox{selection B}. The invariant mass distribution of {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace candidates is shown in Fig.~\ref{fig:selectionB_fit} with the fit under the background-only hypothesis.
\begin{figure}[tb]
\centering
\includegraphics[width=0.7\linewidth]{Figs/Fig3.pdf}
\caption{Local $p$-value at different $m({\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace)$ values
evaluated with the likelihood ratio test. Lines indicating one, two and three standard deviations ($\sigma$) of local significance are also shown.}
\label{fig:pvalue}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=0.70\linewidth]{Figs/Fig4.pdf}
\caption{Invariant mass $m({\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace)$ distribution of selected {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace candidates (black points) with selection B, only background fit is shown.}
\label{fig:selectionB_fit}
\end{figure}
The measured production ratio is a function of single-event sensitivity $\alpha$ and $N_{\text{sig}}$, as shown in Eq.~\ref{eq:alphaN}. The parameter $\alpha$ is calculated using the yield of the normalisation mode $N_{\text{norm}}$ multiplied by the efficiency ratio between the normalisation and signal modes, while $N_{\text{sig}}$ is extracted by fitting the data of the signal mode.
The ${\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace$ yields, $N_{\text{norm}}$, are determined by performing an extended unbinned maximum likelihood fit to the invariant mass in the two trigger categories. The invariant mass distribution $m({\ensuremath{{\ensuremath{\PLambda}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace\pip)$ is defined as the difference of the reconstructed mass of the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace and {\ensuremath{\Lz^+_\cquark}}\xspace candidates plus the known {\ensuremath{\Lz^+_\cquark}}\xspace mass~\cite{PDG2020}.
For illustration,
the $m({\ensuremath{\Lz^+_\cquark}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace\pip)$ distributions for the 2018 data set
are shown in Fig.~\ref{fig:yield_xicc} together with the associated fit projections.
The mass shapes of the normalisation mode are a sum of a Gaussian function and a modified Gaussian function with power-law tails on both sides for signal and a second-order Chebyshev polynomial for background, which is the same as used in the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{+}}}\xspace search~\cite{LHCb-PAPER-2019-035}.
The {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace yields are summarised in Table~\ref{tab:yield_summary}, where the TOS refers to the trigger on signal and the exTIS refers to exclusive trigger independently of signal.
\begin{figure}[tb]
\centering
\includegraphics[width=0.98\linewidth]{Figs/Fig5.pdf}
\caption{Distribution of invariant mass $m({\ensuremath{\Lz^+_\cquark}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace\pip)$ for selected ${\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace$ candidates in different categories: (a)~triggered by one of the {\ensuremath{{\ensuremath{\PLambda}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace decay products and (b)~triggered exclusively by particles unrelated to the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace decay products, in the 2018 data set. The fit results are superimposed.}
\label{fig:yield_xicc}
\end{figure}
\begin{table}[bt]
\centering
\caption{Signal yields for the ${\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace\ensuremath{\rightarrow}\xspace{\ensuremath{\Lz^+_\cquark}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace\pip$ normalisation mode $N_{\text{norm}}$ for both trigger categories and different data-taking periods with the corresponding integrated luminosity $\mathcal{L}$. The uncertainties are statistical only.}
\label{tab:yield_summary}
\begin{tabular}{cccc}
\toprule
\multirow{2}*{Year} & \multirow{2}*{$\mathcal{L}$ $[\ensuremath{\fb^{-1}}\xspace]$} &\multicolumn{2}{c}{$N_{\text{norm}}$} \\
\cline{3-4}
~ & ~ & \multirow{1.2}*{TOS} & \multirow{1.2}*{exTIS}\\
\midrule
2016& 1.7& $126\pm21$ & $165\pm23$ \\
2017& 1.6& $145\pm21$ & $255\pm26$ \\
2018& 2.1& $164\pm21$ & $349\pm30$ \\
\bottomrule
\end{tabular}
\end{table}
\section{Efficiency ratio estimation}%
\label{sec:efficiency_ratio}
The efficiency ratio between the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace mode and {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace mode, defined as ${\ensuremath{\varepsilon}}\xspace_{\rm norm}/{\ensuremath{\varepsilon}}\xspace_{\rm sig}$, is determined from simulation,
where kinematic distributions are weighted to match those in data.
The tracking and PID efficiencies for both normalisation and signal modes are corrected using calibration data samples~\cite{LHCb-DP-2013-002,LHCb-PUB-2016-021,LHCb-DP-2018-001}.
The efficiency ratio of both trigger categories for different data-taking periods are summarised in Table~\ref{tab:eff_ratio}. Since there is an additional track in the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace decay when compared to the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace decay, the reconstruction and selection efficiency of {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace candidates is significantly lower. The increase in the efficiency ratio for the 2017 and 2018 data is due to the optimisation of the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace online selection, following the observation of the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace baryon~\cite{LHCb-PAPER-2017-018}.
\begin{table}[tb]
\centering
\caption{Efficiency ratios ${\ensuremath{\varepsilon}}\xspace_{\rm norm}/{\ensuremath{\varepsilon}}\xspace_{\rm sig}$ between normalisation and signal modes for both trigger categories for
different data-taking periods, where the TOS refers to the trigger on signal and the exTIS refers to exclusive trigger independently of signal. The uncertainties are statistical only.}
\label{tab:eff_ratio}
\begin{tabular}{ccc}
\toprule
\multirow{2}*{Year} & \multicolumn{2}{c}{${\ensuremath{\varepsilon}}\xspace_{\rm norm}/{\ensuremath{\varepsilon}}\xspace_{\rm sig}$} \\
\cline{2-3}
~ & \multirow{1.2}*{TOS} & \multirow{1.2}*{exTIS}\\
\midrule
2016& 0.32 $\pm$ 0.03 & 0.28 $\pm$ 0.02\\
2017& 0.55 $\pm$ 0.03 & 0.71 $\pm$ 0.02\\
2018& 0.61 $\pm$ 0.04 & 0.69 $\pm$ 0.02\\
\bottomrule
\end{tabular}
\end{table}
In order to take into account the dependence of the selection efficiency upon the unknown value of the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace lifetime, simulated {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace events
are weighted to reproduce different exponential decay time distributions corresponding to lifetimes of 40\ensuremath{\aunit{fs}}\xspace, 80\ensuremath{\aunit{fs}}\xspace, 120\ensuremath{\aunit{fs}}\xspace, 160\ensuremath{\aunit{fs}}\xspace , and 200\ensuremath{\aunit{fs}}\xspace. This method is used to estimate the change in the efficiency.
The single-event sensitivities are calculated by the ratio of {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace efficiency to the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace efficiency with different lifetime hypotheses, as shown in Tables~\ref{tab:alpha_TOS} and~\ref{tab:alpha_TIS}, for both trigger categories.
\begin{table}[tb]
\centering
\caption{Single-event sensitivity $\alpha({\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace)$ [$10^{-2}$] of the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace normalisation mode triggered by one of the {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace ({\ensuremath{{\ensuremath{\PLambda}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace) products for different lifetime hypotheses of the ${\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace$ baryon for different data-taking periods.
The uncertainties are due to the limited size of the simulated samples
and the statistical uncertainties on the measured {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace baryon yields.}
\label{tab:alpha_TOS}
\begin{tabular}{c rrrrr}
\toprule
\multirow{2}*{Year} & \multicolumn{5}{c}{$\alpha$ [$10^{-2}$]} \\
\cline{2-6}
~ & \multirow{1.2}*{$\tau=40\ensuremath{\aunit{fs}}\xspace$} & \multirow{1.2}*{$\tau=80\ensuremath{\aunit{fs}}\xspace$} & \multirow{1.2}*{$\tau=120\ensuremath{\aunit{fs}}\xspace$} &
\multirow{1.2}*{$\tau=160\ensuremath{\aunit{fs}}\xspace$} & \multirow{1.2}*{$\tau=200\ensuremath{\aunit{fs}}\xspace$}\\
\midrule
2016& 0.86 $\pm$ 0.17 & 0.46 $\pm$ 0.09 & 0.32 $\pm$ 0.06 & 0.25 $\pm$ 0.05 & 0.22 $\pm$ 0.04\\
2017& 1.29 $\pm$ 0.20 & 0.69 $\pm$ 0.11 & 0.48 $\pm$ 0.07 & 0.38 $\pm$ 0.06 & 0.33 $\pm$ 0.05\\
2018& 1.26 $\pm$ 0.18 & 0.67 $\pm$ 0.10 & 0.47 $\pm$ 0.07 & 0.37 $\pm$ 0.05 & 0.32 $\pm$ 0.05\\
\bottomrule
\end{tabular} \end{table}
\begin{table}[!tb]
\centering
\caption{Single-event sensitivity $\alpha({\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace)$ [$10^{-2}$] of the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace normalisation mode triggered exclusively by particles unrelated to the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace ({\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace) decay products for different lifetime hypotheses of the ${\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace$ baryon in the different data-taking periods.
The uncertainties are due to the limited size of the simulated samples
and the statistical uncertainty on the measured {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace baryon yield.}
\label{tab:alpha_TIS}
\begin{tabular}{c rrrrr}
\toprule
\multirow{2}*{Year} & \multicolumn{5}{c}{$\alpha$ [$10^{-2}$]} \\
\cline{2-6}
~ & \multirow{1.2}*{$\tau=40\ensuremath{\aunit{fs}}\xspace$} & \multirow{1.2}*{$\tau=80\ensuremath{\aunit{fs}}\xspace$} & \multirow{1.2}*{$\tau=120\ensuremath{\aunit{fs}}\xspace$} &
\multirow{1.2}*{$\tau=160\ensuremath{\aunit{fs}}\xspace$} & \multirow{1.2}*{$\tau=200\ensuremath{\aunit{fs}}\xspace$}\\
\midrule
2016& 0.71 $\pm$ 0.11 & 0.35 $\pm$ 0.06 & 0.22 $\pm$ 0.04 & 0.17 $\pm$ 0.03 & 0.14 $\pm$ 0.02\\
2017& 1.16 $\pm$ 0.12 & 0.57 $\pm$ 0.06 & 0.37 $\pm$ 0.04 & 0.28 $\pm$ 0.03 & 0.23 $\pm$ 0.02\\
2018& 0.82 $\pm$ 0.08 & 0.41 $\pm$ 0.04 & 0.26 $\pm$ 0.02 & 0.20 $\pm$ 0.02 & 0.17 $\pm$ 0.02\\
\bottomrule
\end{tabular} \end{table}
The {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace mass is also unknown. To test the effects of different mass hypotheses,
two simulated samples are generated with $m({\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace)=3638\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace$ and $m({\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace)=3838\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace$.
These samples are used to weight the \ensuremath{p_{\mathrm{T}}}\xspace distributions of final states in the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace decay
to match those in the other mass hypotheses, and the efficiency is recalculated with the weighted samples. When varying the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace mass, it is found that the efficiency is constant; therefore, the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace mass dependence is neglected in the evaluation of the single-event sensitivities.
\section{Systematic uncertainties}
The sources of systematic uncertainties on the production ratio $R$ are listed in Table~\ref{tab:sys_fit}, where individual sources are assumed to be independent and summed in quadrature to compute the total systematic uncertainty.
The choice of the mass models used to fit the invariant mass distribution affects the normalisation yields and therefore affects the calculation of single-event sensitivities. The related systematic uncertainty is studied by using alternative functions to describe the signal and background shapes of the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace mode.
The sum of two Gaussian functions is chosen as an alternative signal model and a second-order polynomial function is chosen to substitute the background model. The difference in the signal yields obtained by changing models is assigned as the systematic uncertainty.
The systematic uncertainty associated with the trigger efficiency is evaluated using a tag-and-probe method~\cite{LHCb-DP-2012-004}.
The size of the normalisation sample is insufficient to derive this systematic uncertainty. Instead, $b$-flavoured hadrons decaying with similar final-state topologies are used.
For the TOS category,
{\ensuremath{\Lz^0_\bquark}}\xspace\ensuremath{\rightarrow}\xspace{\ensuremath{\Lz^+_\cquark}}\xspace{\ensuremath{\pion^+}}\xspace{\ensuremath{\pion^-}}\xspace\pim and
{\ensuremath{\Lz^0_\bquark}}\xspace\ensuremath{\rightarrow}\xspace{\ensuremath{\Lz^+_\cquark}}\xspace{\ensuremath{\pion^-}}\xspace candidates can be
triggered by the energy deposit in the calorimeter by one of the {\ensuremath{\Lz^+_\cquark}}\xspace decay products, which are similar to
the ${\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace \ensuremath{\rightarrow}\xspace {\ensuremath{\Lz^+_\cquark}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace\pip$ and ${\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace \ensuremath{\rightarrow}\xspace {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace$ decays.
The efficiency ratio of these two {\ensuremath{\Lz^0_\bquark}}\xspace modes is estimated and the difference of the ratio between data and simulation is assigned as a systematic uncertainty.
For the exTIS category, the {\ensuremath{\B_\cquark^+}}\xspace\ensuremath{\rightarrow}\xspace{\ensuremath{{\PJ\mskip -3mu/\mskip -2mu\Ppsi\mskip 2mu}}}\xspace{\ensuremath{\pion^+}}\xspace decay, which has two heavy-flavour particles ($b$- and $c$-hadrons) and is similar to the signal topology, is used to study the trigger efficiency with particle candidates that are independent and unrelated to the signal. The systematic uncertainty for the exTIS trigger category is assigned as the difference in the efficiency ratio of {\ensuremath{\Lz^0_\bquark}}\xspace\ensuremath{\rightarrow}\xspace{\ensuremath{\Lz^+_\cquark}}\xspace{\ensuremath{\pion^+}}\xspace{\ensuremath{\pion^-}}\xspace\pim mode to {\ensuremath{\B_\cquark^+}}\xspace\ensuremath{\rightarrow}\xspace{\ensuremath{{\PJ\mskip -3mu/\mskip -2mu\Ppsi\mskip 2mu}}}\xspace{\ensuremath{\pion^+}}\xspace mode in data and in simulation.
The tracking efficiency is corrected with calibration data samples~\cite{LHCb-DP-2013-002}, and is affected by three sources of systematic uncertainties.
First, the inaccuracy of the simulation in terms of detector occupancy, which is assigned as 1.5\% and 2.5\% for kaons and pions, does not cancel in the ratio. An additional systematic uncertainty arises from the calibration method which provides a 0.8\% uncertainty per track~\cite{LHCb-DP-2013-002}. The third uncertainty is due to the limited size of the calibration samples and studied by pseudoexperiments. The tracking efficiency is corrected by the pseudoexperiments and the Gaussian width of the newly obtained distribution of the efficiency ratio is assigned as the systematic uncertainty.
The PID efficiency is determined in intervals of particle momentum,
pseudorapidity and event multiplicity using calibration data samples. The corresponding sources of systematic uncertainty are due to the limited size of the calibration samples and the binning scheme used. To study their effects, a large number of pseudoexperiments are performed, and the binning scheme is varied.
The {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace lifetime is measured with limited precision, $\ensuremath{256\,^{+24}_{-22}\,{\rm(stat)\,}\pm
14\,{\rm(syst)}\,\ensuremath{\aunit{fs}}\xspace}$~\cite{LHCb-PAPER-2018-019}, which is propagated to the systematic uncertainty in the efficiency.
As the agreeemnt between data and simulation is limited, a difference of 5.0\% is found among different periods of data-taking, which is taken as systematic uncertainty.
\begin{table}[tp]
\centering
\caption{Systematic uncertainties on the production ratio $R$.}
\label{tab:sys_fit}
\begin{tabular}{lr}
\toprule
Source & $R$~[\%] \\
\midrule
Fit model & 3.5 \\
Hardware trigger & 11.2 \\
Tracking & 2.7 \\
PID & 0.9 \\
{\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace lifetime & 12.0\\
Simulation/data difference & 5.0 \\
\midrule
Total & 17.7 \\
\bottomrule
\end{tabular}
\end{table}
\section{Results}%
\label{sec:results}
Upper limits on the production ratio $R$ are set with a simultaneous fit to the $m({\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace)$ distributions of different trigger categories for all the data sets from 2016 to 2018,
following the strategy described in Sec.~\ref{sec:yield} for the normalisation mode.
The upper limit values are calculated by setting different {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace mass hypotheses in the fit within the $m({\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace)$ mass range from 3600 to 4000\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace with a step of 2\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace, for five different lifetime hypotheses, 40\ensuremath{\aunit{fs}}\xspace, 80\ensuremath{\aunit{fs}}\xspace, 120\ensuremath{\aunit{fs}}\xspace, 160\ensuremath{\aunit{fs}}\xspace , and 200\ensuremath{\aunit{fs}}\xspace.
For each {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace mass and lifetime hypothesis,
the likelihood profile
is determined as a function of $R$.
It is then convolved with a Gaussian distribution whose width is equal to the square root of the quadratic combination of the
statistical and systematic uncertainties on the single-event sensitivity.
The upper limit at 95\% credibility level is defined as
the value of $R$ at which the integral of the profile likelihood equals 95\% of the total area. Figure~\ref{fig:mass_scan_tau_run2}
shows the 95\% credibility level upper limits at different mass hypotheses for five different lifetimes. The upper limits on $R$ decrease when increasing the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace lifetime. Considering the whole explored mass range, the highest upper limit on $R$ is $0.11$ obtained under lifetime hypothesis of 40 fs while the lowest is $0.5 \times 10^{-2}$ obtained under lifetime hypothesis of 200 fs.
\begin{figure}[tb]
\centering
\includegraphics[width=0.70\linewidth]{Figs/Fig6.pdf}
\caption{Upper limits on the production ratio $R$
at 95\% credibility level as a function of $m({\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace)$ at
$\sqrt{s}=13\aunit{Te\kern -0.1em V}\xspace$,
for five ${\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace$ lifetime hypotheses.}
\label{fig:mass_scan_tau_run2}
\end{figure}
\section{Conclusion}%
\label{sec:conclusion}
A search for the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace baryon through the {\ensuremath{{\ensuremath{\PXi}}\xspace^+_{\ensuremath{\Pc}}\xspace}}\xspace{\ensuremath{\kaon^-}}\xspace{\ensuremath{\pion^+}}\xspace decay is performed, using $pp$ collision data collected by the \mbox{LHCb}\xspace experiment
from 2016 to 2018 at a centre-of-mass energy of 13\aunit{Te\kern -0.1em V}\xspace, corresponding to an integrated luminosity of 5.4\ensuremath{\fb^{-1}}\xspace.
No significant signal is observed in the mass range of 3.6 to 4.0\ensuremath{\aunit{Ge\kern -0.1em V\!/}c^2}\xspace.
Upper limits are set at $95\%$ credibility level
on the ratio of the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace production cross-section times
the branching fraction to that of the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{++}}}\xspace baryon as a function of the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace mass and for different lifetime hypotheses,
in the rapidity range of 2.0 to 4.5 and the transverse momentum range of 4 to 15\ensuremath{\aunit{Ge\kern -0.1em V\!/}c}\xspace.
The upper limits depend strongly on the mass and lifetime hypotheses of the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace, and vary from $1.1 \times 10^{-1}$ to $0.5 \times 10^{-2}$ for 40\ensuremath{\aunit{fs}}\xspace to 200\ensuremath{\aunit{fs}}\xspace, respectively.
Future searches by the \mbox{LHCb}\xspace experiment with upgraded detector, improved trigger conditions, additional
${\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace$ decay modes, and larger data samples will further increase the {\ensuremath{\Omegares^{+}_{\cquark\cquark}}}\xspace signal sensitivity.
\section{Variation of efficiency with mass and lifetime}%
\label{sec:variation_of_efficiency_with_mass_and_lifetime}
The efficiency depends strongly on the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{+}}}\xspace lifetime,
due to the use of lifetime-related variables in the selection.
Unfortunately, the prediction of {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{+}}}\xspace lifetime has a large uncertainty.
Therefore, the upper limits vary under different lifetime hypotheses.
To take this effect into account, the efficiency is re-calculated under each
different lifetime hypothesis
(a discrete set of $40\ensuremath{\aunit{fs}}\xspace$, $80\ensuremath{\aunit{fs}}\xspace$, $120\ensuremath{\aunit{fs}}\xspace$, and $160\ensuremath{\aunit{fs}}\xspace$ is considered).
For the non-zero lifetime hypotheses,
the MC sample generated with $\tau_0=333\ensuremath{\aunit{fs}}\xspace$ is re-weighted
according to the TRUE decay time $t$ with the weight $w(t)$ defined as
$$ w(t) = \frac{\frac{1}{\tau}\text{exp}(-\frac{t}{\tau})}{\frac{1}{\tau_0}\text{exp}(-\frac{t}{\tau_0})}, $$
where $\tau$ is the lifetime hypothesis under study.
Hence, the efficiency for a given selection is
$$ \varepsilon = \frac{\sum_{\text{pass}} w_i}{\sum_{\text{before}} w_j}, $$
where the sum $i$ runs over the events that pass the selection and
$j$ runs over all the events before the selection.
For the zero lifetime hypothesis,
a dedicated MC sample generated with $\tau=0\ensuremath{\aunit{fs}}\xspace$ is used to calculate the efficiency.
The results are summarised in Tables~\ref{tab:eff_vary_tau} and \ref{tab:eff_vary_tau_12},
and are visualised in Fig.~\ref{fig:eff_vary_tau}.
A roughly linear dependence of the efficiency on lifetime hypotheses is observed.
\begin{table}
\centering
\caption{Efficiencies at different lifetime hypotheses for 2016 data.}
\label{tab:eff_vary_tau}
\input{tables/Efficiency_VaryTau_16_dalitz}
\end{table}
\begin{table}
\centering
\caption{Efficiencies at different lifetime hypotheses for 2012 data.}
\label{tab:eff_vary_tau_12}
\input{tables/Efficiency_VaryTau_12_dalitz}
\end{table}
\begin{figure}[tb]
\centering
\includegraphics[width=0.49\linewidth]{efficiency/Efficiency_VaryTau.pdf}
\includegraphics[width=0.49\linewidth]{efficiency/Efficiency_VaryTau_12.pdf}
\caption{Efficiencies at different lifetime hypotheses for (left) 2016 and (right) 2012 data.}
\label{fig:eff_vary_tau}
\end{figure}
The kinematics of final tracks depend on the mass of {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{+}}}\xspace baryons.
Therefore, the efficiency may vary as a function of the mass.
To take into account this effect, a weighting procedure is developed.
Firstly, produce generator-level MC samples with different {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{+}}}\xspace mass hypotheses
(here the values of $3518.7\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace$ and $3700.0\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace$ are considered.)
Secondly, re-weight the full-simulated sample (generated with $m_0 = 3621.4\ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace$)
according to the generator-level \ensuremath{p_{\mathrm{T}}}\xspace distributions of {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{+}}}\xspace secondaries ({\ensuremath{\Lz^+_\cquark}}\xspace, {\ensuremath{\kaon^-}}\xspace and {\ensuremath{\pion^+}}\xspace from {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{+}}}\xspace),
the efficiency is then re-calculated at each mass hypothesis.
The weights are acquired with the \texttt{GBReweightor} from the \texttt{hep\_ml} python module,
a reweightor algorithm based on ensemble of regression trees~\cite{gbreweight:1}.
The results are shown in Table~\ref{tab:eff_vary_mass} and visualized in Fig.~\ref{fig:eff_vary_mass}.
The dependence of the total efficiency on the {\ensuremath{{\ensuremath{\PXi}}\xspace_{{\ensuremath{\Pc}}\xspace\cquark}^{+}}}\xspace mass hypotheses is weak in a wide range.
We tend to not include this effect when calculating the upper limits at different mass hypotheses.
\begin{table}[tb]
\centering
\caption{Efficiencies at different mass hypotheses.}
\label{tab:eff_vary_mass}
\input{tables/Efficiency_VaryMass}
\end{table}
\begin{figure}[tb]
\centering
\includegraphics[width=0.6\linewidth]{efficiency/Efficiency_VaryMass.pdf}
\caption{Efficiency as a function of mass hypotheses.}
\label{fig:eff_vary_mass}
\end{figure}
\section{Typography}
\label{sec:typography}
The use of the \mbox{\LaTeX}\xspace typesetting symbols defined in the file
\texttt{lhcb-symbols-def.tex} and detailed in the appendices of this
document is strongly encouraged as it will make it much easier to
follow the recommendation set out below.
\begin{enumerate}
\item \mbox{LHCb}\xspace is typeset with a normal (roman) lowercase b.
\item Titles are in bold face, and usually only the first word is
capitalised.
\item Mathematical symbols and particle names should also be typeset
in bold when appearing in titles.
\item Units are in roman type, except for constants such as $c$ or $h$
that are italic: \aunit{Ge\kern -0.1em V}\xspace, \ensuremath{\aunit{Ge\kern -0.1em V\!/}c^2}\xspace. The unit should be separated from
the value with a thin space (``\verb!\,!''),
and they should not be broken over two lines.
Correct spacing is automatic when using predefined units inside math mode: \verb!$3.0\aunit{Ge\kern -0.1em V}\xspace$! $\ensuremath{\rightarrow}\xspace 3.0\aunit{Ge\kern -0.1em V}\xspace$.
Spacing goes wrong when using predefined units outside math mode AND forcing extra space:
\verb!3.0\,\aunit{Ge\kern -0.1em V}\xspace! $\ensuremath{\rightarrow}\xspace$ 3.0\,\aunit{Ge\kern -0.1em V}\xspace or worse: \verb!3.0~\aunit{Ge\kern -0.1em V}\xspace! $\ensuremath{\rightarrow}\xspace$ 3.0~\aunit{Ge\kern -0.1em V}\xspace.
\item If factors of $c$ are kept, they should be used both for masses and
momenta, \mbox{\itshape e.g.}\xspace $p=5.2\ensuremath{\aunit{Ge\kern -0.1em V\!/}c}\xspace$ (or $\aunit{Ge\kern -0.1em V}\xspace c^{-1}$), $m = 3.1\ensuremath{\aunit{Ge\kern -0.1em V\!/}c^2}\xspace$ (or $\aunit{Ge\kern -0.1em V}\xspace c^{-2}$). If they are dropped this
should be done consistently throughout, and a note should be added
at the first instance to indicate that units are taken with $c=1$.
Note that there is no consensus on whether decay widths $\Gamma$ are
in \aunit{Me\kern -0.1em V}\xspace or \ensuremath{\aunit{Me\kern -0.1em V\!/}c^2}\xspace (the former is more common).
Both are accepted if consistent.
\item The \% sign should not be separated from the number that precedes it: 5\%, not 5 \%.
A thin space is also acceptable: 5\,\%, but should be applied consistently throughout the paper.
\item Ranges should be formatted consistently. The recommended form is to use a dash with no spacing around it:
7--8\aunit{Ge\kern -0.1em V}\xspace, obtained as \verb!7--8\aunit{Ge\kern -0.1em V}\xspace!. Another possibility is ``7 to 8\aunit{Ge\kern -0.1em V}\xspace''.
\item Italic is preferred for particle names (although roman is
acceptable, if applied consistently throughout). Particle Data
Group conventions should generally be followed: {\ensuremath{\B^0}}\xspace (no need for a
``d'' subscript), \decay{{\ensuremath{\B^0_\squark}}\xspace}{{\ensuremath{{\PJ\mskip -3mu/\mskip -2mu\Ppsi\mskip 2mu}}}\xspace\phi}, {\ensuremath{\Bbar{}^0_\squark}}\xspace,
(note the long bar, obtained with \verb!\overline!, in contrast to the discouraged short \verb!\bar{B}! resulting in $\bar{B}$), {\ensuremath{\kaon^0_{\mathrm{S}}}}\xspace (note the
uppercase roman type ``S'').
This is most easily achieved by using the predefined symbols described in
Appendix~\ref{sec:listofsymbols}.
Italic is also used for particles whose name is an uppercase Greek letter:
{\ensuremath{\PUpsilon}}\xspace, {\ensuremath{\PDelta}}\xspace, {\ensuremath{\PXi}}\xspace, {\ensuremath{\PLambda}}\xspace, {\ensuremath{\PSigma}}\xspace, {\ensuremath{\POmega}}\xspace, typeset as
\verb!{\ensuremath{\PUpsilon}}\xspace!, \verb!{\ensuremath{\PDelta}}\xspace!, \verb!{\ensuremath{\PXi}}\xspace!, \verb!{\ensuremath{\PLambda}}\xspace!, \verb!{\ensuremath{\PSigma}}\xspace!, \verb!{\ensuremath{\POmega}}\xspace! (or with the appropriate macros adding charge and subscripts). Paper titles in the bibliography must be adapted accordingly.
Note that the {\ensuremath{\PLambda}}\xspace baryon has no zero, while the {\ensuremath{\Lz^0_\bquark}}\xspace baryon has one. That's historical.
\item Unless there is a good reason not to, the charge of a particle should be
specified if there is any possible ambiguity
($m({\ensuremath{\kaon^+}}\xspace{\ensuremath{\kaon^-}}\xspace)$ instead of $m(KK)$, which could refer to neutral kaons).
\item Decay chains can be written in several ways, depending on the complexity and the number of times it occurs. Unless there is a good reason not to, usage of a particular type should be consistent within the paper.
Examples are:
\decay{{\ensuremath{\D^+_\squark}}\xspace}{\phi{\ensuremath{\pion^+}}\xspace}, with \decay{\phi}{{\ensuremath{\kaon^+}}\xspace{\ensuremath{\kaon^-}}\xspace};
\decay{{\ensuremath{\D^+_\squark}}\xspace}{\phi{\ensuremath{\pion^+}}\xspace} (\decay{\phi}{{\ensuremath{\kaon^+}}\xspace{\ensuremath{\kaon^-}}\xspace});
\decay{{\ensuremath{\D^+_\squark}}\xspace}{\phi(}{{\ensuremath{\kaon^+}}\xspace{\ensuremath{\kaon^-}}\xspace){\ensuremath{\pion^+}}\xspace}; or
\decay{{\ensuremath{\D^+_\squark}}\xspace}{[{\ensuremath{\kaon^+}}\xspace{\ensuremath{\kaon^-}}\xspace]_\phi{\ensuremath{\pion^+}}\xspace}.
\item Variables are usually italic: $V$ is a voltage (variable), while
1 V is a volt (unit). Also in combined expressions: $Q$-value, $z$-scale, $R$-parity \mbox{\itshape etc.}\xspace
\item Subscripts and superscripts are roman type when they refer to a word (such as T
for transverse) and italic when they refer to a variable (such as
$t$ for time): \ensuremath{p_{\mathrm{T}}}\xspace, {\ensuremath{\Delta m_{{\ensuremath{\Ps}}\xspace}}}\xspace, $t_{\mathrm{rec}}$.
\item Standard function names are in roman type: \mbox{\itshape e.g.}\xspace $\cos$, $\sin$
and $\exp$.
\item Figure, Section, Equation, Chapter and Reference should be
abbreviated as Fig., Sect. (or alternatively Sec.), Eq., Chap.\ and
Ref.\ respectively, when they refer to a particular (numbered) item,
except when they start a sentence. Table and Appendix are not
abbreviated. The plural form of abbreviation keeps the point after
the s, \mbox{\itshape e.g.}\xspace Figs.~1 and~2. Equations may be referred to either with
(``Eq.~(1)'') or without (``Eq.~1'') parentheses,
but it should be consistent within the paper.
\item Common abbreviations derived from Latin such as ``for example''
(\mbox{\itshape e.g.}\xspace), ``in other words'' (\mbox{\itshape i.e.}\xspace), ``and so forth'' (\mbox{\itshape etc.}\xspace), ``and
others'' (\mbox{\itshape et al.}\xspace), ``versus'' (\mbox{\itshape vs.}\xspace) can be used, with the typography
shown, but not excessively; other more esoteric abbreviations should be avoided.
\item Units, material and particle names are usually lower case if
spelled out, but often capitalised if abbreviated: amps (A), gauss
(G), lead (Pb), silicon (Si), kaon ({\ensuremath{\PK}}\xspace), but proton ({\ensuremath{\Pp}}\xspace).
\item Counting numbers are usually written in words if they start a
sentence or if they have a value of ten or below in descriptive
text (\mbox{\itshape i.e.}\xspace not including figure numbers such as ``Fig.\ 4'', or
values followed by a unit such as ``4\,cm'').
The word 'unity' can be useful to express the special meaning of
the number one in expressions such as:
``The BDT output takes values between zero and unity''.
\item Numbers larger than 9999 have a small space between
the multiples of thousand: \mbox{\itshape e.g.}\xspace 10\,000 or 12\,345\,678. The decimal
point is indicated with a point rather than a comma: \mbox{\itshape e.g.}\xspace 3.141.
\item We apply the rounding rules of the
PDG~\cite{PDG2018}. The basic rule states that if the three
highest order digits of the uncertainty lie between 100 and 354, we round
to two significant digits. If they lie between 355 and 949, we round
to one significant digit. Finally, if they lie between 950 and 999,
we round up and keep two significant digits. In all cases,
the central value is given with a precision that matches that of the
uncertainty. So, for example, the result $0.827 \pm 0.119$ should be
written as $0.83\pm 0.12$, $0.827\pm 0.367$ should turn into
$0.8\pm 0.4$, and $14.674\pm0.964$ becomes $14.7\pm1.0$.
When writing numbers with uncertainty components from
different sources, \mbox{\itshape i.e.}\xspace statistical and systematic uncertainties, the rule
applies to the uncertainty with the best precision, so $0.827\pm
0.367\aunit{(stat)}\xspace\pm 0.179\aunit{(syst)}\xspace$ goes to $0.83\pm 0.37\aunit{(stat)}\xspace\pm 0.18\aunit{(syst)}\xspace$ and
$8.943\pm 0.123\aunit{(stat)}\xspace\pm 0.995\aunit{(syst)}\xspace$ goes to $8.94\pm 0.12\aunit{(stat)}\xspace\pm
1.00\aunit{(syst)}\xspace$.
\item When rounding numbers, it should be avoided to pad with zeroes
at the end. So $51237 \pm 4561$ should be rounded as $(5.12 \pm 0.46)
\times 10^4$ rather than $51200 \pm 4600$. Zeroes are accepted for yields.
\item When rounding numbers in a table, some variation of the rounding
rules above may be required to achieve uniformity.
\item Hyphenation should be used where necessary to avoid ambiguity,
but not excessively. For example: ``big-toothed fish'' (to indicate
that big refers to the teeth, not to the fish), but ``big white
fish''. A compound modifier often requires hyphenation
({\ensuremath{C\!P}}\xspace-violating observables, {\ensuremath{\Pb}}\xspace-hadron decays, final-state
radiation, second-order polynomial), even if the same combination in
an adjective-noun combination does not (direct {\ensuremath{C\!P}}\xspace violation, heavy
{\ensuremath{\Pb}}\xspace hadrons, charmless final state). Adverb-adjective
combinations are not hyphenated if the adverb ends with 'ly':
oppositely charged pions, kinematically similar decay. Words
beginning with ``all-'', ``cross-'', ``ex-'' and ``self-'' are
hyphenated \mbox{\itshape e.g.}\xspace\ cross-section and cross-check. ``two-dimensional''
is hyphenated. Words beginning with small prefixes (like ``anti'',
``bi'', ``co'', ``contra'', ``counter'', ``de'', ``extra'', ``infra'',
``inter'', ``intra'', ``micro'', ``mid'', ``mis'', ``multi'', ``non'', ``over'',
``peri'', ``post'', ``pre'', ``pro'', ``proto'', ``pseudo'', ``re'', ``semi'',
``sub'', ``super'', ``supra'', ``trans'', ``tri'', ``ultra'', ``un'', ``under'' and
``whole'') are single words and should not be hyphenated
\mbox{\itshape e.g.}\xspace\ semileptonic, pseudorapidity, pseudoexperiment, multivariate,
multidimensional, reweighted,\footnote{Note that we write weighted unless it's the second weighting} preselection, nonresonant, nonzero,
nonparametric, nonrelativistic, antiparticle,
misreconstructed and misidentified.
\item Minus signs should be in a proper font ($-1$), not just hyphens
(-1); this applies to figure labels as well as the body of the text.
In \mbox{\LaTeX}\xspace, use math mode (between \verb!$$!'s) or make a dash (``\verb!--!'').
In ROOT, use \verb!#minus! to get a normal-sized minus sign.
\item Inverted commas (around a title, for example) should be a
matching set of left- and right-handed pairs: ``Title''. The use of
these should be avoided where possible.
\item Single symbols are preferred for variables in equations, \mbox{\itshape e.g.}\xspace\
{\ensuremath{\mathcal{B}}}\xspace\ rather than BF for a branching fraction.
\item Parentheses are not usually required around a value and its
uncertainty, before the unit, unless there is possible ambiguity: so
\mbox{${\ensuremath{\Delta m_{{\ensuremath{\Ps}}\xspace}}}\xspace = 20 \pm 2\ensuremath{\ps^{-1}}\xspace$} does not need parentheses,
whereas \mbox{$f_d = (40 \pm 4)$\%} or \mbox{$x=(1.7\pm0.3)\times 10^{-6}$} does.
The unit does not need to be repeated in
expressions like \mbox{$1.2 < E < 2.4\aunit{Ge\kern -0.1em V}\xspace$}.
\item The same number of decimal places should be given for all values
in any one expression (\mbox{\itshape e.g.}\xspace \mbox{$5.20 < m_B < 5.34\ensuremath{\aunit{Ge\kern -0.1em V\!/}c^2}\xspace$}).
\item Apostrophes are best avoided for abbreviations: if the abbreviated term
is capitalised or otherwise easily identified then the plural can simply add
an s, otherwise it is best to rephrase: \mbox{\itshape e.g.}\xspace HPDs, pions, rather
than HPD's, {\ensuremath{\pion^0}}\xspace's, ${\ensuremath{\Ppi}}\xspace$s.
\item Particle labels, decay descriptors and mathematical functions are not nouns, and need often to be followed by a noun.
Thus ``background from \decay{{\ensuremath{\B^0}}\xspace}{{\ensuremath{\pion^+}}\xspace{\ensuremath{\pion^-}}\xspace} decays'' instead of ``background from \decay{{\ensuremath{\B^0}}\xspace}{{\ensuremath{\pion^+}}\xspace{\ensuremath{\pion^-}}\xspace}'',
and ``the width of the Gaussian function'' instead of ``the width of the Gaussian''.
\item In equations with multidimensional integrations or differentiations, the differential terms should be separated by a thin space and the $\rm d$ should be in roman.
Thus $\int f(x,y) {\rm d}x\,{\rm d}y$ instead $\int f(x,y) {\rm d}x{\rm d}y$ and
$\frac{{\rm d}^2\Gamma}{{\rm d}x\,{\rm d}Q^2}$ instead of $\frac{{\rm d}^2\Gamma}{{\rm d}x{\rm d}Q^2}$.
\item Double-barrelled names are typeset with a hyphen (\verb!-!), as in Gell-Mann, but joined named use an n-dash (\verb!--!), as in Breit--Wigner.
\item Avoid gendered words. Mother is rarely needed. Daughter can be a decay product or a final-state particle. Bachelor can be replaced by companion.
\end{enumerate}
| 22,922 |
\section{Introduction}
The problem of completing a low rank matrix by sampling only a few of its entries is a well studied problem which finds its application in variety of areas a few of which include Euclidean distance matrix completion \cite{localizationedm}, environmental monitoring using sensors \cite{environmental} \cite{lowcostwsnmc}, array signal processing \cite{arraysignal}, beamforming \cite{beamforming} and wireless channel estimation \cite{mimomc}. In \cite{exact} the authors showed that its possible to perfectly complete low rank matrix by observing only a few entries and by solving a convex optimization problem.
The generalized low rank matrix completion problem in which a number of linear combinations of its entries are measured as opposed to sampling the entries directly has also attracted a lot of attention. The problem of reconstructing finite dimensional quantum states is naturally a generalized matrix completion problem \cite{gross}, \cite{sevenqubit}. In \cite{arm}, the affine rank minimization problem which minimizes the rank of the matrix with affine constraints was proposed for solving this generalized matrix completion. The work in \cite{lowrankanybasisgross} quantified the number of measurements required for the success of matrix recovery from its linear measurements. In \cite{powerfactorization} authors proposed the power factorization algorithm for generalized matrix completion and the idea was further extended in \cite{als}. In \cite{saresepowerfactorization}, the authors extended the power factorization algorithm to estimate sparse low rank matrix whose left and right singular matrices are sparse. The recovery of positive semidefinite matrix in the context of generalized matrix completion has been analyzed in \cite{rankonemeas} and \cite{psdrlm}. The singular value thresholding (SVT) \cite{svt} algorithm is a very popular algorithm used for performing matrix completion and generalized matrix completion.
The capability of computing machines to store and process a huge amount of data and the capability of neural networks to learn very complicated functions have enabled the deep networks to find its roots in almost all fields. However, it has been very hard to interpret these deep neural networks. In this context, unrolled algorithms which are deep neural networks whose architecture is inspired from interpretable classical algorithms have attracted considerable attention recently \cite{unrollingsurvey}. We present a deep neural network inspired from SVT algorithm for general matrix completion which reconstructs matrix with significantly lower MSE and is more robust to the parameters that need to be carefully chosen in SVT.
\subsection{Our contributions and the outline of work}
In this paper, we design a trainable deep neural network to perform affine rank minimization by unrolling the SVT algorithm. Each layer of the deep network is similar to a single iteration of the SVT algorithm except that the parameters such as measurement matrices, the step sizes, threshold values used in the SVT are now learnable. We term the proposed method Learned SVT (LSVT). The advantages of the proposed method are as follows. Firstly, the proposed LSVT outperforms SVT meaning that our network with $T$ layers reconstructs the matrix with lesser mean squared error (MSE) compared with the MSE incurred by SVT with fixed (same $T$) number of iterations. Secondly, LSVT seems more robust to the initialization than SVT in all our empirical results.
\vspace{-3mm}
\subsection{Notations used}
Matrices, vectors and scalars are represented by upper case, bold lower case and lower case respectively. $\tr[A], \norm{A}_F$ and $\norm{A}_{tr}$ denote trace, Frobenius norm and nuclear norm of matrix $A$ respectively where nuclear norm is the sum of singular values of $A$. The $i^{th}$ element of $\bm b$ is denoted by $b_i$ and $\norm{\bm b}_2$ denotes the Euclidean norm of $\bm b$.
\section{Problem formulation} \label{sec:problem}
Let $X \in \mathbb{R}^{d \times d}$ be the true matrix to be recovered and $r$ be the rank of $X$. Note that, in general, we need $d^2$ measurements of the matrix $X$ to get the complete information about the matrix $X$. This holds when the matrix is of full rank. But the low rank structure enables the recovery of $X$ from its fewer than $d^2$ measurements. For a rank-$r$ matrix, the degree of freedom reduces from $d^2$ to $r \times (2d-r)$. Let $\{A_i \in \mathbb{R}^{d \times d}\}_{i=1}^{m}$ be a set of $m$ ($m<d^2$) measurement matrices such that $\tr[A_i^T A_j] = \delta_{ij}$. Let $\bm b \in \mathbb{R}^m$ be $m$ linear measurements of the matrix $X$ which are given as
\begin{equation}
b_i = \tr[A_i X], \quad \forall \, i = 1, \dots m \label{eqn:meas}
\end{equation}
In other words, $\bm b$ is the linear function of the unknown matrix $X$ where the linear map $\mathcal{A}: \mathbb{R}^{d \times d} \to \mathbb{R}^m$ (also called the sampling operator) is defined as
\begin{equation}
\mathcal{A}(X) = \begin{bmatrix} \tr[A_1 X] & \tr[A_2 X] & \dots & \tr[A_m X] \end{bmatrix}^T \label{eqn:lmap}
\end{equation}
The problem that we consider in this paper is to recover the matrix $X$ from the measurements $\bm b$ assuming that the measurement matrices are known. The matrix recovery is done by solving the affine rank minimization problem given as
\begin{subequations}
\begin{alignat}{2}
&\!\min_{X \in \mathbb{R}^{d \times d}} &\qquad& \text{Rank}(X) \\
&\text{such that} & & \mathcal{A}(X)=\bm b
\end{alignat} \label{eqn:arm}
\end{subequations}
where $\mathcal{A}$ is the linear map as defined in \eqref{eqn:lmap}. The affine rank minimization problem \eqref{eqn:arm} minimizes the rank of the matrix within an affine constraint set $\{X: \mathcal{A}(X)=\bm b\}$ which is a level set of the linear map $\mathcal{A}$.
We design a deep neural network inspired from SVT \cite{svt} algorithm to solve the affine rank minimization formulated in \eqref{eqn:arm}. First, we present the SVT algorithm in the next section then in subsequent sections we design a neural network by unrolling the iterations of the SVT algorithm.
\begin{figure*}[ht!]
\hspace{9mm}
\begin{subfigure}[H]{.5\textwidth}
\centering
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\draw (171.38,185.73) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] [align=left] {Hidden layer - $\displaystyle t$};
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\draw (244.3,152.94) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] [align=left] {$\displaystyle \mathcal{D}_{\textcolor[rgb]{0.82,0.01,0.11}{\tau }\textcolor[rgb]{0.82,0.01,0.11}{_{t}}}( .) \ $};
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\end{tikzpicture}
\caption{Hidden layers of the Learned SVT network. Two hidden layers connected sequentially are depicted.}
\label{fig:hiddenlayers}
\end{subfigure} \\
\hspace{12mm}
\begin{subfigure}[H]{.5\textwidth}
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
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\draw (458,109) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] [align=left] {$\displaystyle X_{out}$};
\draw (289,149) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] [align=left] {Output layer};
\draw (170,82) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] [align=left] {$\displaystyle b$};
\end{tikzpicture}
\caption{Output layer of the Learned SVT network.}
\label{fig:outputlayer}
\end{subfigure}
\hspace{1mm}
\begin{subfigure}[H]{.5\textwidth}
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
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\begin{center}
Hidden \\layer-1
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\begin{center}
Hidden \\layer-T
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\draw (73.48,66.43) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] [align=left] {$\displaystyle \textcolor{red}{\delta_0} b$};
\draw (293.7,59.34) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] [align=left] {$\displaystyle X_{out}$};
\draw (249.94,110.75) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] [align=left] {\begin{minipage}[lt]{26.123356pt}\setlength\topsep{0pt}
\begin{center}
Output \\layer
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\end{minipage}};
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\end{tikzpicture}
\caption{The architechture of the LSVT network}
\label{fig:architechture}
\end{subfigure}
\caption{Deep neural network for the Learned SVT. The variables in red are learnable parameters.}
\label{fig:DNN}
\end{figure*}
\section{SVT algorithm} \label{sec:svt}
SVT algorithm solves affine rank minimization formulated in \eqref{eqn:arm} by minimizing the nuclear norm of the matrix which is a surrogate to the rank function. Specifically, the convex optimization problem that the SVT solves is given as \cite{svt}
\begin{subequations}
\begin{alignat}{2}
&\!\min_{X \in \mathbb{R}^{d \times d}} &\qquad& \tau \norm{X}_{tr} + \frac{1}{2} \norm{X}_F^2 \\
&\text{such that} & & \mathcal{A}(X)=\bm b
\end{alignat} \label{eqn:svtproblem}
\end{subequations} where $\tau>0$ is a constant and the map $\mathcal{A}$ is as defined in \eqref{eqn:lmap}. In theorem 3.1 of \cite{svt}, authors proved that in the limit $\tau$ tending to infinity, the solutions to \eqref{eqn:svtproblem} converge to the matrix with minimum trace norm that is also consistent with the measurements i.e., ($\mathcal{A}(X) = \bm b$). In \cite{arm} authors proved that minimizing the nuclear norm yields the minimum rank solution with high probability (see theorem 3.3 and theorem 4.2 of \cite{arm}). Hence SVT minimizes rank formulated in \eqref{eqn:arm} by solving \eqref{eqn:svtproblem}.
Note that the problem \eqref{eqn:svtproblem} is a contrained convex optimization problem and the strong duality holds for the problem \cite{boyd}. The Lagrangian function for the problem is given as
\begin{equation}
\mathcal{L}(X, \bm y) = \tau \norm{X}_{tr} + \frac{1}{2} \norm{X}_F^2 + \bm y^T (\bm b - \mathcal{A}(X))
\end{equation}
where $X$ is the optimization variable and $\bm y$ is the Lagrangian variable. Since strong duality holds for the problem, finding the saddle point $(X^*, \bm y^*)$ of the Lagrangian $\mathcal{L}(X, \bm y)$ gives the solution ($X^*$) to \eqref{eqn:svtproblem}. The saddle point is written as
\begin{equation}
\underset{\bm y}{\sup} \, \underset{X}{\inf} \, \mathcal{L}(X, \bm y) = \mathcal{L}(X^*,\bm y^*) = \underset{X}{\inf} \, \underset{\bm y}{\sup} \, \mathcal{L}(X, \bm y)
\end{equation}
Authors in \cite{svt} used Uzawa's iterations \cite{uzawa} to find the saddle point of the Lagrangian. Uzawa's iterations starts with an initial $\bm y^0$ and repeats two steps until convergence. In the first step, the minimizer (say $X^k$) of $\mathcal{L}(., \bm y)$ for the given $\bm y$ is found. In the second step, a gradient ascent step is taken along the direction $\bm y$ for the given $X^k$ found in the previous step. These steps are given as follows
\begin{align}
\begin{cases}
X^k &= \underset{X}{\arg \min} \, \mathcal{L}(X, \bm y^{k-1}) \\
\bm y^k &= \bm y^{k-1} + \delta_k \nabla_{\bm y} \mathcal{L}(X^k , \bm y)
\end{cases} \label{eqn:uzawaiterations}
\end{align} where $\delta_k > 0$ is the step size used for gradient ascent in the $k^{th}$ iteration. The gradient $\nabla_{\bm y} \mathcal{L}(X^k , \bm y)$ is given as $\mathcal{A}(X^k) - \bm b$. The closed form solution to the minimization problem in \eqref{eqn:uzawaiterations} is given as
\begin{equation}
\underset{X}{\arg \min} \, \mathcal{L}(X, \bm y^{k-1}) = \mathcal{D}_{\tau}( \mathcal{A}^*(\bm y^{k-1})) \label{eqn:minalongX}
\end{equation}
where the operator $\mathcal{A}^*:\mathbb{R}^m \to \mathbb{R}^{d \times d}$ is the adjoint of the linear map $\mathcal{A}$ and is defined as
\begin{equation}
\mathcal{A}^*(\bm y) = \sum_{i=1}^m y_i A_i^T \label{eqn:adjoint}
\end{equation} and the operator $\mathcal{D}_\tau:\mathbb{R}^{d \times d} \to \mathbb{R}^{d \times d}$ is the singular value thresholding operator and is defined as
\begin{equation}
\mathcal{D}_{\tau}(X) = U \mathcal{D}_{\tau}(\Sigma)V^T \label{eqn:dtau}
\end{equation}
where the singular value decomposition of $X$ is given as $X = U \Sigma V^T$, $\Sigma$ is the diagonal matrix with the singular values of $X$ in its diagonal positions which we write as $\Sigma = \text{diag}(\sigma_1(X), \dots \sigma_r(X))$ where $r$ represents the rank of $X$. $\mathcal{D}_{\tau}(\Sigma)$ is a diagonal matrix whose non zero entries are found by soft thresolding the entries of $\Sigma$ and is given as $\mathcal{D}_{\tau}(\Sigma) = \text{diag}(\max(\sigma_1(X) - \tau, 0), \dots \max(\sigma_r(X) - \tau, 0))$.
Using the gradient of $\mathcal{L}$ along $\bm y$ and the equations \eqref{eqn:minalongX}, \eqref{eqn:adjoint} and \eqref{eqn:dtau} Uzawa's iterations \eqref{eqn:uzawaiterations} can be rewritten as
\begin{align}
\begin{cases}
X^k &= \mathcal{D}_{\tau}( \mathcal{A}^*(\bm y^{k-1}))\\
\bm y^k &= \bm y^{k-1} + \delta_k
(\bm b - \mathcal{A}(X^k))
\end{cases} \label{eqn:uzawaiterationsfinal}
\end{align}
where $\delta_k > 0$ is the stepsize to do gradient ascent, $\tau > 0$ is the threshold value used in singular value thresholding operator and the maps $\mathcal{A}$ and $\mathcal{A}^*$ are defined using the measurement matrices $A_1$ through $A_m$ (given in \eqref{eqn:lmap} and \eqref{eqn:adjoint}).
SVT algorithm repeatedly performs the steps in \eqref{eqn:uzawaiterationsfinal} which requires the knowledge of the right choice of $\delta_k$ and $\tau$ to solve the affine rank minimization problem \eqref{eqn:arm}. In the subsequent sections, we design a trainable deep network based on the steps in \eqref{eqn:uzawaiterationsfinal} for performing affine rank minimization that demands no such tunable parameters.
\section{Learned SVT} \label{sec:LSVT}
In this section, we design a deep neural network based on the SVT algorithm discussed in the previous section and present a training method to train the network for performing affine rank minimization.
\subsection{Network Architecture} \label{sec:architecture}
Recall that in each iteration of SVT \eqref{eqn:uzawaiterationsfinal}, two steps are performed. We first design a single hidden layer of our network which performs these two steps. To do so, we see that the steps \eqref{eqn:uzawaiterationsfinal} can be rewritten as
\begin{equation}
\bm y_t = \bm y_{t-1} + \delta_t \left[\bm b - \mathcal{A}(\mathcal{D}_\tau(\mathcal{A}^*(\bm y_{t-1}))) \right] \label{eqn:svtunrolledeqn}
\end{equation} where $\bm y_t$ would represent the output of the $t^{th}$ hidden layer. It can be seen from this reformulation that performing \eqref{eqn:svtunrolledeqn} $T$ times is equivalent to running the SVT algorithm for $T$ iterations. We design the deep network wherein each hidden layer perform \eqref{eqn:svtunrolledeqn} \footnote{$\mathcal{D}_{\tau}(.)$ in \eqref{eqn:svtunrolledeqn} is the singular value thresholding (SVT) operator. Note that the recent work on RPCA \cite{RPCA} also designed a neural network which uses SVT in its layers and backpropagates through it}. To obtain a matrix as the output, we add an output layer at the end of the network which performs the first step of \eqref{eqn:uzawaiterationsfinal}.
To unroll a fixed number (say $T$) of SVT iterations, we build $T-1$ hidden layers and an output layer and connect them sequentially. Hence, a $T$ layered unrolled network and the SVT algorithm that runs for a fixed $T$ number of iterations are comparable and they are one and the same when the maps $\mathcal{A}, \mathcal{A}^*$ and the parameters $\delta_t$, $\tau_t$ used in both algorithms are same.
We denote by $W_t$ the measurements matrices $A_1$ through $A_m$ used in the maps $\mathcal{A}$ and $\mathcal{A}^*$ and make it learnable. Here $W_t$ is an $m \times d^2$ matrix whose $i^{th}$ row is formed from the entries of $A_i$ such that the measurement vector $\bm b$ given in \eqref{eqn:meas} can be written as $W_t vec(X)$. By making this pair $\{\mathcal{A},\mathcal{A}^*\}$ learnable we try to leverage the power of deep learning to obtain an unrolled variant of SVT which performs better than SVT (which has fixed known pair of $\{\mathcal{A}, \mathcal{A}^*\}$. We denote by $\delta_t$ the step size used in the $t^{th}$ layer and by $\tau_t$ the threshold used in the $t^{th}$ layer and we also make these learnable. With these learnable parameters, the complete network architecture is depicted in Fig. \ref{fig:DNN} where Fig. \ref{fig:hiddenlayers} depicts the hidden layers of our unrolled network and Fig. \ref{fig:outputlayer} depicts the output layer of our network.
\subsection{Training the network} \label{sec:training}
Consider a Learned SVT network with $T$ layers as discussed in previous subsection. The input to this network is $\bm y_0 \in \mathbb{R}^m$ and the output of the network is a matrix $\hat X \in \mathbb{R}^{d \times d}$. From \eqref{eqn:svtunrolledeqn} it can be seen that if $\bm y_{0}$ was a zero vector, $\bm y_1$ would then be $\delta \bm b$. Hence we feed the network with $\bm y_0 = \delta_0 \bm b$ to recover the corresponding matrix $X$. Note that we can also feed the network with zero vector and compare its performance with SVT whose input is also zero vector. We denote by $\Theta$ the set of all learnable parameters in the network i.e., $\Theta = \{ W_1, \dots W_{T}, \delta_0, \delta_1, \dots \delta_{T}, \tau_1, \dots \tau_T \}$. With this notation, the output of the network for a given measurement vector (say $\bm b$) is written as
\begin{equation}
\hat X = f_{\Theta}(\bm b)
\end{equation}
We denote the training dataset by $\{X^{(i)}, \bm b^{(i)}\}_{i=1}^M$ where $\bm b^{(i)} \in \mathbb{R}^m$ is the measurement vector of the matrix $X^{(i)} \in \mathbb{R}^{d \times d}$ obtained using the known measurement matrices $A_1$ through $A_m$ as described in \eqref{eqn:meas}. For our numerical simulations, which we discuss in the next section, we generate the measurement matrices and $\{X^{(i)}\}$ synthetically. Then we obtain $\{\bm b^{(i)}\}$ using \eqref{eqn:lmap} as $\bm b^{(i)} = \mathcal{A}(X^{(i)} ; A_1, \dots A_m)$. We train our network to minimize the mean squared error (MSE) between the matrices $\{X^{(i)}\}$ in the training dataset and the estimated matrices $f_{\Theta}(\bm b^{(i)})$. The MSE loss is given as
\begin{equation}
\ell \{ \Theta;\{X^{(i)}, \bm b^{(i)}\} \} = \frac{1}{M} \sum_{i=1}^M \norm{ X^{(i)} - f_{\Theta}(\bm b^{(i)}) }_F^2 \label{eqn:mse}
\end{equation}
\begin{table*}[ht!]
\centering
\resizebox{13cm}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\textbf{\begin{tabular}[c]{@{}c@{}}Iterations/\\ Layers\end{tabular}} & \multicolumn{2}{c|}{\textbf{2}} & \multicolumn{2}{c|}{\textbf{3}} & \multicolumn{2}{c|}{\textbf{4}} & \multicolumn{2}{c|}{\textbf{5}} & \multicolumn{2}{c|}{\textbf{6}} \\ \hline
\textbf{Rank(r)} & \textbf{SVT} & \textbf{LSVT} & \textbf{SVT} & \textbf{LSVT} & \textbf{SVT} & \textbf{LSVT} & \textbf{SVT} & \textbf{LSVT} & \textbf{SVT} & \textbf{LSVT} \\ \hline
1 & 2.3166 & \textbf{0.2234} & 0.8655 & \textbf{0.0687} & 0.4269 & \textbf{0.0300} & 0.2240 & \textbf{0.0174} & 0.1460 & \textbf{0.0158} \\ \hline
2 & 4.0590 & \textbf{0.1594} & 1.2707 & \textbf{0.0268} & 0.4320 & \textbf{0.0086} & 0.1826 & \textbf{0.0053} & 0.0969 & \textbf{0.0013} \\ \hline
3 & 5.3233 & \textbf{0.4035} & 1.8576 & \textbf{0.0913} & 0.8466 & \textbf{0.0370} & 0.4532 & \textbf{0.0184} & 0.2748 & \textbf{0.0107} \\ \hline
\end{tabular}
}
\caption{MSE in estimating 10 x 10 matrix by SVT and LSVT for different layers.}
\label{tab:mse10}
\end{table*}
\begin{table*}[ht!]
\centering
\resizebox{13cm}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\textbf{\begin{tabular}[c]{@{}c@{}}Iterations/\\ Layers\end{tabular}} & \multicolumn{2}{c|}{\textbf{2}} & \multicolumn{2}{c|}{\textbf{3}} & \multicolumn{2}{c|}{\textbf{4}} & \multicolumn{2}{c|}{\textbf{5}} & \multicolumn{2}{c|}{\textbf{6}} \\ \hline
\textbf{Rank(r)} & \textbf{SVT} & \textbf{LSVT} & \textbf{SVT} & \textbf{LSVT} & \textbf{SVT} & \textbf{LSVT} & \textbf{SVT} & \textbf{LSVT} & \textbf{SVT} & \textbf{LSVT} \\ \hline
2 & 4.6028 & \textbf{0.5565} & 1.4991 & \textbf{0.1630} & 0.7064 & \textbf{0.0693} & 0.3738 & $\mathbf{0.2690}^@$ & 0.2538 & $\mathbf{0.1364}^@$ \\ \hline
4 & 8.3921 & \textbf{0.4007} & 2.5227 & \textbf{0.0784} & 0.8701 & \textbf{0.0893} & 0.3832 & \textbf{0.0942} & 0.2224 & \textbf{0.0097} \\ \hline
6 & 10.9441 & \textbf{1.008} & 3.9163 & \textbf{0.2498} & 1.7994 & \textbf{0.2904} & 0.9914 & \textbf{0.0471} & 0.6328 & \textbf{0.0564} \\ \hline
\end{tabular}
}
\caption{MSE in estimating 20 x 20 matrix by SVT and LSVT for different layers.}
\label{tab:mse20}
\end{table*}
\subsection{Initialization and training}
We initialize the network's trainable parameters $\Theta$ with those used in SVT algorithm. We initialize each of $\{W_t\}$ with the measurement matrices $A_1$ through $A_m$. Each of the stepsizes $\{\delta_0, \delta_1, \dots \delta_T\}$ is initialized to $1.2 \times \frac{d^2}{m}$ and each of the thresholds $\{\tau_1, \dots \tau_T\}$ is initializes to $5 \times d$ as these values were adapted by the authors of SVT for better results. Since we initialize our network with the parameters used in SVT algorithm, our networks performs similar to SVT (with fixed $T$ number of iterations) initially. We also initialize the parameters $\{\delta_0, \delta_1, \dots \delta_T \dots \tau_1 \dots \tau_T\}$ with values other than the ones used by authors of SVT, to see the dependence of SVT and LSVT on these parameters. The initial error that the network incurs is same as that incurred by SVT. We use gradient based optimizer ADAM \cite{adam} to minimize the loss \eqref{eqn:mse} and train the network.
\section{Numerical simulation} \label{sec:simulations}
We design and train the proposed Learned SVT network to estimate $d \times d$ matrices of rank $r$ from its $m$ linear measurements. The linear map $\mathcal{A}$ that defines the linear measurements is synthetically generated and is fixed through out the experiments. To do this we randomly generate the matrices $A_1$ through $A_m$ such that $\tr[A_i^T A_j] = \delta_{ij}$. We generate 61,000 ground truth data $\{X^{(i)} \in \mathbb{R}^{d \times d}, \bm b^{(i)} \in \mathbb{R}^m \}_{i=1}^{61,000}$ where $\{X^{(i)}\}$ are of rank-r and $\{\bm b^{(i)}\}$ are the corresponding measurement vectors. To do this, we generate $P^{(i)} \in \mathbb{R}^{d \times r}$ and $Q^{(i)} \in \mathbb{R}^{r \times d}$ randomly with each entry of $P^{(i)}$ and $Q^{(i)}$ generated from $\mathcal{N}(0,2)$ then we get $X^{(i)}$ by multiplying $P^{(i)}$ and $Q^{(i)}$. $\bm b^{(i)}$ are obtained by measuring $X^{(i)}$ using the measurement map generated. We use PyTorch to design and train our LSVT network. PyTorch automatically computes the gradients of the loss function with respect to the network parameters using computational graphs and Autograd functionality. We use stochastic gradient descent based ADAM optimizer with a learning rate of $10^{-4}$ to train the network. We perform mini batch training with a minibatch size of 1000. Of the 61,000 ground truth data, we use 50,000 to train the network, 10,000 to validate the network. We validate the network every time the network parameter gets updated and we stop the training process when this validation loss doesn't decrease over the course of training. Once the training is over, we freeze the network parameters and test the network with the remaining 1000 ground truth data and the resulted MSE are reported.
LSVT is designed and trained to estimate $10 \times 10$ matrices of rank 1, 2 and 3 with different layers. We sampled the rank-1 matrix with a oversampling ratio of $3$. For rank-2 and rank-3 matrix, 90 linear measurements were obtained. The MSE in estimating the matrices by both SVT and LSVT algorithms are compared in Table \ref{tab:mse10}. We also estimated $20 \times 20$ matrices of ranks 2, 4 and 6. Rank-2 matrices were sampled with oversampling ratio 3. 350 linear measurements were obtained for rank-4 and rank-6 matrices. The corresponding MSE values are compared in Table \ref{tab:mse20}. The learning rate of $10^{-4}$ is used to train all the networks except for the values super-scripted by $@$ where $10^{-5}$ was used. All MSE values are reported by averaging over 1000 instances. It can be seen from Tables \ref{tab:mse10} and \ref{tab:mse20} that the Learned LSVT performs much better in terms of MSE in estimating the matrices compared with the original SVT algorithm.
To study the dependence of SVT and LSVT on the threshold value ($\tau$) and stepsize $(\delta)$ both algorithms are simulated with different thresholds ($\tau$) and stepsizes ($\delta$) to estimate $20 \times 20$ matrices of rank-2 with a oversampling ratio of 3. SVT uses these parameters and performs 4 SVT iterations, while 4-layered LSVT is initialized with these parameters and trained. The corresponding MSE values are tabulated in Table \ref{tab:svtlsvtmultiparas} where the $*$ denoted values are the ones used by the authors of SVT. From Table \ref{tab:svtlsvtmultiparas} it can be seen that even with different initial parameters ($\tau ,\delta$) LSVT always performs better than SVT in terms of MSE.
\begin{table}[h!]
\centering
\resizebox{6cm}{!}{
\begin{tabular}{|c|c|c|}
\hline
\textbf{Parameters} & \textbf{SVT} & \textbf{LSVT} \\ \hline
$\tau = 5, \delta = 1$ & 2.2477 & \textbf{0.1831} \\ \hline
$\tau = 50, \delta = 0.5$ & 6.4043 & \textbf{0.0479} \\ \hline
$\tau = 50, \delta = 2.10^*$ & 0.2536 & \textbf{0.0483} \\ \hline
$\tau = 100^*, \delta = 2.10^*$ & 0.7064 & \textbf{0.0693} \\ \hline
$\tau = 200, \delta = 5$ & 7.5677 & \textbf{0.4287} \\ \hline
$\tau = 300, \delta = 5$ & 8.1711 & \textbf{1.2766} \\ \hline
\end{tabular}
}
\caption{Comparing MSE in estimating matrices by SVT and LSVT when different parameters are used.}
\label{tab:svtlsvtmultiparas}
\end{table}
\vspace{-5mm}
\section{Conclusion} \label{sec:conclusion}
We designed a trainable deep neural network called LSVT by appropriately unrolling the SVT algorithm. The proposed LSVT with fixed $T$ layers estimates matrices with significantly lesser mean squared error (MSE) compared with MSE incurred by SVT with fixed $T$ iterations. We also showed that LSVT outperforms SVT even when both algorithms use different values for the step sizes and thresholds rather than those suggested by the authors of SVT \cite{svt}.
| 18,135 |
\section{Introduction}
Despite of the amazing success of the Standard Model (SM) of particle physics,
new physics (NP) beyond the SM is ineluctable because compelling evidence showing that at least two active neutrinos are massive\cite{Zyla:2020zbs}.
In addition to the nonzero neutrino masses, several recent experimental measurements prominently deviated from the SM predictions are also suggestive to NP.
In particular,
\bi
\item
The measured anomalous magnetic moment of muon $(g-2)_\mu$\cite{Zyla:2020zbs, Bennett:2006fi} differs from the most recent SM prediction\cite{ Blum:2018mom} by an amount of $\sim 3.7\sigma$:
\beq
\tri a_\mu =a_\mu^{exp} -a_\mu^{SM} \simeq (27.4\pm7.3)\times 10^{-10}\,,
\eeq
where the uncertainty is the quadratic combination of the experimental and theoretical ones.
The most recent $0.46 ppm$ measurement conducted at Fermilab\cite{Abi:2021gix} gives
\beq
a_\mu^{exp}= 116 592 040(54)\times 10^{-11}\,,
\eeq
which agrees with the previous measurements.
And the new experimental average of $a_\mu^{exp}= 116 592 061(41) \times 10^{-11}$
drives the deviation to $4.2\sigma$.
Another new measurement at the J-PARC\cite{Saito:2012zz}
is also expected to improve the experimental uncertainty in the near future.
\item
With the new determination of the fine-structure constant\cite{Parker:2018vye}, the measured electron $(g-2)_e$ \cite{Hanneke:2010au} shows a $\sim 2.4 \sigma$ discrepancy from the SM prediction\cite{Aoyama:2017uqe}:
\beq
\tri a_e =a_e^{exp} -a_e^{SM} \simeq (-8.7\pm 3.6)\times 10^{-13}\,.
\eeq
Note that $\tri a_e$ and $\tri a_\mu$ have opposite signs. New models\cite{Liu:2018xkx, Crivellin:2018qmi, Endo:2019bcj, Bauer:2019gfk, Badziak:2019gaf, Abdullah:2019ofw, Hiller:2019mou, Cornella:2019uxs, Haba:2020gkr, Bigaran:2020jil, Jana:2020pxx, Calibbi:2020emz, Yang:2020bmh, Chen:2020jvl, Hati:2020fzp, Dutta:2020scq, Chen:2020tfr,Chun:2020uzw,Li:2020dbg,Dorsner:2020aaz,Keung:2021rps} have been constructed to accommodate both $\tri a_e$ and $\tri a_\mu$.
Moreover, \cite{Arbelaez:2020rbq,Jana:2020joi,Escribano:2021css} also attempt to incorporate the neutrino mass generation with the observed $\tri a_{e,\mu}$.
\item
From the global fits\cite{Capdevila:2017bsm, Altmannshofer:2017yso, DAmico:2017mtc, Hiller:2017bzc, Ciuchini:2017mik, Geng:2017svp, Hurth:2017hxg, Alok:2017sui, Alguero:2019ptt, Aebischer:2019mlg, Ciuchini:2019usw} to various $b\ra s l^+ l^-$ data\cite{Aaij:2019wad, Aaij:2017vbb, Abdesselam:2019wac, Abdesselam:2019lab, Aad:2014fwa,Aaij:2015esa, Wei:2009zv, Aaltonen:2011ja, Khachatryan:2015isa, Abdesselam:2016llu, Aaij:2015oid, Wehle:2016yoi, Sirunyan:2017dhj, Aaboud:2018krd, Lees:2015ymt}, the discrepancy is more than $5\sigma$ from the SM predictions. The new result\cite{Aaij:2021vac} further strengthens the lepton flavor universality violation.
This anomaly alone convincingly indicates NP, and stimulates many investigations to address this deviation.
For example, a new gauge sector was introduced in \cite{Capdevila:2020rrl, Altmannshofer:2019xda, Gauld:2013qja},
leptoquark has been employed in \cite{Bauer:2015knc, Angelescu:2018tyl, Crivellin:2019dwb, Fuentes-Martin:2020bnh, Saad:2020ucl, Balaji:2019kwe}, assisted by the 1-loop contributions from exotic particles in \cite{Gripaios:2015gra, Hu:2019ahp, Arnan:2016cpy, Arnan:2019uhr},
and more references can be found in \cite{Capdevila:2017bsm, Li:2018lxi, Bifani:2018zmi}.
\item
The so-called Cabibbo angle anomaly refers to the unexpected shortfall in the first row Cabibbo-Kobayashi-Maskawa(CKM) unitarity\cite{Zyla:2020zbs},
\beq
|V_{ud}|^2+|V_{us}|^2+|V_{ub}|^2 = 0.9985\pm 0.0005\,.
\label{eq:PDGCKM}
\eeq
The above value is smaller than one, and the inconsistence is now at the level of $\simeq 2-4 \sigma$ level\cite{Grossman:2019bzp, Seng:2020wjq}.
There are tensions among different determinations of the Cabibbo angle from tau decays\cite{Amhis:2019ckw}, kaon decays\cite{Aoki:2019cca}, and super-allowed $\beta$ decay (by using CKM unitarity and the theoretical input\cite{Seng:2018yzq, Czarnecki:2019mwq}).
The potential NP involving vector quarks or the origins of lepton flavor universality violation have been discussed in \cite{Belfatto:2019swo, Cheung:2020vqm, Crivellin:2020oup, Coutinho:2019aiy, Crivellin:2020lzu, Crivellin:2020ebi, Kirk:2020wdk, Alok:2020jod}.
\ei
Whether these $2-5\, \sigma$ anomalies will persist is not predictable; the future improvement on the theoretical predictions and the experimental measurements will be the ultimate arbiters.
However, at this moment, it is interesting to speculate whether all the above mentioned anomalies and the neutrino mass can be explained simultaneously.
In this paper, we point out one of such resolutions.
With the addition of three scalar leptoquarks, $T(3,3,1/3)$, $D(3,2,1/6)$, and $S(3,1,2/3)$, and a pair of vector fermion, $b'_{L,R}(3,1,-1/3)$, the plethoric new parameters ( mostly the Yukawa couplings) allow one to reconcile all data contemporarily.
Parts of the particle content of this model had been employed in the past to accommodate some of the anomalies.
However, to our best knowledge, this model as whole is new to comprehensively interpret all the observed deviations from the SM.
The paper is laid out as follows: in Sec.\ref{sec:model} we spell out the model and the relevant ingredients. Following that in Sec.\ref{sec:howitworks} we explain how each anomaly and the neutrino mass generation works in our model. Next we discuss the various phenomenological constraint and provide some model parameter samples in Sec.\ref{sec:pheno} . In Sec.\ref{sec:discussion} we discuss some phenomenological consequences, and the UV origin of the flavor pattern of the parameter space. Then comes our conclusion in Sec.\ref{sec:conclusion}. Some details of our notation and the low energy effective Hamiltonian are collected in the Appendix.
\section{The Model}
\label{sec:model}
In this model, three scalar leptoquarks, $T(3,3,1/3)$, $D(3,2,1/6)$, and $S(3,1,2/3)$\footnote{ In the literature\cite{Buchmuller:1986zs}, the corresponding notations for $D(3,2,1/6)$ and $S(3,1,2/3)$
are $\wt{R}_2$ and $(\bar{S}_1)^*$, respectively. The one closely related to our $T(3,3,1/3)$ is $S_3(\bar{3},3,1/3)$. }, and a pair of down-quark-like vector fermion, $b'_{L,R}(3,1,-1/3)$, are augmented on top of the SM. Our notation for the SM particle content and the exotics are listed in Tab.\ref{table:SMparticle} and Tab.\ref{table:newparticle}, respectively.
\begin{table
\begin{center}
\begin{tabular}{|c||ccccc|c|}\hline
& \multicolumn{5}{c|}{ SM Fermion} & \multicolumn{1}{c|}{ SM Scalar} \\\hline
Fields & $Q_L= \begin{pmatrix} u_L \\ d_L\end{pmatrix}
$ &$u_R$ &$d_R$ & $L_L=\begin{pmatrix} \nu_L \\ e_L\end{pmatrix}$
&$e_R$ & $ H=\begin{pmatrix} H^+ \\ H^0\end{pmatrix}$
\\\hline
$SU(3)_c$ & $3$ &$3$ &$3$ & $1$ & $1$ & $1$ \\%\hline
$SU(2)_L$ & $2$ & $1$ & $1$ & $2$ & $1$ & $2$ \\%\hline
$U(1)_Y$ & $\frac{1}{6}$ & $\tth$ &$-\ot$ & $-\frac{1}{2}$ &$-1$ & $\frac{1}{2}$ \\\hline
\end{tabular}
\caption{The SM field content and quantum number assignment under the SM gauge symmetries $SU(3)_c\otimes SU(2)_L \otimes U(1)_Y $, where $L,R$ stand for the chirality of the fermion. For simplicity, all the generation indices associated with the fermions are suppressed. }
\label{table:SMparticle}
\end{center}
\end{table}
\begin{table
\begin{center}
\begin{tabular}{|c||c|ccc|}\hline
& \multicolumn{1}{c|}{ New Fermion} & \multicolumn{3}{c|}{ New Scalar}\\\hlin
Fields & $b'_{L,R}$
& $T=\begin{pmatrix} T^\tth \\ T^{-\ot} \\ T^{-\ft}\end{pmatrix}$
& $D=\begin{pmatrix} D^\tth \\ D^{-\ot}\end{pmatrix}$ & $S^{\tth}$
\\\hlin
$SU(3)_c$ & $3$ & $3$ & $3$ & $3$ \\%\hline
$SU(2)_L$ & $1$ & $3$ & $2$ & $1$ \\%\hline
$U(1)_Y$ & $-\ot$ & $-\ot$ & $\frac{1}{6}$ & $\tth$\\
lepton number & $0$ & $1$ & $-1$ & $-1$ \\
baryon number & $\ot$ & $\ot$ & $\ot$ & $\ot$\\
\hline
\end{tabular}
\caption{New field content and quantum number assignment under the SM gauge symmetries $SU(3)_c\otimes SU(2)_L \otimes U(1)_Y $, and the global lepton/baryon numbers. }
\label{table:newparticle}
\end{center}
\end{table}
Like most models beyond the SM, the complete Lagrangian is lengthy, and not illuminating. In this work, we only focus on the new gauge invariant interactions relevant to addressing the flavor anomalies.
For simplicity, we also assume the model Lagrangian respects the global baryon number symmetry, and both $T$ and $S$ carry one third of baryon-number to avoid their possible di-quark couplings.
Moreover, we do not consider the possible CP violating signals in this model.
For the scalar couplings, we have\footnote{To simplify the notation, we use ``$\{,\}$'' and ``$[,]$'' to denote the $SU(2)_L$ triplet and singlet constructed from two given $SU(2)_L$ doublets, respectively. Also, ``$\odot$'' means forming an $SU(2)_L$ singlet from two given triplets; see Appendix for the details.}
\beqa
{\cal L} &\supset& \mu_3 \left\{H ,\tilde{D} \right\}\odot T+ \mu_1 \left[ H ,D \right] S^{-\tth} +H.c.\\
&=& \mu_3\left[ H^+ D^{\ot}T^{-\ft} -\frac{1}{\sqrt{2}}\left(H^0D^{\ot}-H^+D^{-\tth}\right)T^{-\ot}-H^0D^{-\tth}T^{\tth}\right]\nonr\\
&& - \mu_1 \frac{1}{\sqrt{2}} \left(H^0 D^{\tth}- H^+D^{-\ot}\right)S^{-\tth}+H.c.
\eeqa
The couplings $\mu_1$ and $\mu_3$ are unknown dimensionful parameters.
Note that the $\mu_3$ coupling softly breaks the global lepton number by two units, which is crucial for the neutrino mass generation.
On the other hand, the lepton-number conserving $\mu_1$ triple scalar interaction is essential for explaining $\tri a_e$ and $\tri a_\mu$ ( to be discussed in the following sections).
As it will be clear later, to fit all the data, $\mu_3$ turns out to be very small, $\sim {\cal O}(0.2\mkev)$, and $\mu_{1}\sim {\cal O}(\mtev)$.
After electroweak spontaneous symmetry breaking (SSB), $\langle H^0 \rangle = v_0/\sqrt{2}$ and the Goldstone $H^\pm$ are eaten by the $W^\pm$ bosons. Below the electroweak scale, it becomes:
\beq
-\frac{\mu_3 v_0}{2} D^{\ot} T^{-\ot} -\frac{\mu_3 v_0}{\sqrt{2}} D^{-\tth} T^{\tth} - \frac{\mu_1 v_0}{2} D^{\tth} S^{-\tth} +H.c.
\eeq
Comparing to their tree-level masses, $\widetilde{M}_{T,D,S} \simeq M_{LQ}$\footnote{ Our notation is ${\cal L} \supset -\tilde{M}^2_T T^\dag T -\tilde{M}^2_D D^\dag D -\tilde{M}^2_S S^\dag S $. In order to preserve the $SU(3)_c$ symmetry, $T,D,S$ cannot develop nonzero vacuum expectation values. }, we expect the mixings are small and suppressed by the factor of ${\cal O}( \mu_{LQ} v_0/M_{LQ}^2)$. However, these mixings break the isospin multiplet mass degeneracy of $T$ and $D$. After the mass diagonalization, we have two charge-$\ot$, three charge-$\tth$, and one charge-$\ft$ physical scalar leptoquarks.
In addition to the SM Yukawa interactions in the form of $\bar{Q}d_R H$, $\bar{Q}u_R \tilde{H}$, and $\bar{L}e_R H$, this model has the following new Yukawa couplings ( in the interaction basis):
\beqa
{\cal L} &\supset& -\widetilde{\lambda}_T T^\dag\cdot \left\{\bar{L}^c , Q \right\} - \widetilde{\lambda}_D \bar{d}_R\left[ L ,D \right] - \widetilde{\lambda}'_D \bar{b'}_R\left[ L ,D \right
- \widetilde{\lambda}_S \bar{e}_R b'_L S^{-\tth} -\widetilde{Y}_d' \bar{Q}b'_R H +H.c.\\
&=& -\widetilde{\lambda}_T \left[ \bar{\nu}^c u_L T^{-\tth}+ \left(\bar{\nu}^c d_L +\bar{e}^c u_L\right)\frac{ T^{\ot} }{\sqrt{2}}+ \bar{e}^c d_L T^{\ft}\right] - \widetilde{Y}_d' (\bar{u}_L H^++\bar{d}_L H^0) b'_R \nonr\\
&&- \widetilde{\lambda}_D \frac{ \bar{d}_R}{\sqrt{2}} \left(\nu_L D^{-\ot}-e_L D^{\tth}\right)
- \widetilde{\lambda}'_D \frac{ \bar{b'}_R}{\sqrt{2}} \left(\nu_L D^{-\ot}-e_L D^{\tth}\right
- \widetilde{\lambda}_S \bar{e}_R b'_L S^{-\tth} +H.c.\,,
\label{eq:LQYukawa_coupling}
\eeqa
where all the generation indices are suppressed to keep the notation simple and it should be understood that all the Yukawa couplings are matrices.
Moreover, the model allows two kinds of tree-level Dirac mass term:
\beq
{\cal L} \supset M_1 \bar{b}'_R b'_L + M_2 \bar{d}_R b'_L + H.c.
\eeq
With the introduction of $b'$, the mass matrix for down-quark-like fermions after the electroweak SSB becomes:
\beq
{\cal L} \supset - (\bar{d_R},\bar{b'_R}) {\cal M}^d \left(\begin{array}{c} d_L\\ b'_L\end{array} \right)+H.c.\,,\,\,
{\cal M}^d = \left(
\begin{array}{cc}
\frac{\widetilde{Y}_d v_0}{\sqrt{2}} & M_2 \\
\frac{\widetilde{Y}'_d v_0}{\sqrt{2}} & M_1 \\
\end{array}
\right)\,,
\eeq
where $\widetilde{Y}_d$ is the SM down-quark three-by-three Yukawa coupling matrix in the interaction basis. Note that ${\cal M}^d$ is now a four-by-four matrix.
This matrix can be diagonalized by the bi-unitary transformation, ${\cal M}^d_{diag}=U^d_R {\cal M}^d (U^d_L)^\dag=\mbox{diag}(m_d,m_s,m_b,M_{b'})$, and
\beqa
&&U^d_R {\cal M}^d ({\cal M}^d)^\dag (U^d_R)^\dag =
U^d_L ({\cal M}^d)^\dag{\cal M}^d (U^d_L )^\dag=
({\cal M}_{diag}^d)^2\,,\\
&&
( d_1,d_2,d_3, d_4)_{L/R}= ( d, s,b,b')_{L/R} (U^d_{L/R})^*\,,
\eeqa
where $( d_1, d_2, d_3, d_4)$ and $(d,s,b,b')$ stand for the interaction and mass eigenstates, respectively.
The new notation, $d_4$, is designated for the interaction basis of the singlet $b'_{L,R}$, and $b'$ is recycled to represent the heaviest mass eigenstate of down-type quark. One will see that the mass and interaction eigenstates of $b'$ are very close to each other from the later phenomenology study.
Similarly, the SM up-type quarks and the charged leptons can be brought to their mass eigenstates by $U^u_{L/R}$ and $U^e_{L/R}$, respectively\footnote{Note that the SM neutrinos are still massless at the tree-level.}.
Since $\widetilde{\lambda}$'s are unknown in the first place, one can focus on the couplings in the charged fermion mass basis,
denoted as $\lambda^{T,D,S}$, which are more phenomenologically useful.
However, note that the mass diagonalization matrices are in general different for the left-handed (LH) up- and LH down-quark sectors.
If we pick the flavor indices of $\lambda^T$ to label the charged lepton and down quark mass states,
the up-type quark in the triplet leptoquark coupling will receive an extra factor to compensate the difference between $U^d_L$ and $U^u_L$.
Explicitly,
\beqa
{\cal L} \supset &-& \sum_{l=e,\mu,\tau}\sum_{p=d,s,b,b'}(\lambda^T)_{l p} \sum_{r=u,c,t}\tilde{A}^\dag_{pr} \left[ \bar{\nu}_l^c T^{-\tth}+ \bar{e}_l^c \frac{ T^{\ot} }{\sqrt{2}}\right]u_{L,r}\nonr\\
&-& \sum_{l=e,\mu,\tau}\sum_{p=d,s,b,b'}(\lambda^T)_{l p} \left[ \bar{e}_l^c T^{\ft}+\bar{\nu}_l^c \frac{ T^{\ot} }{\sqrt{2}} \right]d_{L,p} + H.c.
\eeqa
with the four-by-three matrix
\beq
\tilde{A}^\dag_{pr}= \sum_{j=1}^4 (U^d_L)_{p j} (U^u_L)^\dag_{j r}\,.
\eeq
As will be discussed in below, $\tilde{A}$ is the extended CKM rotation matrix, $\tilde{V}$, and $\tilde{A}\ra (V_{CKM})$ if $b'_L$ decouples. Now, all $\lambda^T$, $\lambda^D$, and $\lambda^S$ are three-by-four matrices.
In the interaction basis, only the LH doublets participate in the charged-current(CC) interaction.
Thus, the SM $W^\pm$ interaction for the quark sector is
\beq
{\cal L} \supset \frac{g_2}{\sqrt{2}} \sum_{i=1}^3 \left(\bar{u}_i \gamma^\alpha \PL d_i \right)W^+_\alpha +H.c.
\eeq
However, the singlet $b'_{L}$ mixes with other LH down-type-quarks and change the SM CC interaction.
In the mass basis, it becomes
\beq
{\cal L} \supset \frac{g_2}{\sqrt{2}} (\bar{u},\bar{c},\bar{t})\gamma^\alpha \PL
\widetilde{V}\left( \begin{array}{c} d \\s\\b\\b' \end{array} \right) W^+_\alpha +H.c.
\eeq
where
\beq
\widetilde{V}=\left(
\begin{array}{cccc}
\wt{V}_{ud} & \wt{V}_{us} & \wt{V}_{ub} & \wt{V}_{u b'} \\
\wt{V}_{cd} & \wt{V}_{cs} & \wt{V}_{cb} & \wt{V}_{c b'} \\
\wt{V}_{td} & \wt{V}_{ts} & \wt{V}_{tb} & \wt{V}_{t b'} \\
\end{array}
\right)\,,\; \mbox{ and }\,\,
\widetilde{V}_{pq}\equiv \sum_{i=1}^3 (U^u_L)_{pi}(U^d_L)^\dag_{iq} \,.
\eeq
Therefore, the SM three-by-three unitary CKM matrix changes into a three-by-four matrix in our model.
When the $b'_L$ decouples, the coupling matrix $\widetilde{V}$ reduces to the SM $V_{CKM}$.
Instead of dealing with a three-by-four matrix, it is helpful to consider an auxiliary unitary four-by-four matrix
\beq
\wt{V}_4 \equiv \left( \begin{array}{cc} U^u_L & 0 \\ 0 & 1 \end{array} \right)
\cdot (U^d_L)^\dag =
\left(
\begin{array}{cccc}
\wt{V}_{ud} & \wt{V}_{us} & \wt{V}_{ub} & \wt{V}_{u b'} \\
\wt{V}_{cd} & \wt{V}_{cs} & \wt{V}_{cb} & \wt{V}_{c b'} \\
\wt{V}_{td} & \wt{V}_{ts} & \wt{V}_{tb} & \wt{V}_{t b'} \\
(U^d_L)_{d4}^* & (U^d_L)_{s4}^* &(U^d_L)_{b4}^* &(U^d_L)_{b'4}^*\\
\end{array}
\right)\,.
\label{eq:V4}
\eeq
To quantify the NP effect, one can parameterize the four-by-four unitary matrix $U^d_L$ by a unitary three-by-three sub-matrix,
$U^d_{L3}$, and three rotations as:
\beq
(U^d_L)^\dag =\left(
\begin{array}{cc}
(U^d_{L3})^\dag &0 \\
0& 1 \\
\end{array}
\right)\cdot R_4\,,\;\;\mbox{where}\;\;
R_4 =
\left(
\begin{array}{cccc}
c_1 & 0 & 0 & s_1\\
-s_1 s_2 & c_2 & 0 & c_1 s_2 \\
-s_1 c_2 s_3 & -s_2 s_3 & c_3 & c_1 c_2 s_3\\
-s_1 c_2 c_3 & -s_2 c_3 & -s_3 & c_1 c_2 c_3\\
\end{array}
\right)\,,
\label{eq:R4}
\eeq
where $s_i(c_i)$ stands for $\sin \theta_i (\cos \theta_i) $, and $\theta_i$ is the mixing angle between $d_{iL}$ and $b'_L$.
In this work, we assume there is no new CP violation phase beyond the SM CKM phase for simplicity.
Now, Eq.(\ref{eq:V4}) can be parameterized as
\beq
\wt{V}_4 = \left(
\begin{array}{cc}
V_{CKM} &0 \\
0& 1 \\
\end{array}
\right)\cdot R_4\,.
\eeq
Again, we use $d,s,b,b'$ to denote the mass eigenstates with $m_d\simeq 4.7\mmev$, $m_s\simeq 96\mmev$, $m_b\simeq 4.18\mgev$, and $M_{b'}$, the mass of $b'$, unknown.
The null result of direct searching for the singlet $b'$ at ATLAS sets a limit that $M_{b'} > 1.22 TeV$ \cite{Aaboud:2018pii} (by assuming only three 2-body decays: $b'\ra Wt,bZ,bH$ ), and similar limits have obtained by CMS\cite{Sirunyan:2018qau,Sirunyan:2019sza}. We take $M_{b'}=1.5\mtev$ as a reference in this paper.
Moreover, all the direct searches for the scalar leptoquarks at the colliders strongly depend on the assumption of their decay modes.
Depending on the working assumptions, the exclusion limits range from $\sim 0.5\mtev$ to $\sim 1.6\mtev$\cite{Zyla:2020zbs}.
Instead of making simple assumptions, it will be more motivated to associate the leptoquark branching ratios to neutrino mass generation\cite{Chang:2016zll} or the $b$-anomalies \cite{Diaz:2017lit,Schmaltz:2018nls}.
In this paper, we take $m_{T,D,S}\sim M_{LQ} = 1\mtev$ as the reference point. And the constraint we obtained can be easily scaled for a different $M_{LQ}$ or $M_{b'}$.
Since all the new color degrees of freedom are heavier than $\gtrsim TeV$, it is straightforward to integrate them out and perform the Fierz transformation to get the low energy effective Hamiltonian, see Appendix \ref{sec:H_eff}.
\section{Explaining the anomalies}
\label{sec:howitworks}
\begin{table
\begin{center}
\begin{tabular}{|c|cccc|c|}
\hline
Anomaly $\backslash$ Field & $T(3,3,\ot)$ & $D(3,2,\frac{1}{6})$ & $S(3,1,\tth)$ & $b'(3,1,-\ot)$ & Remark\\
\hline
Neutrino mass & $\checkmark$ & $\checkmark$ & - & $\checkmark$ & 1-loop\\
Cabibbo angle anomaly & - & - & - & $\checkmark$ & extended CKM \\
$\tri a_e$ & $\times$ &$\checkmark$ &$\checkmark$ & $\checkmark$ & 1-loop \\
$\tri a_\mu$ & $\hchecked$ &$\checkmark$ &$\checkmark$ & $\checkmark$ & 1-loop \\
$b\ra sl^+l^-$ & $\checkmark$ & - & - & $\checkmark$ & box diagram \\
\hline
\end{tabular}
\caption{The anomalies and the fields to accommodate them in this model. The meaning of the legends: $\checkmark$: essential, $\hchecked$: helpful but not important or required, $\times$: negative effect, $-$: irrelevant. }
\end{center}
\end{table}
\subsection{Neutrino mass}
\begin{figure}[htb]
\centering
\includegraphics[width=0.6\textwidth]{nu_mass.eps}
\caption{ The Feynman diagrams for
the neutrino mass generation. (a) General case, where $\phi_{1,2}$ are in their interaction basis, and $F$ is in its mass eigenstate, and (b) for this model, where the fields are in the interaction basis. Here all the flavor indices are omitted.}
\label{fig:nu_mass}
\end{figure}
Instead of using the bi-lepton $SU(2)$ singlet and a charged scalar without lepton number as first proposed in Ref.\cite{Zee:1980ai}, we employ two leptoquarks which carry different lepton numbers to break the lepton number and generate the neutrino mass radiactively.
We start with a general discussion on the 1-loop neutrino mass generation.
If there are two scalars $\phi_{1,2}$ which interact with fermion $F_k$ and the neutrino via a general Yukawa coupling parameterized as
\beq
{\cal L}\supset \lambda_{ij} \bar{F}_j \nu_{L i}\phi_1 + \kappa_{ij} \bar{F}_j \nu^c_i \phi_2 + H.c.\,,
\eeq
where the fermion $F$ can carry arbitrary lepton number and baryon number $(L_F, B_F)$.
If the two scalars do not mix, $\phi_1/ \phi_2$ can be assigned with the lepton-number and baryon number $ (L_F-1, L_B)/(L_F+1, L_B)$ and the Lagrangian enjoys both the global lepton-number $U(1)_L$ and the global baryon-number $U(1)_B$ symmetries\footnote{ For the discussion of the pure leptonic gauge symmetry $U(1)_L$, see for example \cite{Schwaller:2013hqa, Chao:2010mp, Chang:2018nid, Chang:2018wsw, Chang:2018vdd}.}.
Without losing the generality, $F_k$ is assumed to be in its mass eigenstate with a mass $m_k$.
If $\phi_{1,2}$ can mix with each other, the lepton number is broken by two units, and the Weinberg operator\cite{Weinberg:1979sa} can be generated radiatively.
Let's denote $\phi_{h(l)}$ as the heavier(lighter) mass state with mass $m_h(m_l)$, and parameterize their mixing as
$\phi_1 = c_\alpha \phi_l +s_\alpha \phi_h$ and $\phi_2 = -s_\alpha \phi_l +c_\alpha \phi_h$, where $s_\alpha(c_\alpha)$ is the shorthand notation for $\sin\alpha(\cos\alpha)$ and $\alpha$ is the mixing angle.
The resultant neutrino mass from Fig.\ref{fig:nu_mass}(a) can be calculated as
\beq
M^\nu_{ij}= \sum_k {N^F_c m_k \over 16\pi^2} s_\alpha c_\alpha (\kappa_{ik}\lambda_{jk}+\kappa_{jk}\lambda_{ik})
\left[ {m_h^2 \over m_h^2 -m_k^2}\ln \frac{m_h^2}{m_k^2} - {m_l^2 \over m_l^2 -m_k^2}\ln \frac{m_l^2}{m_k^2} \right]\,,
\eeq
which is exact and free of divergence. Note that for the diagonal element, the combination in the bracket should be replaced by
$2 \mbox{Re}(\kappa_{ik}\lambda_{ik})$.
When the mixing is small, this result can also be approximately calculated in the interaction basis of $\phi_1$ and $\phi_2$.
In our model, the mass eigenstate $F$ can be the SM down-type quark or the exotic $b'$, and $D^{\ot}/T^\ot$ plays the role of $\phi_1/\phi_2$, as depicted in Fig.\ref{fig:nu_mass}(b). Assume the $D\mhyphen T$ mixing is small, then
\beq
M^\nu_{ij} \simeq \sum_{k=d,s,b,b'} {3 m_k \over 32\pi^2} (\lambda^T_{ik}\lambda^D_{jk}+\lambda^T_{jk}\lambda^D_{ik})
{\mu_3 v_0 \over M_D^2-M_T^2}\ln \frac{M_T^2}{M_D^2}
\label{eq:nu_mass}
\eeq
for $i\neq j$, and $2 \mbox{Re}(\lambda^T_{ik}\lambda^D_{ik})$ should be used in the bracket for the diagonal elements.
To have sub-eV neutrino masses, we need roughly
\beq
\mu_3 m_b\lambda^D \lambda^T\,,\, \mu_3 M_{b'}\lambda^D \lambda^T \simeq {\cal O}(10^{-5})\times\left( \frac{M_{LQ}}{\mbox{TeV}}\right)^2 (\mgev)^2
\eeq
if $b^{(')}$-quark contribution dominates. More comprehensive numerical consideration with other phenomenology will be given in section \ref{sec:pheno}.
\subsection{$(g-2)$ of charged leptons }
\label{sec:g-2}
\begin{figure}[htb]
\centering
\includegraphics[width=0.57\textwidth]{g-2_general.eps}
\caption{ The Feynman diagrams in the mass basis for
the anomalous magnetic dipole moment of charged lepton in general cases. }
\label{fig:g-2G}
\end{figure}
We also start with a general discussion on the 1-loop contribution to $(g-2)_\ell$ by adding a fermion, $F$, and a charged scalar, $h$.
The $F\mhyphen \ell \mhyphen h$ Yukawa interaction can be parameterized as
\beq
{\cal L}\supset \bar{F} (y^l_R \hat{R} + y^l_L \hat{L}) \ell h + H.c.
\label{eq:gen_eF_Yukawa}
\eeq
where both $F$ and $\ell$ are in their mass eigenstates.
Here, we have suppressed the flavor indices but it should be understood that both $y_R$ and $y_L$ are in general flavor dependent.
Then, the resulting 1-loop anomalous magnetic moment depicted in Fig.\ref{fig:g-2G}(a,b) can be calculated as
\beqa
\tri a^h_l &=& \frac{- N^F_c(1+Q_F) m_l^2}{8\pi^2}\int^1_0 dx\, x(1-x){ x \frac{ |y^l_L|^2+|y^l_R|^2 }{2} + \frac{m_F}{m_l}\Re[(y^l_R)^* y^l_L]
\over x^2 m_l^2 + x ( m_h^2-m_l^2) +(1-x) m_F^2}\,,\\
\tri a^F_l &=& \frac{- N^F_c Q_F m_l^2}{8\pi^2}\int^1_0 dx\, x^2 { (1-x) \frac{ |y^l_L|^2+|y^l_R|^2 }{2} + \frac{m_F}{m_l}\Re[(y^l_R)^* y^l_L]
\over x^2 m_l^2+ x ( m_F^2-m_l^2)+(1-x)m_h^2 }\,,
\eeqa
where $Q_F$ is the electric charge of $F$, and $\tri a_l^F (\tri a_l^h)$ is the contribution with the external photon attached to the fermion (scalar) inside the loop.
We keep $\tri a_l^F (\tri a_l^h)$ in the integral form since the analytic expression of resulting integration is not illuminating at all.
The physics is also clear from the above expression that one needs $m_F\gg m_l$ also both $y^l_R$ and $y^l_L$ nonzero to make $\tri a_e$ and $\tri a_\mu$ of opposite sign.
For $m_F\gg m_l$, we have
\beqa
\tri a_l^h &\simeq& \Re[(y^l_R)^* y^l_L]\left( \frac{m_l}{ m_F}\right) \frac{- N^F_c(1+Q_F)}{8\pi^2}\int^1_0 dx\,{ x(1-x)
\over x +(1-x)\frac{m_h^2}{m_F^2} }\,,\\
\tri a_l^F &\simeq& \Re[(y^l_R)^* y_L] \left( \frac{m_l}{ m_F}\right) \frac{-N^F_c Q_F }{8\pi^2}\int^1_0 dx\, { x^2
\over x + (1- x) \frac{m_h^2}{m_F^2} }\,.
\eeqa
Namely,
\beq
\tri a_l = \tri a_l^F+ \tri a_l^h \simeq -\frac{N^F_c \Re[(y^l_R)^* y_L] }{8\pi^2} \left( \frac{m_l}{m_F} \right) {\cal J}_{Q_F}\left(\frac{m_h^2}{m_F^2}\right)\,,
\eeq
where
\beqa
{\cal J}_Q(\alpha )&=&\int^1_0 dx\,{ x(1-x) +x\, Q \over x +(1-x)\alpha }\nonr\\
&=&{ 2 Q(1-\alpha)(1-\alpha+\alpha\ln\alpha)+ (1-\alpha^2+2\alpha\ln\alpha) \over 2(1-\alpha)^3 }\,.
\label{eq:a2J}
\eeqa
From Eq.(\ref{eq:a2J}), it is clear that ${\cal J}_Q(0)=(1+2Q)/2$, ${\cal J}_Q(1)=(1+3Q)/6$,
and ${\cal J}_Q(\alpha )\ra ( Q\ln\alpha-1/2)/\alpha$ for $\alpha \gg 1$.
Similar calculation leads to a $l\ra l' \gamma$ transition amplitude:
\beq
i{\cal M} \simeq i e \frac{m_l N_c^F}{16\pi^2 m_F} {\cal J}_{Q_F}(\beta_h )\times
\overline{u_{l'}}(p-k)\left[\frac{i \sigma^{\alpha\beta} k_\beta \epsilon^*_\alpha}{m_l}\left(A^{l l'}_M+A^{l l'}_E\gamma^5\right)\right ] u_l(p)\,,
\eeq
where $\beta_h=(m_h/m_F)^2 $, $\epsilon$ is the polarization of the photon, and
\beq
A^{l l'}_M = \frac{1}{2}\left[(y^{l'}_R)^* y^l_L + (y^{l'}_L)^* y^l_R \right]\,,\;
A^{l l'}_E = \frac{1}{2}\left[(y^{l'}_R)^* y^l_L - (y^{l'}_L)^* y^l_R \right]\,.
\eeq
For $l=\mu$ and $l'=e$, the above transition amplitude results in the $\mu\ra e \gamma$ branching ratio\cite{Chang:2005ag}
\beq
Br(\mu\ra e \gamma)= {3 \alpha (N^F_c)^2 \over 8 \pi G_F^2 m_F^2 m_\mu^2 }
\left( \left|A^{\mu e}_M\right|^2 +\left|A^{\mu e}_E\right|^2 \right)\,,
\label{eq:brmeg}
\eeq
and it must complies with the current experimental limit, $Br(\mu\ra e \gamma)< 4.2\times 10^{-13}$\cite{TheMEG:2016wtm}, or $|A^{\mu e}_{E,M}| \lesssim {\cal O}(10^{-8})$.
Moreover, if the dipole transition is dominate, then
\beq
{Br(\mu \ra 3 e) \over Br(e\ra e \gamma)}= \frac{2 \alpha}{3\pi}\left[\ln \frac{m_\mu}{m_e}-\frac{11}{8}\right] \simeq 6.12\times 10^{-3}\,,
\eeq
thus can be ignored.
\begin{figure}[htb]
\centering
\includegraphics[width=0.57\textwidth]{g-2.eps}
\caption{ The Feynman diagrams, in the interaction basis, for
the anomalous magnetic dipole moment of charged lepton in this model.
Here all the flavor indices are omitted. The external photon (not shown ) can be attached to any charged particle in the loop. }
\label{fig:g-2}
\end{figure}
In our model, the vector fermion $b'_{L,R}(3,1,-1/3)$, Fig.\ref{fig:g-2}(a), and/or the SM b-quark, Fig.\ref{fig:g-2}(b), can play the role of $F$ both carrying an electric charge $-\ot$.
The function ${\cal J}_{-\ot}(\alpha )$ takes a value in the range from $-0.022$ to $0.087$ for $\alpha\in[0.1,10.0]$.
In the interaction basis, $T(3,3,1/3)$ does not couple to $b'$, $D(3,2,1/6)$ only couples to left-handed charged lepton, and $S(3,1,2/3)$ only couples to the right-handed charged lepton. Due to the $D\mhyphen S$ and $D\mhyphen T$ mixings, the three charge-$2/3$ physical mass states acquire both the LH and RH Yukawa couplings as shown in Eq.(\ref{eq:gen_eF_Yukawa}). However, the physical state dominated by the $T$ component gets double suppression form $D\mhyphen T$ and $d\mhyphen b'$ mixings, thus not important here.
Assuming small $D\mhyphen S$ mixing in our model, the anomalous magnetic moment of charged lepton becomes
\beqa
\tri a_l &\simeq & ( \lambda^D_{l b'} \lambda^S_{l b'} ) {3 \mu_1 v_0\over 16\sqrt{2}\pi^2}\frac{m_l}{M^3_{b'}}
\times {\cal K}\left(\frac{M_D^2}{M^2_{b'}}, \frac{M_S^2}{M^2_{b'}}\right)\nonr\\
&+ & ( \lambda^D_{l b} \lambda^S_{l b} ) {3 \mu_1 v_0\over 16\sqrt{2}\pi^2}\frac{m_l}{m^3_{b}}
\times {\cal K}\left(\frac{M_D^2}{m^2_{b}}, \frac{M_S^2}{m^2_{b}}\right) \,,
\eeqa
where
\beq
{\cal K}(a,b)\equiv {{\cal J}_{-\ot}(a)- {\cal J}_{-\ot}(b) \over b-a }\,.
\eeq
When $a \simeq b$, the function ${\cal K}$ takes a limit
\beq
{\cal K}(a,b) \stackrel{b\ra a}{\Rightarrow} - \left. \frac{d }{d\alpha}{\cal J}_{-\ot}(\alpha )\right|_{\alpha=a}
= -{11-4 a-7 a^2+2[2+a(6+a)]\ln a \over 6(1-a)^4}\,.
\eeq
For $a \simeq b \simeq 1$, it can be approximated by ${\cal K}(a,b) \simeq 1/36-(a+b-2)/45$, and
${\cal K}(a,b) \simeq -\ln a/ (3 a^2)$ for $ a\simeq b \gg 1$.
If factoring out the $M_F=M_{b'}$, the dipole transition coefficients in Eq.(\ref{eq:brmeg}) are given by
\beq
A^{\mu e}_{M/E} \simeq -{\mu_1 v_0 \over 4\sqrt{2} M_{b'}^2}\left\{
\left[ ( \lambda^S_{e b'})^*\lambda^D_{\mu b'} \pm (\lambda^D_{e b'})^*\lambda^S_{\mu b'} \right]{\cal K}(\beta_D,\beta_S)+ \frac{M_{b'}^3}{m_b^3}\left[ (\lambda^S_{e b})^*\lambda^D_{\mu b} \pm (\lambda^D_{e b})^*\lambda^S_{\mu b} \right]{\cal K}(b_D,b_S) \right\}\,,
\eeq
where $b_{D,S}\equiv ( M_{D,S}/m_b )^2$, and $\beta_{D,S}= (M_{D,S}/M_{b'})^2$.
The current upper bound of $Br(\mu\ra e \gamma)$ amounts to a stringent limit that the relevant $|\lambda^S \lambda^D| \lesssim 10^{-5}$.
Instead of making the product of Yukawa couplings small, the $\mu\ra e \gamma$ transition from $D\mhyphen S$ mixing, Fig.\ref{fig:g-2}, can be simply arranged to vanish if muon/electron only couples to $b'/b$ or the other way around.
Modulating by the leptoquark masses,
numerically we have either
\beqa
\mbox{Sol-1} &:&\nonr\\
\tri a_e & \simeq & 2.28\times 10^{-5}\times [\lambda^D_{e b} \lambda^S_{e b} ]\times \left( \frac{\mu_1}{\mbox{GeV}}\right) \times {\cal K}(b_D,b_S)\,,\nonr\\
\tri a_\mu & \simeq & 1.03\times 10^{-10} \times [\lambda^D_{\mu b'} \lambda^S_{\mu b'} ]\times \left( \frac{\mu_1}{\mbox{GeV}}\right) \left( \frac{1.5 \mtev}{M_{b'}}\right)^3 \times {\cal K}(\beta_D,\beta_S)\,,
\label{eq:g2sol1}
\eeqa
or
\beqa
\mbox{Sol-2} &:&\nonr\\
\tri a_e & \simeq & 5.00 \times 10^{-13} \times [\lambda^D_{e b'} \lambda^S_{e b'} ]\times \left( \frac{\mu_1}{\mbox{GeV}}\right) \left( \frac{1.5 \mtev}{M_{b'}}\right)^3 \times {\cal K}(\beta_D,\beta_S)\,,\nonr\\
\tri a_\mu & \simeq & 4.71\times 10^{-3} \times [\lambda^D_{\mu b} \lambda^S_{\mu b} ]\times
\left( \frac{\mu_1}{\mbox{GeV}}\right) \times {\cal K}(b_D,b_S)\,.
\label{eq:g2sol2}
\eeqa
For $M_{b'}=1.5 \mtev$ and $M_{LQ} \simeq 1 \mtev$, then either $\left\{ \mu_1 \lambda^D_{e b} \lambda^S_{e b}\,,\; \mu_1 \lambda^D_{\mu b'} \lambda^S_{\mu b'} \right \} \simeq \{ 49, 305 \} \mgev $ for (Sol-1), or $\left\{ \mu_1 \lambda^D_{e b'} \lambda^S_{e b'}\,,\; \mu_1 \lambda^D_{\mu b} \lambda^S_{\mu b} \right \} \simeq \{ -20, -753 \} \mbox{GeV}$ for (Sol-2) can accommodate
the observed central values of $\tri a_{e,\mu}$ simultaneously with vanishing $Br(\mu\ra e \gamma)$. However, as will be discussed later, only Sol-2 is viable to simultaneously accommodate the neutrino data.
\subsection{$b\ra s l^+ l^-$}
\begin{figure}[htb]
\centering
\includegraphics[width=0.65\textwidth]{diag_RK_tree.eps}
\caption{ The potential tree-level Feynman diagrams for $b\ra s\mu\bar{\mu}$ transition. }
\label{fig:RK_tree}
\end{figure}
The $b\ra s \mu\bar{\mu}$ transition can be generated by tree-level diagrams mediated by $T^\ft$, $D^\tth$, $S^\tth$, and the one from $D\mhyphen S$ mixing, see Fig.\ref{fig:RK_tree}. In Fig.\ref{fig:RK_tree}(c), the crosses represent the mixing between the $b'_L$ and the physical $b$ and $s$ quarks, because $S$ only couples to $b'_L$ in the interaction basis.
From Eq.(\ref{eq:H_eff}), we see that this model can yield $b\ra s \mu\bar{\mu} $ operators in the vector, scalar, and tensor forms.
However, we failed to find a viable parameter space to explain the $b\ra s l^+ l^-$ anomaly and simultaneously comply with other experimental constraints\footnote{On the other hand, we cannot rule out the possibility of finding such a solution with fine-tuning. }, see Sec.\ref{sec:2q2lHeff}.
Instead, to bypass the stringent experimental bounds and the fine-tuning of the parameters, we go for the 1-loop box diagram contribution, as shown in Fig.\ref{fig:RK}, which requires only four nonzero triplet Yukawa couplings $\lambda^T_{\tau s}, \lambda^T_{\tau b}, \lambda^T_{\mu b}, \lambda^T_{\mu b'}$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.65\textwidth]{diag_RK_2.eps}
\caption{ The Feynman diagram for $b\ra s\mu\bar{\mu}$ transition. }
\label{fig:RK}
\end{figure}
In the usual convention, the transition is described by a low energy effective Hamiltonian
\beq
{\cal H}^{b\ra s \mu\mu}_{eff} =-\frac{G_F}{\sqrt{2}} \wt{V}_{tb}\wt{V}_{ts}^* \frac{\alpha}{\pi}\sum_i {\cal C}_i {\cal O}_i + H.c.
\eeq
with
\beqa
{\cal O}_9 =\left(\bar{s}\gamma^\alpha \hat{L} b \right) \left(\bar{\mu}\gamma_\alpha \mu\right)\,,\,\,
{\cal O}_{10} =\left(\bar{s}\gamma^\alpha \hat{L} b \right) \left(\bar{\mu}\gamma_\alpha \gamma^5\mu\right)\,,\\
{\cal O}'_9 =\left(\bar{s}\gamma^\alpha \hat{R} b \right) \left(\bar{\mu}\gamma_\alpha \mu\right)\,,\,\,
{\cal O}'_{10} =\left(\bar{s}\gamma^\alpha \hat{R} b \right) \left(\bar{\mu}\gamma_\alpha \gamma^5\mu\right)\,.
\eeqa
Ignoring the tau mass in the loop, Fig.\ref{fig:RK}(a), the effective Hamiltonian generated by the box-diagram can be easily calculated as
\beq
{\cal H}^{b\ra s \mu\mu}_{eff (a) } \simeq
- { \lambda^T_{\tau b} (\lambda^T_{\tau s})^* \over 64\pi^2} \left( \frac{|\lambda^T_{\mu b'}|^2}{M_{b'}^2} {\cal G}(\beta'_T)
+\frac{|\lambda^T_{\mu b}|^2}{m_b^2} {\cal G}(\beta_T) \right)
\left(\bar{s}\gamma^\alpha \PL b \right) \left(\bar{\mu}\gamma_\alpha \PL \mu\right) +H.c.\,,
\eeq
where $\beta'_T\equiv (M_T/M_{b'})^2$, $\beta_T\equiv (M_T/m_{b})^2$, and
\beq
{\cal G}(x)= \left[ \frac{1}{1-x}+ \frac{\ln x}{(1-x)^2}\right]\,.
\eeq
The function has a limit ${\cal G}(x=1)= -1/2$, and ${\cal G}\ra -1/x$ when $x\gg 1$.
The second contribution from the box diagram with $T^{\pm \ot}$ and up-type quark running in the loop yields
\beq
{\cal H}^{b\ra s \mu\mu}_{eff (b)} \simeq
{ \lambda^T_{\tau b} (\lambda^T_{\tau s})^* \over 64\pi^2}
\frac{1}{4 M_T^2}
\left[ |\lambda^T_{\mu b'}|^2 ( s_1^2+s_2^2+s_3^2) + |\lambda^T_{\mu b}|^2 \right]
\left(\bar{s}\gamma^\alpha \PL b \right) \left(\bar{\mu}\gamma_\alpha \PL \mu\right) +H.c.\,.
\eeq
In arriving the above expression, we have made use of the unitarity of $\wt{V}$, namely,
\beq
|\wt{V}_{ub'}|^2+ |\wt{V}_{cb'}|^2+ |\wt{V}_{tb'}|^2 =1 -c_1^2 c_2^2 c_3^2 \simeq s_1^2+s_2^2+s_3^2 \ll 1\,,
\eeq
and
\beq
|\wt{V}_{ub}|^2+ |\wt{V}_{cb}|^2+ |\wt{V}_{tb}|^2 \simeq 1\,.
\eeq
It is clear that the contribution from Fig.\ref{fig:RK}(b) is dominated by $\lambda^T_{\mu b}$.
And the relevant Wilson coefficients are determined to be
\beqa
{\cal C}_9 &=& -{\cal C}_{10} \simeq \frac{\sqrt{2} }{ 128 \pi\alpha} { \lambda^T_{\tau b} (\lambda^T_{\tau s})^* \over V_{tb}V_{ts}^* G_F M_T^2 }
\left[ |\lambda^T_{\mu b'}|^2 \beta'_T {\cal G}(\beta'_T)-\frac{5}{4} |\lambda^T_{\mu b}|^2 \right]\,,\nonr\\
{\cal C}'_9 &=& -{\cal C}'_{10}=0\,.
\label{eq:C9_10}
\eeqa
For a typical value of $\beta'_T=(1.0 \mbox{TeV}/1.5 \mbox{TeV})^2$, $\beta'_T {\cal G}(\beta'_T)=-0.3677$.
We use the following values,
\beq
({\cal C}_9)^\mu = -({\cal C}_{10})^\mu \simeq -0.55\pm 0.10\,,
\eeq
for muon, and
\beq
({\cal C}_9)^e \simeq ({\cal C}_{10})^e \simeq ({\cal C}'_9)^e \simeq ({\cal C}'_{10})^e \simeq 0
\eeq
for the electron counter part,
from the global fit to the $b\ra s l^+ l^-$ data \cite{Aebischer:2019mlg}
\footnote{There are other suggestion by the recent study of\cite{Datta:2019zca}. However, to only produce $C_9^{\mu\mu}$ in our model requires large both $b\mhyphen b'$ and $b'\mhyphen s$ mixings and the tree-level processes, which we discard. }.
If we take $\wt{V}_{tb} \wt{V}_{ts}^* =-0.03975$, then it amounts to
\beq
\lambda^T_{\tau b} (\lambda^T_{\tau s})^*
\left[ - |\lambda^T_{\mu b'}|^2 \beta'_T {\cal G}(\beta'_T) +\frac{5}{4} |\lambda^T_{\mu b}|^2 \right]
\simeq -(0.529 \pm 0.096) \left(\frac{M_T}{\mbox{TeV}}\right)^2 \,.
\eeq
Since the combination in the squared bracket is positive, the product $ \lambda^T_{\tau b} (\lambda^T_{\tau s})^* $ has to be negative.
The constraints from $B_s\mhyphen \overline{B}_s$ mixing and $b\ra s \gamma$ will be carefully discussed in Sec.\ref{sec:pheno}.
\subsection{Cabibbo-angle anomaly}
From the unitarity of $\wt{V}_4$, it is clear that
\beq
|\widetilde{V}_{ud}|^2+|\widetilde{V}_{us}|^2+|\widetilde{V}_{ub}|^2=1 -|\wt{V}_{u b'}|^2\leq 1\,,
\label{eq:V4deficit}
\eeq
and the Cabibbo-angle anomaly(CAA) is naturally embedded in this model.
Moreover, the most commonly discussed unitarity triangle becomes
\beq
\widetilde{V}_{ud}\widetilde{V}_{ub}^*+\widetilde{V}_{cd}\widetilde{V}_{cb}^*+
\widetilde{V}_{td}\widetilde{V}_{tb}^* = -(U^d_L)_{d4}^* (U^d_L)_{b4}\,.
\eeq
Similarly, this model also predicts that
\beqa
&&|\widetilde{V}_{cd}|^2+|\widetilde{V}_{cs}|^2+|\widetilde{V}_{cb}|^2=1 -|\wt{V}_{c b'}|^2\,,\\
&&|\widetilde{V}_{td}|^2+|\widetilde{V}_{ts}|^2+|\widetilde{V}_{tb}|^2=1 -|\wt{V}_{t b'}|^2\,,\\
&&|\widetilde{V}_{u d}|^2+|\widetilde{V}_{c d}|^2+|\widetilde{V}_{t d}|^2=1-|(U^d_L)_{d4}|^2\,, \\
&&|\widetilde{V}_{u s}|^2+|\widetilde{V}_{c s}|^2+|\widetilde{V}_{t s}|^2=1-|(U^d_L)_{s4}|^2\,, \\
&&|\widetilde{V}_{u b}|^2+|\widetilde{V}_{c b}|^2+|\widetilde{V}_{t b}|^2=1-|(U^d_L)_{b4}|^2\,,
\eeqa
and all the other SM CKM unitary triangles are no more closed in general.
The matrix elements are easy to read. For example, we have
\beqa
&& \wt{V}_{us} = c_2 V_{us}-s_2 s_3 V_{ub}\,,\;\wt{V}_{u b'} = s_1 V_{ud} +c_1 s_2 V_{us} +c_1 c_2 s_3 V_{ub}\,,\\
&& \left\{ (U^d_L)_{d4},(U^d_L)_{s4},(U^d_L)_{b4},(U^d_L)_{b'4} \right\}= \left\{ -s_1 c_2 c_3,-s_2 c_3, -s_3, c_1 c_2 c_3\right\}\,.
\eeqa
The mixing $\theta_i$ is expected to be small, so a smaller universal
\beq
\left|\wt{V}_{us}\right| \simeq \left|V_{us} \right| \left(1-\frac{\theta_2^2}{2}\right)
\eeq
to leading order is expected as well.
By using the Wolfenstein parameterization and the central values from global fit\cite{Zyla:2020zbs}, we have
\beq
\wt{V}_{u b'} \simeq 0.9740 s_1 + 0.2265 c_1 s_2 +0.0036 c_1 c_2 s_3 e^{1.196 i} \,.
\eeq
Therefore, to accommodate the deficit of 1st row CKM unitarity (Eq.(\ref{eq:PDGCKM}) and Eq.(\ref{eq:V4deficit}))
we have
\beq
\left| s_1 + 0.233 s_2 \right| \simeq 0.039(7)\,.
\label{eq:CKMA_req}
\eeq
\section{Constraints and parameter space}
\label{sec:pheno}
As discussed in the previous section, this model is capable to address neutrino mass generation, $\tri a_{e,\mu}$, $b\ra s\mu\mu$, and the CAA. For readers' convenience, all the requirements are collected and displayed in Table \ref{tab:req_sum}.
\begin{table
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Anomaly & Requirement & Remark\\
\hline
$m_\nu$ &
$\mu_3 m_{b^{(')}}\lambda^D \lambda^T \simeq {\cal O}(10^{-5})(\mgev)^2 $ & Eq.(\ref{eq:nu_mass})\\
$\tri a_{e,\mu}$ ( Sol-1 ) &
$\left\{ \mu_1 \lambda^D_{e b} \lambda^S_{e b}\,,\; \mu_1 \lambda^D_{\mu b'} \lambda^S_{\mu b'} \right \} \simeq \{ (49\pm 20), (305\pm 81) \} \mbox{GeV}$
& Eq.(\ref{eq:g2sol1}) \\
$\tri a_{e,\mu}$ ( Sol-2 ) &
$\left\{ \mu_1 \lambda^D_{e b'} \lambda^S_{e b'}\,,\; \mu_1 \lambda^D_{\mu b} \lambda^S_{\mu b} \right \} \simeq \{ -(20\pm8), -(752\pm200) \} \mbox{GeV}$
& Eq.(\ref{eq:g2sol2}) \\
$b\ra sl^+l^-$ &
$\lambda^T_{\tau b} (\lambda^T_{\tau s})^* ( |\lambda^T_{\mu b'}|^2 +3.39|\lambda^T_{\mu b}|^2 ) \simeq -(1.43 \pm 0.27)$
& Eq.(\ref{eq:C9_10}) \\
Cabibbo angle anomaly & $\left| s_1 + 0.233 s_2 \right| \simeq 0.039(7)$
& Eq.(\ref{eq:V4deficit}) \\
\hline
\end{tabular}
\caption{The requirement for explaining each mechanism/anomaly. For illustration, we take the following values: $ \wt{V}_{tb} \wt{V}_{ts}^*=-0.03975$, $M_{LQ}=1.0$ TeV, and $M_{b'}=1.5$ TeV. }
\end{center}
\label{tab:req_sum}
\end{table}
In this section, we should carefully scrutinize all the existing experimental limits and try to identify the viable model parameter at the end.
\subsection{Low energy $2q2l$ effective Hamiltonian}
\label{sec:2q2lHeff}
\begin{table
\begin{center}
\begin{tabular}{|lllr|}
\hline
${(\bar{q}_k \gamma^\mu \PL q_l)(\bar{e}_i \gamma_\mu \PL e_j)\over 4 M_T^2}$ & Wilson Coef. & Constraint& Model \\ \hline
$bb\mu\mu$ & $2 |\lambda^T_{\mu b}|^2 $ & $211.1$\cite{Carpentier:2010ue}&$1.57$ \\
$sb\tau\tau$ & $2 \lambda^T_{\tau b} (\lambda^T_{\tau s})^* $ & - & $-0.15$ \\
$sb\mu\mu$ & $0\footnote{There is no such effective operator at tree-level.}$ & - & $0$ \\
$sb\mu\tau$& $0$ & - & $0$ \\
$sb\tau\mu$ & $2 \lambda^T_{\mu b} (\lambda^T_{\tau s})^* $ & $0.199\footnote{ We update this value by using the new data ${\cal B}(B^+\ra K^+\mu^+ \tau^-)< 4.5\times 10^{-5}$\cite{Zyla:2020zbs}.}$\cite{Carpentier:2010ue}& $0.16$ \\ \hline
$u u \tau\mu$ & $ \left( \wt{V}_{ub}(\lambda^T_{\tau b})^*+ \wt{V}_{ub'}(\lambda^T_{\tau b'})^*+\wt{V}_{us}(\lambda^T_{\tau s})^*\right)
\times \left(\wt{V}^*_{ub}\lambda^T_{\mu b}+ \wt{V}^*_{ub'}\lambda^T_{\mu b'} \right] $ & $0.13$\cite{Carpentier:2010ue}&$0.0076$ \\
$u u \mu\mu$ & $ \left| \wt{V}_{ub}(\lambda^T_{\mu b})^*+ \wt{V}_{ub'}(\lambda^T_{\mu b'})^* \right|^2 $&$1.03$\cite{Carpentier:2010ue}&$0.025$ \\ \hline
$u c \mu\mu$ & $ \left( \wt{V}_{ub}(\lambda^T_{\mu b})^*+ \wt{V}_{ub'}(\lambda^T_{\mu b'})^*\right)
\times \left(\wt{V}^*_{cb}\lambda^T_{\mu b}+ \wt{V}^*_{cb'}\lambda^T_{\mu b'} \right) $ & $0.11\footnote{ We update this value by using the new data ${\cal B}(D^+\ra \pi^+\mu^+ \mu^-)< 7.3\times 10^{-8}$\cite{Zyla:2020zbs}.}
$\cite{Carpentier:2010ue}&$0^*$ \\ \hline
$cc \mu\mu$ & $ \left| \wt{V}_{cb}(\lambda^T_{\mu b})^*+ \wt{V}_{cb'}(\lambda^T_{\mu b'})^* \right|^2 $ & $52.8$\cite{Carpentier:2010ue}&$0^*$ \\
$cc \tau\mu$ & $ \left( \wt{V}_{cb}(\lambda^T_{\tau b})^*+ \wt{V}_{cb'}(\lambda^T_{\tau b'})^*+\wt{V}_{cs}(\lambda^T_{\tau s})^*\right)
\times \left(\wt{V}^*_{cb}\lambda^T_{\mu b}+ \wt{V}^*_{cb'}\lambda^T_{\mu b'} \right) $ & $211.1$\cite{Carpentier:2010ue}&$0^*$\\ \hline
$tc\mu\mu$ & $ \left( \wt{V}_{tb}(\lambda^T_{\mu b})^*+ \wt{V}_{tb'}(\lambda^T_{\mu b'})^*\right)
\times \left(\wt{V}^*_{cb}\lambda^T_{\mu b}+ \wt{V}^*_{cb'}\lambda^T_{\mu b'} \right) $ & -&$0^*$ \\
$tc\tau\tau$ & $ \left( \wt{V}_{tb}(\lambda^T_{\tau b})^*+ \wt{V}_{tb'}(\lambda^T_{\tau b'})^*+ \wt{V}_{ts}(\lambda^T_{\tau s})^*\right)
\times \left(\wt{V}^*_{cb}\lambda^T_{\tau b}+ \wt{V}^*_{cb'}\lambda^T_{\tau b'} + \wt{V}^*_{cs}\lambda^T_{\tau s} \right) $ & - &$-0.039$ \\
$tc\tau\mu$ & $ \left( \wt{V}_{tb}(\lambda^T_{\tau b})^*+ \wt{V}_{tb'}(\lambda^T_{\tau b'})^*+ \wt{V}_{ts}(\lambda^T_{\tau s})^*\right)
\times \left(\wt{V}^*_{cb}\lambda^T_{\mu b}+ \wt{V}^*_{cb'}\lambda^T_{\mu b'} \right) $ & $11.35\footnote{We obtain the limit by using the top quark decay width, $\Gamma_t=1.42 \mgev$, and ${\cal B}(t\ra q l l')<1.86\times 10^{-5}$\cite{ATLAS:2018avw}.}$&$0^*$ \\
$tc\mu\tau$ & $ \left( \wt{V}_{tb}(\lambda^T_{\mu b})^*+ \wt{V}_{tb'}(\lambda^T_{\mu b'})^* \right)
\times \left(\wt{V}^*_{cb}\lambda^T_{\tau b}+ \wt{V}^*_{cb'}\lambda^T_{\tau b'} + \wt{V}^*_{cs}\lambda^T_{\tau s} \right) $ & $11.35$&$0.04$ \\ \hline
$tu\mu\mu$ & $ \left( \wt{V}_{tb}(\lambda^T_{\mu b})^*+ \wt{V}_{tb'}(\lambda^T_{\mu b'})^*\right)
\times \left(\wt{V}^*_{ub}\lambda^T_{\mu b}+ \wt{V}^*_{ub'}\lambda^T_{\mu b'} \right) $ & - &$0.14$ \\
$tu\tau\tau$ & $ \left( \wt{V}_{tb}(\lambda^T_{\tau b})^*+ \wt{V}_{tb'}(\lambda^T_{\tau b'})^*+ \wt{V}_{ts}(\lambda^T_{\tau s})^*\right)
\times \left(\wt{V}^*_{ub}\lambda^T_{\tau b}+ \wt{V}^*_{ub'}\lambda^T_{\tau b'} + \wt{V}^*_{us}\lambda^T_{\tau s} \right) $ & -&$-0.04$ \\
$tu\tau\mu$ & $ \left( \wt{V}_{tb}(\lambda^T_{\tau b})^*+ \wt{V}_{tb'}(\lambda^T_{\tau b'})^*+ \wt{V}_{ts}(\lambda^T_{\tau s})^*\right)
\times \left(\wt{V}^*_{ub}\lambda^T_{\mu b}+ \wt{V}^*_{ub'}\lambda^T_{\mu b'} \right) $ & $11.35$&$-0.13$ \\
$tu\mu\tau$ & $ \left( \wt{V}_{tb}(\lambda^T_{\mu b})^*+ \wt{V}_{tb'}(\lambda^T_{\mu b'})^* \right)
\times \left(\wt{V}^*_{ub}\lambda^T_{\tau b}+ \wt{V}^*_{ub'}\lambda^T_{\tau b'} + \wt{V}^*_{us}\lambda^T_{\tau s} \right) $ & $11.35$&$0.04$ \\ \hline
\end{tabular}
\caption{The tree-level NC operators and their Wilson coefficients.
We take $M_T=1\mtev$ for illustration, and the values in last two columns scale as $ (M_T/1 \mtev)^2$.
By using the parameter set example of Eq.(\ref{eq:num_result}), the model predictions, with the signs kept, are displayed in the last column. In the table, $0^*$ stems from choosing $\wt{V}^*_{cb}\lambda^T_{\mu b}+ \wt{V}^*_{cb'}\lambda^T_{\mu b'} =0$ to retain the $\mu\mhyphen e$ universality in $b\ra c l\nu$ transition as discussed in the text.
}
\label{tab:NCTL_list}
\end{center}
\end{table}
\begin{table
\begin{center}
\begin{tabular}{|lllr|}
\hline
${(\bar{d}_k \gamma^\mu \PL u_l)(\bar{\nu}_i \gamma_\mu \PL e_j)\over 4 M_T^2}$ & Wilson Coef. & Constraint & Model \\ \hline
$su\nu_\mu \mu$ & $0 $ & & $0$ \\
$su\nu_\tau \mu$ & $(\lambda^T_{\tau s})^* \left(\wt{V}^*_{ub}\lambda^T_{\mu b}+ \wt{V}^*_{ub'}\lambda^T_{\mu b'} \right) $ & $3.96 $&$ 0.014$ \\
$su\nu_\tau \tau$ & $(\lambda^T_{\tau s})^* \left(\wt{V}^*_{ub}\lambda^T_{\tau b}+ \wt{V}^*_{ub'}\lambda^T_{\tau b'}+ \wt{V}^*_{us}\lambda^T_{\tau s} \right) $ & $0.79$&$0.004$ \\
$su\nu_\mu \tau$ & $0$ & & $0$ \\ \hline
$sc\nu_\mu \mu$ & $0 $ & & $0$\\
$sc\nu_\tau \mu$ & $(\lambda^T_{\tau s})^* \left(\wt{V}^*_{cb}\lambda^T_{\mu b}+ \wt{V}^*_{cb'}\lambda^T_{\mu b'} \right) $ & $31.7$& $0^*$ \\
$sc\nu_\tau \tau$ & $(\lambda^T_{\tau s})^* \left(\wt{V}^*_{cb}\lambda^T_{\tau b}+ \wt{V}^*_{cb'}\lambda^T_{\tau b'} + \wt{V}^*_{cs}\lambda^T_{\tau s} \right) $ & $15.8$&$0.004$ \\
$sc\nu_\mu \tau$ & $0$ & & $0$ \\ \hline
$b u \nu_\mu\mu$ & $(\lambda^T_{\mu b})^* \left(\wt{V}^*_{ub}\lambda^T_{\mu b}+ \wt{V}^*_{ub'}\lambda^T_{\mu b'} \right) $ & $0.51$&$0.14$ \\
$b u \nu_\tau\mu$ & $(\lambda^T_{\tau b})^* \left(\wt{V}^*_{ub}\lambda^T_{\mu b}+ \wt{V}^*_{ub'}\lambda^T_{\mu b'} \right) $ & $0.51$&$-0.13$ \\
$b u \nu_\tau\tau$ & $(\lambda^T_{\tau b})^* \left(\wt{V}^*_{ub}\lambda^T_{\tau b}+ \wt{V}^*_{ub'}\lambda^T_{\tau b'}+\wt{V}^*_{us}\lambda^T_{\tau s} \right) $ & $0.51$&$-0.04$ \\
$b u \nu_\mu \tau$ & $(\lambda^T_{\mu b})^* \left(\wt{V}^*_{ub}\lambda^T_{\tau b}+ \wt{V}^*_{ub'}\lambda^T_{\tau b'}+ \wt{V}^*_{us}\lambda^T_{\tau s} \right) $ & $0.51$&$0.04$ \\ \hline
$b c \nu_\mu\mu$ & $(\lambda^T_{\mu b})^* \left(\wt{V}^*_{cb}\lambda^T_{\mu b}+ \wt{V}^*_{cb'}\lambda^T_{\mu b'} \right) $ & $5.41$&$0^*$ \\
$b c \nu_\tau\mu$ & $(\lambda^T_{\tau b})^* \left(\wt{V}^*_{cb}\lambda^T_{\mu b}+ \wt{V}^*_{cb'}\lambda^T_{\mu b'} \right) $ & $5.41$&$0^*$ \\
$b c \nu_\tau\tau$ & $(\lambda^T_{\tau b})^* \left(\wt{V}^*_{cb}\lambda^T_{\tau b}+ \wt{V}^*_{cb'}\lambda^T_{\tau b'}+\wt{V}^*_{cs}\lambda^T_{\tau s} \right) $ & $5.41$&$-0.04\footnote{This is the effective operator to address the $R(D^{(*)})$ anomaly.}$\\
$b c \nu_\mu \tau$ & $(\lambda^T_{\mu b})^* \left(\wt{V}^*_{cb}\lambda^T_{\tau b}+ \wt{V}^*_{cb'}\lambda^T_{\tau b'}+ \wt{V}^*_{cs}\lambda^T_{\tau s} \right) $ & $5.41$&$0.04$ \\ \hline
\end{tabular}
\caption{The tree-level CC operators and their Wilson coefficients.
All the constraints are taken and derived from\cite{Carpentier:2010ue}. By using the parameter set example of Eq.(\ref{eq:num_result}), the model predictions, with the signs kept, are displayed in the last column. We take $M_T=1\mtev$ for illustration, and the values in last two columns scale as $ (M_T/1 \mtev)^2$.
Note that the coefficients for $su(c)\nu_\mu \tau$ and $su(c)\nu_\mu \mu$ are zero at tree-level. In the table, $0^*$ stems from choosing $\wt{V}^*_{cb}\lambda^T_{\mu b}+ \wt{V}^*_{cb'}\lambda^T_{\mu b'} =0$ to retain the $\mu\mhyphen e$ universality in $b\ra c l\nu$ transition as discussed in the text. }
\label{tab:CCTL_list}
\end{center}
\end{table}
In this model, the minimal set ($\mbox{MinS}_T$) of $\lambda^T$ parameters for addressing all the anomalies and neutrino mass consists of five elements:
\beq
\mbox{MinS}_T =\{ \lambda^T_{\tau b}\,,\; \lambda^T_{\tau s}\,,\; \lambda^T_{\tau b'}\,,\; \lambda^T_{\mu b'}\,,\; \lambda^T_{\mu b}\}\,.
\eeq
The following are the consequences of adding other Triplet Yukawa couplings outside the $\mbox{MinS}_T$ :
(1) At tree-level, $\lambda^T_{ed}$ leads to $B^+\ra \pi^+ e\mu$, $B^0\ra \bar{e}\tau$, $\tau\ra e K$, and $\mu\mhyphen e$ conversion.
Then $\lambda^T_{ed}\lesssim 10^{-2}$ must be satisfied if all other $\lambda^T$'s are around ${\cal O}(1)$.
(2) At tree-level, $\lambda^T_{es}$ leads to $B^+\ra K^+ e\mu$ and $\mu\mhyphen e$ conversion.
Then $\lambda^T_{ed}\lesssim 10^{-2}$ is also required if all other $\lambda^T$'s are around ${\cal O}(1)$.
(3) At tree-level $\lambda^T_{eb}$ also leads to $\mu\mhyphen e$ conversion, but the constraint is weak due to the $|\wt{V}_{ub}|^2$ suppression. On the other hand, at 1-loop level, it generates the unfavored $b\ra s e e$ transition.
Also, note that $\lambda^T_{eb}\neq 0$ is not helpful for generating $M^\nu_{ee}$, which is crucial for the neutrinoless double beta decay. (4) Together with $\lambda^T_{\tau s}$, required for $b\ra s \mu\mu$, any nonzero $\lambda^T_{\ell_i d} (i=e, \mu, \tau)$ gives rise to $K^+\ra \pi^+ \nu \nu$ at the tree-level, and thus strongly constrained.
Moreover, $\lambda^T_{\tau s}$ and $\lambda^T_{\tau d}$ generate the $K\mhyphen\bar{K}$ mixing via the 1-loop box diagram, and thus
stringently limited.
(5) Together with $\mbox{MinS}_T$, the presence of any of $\lambda^T_{e d_i}, (i=d,s,b,b')$ leads to $l\ra l' \gamma$ transition at the one-loop level.
(6) On the other hand, the introduction of $\lambda^T_{\tau d}$ generates $s\ra d\mu\mu$ transition via the box-diagram
which is severely constraint by the $K_L\ra \mu\mu $ data. So it has to be small too.
(7) In general, adding $\lambda^T_{\mu s}$ will cause conflict with the precision Kaon data.
From the above discussion, adding any $\lambda^T \not\in \mbox{MinS}_T$ requires fine tuning the parameters.
For simplicity, we set any triplet Yukawa couplings outside the $\mbox{MinS}_T$ to zero.
However, we still need to scrutinize all the phenomenological constraint on the minimal set of parameters.
All the potential detectable effective operators from tree-level contribution of $\mbox{MinS}_T$ are listed in Table \ref{tab:NCTL_list} and Table \ref{tab:CCTL_list}. And one has to make sure all the constraints have to be met.
In addition to the limits considered in Ref\cite{Carpentier:2010ue}, one needs to take into account the
constraint from the lepton universality tests in B decays\cite{Bifani:2018zmi}.
In particular, the $\mu\mhyphen e$ universality in the $b\ra c l_i\nu (i=e,\mu)$ transition has been tested to $\simeq 1\%$ level\cite{Jung:2018lfu}. The $\mbox{MinS}_T$ of $\lambda^T$ introduces two operators,
$(\bar{b}\gamma^\alpha\PL c)(\bar{\nu}_\mu \gamma_\alpha \PL \mu)$ and $(\bar{b}\gamma^\alpha\PL c)(\bar{\nu}_\tau \gamma_\alpha \PL \mu)$, where the first one interferes with the SM CC interaction while the second one does not. On the other hand, there are no electron counter parts.
Therefore, it is required that the modification to the $b\ra c \mu \nu_j$ transition rate due to the two new operators is less than $\sim 2\%$.
Their Wilson coefficients, the third and the fourth entities from the end in Table \ref{tab:CCTL_list}, are both proportional to
$\left(\wt{V}^*_{cb}\lambda^T_{\mu b}+ \wt{V}^*_{cb'}\lambda^T_{\mu b'} \right)$. For simplicity, we artificially set this combination to zero to make sure the perfect $\mu\mhyphen e$ universality in $b\ra c l \nu$ at tree level, such that the ratio of $\lambda^T_{\mu b}/\lambda^T_{\mu b'}$ is fixed as well. However, if more parameter space is wanted, this strict relationship can be relaxed as long as the amount of $\mu\mhyphen e$ universality violation is below the experimental precision.
Finally, due to the QCD corrections, the semi-leptonic effective vector operator for addressing the $b\ra s\mu\mu$ anomaly gets about $\sim +10\%$ enhancement at low energy\cite{Aebischer:2018acj}.
However, all the tree-level $2q2l$ vectors operators listed in Table \ref{tab:NCTL_list} and Table \ref{tab:CCTL_list}, as the constraint, also get roughly the same enhancement factor. Therefore, we do not consider this RGE running factor at this moment.
Next, we move on to consider the tree-level effects from the doublet leptoquark.
The non-zero $\lambda^D_{\tau b}$ and $\lambda^D_{\mu b}$, required for addressing $\tri a_{e,\mu}$ and neutrino data, lead to the following relevant low energy effective Hamiltonian,
\beq
{\cal H}^{D}_{eff} \supset \frac{[ \bar{b}\gamma^\alpha\PR b ]}{4M_D^2}\left[ |\lambda^D_{\tau b}|^2 (\bar{\tau}\gamma_\alpha \PL \tau)
|\lambda^D_{\mu b}|^2 (\bar{\mu}\gamma_\alpha\PL \mu)+ \lambda^D_{\mu b}(\lambda^D_{\tau b})^* (\bar{\tau}\gamma_\alpha \PL\mu)
+ \lambda^D_{\tau b}(\lambda^D_{\mu b})^* (\bar{\mu}\gamma_\alpha\PL \tau)
\right]\,,
\eeq
and its neutrino counter part as well, see Eq.(\ref{eq:H_eff}). However, the constraint on these operators are rather weak\cite{Carpentier:2010ue} and can be ignored.
\subsection{SM $Z^0$ couplings }
\label{sec:Zcoupling}
Because $b'_{L,R}$ are charged under $U(1)_Y$ hypercharge, they interact with the $Z^0$ boson.
In the interaction basis\footnote{Here we temporarily switch back to earlier notation that $b'_{L,R}$ represent the interaction basis. }, the SM $Z^0$ interaction for the down quark sector is
\beq
{\cal L} \supset \frac{g_2}{c_W} \left[ g_L^{SM} \sum_{i=1}^3 \bar{d}_{Li} \gamma^\alpha d_{L i}
+g_R^{SM}\sum_{i=1}^3 \bar{d}_{Ri} \gamma^\alpha d_{R i} +
g_R^{SM}(\bar{b}'_L \gamma^\alpha b'_L + \bar{b}'_R \gamma^\alpha b'_R) \right] Z_\alpha\,,
\eeq
where $g_R^{SM}=\frac{s_W^2}{3}\simeq 0.077$, $g_L^{SM}=(-\frac{1}{2}+\frac{s_W^2}{3})\simeq -0.423$, $s_W= \sin\theta_W$, and $\theta_W$ is the Weinberg angle. If we denote $b'$ as $d_4$, then the above expression can be neatly written as
\beq
\frac{g_2}{c_W} \left[ g_L^{SM} \sum_{i=1}^4 \bar{d}_{Li} \gamma^\alpha d_{L i}
+g_R^{SM}\sum_{i=1}^4 \bar{d}_{Ri} \gamma^\alpha d_{R i}
+\frac{1}{2}(\bar{b}'_L \gamma^\alpha b'_L) \right] Z_\alpha\,.
\eeq
When rotating into the mass basis, due to the unitarity of the four-by-four $U^d_{L,R}$, it becomes
\beq
\frac{g_2}{c_W} \left[ \sum_{\alpha=s,d,b,b'} \bar{d}_{\alpha} \gamma^\alpha (g_L^{SM} \hat{L}+ g_R^{SM} \hat{R} ) d_{\alpha}\right] Z_\mu + \frac{g_2}{2 c_W} \sum_{\alpha,\beta=s,d,b,b'}
\kappa_{\alpha\beta}\left[ (\bar{d}_\alpha \gamma^\alpha \hat{L} d_\beta) \right] Z_\alpha\,,
\eeq
where $\PL=(1-\gamma^5)/2$, $\PR=(1+\gamma^5)/2$, and
\beq
\kappa_{\alpha\beta}\equiv (U^d_L)_{\alpha 4}[(U^d_L)_{\beta 4}]^* \,.
\eeq
Using the CP-conserving parametrization introduced in Eq.(\ref{eq:R4}), we have
\beq
\kappa_{sd} = \kappa_{ds}= s_1 s_2 c_2 c_3^2\,,\;
\kappa_{sb} = \kappa_{bs}= s_2 s_3 c_3\,,\;
\kappa_{bd} = \kappa_{db}= s_1 s_3 c_2 c_3\,.
\eeq
It is clear that, with the presence of $b'_L$, the tree-level Flavor-Changing-Neutral-Current (FCNC) in the down sector is inevitable
unless at most one of $\theta_{1,2,3}$ being sizable. For simplicity, we assume one nonvanishing $(U^d_L)_{4d_F}$, where $F$ could be one of $d,s,b$, and all the others are zero.
Let's focus on that specific non-zero flavor diagonal $Z\mhyphen d_F\mhyphen\bar{d_F}$ coupling. The mixing with $b'$ leads to
\beq
g^{SM}_{d_F,R} \Rightarrow g^{SM}_{d_i,R}\,,\,\,
g^{SM}_{d_F,L} \Rightarrow g^{SM}_{d_i L} +\frac{1}{2} \left| (U^d_L)_{4d_F}\right|^2\,,
\eeq
The introduction of $b'_{L,R}$ leads to a robust prediction that $(g_{d_F L})^2<(g^{SM}_{d_F L})^2 $ and $(g_{d_F R})^2=(g^{SM}_{d_F R})^2$ for that down-type quark at the tree-level.
Namely, in this model, $A_F$ and $A_F^{FB}$ ( both $\propto [(g_{d_F L})^2 - (g_{d_F R})^2]$ ), and $\Gamma_{d_F}$ ($\propto [(g_{d_F L})^2 + (g_{d_F R})^2]$) are smaller than the SM prediction. This remind us the long standing puzzle of the bottom-quark forward-backward asymmetry, $A^b_{FB}$, which is $2.3 \sigma$ below the SM value\cite{Zyla:2020zbs}. However, if we pick $\theta_3$ to be nonzero, then the CAA cannot be addressed, see Eq.(\ref{eq:CKMA_req}).
Moreover, from our numerical study, only $\theta_1\neq 0$ is viable to satisfy all experimental limits. Thus we set $\theta_2=\theta_3=0$.
From Eq.(\ref{eq:CKMA_req} ), we have
\beq
| s_1 | \simeq 0.039(7)\,,
\label{eq:theta2}
\eeq
and
\beq
\wt{V}_{ub'}=s_1 \wt{V}_{ud} \simeq 0.03798\,,\,
\wt{V}_{cb'}=s_1 \wt{V}_{cd} \simeq -0.00883\,,\,
\wt{V}_{tb'}=s_1 \wt{V}_{td} \simeq 0.00033\,,
\eeq
if we take $\theta_1$ to be positive.
This predicts $g_{dL}= g_{dL}^{SM}+s_1^2/2$ at tree-level, but with negligible effect.
On the other hand, one may wonder whether the introduction of
$\lambda^T_{\tau b}, \lambda^T_{\mu b}$ and $\lambda^D_{\tau b}$ can lead to sizable non-oblique radiactive
$Z\mhyphen b\mhyphen \bar{b}$ vertex corrections and address both the $A^b_{FB}$ anomaly and $R_b$ with the later one agrees with the SM prediction.
To address the $A^{FB}_b$ anomaly and $R_b$ simultaneously, one needs to increase $g_{bR}^2$ and decrease $g_{bL}^2$ at the same time.
We perform the 1-loop calculation in the $\overline{MS}$ scheme and the on-shell renormalization, and obtain the UV-finite result:
\beqa
\delta g^b_L &\simeq & {|\lambda^T_{\tau b}|^2+|\lambda^T_{\mu b}|^2\over 64\pi^2}\left[ \left(-1+\frac{5}{3}s_W^2\right)\frac{1}{9 \beta_Z}
-s_W^2{ 2\ln \beta_Z +\ot+i\pi/2 \over 3 \beta_Z} \right]\,,\nonr\\
\delta g^b_R &\simeq & {|\lambda^D_{\tau b}|^2 \over 64\pi^2}\left[ \left(-\frac{1}{3}s_W^2\right)\frac{1}{9 \beta_Z}
+s_W^2{ 2\ln \beta_Z +\ot+i\pi/2 \over 3 \beta_Z}\right]\,,
\eeqa
where $\beta_Z=(M_{LQ}/m_Z)^2$. Note the diagrams with $Z$ attached to the lepton in the loop have imaginary parts, and this is due to that the lepton pair can go on-shell. Unfortunately, these loop corrections are too small, $|\delta g^b_{L,R}|\sim {\cal O}(10^{-5})\times |\lambda^{T,D}|^2$, to be detectable. From the above, we conclude that, barring the tree-level FCNC $Z$ coupling, both $A_{FB}^b$ and $R_b$ receive no significant modification in this model.
Of course, future Z-pole electroweak precision measurements\cite{Abada:2019zxq, Baer:2013cma, CEPCStudyGroup:2018ghi} will remain the ultimate judge.
If the $A_{FB}^b$ deviation endures, one must go beyond this model.
We note by passing that more complicated model constructions are possible to address the $A^b_{FB}$ anomaly.
For example, this anomaly can be addressed by adding an anomaly-free set of chiral exotic quarks and leptons\cite{Chang:1999zc,Chang:1998pt}, or the vector-like quarks\cite{Choudhury:2001hs, Cheung:2020vqm, Crivellin:2020oup} to the SM.
\subsection{$B_s-\overline{B_s}$ mixing }
\label{sec:BSBSmixing}
One important constraint on the parameters related to $b\ra s \mu\mu$ transition comes from the $B_s\mhyphen \overline{B}_s$ mixing.
In our model, the box diagrams with leptoquark $T$ and lepton running in the loop give a sole effective Hamiltonian
\beq
{\cal H}^{B\bar{B}}_{eff}= {\cal C}_{B\bar{B}} \left(\bar{s}\gamma^\alpha \PL b \right) \left(\bar{s}\gamma_\alpha \PL b \right)\, +H.c.
\eeq
The Wilson coefficient can be easily calculated to be
\beq
{\cal C}_{B\bar{B}}\simeq { | \lambda^T_{\tau b}|^2 |\lambda^T_{\tau s}|^2 \over 128\pi^2 M_T^2} \left(1+\frac{1}{4}\right)\,,
\eeq
where the one-forth in the parenthesis is the contribution from $T^{\pm \ot}$.
Note that this ${\cal C}_{B\bar{B}}$ and the SM one are of the same sign, and it increases $\tri M_s$, the mass difference between $B_s$ and $\bar{B}_s$. But, the central value of the precisely measured $\tri M_s=17.757(21) \mbox{ps}^{-1}$\cite{Amhis:2019ckw} is smaller than the SM one. On the other hand, the SM prediction has relatively large, $\sim 10\%$\cite{Bona:2006sa, Altmannshofer:2020axr}, uncertainties arising from the hadronic matrix elements.
If putting aside the hadronic uncertainty, this tension could be alleviated in this model by the extended CKM, $V^*_{ts}V_{tb}\Rightarrow \wt{V}^*_{ts}\wt{V}_{tb}=(V^*_{ts}c_2-V^*_{tb}s_2s_3)V_{tb}c_3$, which reduces the SM prediction.
However, it does not work because we set $\theta_3=\theta_2=0$ as discussed in Sec.\ref{sec:Zcoupling}. Instead, we use the 2$\sigma$ range to constraint the model parameters.
Following Refs.\cite{Arnan:2019uhr,Huang:2020ris}, the NP contribution can be constrained to be
\beq
\left| 1+\frac{0.8 {\cal C}_{B\bar{B}}(\mu_{LQ}) }{{\cal C}^{SM}_{B\bar{B}}(\mu_b)}\right|-1 = -0.09\pm 0.08\,,\; \mbox{at 1 $\sigma$ C.L.}\,,
\label{eq:delMS}
\eeq
where the factor $0.8$ is the RGE running effect from $\mu_{LQ}\simeq 1\mtev$ to $\mu_b$, and ${\cal C}^{SM}_{B\bar{B}}(\mu_b)\simeq 7.2\times 10^{-11}\mgev^{-2}$ is the SM value at the scale $\mu_b$.
From the above, we obtain
\beq
| \lambda^T_{\tau b} (\lambda^T_{\tau s})^*| <0.0798 \left( M_T \over \mbox{TeV}\right)\,,
\eeq
so that Eq.(\ref{eq:delMS}) can be inside the $2\sigma$ confidence interval.
\subsection{$\tau \ra \mu(e) \gamma$}
\label{sec:tau_mu_p}
Since we also set $\lambda^T_{e d_i}=0 (d_i=d,s,b,b')$, there is no $\mu\ra e \gamma$ transition at 1-loop level by default. Therefore, we only focus on the constraint from $\tau \ra \mu \gamma$ and $\tau \ra e \gamma$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.65\textwidth]{b_s_gamma.eps}
\caption{ The Feynman diagram for (a) $\tau \ra \mu \gamma$, and (b,c) $b\ra s \gamma(g)$ transition. The external photon (gluon), which is not shown in the illustration, can attach to any charged(color) line in the loop. }
\label{fig:bsg}
\end{figure}
The rare $\tau \ra \mu \gamma$ transition can be induced when both $\lambda^T_{\tau b'}$ and $\lambda^T_{\mu b'}$ are nonzero.
The 1-loop diagram are shown in Fig.\ref{fig:bsg}(a).
The dipole $\tau \ra \mu \gamma$ amplitude can be parameterized as
\beq
i {\cal M}^\mu = i\, \left[ \bar{\mu}\left( i \sigma^{\mu\nu}k_\nu \right) ( d^{\tau \mu}_R \PR +d^{\tau \mu}_L \PL )\tau \right]\,,
\eeq
where $k$ is the photon momentum transfer.
If ignoring the muon mass, the partial decay width is given as\cite{Chang:2005ag}
\beq
\Gamma(\tau \ra \mu \gamma) \simeq \frac{m_\tau^3}{16 \pi} ( |d^{\tau \mu}_R|^2 + |d^{\tau \mu}_L|^2)\,.
\eeq
Since the leptoquark $T(D)$ only couples to the LH(RH) charged leptons, it contributes solely to $d^{\tau \mu}_{R(L)}$.
If ignoring the charged lepton masses in the loop, the dipole transition coefficient can be easily calculated to be
\beqa
d^{\tau \mu}_R = { e N_c m_\tau \over 16 \pi^2 M_T^2 }\{ && \lambda^T_{\tau b'} (\lambda^T_{\mu b'})^* \beta'_T
[ - Q_T R_S(\beta'_T) +Q_{(b')^c} R_F(\beta'_T)]\nonr\\
&+& \lambda^T_{\tau b} (\lambda^T_{\mu b})^* \beta_T
[ - Q_T R_S(\beta_T) +Q_{(b)^c} R_F(\beta_T)]\; \}\,.
\eeqa
In the above, $\beta'_T \equiv (M_T/M_{b'})^2$, $\beta_T \equiv (M_T/m_b)^2$, $Q_T$($Q_{(b^{(')})^c}$) is the electric charge of the scalar(fermion) in the loop,
and the loop functions,
\beqa
R_S(x) &=& {2+3x-6x^2+x^3+6 x \ln x \over 12(1-x)^4}\,,\;\mbox{and}\nonr\\
R_F(x)&=& \frac{R_S(1/x)}{x}={1-6x+3x^2+2x^3- 6 x^2 \ln x \over 12(1-x)^4}\,,
\label{eq:mueg_loop}
\eeqa
correspond to the contributions where the external photon attached to the scalar and fermion line in the loop, respectively.
Both functions have the same limit $1/24$ when $x\ra 1$.
When $x\gg 1$, $R_S(x)\ra 1/12x$ and $R_F(x)\ra 1/6x $.
Note the fermionic and bosonic contributions have opposite signs, and the charged fermion in the loop is the anti-$b^{(')}$.
Similarly, the contribution from the diagram with leptoquark $D^{-\tth}$ and $b^{(')}$ running in the loop yields
\beqa
d^{\tau \mu}_L = { e N_c m_\tau \over 32 \pi^2 M_D^2 } && \left\{ \lambda^D_{\tau b'} (\lambda^D_{\mu b'})^* \beta'_D
\left[ +\tth R_S(\beta'_D) - \ot R_F(\beta'_D)\right]\right.\nonr\\
&+& \left. \lambda^D_{\tau b} (\lambda^D_{\mu b})^* \beta_D
\left[ +\tth R_S(\beta_D) -\ot R_F(\beta_D)\right]\; \right\}\,.
\eeqa
As discussed in Sec.\ref{sec:g-2}, we set $\lambda^D_{\mu b'}=0$ in ( Sol-2) to avoid the dangerous $\mu\ra e \gamma$ transition\footnote{It will be clear that this is the case for fitting neutrino oscillation data successfully.}.
Then, we obtain
\beq
d^{\tau \mu}_L \simeq { e N_c m_\tau \over 32 \pi^2 M_D^2 }\lambda^D_{\tau b} (\lambda^D_{\mu b})^*\beta_D\left[\tth R_S(\beta_D)-\ot R_F(\beta_D) \right] \simeq 0\,,
\eeq
and due to the accidental cancellation in the squared bracket, it vanishes in the limit of $m_b \ll m_D$ in this case. Therefore, only $d^{\tau \mu}_R$ needs to be taken into account.
From $\tau_\tau=(290.3\pm 0.5)\times 10^{-15}s$\cite{Zyla:2020zbs}, the branching ratio of this rare process is
\beqa
{\cal B}(\tau\ra \mu \gamma)&=& \Gamma(\tau \ra \mu \gamma)/\Gamma_\tau \nonr\\
&\simeq& 5.14\times 10^{-6}
\left| \lambda^T_{\tau b'} (\lambda^T_{\mu b'})^* \frac{\beta'_T}{3}[4R_S(\beta'_T)+ R_F(\beta'_T)] + \frac{\lambda^T_{\tau b} (\lambda^T_{\mu b})^*}{6} \right|^2
\times \left({\mbox{TeV}\over M_T}\right)^4\,.
\eeqa
The current experimental bound, ${\cal B}(\tau\ra \mu \gamma)< 4.4\times 10^{-8}$\cite{Aubert:2009ag}, sets an upper bound
\beq
\left| \lambda^T_{\tau b'} (\lambda^T_{\mu b'})^* \frac{4\beta'_T}{3}[R_S(\beta'_T)+ R_F(\beta'_T)] + \frac{\lambda^T_{\tau b} (\lambda^T_{\mu b})^*}{6} \right|
< 0.092 \times \left({ M_T \over \mbox{TeV}}\right)^2\,.
\eeq
On the other hand, if both $\lambda^D_{\tau b'}$ and $\lambda^D_{e b'}$ present, we have
\beq
d^{\tau e}_L \simeq { e N_c m_\tau \over 16 \pi^2 M_D^2 } {\lambda^D_{\tau b'} (\lambda^D_{\mu b'})^* \beta'_D \over 6}
\left[ 2 R_S(\beta'_D) - R_F(\beta'_D)\right]
\eeq
for $\tau\ra e\gamma$ transition.
From ${\cal B}(\tau\ra e \gamma)< 3.3 \times 10^{-8}$ \cite{Aubert:2009ag}, it gives a weak bound
\beq
\left| \lambda^D_{\tau b'} (\lambda^D_{e b'})^*\right|< 16.93 \times \left({ M_D \over \mtev }\right)^2
\eeq
for $\beta'_D=(1.0/1.5)^2$.
\subsection{$b\ra s \gamma$ }
\label{sec:bsg}
Similar to the previous discussion on $\tau\ra \mu\gamma$, the $b \ra s \gamma (g)$ transition can be induced when both $\lambda^T_{\tau b}$ and $\lambda^T_{\tau s}$ are nonzero, see Fig.\ref{fig:bsg}(b,c).
Moreover, the fermion masses, $m_\tau$ or $m_{\nu_\tau}$, can be ignored, which corresponds to the $\beta_T\gg 1$ limit.
From Eq.(\ref{eq:mueg_loop}), it is easy to see that $R_S(x) \ra 1/(12 x)$ and $R_F(x)\ra 1/(6x)$ when $x\gg 1$.
Therefore, by plugging in the electric charges in the loop, the $b\ra s \gamma$ transition amplitude can readily read as
\beq
i{\cal M}^\mu(b\ra s \gamma) \simeq
i {e m_b \over 16\pi^2} {\lambda^T_{\tau b} (\lambda^T_{\tau s})^* \over 12 m_T^2}\left[
-\frac{1}{2}\left(-\ot\right) -\left(-\fth \right) + 2(1)\right] \left[ \bar{s}\left( i \sigma^{\mu\nu}k_\nu \right)\PR b \right]\,.
\eeq
The first one-half factor comes from the $T^{-\ot}$ Yukawa couplings which associate with the $(1/\sqrt{2})$ normalization, see Eq.(\ref{eq:LQYukawa_coupling}), and the factor 2 in the last term comes from the anti-tau contribution.
We also need to consider $b\ra s g$ transition because the RGE running will generate the $b\ra s \gamma$ operator at the low energy.
Because the gluon can only couple to the leptoquark, the amplitude reads
\beq
i{\cal M}^\mu(b\ra s g) \simeq
i {g_s m_b \over 16\pi^2} {\lambda^T_{\tau b} (\lambda^T_{\tau s})^* \over 12 m_T^2}\left[
-\frac{1}{2} -\left( 1 \right)\right] \left[ \bar{s}\left( i T^{(a)}\sigma^{\mu\nu}k_\nu \right)\PR b \right]\,.
\eeq
where $T^{(a)}$ is the $SU(3)_c$ generator.
Conventionally, the relevant effective Hamiltonian is given as
\beq
{\cal H}^{b\ra s\gamma}_{eff} = -\frac{4 G_F}{\sqrt{2}} V_{tb} V_{ts}^* ( {\cal C}_7 {\cal O}_7 + {\cal C}_8 {\cal O}_8)\,,
\eeq
with
\beqa
{\cal O}_7 &=& \frac{e}{16\pi^2} m_b \bar{s} \sigma^{\mu\nu}\PR b F_{\mu\nu}\,,\nonr\\
{\cal O}_8 &=& \frac{g_s}{16\pi^2} m_b \bar{s}_\alpha \sigma^{\mu\nu}\PR T^{(a)}_{\alpha\beta} b_\beta G^{(a)}_{\mu\nu}\,,
\eeqa
In our model, the Wilson coefficients for ${\cal O}_7$ and ${\cal O}_8$ can be identified as
\beqa
{\cal C}_7 \simeq {1\over 2\sqrt{2} G_F V_{tb} V_{ts}^* } {7 \lambda^T_{\tau b} (\lambda^T_{\tau s})^* \over 48 M_T^2}\,,\nonr\\
{\cal C}_8 \simeq -{1\over 2\sqrt{2} G_F V_{tb} V_{ts}^* } { \lambda^T_{\tau b} (\lambda^T_{\tau s})^* \over 8 M_T^2}\,.
\eeqa
The current experimental measurement, $Br^{Exp}(b\ra s\gamma)=(3.32\pm 0.15)\times 10^{-4}$\cite{Amhis:2016xyh}, and the SM prediction, $Br^{SM}(b\ra s\gamma)=(3.36\pm 0.23)\times 10^{-4}$\cite{Misiak:2015xwa,Misiak:2017woa}, agree with each other and set constraints on the NP contribution. Following Refs.\cite{Arnan:2016cpy,Arnan:2019uhr,Huang:2020ris}, we adopt the $2\sigma$ bound for the NP that
$|{\cal C}_7 +0.19 {\cal C}_8|\lesssim 0.06$, which leads to $|\lambda^T_{\tau b} (\lambda^T_{\tau s})^*|< 0.55$. This limit is much weaker than the one obtained from $\tri M_s$.
\subsection{Neutrino oscillation data }
\label{sec:num_fit}
As discussed before, the vanishing $\lambda^T_{e d_i}$ are preferred by phenomenological consideration.
It is then followed by a robust prediction that ${\cal M}_{ee}=0$, and the neutrinoless double beta decay mediated by ${\cal M}_{ee}$ vanishes as well. Therefore, the neutrino mass is predicted to be the normal hierarchical(NH).
Later we should discuss the consequence if this vanishing-$\lambda^T_{e d_i}$ assumption is relaxed.
A comprehensive numerical fit to the neutrino data is unnecessary to understand the physics, and it is beyond the scope of this paper as well. For simplicity, we assume all the Yukawa couplings are real, and thus all the CP phases in the neutrino mixings vanish.
However, this model has no problem to fit the CP violation phases of any values once the requirement of
all the Yukawa couplings being real is lifted.
Moreover, to adhere to the philosophy of using the least number of real parameters, only two more Yukawa couplings,
$\lambda^D_{\tau b'}$ and $\lambda^D_{\tau b}$, are introduced to fit the neutrino data\footnote{Note that we have not employed $\lambda^T_{\tau s}$ in the neutrino data fitting yet.}. Together with $\mbox{MinS}_T$, we make use of eleven Yukawa couplings, and the complete minimal set of parameter is
\beq
\mbox{MinS} = \mbox{MinS}_T \cup \left\{ \lambda^D_{e b'}, \lambda^D_{\tau b'}, \lambda^D_{\mu b}, \lambda^D_{\tau b}, \lambda^S_{e b'}, \lambda^S_{\mu b} \right\}\,.
\eeq
Assuming that $M_D\simeq M_T\simeq M_{LQ}$, the neutrino mass matrix takes the form
\beq
{\cal M}^\nu \simeq N^\nu
\left( \begin{array}{ccc}
0 & \lambda^D_{eb'} \lambda^T_{\mu b'} & \lambda^D_{eb'} \lambda^T_{\tau b'} \\
\lambda^D_{eb'} \lambda^T_{\mu b'} & 2 \rho_b \lambda^D_{\mu b} \lambda^T_{\mu b} &
\rho_b(\lambda^D_{\mu b}\lambda^T_{\tau b}+ \lambda^D_{\tau b}\lambda^T_{\mu b})+ \lambda^D_{\mu b'} \lambda^T_{\tau b'} \\
\lambda^D_{eb'} \lambda^T_{\tau b'} & \rho_b (\lambda^D_{\mu b}\lambda^T_{\tau b}+ \lambda^D_{\tau b}\lambda^T_{\mu b})+ \lambda^D_{\mu b'} \lambda^T_{\tau b'} &
2 \rho_b\lambda^D_{\tau b} \lambda^T_{\tau b}+ 2 \lambda^D_{\tau b'} \lambda^T_{\tau b'} \\
\end{array} \right)\,,
\label{eq:nuM_expression}
\eeq
where $\rho_b = m_b/M_{b'}$ and $N^\nu= { 3 \mu_3 v_0 M_{b'}\over 32 \pi^2 M^2_{LQ}} $.
Note that the leptoquark Yukawa couplings are tightly entangled with the neutrino mass matrix.
For instance, $\lambda^T_{\tau b'}/\lambda^T_{\mu b'} = {\cal M}^\nu_{e\tau}/{\cal M}^\nu_{e\mu}$ is required by this minimal assumption.
For illustration, we consider an approximate neutrino mass matrix\footnote{ It is just a randomly generated example for illustration. There are infinite ones with the similar structure. }
\beq
{\cal M}^\nu \simeq \left(
\begin{array}{ccc}
0 & 0.90373& 0.17912 \\
0.90373 & -2.4027 & -2.7685 \\
0.17912 & -2.7685 & -2.0147 \\
\end{array}
\right)\times 10^{-2} \mev\,,
\label{eq:Mnu_num}
\eeq
with all elements being real.
It leads to the following mixing angles and mass squared differences
\beqa
&&\theta_{12} \simeq 32.6 ^\circ\,,\;
\theta_{23}\simeq 47.8^\circ\,,\;
\theta_{13}\simeq 8.8^\circ\,,\;
\delta_{CP}=0^\circ\,,\nonr\\
&&\tri m_{21}^2 \sim 7.28 \times10^{-5} \mbox{eV}^2\,,\;
\tri m_{31}^2 \sim 2.53\times 10^{-3} \mbox{eV}^2\,.
\eeqa
Note all of the above values are inside the $3\sigma$ best fit (with SK atmospheric data) range for the normal ordering given by \cite{Esteban:2020cvm},
\beqa
&&\theta_{12} \in ( 31.27- 35.68)^\circ\,,\;
\theta_{23} \in ( 40.1- 51.7)^\circ\,,\;
\theta_{13} \in ( 8.20- 8.93)^\circ\,,\;
\delta_{CP} \in (120-369)^\circ\,,\nonr\\
&&\tri m_{21}^2 \in (6.82-8.04)\times 10^{-5} \mbox{eV}^2\,,\;
\tri m_{31}^2 \in (2.435-2.598)\times 10^{-3} \mbox{eV}^2\,,
\eeqa
and very close to the $1\sigma$ range.
This example neutrino mass matrix captures the essential features of the current neutrino data.
A better fitting to the neutrino oscillation data, including the phase, by using more(complex) model parameters is expected.
In order to reproduce the neutrino mass matrix, the second solution to $\tri a_{e,\mu}$, Eq.(\ref{eq:g2sol2}), must be adopted.
Because the first solution $\tri a_{e,\mu}$, Eq.(\ref{eq:g2sol1}), requires $\mu_1 \sim 50 \mtev$ to render mass matrix elements of about the same order, as shown in Eq.(\ref{eq:Mnu_num}).
All the best fit central values for $b\ra s\mu\mu$, $\tri a_{e,\mu}$, CAA, and the approximate neutrino mass matrix shown in Eq.(\ref{eq:Mnu_num}) can be easily accommodated with the specified non-vanishing parameters in the model.
As an example, below is one of the viable set of model parameters:
\beqa
M_{LQ}=1.0\, \mtev\,,\, M_{b'}=1.5\, \mtev\,,\, \mu_1=1.0\,\mtev\,,\, \mu_3= 0.211\,\mkev\,,s_1= 0.039\nonr\\
\lambda^T_{\mu b'}=-4.0676\,,\,
\lambda^T_{\tau b'}=-0.80622\,,\,
\lambda^T_{\mu b}=-0.88583\,,\nonr\\
\lambda^T_{\tau b}=0.82616\,,\,
\,\lambda^T_{\tau s}=-0.09064\,,\nonr\\
\lambda^D_{e b'}=0.003\,,\,
\lambda^D_{\mu b}=-6.5401\,,\,
\lambda^D_{\tau b'}=-0.0136\,,\,
\lambda^D_{\tau b}=1.1388\,,\nonr\\
\lambda^S_{e b'}= -6.6667\,,\,
\lambda^S_{\mu b}=0.11498\,,\,
\label{eq:num_result}
\eeqa
and all the other Yukawa couplings are set to zero.
This specific set of model parameters also predicts $\tri M_s \simeq 1.06\, \tri M_s^{SM}$,
\beq
{\cal B}(\tau \ra \mu \gamma)=4.5\times 10^{-9}\,,\;
{\cal B}(\tau \ra e \gamma)=1.9 \times 10^{-19}\,,
{\cal B}(B^+\ra K^+\mu^+ \tau^-) =2.9\times 10^{-5}\,,
\eeq
and pass all experimental limits we have considered, the last column in Tables \ref{tab:NCTL_list} and \ref{tab:CCTL_list}.
Note that three of the Yukawa couplings in Eq.(\ref{eq:num_result}) are larger than $2.0$ but below the nonperturbative limit $4\pi$.
However, this is because we want to use the minimal number of model parameters.
For instance, if the complex Yukawa is allowed, the degrees of freedom are doubled.
Also, the fitted values strongly depend on the goal. If aiming for $1\sigma$ intervals instead the central values of $\tri a_{e,\mu}$, both $\lambda^S_{e b'}$ and $\lambda^D_{\tau b}$ will be reduced by $\sim 50\%$. Lowering $M_{b'}$ and increasing $\mu_1$ can both
make the above values smaller as well. We have no doubt that a better fitting can be achieved in this model by using more (complex) free parameters. But we are content with the demonstration about the ability of this model to accommodate all the observed anomalies and explain the pattern of the observed neutrino data with minimal number of real parameters.
\subsection{$0\nu\beta\beta$ decay}
\begin{figure
\centering
\includegraphics[width=0.5\textwidth]{0nu2beta.eps}
\caption{ The Feynman diagrams for $0\nu\beta\beta$ decay (a) mediated by the $T\mhyphen D$ mixing, and (b) mediated by the neutrino Majorana mass. }
\label{fig:0n2b}
\end{figure}
The mixing between $T$ and $D$ breaks the global lepton number, see Appendix \ref{sec:H_eff}. Here, we consider whether the neutrinoless double beta decay can be generated beyond the contribution from the neutrino Majorana mass.
From the low energy effective Hamiltonian, together with the SM CC interaction, the lepton number violating charged current operators can induce the $0\nu\beta\beta$ process via the diagram, see Fig.\ref{fig:0n2b}(a).
By order of magnitude estimation, the absolute value of the amplitude strength relative to the usual one mediated by the neutrino Majorana mass, Fig.\ref{fig:0n2b}(b), is given by
\beq
\left| { {\cal M}_{TD} \over {\cal M}_{m_\nu} }\right| \simeq
{\lambda_T \lambda_D \mu_3 v_0 \over M_T^2 M_D^2} \frac{M_W^2 \langle p\rangle}{g_2^2 {\cal M}^\nu_{ee}}
\simeq 10^{-5} \left( \frac{1\, \mbox{TeV}}{M_{LQ}}\right)^4 \ll 1\,.
\eeq
In arriving the above value, we take $\mu_3=0.2\mkev$, $\langle p\rangle \sim 1\, \mmev$ for the typical average momentum transfer in the $0\nu\beta\beta$ process and assume ${\cal M}^\nu_{ee} \sim 0.01\mev$ for comparison.
Thus, this tree-level process mediated by $T$ and $D$ can be ignored.
\subsection{ A recap}
After taking into account all the phenomenological limits, we found the following simple assignment with minimal number of real parameters,
\beqa
&& M_{LQ}\simeq 1.0\, \mtev\,,M_{b'}= 1.5\, \mtev\,,\mu_1 = 1.0\, \mtev\,, \mu_3 \simeq 0.2\, \mkev\, \nonr\\
&& \mu_1 \lambda^S_{e b'}\lambda^D_{e b'} = -(20\pm 8) \mgev\,,\;
\mu_1 \lambda^S_{\mu b}\lambda^D_{\mu b}= -(752\pm 200) \mgev\,,\nonr\\
&& \theta_2 =\theta_3=0\,,\;\sin\theta_1 = 0.039(7)\,,\nonr\\
&&\lambda^T_{\mu b'} \simeq -4.1 \,,\lambda^T_{\tau b'} \simeq -0.81 \,,\lambda^T_{\mu b} \simeq- 0.89 \,,
\lambda^T_{\tau b}\simeq 0.83\,,\lambda^T_{\tau s} \simeq -0.09 \,,\nonr\\
&& \lambda^D_{e b'}\simeq 0.003\,, \lambda^D_{\mu b}\simeq -6.54\,,\lambda^D_{\tau b'}\simeq -0.014\,, \lambda^D_{\tau b}\simeq 1.14\,,
\label{eq:recap_range}
\eeqa
is able to yield the best fit solutions to accommodate $\tri a_{e,\mu}$, the Cabibbo angle, and $b\ra s \mu\mu$ anomalies
simultaneously. Moreover, the resulting neutrino mass pattern is very close to the observed one.
\section{Discussion}
\label{sec:discussion}
\subsection{Neutrino mass hierarchy and neutrinoless double beta decay}
\label{sec:parameter}
Because we set $\lambda^T_{e d_i}=0 (i=d,s,b,b')$, the neutrino mass element $M^\nu_{ee}$ vanishes and the neutrino mass is of the NH type.
Since we have used $\lambda^S_{e b'}$ to explain the observed $\tri a_e$, adding $\lambda^T_{e b'}$ is the minimal extension to yield a non-zero $M^\nu_{ee}$.
Together with $\lambda^T_{\mu b}$ and $\lambda^T_{\mu b'}$, the augmentation of $\lambda^T_{e b'}$ leads to an effective Hamiltonian,
\beq
H \supset \frac{{\cal C}_{\mu e}}{4M_T^2}(\bar{u}\gamma^\mu \PL u)(\bar{\mu}\gamma_\mu \PL e) +H.c.\,,
\eeq
where
\beq
{\cal C}_{\mu e} = \lambda^T_{eb'} (\wt{V}_{ub'})^*\left[ \wt{V}_{ub}(\lambda^T_{\mu b})^* +\wt{V}_{ub'}(\lambda^T_{\mu b'})^*\right]\,. \eeq
Numerically,
\beq
{\cal C}_{\mu e} \simeq 1.44\times10^{-3} \lambda^T_{eb'}(\lambda^T_{\mu b'})^*+ 1.37\times10^{-4} \lambda^T_{eb'}(\lambda^T_{\mu b})^* \,,
\label{eq:mu-e-con-C}
\eeq
if we set $s_1=0.039$ and $\wt{V}_{ub}=0.00361$.
The Wilson coefficient is severely constrained, ${\cal C}_{\mu e}<9.61\times 10^{-5}$, from the experimental limit of $\mu\mhyphen e$ conversion rate\cite{Dohmen:1993mp, Carpentier:2010ue}.
Namely, $\lambda^T_{eb'}(\lambda^T_{\mu b'})^* \lesssim 0.07$ unless the cancellation is arranged in Eq.(\ref{eq:mu-e-con-C}).
Moreover, in this model, the ratio of neutrino mass element $M^\nu_{ee}$ to $M^\nu_{e\mu}$, see Eq.(\ref{eq:nuM_expression}),
\beq
\frac{M^\nu_{ee}}{M^\nu_{e\mu}} ={ 2 M_{b'} \lambda^D_{eb'}\lambda^T_{eb'} \over M_{b'} \lambda^D_{eb'}\lambda^T_{\mu b'} }
= \frac{2 \lambda^T_{eb'}}{\lambda^T_{\mu b'}}\,,
\eeq
should be around $\sim {\cal O}(1)$ and $\sim {\cal O}(10)$ for the NH and Inverted Hierarchy (IH) type, respectively.
Since we need $\lambda^T_{\mu b'}\sim {\cal O}(1)$ to accommodate the b-anomalies, that implies $\frac{M^\nu_{ee}}{M^\nu_{e\mu}} \lesssim 0.07$. Thus, even if we include a non-zero $\lambda^T_{eb'}$ to generate the $ee$-component of $M_\nu$, the neutrino mass is still of the NH type, and roughly $|M^\nu_{ee}| \lesssim 3\times 10^{-4} \mev$. The precision required is beyond the capabilities of the near future experiments\cite{Tornow:2014vta}.
\subsection{Some phenomenological consequences at the colliders}
The smoking gun signature of this model will be the discovery of $b'$ and the three scalar leptoquarks.
Once their quantum numbers are identified, the gauge invariant allowed Yukawa couplings and the mechanisms to address the anomalies discussed in the paper follow automatically.
The collider physics of leptoquarks have been extensively studied before, and thus we do not have much to add. The readers interested in this topic are referred to the comprehensive review \cite{Dorsner:2016wpm} and the references therein.
In the paper, we should concentrate on the flavor physics at around or below the Z pole.
However, it is worthy to point out the nontrivial decay branching ratios of the exotic color states.
If we assume the mixings among the leptoquarks are small, their isospin members should be approximately degenerate in mass.
Then, the decays are dominated by 2-body decay with two SM fermions in the final states.
From the Yukawa couplings shown in Eq.(\ref{eq:num_result}), the decay branching ratio of leptoquarks can be easily read.
For $T^{-\ot}$, its decay branching ratios are
\beqa
&&{\cal B}(T^{-\ot} \ra b \nu_{\mu}) \simeq 26.6\%\,,\;
{\cal B}(T^{-\ot} \ra b \nu_{\tau}) \simeq 23.2\%\,,\;
{\cal B}(T^{-\ot} \ra s \nu_{\tau}) \simeq 2.8\times 10^{-3}\,,\nonr\\
&&{\cal B}(T^{-\ot} \ra \tau t ) \simeq 23.1\%\,,\;
{\cal B}(T^{-\ot} \ra \mu t ) \simeq 26.5\%\,,\;
{\cal B}(T^{-\ot} \ra \tau c) \simeq 2.6\times 10^{-3}\,.
\eeqa
For $T^{\tth}$ and $T^\ft$, the corresponding decay branching ratios are
\beqa
&&{\cal B}(T^\tth \ra t \nu_{\mu}) \simeq 53.2\%\,,\;
{\cal B}(T^\tth \ra t \nu_{\tau}) \simeq 46.3\%\,,\;
{\cal B}(T^\tth \ra c \nu_{\tau}) \simeq 0.5\%\,,\nonr\\
&&{\cal B}(T^{-\ft} \ra b \mu^- ) \simeq 53.2\%\,,\;
{\cal B}(T^{-\ft} \ra b \tau^-) \simeq 46.3\%\,,\;
{\cal B}(T^{-\ft} \ra s \tau^-) \simeq 0.5\times 10^{-4}\,.
\eeqa
Finally, we have
\beqa
&&{\cal B}(D^{-\ot} \ra b \bar{\nu}_{\mu}) \simeq 97.1\%\,,\;
{\cal B}(D^{-\ot} \ra b \bar{\nu}_{\tau}) \simeq 2.9\%\,,\nonr\\
&&{\cal B}(D^{\tth} \ra b \mu^+ ) \simeq 97.1\%\,,\;{\cal B}(D^{\tth} \ra b \tau^+) \simeq 2.9\%\,,
\nonr\\
&&{\cal B}(S^{\tth} \ra b \mu^+) \simeq 100\%\,,
\eeqa
for $D$ and $S$ leptoquarks.
The dominate decay modes of $b'$ are
$b'\ra LQ + l$, and $ b'\ra W^- u_i (u_i=u,c,t)$ through the mixing of $\wt{V}_{u_i b'}$.
Comparing to $M_{b'}$, the masses of final state particles can be ignored.
The width for $ b'\ra W u_i$ is simply given by
\beq
\Gamma(b'\ra u_i W^-)\simeq \frac{G_F |\wt{V}_{u_i b'}|^2 M_{b'}^3}{8\sqrt{2} \pi}\,,
\eeq
and
\beqa
\Gamma(b'\ra \bar{\nu_i} T^{-\ot})\simeq \frac{|\lambda^T_{l_i b'}|^2 }{ 64 \pi}M_{b'}\left(1-\frac{M_T^2}{M_{b'}^2}\right)^2\,,\;
\Gamma(b'\ra \ell^+_i T^{-\ft})\simeq \frac{|\lambda^T_{l_i b'}|^2}{ 32 \pi}M_{b'}\left(1-\frac{M_T^2}{M_{b'}^2}\right)^2\,,\nonr\\
\Gamma(b'\ra \nu_i D^{-\ot})\simeq \Gamma(b'\ra \ell^-_i D^{\tth})\simeq \frac{|\lambda^D_{l_i b'}|^2 }{ 64 \pi}M_{b'}\left(1-\frac{M_D^2}{M_{b'}^2}\right)^2\,,\nonr\\
\Gamma(b'\ra \ell_i S^{\tth})\simeq \frac{|\lambda^S_{l_i b'}|^2 }{ 32 \pi}M_{b'}\left(1-\frac{M_S^2}{M_{b'}^2}\right)^2\,.\;
\eeqa
The Yukawa coupling between $b'$ and the SM Higgs is through the $b'\mhyphen d$ mixing. So the resulting Yukawa coupling gets double suppression from the small $\theta_1$ and the ratio of $m_d/v_0$, and so the $b'\ra H d$ decay can be ignored. Similarly the decays of $b'\ra Z^0 d_i$ can be ignored as well.
By plugging in the parameters we found, the total decay width of $b'$ is $\Gamma_{b'}\simeq 325.1\mgev$, and the branching ratios are
\beqa
{\cal B}( b'\ra u W^- ) \simeq 0.5\%\,,\;
{\cal B}( b'\ra c W^- ) \simeq 2\times 10^{-4}\,,\;
{\cal B}( b'\ra t W^- ) \simeq 3\times 10^{-7}\,,\nonr\\
{\cal B}( b'\ra \bar{\nu} T^{-\ot} ) \simeq 12.2\%\,,\;
{\cal B}( b'\ra \mu^+ T^{-\ft} ) \simeq 23.4\%\,,\nonr\\
{\cal B}( b'\ra \tau^+ T^{-\ft} ) \simeq 0.9\%\,,\;
{\cal B}( b'\ra e S^{\tth} ) \simeq 62.9\%\,,
\eeqa
for $M_{LQ}=1\mtev$ and $M_{b'}=1.5\mtev$.
We stress that the above decay branching ratios are the result of using the example parameter set given in Eq.(\ref{eq:num_result}).
The decay branching ratios depend strongly on the model parameters, and the branching ratio pattern varies dramatically from one neutrino mass matrix to another\footnote{ In particular, in the minimal setup one has $\lambda^T_{\tau b'}/\lambda^T_{\mu b'} = {\cal M}^\nu_{e\tau}/{\cal M}^\nu_{e\mu}$.}. However, one can see that the decay pattern of the heavy exotic color states in this example solution is
very different from the working assumption of 100\% $b'\ra t W, Z b, H b$ used for singlet $b'$ and other assumption used for the leptoquark searches at the colliders.
Before closing this section, we want to point out some potentially interesting FCNC top 3-body decays in this framework.
From the example solution, we have
\beq
{\cal B}(t\ra u \tau^+ \mu^- ) \simeq {\cal B}(t\ra u \mu^+ \mu^-)\simeq 2.8\times 10^{-9}\,.
\eeq
With an integrated luminosity of $3\mbox{ab}^{-1}$ and CM energy at $13\mtev$, about $2.5\times 10^9$ top quark pair events will be produced at the LHC. Therefore, only $\sim 7$ events which include at least one top decaying in these 3-body FCNC are expected.
However, the 3-body FCNC $t\ra c l_i l_j$ branching ratios change if adopting a different solution.
During our numerical study, we observe that in some cases there are one or two of them
at the ${\cal O}(10^{-7})$ level, which might be detectable at the LHC.
See \cite{Kim:2018oih} for the prospect of studying these potentially interesting 3-body top decay modes at the LHC.
\subsection{Flavor violating neutral current processes }
A few comments on the data fitting are in order: From Table \ref{tab:NCTL_list}, one sees that the fit almost saturates, $\sim 80\%$, of the current limit on decay branching ratio $B^+\ra K^+ \mu^+\tau^- $. Moreover, the fit is very close to the $2\sigma$ limit from $B_s-\overline{B}_s$ mixing.
The solution seems to be stretched to the limit, and the discovery of lepton flavor violating signals are around the corners. However, this is the trade-off of using minimal number of parameters to reproduce all the central values. If instead aiming for the $1\sigma$ values, both can be reduced by half. In addition, the Yukawa couplings are tightly connected with the neutrino mass matrix. During our numerical study, we observe that $D^+\ra \pi^+\mu^+ \mu^-$ and $B^+\ra K^+ \mu^+\tau^- $ can be far below their experimental upper bounds while $\tau \ra \mu \gamma$ close to the current experimental limit if using some different neutrino mass matrix or relaxing the strict relationship that $\wt{V}^*_{cb}\lambda^T_{\mu b}+ \wt{V}^*_{cb'}\lambda^T_{\mu b'} =0$. Therefore, the model has vast parameter space to accommodate the anomalies with diversified predictions, and we cannot conclusively predict the pattern of the rare process rates at the moment.
However, because we need $ \lambda^T_{\tau b}\lambda^T_{\tau s}\neq 0$ to explain the $b\ra s \mu\mu$ anomaly, the $b\ra s \tau\tau$ transition will be always generated at the tree-level.
From the example solution, we have
\beqa
{\cal H}^{b\ra s \tau\tau}_{eff}& \simeq& -\frac{G_F}{\sqrt{2}} \wt{V}_{tb}\wt{V}_{ts}^* \frac{\alpha}{\pi}
{\cal C}^{bs\tau\tau}\left[\bar{s}\gamma^\alpha \PL b \right] \left[\bar{\tau}\gamma_\alpha (1-\gamma^5) \tau\right] +H.c.\,,\nonr\\
{\cal C}^{bs\tau\tau} &\simeq & \frac{\sqrt{2} \pi }{ 4 \alpha} { \lambda^T_{\tau b} (\lambda^T_{\tau s})^* \over \wt{V}_{tb}\wt{V}_{ts}^* G_F M_T^2 } = 24.6 \times \left({\mtev \over M_T}\right)^2\,,
\eeqa
if taking $\wt{V}_{tb}\wt{V}_{ts}^*=-0.03975$.
The additional 1-loop contribution via the box diagram similar to that of $b\ra s\mu\mu$ can be ignored.
This Wilson coefficient is roughly six times larger than the SM prediction that ${\cal C}^{bs\tau\tau}_{SM} \sim -4.3$\cite{Bobeth:1999mk,Huber:2005ig,DescotesGenon:2011yn}, and push the decay branching ratio to $Br(B_s\ra \tau^+\tau^-)\simeq 1.8\times 10^{-5}$.
Although the above value is still two orders below the relevant experimental upper limit, $\sim {\cal O}(10^{-3})$, for $B_s\ra \tau^+\tau^-$ at LHCb\cite{Aaij:2017xqt} and $B^+\ra K^+\tau^+\tau^-$ at BaBar\cite{TheBaBar:2016xwe},
this interesting $b\ra s \tau\tau$ transition could be potentially studied at the LHCb and Belle II\cite{Capdevila:2017iqn}, or at the Z-pole\cite{Li:2020bvr}.
\subsection{$B\ra D^{(*)}\tau \bar{\nu}$}
\begin{figure
\centering
\includegraphics[width=0.35\textwidth]{diag_RD.eps}
\caption{ The Feynman diagram for $b\ra c \tau \nu$ transition. }
\label{fig:RD}
\end{figure}
Alongside the $b\ra s l^+ l^-$ anomaly, the global analysis\cite{Amhis:2016xyh, Murgui:2019czp, Shi:2019gxi, Blanke:2019qrx, Kumbhakar:2019avh} of the $R(D^{(*)})={\cal B}(B\ra D^{(*)}\tau \nu)/{\cal B}(B\ra D^{(*)}\mu \nu) $ data \cite{Lees:2012xj, Lees:2013uzd, Aaij:2015yra, Aaij:2017deq, Aaij:2017uff, Abdesselam:2019dgh}
also point to the $\tau\mhyphen\mu$ lepton flavor universality violation with a significance of $\geq 3 \sigma$.
In this mode, the $\mbox{MinS}_T$ of $\lambda^T$ contains the needed tree-level $b\ra c\tau \nu$ operator, Fig.\ref{fig:RD}, to address the $R(D^{(*)})$ problem, and
\beq
{\cal H}^{CC}_{eff} \supset -\left[{\lambda^T_{\tau s} (\lambda^T_{\tau b})^*\wt{V}_{cs}+ \lambda^T_{\tau b} (\lambda^T_{\tau b'})^*\wt{V}_{c b'} + |\lambda^T_{\tau b}|^2 \wt{V}_{cb} \over 4 M_T^2}\right]\left (\bar{c} \gamma^\alpha \PL b \right)\left(\bar{\tau} \gamma_\alpha\PL \nu_\tau \right) + H.c.
\label{eq:CC_bctaunu}
\eeq
This is to compare with the standard effective Hamiltonian
\beq
{\cal H}_{eff}^{b\ra cl\nu} = \frac{4 G_F}{\sqrt{2}} V_{cb} [ (1+ {\cal C}_{V_L}){\cal O}_{V_L}
+ {\cal C}_{V_R} {\cal O}_{V_R} +{\cal C}_{S_L} {\cal O}_{S_L} +{\cal C}_{S_R} {\cal O}_{S_R}+{\cal C}_{T} {\cal O}_{T}
] +H.c.
\eeq
where
\beqa
{\cal O}_{V_{L,R}} &=& (\bar{c}\gamma^\mu b_{L,R})(\bar{l}_L\gamma_\mu \nu_{lL})\,,\nonr\\
{\cal O}_{S_{L,R}} &=& (\bar{c} b_{L,R})(\bar{l}_R \nu_{lL})\,,\;
{\cal O}_{T} = (\bar{c}\sigma^{\mu\nu} b_{L})(\bar{l}_R \sigma_{\mu\nu} \nu_{lL})\,.
\eeqa
In this model, the only CC operator can be generated at tree-level is ${\cal O}_{V_L}$, thus ${\cal C}_{V_{R}}=0$, ${\cal C}_{S_{R,L}}=0$, and ${\cal C}_{T}=0$.
From the global fit with single operator\cite{Murgui:2019czp}, the $R(D^{(*)})$ anomaly can be well addressed if ${\cal C}_{V_L}\simeq 0.08$.
However, the requirement to retain the $\mu\mhyphen e$ universality in $b\ra c l \nu$ strictly limits the parameter space.
By using the real MinS of parameters, we found the model can render at most ${\cal C}^{NP}_{V_L}\lesssim 0.01$.
On the other hand, we cannot rule out the possibility that the model has the viable complex number parameter space to accommodate this anomaly with others simultaneously.
On the other hand, this model predicts the lepton universality violation in the $b\ra u l_i \nu_j (i,j=\mu,\tau)$ transition, see the relevant Wilson coefficients in Table \ref{tab:CCTL_list}. Again, there is no electron counter parts if we use the $\mbox{MinS}_T$.
From our example solution, the rate of $b\ra u \mu \nu$ could deviate from the SM one by $\sim 40\%$ due to the smallness of $\wt{V}_{ub}$ and the interference between the NP and the SM weak interaction. More insights of the intriguing flavor problem are expected if the better experimental measurements on the $b\ra u l\nu$ transitions are available\cite{Colangelo:2020jmb}.
\subsection{Origin of the flavor structure}
Not only the subset of parameters are required to explain the anomalies, the nearly vanishing entities in the Yukawa matrices play vital roles to bypass the strong flavor-changing experimental constraints.
The next question is how to understand the origin of this staggering flavor pattern.
Usually, the flavor pattern is considered within the framework of flavor symmetries.
It is highly nontrivial to embed the flavor pattern we found into a flavor symmetry, and it is beyond the scope of this paper.
Alternatively, we discuss the possible geometric origin of the flavor pattern in the extra-dimensional theories\cite{ArkaniHamed:1998rs, Antoniadis:1998ig,Randall:1999ee,ArkaniHamed:1999dc}.
A comprehensive fitting, including all the SM fermion masses and mixings, like \cite{Chang:2002ww, Chang:2008vx, Chang:2008zx, Chang:2010ic} is also beyond the scope of this paper.
Instead, here we only consider how to generate the required flavor pattern of the leptoquark Yukawa coupling shown in Eq.(\ref{eq:recap_range}).
To illustrate, we consider a simple split fermion toy model\cite{ArkaniHamed:1999dc}. We assume all the chiral fermion wave functions are Gaussian locating in a small region, but at different positions, in the fifth dimension.
Moreover, all the 5-dim Gaussian distributions are assumed to share a universal width, $\sigma_{SF}$.
As for the three leptoquarks, we assume they do not have the zero mode such that their first Kaluza-Klein(KK) mode are naturally heavy.
More importantly, this setup forbids the leptoquark to develop VEV and break the $SU(3)_c$ symmetry.
In addition, the wavefunctions of the first leptoquark KK mode are assumed to be slowly varying in the vicinity of the fermion cluster,
and can be approximated as constants. On the other hand, the SM Higgs must acquires the zero mode so that it can develop $v_0$ and breaks the SM electroweak symmetry.
Then, the 4-dim effective theory is obtained after integrating out the fifth dimension.
The effective $\lambda^{s}_{ij}(s=T,D,S)$ Yukawa couplings is determined by the overlapping of two chiral fermions' 5-dim wave functions times the product of the scalar-specific 5-dim Yukawa coupling and the scalar's 5-dim wavefunction, denoted as $N_s$, in the vicinity where the fermions locate. The 4-dim Yukawa coupling is given by
\beq
\lambda^{s}_{ij} = N_s\, \mbox{Exp}\left[- \frac{(z_i-z_j)^2}{2 \sigma_{SF}^2 } \right]\,,\; (s= T,D,S)\,,
\eeq
where $z_i$ is the center location in the fifth dimension of the Gaussian wave function of particle$-i$.
It is clear that only the relative distances matter, so we arbitrarily set $z_{\tau_L}=0$ for the LH tau.
Note that different fermion chiralities are involved for different scalar leptoquark Yukawa couplings.
For example, $\lambda^T_{ij}$ is determined by the separation between the corresponding LH down-quark($z_{d_{jL}}$) and the LH lepton($z_{\ell_{iL}}$), but $z_{l_{iR}}$ and $z_{d_{jL}}$ are involved for $\lambda^S_{ij}$.
For simplicity, we assume the mass and interaction eigenstates coincide for the down-type quarks and the SM charged leptons.
We found the flavor structure can be excellently reproduced
if the chiral fermion locations in the fifth dimension are
\beqa
\{ z_{\tau_L},z_{\mu_L}, z_{e_L}, z_{\mu_R}, z_{e_R}\}=\{0, -0.57, 7.52, -5.00, -1.23\}\times \sigma_{SF} \,,\nonr\\
\{ z_{d_L},z_{s_L}, z_{b_L}, z_{b'_L}, z_{b_R}, z_{b'_R}\}=\{-4.38, 2.83, -2.19, -1.62, -1.70, 3.53\}\times \sigma_{SF} \,,
\eeqa
and $\{N_T,N_D,N_S\}=\{5.0, 7.0, 7.0 \}$.
The above configuration of split fermion locations results in
\beqa
|\lambda^T_{\mu b'}|=2.86\,,\,
|\lambda^T_{\tau b'}|=1.34\,,\,
|\lambda^T_{\mu b}|=1.34\,,\,
|\lambda^T_{\tau b}|=0.45\,,\,
|\lambda^T_{\tau s}|=8.9\times 10^{-2}\,,\,\nonr\\
|\lambda^D_{e b'}|= 2.5\times 10^{-3}\,,\,
|\lambda^D_{\mu b}|= 3.67\,,\,
|\lambda^D_{\tau b'}|= 1.4\times 10^{-2}\,,\,
|\lambda^D_{\tau b}|=1.64\,,\,
\nonr\\
|\lambda^S_{e b'}|= 6.47\,,\,
|\lambda^S_{\mu b}|=0.13\,.
\label{eq:num_SFfit}
\eeqa
Comparing to Eq.(\ref{eq:num_result}), one can see that all the Yukawa coupling magnitudes agree with the fitted values within $\lesssim 60\%$.
Moreover, the parameters we set to zero to evade the stringent constraints from Koan and muon data are indeed very small,
\beqa
|\lambda^D_{e b}|= 2.3 \times 10^{-18}\,,\,
|\lambda^D_{\mu b'}|= 1.6 \times 10^{-3}\,,\nonr\\
|\lambda^T_{e d}|=9.1 \times 10^{-31}\,,\,
|\lambda^T_{e s}|=8.6 \times 10^{-5}\,,\,
|\lambda^T_{e b}|=1.6 \times 10^{-20}\,,\,\nonr\\
|\lambda^T_{e b'}|=3.4 \times 10^{-18}\,,\,
|\lambda^T_{\mu d}|=3.5 \times 10^{-3}\,,\,
|\lambda^T_{\tau d}|= 3.5 \times 10^{-4}\,,
\eeqa
in this given split fermion location configuration.
Finally, the lepton number symmetry is broken if $\mu_3\neq 0$. The phenomenological solution we found only calls for a very tiny
$\mu_3\sim {\cal O}(0.2) \mkev$.
The smallness of $\mu_3$ can be arranged by assigning different orbiforlding parities to $T$ and $D$ such that their 5-D wave functions are almost orthogonal to each other and leads to the tiny 4D effective mixing\footnote{For instance, one takes $(+-)$ and the other takes $(-+)$ Kaluza-Klein parity on the $S_1/(Z_2\times Z_2)$ orbiford.}. Contrarily, $T$ and $D$ should share the same orbifolding parities such that the maximal mixing yields a large effective 4D mixing $\mu_1\sim {\cal O}(\mtev)$.
On the other hand, in terms of flavor symmetry, the smallness of $\mu_3$ seems to indicate the global/gauged lepton number symmetry is well preserved and only broken very softly or radiatively.
\section{Conclusion}
\label{sec:conclusion}
We proposed a simple scenario with the addition of three scalar leptoquarks $T(3,3,-1/3)$, $D(3,2,1/6)$, $S(3,1,2/3)$, and one pair of down-quark-like vector fermion $b'_{L,R}(3,1,-1/3)$ to the SM particle content. The global baryon number $U(1)_B$ is assumed for simplicity.
This model is able to accommodate the observed $\tri a_{e,\mu}$, $R(K)$, Cabibbo angle anomalies, and pass all experimental limits simultaneously. Moreover, the right pattern of neutrino oscillation data can be reproduced as well.
We have shown the existence of phenomenologically viable model parameter set by furnishing one example configuration with the minimal number of real Yukawa couplings.
For the possible UV origin, we provided a split fermion toy model to explain the flavor structure embedded in the viable model parameter set. It will be interesting to reproduce the flavor pattern by nontrivial flavor symmetry. However, the tiny lepton number violating parameter, $\mu_3 \sim {\cal O}(0.2) \mkev$, seems to indicate the possible link of global/gauged lepton number and the unknown underling flavor symmetry.
In addition to the smoking gun signatures, the discovery of these new color states, this model robustly predicts the neutrino mass is of the normal hierarchy type with ${\cal M}^\nu_{ee}\lesssim 3\times 10^{-4} \mev$.
The $R(D^{(*)})$ anomaly can only be partially addressed in this model if one employs the minimal number of real Yukawa couplings.
However, we cannot rule out the possibility that could be achieved by using more (complex) parameters.
From the parameter set example, more motivated heavy color state decay branching ratios should be taken into account in their collider searches.
\section*{Acknowledgments}
WFC thanks Prof. John Ng for his comments on the draft. This work is supported by the Ministry of Science and Technology (MOST) of Taiwan under Grant No.~MOST-109-2112-M-007-012.
| 44,707 |
\section{Introduction} \label{sec:introduction}
The availability of unused frequency resources at millimeter-wave (mmWave) frequencies and the ever-increasing demand for higher data rates make mmWave MIMO systems a natural choice for the next generation wireless communications~\cite{rappaport2013millimeter,ghosh2014millimeter}. One of the major challenges in operating at mmWave frequencies is the heavy path loss with weak or no line-of-sight (LoS) components~\cite{wan2020broadband,mo2014channel}. Therefore ensuring a strong LoS component is necessary to establish a reliable communication link.
Reconfigurable intelligent surfaces (RISs) can be used to control the propagation environment and ensure a reasonable LoS communication link~\cite{basar2019wireless, ozdogan2019intelligent, ozdogan2020using, bjornson2020intelligent, najafi2020physicsbased, arun2019rfocus}. RIS is a two-dimensional structure consisting of many passive sub-wavelength elements, which act as diffuse scatterers. By controlling the surface impedance of these elements, we can steer and focus the energy of the electromagnetic wave impinging on the RIS to any desired direction~\cite{najafi2020physicsbased}. We can view these elements as phase shifters, each of which can be controlled independently and remotely via a low-rate link~\cite{basar2019wireless}. The channel estimation problem in a RIS-assisted MIMO system amounts to estimating the MIMO channel between the transmitter, e.g., a user equipment~(UE), and the RIS and the MIMO channel between the RIS and the receiver, e.g., a base station~(BS). When the channel is known, the RIS phase shifts can be designed to form beams in any desired direction. However, due to the passive nature of the RIS with no pilot decoding or transmission capabilities, we cannot readily use conventional pilot-based channel estimation techniques.
\subsection{Related prior works} \label{subsec:intro:priorworks}
Recent works on channel estimation in RIS-assisted mmWave MIMO systems can be broadly classified as follows.
\begin{itemize}
\item {\bf RIS with a few active elements}: Interleaving RISs with a small number of active elements to decode pilots allows a separate estimation of the UE-RIS and the RIS-BS channels. With such a RIS configuration, channel estimation algorithms for non-parametric and angular channel models were proposed in~\cite{taha2019enabling} and~\cite{xiao2021low_complexity}, respectively.
\item {\bf RIS with passive-only elements}: RIS with passive-only elements are desirable because of their lower power consumption and hardware costs. Existing techniques estimate a cascaded channel, i.e., the overall channel between the UE and BS via the RIS exploiting the inherent sparsity in the angular domain in mmWave MIMO channels~\cite{wang2020compressed,he2020channelirw,he2020channelanm}. In~\cite{he2020cascaded}, a bilinear matrix factorization technique was proposed to factorize the cascaded channel matrix into low-rank UE-RIS and RIS-BS channel matrices.
\end{itemize}
Uniquely estimating the UE-RIS and RIS-BS channels is challenging because of the ambiguity involved in resolving the complex path gains and angles (see Sec.~\ref{sec:uniqueness} for details). Unlike the RIS-BS channel, which is mostly time-invariant due to the fixed locations of the RIS and the BS, the UE-RIS channel can be time-varying due to the mobility of the UE~\cite{xiao2021low_complexity}. Separately estimating the UE-RIS and RIS-BS channels allows us to exploit any difference in the time scales of variation and avoid unnecessary estimation of the cascaded channel or its factorization each time. In addition, in a multi-user setting, estimating only the UE-RIS channels with a shared RIS-BS channel is computationally less expensive. The RIS-BS channel is usually LoS and is completely characterized (up to a complex path gain) by the known locations of the RIS and BS~\cite{wan2020broadband}. Therefore, the channel estimation amounts to estimating the UE-RIS channel, or equivalently estimating the complex path gain and localizing the UE, which are the main goals of this work.
\subsection{Main results and contributions} \label{subsec:intro:mainresults}
In this work, we propose a pilot-based uplink channel estimation algorithm to estimate the unknown channel parameters, namely, the angle of departure (AoD) at the UE, azimuth and elevation angles of arrival (AoA) at the RIS, and the overall complex path gain of the UE-RIS-BS link.
One of the main results of this work is, we show that at least two channel soundings with different RIS phase shifts are required to estimate the complex path gain and the angles uniquely, and thus the UE-RIS channel. The channel estimation is carried out entirely at the BS without any direct link (low-rate feedback or communication) between the BS and UE. Through simulations, we demonstrate that the proposed technique performs similarly in terms of normalized mean squared error and average spectral efficiency to an oracle estimator, which assumes perfect knowledge of the angles associated with the UE-RIS and RIS-BS links.
\begin{table}[t]
\begin{center}
\small
\begin{tabular}{l l}
\hline
Symbol & Definition \\ [0.5ex]
\hline\hline
$\theta_{\rm u}$ &AoD at the UE \\
$(\phi_{\rm u}, \psi_{\rm u})$ & Elevation and azimuth AoA at the RIS from the UE \\
$\theta_{\rm b}$ & AoA at the BS \\
$(\phi_{\rm b}, \psi_{\rm b})$ & Elevation and azimuth AoD from the RIS to the BS \\
$g$ & Complex path gain \\
\hline
\end{tabular}
\end{center}
\vspace{-2mm}
\caption{Channel parameters.}
\label{table:1}
\vspace*{-7mm}
\end{table}
\section{RIS-assisted MIMO channel}~\label{sec:prob_model}
In this section, we introduce the signal model and describe the main challenge involved in estimating the UE-RIS channel.
\subsection{Angular MIMO channel model}
Consider a MIMO communication system having a BS with a uniform linear array (ULA) of $N_{\rm{b}}$ antennas, a UE with a ULA of $N_{\rm{u}}$ antennas, and a RIS with a uniform planar array (UPA) of $N_{\rm{r}}$ passive phase shifters. Let us define the array response vector of a ULA with $N$ antennas as
$\mathbf{a}(u,N) = \begin{bmatrix} 1 & e^{-j 2\pi d u/\lambda} & \cdots & e^{-j (N-1) 2\pi d u/\lambda} \end{bmatrix}^T.$ Here, $u$ is the direction cosine, $d$ is the inter-element spacing, and $\lambda$ is the signal wavelength. Then the array response vectors of the arrays at the BS, UE, and RIS are, respectively, defined as $\mathbf{a}_{\rm B}(\theta) = \mathbf{a}({\rm sin(\theta)}, N_{\rm b})$, $\mathbf{a}_{\rm U}(\theta) = \mathbf{a}({\rm sin(\theta)}, N_{\rm u})$, and $\mathbf{a}_{\rm R}(\phi,\psi) = \mathbf{a}({\rm sin(\phi) sin(\psi)} , N_{\rm x}) \otimes \mathbf{a}({\rm sin(\phi) cos(\psi)} , N_{\rm y})$. Here, $N_{\rm x}$ and $N_{\rm y}$ denote the number of RIS elements in the horizontal and vertical directions, respectively, so that $N_{\rm r} = N_{\rm x}N_{\rm y}$, and $\otimes$ denotes the Kronecker product.
We assume that the direct path between the BS and UE is blocked and that the RIS is used to establish a non-direct UE-BS link. Let us denote the UE-RIS and RIS-BS MIMO channel matrices by $\mathbf{H}_{\rm{ur}} \in \mathbb{C}^{N_{\rm r} \times N_{\rm u}}$ and $\mathbf{H}_{\rm{rb}} \in \mathbb{C}^{N_{\rm b} \times N_{\rm r}}$, respectively. Let us collect the phase shifts in the diagonal matrix $\boldsymbol{\Omega} \in \mathbb{C}^{N_{\rm r} \times N_{\rm r}}$. Usually, the BS and RIS are situated in environments with limited local scattering, for which $\mathbf{H}_{\rm rb }$ will be an LoS channel matrix~\cite{wan2020broadband}. We also assume that the UE-RIS link is LoS. Although this is a simplifying assumption, it is reasonable because of the excessive path loss at mmWave frequencies~\cite{wan2020broadband,mo2014channel}. Under these assumptions, both $\mathbf{H}_{\rm rb}$ and $\mathbf{H}_{\rm ur}$ are rank-1 matrices defined as
\[
\mathbf{H}_{\rm{ur}} = g_{\rm{ur}}\mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u})\mathbf{a}_{\rm U}^{H}(\theta_{\rm u}),
\,\,\, \mathbf{H}_{\rm{rb}} = g_{\rm{rb}}\mathbf{a}_{\rm B}(\theta_{\rm b})\mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b}),
\]
where $g_{\rm{ur}}$ and $g_{\rm{rb}}$ denote the complex path gains, $\theta_{\rm u}$ is the AoD at the UE, and $\phi_{\rm u}$ and $\psi_{\rm u}$ are, respectively, the elevation and azimuth angles at the RIS made by the LoS path arriving from the UE, $\theta_{\rm b}$ is the AoA at the BS, and $\phi_{\rm b}$ and $\psi_{\rm b}$ are the angles made by the LoS path departing from the RIS towards the BS in the elevation and azimuth directions, respectively. Then the cascaded MIMO channel matrix is given by
\begin{equation} \label{eq:model_1}
\mathbf{H} = g \mathbf{a}_{\rm B}(\theta_{\rm b})\mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b}) \boldsymbol{\Omega} \mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u})\mathbf{a}_{\rm U}^{H}(\theta_{\rm u}),
\end{equation}
where $g = g_{\rm{rb}}g_{\rm{ur}}$ is the overall complex path gain.
The channel parameters, which we frequently refer, are summarized in~Table~\ref{table:1}.
The main aim of this work is to estimate the parameters $\{\theta_{\rm u}, \phi_{\rm u}, \psi_{\rm u}, g\}$ assuming that the locations of the BS and RIS are known. In other words, we localize the UE by finding the directions $\theta_{\rm u}$ and $(\phi_{\rm u},\psi_{\rm u})$.
\subsection{Uniqueness} \label{sec:uniqueness}
Let us decompose the channel matrix as ${\mathbf{H}} = c \mathbf{a}_{\rm B}(\theta_{\rm b}) \mathbf{a}_{\rm U}^{H}(\theta_{\rm u})$
with $c = g\mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b}) \boldsymbol{\Omega} \mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u})$.
Since the scalar parameter $c$ is a product of two complex numbers $\mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b}) \boldsymbol{\Omega} \mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u})$ and $g$, it is not possible to uniquely identify $(\phi_{\rm u},\psi_{\rm u})$ and $g$ from ${\bf H}$ as there are different set of angles $(\phi_{\rm u},\psi_{\rm u})$ and $g$ that result in the same product $c$. This ambiguity makes the UE-RIS channel estimation challenging.
To resolve this ambiguity, suppose we sound the channel with two different phase shifts $\boldsymbol{\Omega}_{1}$ and $\boldsymbol{\Omega}_{2}$ to obtain
\begin{align}
c_{i} {=} \,\, g \mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b}) \boldsymbol{\Omega}_{i} \mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u}), \quad\label{eq:ci}
\end{align}
for $ i=1,2$. By computing $c_{1}/c_{2}$, we can eliminate $g$ in \eqref{eq:ci} as
\begin{equation} \label{eq:modified_stage2}
\mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b})( c_{1}\boldsymbol{\Omega}_{2} - c_{2}\boldsymbol{\Omega}_{1})\mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u}) = 0.
\end{equation}
Therefore, it is immediately clear that {\it at least two} channel soundings are required to resolve the ambiguity. For robustness against noise, we extend~\eqref{eq:modified_stage2} to {\it multiple channel soundings} with different RIS phase shift matrices and uplink pilots as discussed next.
\subsection{Uplink training} \label{sec:signal_model}
Let $\mathbf{S} \in \mathbb{C}^{N_{\rm u} \times M}$ be the uplink pilot matrix transmitted from the UE with $M$ denoting the number of channel uses. We perform multiple channel soundings with the same $\mathbf{S}$ over $L$ training blocks, but with different phase shifts $\boldsymbol{\Omega}_{i}$ for $i=1,2,\ldots,L$.
The signal received at the BS during the $i$th training block, $\mathbf{X}_i \in \mathbb{C}^{N_{\rm b}\times M}$, is given by
\begin{equation} \label{symb1}
\mathbf{X}_i = \mathbf{H}_i \mathbf{S} + \mathbf{N}_i, \quad i=1,\ldots, L,
\end{equation}
where $\mathbf{H}_{i} = g \mathbf{a}_{\rm B}(\theta_{\rm b})\mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b}) \boldsymbol{\Omega}_{i} \mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u})\mathbf{a}_{\rm U}^{H}(\theta_{\rm u})$, $\mathbf{N}_i \in \mathbb{C}^{N_{\rm b}\times M} $ is the noise matrix whose each entry follows a complex Gaussian distribution as $[\mathbf{N}_i]_{mn} \sim \mathcal{C}\mathcal{N}(0,1)$.
Without loss of generality, we consider the pilot matrix to be orthogonal with $M = N_{\rm u}$ so that $\mathbf{S}\mathbf{S}^{H} = \mathbf{S}^{H}\mathbf{S} = PN_{\rm u}^{-1}\mathbf{I}_{N_{\rm u}}$. Here, $P$ is the signal-to-noise ratio (SNR).
\section{Proposed algorithm} \label{sec:ch_estm_algo}
In this section, we develop an algorithm to estimate the UE-RIS channel based on the observations from multiple channel soundings. The algorithm comprises of three stages to estimate the {\it AoD at the UE}, {\it AoA from UE at RIS}, and the {\it path gain}. Before describing the algorithm, let us first obtain a coarse estimate of the channel in each block as
\begin{equation} \label{eq:hnoisegen}
\Tilde{\mathbf{H}}_{i} = \frac{N_{\rm u}}{P}\mathbf{X}_{i}\mathbf{S}^{H} = \mathbf{H}_{i} + \mathbf{W}_{i}, \quad i=1,2,\ldots,L,
\end{equation}
where $\mathbf{W}_{i} = N_{\rm u} P^{-1}\mathbf{N}_{i}\mathbf{S}^{H}$ is the noise term after pilot removal.
\subsection{Estimation of the AoD at the UE}
To estimate the AoD at the UE, i.e., $\theta_{\rm u}$, we use MUSIC~\cite{vantrees2002optimum}, which is a subspace-based direction finding method. In the absence of noise, we have $\mathcal{R}(\Tilde{\mathbf{H}}_{i}^{H}) = \mathcal{R}(\mathbf{a}_{\rm U}(\theta_{\rm u})) \> \forall \> i=1,2,\ldots,L$. Here, $\mathcal{R}({\bf H})$ denotes the column span of ${\bf H}$. We can estimate $\mathcal{R}(\mathbf{a}_{\rm U}(\theta_{\rm u}))$ from the observations from all the $L$ blocks by forming the rank-1 matrix ${\mathbf{G}} \in \mathbb{C}^{N_{\rm u} \times LN_{\rm b}}$ as
\begin{equation}
\mathbf{G} = \begin{bmatrix} \Tilde{\mathbf{H}}_{1}^{H} & \Tilde{\mathbf{H}}_{2}^{H} & \ldots & \Tilde{\mathbf{H}}_{L}^{H} \end{bmatrix},
\end{equation}
where $\mathcal{R}(\mathbf{G}) = \mathcal{R}(\mathbf{a}_{\rm U}(\theta_{\rm u}))$. Let us denote the subspace orthogonal to the one-dimensional subspace $\mathcal{R}(\mathbf{a}_{\rm U}(\theta_{\rm u}))$
by $\mathbf{U}$ so that $\mathbf{U}^H\mathbf{a}_{\rm U}(\theta_{\rm u}) = {\bf 0}$. We may compute $\mathbf{U}$ using the singular value decomposition (SVD) of $\mathbf{G}$ and by considering the left singular vectors corresponding to all but its largest singular value. In the presence of noise, we minimize $\vert \vert \mathbf{U}
^{H}\mathbf{a}_{\rm U}(\theta) \vert \vert^{2}$ with respect to $\theta$. The estimate of the AoD at the UE, denoted by $\hat{\theta}_{\rm u}$, can be obtained from the peak of the pseudo spectrum
\begin{equation} \label{eq:theta_est}
\mathcal{P}_{\rm UE}(\theta) = {\vert \vert \mathbf{U}^{H}\mathbf{a}_{\rm U}(\theta) \vert \vert^{-2}}
\end{equation}
by sweeping over $\theta$.
Although $\theta_{\rm u}$ may also be estimated using any other standard direction finding technique (e.g., sparse recovery or maximum likelihood estimators), the estimation of $(\phi_{\rm u},\psi_{\rm u})$ that we discuss next is more challenging as we do not have access to the measurements at the RIS.
\subsection{Estimation of the AoA from the UE at the RIS}
Given $\hat{\theta}_{\rm u}$, we can obtain noisy measurements of $c_{i}$ by multiplying $\tilde{\mathbf{H}}_{i}$ in \eqref{eq:hnoisegen} from the left with $\mathbf{a}_{\rm B}^H(\theta_{\rm b})$ and from the right with $\mathbf{a}_{\rm U}(\hat{\theta}_{\rm u})$ as $\mathbf{a}_{\rm B}^H(\theta_{\rm b})\tilde{\mathbf{H}}_{i}\mathbf{a}_{\rm U}(\hat{\theta}_{\rm u})$ to obtain [cf. \eqref{eq:ci}]
\begin{equation}
\mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b})( c_{i}\boldsymbol{\Omega}_{j} - c_{j}\boldsymbol{\Omega}_{i})\mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u}) = e_i,
\label{eq:noisyci}
\end{equation}
for $i,j=1,2, \ldots, L$. Here, $e_i$ is the error due to the noise~$\mathbf{W}_i$.
Using \eqref{eq:noisyci}, we pose the problem of estimating $(\phi_{\rm u}, \psi_{\rm u})$ as the problem of computing the solution to a system of non-linear equations. Although there are multiple ways to form the required system of equations, we may stack \eqref{eq:noisyci} for $i=1$ and $j=2,3,\ldots, L$ to form
\begin{equation} \label{phipsicombined}
\underbrace{ \begin{bmatrix} \mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b})( c_{1}\boldsymbol{\Omega}_{2} - c_{2}\boldsymbol{\Omega}_{1}) \\
\mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b})( c_{1}\boldsymbol{\Omega}_{3} - c_{3}\boldsymbol{\Omega}_{1}) \\
\vdots \\
\mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b})( c_{1}\boldsymbol{\Omega}_{L} - c_{L}\boldsymbol{\Omega}_{1})\end{bmatrix} }_\text{$\Tilde{\mathbf{A}} \in \mathbb{C}^{(L-1)\times N_{\rm r}}$} \mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u}) = {\bf e},\end{equation}
where ${\bf e}$ denotes the error vector. Since we do not know the direction of the UE during the channel estimation phase, $\boldsymbol{\Omega}_{i}$, $i=1,2,\ldots,L$ are randomly chosen such that all the directions are excited. Thus the matrix $\Tilde{\mathbf{A}}$ will be full row rank.
Let us denote the basis for $\mathcal{R}(\Tilde{\mathbf{A}}^{H})$ as $\mathbf{Q} \in \mathbb{C}^{N_{\rm r}\times (L-1)}$, which may be obtained using the SVD of $\Tilde{\mathbf{A}}^{H}$ and by considering the left singular vectors corresponding to the $(L-1)$ largest singular values of $\Tilde{\mathbf{A}}^{H}$. In the noiseless setting with ${\bf e} = {\bf 0}$ (as in \eqref{eq:ci}), $\mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u})$ lies in the null space of $\Tilde{\bf A}$ so that $\vert\vert \mathbf{Q}^{H}\mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u}) \vert\vert^{2} = 0$. Therefore, in presence of noise, minimizing $\vert\vert \mathbf{U}_{s}^{H}\mathbf{a}_{\rm R}(\phi,\psi) \vert\vert^{2}$ with respect to $(\phi,\psi)$ amounts to solving the system of non-linear equations in \eqref{phipsicombined} in the least squares sense. Thus, we can obtain the estimates of the AoA at the RIS, $(\hat{\phi}_{\rm u},\hat{\psi}_{\rm u})$, by computing the locations of the peaks of the pseudo spectrum
\begin{equation} \label{eq:phi_psi_est}
\mathcal{P}(\phi,\psi) = {\vert\vert \mathbf{U}_{s}^{H}\mathbf{a}_{\rm R}(\phi,\psi) \vert\vert^{-2}},
\end{equation}
by sweeping over $(\phi,\psi)$
\subsection{Estimation of the path gain} \label{sec:estm_g}
Once we have the estimates of the AoA and AoD available, estimating the path gain $g$ can be done using least squares. Let us define the following $N_{\rm b}$-length vectors
$\mathbf{b}_{i} = \Tilde{\mathbf{H}}_{i}\mathbf{a}_{{\rm U}}({\theta}_{\rm u})$ and
${\bf v}_i = N_{\rm u}\mathbf{a}_{\rm B}(\theta_{\rm b})\mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b}) \boldsymbol{\Omega}_{i} \mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u})$. Then, from \eqref{eq:model_1} and \eqref{eq:hnoisegen}, we have
\begin{align} \label{bdef1}
\mathbf{b}_{i} &= g{\bf v}_i + \mathbf{z}_{i}, \> i=1,2,\ldots,L,
\end{align}
where $\mathbf{z}_{i} = \mathbf{W}_{i}\mathbf{a}_{{\rm U}}({\theta}_{\rm u})$ is the noise vector.
By defining the length-$LN_{\rm b}$ vectors
$\mathbf{b} = [{\bf b}_1^T,\ldots, {\bf b}_L^T]^T$, ${\mathbf{v}} = [{\bf v}_1^T,\ldots, {\bf v}_L^T]^T$, and ${\bf z} = [{\bf z}_{1}^{T},\ldots,{\bf z}_{L}^{T}]^{T}$, we can estimate the complex path gain in the least squares sense as
\begin{equation} \label{gamma_est}
\hat{g} = \frac{{\mathbf{v}}^{H}\mathbf{b}}{\vert \vert {\mathbf{v}} \vert \vert ^{2}},
\end{equation}
where we use the angle estimates $({\phi}_{\rm b},{\psi}_{\rm b}) = (\hat{\phi}_{\rm b},\hat{\psi}_{\rm b})$ to compute ${\bf v}$ and ${\theta}_{\rm u} = \hat{\theta}_{\rm u}$ to compute ${\bf b}$.
\section{Numerical experiments} \label{sec:numerical_simulations}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\columnwidth]{Images/fig1_spectrum.eps}
\vspace*{-2mm}
\caption{Pseudo spectrum~\eqref{eq:phi_psi_est} with $L= 5$, $N_{\rm r} = 16 \times 16$, ${\rm SNR} = -10~{\rm dB}$.}
\label{fig1:phi_psi_spec}
\vspace*{-6mm}
\end{figure}
\begin{figure*}[t]
\begin{subfigure}[c]{0.31\columnwidth}\centering
\includegraphics[width=\columnwidth]{Images/fig2_MSE_v_SNR.eps}
\caption{}
\label{fig2:MSEvssnr}
\end{subfigure}
~
\begin{subfigure}[c]{0.31\columnwidth} \centering
\includegraphics[width=\columnwidth]{Images/fig3_SE_v_L.eps}
\caption{}
\label{fig3:MSEvsL}
\end{subfigure}
~
\begin{subfigure}[c]{0.31\columnwidth} \centering
\includegraphics[width=\columnwidth]{Images/fig4_SE_v_SNR.eps}
\caption{}
\label{fig4:SEvsSNR}
\end{subfigure}
\vspace*{-2mm}
\caption{ (a) NMSE for different SNR with $L= 5$.
(b) NMSE for different $L$ at ${\rm SNR} = -10~{\rm dB}$. (c) Average SE for different SNR for $L= 2$.}
\vspace*{-6mm}
\end{figure*}
In this section, we present simulation results to demonstrate
the effectiveness of the proposed channel estimation technique by computing the normalized mean square error (NMSE) and average spectral efficiency (SE).
For comparison, we use a scheme that assumes a perfect knowledge of the angles at the UE, RIS, and BS, thereby reducing the problem of channel estimation to that of estimating the overall complex path gain $g$. This method is referred to as the \textit{oracle least squares}~(\texttt{OracleLS}) estimator, which also forms the benchmark for compressed sensing based parametric channel estimation schemes~\cite{wang2020compressed,he2020channelirw,he2020channelanm}.
We define the NMSE as
\begin{equation}
{\rm NMSE} = \frac{\mathbb{E}\left[\| \mathbf{H}_{L} - \hat{\mathbf{H}}_{L} \|_{F}^{2}\right]}{\mathbb{E}\left[\| \mathbf{H}_{L} \|_{F}^{2} \right]},
\end{equation}
where $\hat{\mathbf{H}}_{L} = \hat{g} \mathbf{a}_{\rm B}(\theta_{\rm b})\mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b}) \boldsymbol{\Omega}_{L} \mathbf{a}_{\rm R}(\hat{\phi}_{\rm u},\hat{\psi}_{\rm u})\mathbf{a}_{\rm U}^{H}(\hat{\theta}_{\rm u})$.
Average SE indicates the performance of a RIS-assisted MIMO communication system, where the precoders at the UE and the phase shift matrix at the RIS are selected based on the estimated angles $\hat{\theta}_{\rm u}$, $\hat{\phi}_{\rm u}$, and $\hat{\psi}_{\rm u}$ to obtain the maximum transmit diversity and best possible reflection, respectively. The optimum choice of the phase shift matrix $\boldsymbol{\Omega} = {\rm diag}(\boldsymbol{\omega})$ to obtain the best possible reflection to focus the signal from the transmitter to the receiver is given by
$ {\rm angle}([{\boldsymbol{\omega}}]_{i}) = {\rm angle}([\mathbf{a}_{\rm R}({\phi}_{\rm u},{\psi}_{\rm u})]_{i}) - {\rm angle}([\mathbf{a}_{\rm R}(\phi_{\rm b},\psi_{\rm b})]_{i})$~\cite{he2020channelirw}. We now define the average ${\rm SE} = \mathbb{E}\left[ \log(1 + \vert \mathbf{w}^{H} \mathbf{H}(\hat{\boldsymbol{\Omega}})\mathbf{f} \vert ^2 P) \right]$,
where $\mathbf{w} = \frac{1}{\sqrt{N_{\rm b}}}\mathbf{a}_{\rm B}(\theta_{\rm b})$ is the combiner at the BS, $\mathbf{f} = \frac{1}{\sqrt{N_{\rm u}}}\mathbf{a}_{\rm U}(\hat{\theta}_{\rm u})$ is the optimum precoder at the UE, $\mathbf{H}(\hat{\boldsymbol{\Omega}}) = g \mathbf{a}_{\rm B}(\theta_{\rm b})\mathbf{a}_{\rm R}^{H}(\phi_{\rm b},\psi_{\rm b}) \hat{\boldsymbol{\Omega}} \mathbf{a}_{\rm R}(\phi_{\rm u},\psi_{\rm u})\mathbf{a}_{\rm U}^{H}(\theta_{\rm u})$, and $\hat{\boldsymbol{\Omega}} = {\rm diag}(\hat{\boldsymbol{\omega}})$ is the RIS phase matrix computed from the estimated angles.
We consider $N_{\rm u} = 8$ and $N_{\rm b} = 12$ throughout the simulations and the results are obtained by averaging over 1000 independent realizations of the receiver noise and unit-modulus complex path gain. An inter-element spacing of ${\lambda}/{2}$ is considered for the ULAs at the BS and UE. We have selected a sub-wavelength~\cite{najafi2020physicsbased} inter-element spacing of ${\lambda}/{5}$ for the UPA at the RIS. The phase shifts are selected randomly by uniformly sampling the unit circle. We considered a setup with $\theta_{\rm b} = 40^{0}$, $\theta_{\rm u} =40 ^{0}$, $\phi_{\rm b} =50 ^{0}$, $\psi_{\rm b} = 65^{0}$, $\phi_{\rm u} = 50^{0}$, and $\psi_{\rm u} =30 ^{0}$. The angles $\theta_{\rm b}$, $\phi_{\rm b}$ and $\psi_{\rm b}$ are considered to be known from the knowledge of the positions of the RIS and BS. The remaining angles as well as the complex path gain are estimated.
In Fig. \ref{fig1:phi_psi_spec}, we illustrate the pseudo spectrum $\mathcal{P}(\phi,\psi)$ in~\eqref{eq:phi_psi_est} for an SNR of $-10~{\rm dB}$ and for $L=5$. We can see that with multiple soundings, the pseudo spectrum results in a reasonable estimate of $(\phi_{u},\psi_{u})$ with a peak at the true location. In Fig.~\ref{fig2:MSEvssnr}, we show NMSE for different SNRs and for different number of elements in the RIS, where we have used $L=5$.
We can see that NMSE reduces with an increase in the number of RIS elements since the direction estimates improve with an increase in the array aperture.
In Fig. \ref{fig3:MSEvsL}, we show NMSE for different number of channel soundings, $L$, where we fix the SNR to $-10~\rm{dB}$. We can see that the channel estimation performance improves as $L$ increases. More importantly, for the considered setup, we can see that the algorithm performs similar to \texttt{OrcaleLS} for SNR above $-15~{\rm dB}$ and for $L>4$.
In Fig. \ref{fig4:SEvsSNR}, we show the average SE of \texttt{OrcaleLS} for different SNRs and different values of $N_{\rm r}$. Since the design of an optimal precoder, combiner, and phase shift matrix depend only on the angles and not on the path gains, the SE of the \texttt{OracleLS} estimator is the same as the maximum spectral efficiency that can be achieved using a perfect channel state information. The proposed scheme achieves average SE as that of \texttt{OrcaleLS} for low SNRs (around $0~ {\rm dB}$) even when $L=2$. This means that the error in estimating the angles $\{\theta_{\rm u}, \phi_{\rm u}, \psi_{\rm u}\}$ is very less.
We can also observe that the performance of the proposed channel estimation scheme is improving with an increase in the number of elements (thus the aperture) of the RIS. We observe a 6 dB and 3 dB gain in the NMSE and SE, respectively, when we quadruple the number of RIS elements from $N_{\rm r} = 8 \times 8$ to $N_{\rm r} = 16 \times 16$. This improvement in the NMSE and SE is due to the improved angle estimates and higher beamforming gain with the larger RIS array.
\section{Conclusions} \label{sec:conclusions}
We have presented a pilot-based uplink channel estimation algorithm for RIS-assisted mmWave MIMO systems. We have assumed that the RIS-BS channel is known up to a complex path gain and considered an LoS angular channel model for the UE-RIS link. To resolve the ambiguity and uniquely estimate the complex path gain and the angles at the RIS, we have proposed a multiple channel sounding technique in which we observe the channel through different RIS phase shifts. We have presented an algorithm to estimate the RIS-UE channel parameters at the BS. Through numerical simulations, we have demonstrated that the proposed algorithm performs on par with a method that perfectly knows the locations of the BS, UE, and RIS.
\bibliographystyle{IEEEtran}
| 9,673 |
\section{Introduction}
\label{sec:intro}
Panel data, represented as $X\in \mathbb{R}^{N\times T\times d}$ and $Y\in \mathbb{R}^{N\times T}$ where $N$ is the number of entities/individuals, $T$ is the number of time periods and $d$ is the number of features is widely used in statistics and applied machine learning.
Such data track features of a cross-section of entities (e.g., customers) longitudinally over time.
Such data are widely preferred in supervised machine learning for more accurate prediction and unbiased inference of relationships between variables relative to cross-sectional data (where each entity is observed only once) \cite{hsiao2003analysis, baltagi2008econometric}.
The most common method for inferring relationships between variables using observational data involves solving regression problems on panel data.
The main difference between regression on panel data when compared to cross-sectional data is that there may exist correlations within observations associated with entities over time periods.
Consequently, the regression problem for panel data is the following optimization problem over regression variables $\beta \in \mathbb{R}^d$ and the covariance matrix $\Omega$ that is induced by the abovementioned correlations:
$\min_{\beta\in \mathbb{R}^d, \Omega\in \mathbb{R}^{T\times T}} \sum_{i\in [N]} (y_i - X_i \beta)^\top \Omega^{-1} (y_i - X_i \beta).$
Here $X_i\in \mathbb{R}^{T\times d}$ denotes the observation matrix of entity $i$ whose $t$-th row is $x_{it}$ and $\Omega$ is constrained to have largest eigenvalue at most 1 where $\Omega_{t t'}$ represents the correlation between time periods $t$ and $t'$.
This regression model is motivated by the random effects model (Eq.~\eqref{eq:linear} and Appendix~\ref{sec:discussion}), common in the panel data literature~\cite{hoechle2007robust,griffiths1985theory,frees2004longitudinal}.
A common way to define the correlation between observations is an autocorrelation structure $\mathsf{AR}(q)$~\cite{haddad1998simple,lesage1999theory} whose covariance matrix $\Omega$ is induced by a vector $\rho\in \mathbb{R}^{q}$ (integer $q\geq 1$).
This type of correlation results in the generalized least-squares estimator (GLSE), where the parameter space is ${\mathcal{P}} = R^{d+q}$.
As the ability to track entities on various features in real-time has grown, panel datasets have grown massively in size.
However, the size of these datasets limits the ability to apply standard learning algorithms due to space and time constraints.
Further, organizations owning data may want to share only a subset of data with others seeking to gain insights to mitigate privacy or intellectual property related risks.
Hence, a question arises: {\em can we construct a smaller subset of the panel data on which we can solve the regression problems with performance guarantees that are close enough to those obtained when working with the complete dataset?}
One approach to this problem is to appeal to the theory of ``coresets.''
Coresets, proposed in \cite{agarwal2004approximating}, are weighted subsets of the data that allow for fast approximate inference for a large dataset by solving the problem on the smaller coreset.
Coresets have been developed for a variety of unsupervised and supervised learning problems; for a survey, see \cite{phillips2016coresets}.
But, thus, far coresets have been developed only for $\ell_2$-regression cross-sectional data \cite{drineas2006sampling,li2013iterative,boutsidis2013near,cohen2015uniform,jubran2019fast}; no coresets have been developed for regressions on panel data -- an important limitation, given their widespread use and advantages.
Roughly, a coreset for cross-sectional data is a weighted subset of observations associated with entities that approximates the regression objective for every possible choice of regression parameters.
An idea, thus, is to construct a coreset for each time period (cross-section) and output their union as a coreset for panel data.
However, this union contains at least $T$ observations which is undesirable since $T$ can be large.
Further, due to the covariance matrix $\Omega$, it is not obvious how to use this union to approximately compute regression objectives.
With panel data, one needs to consider both how to sample entities, and within each entity how to sample observations across time.
Moreover, we also need to define how to compute regression objectives on such a coreset consisting of entity-time pairs.
\noindent
\textbf{Our contributions.}
We initiate the study of coresets for versions of $\ell_2$-regression with panel data, including the ordinary least-squares estimator (OLSE; Definition~\ref{def:olse}), the generalized least-squares estimator (GLSE; Definition~\ref{def:glse}), and a clustering extension of GLSE (GLSE$_k$; Definition~\ref{def:glsek}) in which all entities are partitioned into $k$ clusters and each cluster shares the same regression parameters.
Overall, we formulate the definitions of coresets and propose efficient construction of $\varepsilon$-coresets of sizes independent of $N$ and $T$. Our key contributions are:
\begin{enumerate}
\item We give a novel formulation of coresets for GLSE (Definition~\ref{def:coreset_glse}) and GLSE$_k$ (Definition~\ref{def:coreset_glsek}).
We represent the regression objective of GLSE as the sum of $NT$ sub-functions w.r.t. entity-time pairs, which enables us to define coresets similar to the case of cross-sectional data.
For GLSE$_k$, the regression objective cannot be similarly decomposed due to the $\min$ operations in Definition~\ref{def:glsek}.
To deal with this issue, we define the regression objective on a coreset $S$ by including $\min$ operations.
\item Our coreset for OLSE is of size $O(\min\{\varepsilon^{-2}d, d^2\})$ (Theorems~\ref{thm:olse} and~\ref{thm:OLS_acc}), based on a reduction to coreset for $\ell_2$-regression with cross-sectional data.
\item Our coreset for GLSE consists of at most $\tilde{O}(\varepsilon^{-2}\max\{q^4d^2, q^3 d^3\})$ points (Theorem~\ref{thm:coreset_glse}), independent of $N$ and $T$ as desired.
\item
Our coreset for GLSE$_k$ is of size $\operatorname{poly}(M,k,q,d,1/\varepsilon)$ (Theorem~\ref{thm:coreset_glsek}) where $M$ upper bounds the gap between the maximum individual regression objective of OLSE and the minimum one (Definition~\ref{def:bounded_dataset_main}).
We provide a matching lower bound $\Omega(N)$ (Theorem~\ref{thm:lower_main}) for $k,q,d\leq 2$, indicating that the coreset size should contain additional factors than $k,q,d,1/\varepsilon$, justifying the $M$-bounded assumption.
\end{enumerate}
\noindent
Our coresets for GLSE/GLSE$_k$ leverage the Feldman-Langberg (FL) framework~\cite{feldman2011unified} (Algorithms~\ref{alg:glse} and~\ref{alg:glsek}).
The $\rho$ variables make the objective function of GLSE non-convex in contrast to the cross-sectional data setting where objective functions are convex.
Thus, bounding the ``sensitivity'' (Lemma~\ref{lm:sen_glse}) of each entity-time pair for GLSE, which is a key step in coreset construction using the FL framework, becomes significantly difficult.
We handle this by upper-bounding the maximum effect of $\rho$, based on the observation that the gap between the regression objectives of GLSE and OLSE with respect to the same $\beta\in \mathbb{R}^d$ is always constant, which enables us to reduce the problem to the cross-sectional setting.
For GLSE$_k$, a key difficulty is that the clustering centers are \textit{subspaces} induced by regression vectors, instead of \textit{points} as in Gaussian mixture models or $k$-means.
Hence, it is unclear how GLSE$_k$ can be reduced to projective clustering used in Gaussian mixture models; see~\cite{feldman2019coresets}.
To bypass this, we consider observation vectors of an individual as one entity and design a two-staged framework in which the first stage selects a subset of individuals that captures the $\min$ operations in the objective function and the second stage applies our coreset construction for GLSE on each selected individuals.
As in the case of GLSE, bounding the ``sensitivity'' (Lemma~\ref{lm:sen_glsek}) of each entity for GLSE$_k$ is a key step at the first stage.
Towards this, we relate the total sensitivity of entities to a certain ``flexibility'' (Lemma~\ref{lm:sen_olsek}) of each individual regression objective which is, in turn, shown to be controlled by the $M$-bounded assumption (Definition~\ref{def:bounded_dataset_main}).
We implement our GLSE coreset construction algorithm and test
it on synthetic and real-world datasets while varying $\varepsilon$. Our coresets perform well relative to uniform samples on multiple datasets with different generative distributions. Importanty, the relative performance is robust and better on datasets with outliers.
The maximum empirical error of our coresets is always below the guaranteed $\varepsilon$ unlike with uniform samples. Further, for comparable levels of empircal error, our coresets perform much better than uniform sampling in terms of sample size and coreset construction speed.
%
\subsection{Related work}
\label{sec:related}
With panel data, depending on different generative models, there exist several ways to define $\ell_2$-regression~\cite{hoechle2007robust,griffiths1985theory,frees2004longitudinal}, including the pooled model, the fixed effects model, the random effects model, and the random parameters model.
%
In this paper, we consider the random effects model (Equation~\eqref{eq:linear}) since the number of parameters is independent of $N$ and $T$ (see Section~\ref{sec:discussion} for more discussion).
%
For cross-sectional data, there is more than a decade of extensive work on coresets for regression; e.g., $\ell_2$-regression~\cite{drineas2006sampling,li2013iterative,boutsidis2013near,cohen2015uniform,jubran2019fast}, $\ell_1$-regression~\cite{clarkson2005subgradient,sohler2011subspace,clarkson2016fast}, generalized linear models~\cite{huggins2016coresets,molina2018core} and logistic regression~\cite{reddi2015communication,huggins2016coresets,munteanu2018coresets,Tolochinsky2018GenericCF}.
%
The most relevant for our paper is $\ell_2$-regression (least-squares regression), which admits an $\varepsilon$-coreset of size $O(d/\varepsilon^2)$~\cite{boutsidis2013near} and an accurate coreset of size $O(d^2)$~\cite{jubran2019fast}.
%
With cross-sectional data, coresets have been developed for a large family
of problems in machine learning and statistics, including clustering~\cite{feldman2011unified,feldman2013turning,huang2020coresets}, mixture model~\cite{lucic2017training}, low rank approximation~\cite{cohen2017input}, kernel regression~\cite{zheng2017coresets} and logistic regression~\cite{munteanu2018coresets}.
%
We refer interested readers to recent surveys~\cite{munteanu2018survey,feldman2020survey}.
%
It is interesting to investigate whether these results can be generalized to panel data.
%
{There exist other variants of regression sketches beyond coreset, including weighted low rank approximation~\cite{Clarkson2017LowRankAA}, row sampling~\cite{cohen2015lp}, and subspace embedding~\cite{sohler2011subspace,meng2013low}. These methods mainly focus on the cross-sectional setting. It is interesting to investigate whether they can be adapted to the panel data setting that with an additional covariance matrix.}
\section{$\ell_2$-regression with panel data}
\label{sec:pre}
We consider the following generative model of $\ell_2$-regression: for $(i,t)\in [N]\times [T]$,
\begin{align}
\label{eq:linear}
\textstyle y_{it} = x_{it}^\top \beta_i + e_{it},
\end{align}
where $\beta_i\in \mathbb{R}^d$ and $e_{it}\in \mathbb{R}$ is the error term drawn from a normal distribution.
Sometimes, we may include an additional entity or individual specified effect $\alpha_i\in \mathbb{R}$ so that the outcome can be represented by $y_{it} = x_{it}^\top \beta_i+ \alpha_i + e_{it}$.
This is equivalent to Equation~\eqref{eq:linear} by appending an additional constant feature to each observation $x_{it}$.
\begin{remark}
Sometimes, we may not observe individuals for all time periods, i.e., some observation vectors $x_{it}$ and their corresponding outcomes $y_{it}$ are missing.
%
One way to handle this is to regard those missing individual-time pairs as $(x_{it},y_{it})=(0,0)$.
%
Then, for any vector $\beta\in \mathbb{R}^d$, we have $y_{it}-x_{it}^\top \beta = 0$ for each missing individual-time pairs.
\end{remark}
\noindent
As in the case of cross-sectional data, we assume there is no correlation between individuals.
Using this assumption, the $\ell_2$-regression function can be represented as follows: for any regression parameters $\zeta\in {\mathcal{P}}$ (${\mathcal{P}}$ is the parameter space),
$
\psi(\zeta) = \sum_{i\in [N]} \psi_i(\zeta),
$
where $\psi_i$ is the individual regression function.
Depending on whether there is correlation within individuals and whether $\beta_i$ is unique, there are several variants of $\psi_i$.
The simplest setting is when all $\beta_i$s are the same, say $\beta_i=\beta$, and there is no correlation within individuals.
This setting results in the ordinary least-squares estimator (OLSE); summarized in the following definition.
\begin{definition}[\bf{Ordinary least-squares estimator (OLSE)}]
\label{def:olse}
For an ordinary least-squares estimator (OLSE), the parameter space is $\mathbb{R}^d$ and for any $\beta\in \mathbb{R}^d$ the individual objective function is
\[
\textstyle \psi^{(O)}_i(\beta):=\sum_{t\in [T]} \psi^{(O)}_{it}(\beta) =\sum_{t\in [T]} (y_{it}-x_{it}^\top \beta)^2.
\]
\end{definition}
\noindent
Consider the case when $\beta_i$ are the same but there may be correlations between time periods within individuals.
A common way to define the correlation is called autocorrelation $\mathsf{AR}(q)$~\cite{haddad1998simple,lesage1999theory}, in which there exists $\rho\in B^q$, where $q\geq 1$ is an integer and $B^q = \left\{x\in \mathbb{R}^q: \|x\|_2<1\right\}$, such that
\begin{align}
\label{eq:error}
\textstyle e_{it} = \sum_{a=1}^{\min\left\{t-1,q\right\}} \rho_a e_{i, t-a} + N(0,1).
\end{align}
This autocorrelation results in the generalized least-squares estimator (GLSE).
\begin{definition}[\bf{Generalized least-squares estimator (GLSE)}]
\label{def:glse}
For a generalized least-squares estimator (GLSE) with $\mathsf{AR}(q)$ (integer $q\geq 1$), the parameter space is $\mathbb{R}^d\times B^q$ and for any $\zeta=(\beta,\rho)\in \mathbb{R}^{d}\times B^q$ the individual objective function is $ \psi^{(G,q)}_i(\zeta):= \sum_{t\in [T]} \psi^{(G,q)}_{it}(\zeta)$ equal to
\begin{eqnarray*}
\textstyle (1-\|\rho\|_2^2) (y_{i1}-x_{i1}^\top \beta)^2
+ \sum_{t=2}^{T} \left((y_{it}-x_{it}^\top \beta)-\sum_{j=1}^{\min\left\{t-1,q\right\}} \rho_j (y_{i,t-j}-x_{i,t-j}^\top \beta)\right)^2.
\end{eqnarray*}
%
\end{definition}
\noindent
The main difference from OLSE is that a sub-function $\psi^{(G,q)}_{it}$ is not only determined by a single observation $(x_{it},y_{it})$; instead, the objective of $\psi^{(G,q)}_{it}$ may be decided by up to $q+1$ contiguous observations $(x_{i,\max\left\{1,t-q\right\}},y_{i,\max\left\{1,t-q\right\}}),\ldots,(x_{it},y_{it})$.
Motivated by $k$-means clustering~\cite{tan2006cluster}, we also consider a generalized setting of GLSE, called GLSE$_k$ ($k\geq 1$ is an integer), in which all individuals are partitioned into $k$ clusters and each cluster corresponds to the same regression parameters with respect to some GLSE.
\begin{definition}[\bf{GLSE$_k$: an extention of GLSE}]
\label{def:glsek}
Let $k,q\geq 1$ be integers.
%
For a GLSE$_k$, the parameter space is $\left(\mathbb{R}^{d}\times B^q\right)^k$ and for any $ \zeta=(\beta^{(1)},\ldots,\beta^{(k)}, \rho^{(1)},\ldots,\rho^{(k)})\in \left(\mathbb{R}^{d}\times B^q\right)^k$ the individual objective function is
$
\psi^{(G,q,k)}_i(\zeta):=\min_{l\in [k]}\psi^{(G,q)}_i(\beta^{(l)},\rho^{(l)}).
$
%
\end{definition}
\noindent
GLSE$_k$ is a basic problem with applications in many real-world fields; as accounting for \textit{unobserved heterogeneity} in panel regressions is critical for unbiased estimates~\cite{arellano2002panel,halaby2004panel}.
Note that each individual selects regression parameters $(\beta^{(l)}, \rho^{(l)})$ ($l\in [k]$) that minimizes its individual regression objective for GLSE.
Note that GLSE$_1$ is exactly GLSE.
Also note that GLSE$_k$ can be regarded as a generalized version of clustered linear regression~\cite{ari2002clustered}, in which there is no correlation within individuals.
\section{Our coreset definitions for panel data}
\label{sec:coreset}
In this section, we show how to define coresets for regression on panel data, including OLSE and GLSE.
Due to the additional autocorrelation parameters, it is not straightforward to define coresets for GLSE as in the cross-sectional setting.
One way is to consider all observations of an individual as an indivisible group and select a collection of individuals as a coreset.
However, this construction results in a coreset of size depending on $T$, which violates the expectation that the coreset size should be independent of $N$ and $T$.
To avoid a large coreset size, we introduce a generalized definition: coresets of a query space, which captures the coreset definition for OLSE and GLSE.
\begin{definition}[\bf{Query space~\cite{feldman2011unified,braverman2016new}}]
\label{def:query_space}
Let ${\mathcal{X}}$ be a index set together with a weight function $u: {\mathcal{X}}\rightarrow \mathbb{R}_{\geq 0}$.
Let ${\mathcal{P}}$ be a set called queries, and $\psi_x:{\mathcal{P}}\rightarrow \mathbb{R}_{\geq 0}$ be a given loss function w.r.t. $x\in {\mathcal{X}}$.
The total cost of ${\mathcal{X}}$ with respect to a query $\zeta\in {\mathcal{P}}$ is
$
\psi(\zeta) := \sum_{x\in {\mathcal{X}}} u(x)\cdot \psi_x(\zeta).
$
The tuple $({\mathcal{X}},u,{\mathcal{P}},\psi)$ is called a query space.
Specifically, if $u(x)=1$ for all $x\in {\mathcal{X}}$, we use $({\mathcal{X}},{\mathcal{P}},\psi)$ for simplicity.
\end{definition}
\noindent
Intuitively, $\psi$ represents a linear combination of weighted functions indexed by ${\mathcal{X}}$, and ${\mathcal{P}}$ represents the ground set of $\psi$.
Due to the separability of $\psi$, we have the following coreset definition.
\begin{definition}[\bf{Coresets of a query space~\cite{feldman2011unified,braverman2016new}}]
\label{def:coreset_query}
Let $({\mathcal{X}},u,{\mathcal{P}},\psi)$ be a query space and $\varepsilon \in (0,1)$ be an error parameter.
An $\varepsilon$-coreset of $({\mathcal{X}},u,{\mathcal{P}},\psi)$ is a weighted set $S\subseteq {\mathcal{X}}$ together with a weight function $w: S\rightarrow \mathbb{R}_{\geq 0}$ such that for any $\zeta\in {\mathcal{P}}$,
$
\psi_S(\zeta):=\sum_{x\in S} w(x)\cdot \psi_x(\zeta) \in (1\pm \varepsilon)\cdot \psi(\zeta).
$
\end{definition}
\noindent
Here, $\psi_S$ is a computation function over the coreset that is used to estimate the total cost of ${\mathcal{X}}$.
By Definitions~\ref{def:olse} and~\ref{def:glse}, the regression objectives of OLSE and GLSE can be decomposed into $NT$ sub-functions.
Thus, we can apply the above definition to define coresets for OLSE and GLSE.
Note that OLSE is a special case of GLSE for $q=0$.
Thus, we only need to provide the coreset definition for GLSE.
We let $u=1$ and ${\mathcal{P}} = \mathbb{R}^d\times B^q$.
The index set of GLSE has the following form:
\begin{align*}
Z^{(G,q)} = \left\{z_{it}=\left(x_{i,\max\left\{1,t-q\right\}}, y_{i,\max\left\{1,t-q\right\}}, \ldots x_{it},y_{it}\right) : (i,t)\in [N]\times [T] \right\},
\end{align*}
where each element $z_{it}$ consists of at most $q+1$ observations.
Also, for every $z_{it}\in Z^{(G,q)}$ and $\zeta=(\beta,\rho)\in {\mathcal{P}}$, the cost function $\psi_{it}$ is: if $t=1$,
$
\psi^{(G,q)}_{it}(\zeta) = (1-\|\rho\|_2^2)\cdot (y_{i1}-x_{i1}^\top \beta)^2;
$
and if $t\neq 1$,
$
\psi^{(G,q)}_{it}(\zeta) =\left((y_{it}-x_{it}^\top \beta)-\sum_{j=1}^{\min\left\{t-1,q\right\}} \rho_j (y_{i,t-j}-x_{i,t-j}^\top \beta)\right)^2.
$
Thus, $(Z^{(G,q)},{\mathcal{P}},\psi^{(G,q)})$ is a query space of GLSE.\footnote{Here, we slightly abuse the notation by using $\psi^{(G,q)}_{it}(\zeta)$ instead of $\psi^{(G,q)}_{z_{it}}(\zeta)$.}
Then by Definition~\ref{def:coreset_query}, we have the following coreset definition for GLSE.
\begin{definition}[\bf{Coresets for GLSE}]
\label{def:coreset_glse}
Given a panel dataset $X\in \mathbb{R}^{N\times T\times d}$ and $Y\in \mathbb{R}^{N\times T}$, a constant $\varepsilon \in (0,1)$, integer $q\geq 1$, and parameter space ${\mathcal{P}}$, an $\varepsilon$-coreset for GLSE is a weighted set $S\subseteq [N]\times [T]$ together with a weight function $w: S\rightarrow \mathbb{R}_{\geq 0}$ such that for any $\zeta=(\beta, \rho)\in {\mathcal{P}}$,
\begin{align*}
\psi^{(G,q)}_S(\zeta):=\sum_{(i,t)\in S} w(i,t)\cdot \psi^{(G,q)}_{it}(\zeta) \in (1\pm \varepsilon)\cdot \psi^{(G,q)}(\zeta).
\end{align*}
\end{definition}
\noindent
The weighted set $S$ is exactly an $\varepsilon$-coreset of the query space $(Z^{(G,q)},{\mathcal{P}},\psi^{(G,q)})$.
Note that the number of points in this coreset $S$ is at most $(q+1)\cdot|S|$.
Specifically, for OLSE, the parameter space is $\mathbb{R}^d$ since $q=0$, and the corresponding index set is
$
Z^{(O)} = \left\{z_{it}=(x_{it},y_{it}) : (i,t)\in [N]\times [T] \right\}.
$
Consequently, the query space of OLSE is $(Z^{(O)},\mathbb{R}^d,\psi^{(O)})$.
\paragraph{Coresets for GLSE$_k$}
Due to the $\min$ operation in Definition~\ref{def:glsek}, the objective function $\psi^{(G,q,k)}$ can only be decomposed into sub-functions $\psi^{(G,q,k)}_i$ instead of individual-time pairs.
Then let $u=1$, ${\mathcal{P}}^k = \left(\mathbb{R}^{d}\times B^q\right)^k$, and
$
Z^{(G,q,k)}=\left\{z_i=(x_{i1},y_{i1},\ldots, x_{iT},y_{iT}): i\in [N]\right\}.
$
We can regard $(Z^{(G,q,k)},{\mathcal{P}}^k,\psi^{(G,q,k)})$ as a query space of GLSE$_k$.
By Definition~\ref{def:coreset_query}, an $\varepsilon$-coreset of $(Z^{(G,q,k)},{\mathcal{P}}^k,\psi^{(G,q,k)})$ is a subset $I_S\subseteq [N]$ together with a weight function $w': I_S\rightarrow \mathbb{R}_{\geq 0}$ such that for any $\zeta\in {\mathcal{P}}^k$,
\begin{eqnarray}
\label{eq:I_S}
\sum_{i\in I_S} w'(i)\cdot \psi^{(G,q,k)}_{i}(\zeta) \in (1\pm \varepsilon)\cdot \psi^{(G,q,k)}(\zeta).
\end{eqnarray}
However, each $z_i\in Z^{(G,q,k)}$ consists of $T$ observations, and hence, the number of points in this coreset $S$ is $T\cdot |S|$.
To avoid the size dependence of $T$, we propose a new coreset definition for GLSE$_k$.
The intuition is to further select a subset of time periods to estimate $\psi^{(G,q,k)}_i$.
Given $S\subseteq [N]\times [T]$, we denote $I_S := \left\{i\in [N]: \exists t\in [T], s.t., (i,t)\in S \right\}$ as the collection of individuals that appear in $S$.
Moreover, for each $i\in I_S$, we denote $J_{S,i}:=\left\{t\in [T]: (i,t)\in S\right\}$ to be the collection of observations for individual $i$ in $S$.
\begin{definition}[\bf{Coresets for GLSE$_k$}]
\label{def:coreset_glsek}
Given a panel dataset $X\in \mathbb{R}^{N\times T\times d}$ and $Y\in \mathbb{R}^{N\times T}$, constant $\varepsilon \in (0,1)$, integer $k,q\geq 1$, and parameter space ${\mathcal{P}}^k$, an $\varepsilon$-coreset for GLSE$_k$ is a weighted set $S\subseteq [N]\times [T]$ together with a weight function $w: S\rightarrow \mathbb{R}_{\geq 0}$ such that for any $\zeta=(\beta^{(1)},\ldots,\beta^{(k)},\rho^{(1)},\ldots,\rho^{(k)})\in {\mathcal{P}}^k$,
\begin{align*}
\psi^{(G,q,k)}_S(\zeta):=\sum_{i\in I_S} \min_{l\in [k]} \sum_{t\in J_{S,i}} w(i,t)\cdot \psi^{(G,q)}_{it}(\beta^{(l)},\rho^{(l)}) \in (1\pm \varepsilon)\cdot \psi^{(G,q,k)}(\zeta).
\end{align*}
\end{definition}
\noindent
The key is to incorporate $\min$ operations in the computation function $\psi^{(G,q,k)}_S$ over the coreset.
Similar to GLSE, the number of points in such a coreset $S$ is at most $(q+1)\cdot|S|$.
\section{Coresets for GLSE}
\label{sec:alg}
{
In this section, we show how to construct coresets for GLSE.
We let the parameter space be ${\mathcal{P}}_\lambda = \mathbb{R}^d\times B^q_{1-\lambda}$ for some constant $\lambda\in (0,1)$ where
$B^q_{1-\lambda}=\left\{\rho\in \mathbb{R}^q: \|\rho\|_2^2\leq 1-\lambda \right\}$.
The assumption of the parameter space $B^q_{1-\lambda}$ for $\rho$ is based on the fact that $\|\rho\|_2^2< 1$ ($\lambda\rightarrow 0$) is a stationary condition for $\mathsf{AR}(q)$~\cite{lesage1999theory}.
\begin{theorem}[\bf{Coresets for GLSE}]
\label{thm:coreset_glse}
There exists a randomized algorithm that, for a given panel dataset $X\in \mathbb{R}^{N\times T\times d}$ and $Y\in \mathbb{R}^{N\times T}$, constants $\varepsilon,\delta,\lambda \in (0,1)$ and integer $q\geq 1$, with probability at least $1-\delta$, constructs an $\varepsilon$-coreset for GLSE of size
\[
O\left(\varepsilon^{-2} \lambda^{-1} q d\left(\max\left\{q^2d, qd^2\right\}\cdot \log \frac{d}{\lambda}+\log \frac{1}{\delta}\right) \right)
\]
and runs in time $O(NTq+NTd^2)$.
\end{theorem}
\noindent
\sloppy
Note that the coreset in the above theorem contains at most
$
(q+1)\cdot O\left(\varepsilon^{-2}\lambda^{-1} qd\left(\max\left\{q^2d, qd^2\right\}\cdot \log \frac{d}{\lambda}+\log \frac{1}{\delta}\right) \right)
$
points $(x_{it},y_{it})$, which is independent of both $N$ and $T$.
Also note that if both $\lambda$ and $\delta$ are away from 0, e.g., $\lambda=\delta=0.1$ the number of points in the coreset can be further simplified:
$
O\left(\varepsilon^{-2} \max\left\{q^4 d^2, q^3 d^3\right\}\cdot \log d\right) = \operatorname{poly}(q,d,1/\varepsilon).
$
\subsection{Algorithm for Theorem~\ref{thm:coreset_glse}}
\label{sec:algorithm_glse}
We summarize the algorithm of Theorem~\ref{thm:coreset_glse} in Algorithm~\ref{alg:glse}, which takes a panel dataset $(X,Y)$ as input and outputs a coreset $S$ of individual-time pairs.
The main idea is to use importance sampling (Lines 6-7) leveraging the Feldman-Langberg (FL) framework~\cite{feldman2011unified,braverman2016new}.
The key new step appears in Line 5, which computes a sensitivity function $s$ for GLSE that defines the sampling distribution.
Also note that the construction of $s$ is based on another function $s^{(O)}$ (Line 4), which is actually a sensitivity function for OLSE that has been studied in the literature~\cite{boutsidis2013near}.
\begin{algorithm}[ht!]
\caption{$\mathbf{CGLSE}$: Coreset construction of GLSE}
\label{alg:glse}
\begin{algorithmic}[1]
\REQUIRE {$X\in \mathbb{R}^{N\times T\times d}$, $Y\in \mathbb{R}^{N\times T}$, constant $\varepsilon,\delta,\lambda \in (0,1)$, integer $q\geq 1$ and parameter space ${\mathcal{P}}_\lambda$.}
\ENSURE {a subset $S\subseteq [N]\times [T]$ together with a weight function $w:S\rightarrow \mathbb{R}_{\geq 0}$.}
%
\STATE
$M\leftarrow O\left(\varepsilon^{-2} \lambda^{-1} q d\left(\max\left\{q^2d, qd^2\right\}\cdot \log \frac{d}{\lambda}+\log \frac{1}{\delta}\right) \right)$.
%
\STATE Let $Z\in \mathbb{R}^{NT\times (d+1)}$ be whose $(iT-T+t)$-th row is $z_{it}=(x_{it},y_{it})\in \mathbb{R}^{d+1}$ for $(i,t)\in [N]\times [T]$.
%
\STATE Compute $A\subseteq \mathbb{R}^{NT\times d'}$ whose columns form a unit basis of the column space of $Z$.
%
\STATE For each $(i,t)\in [N]\times [T]$, $s^{(O)}(i,t)\leftarrow \|A_{iT-T+t}\|_2^2$.
%
\STATE For each pair $(i,t)\in [N]\times [T]$, $s(i,t)\leftarrow
\min\left\{1, 2\lambda^{-1} \left(s^{(O)}(i,t)+\sum_{j=1}^{\min\left\{t-1,q\right\}} s^{(O)}(i,t-j) \right)\right\}.
$
%
\STATE Pick a random sample $S\subseteq [N]\times [T]$ of $M$ pairs, where each $(i,t)\in S$ is selected with probability $\frac{s(i,t)}{\sum_{(i',t')\in [N]\times [T]}s(i',t')}$.
%
\STATE For each $(i,t)\in S$, $w(i,t)\leftarrow \frac{\sum_{(i',t')\in [N]\times [T]}s(i',t')}{M\cdot s(i,t)}$.
%
\STATE Output $(S,w)$.
\end{algorithmic}
\end{algorithm}
\subsection{Useful notations and useful facts for Theorem~\ref{thm:coreset_glse}}
\label{sec:technical}
Feldman and Langberg~\cite{feldman2011unified} show how to construct coresets by importance sampling and the coreset size has been improved by~\cite{braverman2016new}.
%
\begin{theorem}[\bf{FL framework~\cite{feldman2011unified,braverman2016new}}]
\label{thm:fl11}
Let $\varepsilon,\delta\in (0,1)$.
%
Let $\dim$ be an upper bound of the pseudo-dimension.
%
Suppose $s:[N]\times [T]\rightarrow \mathbb{R}_{\geq 0}$ is a sensitivity function satisfying that for any $(i,t)\in [N]\times [T]$,
$
s(i,t) \geq \sup_{\zeta\in {\mathcal{P}}_\lambda} \frac{\psi^{(G,q)}_{it}(\zeta)}{\psi^{(G,q)}(\zeta)},
$
and ${\mathcal{G}} := \sum_{(i,t)\in [N]\times [T]} s(i,t)$.
%
Let $S\subseteq {\mathcal{X}}$ be constructed by taking
\[
O\left(\varepsilon^{-2} {\mathcal{G}} (\dim \cdot \log {\mathcal{G}} +\log(1/\delta))\right)
\]
samples, where each sample $x\in {\mathcal{X}}$ is selected with probability $\frac{s(x)}{{\mathcal{G}}}$ and has weight $w(x):= \frac{{\mathcal{G}}}{|S|\cdot s(x)}$.
%
Then, with probability at least $1-\delta$, $S$ is an $\varepsilon$-coreset for GLSE.
%
\end{theorem}
\noindent
%
Here, the sensitivity function $s$ measures the maximum influence for each $x_{it}\in X$.
%
%
Note that the above is an importance sampling framework that takes samples from a distribution proportional to sensitivities.
%
The sample complexity is controlled by the total sensitivity ${\mathcal{G}}$ and the pseudo-dimension $\dim$.
%
Hence, to apply the FL framework, we need to upper bound the pseudo-dimension and construct a sensitivity function.
\subsection{Proof of Theorem~\ref{thm:coreset_glse}}
\label{sec:proof_glse}
Algorithm~\ref{alg:glse} applies the FL framework (Feldman and Langberg~\cite{feldman2011unified}) that constructs coresets by importance sampling and the coreset size has been improved by~\cite{braverman2016new}.
%
The key is to verify the “pseudo-dimension” (Lemma~\ref{lm:dim_glse}) and “sensitivities” (Lemma~\ref{lm:sen_glse}) separately; summarized as follows.
%
\paragraph{Upper bounding the pseudo-dimension.}
We have the following lemma that upper bounds the pseudo-dimension of $(Z^{(G,q)},{\mathcal{P}}_\lambda,\psi^{(G,q)})$.
%
\begin{lemma}[\bf{Pseudo-dimension of GLSE}]
\label{lm:dim_glse}
\sloppy
The pseudo-dimension of any query space $(Z^{(G,q)},u,{\mathcal{P}}_\lambda,\psi^{(G,q)})$ over weight functions $u: [N]\times [T]\rightarrow \mathbb{R}_{\geq 0}$ is at most $ O\left((q+d)qd \right)$.
\end{lemma}
\noindent
The proof can be found in Section~\ref{sec:dim}.
%
The main idea is to apply the prior results~\cite{anthony2009neural,vidyasagar2002theory} which shows that the pseudo-dimension is polynomially dependent on the number of regression parameters ($q+d$ for GLSE) and the number of operations of individual regression objectives ($O(qd)$ for GLSE).
%
Consequently, we obtain the bound $O\left((q+d)qd \right)$ in Lemma~\ref{lm:dim_glse}.
%
\paragraph{Constructing a sensitivity function.}
Next, we show that the function $s$ constructed in Line 5 of Algorithm~\ref{alg:glse} is indeed a sensitivity function of GLSE that measures the maximum influence for each $x_{it}\in X$; summarized by the following lemma.
%
\begin{lemma}[\bf{Total sensitivity of GLSE}]
\label{lm:sen_glse}
Function $s:[N]\times [T]\rightarrow \mathbb{R}_{\geq 0}$ of Algorithm~\ref{alg:glse} satisfies that for any $(i,t)\in [N]\times [T]$, $s(i,t) \geq \sup_{\zeta\in {\mathcal{P}}} \frac{\psi^{(G,q)}_{it}(\zeta)}{\psi^{(G,q)}(\zeta)}$ and ${\mathcal{G}}:=\sum_{(i,t)\in [N]\times [T]} s(i,t) = O(\lambda^{-1}qd)$.
%
Moreover, the construction time of function $s$ is $O(NTq+NTd^2)$.
%
\end{lemma}
\noindent
Intuitively, if the sensitivity $s(i,t)$ is large, e.g., close to 1, $\psi^{(G,q)}_{it}$ must contribute significantly to the objective with respect to some parameter $\zeta\in {\mathcal{P}}_\lambda$.
The sampling ensures that we are likely to include such pair $(i,t)$ in the coreset for estimating $\psi(\zeta)$.
%
Due to the fact that the objective function of GLSE is non-convex which is different from OLSE, bounding the sensitivity of each individual-time pair for GLSE becomes significantly difficult.
%
To handle this difficulty, we develop a reduction of sensitivities from GLSE to OLSE (Line 5 of Algorithm~\ref{alg:glse}), based on the relations between $\psi^{(G,q)}$ and $\psi^{(O)}$, i.e., for any $\zeta=(\beta,\rho)\in {\mathcal{P}}_\lambda$ we prove that
$
\psi^{(G,q)}_i(\zeta) \geq \lambda \cdot \psi^{(O)}_i(\beta) \text{ and }
\psi^{(G,q)}_{it}(\zeta) \leq 2\cdot\left(\psi^{(O)}_{it} (\beta) +\sum_{j=1}^{\min\left\{t-1,q\right\}} \psi^{(O)}_{i,t-j}(\beta)\right).
$
%
The first inequality follows from the fact that the smallest eigenvalue of $\Omega_\rho^{-1}$ (the inverse covariance matrix induced by $\rho$) is at least $\lambda$.
%
The intuition of the second inequality is from the form of function $\psi^{(G,q)}_{it}$, which relates to $\min\left\{t, q+1\right\}$ individual-time pairs, say $(x_{i,\min\left\{1,t-q\right\}}, y_{i,\min\left\{1, t-q\right\}}), \ldots, (x_{it},y_{it})$.
%
Combining these two inequalities, we obtain a relation between the sensitivity function $s$ for GLSE and the sensitivity function $s^{(O)}$ for OLSE,
based on the following observation: for any $\zeta=(\beta,\rho)\in {\mathcal{P}}_\lambda$,
\begin{align*}
\frac{\psi^{(G,q)}_{it}(\zeta)}{\psi^{(G,q)}(\zeta)}
&\leq &&\frac{2\cdot\left(\psi^{(O)}_{it} (\beta) + \sum_{j=1}^{\min\left\{t-1,q\right\}} \psi^{(O)}_{i,t-j}(\beta)\right)}{\lambda \cdot \psi^{(O)}(\beta)} \\
&\leq && 2\lambda^{-1}\cdot \left(s^{(O)}(i,t)+{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} s^{(O)}(i,t-j) \right) \\
&= && s(i,t).
\end{align*}
which leads to the construction of $s$ in Line 5 of Algorithm~\ref{alg:glse}.
%
Then it suffices to construct $s^{(O)}$ (Lines 2-4 of Algorithm~\ref{alg:glse}), which reduces to the cross-sectional data setting and has total sensitivity at most $d+1$ (Lemma~\ref{lm:sen_olse}).
%
Consequently, we conclude that the total sensitivity ${\mathcal{G}}$ of GLSE is $O(\lambda^{-1}qd)$ by the definition of $s$.
%
Now we are ready to prove Theorem~\ref{thm:coreset_glse}.
%
\begin{proof}[Proof of Theorem~\ref{thm:coreset_glse}]
By Lemma~\ref{lm:sen_glse}, the total sensitivity ${\mathcal{G}}$ is $O(\lambda^{-1} q d)$.
%
By Lemma~\ref{lm:dim_glse}, we let $\dim = O\left((q+d)qd \right)$.
%
Pluging the values of ${\mathcal{G}}$ and $\dim$ in the FL framework~\cite{feldman2011unified,braverman2016new}, we prove for the coreset size.
%
For the running time, it costs $O(NTq+NTd^2)$ time to compute the sensitivity function $s$ by Lemma~\ref{lm:sen_glse}, and $O(NTd)$ time to construct an $\varepsilon$-coreset.
%
This completes the proof.
%
\end{proof}
\subsection{Proof of Lemma~\ref{lm:dim_glse}: Upper bounding the pseudo-dimension}
\label{sec:dim}
Our proof idea is similar to that in~\cite{lucic2017training}.
For preparation, we need the following lemma which is proposed to bound the pseudo-dimension of feed-forward neural networks.
\begin{lemma}[\bf{Restatement of Theorem 8.14 of~\cite{anthony2009neural}}]
\label{lm:dim_bound}
\sloppy
Let $({\mathcal{X}},u,{\mathcal{P}},f)$ be a given query space where $f_x(\zeta)\in \left\{0,1\right\}$ for any $x\in {\mathcal{X}}$ and $\zeta\in {\mathcal{P}}$, and ${\mathcal{P}}\subseteq \mathbb{R}^{m}$.
%
Suppose that $f$ can be computed by an algorithm that takes as input the pair $(x,\zeta)\in {\mathcal{X}}\times {\mathcal{P}}$ and returns $f_x(\zeta)$ after no more than $l$ of the following operations:
\begin{itemize}
\item the arithmetic operations $+,-,\times$, and $/$ on real numbers.
%
\item jumps conditioned on $>,\geq,<,\leq,=$, and $\neq$ comparisons of real numbers, and
%
\item output 0,1.
\end{itemize}
%
Then the pseudo-dimension of $({\mathcal{X}},u,{\mathcal{P}},f)$ is at most $O(ml)$.
%
\end{lemma}
\noindent
Note that the above lemma requires that the range of functions $f_x$ is $[0,1]$.
We have the following lemma which can help extend this range to $\mathbb{R}$.
\begin{lemma}[\bf{Restatement of Lemma 4.1 of~\cite{vidyasagar2002theory}}]
\label{lm:dim_range}
Let $({\mathcal{X}},u,{\mathcal{P}},f)$ be a given query space.
%
Let $g_x:{\mathcal{P}}\times \mathbb{R}\rightarrow \left\{0,1\right\}$ be the indicator function satisfying that for any $x\in {\mathcal{X}}$, $\zeta\in {\mathcal{P}}$ and $r\in \mathbb{R}$,
\[
g_x(\zeta,r) = I\left[u(x)\cdot f(x,\zeta)\geq r\right].
\]
%
Then the pseudo-dimension of $({\mathcal{X}},u,{\mathcal{P}},f)$ is precisely the pseudo-dimension of the query space $({\mathcal{X}},u,{\mathcal{P}}\times \mathbb{R},g_f)$.
\end{lemma}
\noindent
Now we are ready to prove Lemma~\ref{lm:dim_glse}.
\begin{proof}[Proof of Lemma~\ref{lm:dim_glse}]
Fix a weight function $u: [N]\times [T]\rightarrow \mathbb{R}_{\geq 0}$.
%
For every $(i,t)\in [N]\times [T]$, let $g_{it}: {\mathcal{P}}_\lambda\times \mathbb{R}_{\geq 0}\rightarrow \left\{0,1\right\}$ be the indicator function satisfying that for any $\zeta\in {\mathcal{P}}_\lambda$ and $r\in \mathbb{R}_{\geq 0}$,
\[
g_{it}(\zeta,r) := I\left[u(i,t)\cdot \psi^{(G,q)}_{it}(\zeta)\geq r\right].
\]
%
We consider the query space $(Z^{(G,q)},u,{\mathcal{P}}_\lambda\times \mathbb{R}_{\geq 0},g)$.
%
By the definition of ${\mathcal{P}}_\lambda$, the dimension of ${\mathcal{P}}_\lambda\times \mathbb{R}_{\geq 0}$ is $m=q+1+d$.
%
By the definition of $\psi^{(G,q)}_{it}$, $g_{it}$ can be calculated using $l=O(qd)$ operations, including $O(qd)$ arithmetic operations and a jump.
%
Pluging the values of $m$ and $l$ in Lemma~\ref{lm:dim_bound}, the pseudo-dimension of $(Z^{(G,q)},u,{\mathcal{P}}_\lambda\times \mathbb{R}_{\geq 0},g)$ is $O\left((q+d)qd\right)$.
%
Then by Lemma~\ref{lm:dim_range}, we complete the proof.
\end{proof}
\subsection{Proof of Lemma~\ref{lm:sen_glse}: Bounding the total sensitivity}
\label{sec:sen}
We prove Lemma~\ref{lm:sen_glse} by relating sensitivities between GLSE and OLSE.
For preparation, we give the following lemma that upper bounds the total sensitivity of OLSE.
Given two integers $a,b\geq 1$, denote $T(a,b)$ to be the computation time of a column basis of a matrix in $\mathbb{R}^{a\times b}$.
For instance, a column basis of a matrix in $\mathbb{R}^{a\times b}$ can be obtained by computing its SVD decomposition, which costs $O(\min\left\{a^2 b, ab^2\right\})$ time by~\cite{cline2006computation}.
\begin{lemma}[\bf{Total sensitivity of OLSE}]
\label{lm:sen_olse}
Function $s^{(O)}:[N]\times [T]\rightarrow \mathbb{R}_{\geq 0}$ of Algorithm~\ref{alg:glse} satisfies that for any $(i,t)\in [N]\times [T]$,
\begin{align}
\label{ineq:sen_olse}
s^{(O)}(i,t) \geq \sup_{\beta\in \mathbb{R}^d} \frac{\psi^{(O)}_{it}(\beta)}{\psi^{(O)}(\beta)},
\end{align}
and ${\mathcal{G}}^{(O)} := \sum_{(i,t)\in [N]\times [T]} s^{(O)}(i,t)$ satisfying ${\mathcal{G}}^{(O)} \leq d+1$.
%
Moreover, the construction time of function $s^{(O)}$ is $T(NT,d+1)+O(NTd)$.
%
\end{lemma}
\begin{proof}
The proof idea comes from~\cite{varadarajan2012sensitivity}.
%
By Line 3 of Algorithm~\ref{alg:glse}, $A\subseteq \mathbb{R}^{NT\times d'}$ is a matrix whose columns form a unit basis of the column space of $Z$.
%
We have $d'\leq d+1$ and hence $\|A\|_2^2 = d'\leq d+1$.
%
Moreover, for any $(i,t)\in [N]\times [T]$ and $\beta'\in \mathbb{R}^{d'}$, we have
\[
\|\beta'\|_2^2 \leq \|A \beta'\|_2^2,
\]
%
Then by Cauchy-Schwarz and orthonormality of $A$, we have that for any $(i,t)\in [N]\times [T]$ and $\beta'\in \mathbb{R}^{d+1}$,
\begin{align}
\label{ineq:sen}
|z_{it}^\top \beta'|^2 \leq \|A_{iT-T+t}\|_2^2\cdot \|Z \beta'\|_2^2,
\end{align}
where $A_{iT-T+t}$ is the $(iT-T+t)$-th row of $A$.
%
For each $(i,t)\in [N]\times [T]$, we let $s^{(O)}(i,t):=\|A_{iT-T+t}\|_2^2$.
%
Then ${\mathcal{G}}^{(O)} = \|A\|_2^2 =d'\leq d+1$.
%
Note that constructing $A$ costs $T(NT,d+1)$ time and computing all $\|A_{iT-T+t}\|_2^2$ costs $O(NTd)$ time.
%
Thus, it remains to verify that $s^{(O)}(i,t)$ satisfies Inequality~\eqref{ineq:sen_olse}.
%
For any $(i,t)\in [N]\times [T]$ and $\beta\in \mathbb{R}^d$, letting $\beta'=(\beta,-1)$, we have
\begin{eqnarray*}
\begin{split}
\psi^{(O)}_{it}(\beta) &= && |z_{it}^\top \beta'|^2 && (\text{Defn. of $\psi^{(O)}_{it}$}) \\
& \leq && \|A_{iT-T+t}\|_2^2\cdot \|Z \beta'\|_2^2 && (\text{Ineq.~\eqref{ineq:sen}}) \\
& = && \|A_{iT-T+t}\|_2^2\cdot \psi^{(O)}(\beta). && (\text{Defn. of $\psi^{(O)}$})
\end{split}
\end{eqnarray*}
%
This completes the proof.
%
\end{proof}
\noindent
Now we are ready to prove Lemma~\ref{lm:sen_glse}.
\begin{proof}[Proof of Lemma~\ref{lm:sen_glse}]
For any $(i,t)\in [N]\times [T]$, recall that $s(i,t)$ is defined by
\[
s(i,t):=\min \left\{1, 2\lambda^{-1}\cdot \left(s^{(O)}(i,t)+{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} s^{(O)}(i,t-j) \right)\right\}.
\]
%
We have that
\begin{eqnarray*}
\begin{split}
\sum_{(i,t)\in [N]\times [T]} s(i,t) &\leq && \sum_{(i,t)\in [N]\times [q]} 2\lambda^{-1} \times \left(s^{(O)}(i,t)+{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}}s^{(O)}(i,t-j) \right) & (\text{by definition})\\
& \leq && 2\lambda^{-1}\cdot {\textstyle\sum}_{(i,t)\in [N]\times [T]} (1+q)\cdot s^{(O)}(i,t) && \\
& \leq && 2\lambda^{-1} (q+1)(d+1). & (\text{Lemma~\ref{lm:sen_olse}})
\end{split}
\end{eqnarray*}
%
Hence, the total sensitivity ${\mathcal{G}} = O(\lambda^{-1}qd)$.
%
By Lemma~\ref{lm:sen_olse}, it costs $T(NT,d+1)+O(NTd)$ time to construct $s^{(O)}$.
%
We also know that it costs $O(NTq)$ time to compute function $s$.
%
Since $T(NT,d+1) = O(NTd^2)$, this completes the proof for the running time.
%
Thus, it remains to verify that $s(i,t)$ satisfies that
\[
s(i,t) \geq \sup_{\zeta\in {\mathcal{P}}} \frac{\psi^{(G,q)}_{it}(\zeta)}{\psi^{(G,q)}(\zeta)}.
\]
%
Since $\sup_{\beta\in \mathbb{R}^d} \frac{\psi^{(O)}_{it}(\beta)}{\psi^{(O)}(\beta)}\leq 1$ always holds, we only need to consider the case that
\[
s(i,t)=2\lambda^{-1}\cdot \left(s^{(O)}(i,t)+{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} s^{(O)}(i,t-j) \right).
\]
%
We first show that for any $\zeta=(\beta,\rho)\in {\mathcal{P}}_\lambda$,
\begin{align}
\label{ineq:relation}
\psi^{(G,q)}(\zeta) \geq \lambda \cdot \psi^{(O)}(\beta).
\end{align}
%
Given an autocorrelation vector $\rho\in \mathbb{R}^q$, the induced covariance matrix $\Omega_\rho$ satisfies that $\Omega_\rho^{-1}=P_\rho^\top P_\rho$ where
\begin{eqnarray}
\label{eq:cov_glse}
\begin{split}
& P_\rho =
\begin{bmatrix}
\sqrt{1-\|\rho\|_2^2} & 0 & 0 & \ldots & \ldots& \ldots & 0 \\
-\rho_1 & 1 & 0 & \ldots & \ldots& \ldots & 0 \\
-\rho_2 & -\rho_1 & 1 & \ldots & \ldots& \ldots & 0 \\
\ldots & \ldots & \ldots & \ldots & \ldots& \ldots & \ldots \\
0 & 0 & 0 & -\rho_q & \ldots& -\rho_1 & 1
\end{bmatrix}.
\end{split}
\end{eqnarray}
%
Then by Equation~\eqref{eq:cov_glse}, the smallest eigenvalue of $P_\rho$ satisfies that
\begin{eqnarray}
\label{ineq:eigenvalue}
\begin{split}
\lambda_{\min} &=&& \sqrt{1-\|\rho\|_2^2} && (\text{Defn. of $P_\rho$})\\
&\geq&& \sqrt{\lambda}. && (\rho\in B^q_{1-\lambda})
\end{split}
\end{eqnarray}
%
Also we have
\begin{eqnarray*}
\begin{split}
\psi^{(G,q)}(\zeta) & = && \sum_{i\in [N]} (y_i-X_i \beta)^\top \Omega_\rho^{-1} (y_i-X_i \beta) & (\text{Program (GLSE)}) \\
& = && \sum_{i\in [N]} \|P_\rho (y_i-X_i \beta)\|_2^2 &(P_\rho^\top P_\rho = \Omega_\rho^{-1}) \\
& \geq && \sum_{i\in [N]} \lambda \cdot \|(y_i-X_i \beta)\|_2^2 & (\text{Ineq.~\eqref{ineq:eigenvalue}}) \\
& = && \lambda\cdot \psi^{(O)}(\beta), & (\text{Defns. of $\psi^{(O)}$})
\end{split}
\end{eqnarray*}
which proves Inequality~\eqref{ineq:relation}.
%
We also claim that for any $(i,t)\in [N]\times [T]$,
\begin{align}
\label{ineq:relation2}
\psi^{(G,q)}_{it}(\zeta) \leq 2\cdot\left(\psi^{(O)}_{it} (\beta) +{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} \psi^{(O)}_{i,t-j}(\beta)\right).
\end{align}
%
This trivially holds for $t=1$.
%
For $t\geq 2$, this is because
\begin{eqnarray*}
\begin{split}
& && \psi^{(G,q)}_{it}(\zeta)& \\
&= && \left((y_{it}-x_{it}^\top \beta)-{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} \rho_j\cdot (y_{i,t-j}-x_{i,t-j}^\top \beta)\right)^2 & (t\geq 2) \\
& \leq && \left(1+{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} \rho_j^2\right) \times \left((y_{it}-x_{it}^\top \beta)^2+{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} (y_{i,t-j}-x_{i,t-j}^\top \beta)^2 \right) & (\text{Cauchy-Schwarz}) \\
& =&& 2\cdot\left(\psi^{(O)}_{it} (\beta) + {\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} \psi^{(O)}_{i,t-j}(\beta)\right). & (\|\rho\|_2^2\leq 1)
\end{split}
\end{eqnarray*}
%
Now combining Inequalities~\eqref{ineq:relation} and~\eqref{ineq:relation2}, we have that for any $\zeta=(\beta,\rho)\in {\mathcal{P}}_\lambda$,
\begin{align*}
\frac{\psi^{(G,q)}_{it}(\zeta)}{\psi^{(G,q)}(\zeta)}
&\leq &&\frac{2\cdot\left(\psi^{(O)}_{it} (\beta) + {\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} \psi^{(O)}_{i,t-j}(\beta)\right)}{\lambda \cdot \psi^{(O)}(\beta)} \\
&\leq && 2\lambda^{-1}\cdot \left(s^{(O)}(i,t)+{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} s^{(O)}(i,t-j) \right) \\
&= && s(i,t).
\end{align*}
%
This completes the proof.
\end{proof}
\section{Coresets for GLSE$_k$}
\label{sec:glsek}
Following from Section~\ref{sec:alg}, we assume that the parameter space is ${\mathcal{P}}_\lambda^k = (\mathbb{R}^d \times B^q_{1-\lambda})^k$ for some given constant $\lambda\in (0,1)$.
%
Given a panel dataset $X\in \mathbb{R}^{N\times T\times d}$ and $Y\in \mathbb{R}^{N\times T}$, let $Z^{(i)}\in \mathbb{R}^{T\times (d+1)}$ denote a matrix whose $t$-th row is $(x_{it},y_{it})\in \mathbb{R}^{d+1}$ for all $t\in [T]$ ($i\in [N]$).
%
Assume there exists constant $M\geq 1$ such that the input dataset satisfies the following property.
%
\begin{definition}[\bf{$M$-bounded dataset}]
\label{def:bounded_dataset_main}
Given $M\geq 1$, we say a panel dataset $X\in \mathbb{R}^{N\times T\times d}$ and $Y\in \mathbb{R}^{N\times T}$ is $M$-bounded if for any $i\in [N]$, the condition number of matrix $(Z^{(i)})^\top Z^{(i)}$ is at most $M$, i.e.,
$
\max_{\beta\in \mathbb{R}^d} \frac{\psi^{(O)}_i(\beta)}{\|\beta\|_2^2+1} \leq M\cdot \min_{\beta\in \mathbb{R}^d} \frac{\psi^{(O)}_i(\beta)}{\|\beta\|_2^2+1}.
$
\end{definition}
\noindent
If there exists $i\in [N]$ and $\beta\in \mathbb{R}^d$ such that $\psi^{(O)}_i(\beta)=0$, we let $M=\infty$.
%
Specifically, if all $(Z^{(i)})^\top Z^{(i)}$ are identity matrix whose eigenvalues are all 1, i.e., for any $\beta$, $\psi^{(O)}_i(\beta) = \|\beta\|_2^2+1$, we can set $M=1$.
%
Another example is that if $n\gg d$ and all elements of $Z^{(i)}$ are independently and identically distributed standard normal random variables, then the condition number of matrix $(Z^{(i)})^\top Z^{(i)}$ is upper bounded by some constant with high probability (and constant in expectation)~\cite{chen2005condition,shi2013the}, which may also imply $M = O(1)$.
%
The main theorem is as follows.
%
\begin{theorem}[\bf{Coresets for GLSE$_k$}]
\label{thm:coreset_glsek}
There exists a randomized algorithm that given an $M$-bounded ($M\geq 1$) panel dataset $X\in \mathbb{R}^{N\times T\times d}$ and $Y\in \mathbb{R}^{N\times T}$, constant $\varepsilon,\lambda \in (0,1)$ and integers $q,k\geq 1$, with probability at least 0.9, constructs an $\varepsilon$-coreset for GLSE$_k$ of size
\[
O\left(\varepsilon^{-4} \lambda^{-2} M k^2 \max\left\{q^7 d^4, q^5 d^6\right\} \cdot \log \frac{Mq}{ \lambda} \log \frac{Mkd}{\lambda} \right)
\]
and runs in time $O(NTq+NTd^2)$.
\end{theorem}
\noindent
Similar to GLSE, this coreset for GLSE$_k$ ($k\geq 2$) contains at most
\[
(q+1)\cdot O\left(\varepsilon^{-4} \lambda^{-2} M k^2 \max\left\{q^7 d^4, q^5 d^6\right\} \cdot \log \frac{Mq}{ \lambda} \log \frac{kd}{\lambda} \right)
\]
points $(x_{it},y_{it})$, which is independent of both $N$ and $T$ when $M$ is constant.
%
Note that the size contains an addtional factor $M$ which can be unbounded.
%
Our algorithm is summarized in Algorithm~\ref{alg:glsek} and we outline Algorithm~\ref{alg:glsek} and discuss the novelty in the following.
%
\begin{algorithm}[htp!]
\caption{$\mathsf{CGLSE_k}$: Coreset construction of GLSE$_k$}
\label{alg:glsek}
\begin{algorithmic}[1]
\REQUIRE an $M$-bounded (constant $M\geq 1$) panel dataset $X\in \mathbb{R}^{N\times T\times d}$ and $Y\in \mathbb{R}^{N\times T}$, constant $\varepsilon,\lambda \in (0,1)$, integers $k,q\geq 1$ and parameter space ${\mathcal{P}}_\lambda^k$. \\
\ENSURE a subset $S\subseteq [N]\times [T]$ together with a weight function $w:S\rightarrow \mathbb{R}_{\geq 0}$. \\
%
\% {Constructing a subset of individuals}
\STATE $ \Gamma \leftarrow O\left(\varepsilon^{-2} \lambda^{-1} M k^2 \max\left\{q^4 d^2, q^3 d^3\right\}\cdot \log \frac{Mq}{\lambda}\right)$.
%
\STATE For each $i\in [N]$, let matrix $Z^{(i)}\in \mathbb{R}^{T\times (d+1)}$ be whose $t$-th row is $z^{(i)}_{t}=(x_{it},y_{it})\in \mathbb{R}^{d+1}$.
%
\STATE For each $i\in [N]$, construct the SVD decomposition of $Z^{(i)}$ and compute
\[
u_i:=\lambda_{\max}((Z^{(i)})^\top Z^{(i)}) \text{ and } \ell_i:=\lambda_{\min}((Z^{(i)})^\top Z^{(i)}).
\]
%
\STATE For each $i\in [N]$,
$
s^{(O)}(i)\leftarrow \frac{u_i}{u_i+\sum_{i'\neq i} \ell_{i'}}.
$
%
\STATE For each $i\in [N]$, $s(i)\leftarrow \min\left\{1, \frac{2(q+1)}{\lambda}\cdot s^{(O)}(i)\right\}$.
%
\STATE Pick a random sample $I_S\subseteq [N]$ of size $M$, where each $i\in I_S$ is selected w.p. $\frac{s(i)}{\sum_{i'\in [N]}s(i')}$.
%
\STATE For each $i\in I_S$, $w'(i)\leftarrow \frac{\sum_{i'\in [N]}s(i')}{\Gamma \cdot s(i)}$. \\
%
\% {Constructing a subset of time periods for each selected individual}
%
\STATE For each $i\in I_S$, apply $\mathbf{CGLSE}(X_i,y_i,\frac{\varepsilon}{3},\frac{1}{20\Gamma},\lambda,q$) and construct $J_{S,i}\subseteq [T]$ together with a weight function $w^{(i)}: J_{S,i}\rightarrow \mathbb{R}_{\geq 0}$.
%
\STATE Let $S\leftarrow \left\{(i,t)\in [N]\times [T]: i\in I_S, t\in J_{S,i}\right\}$.
%
\STATE For each $(i,t)\in S$, $w(i,t) \leftarrow w'(i)\cdot w^{(i)}(t)$.
%
\STATE Output $(S,w)$.
\end{algorithmic}
\end{algorithm}
\begin{remark}
\label{remark:framework_glsek}
Algorithm~\ref{alg:glsek} is a two-staged framework, which captures the $\min$ operations in GLSE$_k$.
%
\paragraph{First stage.}
%
We construct an $\frac{\varepsilon}{3}$-coreset $I_S\subseteq [N]$ together with a weight function $w':I_S\rightarrow \mathbb{R}_{\geq 0}$ of the query space $(Z^{(G,q,k)},{\mathcal{P}}^k,\psi^{(G,q,k)})$, i.e., for any $\zeta\in {\mathcal{P}}^k$
\[
\sum_{i\in I_S} w'(i)\cdot \psi^{(G,q,k)}_{i}(\zeta) \in (1\pm \varepsilon)\cdot \psi^{(G,q,k)}(\zeta).
\]
%
The idea is similar to Algorithm~\ref{alg:glse} except that we consider $N$ sub-functions $\psi^{(G,q,k)}_{i}$ instead of $NT$.
%
In Lines 2-4 of Algorithm~\ref{alg:glsek}, we first construct a sensitivity function $s^{(O)}$ of OLSE$_k$.
%
The definition of $s^{(O)}$ captures the impact of $\min$ operations in the objective function of OLSE$_k$ and the total sensitivity of $s^{(O)}$ is guaranteed to be upper bounded by Definition~\ref{def:bounded_dataset_main}.
%
The key is showing that the maximum influence of individual $i$ is at most $\frac{u_i}{u_i+\sum_{j\neq i} \ell_j}$ (Lemma~\ref{lm:sen_olsek}), which implies that the total sensitivity of $s^{(O)}$ is at most $M$.
%
Then in Line 5, we construct a sensitivity function $s$ of GLSE$_k$, based on a reduction from $s^{(O)}$ (Lemma~\ref{lm:sen_glsek}).
%
\paragraph{Second stage.} In Line 8, for each $i\in I_S$, apply $\mathbf{CGLSE}(X_i,y_i,\frac{\varepsilon}{3},\frac{1}{20\cdot |I_S|},\lambda,q$) and construct a subset $J_{S,i}\subseteq [T]$ together with a weight function $w^{(i)}: J_{S,i}\rightarrow \mathbb{R}_{\geq 0}$.
%
Output $S=\left\{(i,t)\in [N]\times [T]: i\in I_S, t\in J_{S,i}\right\}$ together with a weight function $w: S\rightarrow \mathbb{R}_{\geq 0}$ defined as follows: for any $(i,t)\in S$, $ w(i,t) := w'(i)\cdot w^{(i)}(t)$.
\end{remark}
\noindent
We also provide a lower bound theorem which shows that the size of a coreset for GLSE$_k$ can be up to $\Omega(N)$.
%
It indicates that the coreset size should contain additional factors than $k,q,d,1/\varepsilon$, which reflects the reasonability of the $M$-bounded assumption.
%
\begin{theorem}[\bf{Size lower bound of GLSE$_k$}]
\label{thm:lower_main}
Let $T=1$ and $d=k=2$ and $\lambda\in (0,1)$.
%
There exists $X\in \mathbb{R}^{N\times T\times d}$ and $Y\in \mathbb{R}^{N\times T}$ such that any 0.5-coreset for GLSE$_k$ should have size $\Omega(N)$.
%
\end{theorem}
\subsection{Proof overview}
\label{sec:proof_overview}
We first give a proof overview for summarization.
\paragraph{Proof overview of Theorem~\ref{thm:coreset_glsek}.}
For GLSE$_k$, we propose a two-staged framework (Algorithm~\ref{alg:glsek}): first sample a collection of individuals and then run $\mathbf{CGLSE}$ on every selected individuals.
By Theorem~\ref{thm:coreset_glse}, each subset $J_{S,i}$ at the second stage is of size $\operatorname{poly}(q,d)$.
Hence, we only need to upper bound the size of $I_S$ at the first stage.
By a similar argument as that for GLSE, we can define the pseudo-dimension of GLSE$_k$ and upper bound it by $\operatorname{poly}(k,q,d)$, and hence, the main difficulty is to upper bound the total sensitivity of GLSE$_k$.
We show that the gap between the individual regression objectives of GLSE$_k$ and OLSE$_k$ (GLSE$_k$ with $q=0$) with respect to the same $(\beta^{(1)},\ldots,\beta^{(k)})$ is at most $\frac{2(q+1)}{\lambda}$, which relies on $
\psi^{(G,q)}_i(\zeta) \geq \lambda \cdot \psi^{(O)}_i(\beta)$ and an observation that for any $\zeta=(\beta^{(1)},\ldots,\beta^{(k)},\rho^{(1)},\ldots,\rho^{(k)})\in {\mathcal{P}}^k$,
$
\psi^{(G,q,k)}_i(\zeta) \leq 2(q+1)\cdot \min_{l\in [k]} \psi^{(O)}_{i}(\beta^{(l)}).
$
Thus, it suffices to provide an upper bound of the total sensitivity for OLSE$_k$.
We claim that the maximum influence of individual $i$ is at most $\frac{u_i}{u_i+\sum_{j\neq i} \ell_j}$ where $u_i$ is the largest eigenvalue of $(Z^{(i)})^\top Z^{(i)}$ and $\ell_j$ is the smallest eigenvalue of $(Z^{(j)})^\top Z^{(j)}$.
This fact comes from the following observation:
$
\min_{l\in [k]}\|Z^{(i)} (\beta^{(l)},-1)\|_2^2 \leq \frac{u_i}{\ell_j}\cdot \min_{l\in [k]}\|Z^{(j)} (\beta^{(l)},-1)\|_2^2,
$
and results in an upper bound $M$ of the total sensitivity for OLSE$_k$ since
$
\sum_{i\in [N]} \frac{u_i}{u_i+\sum_{j\neq i} \ell_j} \leq \frac{\sum_{i\in [N]} u_i}{\sum_{j\in [N]} \ell_j} \leq M.
$
\paragraph{Proof overview of Theorem~\ref{thm:lower_main}.}
For GLSE$_k$, we provide a lower bound $\Omega(N)$ of the coreset size by constructing an instance in which any 0.5-coreset should contain observations from all individuals.
Note that we consider $T=1$ which reduces to an instance with cross-sectional data.
Our instance is to let $x_{i1}=(4^i,\frac{1}{4^i})$ and $y_{i1}=0$ for all $i\in [N]$.
Then letting $\zeta^{(i)}=(\beta^{(1)},\beta^{(2)},\rho^{(1)},\rho^{(2)})$ where $\beta^{(1)}=(\frac{1}{4^i},0)$, $\beta^{(2)}=(0,4^i)$ and $\rho^{(1)}=\rho^{(2)}=0$, we observe that $\psi^{(G,q,k)}(\zeta^{(i)})\approx \psi^{(G,q,k)}_i(\zeta^{(i)})$.
Hence, all individuals should be contained in the coreset such that regression objectives with respect to all $\zeta^{(i)}$ are approximately preserved.
\subsection{Proof of Theorem~\ref{thm:coreset_glsek}: Upper bound for GLSE$_k$}
\label{sec:proof_glsek_upper}
The proof of Theorem \ref{thm:coreset_glsek} relies on the following two theorems.
The first theorem shows that $I_S$ of Algorithm~\ref{alg:glsek} is an $\frac{\varepsilon}{3}$-coreset of $\left(Z^{G,q,k},{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)}\right)$.
The second one is a reduction theorem that for each individual in $I_S$ constructs an $\varepsilon$-coreset $J_{S,i}$.
\begin{theorem}[\bf{Coresets of $\left(Z^{G,q,k},{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)}\right)$}]
\label{thm:glsek_individual}
For any given $M$-bounded observation matrix $X\in \mathbb{R}^{N\times T\times d}$ and outcome matrix $Y\in \mathbb{R}^{N\times T}$, constant $\varepsilon,\delta,\lambda \in (0,1)$ and integers $q,k\geq 1$, with probability at least 0.95, the weighted subset $I_S$ of Algorithm~\ref{alg:glsek} is an $\frac{\varepsilon}{3}$-coreset of the query space $\left(Z^{G,q,k},{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)}\right)$, i.e., for any $\zeta=(\beta^{(1)},\ldots,\beta^{(k)},\rho^{(1)},\ldots,\rho^{(k)})\in {\mathcal{P}}_\lambda^k$,
\begin{align}
\label{ineq:I_S}
\sum_{i\in I_S} w'(i)\cdot \psi^{(G,q,k)}_{i}(\zeta) \in (1\pm \frac{\varepsilon}{3})\cdot \psi^{(G,q,k)}(\zeta).
\end{align}
Moreover, the construction time of $I_S$ is
\[
N\cdot \mathsf{SVD}(T,d+1)+ O(N).
\]
\end{theorem}
\noindent
We defer the proof of Theorem~\ref{thm:glsek_individual} later.
\begin{theorem}[\bf{Reduction from coresets of $\left(Z^{G,q,k},{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)}\right)$ to coresets for GLSE$_k$}]
\label{thm:reduction}
Suppose that the weighted subset $I_S$ of Algorithm~\ref{alg:glsek} is an $\frac{\varepsilon}{3}$-coreset of the query space $\left(Z^{G,q,k},{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)}\right)$.
%
Then with probability at least 0.95, the output $(S,w)$ of Algorithm~\ref{alg:glsek} is an $\varepsilon$-coreset for GLSE$_k$.
\end{theorem}
\begin{proof}[Proof of Theorem~\ref{thm:reduction}]
Note that $S$ is an $\varepsilon$-coreset for GLSE$_k$ if Inequality~\eqref{ineq:I_S} holds and for all $i\in [N]$, $J_{S,i}$ is an $\frac{\varepsilon}{3}$-coreset of $\left((Z^{(i)})^{(G,q)},{\mathcal{P}}_\lambda,\psi^{(G,q)}\right)$.
%
By condition, we assume Inequality~\eqref{ineq:I_S} holds.
%
By Line 6 of Algorithm~\ref{alg:glsek}, the probability that every $J_{S,i}$ is an $\frac{\varepsilon}{3}$-coreset of $\left((Z^{(i)})^{(G,q)},{\mathcal{P}}_\lambda,\psi^{(G,q)}\right)$ is at least
\[
1-\Gamma\cdot \frac{1}{20\Gamma} = 0.95,
\]
%
which completes the proof.
\end{proof}
\noindent
Observe that Theorem~\ref{thm:coreset_glsek} is a direct corollary of Theorems~\ref{thm:glsek_individual} and~\ref{thm:reduction}.
\begin{proof}
Combining Theorems~\ref{thm:glsek_individual} and~\ref{thm:reduction}, $S$ is an $\varepsilon$-coreset of $\left(Z^{G,q,k},{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)}\right)$ with probability at least 0.9.
%
By Theorem~\ref{thm:coreset_glse}, the size of $S$ is
\[
\Gamma \cdot O\left(\varepsilon^{-2} \lambda^{-1} q d\left(\max\left\{q^2d, qd^2\right\}\cdot \log \frac{d}{\lambda}+\log \frac{\Gamma}{\delta}\right) \right),
\]
%
which satisfies Theorem~\ref{thm:coreset_glsek} by pluging in the value of $\Gamma$.
%
For the running time, it costs $N\cdot \mathsf{SVD}(T,d+1)$ to compute $I_S$ by Theorem~\ref{thm:glsek_individual}.
%
Moreover, by Line 3 of Algorithm~\ref{alg:glsek}, we already have the SVD decomposition of $Z^{(i)}$ for all $i\in [N]$.
%
Then it only costs $O\left(T(q+d)\right)$ to apply $\mathbf{CGLSE}$ for each $i\in I_S$ in Line 8 of Algorithm~\ref{alg:glsek}.
%
Then it costs $O\left(NT(q+d)\right)$ to construct $S$.
%
This completes the proof of the running time.
\end{proof}
\paragraph{Proof of Theorem~\ref{thm:glsek_individual}: $I_S$ is a coreset of $\left(Z^{(G,q,k)},{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)}\right)$.}
It remains to prove Theorem~\ref{thm:glsek_individual}.
Note that the construction of $I_S$ applies the Feldman-Langberg framework.
The analysis is similar to Section~\ref{sec:alg} in which we provide upper bounds for both the total sensitivity and the pseudo-dimension.
We first discuss how to bound the total sensitivity of $(Z^{(G,q,k)},{\mathcal{P}}^k,\psi^{(G,q,k)})$.
Similar to Section~\ref{sec:sen}, the idea is to first bound the total sensitivity of $(Z^{(G,0,k)},{\mathcal{P}}^k,\psi^{(G,0,k)})$ -- we call it the query space of OLSE$_k$ whose covariance matrices of all individuals are identity matrices.
\begin{lemma}[\bf{Total sensitivity of OLSE$_k$}]
\label{lm:sen_olsek}
Function $s^{(O)}:[N]\rightarrow \mathbb{R}_{\geq 0}$ of Algorithm~\ref{alg:glsek} satisfies that for any $i\in [N]$,
\begin{align}
\label{ineq:sen_olsek}
s^{(O)}(i) \geq \sup_{\beta^{(1)},\ldots,\beta^{(k)}\in \mathbb{R}^d} \frac{\min_{l\in [k]}\psi^{(O)}_{i}(\beta^{(l)})}{\sum_{i'\in [N]}\min_{l\in [k]}\psi^{(O)}_{i'}(\beta^{(l)})},
\end{align}
and ${\mathcal{G}}^{(O)} := \sum_{i\in [N]} s^{(O)}(i)$ satisfying that ${\mathcal{G}}^{(O)} = O(M)$.
%
Moreover, the construction time of function $s^{(O)}$ is
\[
N\cdot \mathsf{SVD}(T,d+1)+ O(N).
\]
%
\end{lemma}
\begin{proof}
For every $i\in [N]$, recall that $Z^{(i)}\in \mathbb{R}^{T\times (d+1)}$ is the matrix whose $t$-th row is $z^{(i)}_{t}=(x_{it},y_{it})\in \mathbb{R}^{d+1}$ for all $t\in [T]$.
%
%
By definition, we have that for any $\beta\in \mathbb{R}^d$,
\[
\psi^{(O)}_{i}(\beta) = \|Z^{(i)} (\beta,-1)\|_2^2.
\]
%
Thus, by the same argument as in Lemma~\ref{lm:sen_olse}, it suffices to prove that for any matrix sequences $Z^{(1)},\ldots, Z^{(N)}\in \mathbb{R}^{T\times (d+1)}$,
\begin{eqnarray}
\label{ineq:sen_matrix}
\begin{split}
s^{(O)}(i) &\geq&& \sup_{\beta^{(1)},\ldots,\beta^{(k)}\in \mathbb{R}^{d}} \\
& &&\frac{\min_{l\in [k]}\|Z^{(i)} (\beta^{(l)},-1)\|_2^2}{\sum_{i'\in [N]}\min_{l\in [k]}\|Z^{(i')} (\beta^{(l)},-1)\|_2^2}.
\end{split}
\end{eqnarray}
%
For any $\beta^{(1)},\ldots,\beta^{(k)}\in \mathbb{R}^d$ and any $i\neq j\in [N]$, letting $l^\star = \arg\min_{l\in [k]} \|Z^{(j)} (\beta^{(l)},-1)\|_2^2$, we have
\begin{align*}
& && \min_{l\in [k]}\|Z^{(i)} (\beta^{(l)},-1)\|_2^2 &&\\
& \leq && \|Z^{(i)} (\beta^{(l^\star)},-1)\|_2^2 && \\
& \leq && u_i\cdot (\|\beta^{(l^\star)}\|_2^2+1) && (\text{Defn. of $u_i$}) \\
& \leq && \frac{u_i}{\ell_j}\cdot \|Z^{(j)} (\beta^{(l^\star)},-1)\|_2^2 && (\text{Defn. of $\ell_i$}) \\
& = && \frac{u_i}{\ell_j}\cdot \min_{l\in [k]}\|Z^{(j)} (\beta^{(l)},-1)\|_2^2. && (\text{Defn. of $l^\star$})
\end{align*}
%
Thus, we directly conclude that
\begin{align*}
& && \frac{\min_{l\in [k]}\|Z^{(i)} (\beta^{(l)},-1)\|_2^2}{\sum_{i'\in [N]}\min_{l\in [k]}\|Z^{(i')} (\beta^{(l)},-1)\|_2^2} &&\\
& \leq && \frac{\min_{l\in [k]}\|Z^{(i)} (\beta^{(l)},-1)\|_2^2}{\left(1+\sum_{i'\neq i} \frac{\ell_{i'}}{u_i}\right)\cdot \min_{l\in [k]}\|Z^{(i)} (\beta^{(l)},-1)\|_2^2} && \\
& = && \frac{u_i}{u_i+\sum_{i'\neq i} \ell_{i'}} && \\
& = && s^{(O)}(i). &&
\end{align*}
%
Hence, Inequality~\eqref{ineq:sen_matrix} holds.
%
Moreover, since the input dataset is $M$-bounded, we have
\[
{\mathcal{G}}^{(O)} \leq \sum_{i\in [N]} \frac{u_i}{\sum_{i'\in [N]} \ell_{i'}} \leq M,
\]
which completes the proof of correctness.
For the running time, it costs $N\cdot \mathsf{SVD}(T,d+1)$ to compute SVD decompositions for all $Z^{(i)}$.
%
Then it costs $O(N)$ time to compute all $u_i$ and $\ell_i$, and hence costs $O(N)$ time to compute sensitivity functions $s^{(O)}$.
%
Thus, we complete the proof.
\end{proof}
\noindent
Note that by the above argument, we can also assume
\[
\sum_{i\in [N]}\frac{u_i}{u_i+\sum_{i'\neq i} \ell_{i'}} \leq M,
\]
which leads to the same upper bound for the total sensitivity ${\mathcal{G}}^{(O)}$.
Now we are ready to upper bound the total sensitivity of $(Z^{(G,q,k)},{\mathcal{P}}^k,\psi^{(G,q,k)})$.
\begin{lemma}[\bf{Total sensitivity of GLSE$_k$}]
\label{lm:sen_glsek}
Function $s:[N]\rightarrow \mathbb{R}_{\geq 0}$ of Algorithm~\ref{alg:glsek} satisfies that for any $i\in [N]$,
\begin{align}
\label{ineq:sen_glsek}
s(i) \geq \sup_{\zeta\in {\mathcal{P}}_\lambda^k} \frac{\psi^{(G,q,k)}_{i}(\zeta)}{\psi^{(G,q,k)}(\zeta)},
\end{align}
and ${\mathcal{G}} := \sum_{i\in [N]} s(i)$ satisfying that ${\mathcal{G}} = O(\frac{qM}{\lambda})$.
%
Moreover, the construction time of function $s$ is
\[
N\cdot \mathsf{SVD}(T,d+1)+ O(N).
\]
%
\end{lemma}
\begin{proof}
Since it only costs $O(N)$ time to construct function $s$ if we have $s^{(O)}$, we prove the construction time by Lemma~\ref{lm:sen_olsek}.
%
Fix $i\in [N]$.
%
If $s(i) = 1$ in Line 4 of Algorithm~\ref{alg:glsek}, then Inequality~\eqref{ineq:sen_glsek} trivally holds.
%
Then we assume that $s(i) = \frac{2(q+1)}{\lambda}\cdot s^{(O)}(i)$.
%
We first have that for any $i\in [N]$ and any $\zeta\in {\mathcal{P}}_\lambda^k$,
\begin{align*}
& && \psi^{(G,q,k)}_i(\zeta)&& \\
& = && \min_{l\in [k]} {\textstyle\sum}_{t\in [T]} \psi^{(G,q)}_{it}(\beta^{(l)},\rho^{(l)}) && (\text{Defn.~\ref{def:glsek}}) \\
& \geq && \min_{l\in [k]} {\textstyle\sum}_{t\in [T]} \lambda\cdot \psi^{(O)}_{it}(\beta^{(l)}) && (\text{Ineq.~\eqref{ineq:relation}}) \\
& = && \lambda\cdot \min_{l\in [k]} \psi^{(O)}_{i}(\beta^{(l)}). && (\text{Defn. of $\psi^{(O)}_i$})
\end{align*}
%
which directly implies that
\begin{align}
\label{ineq:sen1}
\psi^{(G,q,k)}(\zeta) \geq \lambda \cdot \sum_{i'\in [N]}\min_{l\in [k]}\psi^{(O)}_{i'}(\beta^{(l)}).
\end{align}
%
We also note that for any $(i,t)\in [N]\times [T]$ and any $(\beta,\rho)\in {\mathcal{P}}_\lambda$,
\begin{eqnarray*}
\begin{split}
& && \psi^{(G,q)}_{it}(\beta,\rho) &\\
& \leq && \left((y_{it}-x_{it}^\top \beta)-{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} \rho_j\cdot (y_{i,t-j}-x_{i,t-j}^\top \beta)\right)^2 & (\text{Defn. of $\psi^{(G,q)}_{it}$}) \\
& \leq && (1+{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} \rho_j^2) \times \left((y_{it}-x_{it}^\top \beta)^2+{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} (y_{i,t-j}-x_{i,t-j}^\top \beta)^2\right) & (\text{Cauchy-Schwarz}) \\
& \leq && 2\left((y_{it}-x_{it}^\top \beta)^2+{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} (y_{i,t-j}-x_{i,t-j}^\top \beta)^2\right). & (\|\rho\|_2^2\leq 1)
\end{split}
\end{eqnarray*}
%
Hence, we have that
\begin{eqnarray}
\label{ineq:sen2}
\frac{1}{2}\cdot \psi^{(G,q)}_{it}(\beta,\rho) \leq (y_{it}-x_{it}^\top \beta)^2+\sum_{j=1}^{\min\left\{t-1,q\right\}} (y_{i,t-j}-x_{i,t-j}^\top \beta)^2.
\end{eqnarray}
%
This implies that
\begin{eqnarray}
\label{ineq:sen3}
\begin{split}
& && \psi^{(G,q,k)}_i(\zeta) & \\
& = && \min_{l\in [k]} {\textstyle\sum}_{t\in [T]} \psi^{(G,q)}_{it}(\beta^{(l)},\rho^{(l)}) \quad &(\text{Defn.~\ref{def:glsek}}) \\
& \leq && \min_{l\in [k]} {\textstyle\sum}_{t\in [T]} 2 \times \left((y_{it}-x_{it}^\top \beta)^2+{\textstyle\sum}_{j=1}^{\min\left\{t-1,q\right\}} (y_{i,t-j}-x_{i,t-j}^\top \beta)^2\right)
& (\text{Ineq.~\eqref{ineq:sen2}})\\
& \leq && 2(q+1)\cdot \min_{l\in [k]}{\textstyle\sum}_{t\in [T]} \psi^{(O)}_{it}(\beta^{(l)}) &\\
& = && 2(q+1)\cdot \min_{l\in [k]} \psi^{(O)}_{i}(\beta^{(l)}). & (\text{Defn. of $\psi^{(O)}_i$})
\end{split}
\end{eqnarray}
%
Thus, we have that for any $i\in [N]$ and $\zeta\in {\mathcal{P}}_\lambda^k$,
\begin{align*}
\frac{\psi^{(G,q,k)}_{i}(\zeta)}{\psi^{(G,q,k)}(\zeta)} & \leq && \frac{2(q+1)\cdot \min_{l\in [k]} \psi^{(O)}_{i}(\beta^{(l)})}{\lambda\cdot \sum_{i\in [N]} \min_{l\in [k]} \psi^{(O)}_{i}(\beta^{(l)})} & (\text{Ineqs.~\eqref{ineq:sen1} and~\eqref{ineq:sen3}}) \\
&\leq && \frac{2(q+1)}{\lambda} \cdot s^{(O)}(i) & (\text{Lemma~\ref{lm:sen_olsek}}) \\
& = && s(i), & (\text{by assumption})
\end{align*}
which proves Inequality~\eqref{ineq:sen_glsek}.
%
Moreover, we have that
\[
{\mathcal{G}} = \sum_{i\in [N]} s(i) \leq \frac{2(q+1)}{\lambda} \cdot {\mathcal{G}}^{(O)} = O(\frac{qM}{\lambda}),
\]
where the last inequality is from Lemma~\ref{lm:sen_olsek}.
%
We complete the proof.
\end{proof}
\noindent
Next, we upper bound the pseudo-dimension of GLSE$_k$.
The proof is similar to that of GLSE by applying Lemmas~\ref{lm:dim_bound} and~\ref{lm:dim_range}.
\begin{lemma}[\bf{Pseudo-dimension of GLSE$_k$}]
\label{lm:dim_glsek}
\sloppy
The pseudo-dimension of any query space $(Z^{(G,q,k)},u,{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)})$ over weight functions $u: [N]\rightarrow \mathbb{R}_{\geq 0}$ is at most
\[
O\left(k^2q^2(q+d)d^2 \right).
\]
\end{lemma}
\begin{proof}
\sloppy
The proof idea is similar to that of Lemma~\ref{lm:dim_glse}.
%
Fix a weight function $u: [N]\rightarrow \mathbb{R}_{\geq 0}$.
%
For every $i\in [N]$, let $g_{i}: {\mathcal{P}}_\lambda^k\times \mathbb{R}_{\geq 0}\rightarrow \left\{0,1\right\}$ be the indicator function satisfying that for any $\zeta=(\beta^{(1)},\ldots,\beta^{(k)},\rho^{(1)},\ldots,\rho^{(k)})\in {\mathcal{P}}_\lambda^k$ and $r\in \mathbb{R}_{\geq 0}$,
\begin{align*}
g_{i}(\zeta,r) &:= && I\left[u(i)\cdot\psi^{(G,q,k)}_{i}(\zeta)\geq r\right] \\
&= && I\left[\forall l\in [k],~ u(i)\cdot \sum_{t\in [T]}\psi^{(G,q)}_{it}(\beta^{(l)},\rho^{(l)})\geq r\right].
\end{align*}
%
We consider the query space $(Z^{(G,q,k)},u,{\mathcal{P}}_\lambda^k\times \mathbb{R}_{\geq 0},g)$.
%
By the definition of ${\mathcal{P}}_\lambda^k$, the dimension of ${\mathcal{P}}_\lambda^k\times \mathbb{R}_{\geq 0}$ is $m=k(q+d)+1$.
%
Also note that for any $(\beta,\rho)\in {\mathcal{P}}_\lambda$, $\psi^{(G,q)}_{it}(\beta,\rho)$ can be represented as a multivariant polynomial that consists of $O(q^2 d^2)$ terms $\rho_{c_1}^{b_1} \rho_{c_2}^{b_2} \beta_{c_3}^{b_3} \beta_{c_4}^{b_4}$ where $c_1,c_2\in [q]$, $c_3,c_4\in [d]$ and $b_1,b_2,b_3,b_4\in \left\{0,1\right\}$.
%
Thus, $g_{i}$ can be calculated using $l=O(kq^2d^2)$ operations, including $O(kq^2d^2)$ arithmetic operations and $k$ jumps.
%
Pluging the values of $m$ and $l$ in Lemma~\ref{lm:dim_bound}, the pseudo-dimension of $(Z^{(G,q,k)},u,{\mathcal{P}}_\lambda^k\times \mathbb{R}_{\geq 0},g)$ is $O\left(k^2q^2(q+d)d^2\right)$.
%
Then by Lemma~\ref{lm:dim_range}, we complete the proof.
\end{proof}
\noindent
Combining with the above lemmas and Theorem~\ref{thm:fl11}, we are ready to prove Theorem~\ref{thm:glsek_individual}.
\begin{proof}[Proof of Theorem~\ref{thm:glsek_individual}]
By Lemma~\ref{lm:sen_glsek}, the total sensitivity ${\mathcal{G}}$ of $(Z^{(G,q,k)},{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)})$ is $O(\frac{qM}{\lambda})$.
%
By Lemma~\ref{lm:dim_glsek}, we can let $\dim = O\left(k^2(q+d)q^2d^2 \right)$ which is an upper bound of the pseudo-dimension of every query space $(Z^{(G,q,k)},u,{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)})$ over weight functions $u: [N]\rightarrow \mathbb{R}_{\geq 0}$.
%
Pluging the values of ${\mathcal{G}}$ and $\dim$ in Theorem~\ref{thm:fl11}, we prove for the coreset size.
%
For the running time, it costs $N\cdot \mathsf{SVD}(T,d+1)+ O(N)$ time to compute the sensitivity function $s$ by Lemma~\ref{lm:sen_glsek}, and $O(N)$ time to construct $I_S$.
%
This completes the proof.
%
\end{proof}
\subsection{Proof of Theorem~\ref{thm:lower_main}: Lower bound for GLSE$_k$}
\label{sec:lower}
Actually, we prove a stronger version of Theorem~\ref{thm:lower_main} in the following.
We show that both the coreset size and the total sensitivity of the query space $(Z^{(G,q,k)},u,{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)})$ may be $\Omega(N)$, even for the simple case that $T=1$ and $d=k=2$.
\begin{theorem}[\bf{Size and sensitivity lower bound of GLSE$_k$}]
\label{thm:lower}
Let $T=1$ and $d=k=2$ and $\lambda\in (0,1)$.
%
There exists an instance $X\in \mathbb{R}^{N\times T\times d}$ and $Y\in \mathbb{R}^{N\times T}$ such that the total sensitivity
\[
\sum_{i\in [N]} \sup_{\zeta\in {\mathcal{P}}_\lambda^k} \frac{\psi^{(G,q,k)}_i(\zeta)}{\psi^{(G,q,k)}(\zeta)} = \Omega(N).
\]
and any 0.5-coreset of the query space $(Z^{(G,q,k)},u,{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)})$ should have size $\Omega(N)$.
%
\end{theorem}
\begin{proof}
We construct the same instance as in~\cite{Tolochinsky2018GenericCF}.
%
Concretely, for $i\in [N]$, let $x_{i1} = (4^i,\frac{1}{4^i})$ and $y_{i1}=0$.
%
We claim that for any $i\in [N]$,
\begin{align}
\label{ineq:sen_fraction}
\sup_{\zeta\in {\mathcal{P}}_\lambda^k} \frac{\psi^{(G,q,k)}_i(\zeta)}{\psi^{(G,q,k)}(\zeta)} \geq \frac{1}{2}.
\end{align}
%
If the claim is true, then we complete the proof of the total sensitivity by summing up the above inequality over all $i\in [N]$.
%
Fix $i\in [N]$ and consider the following $\zeta=(\beta^{(1)},\beta^{(2)},\rho^{(1)},\rho^{(2)})\in {\mathcal{P}}_\lambda^k$ where $\beta^{(1)}=(\frac{1}{4^i},0)$, $\beta^{(2)}=(0,4^i)$ and $\rho^{(1)}=\rho^{(2)}=0$.
%
If $j\leq i$, we have
\begin{align*}
\psi^{(G,q,k)}_j(\zeta) &=&& \min_{l\in [2]} (y_{i1}-x_{i1}^\top \beta^{(l)})^2 \\
&= && \min\left\{\frac{1}{16^{j-i}}, \frac{1}{16^{i-j}}\right\} \\
& =&& \frac{1}{16^{i-j}}.
\end{align*}
Similarly, if $j>i$, we have
\[
\psi^{(G,q,k)}_j(\zeta) = \min\left\{\frac{1}{16^{j-i}}, \frac{1}{16^{i-j}}\right\} = \frac{1}{16^{j-i}}.
\]
%
By the above equations, we have
\begin{align}
\label{eq:lower1}
\psi^{(G,q,k)}(\zeta) = \sum_{j=1}^{i} \frac{1}{16^{i-j}} + \sum_{j=i+1}^{N} \frac{1}{16^{j-i}} < \frac{5}{4}.
\end{align}
%
Combining with the fact that $\psi^{(G,q,k)}_i(\zeta)=1$, we prove Inequality~\eqref{ineq:sen_fraction}.
%
For the coreset size, suppose $S\subseteq [N]$ together with a weight function $w:S\rightarrow \mathbb{R}_{\geq 0}$ is a 0.5-coreset of the query space $(Z^{(G,q,k)},u,{\mathcal{P}}_\lambda^k,\psi^{(G,q,k)})$.
%
We only need to prove that $S=[N]$.
%
Suppose there exists some $i^\star\in S$ with $w(i^\star) > 2$.
%
Letting $\zeta=(\beta^{(1)},\beta^{(2)},\rho^{(1)},\rho^{(2)})$ where $\beta^{(1)}=(\frac{1}{4^{i^\star}},0)$, $\beta^{(2)}=(0,4^{i^\star})$ and $\rho^{(1)}=\rho^{(2)}=0$, we have that
%
\begin{align*}
\sum_{i\in S} w(i)\cdot \psi^{(G,q,k)}_i(\zeta) & > && w(i^\star)\cdot \psi^{(G,q,k)}_{i^\star}(\zeta) &\\
& > && 2 & (w(i^\star)>2 \text{ and Defns. of $\zeta$}) &\\
& > && (1+\frac{1}{2})\cdot \frac{5}{4} &\\
& > && (1+\frac{1}{2})\cdot \psi^{(G,q,k)}(\zeta), & (\text{Ineq.~\eqref{eq:lower1}})
\end{align*}
%
which contradicts with the assumption of $S$.
%
Thus, we have that for any $i\in S$, $w(i)\leq 2$.
%
Next, by contradiction assume that $i^\star\notin S$.
%
Again, letting $\zeta=(\beta^{(1)},\beta^{(2)},\rho^{(1)},\rho^{(2)})$ where $\beta^{(1)}=(\frac{1}{4^{i^\star}},0)$, $\beta^{(2)}=(0,4^{i^\star})$ and $\rho^{(1)}=\rho^{(2)}=0$, we have that
\begin{align*}
\sum_{i\in S} w(i)\cdot \psi^{(G,q,k)}_i(\zeta) & \leq && 2\left(\psi^{(G,q,k)}(\zeta)- \psi^{(G,q,k)}_{i^\star}(\zeta)\right) &\\
& && (w(i)\leq 2) &\\
& \leq && 2 (\frac{5}{4}-1) & (\text{Ineq.~\eqref{eq:lower1}}) \\
& \leq && (1-\frac{1}{2})\cdot 1 &\\
& \leq && (1-\frac{1}{2})\cdot \psi^{(G,q,k)}(\zeta), &
\end{align*}
which contradicts with the assumption of $S$.
%
This completes the proof.
%
\end{proof}
\section{Empirical results}
\label{sec:empirical}
We implement our coreset algorithms for GLSE, and compare the performance with uniform sampling on synthetic datasets and a real-world dataset.
The experiments are conducted by PyCharm on a 4-Core desktop CPU with 8GB RAM.\footnote{Codes are in \url{https://github.com/huanglx12/Coresets-for-regressions-with-panel-data}.}
\noindent\textbf{Datasets.}
We experiment using \textbf{synthetic} datasets with $N=T=500$ ($250k$ observations), $d=10$, $q=1$ and $\lambda = 0.2$.
For each individual $i\in [N]$, we first generate a mean vector $\overline{x}_i\in \mathbb{R}^{d}$ by first uniformly sampling a unit vector $x'_i\in \mathbb{R}^d$, and a length $\tau\in [0,5]$, and then letting $\overline{x}_i = \tau x'_i$.
Then for each time period $t\in [T]$, we generate observation $x_{it}$ from a multivariate normal distribution $N(\overline{x}_i, \|\overline{x}_i\|_2^2\cdot I)$~\cite{tong2012multivariate}.\footnote{The assumption that the covariance of each individual is proportional to $\|\overline{x}_i\|_2^2$ is common in econometrics.
We also fix the last coordinate of $x_{it}$ to be 1 to capture individual specific fixed effects.}
Next, we generate outcomes $Y$.
First, we generate a regression vector $\beta\in \mathbb{R}^d$ from distribution $N(0,I)$.
Then we generate an autoregression vector $\rho\in \mathbb{R}^q$ by first uniformly sampling a unit vector $\rho'\in \mathbb{R}^q$ and a length $\tau\in [0,1-\lambda]$, and then letting $\rho = \tau \rho'$.
Based on $\rho$, we generate error terms $e_{it}$ as in Equation~\eqref{eq:error}. To assess performance robustness in the presence of outliers, we simulate another dataset replacing $N(0,I)$ in Equation~\eqref{eq:error} with the heavy tailed \textbf{Cauchy}(0,2) distribution~\cite{ma2014statistical}.
Finally, the outcome $y_{it} = x_{it}^\top \beta+e_{it}$ is the same as Equation~\eqref{eq:linear}.
We also experiment on a \textbf{real-world} dataset involving the prediction of monthly profits from customers for a credit card issuer as a function of demographics, past behaviors, and current balances and fees.
The panel dataset consisted of 250k observations: 50 months of data ($T=50$) from 5000 customers ($N=5000$) with 11 features ($d=11$).
We set $q=1$ and $\lambda = 0.2$.
\noindent\textbf{Baseline and metrics.}
As a baseline coreset, we use uniform sampling (\textbf{Uni}), perhaps the simplest approach to construct coresets: Given an integer $\Gamma$, uniformly sample $\Gamma$ individual-time pairs $(i,t)\in [N]\times [T]$ with weight $\frac{NT}{\Gamma}$ for each.
Given regression parameters $\zeta$ and a subset $S\subseteq [N]\times [T]$, we define the \emph{empirical error} as $\left| \frac{\psi^{(G,q)}_S(\zeta)}{\psi^{(G,q)}(\zeta)}-1 \right|$.
We summarize the empirical errors $e_1,\ldots, e_n$ by maximum, average, standard deviation (std) and root mean square error (RMSE), where RMSE$= \sqrt{\frac{1}{n}\sum_{i\in [n]}e_i^2}$. By penalizing larger errors, RMSE combines information in both average and standard deviation as a performance metric,.
The running time for solving GLSE on dataset $X$ and our coreset $S$ are $T_X$ and $T_S$ respectively. $T_C$ is the running time for coreset $S$ construction .
\noindent\textbf{Simulation setup.}
We vary $\varepsilon = 0.1, 0.2, 0.3, 0.4, 0.5$ and generate 100 independent random tuples $\zeta=(\beta,\rho)\in \mathbb{R}^{d+q}$ (the same as described in the generation of the synthetic dataset).
For each $\varepsilon$, we run our algorithm $\mathbf{CGLSE}$ and \textbf{Uni} to generate coresets.
We guarantee that the total number of sampled individual-time pairs of $\mathbf{CGLSE}$ and \textbf{Uni} are the same.
We also implement IRLS~\cite{jorgensen2006iteratively} for solving GLSE.
We run IRLS on both the full dataset and coresets and record the runtime.
\noindent\textbf{Results.}
Table~\ref{tab:glse} summarizes the accuracy-size trade-off of our coresets for GLSE for different error guarantees $\varepsilon$.
The maximum empirical error of \textbf{Uni} is always larger than that of our coresets (1.16-793x). Further, there is no error guarantee with \textbf{Uni}, but errors are always below the error guarantee with our coresets. The speed-up with our coresets relative to full data ($\frac{T_X}{T_C+T_S}$) in solving GLSE is 1.2x-108x.
To achieve the maximum empirical error of .294 for GLSE in the real-world data, only 1534 individual-time pairs (0.6\%) are necessary for $\mathbf{CGLSE}$. With \textbf{Uni}, to get the closest maximum empirical error of 0.438, at least 2734 individual-time pairs) (1.1\%) is needed; i.e.., $\mathbf{CGLSE}$ achieves a smaller empirical error with a smaller sized coreset.
Though \textbf{Uni} may sometimes provide lower average error than $\mathbf{CGLSE}$, it \textit{always} has higher RMSE, say 1.2-745x of $\mathbf{CGLSE}$.
When there are outliers as with Cauchy, our coresets perform even better on all metrics relative to \textbf{Uni}.
This is because $\mathbf{CGLSE}$ captures tails/outliers in the coreset, while \textbf{Uni} does not.
Figure~\ref{fig:boxplot} presents the boxplots of the empirical errors.
\begin{table}[t]
\centering
\caption{performance of $\varepsilon$-coresets for GLSE w.r.t. varying $\varepsilon$. We report the maximum/average/standard deviation/RMSE of the empirical error w.r.t. the 100 tuples of generated regression parameters for our algorithm $\mathbf{CGLSE}$ and \textbf{Uni}.
Size is the \# of sampled individual-time pairs, for both $\mathbf{CGLSE}$ and \textbf{Uni}. $T_C$ is construction time (seconds) of our coresets. $T_S$ and $T_X$ are the computation time (seconds) for GLSE over coresets and the full dataset respectively.
%
``Synthetic (G)'' and ``Synthetic (C)'' represent synthetic datasets with Gaussian errors and Cauchy errors respectively.
%
%
}
\label{tab:glse}
\begin{tabular}{ccccccrccc}
\toprule
& \multirow{2}{*}{$\varepsilon$} & \multicolumn{2}{c}{max. emp. err.} & \multicolumn{2}{c}{avg./std./RMSE of emp. err.} & \multirow{2}{*}{size} & \multirow{2}{*}{$T_C$} & \multirow{2}{*}{$T_C + T_S$} & \multirow{2}{*}{$T_X$ (s)} \\
& & $\mathbf{CGLSE}$ & \textbf{Uni} & $\mathbf{CGLSE}$ & \textbf{Uni} & & & & \\
\midrule
\multirow{5}{*}{\rotatebox[origin=c]{90}{synthetic (G)}}
& 0.1 & \textbf{.005} & .015 & .001/.001/.002 & .007/.004/.008 & 116481 & 2 & 372 & 458 \\
& 0.2 & \textbf{.018} & .029 & .006/.004/.008 & .010/.007/.013 & 23043 & 2 & 80 & 458\\
& 0.3 & \textbf{.036} & .041 & .011/.008/.014 & .014/.010/.017 & 7217 & 2 & 29 & 458\\
& 0.4 & \textbf{.055} & .086 & .016/.012/.021 & .026/.020/.032 & 3095 & 2 & 18 & 458\\
& 0.5 & \textbf{.064} & .130 & .019/.015/.024 & .068/.032/.075 & 1590 & 2 & 9 & 458\\
\midrule
\multirow{5}{*}{\rotatebox[origin=c]{90}{synthetic (C)}}
& 0.1 & \textbf{.001} & .793 & .000/.000/.001 & .744/.029/.745 & 106385 & 2 & 1716 & 4430\\
& 0.2 & \textbf{.018} & .939 & .013/.003/.014 & .927/.007/.927 & 21047 & 2 & 346 & 4430\\
& 0.3 & \textbf{.102} & .937 & .072/.021/.075 & .860/.055/.862 & 6597 & 2 & 169 & 4430\\
& 0.4 & \textbf{.070} & .962 & .051/.011/.053 & .961/.001/.961 & 2851 & 2 & 54 & 4430\\
& 0.5 & \textbf{.096} & .998 & .060/.026/.065 & .992/.004/.992 & 472 & 2 & 41 & 4430\\
\midrule
\multirow{5}{*}{\rotatebox[origin=c]{90}{real-world}}
& 0.1 & \textbf{.029} & .162 & .005/.008/.009 & .016/.026/.031 & 50777 & 3 & 383 & 2488 \\
& 0.2 & \textbf{.054} & .154 & .017/.004/.017 & .012/.024/.026 & 13062 & 3 & 85 & 2488 \\
& 0.3 & \textbf{.187} & .698 & .039/.038/.054 & .052/.106/.118 & 5393 & 3 & 24 & 2488\\
& 0.4 & \textbf{.220} & .438 & .019/.033/.038 & .050/.081/.095 & 2734 & 3 & 20 & 2488\\
& 0.5 & \textbf{.294} & 1.107 & .075/.038/.084 & .074/.017/.183 & 1534 & 3 & 16 & 2488\\
\bottomrule
\end{tabular}
\end{table}
\begin{figure}
\includegraphics[width = 0.48\textwidth]{boxplot_synthetic.png}
\quad
\includegraphics[width = 0.48\textwidth]{boxplot_realworld.png}
\caption{Boxplots of empirical errors for GLSE w.r.t. varying $\varepsilon$. \textbf{Uni} has higher average and maximum empirical errors than $\mathbf{CGLSE}$.}
\label{fig:boxplot}
\end{figure}
\section{Conclusion, limitations, and future work}
\label{sec:conclusion}
This paper initiates a theoretical study of coreset construction for regression problems with panel data.
We formulate the definitions of coresets for several variants of $\ell_2$-regression, including OLSE, GLSE, and GLSE$_k$.
For each variant, we propose efficient algorithms that construct a coreset of size independent of both $N$ and $T$, based on the FL framework.
Our empirical results indicate that our algorithms can accelerate the evaluation time and perform significantly better than uniform sampling.
For GLSE$_k$, our coreset size contains a factor $M$, which may be unbounded and result in a coreset of size $\Omega(N)$ in the worst case.
In practice, if $M$ is large, each sensitivity $s(i)$ in Line 5 of Algorithm~\ref{alg:glsek} will be close or even equal to 1.
In this case, $I_S$ is drawn from all individuals via uniform sampling which weakens the performance of Algorithm~\ref{alg:glsek} relative to \textbf{Uni}.
Future research should investigate whether a different assumption than the $M$-bound can generate a coreset of a smaller size.
There are several directions for future work.
Currenly, $q$ and $d$ have a relatively large impact on coreset size; future work needs to reduce this effect. This will advance the use of coresets for machine learning, where $d$ is typically large, and $q$ is large in high frequency data.
This paper focused on coreset construction for panel data with $\ell_2$-regression. The natural next steps would be to construct coresets with panel data for other regression problems, e.g., $\ell_1$-regression, generalized linear models and logistic regression, and beyond regression to other supervised machine learning algorithms.
\vspace{-5mm}
\paragraph{Broader impact.} In terms of broader impact on practice, many organizations have to routinely outsource data processing to external consultants and statisticians.
But a major practical challenge for organizations in doing this is to minimize issues of data security in terms of exposure of their data for potential abuse.
Further, minimization of such exposure is considered as necessary due diligence by laws such as GDPR and CCPA which mandates firms to minimize security breaches that violate the privacy rights of the data owner \cite{shastri2019seven, ke2020privacy}.
Coreset based approaches to sharing data for processing can be very valuable for firms in addressing data security and to be in compliance with privacy regulations like GDPR and CCPA.
Further, for policy and managerial decision making in economics, social sciences and management, obtaining unbiased estimates of the regression relationships from observational data is critical.
Panel data is a critical ingredient for obtaining such unbiased estimates.
As ML methods are being adopted by many social scientists \cite{athey2015machine}, ML scholars are becoming sensitive to these issues and our work in using coreset methods for panel data can have significant impact for these scholars.
A practical concern is that coresets constructed and shared for one purpose or model may be used by the data processor for other kinds of models, which may lead to erroneous conclusions. Further, there is also the potential for issues of fairness to arise as different groups may not be adequately represented in the coreset without incorporating fairness constraints \cite{huang2019coresets}.
These issues need to be explored in future research.
\section*{Acknowledgements}
This research was conducted when LH was at Yale and was supported in part by an NSF CCF-1908347 grant.
| 35,916 |
\section{Introduction}
Motor imagery(MI) is known to be the subconscious
link that instigates the interaction between our brain
and our bodies movements. Physical acts are
triggered with intentional and unintentional
thoughts such as pouring a cup of tea (intentional)
or defending against an opponent’s strike, relying on
pure muscle memory and reaction (unintentional or
accidental).
Primarily, the classification of motor imagery
utilises our intentional thoughts with the aim that a
neural network may identify distinct wave form
patterns and class them into their appropriate labels.
These classifications are turned into commands that
can be used to simply apply the accelerator in a
motor car, moving it in a forward direction or
turning the steering at the desired angle. The
outcome of which could mean that a disabled person
may drive a vehicle purely with their minds alone
rather than their body and mind. Companies such as
Daimler the makers of Mercedes Benz vehicles,
have begun research into allowing disabled persons
control their cars interface using only their thoughts.
A Brain computer interface (BCI) is what is used to
interface between the person and the device it is
trying to control. The BCI hardware on the market
today allow for motor imagery extraction from
motor cortex brain signals, that are filtered, and
feature extracted. The unique features in the signal that are born from the motor cortex are what can be
interpreted as a command. The wearer of the BCI
hardware is in turn able to control a device.
Artificial Intelligence (AI) has shown its capability to apply in various problems \cite{malik2018applications, PHUNG2021107376,PHUNG2020106705}. It also plays the major role in
classifying the extracted signals. Training of a
Convolutional Neural Network (CNN) involves
processing thousands of images of data. This offline
data is fed into the neural network in the form of
training data until it has had enough time to learn at
a realistic rate. An efficient neural network will be
able to differentiate rightly by what is noise and
what is a featured signal of interest.
The outcome of a trained neural network can be
represented as a model. This model can be used to
retrain additional data so that new signals of interest
can be classified, or the model can be used to
classify live or offline data for testing purposes and
ultimately be used to control a device or interface.
Gaps in research into the classification of motor
imagery suggests that there is a reliance to use
offline data for research. There are limited studies
that incorporate online or live data in their papers
and therefore this paper intends to utilise the model
previously implemented, to classify Task1 and with
newly fed samples of EEG signal data to test its
accuracy and usability. Task 1 being the imagery
created when physically opening and closing of both
fists.
\section{Literature Review}
\subsection{Deep learning}
Deep learning falls into three categories; supervised,
unsupervised and semi-supervised learning. When
no labels and classes are known, meaning that the
neural network does not know what its end goal is,
it is a form of unsupervised learning. Machine
learning falls into this category as it relies on
algorithms to work out what it should be looking for
and how. Prior to 2012, the focus of most research was on
unsupervised learning. Semi-supervised learning
requires some data and an algorithm to help it
determine the missing components. Supervised
learning requires input such as labels, classes,
training, testing data and contrastingly, no
algorithms \cite{aa2017deep}.
\subsection{Neural Networks}
CNN’s are considered types of supervised deep
learning architecture models for EEG classification.
Lee et. l \cite{kim2014differences} suggests that they have a
reputation for excellent performance in the field of
image classification that can be used to reduce
‘computation complexities’. Contrastingly though,
Lotte et. al \cite{lotte2018review} argues that their
performances are somewhat held together by their
parameter and architecture combinations. They also
describe the relationship between data size and
architecture, stating that a complex neural
network(NN) with multiple layers require large
datasets for training purposes. They continue to
argue that due to the limited numbers of datasets available for BCI in MI classification, that shallow
neural networks with limited datasets combination
are shown to be more successful.
Other papers suggest that by combining neural
networks, it may produce favourable outcomes, as
each type of model has its distinct advantages over
another {\cite{9406809}. For instance, Sainath et. al \cite{sainath2015convolutional} suggest
that a CNN can help to reduce frequency variations,
Long Short-Term Memory(LSTM) perform better at
temporal modelling and Deep Neural
networks(DNN) are more progressive at mapping
features. Aggarawal and Chugh \cite{aggarwal2019signal} adds that more
recently, CNN’s have been applied for the
classification of multi-classes for motor imagery
tasks by using temporal representations.
The majority of studies point to the lack of datasets
as being the major obstacle in obtaining more
satisfactory results and thereby inhibiting the
research in this field to move forward. Zhang
et. al \cite{zhang2019novel} like many others, have chosen to try and augment their data to try and multiply or artificially
magnify what actually exists. The study used Morlet
Wavelets to transform image signals into three or
four dimensioned tensors, and as a by-product, the
EEG signals are converted to the time frequency
domain.
Both authors in \cite{lotte2018review} and \cite{aggarwal2019signal} agree that more focus should be put on
NN’s to allow them to easily be able to classify
online (non-stationary) data where the sample sizes
would be much smaller and that they should be able
to work with noisier signals.
\section{Methodology}
\subsection{Approach}
A study by Hou et. al \cite{hou2020novel} will be implemented to
validate findings based on their research. They
claim to successfully improve on accuracy scores
attained by other studies in the classification motor
imagery. This research paper will further their
research and use the created and trained CNN deep
learning model to simulate a live testing to
determine if it would be possible to use that model
via a BCI to send a command to a device for the
purpose of mind control.
\subsection{Method of Data Collection}
Acquiring EEG signals is a safe practise according
to (Sanei, Chambers pg. 3 2007), that uses
electrodes of varying types worn using a BCI where
signals are collected in an un invasive manner.
PhysioBank contains multiple databases but the
dataset of interest in this research is from the EEG
Motor Imagery Dataset where the data from 10
patients will be used.
\subsection{Processing the Raw Data}
Utilising an existing EEG database via the
PhysioBank website, raw signal data is to be
imported and processed using Brainstorm software.
Brainstorm is a toolbox that works with MATLAB
as its core to process incoming signals. The raw
EEG data signals are processed to remove unwanted
noise. Such noise can come in the form of high
voltage interference and artefact noise from movement between electrode and the patients head.
Other unwanted signals which is considered noise,
is when the patient is not performing any action or
mentally simulating any action during the signal
recording. Frequencies of interest that correspond
with motor imagery fall between the 5 to 50Hz.This
range of frequencies are what will be kept after
processing has been completed \cite{kim2014differences}.
Processing signal data is an important step that
makes extracting features of interest easier in later
tasks.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/1.png}
\caption{Brainstorm Interface}
\label{fig1}
\end{figure}
A distinct feature of Brainstorm is that it allows the
mapping of the anatomy of a human brain to the
signals of interest created within the motor cortex
region. In fact, regions of interest are created in the
areas of the brain that exhibit the highest intensity
outputs. This can be seen in figure \ref{fig1} where the
colour is intensified indicating that greater signal
strength is present at scouts L2 and L1 regions.
Morlet Wavelets were used to reconfigure the timebased
signal series into a time frequency-based
system for two purposes, the first being that it is a requirement for a neural network to be able to
distinguish the features of interest using this
rearrangement and second, this method augments
the data to artificially increase the training data size.
The final extraction after pre-processing the signals,
contains timeseries from all scout regions in
MATLAB file format. The files created are then
converted into CSV Excel formatted files for input
into the deep learning model.
\subsection{Deep learning Implementation}
The pre-processed data is organised and split into
training and testing data so that a CNN deep learning
model can learn what patterns of waveforms exist in
the thousands of images presented to it.
Approximately 20,000 images are contained as
training data and 2,000 are for testing. The files are
categorized into the following; Test data, training
data, test labels and training labels. The use of
labelled data indicates that the CNN is a form a
supervised learning, where the model is being
instructed what classes to look out for as it is
learning and again when it is being tested.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/2.png}
\caption{CNN Architecture (Hou et.al 2020)}
\label{fig2}
\end{figure}
The CNN’s proposed architecture is as implemented
in \cite{hou2020novel}. It consists of 6 convolutional
layers, 2 max pool layers, 2 flatten layers and 1
SoftMax layer. The implemented architecture in
figure \ref{fig2} will be housed in PyCharm and run.
Training of the neural network will be trialled with
differing parameters until a desired result is
achieved. The parameters that will be modified will
be the batch number, the epoch number and possibly
the learning rate. Testing the neural network will
involve modifying the testing data to extract a much
smaller sample and using that data on a previously saved model that has been trained and has produced
adequate results.
\subsection{Method of Analysis}
The proposed method of analysis will be in the form
of quantitative data analysis. Good results produced
by training the CNN model will be indicated by a
high percentage of accuracy. The accuracy in its
ability to be able to distinguish a pattern within the
thousands of images and resolve them to their
rightful class. Graphs will help to visualize the
journey the neural network has undertaken. Such
graphs may indicate a successful convergence
whereby the training accuracy and validation
accuracy will tend to follow each other’s paths. In
contrast the separation of paths would indicate
overfitting, where it could be described when the
CNN has successfully learned from the training data
but fails to transfer its learning when being tested.
Preventing overfitting may require the model to be
made less complicated or by adding dropout layers
which have been shown to effectively prevent this
symptom \cite{srivastava2014dropout}.
\section{Results and Discussion}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/3.png}
\caption{Ten regions of interests}
\label{fig3}
\end{figure}
\subsection{Scout Time Series}
All 10 scouts are shown in figure \ref{fig3}. Each scout has
successfully extracted the pre-processed extracted
EEG signals. The amplitudes tend to vary in
intensity between each scout. Scouts R1, R2 and L2
(Figures \ref{fig4}, \ref{fig5}) have experienced more intense spikes
than other scouts.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/4.png}
\caption{R5 scout signal extraction}
\label{fig4}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/5.png}
\caption{L2 scout signal extraction}
\label{fig5}
\end{figure}
The research paper by (Hou et.al 2020) concentrated
their efforts on scout R5 and so the results in this
paper will focus on results based on this scout. In
figure 4 the R5 scout is displayed as a single signal.
Approximately 100 seconds of recording can be
seen along with the task labels on the top of the
figure. Labels T0, T1 and T2 are seen at different
time periods as was recorded of the patient. Task T0
in green is the period of inaction and no imagery takes place. Task 2 describes when the patient
imagines opening and closing their left or right fist.
The label classes contained in the R5 scout are found
in the R2 scout also. Over the same time period in
both scouts it seems that the signals are much denser
in the L2 extraction (Figure 5) than that from the R5
scout. This could possibly affect the neural networks
ability to recognise a pattern from the signals and is
possibly one of the reasons that Hou et. al \cite{hou2020novel}
chose to concentrate their research on the R5 scout
signals of interest.
\subsection{Morlet Wavelets}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/6.png}
\caption{R5 Wavelets}
\label{fig6}
\end{figure}
Morlet wavelets were used to extract the time
frequency maps from the scouts as seen in Figure \ref{fig6}.
The frequencies represented, range between 8 and
30 Hz which fits into the range of frequencies that
stem from motor imagery. These features of interest
are then finally extracted into a format recognisable
by the convolutional neural network.
\subsection{Training of the CNN}
Using the PyCharm platform, the Python code was
executed using training data, training labels, testing
data and testing labels. All of which originate from
the R5 scout region that is used as the main source
for training.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/7.png}
\caption{Training Results (Accuracy is
represented as detection probability)}
\label{fig7}
\end{figure}
The CNN was trained with varying parameters until
the training and testing accuracy successfully
converged to produce the results in figure \ref{fig7}. The
results indicated that a class T1 could be recognised
at 100 percent testing accuracy. These results matched
the results obtained in \cite{hou2020novel}.
\subsection{Testing the Trained Model}
This research paper set out to also further the
implementation in \cite{hou2020novel} by then saving
the trained CNN model and using it to test it against
pre-processed sample data. The test data contained
only one image as opposed to approximately 2,000
images when initially training the neural network.
This test would simulate what live data being fed
into an already trained model would look like. This
live data would be equivalent to a person wearing a
BCI device attached to computer waiting for a
command.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/8.png}
\caption{Restored Model}
\label{fig8}
\end{figure}
Figure 8 demonstrates the outcome of training the
restored model with sample test data 1.
The accuracy remains at 100 percent for the duration
of the 32 iterations. Meanwhile the training
successfully converged with the testing accuracy.
Although training the restored model was not
necessary, it was reassuring to see the accuracy
remain constant over the duration proving the
restored model’s ability to recognise what it was
trained to do over a given time.
To be able to determine the restored model’s ability
to accept and recognise different samples
individually, a number of tests were performed
using the image data in figure \ref{fig10}. The results shown
in figure \ref{fig9} reveal that when the restored model is
being fed sample data 1-10, it is 60 percent accurate
in identifying the same T1 class. Even though the
trained model had 100 percent accuracy when it was
trained initially using thousands of test data, the
results here would indicate that possibly the training
was not comprehensive enough. This outcome could
be similar to when a student is preparing for an
exam, they would study certain areas of a topic and
then test themselves on the same information scoring highly, but when they actually sit the test,
the questions could have more depth or variance to
them and therefore the student doesn’t score as
highly because they haven’t varied their studies. In
terms of the trained model, it most likely indicates
that more comprehensive training of the neural
network is required in order for it to perform better
against various samples of data.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/9.png}
\caption{Sample Data Results}
\label{fig9}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/10.png}
\caption{Sample Test Data}
\label{fig10}
\end{figure}
The samples plotted in figure \ref{fig10} are samples 1 and
2. The patterns are offset, and amplitudes do differ
but there are similarities in the pattern waveform. The main difference which is quite noticeable is the
added last spike in sample 2. This was probably
unexpected and therefore unrecognizable to the
trained and restored model thereby excluding
sample 2 from its predictions to be of the T1 class.
\section{Conclusion}
This research paper intended to train a neural
network to identify features of interest and class
them according to their appropriate labels. The
implementation followed a paper by (Hou et.al
2020) up to the point where the training was able to
identify a class of MI. This research paper then
attempted to further their research by restoring a
trained model and using it to classify sample image
data that would simulate live data input. This was an
attempt to challenge the evident research gaps in this
field where offline data is the focus of most of the
research in that area.
The methods used in this paper involved preprocessing
the raw EEG signal data in Brainstorm
and successfully extracting the features of interest
that were then converted into the frequency over
time domain. Using the PyCharm platform, the
CNN was trained until it could accurately class a T1
label. The model was then restored and was fed
sample data to test its ability to recognise what could
be potential live data.
The results indicated that the model would need to
be trained using different and varying parameters so
that it would be able to recognise and class various
forms of sample data. Another method may be to try and reduce the complexity of the CNN’s
architecture as was suggested by Lotte et.al 2018 \cite{lotte2018review}.
Either trials could in turn improve the model’s
ability to produce a higher and more consistent rate
of accuracy, ultimately allowing the CNN model to
be used to control a device or combine with other inputs of human \cite{4586369} to carry out more complicated tasks.
\section{Introduction}
Motor imagery(MI) is known to be the subconscious
link that instigates the interaction between our brain
and our bodies movements. Physical acts are
triggered with intentional and unintentional
thoughts such as pouring a cup of tea (intentional)
or defending against an opponent’s strike, relying on
pure muscle memory and reaction (unintentional or
accidental).
Primarily, the classification of motor imagery
utilises our intentional thoughts with the aim that a
neural network may identify distinct wave form
patterns and class them into their appropriate labels.
These classifications are turned into commands that
can be used to simply apply the accelerator in a
motor car, moving it in a forward direction or
turning the steering at the desired angle. The
outcome of which could mean that a disabled person
may drive a vehicle purely with their minds alone
rather than their body and mind. Companies such as
Daimler the makers of Mercedes Benz vehicles,
have begun research into allowing disabled persons
control their cars interface using only their thoughts.
A Brain computer interface (BCI) is what is used to
interface between the person and the device it is
trying to control. The BCI hardware on the market
today allow for motor imagery extraction from
motor cortex brain signals, that are filtered, and
feature extracted. The unique features in the signal that are born from the motor cortex are what can be
interpreted as a command. The wearer of the BCI
hardware is in turn able to control a device.
Artificial Intelligence (AI) has shown its capability to apply in various problems \cite{malik2018applications, PHUNG2021107376,PHUNG2020106705}. It also plays the major role in
classifying the extracted signals. Training of a
Convolutional Neural Network (CNN) involves
processing thousands of images of data. This offline
data is fed into the neural network in the form of
training data until it has had enough time to learn at
a realistic rate. An efficient neural network will be
able to differentiate rightly by what is noise and
what is a featured signal of interest.
The outcome of a trained neural network can be
represented as a model. This model can be used to
retrain additional data so that new signals of interest
can be classified, or the model can be used to
classify live or offline data for testing purposes and
ultimately be used to control a device or interface.
Gaps in research into the classification of motor
imagery suggests that there is a reliance to use
offline data for research. There are limited studies
that incorporate online or live data in their papers
and therefore this paper intends to utilise the model
previously implemented, to classify Task1 and with
newly fed samples of EEG signal data to test its
accuracy and usability. Task 1 being the imagery
created when physically opening and closing of both
fists.
\section{Literature Review}
\subsection{Deep learning}
Deep learning falls into three categories; supervised,
unsupervised and semi-supervised learning. When
no labels and classes are known, meaning that the
neural network does not know what its end goal is,
it is a form of unsupervised learning. Machine
learning falls into this category as it relies on
algorithms to work out what it should be looking for
and how. Prior to 2012, the focus of most research was on
unsupervised learning. Semi-supervised learning
requires some data and an algorithm to help it
determine the missing components. Supervised
learning requires input such as labels, classes,
training, testing data and contrastingly, no
algorithms \cite{aa2017deep}.
\subsection{Neural Networks}
CNN’s are considered types of supervised deep
learning architecture models for EEG classification.
Lee et. l \cite{kim2014differences} suggests that they have a
reputation for excellent performance in the field of
image classification that can be used to reduce
‘computation complexities’. Contrastingly though,
Lotte et. al \cite{lotte2018review} argues that their
performances are somewhat held together by their
parameter and architecture combinations. They also
describe the relationship between data size and
architecture, stating that a complex neural
network(NN) with multiple layers require large
datasets for training purposes. They continue to
argue that due to the limited numbers of datasets available for BCI in MI classification, that shallow
neural networks with limited datasets combination
are shown to be more successful.
Other papers suggest that by combining neural
networks, it may produce favourable outcomes, as
each type of model has its distinct advantages over
another {\cite{9406809}. For instance, Sainath et. al \cite{sainath2015convolutional} suggest
that a CNN can help to reduce frequency variations,
Long Short-Term Memory(LSTM) perform better at
temporal modelling and Deep Neural
networks(DNN) are more progressive at mapping
features. Aggarawal and Chugh \cite{aggarwal2019signal} adds that more
recently, CNN’s have been applied for the
classification of multi-classes for motor imagery
tasks by using temporal representations.
The majority of studies point to the lack of datasets
as being the major obstacle in obtaining more
satisfactory results and thereby inhibiting the
research in this field to move forward. Zhang
et. al \cite{zhang2019novel} like many others, have chosen to try and augment their data to try and multiply or artificially
magnify what actually exists. The study used Morlet
Wavelets to transform image signals into three or
four dimensioned tensors, and as a by-product, the
EEG signals are converted to the time frequency
domain.
Both authors in \cite{lotte2018review} and \cite{aggarwal2019signal} agree that more focus should be put on
NN’s to allow them to easily be able to classify
online (non-stationary) data where the sample sizes
would be much smaller and that they should be able
to work with noisier signals.
\section{Methodology}
\subsection{Approach}
A study by Hou et. al \cite{hou2020novel} will be implemented to
validate findings based on their research. They
claim to successfully improve on accuracy scores
attained by other studies in the classification motor
imagery. This research paper will further their
research and use the created and trained CNN deep
learning model to simulate a live testing to
determine if it would be possible to use that model
via a BCI to send a command to a device for the
purpose of mind control.
\subsection{Method of Data Collection}
Acquiring EEG signals is a safe practise according
to (Sanei, Chambers pg. 3 2007), that uses
electrodes of varying types worn using a BCI where
signals are collected in an un invasive manner.
PhysioBank contains multiple databases but the
dataset of interest in this research is from the EEG
Motor Imagery Dataset where the data from 10
patients will be used.
\subsection{Processing the Raw Data}
Utilising an existing EEG database via the
PhysioBank website, raw signal data is to be
imported and processed using Brainstorm software.
Brainstorm is a toolbox that works with MATLAB
as its core to process incoming signals. The raw
EEG data signals are processed to remove unwanted
noise. Such noise can come in the form of high
voltage interference and artefact noise from movement between electrode and the patients head.
Other unwanted signals which is considered noise,
is when the patient is not performing any action or
mentally simulating any action during the signal
recording. Frequencies of interest that correspond
with motor imagery fall between the 5 to 50Hz.This
range of frequencies are what will be kept after
processing has been completed \cite{kim2014differences}.
Processing signal data is an important step that
makes extracting features of interest easier in later
tasks.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/1.png}
\caption{Brainstorm Interface}
\label{fig1}
\end{figure}
A distinct feature of Brainstorm is that it allows the
mapping of the anatomy of a human brain to the
signals of interest created within the motor cortex
region. In fact, regions of interest are created in the
areas of the brain that exhibit the highest intensity
outputs. This can be seen in figure \ref{fig1} where the
colour is intensified indicating that greater signal
strength is present at scouts L2 and L1 regions.
Morlet Wavelets were used to reconfigure the timebased
signal series into a time frequency-based
system for two purposes, the first being that it is a requirement for a neural network to be able to
distinguish the features of interest using this
rearrangement and second, this method augments
the data to artificially increase the training data size.
The final extraction after pre-processing the signals,
contains timeseries from all scout regions in
MATLAB file format. The files created are then
converted into CSV Excel formatted files for input
into the deep learning model.
\subsection{Deep learning Implementation}
The pre-processed data is organised and split into
training and testing data so that a CNN deep learning
model can learn what patterns of waveforms exist in
the thousands of images presented to it.
Approximately 20,000 images are contained as
training data and 2,000 are for testing. The files are
categorized into the following; Test data, training
data, test labels and training labels. The use of
labelled data indicates that the CNN is a form a
supervised learning, where the model is being
instructed what classes to look out for as it is
learning and again when it is being tested.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/2.png}
\caption{CNN Architecture (Hou et.al 2020)}
\label{fig2}
\end{figure}
The CNN’s proposed architecture is as implemented
in \cite{hou2020novel}. It consists of 6 convolutional
layers, 2 max pool layers, 2 flatten layers and 1
SoftMax layer. The implemented architecture in
figure \ref{fig2} will be housed in PyCharm and run.
Training of the neural network will be trialled with
differing parameters until a desired result is
achieved. The parameters that will be modified will
be the batch number, the epoch number and possibly
the learning rate. Testing the neural network will
involve modifying the testing data to extract a much
smaller sample and using that data on a previously saved model that has been trained and has produced
adequate results.
\subsection{Method of Analysis}
The proposed method of analysis will be in the form
of quantitative data analysis. Good results produced
by training the CNN model will be indicated by a
high percentage of accuracy. The accuracy in its
ability to be able to distinguish a pattern within the
thousands of images and resolve them to their
rightful class. Graphs will help to visualize the
journey the neural network has undertaken. Such
graphs may indicate a successful convergence
whereby the training accuracy and validation
accuracy will tend to follow each other’s paths. In
contrast the separation of paths would indicate
overfitting, where it could be described when the
CNN has successfully learned from the training data
but fails to transfer its learning when being tested.
Preventing overfitting may require the model to be
made less complicated or by adding dropout layers
which have been shown to effectively prevent this
symptom \cite{srivastava2014dropout}.
\section{Results and Discussion}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/3.png}
\caption{Ten regions of interests}
\label{fig3}
\end{figure}
\subsection{Scout Time Series}
All 10 scouts are shown in figure \ref{fig3}. Each scout has
successfully extracted the pre-processed extracted
EEG signals. The amplitudes tend to vary in
intensity between each scout. Scouts R1, R2 and L2
(Figures \ref{fig4}, \ref{fig5}) have experienced more intense spikes
than other scouts.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/4.png}
\caption{R5 scout signal extraction}
\label{fig4}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/5.png}
\caption{L2 scout signal extraction}
\label{fig5}
\end{figure}
The research paper by (Hou et.al 2020) concentrated
their efforts on scout R5 and so the results in this
paper will focus on results based on this scout. In
figure 4 the R5 scout is displayed as a single signal.
Approximately 100 seconds of recording can be
seen along with the task labels on the top of the
figure. Labels T0, T1 and T2 are seen at different
time periods as was recorded of the patient. Task T0
in green is the period of inaction and no imagery takes place. Task 2 describes when the patient
imagines opening and closing their left or right fist.
The label classes contained in the R5 scout are found
in the R2 scout also. Over the same time period in
both scouts it seems that the signals are much denser
in the L2 extraction (Figure 5) than that from the R5
scout. This could possibly affect the neural networks
ability to recognise a pattern from the signals and is
possibly one of the reasons that Hou et. al \cite{hou2020novel}
chose to concentrate their research on the R5 scout
signals of interest.
\subsection{Morlet Wavelets}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/6.png}
\caption{R5 Wavelets}
\label{fig6}
\end{figure}
Morlet wavelets were used to extract the time
frequency maps from the scouts as seen in Figure \ref{fig6}.
The frequencies represented, range between 8 and
30 Hz which fits into the range of frequencies that
stem from motor imagery. These features of interest
are then finally extracted into a format recognisable
by the convolutional neural network.
\subsection{Training of the CNN}
Using the PyCharm platform, the Python code was
executed using training data, training labels, testing
data and testing labels. All of which originate from
the R5 scout region that is used as the main source
for training.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/7.png}
\caption{Training Results (Accuracy is
represented as detection probability)}
\label{fig7}
\end{figure}
The CNN was trained with varying parameters until
the training and testing accuracy successfully
converged to produce the results in figure \ref{fig7}. The
results indicated that a class T1 could be recognised
at 100 percent testing accuracy. These results matched
the results obtained in \cite{hou2020novel}.
\subsection{Testing the Trained Model}
This research paper set out to also further the
implementation in \cite{hou2020novel} by then saving
the trained CNN model and using it to test it against
pre-processed sample data. The test data contained
only one image as opposed to approximately 2,000
images when initially training the neural network.
This test would simulate what live data being fed
into an already trained model would look like. This
live data would be equivalent to a person wearing a
BCI device attached to computer waiting for a
command.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/8.png}
\caption{Restored Model}
\label{fig8}
\end{figure}
Figure 8 demonstrates the outcome of training the
restored model with sample test data 1.
The accuracy remains at 100 percent for the duration
of the 32 iterations. Meanwhile the training
successfully converged with the testing accuracy.
Although training the restored model was not
necessary, it was reassuring to see the accuracy
remain constant over the duration proving the
restored model’s ability to recognise what it was
trained to do over a given time.
To be able to determine the restored model’s ability
to accept and recognise different samples
individually, a number of tests were performed
using the image data in figure \ref{fig10}. The results shown
in figure \ref{fig9} reveal that when the restored model is
being fed sample data 1-10, it is 60 percent accurate
in identifying the same T1 class. Even though the
trained model had 100 percent accuracy when it was
trained initially using thousands of test data, the
results here would indicate that possibly the training
was not comprehensive enough. This outcome could
be similar to when a student is preparing for an
exam, they would study certain areas of a topic and
then test themselves on the same information scoring highly, but when they actually sit the test,
the questions could have more depth or variance to
them and therefore the student doesn’t score as
highly because they haven’t varied their studies. In
terms of the trained model, it most likely indicates
that more comprehensive training of the neural
network is required in order for it to perform better
against various samples of data.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/9.png}
\caption{Sample Data Results}
\label{fig9}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figure/10.png}
\caption{Sample Test Data}
\label{fig10}
\end{figure}
The samples plotted in figure \ref{fig10} are samples 1 and
2. The patterns are offset, and amplitudes do differ
but there are similarities in the pattern waveform. The main difference which is quite noticeable is the
added last spike in sample 2. This was probably
unexpected and therefore unrecognizable to the
trained and restored model thereby excluding
sample 2 from its predictions to be of the T1 class.
\section{Conclusion}
This research paper intended to train a neural
network to identify features of interest and class
them according to their appropriate labels. The
implementation followed a paper by (Hou et.al
2020) up to the point where the training was able to
identify a class of MI. This research paper then
attempted to further their research by restoring a
trained model and using it to classify sample image
data that would simulate live data input. This was an
attempt to challenge the evident research gaps in this
field where offline data is the focus of most of the
research in that area.
The methods used in this paper involved preprocessing
the raw EEG signal data in Brainstorm
and successfully extracting the features of interest
that were then converted into the frequency over
time domain. Using the PyCharm platform, the
CNN was trained until it could accurately class a T1
label. The model was then restored and was fed
sample data to test its ability to recognise what could
be potential live data.
The results indicated that the model would need to
be trained using different and varying parameters so
that it would be able to recognise and class various
forms of sample data. Another method may be to try and reduce the complexity of the CNN’s
architecture as was suggested by Lotte et.al 2018 \cite{lotte2018review}.
Either trials could in turn improve the model’s
ability to produce a higher and more consistent rate
of accuracy, ultimately allowing the CNN model to
be used to control a device or combine with other inputs of human \cite{4586369} to carry out more complicated tasks.
| 9,391 |
\section{Introduction}
\label{sec:intro}
In model-based RL, a learning agent interacts with its environment to build an increasingly more accurate estimate of the underlying dynamics and rewards, and uses the estimate to progressively improve its policy. This paradigm is attractive as it is amenable to sample-efficiency in both theory~\citep{kearns2002near,brafman2002r,auer2008near,azar2017minimax,sun2019model,foster2021statistical} and practice~\citep{chua2018deep,nagabandi2020deep,schrittwieser2020mastering}. The learned models also offer the possibility of use beyond individual tasks~\citep{ha2018world}, effectively providing learned simulators. Given the importance of this paradigm, it is vital to understand how and when can sample-efficient model-based RL be achieved.
Two key questions any model-based RL agent has to address are: i) how to collect data from the environment, given the current learning state, and ii) how to define the quality of a model, given the currently acquired dataset. In simple theoretical settings such as tabular MDPs, the first question is typically addressed through optimism, typically via an uncertainty bonus, while likelihood of the data under a model is a typical answer to the second. While the empirical literature differs on its answer to the first question in problems with complex state spaces, it still largely adopts likelihood, or a VAE-style approximation~\citep[e.g.][]{chua2018deep,ha2018world,sekar2020planning}, as the measure of model quality. On the other hand, the theoretical literature for rich observation RL is much more varied in the losses used for model fitting, ranging from a direct squared error in parameters~\citep{yang2020reinforcement,kakade2020information} to more complicated divergence measures~\citep{sun2019model,du2021bilinear,foster2021statistical}. Correspondingly, the literature also has a variety of structural conditions which enable model-based RL.
In this paper, we seek to unify the theory of model-based RL under a general statistical and algorithmic framework. We address the aforementioned two questions by always using data likelihood as a model quality estimate, irrespective of the observation complexity, and use an optimistic posterior sampling approach for data collection. For this approach, we define a \emph{Bellman error decoupling} framework which governs the sample complexity of finding a near-optimal policy. Our main result establishes that whenever the decoupling coefficient in a model-based RL setting is small, our \emph{Model-based Optimistic Posterior Sampling\xspace} (\textsc{MOPS}\xspace) algorithm is sample-efficient.
A key conceptual simplification in \textsc{MOPS}\xspace is that the model quality is always measured using likelihood, unlike other general frameworks~\citep{du2021bilinear,foster2021statistical} which need to modify their loss functions for different settings. Practically, posterior sampling is relatively amenable to tractable implementation via ensemble approximations~\citep[see e.g.][]{osband2016deep,lu2017ensemble,chua2018deep,nagabandi2020deep} or sampling methods such as stochastic gradient Langevin dynamics~\citep{welling2011bayesian}. This is in contrast with the version-space based optimization methods used in most prior works.
We further develop broad structural conditions on the underlying MDP and the model class used, under which the decoupling coefficient admits good bounds. This includes many prominent examples, such as
\vspace{0.2cm}
\begin{minipage}{\textwidth}
\begin{multicols}{2}
\begin{itemize}[leftmargin=*]
\item Finite action problems with a small witness rank~\citep{sun2019model}
\item Linear MDPs with infinite actions
\item Small witness rank and linearly embedded backups (\textbf{new})
\item $Q$-type witness rank problems (\textbf{new})
\item Kernelized Non-linear Regulators~\citep{kakade2020information}
\item Linear mixture MDPs~\citep{modi2020sample,ayoub2020model}
\end{itemize}
\end{multicols}
\end{minipage}
\vspace{0.2cm}
Remarkably, \textsc{MOPS}\xspace simultaneously addresses all the scenarios listed here and more, only requiring one change in the algorithm regarding the data collection policy induced as a function of the posterior. The analysis of model estimation itself, which is shared across all these problems, follows from a common online learning analysis of the convergence of our posterior.
Taken together, our results provide a coherent and unified approach to model-based RL, and we believe that the conceptual simplicity of our approach will spur further development in obtaining more refined guarantees, more efficient algorithms, and including new examples under the umbrella of statistical tractability.
\section{Related work}
\label{sec:related}
\mypar{Model-based RL.} There is a rich literature on model-based RL for obtaining strong sample complexity results, from the seminal early works~\citep{kearns2002near,brafman2002r,auer2008near} to the more recent minimax optimal guarantees~\citep{azar2017minimax,zanette2019tighter}. Beyond tabular problems, techniques have also been developed for rich observation spaces with function approximation in linear~\citep{yang2020reinforcement,ayoub2020model} and non-linear~\citep{sun2019model,agarwal2020flambe,uehara2021representation,du2021bilinear} settings. Of these, our work builds most directly on that of~\citet{sun2019model} in terms of the structural properties used. However, the algorithmic techniques are notably different. \citet{sun2019model} repeatedly solve an optimization problem to find an optimistic model consistent with the prior data, while we use optimistic posterior sampling, which scales to large action spaces unlike their use of a uniform randomization over the actions. We also measure the consistency with prior data in terms of the likelihood of the observations under a model, as opposed to their integral probably metric losses. The generality of our design and analysis allows our techniques to apply to models beyond those covered by \citet{sun2019model}. \citet{du2021bilinear} effectively reuse the algorithm of \citet{sun2019model} in the model-based setting, so the same comparison applies. We note that recent model-based feature learning works~\citep{agarwal2020flambe,uehara2021representation} do measure model fit using log-likelihood, and some of their technical analysis shares similarities with our proofs, though there are notable differences in the MLE versus posterior sampling approaches. Finally, we note that a parallel line of work has developed model-free approaches for function approximation settings~\citep{jiang2017contextual,du2021bilinear,jin2021bellman}, and while our structural complexity measure is always smaller than the Bellman rank of~\citet{jiang2017contextual}, our model-based realizability is often a stronger assumption than realizability of just the $Q^\star$ function. For instance, \citet{jin2020provably},~\citet{du2019provably} and~\citet{misra2019kinematic} do not model the entire transition dynamics in linear and block MDP models.
\mypar{Posterior sampling in RL.} Posterior sampling methods for RL, motivated by Thompson sampling (TS)~\citep{thompson1933likelihood}, have been extensively developed and analyzed in terms of their expected regret under a Bayesian prior by many authors~\citep[see e.g.][]{osband2013more,russo2017tutorial,osband2016generalization} and are often popular as they offer a simple implementation heuristic through approximation by bootstrapped ensembles~\citep{osband2016deep}. Worst-case analysis of TS in RL settings has also been done for both tabular~\citep{russo2019worst,agrawal2017posterior} and linear~\citep{zanette2020frequentist} settings. Our work is most closely related to the recent Feel-Good Thompson Sampling strategy proposed and analyzed in~\citep{Zhang2021FeelGoodTS}, and its extensions~\citep{zhang2021provably,agarwal2022non}. Note that our model-based setting does not require the two timestep strategy to solve a minimax problem, as was required in the model-free work of~\citet{agarwal2022non}.
\mypar{Model-based control.} Model-based techniques are widely used in control, and many recent works analyze the Linear Quadratic Regulator~\citep[see e.g.][]{dean2020sample,mania2019certainty,agarwal2019online,simchowitz2020naive}, as well as some non-linear generalizations~\citep{kakade2020information,mania2020active,mhammedi2020learning}. Our framework does capture many of these settings as we demonstrate in Section~\ref{sec:decoupling}.
\mypar{Relationship with \citet{foster2021statistical}.} This recent work studies a broad class of decision making problems including bandits and RL. For RL, they consider a model-based framing, and provide upper and lower bounds on the sample complexity in terms of a new parameter called the decision estimation coefficient (DEC). Structurally, DEC is closely related to the Hellinger decoupling concept introduced in this paper (see Definition~\ref{def:decoupling}). However, while Hellinger decoupling is only used in our analysis (and the distance is measured to the true model), the DEC analysis of \citet{foster2021statistical} measures distance to a plug-in estimator for the true model, and needs complicated algorithms to explicitly bound the DEC. In particular, it is not known if posterior sampling is admissible in their framework, which requires more careful control of a minimax objective. The Hellinger decoupling coefficient, in contrast, admits conceptually simpler optimistic posterior sampling techniques.\iffalse in a fairly broad manner.
For specifically, it was not known that a simple algorithm such as posterior sampling
can be used to control the quantity for general problems.
In comparison, this work showed that posterior sampling can be used to explicitly control the Hellinger decoupling coefficient, as well as the estimation of the model in Hellinger distance.
In summary, while we design a unified algorithmic and analysis framework for model-based RL,
the work of~\citet{foster2021statistical} tried to identify a learning principle and their general purpose algorithm E2D leaves several algorithmic design (and consequently theoretical analysis) knobs unspecified for applications in our setting.\footnote{While they provide a result for Bilinear classes, the estimation algorithm details are only fully instantiated for the $Q$-type scenario.}
Our use of Hellinger decoupling is only in the theoretical analysis with the actual algorithm using the more natural likelihood-based posterior sampling. Our exploration strategy by using optimistic posterior sampling is significantly simpler than the minimax solution to DEC
of \citet{foster2021statistical} both in E2D and in its Bayesian adaptation E2D.Bayes.\fi
\section{Setting and Preliminaries}
\label{sec:setting}
We study RL in an episodic, finite horizon Contextual Markov Decision Process (MDP) that is parameterized as $(\ensuremath{\mathcal{X}}, {\mathcal A}, {\mathcal D}, R_\star, P_\star)$, where $\ensuremath{\mathcal{X}}$ is a state space, ${\mathcal A}$ is an action space, ${\mathcal D}$ is the distribution over the initial context, $R_\star$ is the expected reward function and $P_\star$ denotes the transition dynamics. An agent observes a context $x^1\sim {\mathcal D}$ for some fixed distribution ${\mathcal D}$.\footnote{We intentionally call $x^1$ a context and not an initial state of the MDP as we will soon make certain structural assumptions which depend on the context, but take expectation over the states.} At each time step $h\in\{1,\ldots,H\}$, the agent observes the state $x^h$, chooses an action $a^h$, observes $r^h$ with $\rE[r^h\mid x_h, a_h] = R^h_\star(x_h, a_h)$ and transitions to $x^{h+1}\sim P^h_\star(\cdot \mid x^h, a^h)$. We assume that $x^h$ for any $h > 1$ always includes the context $x^1$ to allow arbitrary dependence of the dynamics and rewards on $x^1$. Following prior works~\citep[e.g.][]{jiang2017contextual,sun2019model}, we assume that $r^h\in[0,1]$ and $\sum_{h=1}^H r^h \in [0,1]$ to capture sparse-reward settings~\citep{jiang2018open}. We make no assumption on the cardinality of the state and/or action spaces, allowing both to be potentially infinite. We use $\pi$ to denote the agent's decision policy, which maps from $\ensuremath{\mathcal{X}}\to\Delta({\mathcal A})$, where $\Delta(\cdot)$ represents probability distributions over a set. The goal of learning is to discover an optimal policy $\pi_\star$, which is always deterministic and conveniently defined in terms of the $Q_\star$ function~\citep[see e.g.][]{puterman2014markov,bertsekas1996neuro}
\begin{equation}
\pi^h_\star(x^h) = \argmax_{a\in{\mathcal A}} Q^h_\star(x^h,a),~~Q^h_\star(x^h,a^h) = \rE[r^h + \max_{a'\in{\mathcal A}}Q_\star^{h+1}(x^{h+1},a')\mid x^h, a^h],
\label{eq:qstar}
\end{equation}
where we define $Q_\star^{H+1}(x,a) = 0$ for all $x,a$. We also define $V^h_\star(x^h) = \max_a Q^h_\star(x^h,a)$.
In the model-based RL setting of this paper, the learner has access to a model class $\ensuremath{\mathcal{M}}$ consisting of tuples $(P_M, R_M)$,\footnote{We use $P_M$ and $R_M$ to denote the set $\{P_M^h\}_{h=1}^H$ and $\{R_M^h\}_{h=1}^H$ respectively.} denoting the transition dynamics and expected reward functions according to the model $M\in\ensuremath{\mathcal{M}}$. For any model $M$, we use $\pi^h_M$ and $V^h_M$ to denote the
optimal policy and value function, respectively, at level $h$ in the model $M$. We assume that like $V^h_\star$, $V^h_M$ also satisfies the normalization assumption $V^h_M(x) \in [0,1]$ for all $M\in\ensuremath{\mathcal{M}}$. We use $\rE^M$ to denote expectations evaluated in the model $M$ and $\rE$ to denote expectations in the true model.
We make two common assumptions on the class $\ensuremath{\mathcal{M}}$.
\begin{assumption}[Realizability]
\label{ass:realizable}
$\exists~M_\star\in\ensuremath{\mathcal{M}}$ such that $P_\star^h = P^h_{M_\star}$ and $R^h_\star = R^h_{M_\star}$ for all $h\in[H]$.
\end{assumption}
The realizability assumption ensures that the model estimation is well-specified. We also assume access to a planning oracle for any fixed model $M\in\ensuremath{\mathcal{M}}$.
\begin{assumption}[Planning oracle]
\label{ass:planning}
There is a planning oracle which given a model $M$, returns its optimal policy $\pi_M=\{\pi^h\}_{h=1}^H$:
$\pi_M =\arg\max_\pi
\rE^{M,\pi}\big[\sum_{h=1}^H R^h_M(x^h,\pi^h(x^h))\big] $,
where $\rE^{M,\pi}$ is the expectation over trajectories obtained by following the policy $\pi$ in the model $M$.
\end{assumption}
Given $M\in\ensuremath{\mathcal{M}}$, we define model-based Bellman error as:
\begin{align}
\ensuremath{{\mathcal E}_B}(M, x^h, a^h) = Q_{M}^h(x^h,a^h) - (P_{\star}^h[r^h+V^{h+1}_{M}])(x^h,a^h), \label{eq:model-bellman}
\end{align}
where for a function $f~:~\ensuremath{\mathcal{X}}\times{\mathcal A}\times[0,1]\times \ensuremath{\mathcal{X}}$, we define $(P_Mf)(x,a) = \rE^M[f(x,a,r,x')|x,a]$.
This quantity plays a central role in our analysis due to the simulation lemma for model-based RL.
\begin{lemma}[Lemma 10 of \citep{sun2019model}]
For any distribution $p\in\Delta(\ensuremath{\mathcal{M}})$ and $x^1$, we have
$\rE_{M\sim p}\left[V^\star(x^1) - V^{\pi_M}(x^1)\right] = \rE_{M\sim p}\left[\sum_{h=1}^H \rE_{x^h, a^h\sim \pi_M|x^1}\ensuremath{{\mathcal E}_B}(M, x^h, a^h) - \Delta V_M(x^1),\right]$,
where $\Delta V_M(x^1) = V_M(x^1) - V^\star(x^1)$.
\label{lemma:simulation}
\end{lemma}
The Bellman error in turn can be related to Hellinger estimation error via the decoupling coefficient in
Definition~\ref{def:decoupling}, which captures the structural properties of the underlying MDP.
We use typical measures of distance between probability distributions to capture the error in dynamcis, and for any two distributions $P$ and $Q$ over samples $z\in\ensuremath{\mathcal{Z}}$, we denote $\mathrm{TV}(P, Q) = 1/2\rE_{z\sim P}|dQ(z)/dP(z) - 1|$, $\mathrm{KL}(P||Q) = \rE_{z\sim P} \ln dP(z)/dQ(z)$ and ${D_H}(P,Q)^2 = \rE_{z\sim P}(\sqrt{dQ(z)/dP(z)} - 1)^2$. We use $\Delta(S)$ to denote the space of all probability distributions over a set $S$ (under a suitable $\sigma$-algebra) and $[H] = \{1,\ldots,H\}$.
\mypar{Effective dimensionality.} We often consider infinite-dimensional feature maps of the form $\chi(z_1, z_2)$ over a pair of inputs $z_1\in\ensuremath{\mathcal{Z}}_1$ and $z_2\in\ensuremath{\mathcal{Z}}_2$. We define the effective dimensionality of such maps as follows.
\begin{definition}[Effective dimension]
Given any measure $p$ over $\ensuremath{\mathcal{Z}}_1\times \ensuremath{\mathcal{Z}}_2$, and feature map $\chi$, define:
\begin{align*}
\Sigma(p,z_1) =& \rE_{z_2\sim p(\cdot | z_1)} \chi(z_1, z_2)\otimes \chi(z_1, z_2),~~
K(\lambda) =& \sup_{p,z_1} {\mathrm{trace}}((\Sigma(p,z_1) + \lambda I)^{-1} \Sigma(p,z_1)).
\end{align*}
For any $\epsilon > 0$, define the effective dimension of $\chi$ as:
$\ensuremath{d_{\mathrm{eff}}}(\chi,\epsilon) = \inf_{\lambda > 0}\left\{K(\lambda)~:~\lambda K(\lambda) \leq \epsilon^2\right\}$.
\label{def:eff-dim}
\end{definition}
If $\dim(\chi) = d$, $\ensuremath{d_{\mathrm{eff}}}(\chi, 0) \leq d$, and $\ensuremath{d_{\mathrm{eff}}}(\chi,\epsilon)$ can more generally be bounded in terms of spectral decay assumptions (see e.g. Proposition 2 in~\citet{agarwal2022non}).
\section{Model-based Optimistic Posterior Sampling\xspace}
\label{sec:algo}
\begin{algorithm}[tb]
\caption{Model-based Optimistic Posterior Sampling\xspace (\textsc{MOPS}\xspace) for model-based RL}
\label{alg:online_TS}
\begin{algorithmic}[1]
\REQUIRE Model class $\ensuremath{\mathcal{M}}$, prior $p_0\in\Delta(\ensuremath{\mathcal{M}})$, policy generator ${\pi_{\mathrm{gen}}}$, learning rates $\eta, \eta'$ and optimism coefficient $\gamma$.
\STATE Set $S_0 = \emptyset$.
\FOR{$t=1,\ldots,T$}
\STATE Observe $x_t^1 \sim {\mathcal D}$ and draw $h_t \sim \{1,\ldots,H\}$ uniformly at random.
\STATE Let $L_s^h(M) = -\eta (R_M^h(x_s^{h},a_s^h)-r_s^h)^2 + \eta' \ln P_M^h(x_s^{h+1} \mid x_s^{h}, a_s^{h})$.\COMMENT{Likelihood function}\label{line:likelihood}
\STATE Define $p_t(M) = p(M|S_{t-1}) \propto p_0(M)\exp(\sum_{s=1}^{t-1}(\gamma V_M(x_s^1) + L_s^{h_s}(M))$ as the posterior. \COMMENT{Optimistic posterior sampling update}\label{line:update}
\STATE Let $\pi_t={\pi_{\mathrm{gen}}}(h_t,p_t)$ \COMMENT{policy generation}\label{line:gen}\\
\STATE Play iteration $t$ using $\pi_{t}$ for $h=1,\ldots,h_t$, and observe $\{(x_t^h, a_t^h, r_t^h, x_t^{h+1})_{h=1}^{h_t}\}$ \label{line:one-step}
\STATE Update $S_t = S_{t-1} \cup \{x_t^h, a_t^h, r_t^h, x_t^{h+1}\}$ for $h = h_t$.
\ENDFOR
\RETURN $(\pi_1,\ldots,\pi_T)$.
\end{algorithmic}
\end{algorithm}
We now describe our algorithm \textsc{MOPS}\xspace for model-based RL in Algorithm~\ref{alg:online_TS}. The algorithm defines an optimistic posterior over the model class and acts according to a policy generated from this posterior. Specifically, the algorithm requires a prior $p_0$ over the model class and uses an optimistic model-error measure to induce the posterior distribution. We now highlight some of the salient aspects of our algorithm design.
\mypar{Likelihood-based dynamics prediction.} At each round, \textsc{MOPS}\xspace computes a likelihood over the space of models as defined in Line~\ref{line:likelihood}. The likelihood of a model $M$ includes two terms. The first term measures the squared error of the expected reward function $R_M$ in predicting the previously observed rewards. The second term measures the loss of the dynamics in predicting the observed states $x_s^{h+1}$, given $x_s^h$ and $a_s^h$, at each previous round $s$. For the dynamics, we use negative log-likelihood as the loss, and the reward and dynamics terms are weighted by respective learning rates $\eta$ and $\eta'$. Prior works of~\citet{sun2019model} and~\citet{du2021bilinear} use an integral probability metric (IPM) divergence, and require a more complicated Scheff\'e tournament algorithmically to handle model fitting under total variation unlike our approach. While the likelihood analysis from more recent works~\citep{agarwal2020flambe,uehara2021representation} can partly address this issue, the witness rank assumption of~\citet{sun2019model,du2021bilinear} is integrally based on the IPM divergence, and the correct structural assumptions needed to analyze the likelihood-based approach in their framework are not clear. The more recent work of~\cite{foster2021statistical} needs to directly incorporate the Hellinger distance in their algorithm. In contrast, we directly learn a good model in the Hellinger distance (and hence total variation) as our analysis shows, by likelihood driven sampling.
\mypar{Optimistic posterior updates.} Prior works in tabular~\citep{agrawal2017posterior}, linear~\citep{zanette2020frequentist} and model-free~\citep{zhang2021provably,agarwal2022non} RL make optimistic modifications to the vanilla posterior to obtain worst-case guarantees, and we perform a similar modification in our algorithm. Concretely, for every model $M$, we add the predicted optimal value in the initial context at all previous rounds $s$ to the likelihood term, weighted by a parameter $\gamma$ in Line~\ref{line:update} and define the posterior using this optimistic likelihood. As learning progresses, the posterior concentrates on models which predict the history well, and whose optimal value function predicts a large average value on the context distribution. Consequently any model sampled from the posterior has an optimal policy that either attains a high value in the true MDP $M^\star$, or visits some parts of the state space not previously explored, as the model predicts the history reasonably well. Incorporating this fresh data into our likelihood further sharpens our posterior, and leads to the typical \emph{exploit-or-learn} behavior that optimistic algorithms manifest in RL.
\mypar{Policy Generator.} Given a sampling distribution $p\in\Delta(\ensuremath{\mathcal{M}})$ (which is taken as optimistic posterior distribution in our approach), and a time step $h$, we assume access to a policy generator ${\pi_{\mathrm{gen}}}$ that takes $h$, $p$, and returns a policy ${\pi_{\mathrm{gen}}}(h,p) : \ensuremath{\mathcal{X}} \to {\mathcal A}$ (line~\ref{line:gen}). \textsc{MOPS}\xspace executes this policy up to a random time $h_t$ (line~\ref{line:one-step}), which we denote as $(x^{h_t},a^{h_t}) |x^1 \sim {\pi_{\mathrm{gen}}}(h_t,p) $.
The MDP then returns the tuple $(x^{h_t}, a^{h_t}, r^{h_t}, x^{h_t+1})$, which is used in our algorithm to update the posterior distribution. The choice of the policy generator plays a crucial role in our sample complexity guarantees, and we shortly present a decoupling condition on the generator which is a vital component of our analysis.
For the examples considered in the paper (see Section~\ref{sec:decoupling}),
policy generators that lead to good regret bounds are given as follows.
\begin{itemize}[leftmargin=*, itemsep=0pt]
\item $Q$-type problems: ${\pi_{\mathrm{gen}}}(h, p)$ follows a sample from the posterior, ${\pi_{\mathrm{gen}}}(h,p) = \pi_M, M \sim p$.
\item V-type problems with finite actions: ${\pi_{\mathrm{gen}}}(h,p)$ generates a trajectory up to $x^h$ using a single sample from posterior $\pi_M$ with $M \sim p$ and then samples $a^h\sim \mathrm{Unif}({\mathcal A})$.\footnote{This can be extended to a more general experimental design strategy as we show in Appendix~\ref{sec:wrank-design-decouple}.}
\item V-type problems with infinite actions: ${\pi_{\mathrm{gen}}}(h,p)$ draws two \emph{independent} samples $M, M' \sim p$ from the posterior. It generates a trajectory up to $x^h$ using $\pi_M$, and samples $a^h$ using $\pi_{M'}(\cdot | x^h)$.
\end{itemize}
It is worth mentioning that for $Q$-type problems, ${\pi_{\mathrm{gen}}}(h,p)$ does not depend on $h$. Hence we can replace random choice of $h_t$ by executing length $H$ trajectories in Algorithm~\ref{alg:online_TS} and using all the samples in the loss. This version of \textsc{MOPS}\xspace can get a better regret bound in terms of $H$ dependency. We do not include its analysis here so that we can focus on a general algorithm (and its analysis) that includes both $Q$-type and $V$-type problems simultaneously.
\iffalse
\mypar{Two samples for statistical decoupling.} A natural strategy, given an optimistic posterior, is to sample a model from this posterior and executing the optimal policy of the model in the true MDP. It is possible to analyze such mechanisms under certain $Q$-type variations of our assumptions (Algorithm~\ref{alg:online_TS-Q} in Section~\ref{sec:Qtype}), but in the low witness rank setting, prior works~\citep{sun2019model,du2021bilinear,foster2021statistical} execute the sampled policy until a random chosen step $h_t$, and then choose the action at $h_t$ from an different exploratory distribution, such as uniform over the action space. The need for this decoupling is evident from the definition~\eqref{eq:old-wit-rank} of witness rank, where the action $a^h$ is selected from the policy $\pi_{M'}$ instead of $\pi_M$. If we execute the same policy for all $h_t$ steps, then we would choose $a^{h_t}$ according to $\pi_M$, and this requires a different $Q$-type low rank assumption, which has been used in model-free settings in prior works~\citep{jin2021bellman,du2021bilinear}. For sample efficiency in our setting, Algorithm~\ref{alg:online_TS} decouples the choice of actions for steps $1,\ldots,h_t-1$ with that at step $h_t$ in Lines~\ref{line:fdraw} and~\ref{line:one-step} by drawing two independent models $M_t$ and $M_t'$ from the posterior. It then chooses the first $h_t-1$ actions according to $\pi_{M_t}$, while that at step $h_t$ is chosen as per $\pi_{M'_t}$. Note that the choice of $a^{h_t}_t$ according to $\pi_{M'_t}$, as opposed to uniformly at random, allows us to scale to large action spaces.
\fi
\section{Main Result}
\label{sec:results}
We now present the main structural condition that we introduce in this paper,
which is used to characterize the quality of generator in Algorithm~\ref{alg:online_TS}.
We will present several examples of concrete models which can be captured by this definition in Section~\ref{sec:decoupling}. The assumption is inspired by prior decoupling conditions~\citep{Zhang2021FeelGoodTS,agarwal2022non} used in the analysis of contextual bandits and some forms of model-free RL.
\begin{definition}[Hellinger Decoupling of Bellman Error]
Let a distribution $p \in \Delta(\ensuremath{\mathcal{M}})$ and a policy $\pi(x^h,a^h|x^1)$ be given.
For any $\epsilon>0$, $\alpha \in (0,1]$ and $h\in[H]$, we define the \emph{Hellinger decoupling coefficient} ${\mathrm{dc}}^h(\epsilon, p,\pi,\alpha)$ of an MDP $M^\star$ as the smallest number $c^h \geq 0$ so that for all $x^1$:
\begin{align*}\textstyle
\rE_{M \sim p}\rE_{\textcolor{red}{(x^h,a^h)\sim \pi_M(\cdot|x^1)}} \ensuremath{{\mathcal E}_B}(M, x^h, a^h)
\leq& \left(c^h \rE_{M \sim p} \rE_{\textcolor{red}{(x^h, a^h)\sim \pi(\cdot | x^1)}} \ell^h(M,x^h,a^h)\right)^{\alpha}
+ \epsilon ,
\end{align*}
where $\ell^h(M,x^h,a^h) = {D_H}(P_M(\cdot | x^h, a^h), P_\star(\cdot | x^h, a^h))^2
+ (R_M(x^h, a^h) - R_\star(x^h, a^h))^2$.
\label{def:decoupling}
\end{definition}
Intuitively, the distribution $p$ in Definition~\ref{def:decoupling} plays the role of our estimate for $M^\star$, and we seek a low regret for the optimal policies of models $M\sim p$, which is closely related to the model-based Bellman error
(by Lemma~\ref{lemma:simulation}), where the samples $x^h, a^h$ are drawn from $\pi^M$. The decoupling inequality relates the Bellman error to the estimation error of $p$ in terms of mean-squared error of the rewards and a Hellinger distance to the true dynamics $P_\star$. However, it is crucial to measure this error under distribution of the data which is used for model-fitting. The policy $\pi={\pi_{\mathrm{gen}}}(h,p)$ plays this role of the data distribution, and is typically chosen in a manner closely related to $p$ in our examples. The decoupling inequality bounds the regret of $p$ in terms of estimation error under $x^h, a^h\sim \pi$, for all $p$, and allows us to find a good distribution $p$ via online learning.
For stating our main result, we define a standard quantity for posterior sampling, measuring how well the prior distribution $p_0$ used in \textsc{MOPS}\xspace covers the optimal model $M_\star$.
\begin{definition}[Prior around true model]
Given $\alpha>0$ and $p_0$ on $\ensuremath{\mathcal{M}}$, define
\[
\ensuremath{\omega}(\alpha,p_0) = \inf_{\epsilon>0}\left[
\alpha \epsilon - \ln p_0(\ensuremath{\mathcal{M}}(\epsilon))\right] ,
\]
where
$\ensuremath{\mathcal{M}}(\epsilon)=
\bigg\{
M \in \ensuremath{\mathcal{M}}: \displaystyle\sup_{x^1} -\Delta V_M(x^1) \leq \epsilon; \displaystyle\sup_{h,x^h,a^h} \tilde{\ell}^h(M, x^h, a^h) \leq \epsilon
\bigg\}$ and $\tilde{\ell}^h(M, x^h, a^h) = \mathrm{KL}(P_\star(\cdot|x^h, a^h)||P_M(\cdot|x^h, a^h)) + (R_M(x^h, a^h) - R_\star(x^h, a^h))^2$.
\label{def:kappa}
\end{definition}
Definition~\ref{def:kappa} implicitly uses model realizability to ensure that $\ensuremath{\mathcal{M}}(\epsilon)$ is non-empty for any $\epsilon > 0$. If $\ensuremath{\mathcal{M}}$ is finite with uniform prior $p_0$ over $\ensuremath{\mathcal{M}}$, and $M^\star \in \ensuremath{\mathcal{M}}$, then $\ensuremath{\omega}(\alpha,p_0) \leq \ln |\ensuremath{\mathcal{M}}|$ for all $\alpha>0$.
However, we note that our bound based on $\ensuremath{\omega}(\alpha,p_0)$ can still be applied even if the model is misspecified, whence the optimization over $\epsilon$ naturally gets limited above the approximation error.
We now state the main result of the paper.
\begin{theorem}[Sample complexity under decoupling]
Under Assumptions~\ref{ass:realizable} and~\ref{ass:planning}, suppose
that there exists $\alpha>0$ such that for all $p$,
${\mathrm{dc}}^h(\epsilon, p,{\pi_{\mathrm{gen}}}(h,p),\alpha) \leq {\mathrm{dc}}^h(\epsilon,\alpha)$.
Define
\[
{\mathrm{dc}}(\epsilon,\alpha)
= \bigg(\frac1H \sum_{h=1}^H
{\mathrm{dc}}^h(\epsilon, \alpha)^{\alpha/(1-\alpha)}\bigg)^{(1-\alpha)/\alpha}.
\]
If we take $\eta =\eta' = 1/6$ and $\gamma \leq 0.5$, then the following bound holds for \textsc{MOPS}\xspace:
\begin{align*}
\sum_{t=1}^T \rE \left[V_\star(x_t^1) - \rE_{M \sim p_t} V_{M} (x_t^1)\right]
\leq
\frac{\ensuremath{\omega}(T,p_0)}\gamma + \frac{\gamma T}{2}
+ H T \left[ \epsilon + (1-\alpha) (20H\gamma\alpha)^{\frac{\alpha}{(1-\alpha)}} {\mathrm{dc}}(\epsilon,\alpha)^{\frac{\alpha}{(1-\alpha)}}\right] .
\end{align*}
\label{thm:main}
\end{theorem}
To simplify the result for easy interpretation, we consider finite model classes with $p_0$ as the uniform prior on $\ensuremath{\mathcal{M}}$. For $\alpha \leq 0.5$, by taking $\gamma =\min(0.5, (\ln |\ensuremath{\mathcal{M}}|/T)^{1-\alpha}{\mathrm{dc}}(\epsilon,\alpha)^{-\alpha}/H)$, we obtain
\begin{equation}
\frac{1}{T} \sum_{t=1}^T \rE \left[V_\star(x_t^1) - \rE_{M \sim p_t} V_{M} (x_t^1)\right]
= O\left( H \left(\frac{{\mathrm{dc}}(\epsilon,\alpha) \ln|\ensuremath{\mathcal{M}}|}{ T}\right)^\alpha
+ \epsilon H T \right) .
\label{eq:regret-simplified}
\end{equation}
We note that in Theorem~\ref{thm:main}, the decoupling coefficient fully characterizes the structural properties of the MDP. Once ${\mathrm{dc}}(\epsilon,\alpha)$ is estimated, Theorem~\ref{thm:main} can be immediately applied. We will instantiate this general result with concrete examples in Section~\ref{sec:decoupling}.
Definition~\ref{def:decoupling} appears related to the decision estimation coefficient (DEC) of~\citet{foster2021statistical}. As expalined in Section~\ref{sec:related}, our definition is only needed in the analysis, and more suitable to posterior sampling based algorithmic design.
The definition is also related to the Bilinear classes model of~\citet{du2021bilinear}, since the bilinear structures can be turned into decoupling results as we will see in our examples. Compared to these earlier results, our definition is more amenable algorithmically.
\subsection*{Proof of Theorem~\ref{thm:main}}
\label{sec:sketch}
We now give a proof sketch for Theorem~\ref{thm:main}.
As in prior works, we start from bounding the regret of any policy $\pi_M$ in terms of a Bellman error term and an optimism gap via Lemma~\ref{lemma:simulation}.
We note that in the definition of Bellman error in Lemma~\ref{lemma:simulation}, model $M$ being evaluated is the same model that also generates the data, and this coupling cannot be handled directly using online learning. This is where the decoupling argument is used, which shows that the coupled Bellman error can be bounded by decoupled loss, where data is generated according to ${\pi_{\mathrm{gen}}}(h_t, p_t)$, while model being evaluated is drawn from $p_t$ independently of data generation.
This intuition is captured in the following proposition, proved in Appendix~\ref{sec:proof-decoupling}.
\begin{proposition}[Decoupling the regret]
Under conditions of Theorem~\ref{thm:main}, the regret of Algorithm~\ref{alg:online_TS} at any round $t$ can be bounded, for any $\mu>0$ and $\epsilon>0$, as
\begin{align*}
\rE \left[V_\star(x_t^1) - \rE_{M \sim p_t} V_{M} (x_t^1)\right]
\leq& \rE\rE_{M \sim p_t}
\left[ \mu H\ell^{h_t}(M,x_t^{h_t},a_t^{h_t})
-\Delta V_M(x^1_t) \right] \\
& + H \left[ \epsilon + (1-\alpha) (\mu/\alpha)^{-\alpha/(1-\alpha)} {\mathrm{dc}}(\epsilon,\alpha)^{\alpha/(1-\alpha)}\right].
\end{align*}
\label{prop:decoupling}
\end{proposition}
The proposition involves error terms involving the observed samples $(x_t^{h_t}, a_t^{h_t}, r_t^{h_t}, x_t^{h_t+1})$, which our algorithm controls via the posterior updates.
Specifically, we expect the regret to be small whenever the posterior has a small average error of models $M\sim p_t$, relative to $M_\star$. This indeed happens as evidenced by our next result, which we prove in Appendix~\ref{sec:proof-online}.
\begin{proposition}[Convergence of online learning]
With $\eta =\eta' = 1/6$ and $\gamma \leq 0.5$, \textsc{MOPS}\xspace ensures:
\begin{align*}
& \sum_{t=1}^T \rE\rE_{M\sim p_t}
\left[ 0.3\eta\gamma^{-1} \ell^{h_t}(M,x_t^{h_t},a_t^{h_t}) -\Delta V_M(x_t^1)\right]
\leq \gamma^{-1}\ensuremath{\omega}(T,p_0) + 0.5 \gamma T.
\end{align*}
\label{prop:online}
\end{proposition}
Armed with Proposition~\ref{prop:decoupling} and
Proposition~\ref{prop:online}, we are ready to prove the main theorem
as follows.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Combining Propositions~\ref{prop:decoupling} and~\ref{prop:online} with $\mu H=\frac{0.3\eta}{\gamma}$ gives the desired result
\end{proof}
\section{MDP Structural Assumptions and Decoupling Coefficients Estimates}
\label{sec:decoupling}
Since Definition~\ref{def:decoupling} is fairly abstract, we now instantiate concrete models where the decoupling coefficient can be bounded in terms of standard problem complexity measures. We give examples of $V$-type and $Q$-type decouplings, a distinction highlighted in many recent works~\citep[e.g.][]{jin2021bellman,du2021bilinear}. The $V$-type setting captures more non-linear scenarios at the expense of slightly higher algorithmic complexity, while $Q$-type is more elegant for (nearly) linear settings.
\subsection{$V$-type decoupling and witness rank}
\label{sec:wrank}
\citet{sun2019model} introduced the notion of witness rank to capture the tractability of model-based RL with general function approximation, building on the earlier Bellman rank work of~\citet{jiang2017contextual} for model-free scenarios. For finite action problems, they give an algorithm whose sample complexity is controlled in terms of the witness rank, independent of the number of states, and show that the witness rank is always smaller than Bellman rank for model-free RL. The measure is based on a quantity called \emph{witnessed model misfit} that captures the difference between two probability models in terms of the differences in expectations they induce over test functions chosen from some class. We next state a quantity closely related to witness rank.
\begin{assumption}[Generalized witness factorization]
Let $\ensuremath{\mathcal{F}} = \{f(x,a,r,x') = r + g(x,a,x')~:~g\in\ensuremath{\mathcal{G}}\}$, with $g(x,a,x')\in[0,1]$, be given. Then there exist maps $\psi^h(M, x^1)$ and $u^h(M, x^1)$, and a constant $\kappa \in (0,1]$, such that for any context $x^1$, level $h$ and models $M, M'\in\ensuremath{\mathcal{M}}$, we have
\begin{align}
\kappa & \ensuremath{{\mathcal E}_B}(M, M', h, x^1) \leq \left|\inner{\psi^h(M, x^1)}{u^h(M', x^1)}\right|\nonumber\\ &\quad\leq \sup_{f\in\ensuremath{\mathcal{F}}}\rE_{x^h\sim \pi_M|x^1} \rE_{a^h\sim \pi_{M'}(x^h)} \left|(P_{M'}^h f)(x^h,a^h) - (P_{\star}^h f)(x^h,a^h)\right|
\tag{Bellman domination}
\end{align}
where $\ensuremath{{\mathcal E}_B}(M, M',h,x^1) = \displaystyle\rE_{x^h\sim \pi_M,a^h\sim \pi_{M'}|x^1}\left[\ensuremath{{\mathcal E}_B}(M', x^h, a^h)\right]$. We assume that $\|u^h(M, x^1)\|_2 \leq B_1$ for all $M$ and $x^1$.
\label{ass:wit-fac}
\end{assumption}
\citet{sun2019model} define a similar factorization, but allow arbitrary dependence of $f$ on the reward to learn the full distribution of rewards, in addition to the dynamics. We focus on only additive reward term, as we only need to estimate the reward in expectation, for which this structure of test functions is sufficient. The form of $g$ for the dynamics roughly corresponds to the discriminator class used in~\citet{sun2019model} for the case of the divergence being TV distance, which is naturally related to our log-likelihood based approach. The dependence on the context $x^1$ allows us to capture contextual RL setups~\citep{hallak2015contextual}. This assumption captures a wide range of structures including tabular, factored, linear and low-rank MDPs (see \citet{sun2019model} for further examples).
The bilinear structure of the factorization enables us to decouple the Bellman error. We begin with the case of finite action sets studied in~\citet{sun2019model}. Let $\pi\circ^h\pi'$ be a non-stationary policy which follows $\pi$ for $h-1$ steps, and chooses $a^h\sim \pi'(\cdot | x^h)$.
\begin{proposition}
Under Assumption~\ref{ass:wit-fac}, suppose further that $|{\mathcal A}| = K$. Let us define $z_1 = x^1, z_2 = M$ and $\chi = \psi^h(M, x^1)$ in Definition~\ref{def:eff-dim}. Then for any $\epsilon > 0$, we have
\begin{equation*}
{\mathrm{dc}}(\epsilon, p, {\pi_{\mathrm{gen}}}(h,p),0.5) \leq \frac{2K}{\kappa^2}\ensuremath{d_{\mathrm{eff}}}\left(\psi^h, \frac{\kappa}{B_1}\epsilon\right), \quad
\mbox{where ${\pi_{\mathrm{gen}}}(h,p)=p\circ^h\mathrm{Unif}({\mathcal A})$} .
\end{equation*}
\label{prop:wrank-finite}
\end{proposition}
The proofs of Proposition~\ref{prop:wrank-finite} and all the other results in this section are in Appendix~\ref{sec:wrank-decouple}.
\mypar{Sample complexity under low witness rank and finite actions.} Plugging Proposition~\ref{prop:wrank-finite} into Theorem~\ref{thm:main} gives a corollary for this setting. For ease of discussion, let $\dim(\psi^h) \leq d$ for all $h\in[H]$ and further assume that the model class is finite, that is, $|\ensuremath{\mathcal{M}}|<\infty$.
\begin{corollary}
Under conditions of Theorem~\ref{thm:main}, suppose further that Assumption~\ref{ass:wit-fac} holds with $\dim(\psi^h) \leq d$ for all $h\in[H]$. Let the model class $\ensuremath{\mathcal{M}}$ an action space ${\mathcal A}$ have a finite cardinality with $|{\mathcal A}| = K$. Then the policy sequence $\pi_1,\ldots,\pi_T$ produced by \textsc{MOPS}\xspace satisfies
\begin{equation*}
\frac{1}{T}\sum_{t=1}^T V^\star(x_t^1) - V^{\pi_t}(x_t^1) \leq \mathcal{O}\left(\sqrt{\frac{H^2d^2 K\ln|\ensuremath{\mathcal{M}}|}{\kappa^2 T}}\right).
\end{equation*}
\label{cor:wrank-finite}
\end{corollary}
With a standard online-to-batch conversion argument~\citep{cesa2004generalization}, this implies a sample complexity bound to find an $\epsilon$-suboptimal policy of $\mathcal{O}\left(\frac{H^2 d K\ln |\ensuremath{\mathcal{M}}|}{\kappa^2\epsilon^2}\right)$, when the contexts are i.i.d. from a distribution.
This bound improves upon those of~\citet{sun2019model} and~\citet{du2021bilinear}, who require $\tilde{\mathcal{O}}\left(\frac{H^3d^2K}{\kappa^2\epsilon^2}\ln\frac{T|\ensuremath{\mathcal{M}}||\ensuremath{\mathcal{F}}|}{\delta}\right)$ samples, where $\ensuremath{\mathcal{F}}$ is a discriminator class explicitly used in their algorithm. In contrast, we only use the discriminators implicitly in our analysis.
\mypar{Factored MDPs} \citet{sun2019model} show an exponential separation between sample complexity of model-based and model-free learning in factored MDPs~\citep{boutilier1995exploiting} by controlling the error of each factor independently. A similar adaptation of our approach to measure the likelihood of each factor separately in the setting of Proposition~\ref{prop:wrank-finite} allows our technique to handle factored MDPs.
Next, we further generalize this decoupling result to large action spaces by making a linear embedding assumption that can simultaneously capture all finite action problems, as well as certain linear settings~\citep{Zhang2021FeelGoodTS,zhang2021provably,agarwal2022non}.
\begin{assumption}[Linear embeddability of backup errors]
\label{ass:linear-embed}
Let $\ensuremath{\mathcal{F}}$ be a function class such that $f(x,a,r,x') = r + g(x,a,x')$ for $g\in\ensuremath{\mathcal{G}}$, with $g(x,a,x')\in[0,1]$, for all $x,a,x'$. There exist functions $\phi^h(x^h, a^h)$ and $w^h(M, f, x^h)$ such that for all $f\in\ensuremath{\mathcal{F}}$, $M\in\ensuremath{\mathcal{M}}$, $h\in[H]$ and $x^h, a^h$:
\begin{equation*}
(P_M^h f)(x^h,a^h) - (P_{\star}^h f)(x^h,a^h) = \inner{w^h(M, f, x^h)}{\phi^h(x^h, a^h)}.
\end{equation*}
We assume that $\|w^h(M, f, x^h)\|_2 \leq B_2$, for all $M, f, x^h$ and $h\in[H]$.
\end{assumption}
Since the weights $w^h$ can depend on both the $f$ and $x^h$, for finite action problems it suffices to choose $\phi^h(x,a) = e_a$ and $w^h(M, f, x^h) = ((P_M^h f)(x^h,a) - (P_{\star}^h f)(x^h,a))_{a=1}^K$. For linear MDPs, the assumption holds with $\phi^h$ being the MDP features and $w^h$ being independent of $x^h$. A similar assumption on Bellman errors has recently been used in the analysis of model-free strategies for non-linear RL scaling to large action spaces~\citep{Zhang2021FeelGoodTS,zhang2021provably,agarwal2022non}. We now state a more general bound on the decoupling coefficient under Assumption~\ref{ass:linear-embed}.
\begin{proposition}
Suppose Assumptions~\ref{ass:wit-fac} and~\ref{ass:linear-embed} hold. For $\phi^h$ in Definition~\ref{def:eff-dim}, we define $z_1 = x^h$, $z_2 = a^h$ and $\chi = \phi^h(x^h, a^h)$, with same choices for $\psi^h$ as Proposition~\ref{prop:wrank-finite}. Then for any $\epsilon > 0$:
\begin{equation*}
{\mathrm{dc}}^h(\epsilon, p, {\pi_{\mathrm{gen}}}(h,p),0.25) \leq 8\kappa^{-4}\ensuremath{d_{\mathrm{eff}}}(\psi^h,\epsilon_1)^2\ensuremath{d_{\mathrm{eff}}}(\phi^h,\epsilon_2),
\quad \mbox{where ${\pi_{\mathrm{gen}}}(h,p)=p\circ^h p$}
\end{equation*}
where $\epsilon_1, \epsilon_2 > 0$ satisfy
\begin{equation*}
\frac{B_1\epsilon_1}{\kappa} + \frac{\sqrt{2B_2\epsilon_2\ensuremath{d_{\mathrm{eff}}}(\psi^h,\epsilon_1)}}{\kappa} = \epsilon.
\end{equation*}
\label{prop:wrank-embed}
\end{proposition}
Compared with Proposition~\ref{prop:wrank-finite}, we see that value of the exponent $\alpha$ worsens to $\alpha = 0.25$ in Proposition~\ref{prop:wrank-embed}. This happens because we now change the action choice at step $h$ to be from $\pi_{M'}$, where $M'\sim p$ independent of $M$. To carry out decoupling for this choice, we need to use both the factorizations in Assumptions~\ref{ass:wit-fac} and~\ref{ass:linear-embed}, which introduces an additional Cauchy-Schwarz step. This change is necessary as no obvious exploration strategy, like the uniform distribution over finite actions in Proposition~\ref{prop:wrank-finite}, is available here.
\mypar{Sample complexity for low witness rank and (unknown) linear embedding.} Similar to Corollary~\ref{cor:wrank-finite}, we can obtain a concrete result for this setting by combining Proposition~\ref{prop:wrank-embed} and Theorem~\ref{thm:main}.
\begin{corollary}
Under conditions of Theorem~\ref{thm:main}, suppose further that Assumptions~\ref{ass:wit-fac} and~\ref{ass:linear-embed} hold, and $|\ensuremath{\mathcal{M}}| < \infty$. For any $\epsilon_1, \epsilon_2 > 0$, let $\ensuremath{d_{\mathrm{eff}}}(\psi^h, \epsilon_1) \leq \ensuremath{d_{\mathrm{eff}}}(\psi,\epsilon_1)$ and $\ensuremath{d_{\mathrm{eff}}}(\phi^h,\epsilon_2) \leq \ensuremath{d_{\mathrm{eff}}}(\phi, \epsilon_2)$, for all $h\in[H]$. Then the policy sequence $\pi_1,\ldots,\pi_T$ produced by \textsc{MOPS}\xspace satisfies
\begin{align*}
&\frac{1}{T}\sum_{t=1}^T V^\star(x_t^1) - V^{\pi_t}(x_t^1) \\&\qquad\qquad\leq
\mathcal{O}\left(\frac{1}{\kappa}\left(\frac{\ensuremath{d_{\mathrm{eff}}}(\psi,\epsilon_1)^2\ensuremath{d_{\mathrm{eff}}}(\phi,\epsilon_2) H^4 \ln|\ensuremath{\mathcal{M}}|}{T}\right)^{1/4} + \left(\frac{B_1\epsilon_1}{\kappa} + \frac{\sqrt{2B_2\epsilon_2\ensuremath{d_{\mathrm{eff}}}(\psi,\epsilon_1)}}{\kappa}\right)H\right).
\end{align*}
\label{cor:wrank-embed}
\end{corollary}
When the maps $\psi^h, \phi^h$ are both finite dimensional with $\dim(\psi^h) \leq d_1$ and $\dim(\psi^h) \leq d_2$ for all $h$, \textsc{MOPS}\xspace enjoys a sample complexity of $\mathcal{O}\left(\frac{8d_1^2d_2H^4\ln|\ensuremath{\mathcal{M}}|}{\kappa^4\epsilon^4}\right)$ in this setting. We are not aware of any prior methods that can handle this setting. The loss of rates in $T$ arises due to the worse exponent of $0.25$ in the decoupling bound of Proposition~\ref{prop:wrank-embed}, which is due to the extra Cauchy-Schwarz step as mentioned earlier. For more general infinite dimensional cases, it is straightforward to develop results based on an exponential or polynomial spectral decay analogous to Proposition 2 in~\citet{agarwal2022non}.
\subsection{$Q$-type decoupling and linear models}
We now give examples of two other structural assumptions, where the decoupling holds pointwise for all $x$ and not just in expectation. As this is somewhat analogous to similar phenomena in $Q$-type Bellman rank~\citep{jin2021bellman}, we call such results $Q$-type decouplings. We begin with the first assumption which applies to linear MDPs as well as certain models in continuous controls, including Linear Quadratic Regulator (LQR) and the Kernelized Nonlinear Regulator model.
\begin{assumption}[$Q$-type witness factorization]
Let $\ensuremath{\mathcal{F}}$ be a function class such that $f(x,a,r,x') = r + g(x,a,x')$ for $g\in\ensuremath{\mathcal{G}}$, with $g(x,a,x')\in[0,1]$, for all $x,a,x'$. Then there exist maps $\psi^h(x^h, a^h)$ and $u^h(M, f)$ and a constant $\kappa > 0$, such that for any $h, x^h, a^h$ and $M\in\ensuremath{\mathcal{M}}$:
\begin{align*}
|(P_M f) (x^h,a^h) - (P^\star f)(x^h,a^h)| \geq \xi(M, f, x^h, a^h) ,~
\sup_{f\in\ensuremath{\mathcal{F}}} \xi(M, f, x^h, a^h) \geq \kappa \ensuremath{{\mathcal E}_B}(M, x^h, a^h),
\end{align*}
with $\xi(M, f, x^h, a^h) = \left|\inner{\psi^h(x^h,a^h)}{u^h(M,f)}\right|$.
We assume $\|u^h(M, f)\|_2 \leq B_1$ for all $M, f, h$.
\label{ass:wit-fac-q}
\end{assumption}
Assumption~\ref{ass:wit-fac-q} is clearly satisfied by a linear MDP when $\phi^h$ are the linear MDP features and $\ensuremath{\mathcal{F}}$ is any arbitrary function class. We show in Appendix~\ref{sec:knr} that this assumption also includes the Kernelized Non-linear Regulator (KNR), introduced in~\citet{kakade2020information} as a model for continuous control that generalizes LQRs to handle some non-linearity. In a KNR, the dynamics follow $x^{h+1} = W^\star \varphi(x^h, a^h)+\epsilon$, with $\epsilon\sim \ensuremath{\mathcal{N}}(0,\sigma^2I)$, and the features $\varphi(x^h, a^h)$ are known and lie in an RKHS. In this case, for an appropriate class $\ensuremath{\mathcal{F}}$, we can show that Assumption~\ref{ass:wit-fac-q} holds with features $\phi^h = \varphi$ from the KNR dynamics.
Under this assumption, we get an immediate decoupling result, with the proof in Appendix~\ref{sec:proof-Qtype-decouple}.
\begin{proposition}
Under Assumption~\ref{ass:wit-fac-q}, let define $z_2 = (x^h, a^h)$, $z_1 = x^1$ and $\chi = \psi^h(x^h, a^h)$ in Definition~\ref{def:eff-dim}.\footnote{We do not explicitly include $x^1$ in $\chi$ as it is assumed to be contained in $x^1$.} Then for any $\epsilon > 0$, we have
\begin{equation*}
{\mathrm{dc}}(\epsilon, p, {\pi_{\mathrm{gen}}}(h,p), 0.5) \leq \frac{2}{\kappa^2}\ensuremath{d_{\mathrm{eff}}}\left(\psi^h, \frac{\kappa}{B_1}\epsilon\right) , \quad\mbox{where ${\pi_{\mathrm{gen}}}(h,p)=\pi_M$
with $M \sim p$.}
\end{equation*}
\label{prop:qtype-control}
\end{proposition}
Here we see that the decoupling coefficient scales with the effective feature dimension, which now simultaneously captures the exploration complexity over both states and actions, consistent with existing results for $Q$-type settings such as linear MDPs.
\mypar{Sample complexity for the KNR model.} We now instantiate a concrete corollary of Theorem~\ref{thm:main} for the KNR model, under the assumption that $x^h\in\ensuremath{\mathbb{R}}^{d_{\ensuremath{\mathcal{X}}}}$, $\varphi(x^h, a^h) \in \ensuremath{\mathbb{R}}^{d_{\varphi}}$ and $\|\varphi(x^h, a^h)\|_2 \leq B$ for all $x^h, a^h$ and $h\in[H]$. A similar result also holds more generally for all problems where Assumption~\ref{ass:wit-fac-q} holds, but we state a concrete result for KNRs to illustrate the handling of infinite model classes, since the natural model class in the KNR setting is $\{W\in \ensuremath{\mathbb{R}}^{d_{\ensuremath{\mathcal{X}}}\times d_{\varphi}}:\|W\|_2 \leq R\}$, where $\|W^\star\|_2 = R$.
\begin{corollary}
Under conditions of Theorem~\ref{thm:main}, suppose further that we apply \textsc{MOPS}\xspace to the KNR model with the model class $\ensuremath{\mathcal{M}}_{KNR} = \{W\in \ensuremath{\mathbb{R}}^{d_{\ensuremath{\mathcal{X}}}\times d_{\phi}}:\|W\|_2 \leq R\}$. Then the policy sequence $\pi_1,\ldots,\pi_T$ produced by \textsc{MOPS}\xspace satisfy:
\begin{equation*}
\frac{1}{T}\sum_{t=1}^T V^\star(x_t^1) - V^{\pi_t}(x_t^1) \leq \mathcal{O}\left(H\sqrt{\frac{d_\varphi^2 d_{\ensuremath{\mathcal{X}}}\log\frac{R\sqrt{d_\varphi}BH}{\epsilon\sigma}}{T\sigma^2}}\right).
\end{equation*}
\label{cor:wit-fac-q}
\end{corollary}
Structurally, the result is a bit similar to Corollary~\ref{cor:wrank-finite}, except that there is no action set dependence any more, since the feature dimension captures both state and action space complexities in the $Q$-type setting as remarked before.
We also do not make a finite model space assumption in this result, as mentioned earlier. To apply Corollary~\ref{cor:wit-fac-q} to this setting, we bound $\ensuremath{\omega}(T, p_0)$ in Lemma~\ref{lemma:knr-cover} in Appendix~\ref{sec:knr}. We notice that Corollary~\ref{cor:wit-fac-q} has a slightly inferior $d_\phi^2d_{\ensuremath{\mathcal{X}}}$ dimension dependence compared to the $d_\phi(d_{\ensuremath{\mathcal{X}}} + d_\phi)$ scaling in \citet{kakade2020information}. It is possible to bridge this gap by a direct analysis of the algorithm in this case, with similar arguments as the KNR paper, but our decoupling argument loses an extra dimension factor. Note that it is unclear how to cast the broader setting of Assumption~\ref{ass:wit-fac-q} in the frameworks of Bilinear classes or DEC.
\mypar{Linear Mixture MDPs.} A slightly different $Q$-type factorization assumption which includes linear mixture MDPs~\citep{modi2020sample,ayoub2020model}, also amenable to decoupling, is discussed in Appendix~\ref{sec:lmm}. For this model, we get an error bound of $\mathcal{O}\left(H\sqrt{\frac{d_\phi\ensuremath{\omega}(T, p_0)}{T\kappa^2}}\right)$. Given our assumption on value functions normalization by 1, this suggests a suboptimal scaling in $H$ factors~\citep{zhou2021nearly}, because our algorithm uses samples from a randomly chosen time step only and our analysis does not currently leverage the Bellman property of variance, which is crucial to a sharper analysis. While addressing the former under the $Q$-type assumptions is easy, improving the latter for general RL settings is an exciting research direction.
\section{Conclusion}
\label{sec:conclusion}
This paper proposes a general algorithmic and statistical framework for model-based RL bassd on optimistic posterior sampling. The development yields state-of-the-art sample complexity results under several structural assumptions. Our techniques are also amenable to practical adaptations, as opposed to some prior attempts relying on complicated constrained optimization objectives that may be difficult to solve. Empirical evaluation of the proposed algorithms would be interesting for future work. As another future direction, our analysis (and that of others in the rich observation setting) does not leverage the Bellman property of variance, which is essential for sharp horizon dependence in tabular~\citep{azar2017minimax} and some linear settings~\citep{zhou2021nearly}. Extending these ideas to general non-linear function approximation is an important direction for future work. More generally, understanding if the guarantees can be more adaptive on a per-instance basis, instead of worst-case, is critical for making this theory more practical.
\section*{Acknowledgements}
The authors thank Wen Sun for giving feedback on an early draft of this work.
\bibliographystyle{plainnat}
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\section{Introduction}
Online reinforcement learning~\cite{NIPS2017_36e729ec} mainly focus on the problem of learning and planning in sequential decision making systems in real time when the interacting environment is partially observed or fully observed. Normally, we could use Markov Decision Process(MDP) to represent such online decision process. At each time step, the system will generate reward and the next state according to a fixed state transition distribution. The decision maker tend to maximize the cumulative reward during its interacting process. Which leads to the trade-off between exploration and exploitation. Many attempts had been made to improve such dilemma~\cite{Kveton2020RandomizedEI}. In this paper, we aims to solve the trade-off problem in finite stochastic games between exploration and exploitation by applying posterior sampling method on policy probability distribution.
Trade-off between exploration and exploitation has been studied extensively in various scenarios such as stochastic games. The goal of exploration is to find as much information as possible of the environment. While the exploitation process aims to maximize the long-term reward based on the known environment. One of the popular way to deal with the trade-off problem is to use the Naive Exploration method such as adaptive $\epsilon$-greedy exploration~\cite{tokic2010adaptive}. It proposed a method that adjust the exploration parameter adaptively depend on the temporal-difference(TD) error observed form value function. Optimistic Initialisation methods have also been studied in factored MDP~\cite{inproceedings}~\cite{10.1162/153244303765208377} to solve trade-off problem. It encourages systematic exploration in early stage. Another common way to handle the exploitation-exploration trade-off is to use the optimism in the face of uncertainty (OFU) principle~\cite{lai1985asymptotically}. Based on this approach, the agent constructs confidence sets to search for optimistic parameters that associate with the maximum reward. Though Many of the optimistic algorithms were shown to have solid theoretical bounds of performance~\cite{auer2002using}~\cite{Hao2019BootstrappingUC}. They can still lock onto suboptimal action during exploration process.
Thompson Sampling(TS),also known as Posterior Sampling has been used in many scenes as an alternative strategy to promote exploration while balancing current reward. Thompson sampling was originally presented for stochastic bandit scenarios\cite{thompson1933likelihood}. Then it's been applied in various MDP contexts~\cite{osband2013more}. A TS algorithm estimate the posterior distribution of the unknown environment based on the prior distribution and experiment process. Theoretically, the TS algorithms tend to have tighter bounds than optimistic algorithms in many different contexts. Empirically, the TS algorithms can easily embedded with other algorithm structures because of its efficiency in computation~\cite{chapelle2011empirical}. The optimistic algorithms requires to solve all MDPs lying within the confident sets while TS algorithms only needs to solve the sampled MDPs to achieve similar results~\cite{russo2014learning}.
In this paper, we propose a sampling method that samples the transition probability distribution and policy distribution at the same time. Traditional posterior sampling method merely concentrate on the transition matrix of the underlying environment. Previous work \emph{UCSG} had given the regret upper bound of $\tilde{\mathcal{O}}\left(\sqrt[3]{D S^{2} A T^{2}}\right)$ on stochastic games~\cite{NIPS2017_36e729ec}. Where $D$ is the diameter of the Stochastic Games(SG). Model-free method has also been used in this area, \emph{Optimistic Q-Learning} achieves a regret bound of $\tilde{\mathcal{O}}\left(T^{2 / 3}\right)$ under infinite-horizon average discounted reward MDP~\cite{wei2020model}. Our approach consists of two optimization sampling method. The first method optimize the long-term policy probability distribution. The other method samples the transition matrix of the unknown environment. We first adopt the previous stopping criterions in Thompson Sampling-based reinforcement learning algorithm with dynamic
episodes (\emph{TSDE})~\cite{10.5555/3294771.3294898}. Then apply the posterior sampling method on both transition matrix and policy probability distribution. During the posterior update process of the policy distribution, we utilize the count-based update approach to represent the importance of each episode we sampled. Based on such approach, we managed to optimize the policy distribution in a time complexity of $\tilde{\mathcal{O}}(\sqrt{T}/S^{2})$ and transition probability distribution in $\tilde{\mathcal{O}}(D\sqrt{SAT})$.
\section{Preliminaries}
\subsection{Notations}
The finite stochastic game(FSG)~\cite{Cui2021MinimaxSC} could be defined by a 4-tuple $M=(\mathcal{S}, \mathcal{A}, r, \theta)$. Denote the size of the state space and the action space as $S=|\mathcal{S}|$ and $A=|\mathcal{A}|$. The reward function is defined as $r: S \times A \rightarrow \mathcal{R}$. And $\theta: S \times A \times S \rightarrow[0,1]$ represents the transition probability such that $\theta\left(s^{\prime} \mid s, a\right)=\mathbbm{P}\left(s_{t+1}=s^{\prime} \mid s_{t}=s, a_{t}=a\right)$. The actual transition probability $\theta_{*}$ is randomly generated before the game start. This probability is then fixed and unknown to agent. The transition probability in epoch $k$ and time step $t$ could be defined as $\theta_{t_{k}}$. After $T$ time step, the periodical transition probability could be represented as $\hat\theta_{k}$. A stationary policy $\pi: S \rightarrow A$ is a deterministic map that maps a state to an action. Therefore, we could define the instantaneous policy under transition probability $\theta_{t_{k}}$ as $\pi_{\theta_{t_{k}}}$. The local optimal policy under sub-optimal transition probability $\hat\theta_{k}$ could be represented as $\pi_{\hat\theta_{k}}$.
And the global optimal policy is defined as $\pi_{\theta_{*}}$ (The notation of the policy will be represented as $\pi_{\theta_{t_{k}}} = \pi_{t_{k}}, \pi_{\hat\theta_{k}} = \pi_{\hat{k}}, \pi_{\theta_{t_{k}}^{*}} = \pi_{t_{k}}^{*}, \pi_{\theta_{*}^{*}} = \pi_{*}$ for the sake of brevity).
In the FSG, the average discounted reward function per time step under stationary policy $\pi$ is defined as:
\begin{equation}
\label{equation1}
J_{\pi}(\theta)=\lim_{T \rightarrow \infty} \frac{1}{T} \mathbbm{E}\left[\gamma\sum_{t=1}^{T} r\left(s_{t}, a_{t}\right)\right]
\end{equation}
$\gamma$ is the discounted factor that satisfies $0 \textless \gamma \textless 1$. Therefore, we could denote the instantaneous average reward return under transition probability $\theta_{t_{k}}$ as $J_{\pi_{t_{k}}}(\theta_{t_{k}})$. Note that the $J_{\pi_{t_{k}}}(\theta_{t_{k}})$ is a theoretical value since its value is simulated under $\theta_{t_{k}}$, $\pi_{\theta_{t_{k}}}$. After $T$ step, the optimal average reward return $J_{\pi_{*}}(\theta_{t_{k}})$ could be deduced by the local optimal policy $\pi_{\hat\theta_{k}}^{*}$. The global optimal average reward return could be represented as $J_{\pi_{*}}(\theta_{*})$.
In the online learning setting, we use total regret to measure the performance of the decision maker. Total regret is defined as the difference between the total optimal game value and the actual game value.
\begin{equation}
Reg = \max_{a}\sum_{t=1}^{T}r(a,s_{t})-\sum_{t=1}^{T}r(a_{t},s_{t})
\end{equation}
Normally, such metric could be hard to calculate. Therefore, we define the \emph{bias vector} $b(\theta,\pi,s)$ as the relative advantage of each states to help us measure the total regret.
\begin{equation}
b(\theta, \pi, s):=E\left[\sum_{t=1}^{\infty} r\left(s_{t}, a_{t}\right)-J(\theta) \mid s_{1}=s, a_{t} \sim \pi(\cdot |s_{t})]\right.
\end{equation}
Under stationary policy $\pi$, the advantage between state $s$ and $s^{\prime}$ is defined as the difference between the accumulated reward with initial state $s$ and $s^{\prime}$.Which will eventually converge to the difference of its bias vectors $b(\theta,\pi,s)-b(\theta,\pi,s^{\prime})$. The bias vector satisfies the Bellman equation. Out of brevity, we denote the the expected total reward under stationary policy $\pi$ as $r(s,\pi) = E_{a \sim \pi(\cdot|s)}[\sum r(s,a)]$. The expected transition probability is denoted as $p_{\theta}\left(s^{\prime} \mid s, \pi\right) = E_{a \sim \pi(\cdot|s)}[p_{\theta}(s^{\prime}|s,a)]$. The Bellman equation is shown in Equation \ref{equation5}.
\begin{equation}
\label{equation5}
J(\theta, \pi, s)+b(\theta, \pi, s)=r(s, \pi)+\sum_{s^{\prime}} p_{\theta}\left(s^{\prime} \mid s, \pi\right) b(\theta, \pi, s^{\prime})
\end{equation}
In order to represent the difference between each state, we define \emph{span(h)} as $sp(b) = max(b) - min(b)$. The regret is strongly connected to $sp(b(\theta_{*},\pi_{\theta_{*}^{*}},\cdot)$. And for any $b(\theta,\pi,\cdot)$, we have $sp(b(\theta,\pi,\cdot) \leq \max_{s,s^{\prime}}T_{s \rightarrow s^{\prime}}^{\pi}(\theta) = D$. This represents the span of vector $b$ is less than or equal to the maximum expected time to reach to state $s^{\prime}$ from state $s$ under transition probability $\theta$ and policy $\pi$.
\subsection{Problem Setting}
When dealing with the non-convex SGs, the global optimal policy may be hard to get. Because they sometimes stuck in local optimal results. The $\epsilon$ tolerance is then introduced to help measure the ability of the algorithm. When the difference between the optimal average return and the current average return is less than constant $\epsilon$. We could consider the current policy as the $\epsilon$-optimal policy.
\begin{assumption}
\label{assumption1}
\textbf{($\epsilon$-Optimal policy)}Under suboptimal and optimal transition probability, if policy $\pi_{t_{k}}$,$\pi_{\hat{k}}$ satisfies
$$
J_{\pi_{*}}(\theta_{t_{k}}) - J_{\pi_{t_{k}}}(\theta_{t_{k}}) \leq \epsilon
$$
$$
J_{\pi_{*}}(\theta_{*}) - J_{\pi_{\hat{k}}}(\theta_{*}) \leq \epsilon
$$
Then, policy $\pi_{t_{k}}$,$\pi_{\hat{k}}$ is $\epsilon$-optimal.
\end{assumption}
Assumption \ref{assumption2} implies that under all circumstances, all the states could be visited in average $D$ steps. When the agent conduct optimal policy under the optimal transition probability, the transition time $T_{s \rightarrow s^{\prime}}^{\pi_{*}}(\theta_{*})$ should be the shortest. Because the agent tend to explore the fewest non-related state with the optimal stationary policy. In a similar fashion, the transition time $T_{s \rightarrow s^{\prime}}^{\pi_{t_{k}}^{*}}(\theta_{t_{k}})$ for agent which conducts optimal policy under suboptimal transition probability should be less than the transition time $T_{s \rightarrow s^{\prime}}^{\pi_{t_{k}}}(\theta_{t_{k}})$ in the normal settings.
\begin{assumption}
\label{assumption2}
\textbf{(Expected transition time)}When conducting stationary policy $\pi$, assume the maximum expected time to reach to state $s^{\prime}$ from state $s$ under suboptimal transition probability and optimal transition probability is less than constant $D$:
$$
\max T_{s \rightarrow s^{\prime}}^{\pi_{*}}(\theta_{*}) \leq \max T_{s \rightarrow s^{\prime}}^{\pi_{t_{k}}^{*}}(\theta_{t_{k}}) \leq \max T_{s \rightarrow s^{\prime}}^{\pi_{t_{k}}}(\theta_{t_{k}}) \leq D
$$
\end{assumption}
Let $e(t):=k$ be the epoch where the time instant $t$ belongs. Define $\mathcal{H}_{s_{1},s_{2}}(k,\pi)$ as the set of all the time instants that the state transition $s_{1} \rightarrow s_{2}$ occurs in the first $k$ epochs when stationary policy $\pi$ was used.
\begin{equation}
\begin{aligned}
&\mathcal{H}_{\left(s_{1}, s_{2}\right)}(k, \pi) \\
&:=\sum_{t=1}^{\infty} \mathbbm{1}\left\{\pi_{e(t)}=\pi,\left(S_{t}, S_{t+1}\right)=\left(s_{1}, s_{2}\right), N(e(t)) \leq k\right\}
\end{aligned}
\end{equation}
Under transition probability $\theta_{t_{k}}$, the expected transition time from state $s$ to state $s^{\prime}$ with stationary policy $\pi_{t_{k}}$ could be denoted as $\tilde\tau_{\pi_{t_{k}}}$, which satisfies $\tilde\tau_{\pi_{t_{k}}} = T_{s \rightarrow s}^{\pi_{t_{k}}}(\theta_{t_{k}})$. Therefore, the posterior probability of the stationary policy $\pi$ could be represented as the difference between the empirical state pair frequency $\frac{\mathcal{H}_{\left(s_{1}, s_{2}\right)}\left(k, \pi\right)}{k}$ and the corresponding expected value $\tilde\tau_{\pi_{t_{k}}}$.
\begin{assumption}
\label{assumption3}
\textbf{(Posterior distribution under suboptimal trajectories)}For any given scalars $e_1,e_2 \geq 0$, there exists $p \equiv p(e_1,e_2) \textgreater 0$ satisfies $\theta_{t_{k}}(\pi_{t_{k}}^{*}) \geq p$ for any epoch index $k$ at which suboptimal transition frequencies have been observed:
$$
\begin{aligned}
&\left|\frac{\mathcal{H}_{\left(s_{1}, s_{2}\right)}\left(k, \pi\right)}{k} - \tilde{\tau}_{\pi_{t_{k}}}\theta \left(s_{1}|s_{2}\right)\right| \leq \sqrt{\frac{e_{1} \log \left(e_{2} \log k\right)}{k}} \\
&\forall s_{1}, s_{2} \in \mathcal{S}, k \geq 1, c \in \mathcal{C}, k=\sum_{\pi \in \Pi} k
\end{aligned}
$$
\end{assumption}
Under finite discounted Markov decision processes, the average discounted return is also finite. So, we define the maximum average discounted reward as $\Gamma$. Which is the maximum reward that an agent could achieve during its exploration in the finite discounted Markov decision processes. The maximum value will be achieved under optimal transition probability with optimal stationary policy.
\begin{assumption}
\label{assumption4}
\textbf{(Upper bound for the average discounted reward)}Under the finite discounted MDP, the maximum average discounted reward is bounded by a constant.
$$
J_{\pi_{*}}(\theta_{*}) \leq \Gamma
$$
\end{assumption}
Based on the upper assumptions, we could then construct our method.
\section{Method}
In this section, we propose the Double Thompson Sampling method. One of the essential parameters under Thompson Sampling setting is the prior distribution. Which is denoted as $\mu_{0}$ in our paper. Note that we generate prior distribution for both transition probability and stationary policy. In each epoch $k$, at each time step $t$, the posterior distribution $\mu_{t_{k}}$ will be updated based on the previous history $h_{t_{k}}$. Let $N_{t}(s,a)$ be the number of visits to any state-action pair $(s,a)$ during a period of time $t$.
\begin{equation}
N_{t}(s, a)=\left|\left\{\tau<t:\left(s_{\tau}, a_{\tau}\right)=(s, a)\right\}\right|
\end{equation}
Therefore, the algorithm could be generated as follows.
\begin{algorithm}[h]
\caption{Double Thompson Sampling}
\label{alg1}
\textbf{Input}: Game Environment, Prior Distribution for transition probability $\mu_{\theta_{0}}$, Prior Distribution for stationary policy $\mu_{\pi_{0}}$, Transition Probability $\theta_{0}$, Initial State $s_{0} \in S$\\
\textbf{Output}: Stationary Policy $\pi_{K}$
\begin{algorithmic}
\FOR{Episode $k = 0,1,2 \dots K$}
\STATE $T_{k-1} \leftarrow t-t_{k}$
\STATE $t_{k} \leftarrow t$
\STATE Generate $\mu_{k}(\hat\theta_{k})$, $\mu_{k}(\hat \pi_{k})$ based on prior distribution
\FOR {$t \leq t_{k}+T_{k-1}$ and $N_{t}(s,a) \leq 2N_{t_{k}}(s,a)$}
\STATE Apply action $a_{t} \sim \pi_{t_{k}}, \pi_{t_{k}} \sim \mu_{t_{k}}(\pi)$
\STATE Observe new state $s_{t+1}$, reward $r_{t+1}$
\STATE Update posterior distribution $\mu_{t+1_{k}}(\pi), \mu_{t+1_{k}}(\theta)$ using EVI
\STATE $t \leftarrow t+1$
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
The Double Thompson Sampling method(Alg \ref{alg1}) is conducted in multiple steps. At the beginning of each epoch $k$, the algorithm estimates the periodical transition probability using the past history $\mu_{k-1}(\hat\theta_{k-1})$(Step 1). This prior distribution satisfies $\mu_{k-1}(\hat\theta_{k-1}) = \mu_{(k-1)_{T}}(\theta_{(k-1)_{T}})$. We set two stopping criterion for our algorithm in order to limit our agent's exploration direction. The first stopping criterion aims to stop meaningless exploration. The second stopping criterion ensures that any state-action pair $(s,a)$ will not be encounter twice during the same epoch. During each epoch $k$, actions are generated from the instantaneous policy $\pi_{t_{k}}$(Step 3). This policy follows a posterior distribution $\mu_{t_{k}}(\pi)$. These actions are then be used by the agent to interact with the environment to observe the next state $s_{t+1}$ and the reward $r_{t+1}$(Step 4). The observation results are then be used to find the optimal posterior distribution for policy $\pi_{t+1_{k}}$ and transition probability $\theta_{t+1_{k}}$(Step 5). If the stopping criterions are not met, the algorithm will start over from Step 2. The whole process will be repeated until the terms of the stopping criterions are met.
\subsection{Update Rule}
In the model-based method, the update method of the transition probability is of great importance. Our method is a Thompson Sampling-based method. The transition probability will be updated based on the prior distribution. Based on the Bayes' rule, the posterior distribution of the transition probability could be represented as :
\begin{equation}
\mu_{t+1}(\theta)=\frac{\theta\left(s_{t+1} \mid s_{t}, a_{t}\right) \mu_{t_{k}}(\theta)}{\sum_{\theta^{\prime} \in \Theta} \theta^{\prime}\left(s_{t+1} \mid s_{t}, a_{t}\right) \mu_{t}\left(\theta^{\prime}\right)}
\end{equation}
The update method of the stationary policy is different from the one of transition probability. In this paper, we introduce the prior policy to guide the current policy. Using the Thompson sampling and the Policy Iteration method(EVI), the algorithm is able to balance between the current optimal action and the history optimal action. This will help our method achieve long-term maximum return. Which is the global optimal value in this scenario. Let $W_{t_{k}}$ be the \emph{posterior weight} in epoch $k$ at time $t$. $J_{\pi_{t}}(\theta)$ and $J_{\pi^{*}}(\theta)$ denotes the instantaneous average discounted return and the local optimal value.
\begin{equation}
W_{t_{k}}(\pi) = \exp \sum_{\pi,s}\mathcal{H}(N_{\pi}(k),\pi)\log \frac{J_{\pi_{t}}(\theta)}{J_{\pi^{*}}(s,\theta)}
\end{equation}
Its value is proportional to the log difference between the average return of the local optimal policy and current policy. Using the posterior factor, we could generate the Policy Iteration method based not only on the current observation but also the historical trajectory.
\begin{algorithm}[h]
\caption{Policy Iteration with Posterior factor}
\label{alg2}
\textbf{Input}: Game Environment, Prior Distribution for stationary policy $\mu_{t}(\pi)$, $0 \textless \gamma \textless 1$\\
\textbf{Output}: Stationary Policy $\pi_{i}$
\begin{algorithmic}
\REPEAT
\STATE $\mu_{t}(\pi) = W_{t} \mu_{t-1}(\pi) + (1-W_{t})\pi_{t}^{*}(s,\theta_{t_{k}})$
\UNTIL {$D_{\theta}(\mu_{*}(\pi)||\mu_{t_{k}}(\pi)) \leq \epsilon$}
\end{algorithmic}
\end{algorithm}
The posterior distribution $\mu(\pi)$ is defined as the transition matrix under time step $t$. Satisfying $\mu_{t}(\pi) = \left(\mathcal{H}_{s_1,s_2}(t,\pi)\right)_{s_1,s_2 \in S}$. In this paper, we could denote the distance between the history optimal policy and the instantaneous policy using the \emph{Marginal Kullback-Leibler Divergence}(Marginal KL Divergence). Marginal KL Divergence is a widely used metric when measuring the difference between two probability distribution. Therefore, the distance could be represented as $D_{\theta}(\mu_{*}(\pi)||\mu_{t_{k}}(\pi))$.
\begin{equation}
\begin{aligned}
D_{\theta}(\mu_{*}(\pi)||\mu_{t_{k}}(\pi)) &:= \sum_{s_{1} \in \mathcal{S}} \theta_{s_{1}}^{\pi} \sum_{s_{2} \in \mathcal{S}} \mu_{*}(\pi) \log \frac{\mu_{*}(\pi)}{\mu_{t_{k}}(\pi)} \\
&=\sum_{s_{1} \in \mathcal{S}} \theta_{s_{1}}^{\pi} \mathbbm{K} \mathbbm{L}\left(\mu_{*}(\pi) \| \mu_{t_{k}}(\pi)\right)
\end{aligned}
\end{equation}
The marginal KL divergence $D_{\pi}(\mu_{*}(\theta)||\mu_{t_{k}}(\theta))$ is a convex combination between the history optimal policy and the instantaneous policy. Parameter $\epsilon$ represents the tolerance between the optimal policy and the instantaneous policy. This posterior policy iteration(PPI) method updates the policy dynamically with the posterior factor. The policy will converge to optimal value after certain amount of iterations under this update method. In the following section, we will be introducing the proof of the astringency of this posterior update method.
\section{Main Results}
\subsection{Astringency of the Update Rule}
\label{section4.1}
In the online learning domain, one of the basic metric of an algorithm is whether it could converge after constant number of steps. So in this section, we provide the proof of the astringency of our posterior update method in order to illustrate the superiority of our method.
The following three Lemmas are meant to prove the convergence of our algorithm. In Lemma \ref{lemma1}, We first prove that for stochastic games $M$, the PPI method converges asymptotically. Then, in Lemma \ref{lemma0}, we demonstrate that the output policy of such policy iteration method updates monotonically towards optimal direction. Which is a vital evidence for the global optimality of our update method. At last, the third lemma(Lemma \ref{lemma10}) proves that under stochastic games $M$, the output policy generated from PPI method would reach $\epsilon$-optimal after constant number of iterations.
\begin{lemma}
\label{lemma1}
Suppose Assumption \ref{assumption2} holds for some stochastic games $M$, then the policy iteration algorithm on $M$ converges asymptotically.
\begin{proof}
If the Assumption \ref{assumption2} holds. From Theorem 4 in ~\cite{72bf2cf9895a47e7be9a668a25215110}, the policy iteration converges.
\end{proof}
\end{lemma}
\begin{lemma}
\label{lemma0}
Under update algorithm PI, the average discounted return should be monotonically increased.
\begin{proof}
From Algorithm \ref{alg2}, we could deduce the update rule of the average discounted return:
\begin{equation}
\begin{aligned}
J_{\pi_{t}}(\theta) - J_{\pi_{t-1}}(\theta) &= (W_{t}-1)J_{t-1}(\theta) + (1-W_{t})J_{\pi^{*}}(s,\theta)\\
&= (1-W_{t})(J_{\pi^{*}}(s,\theta)-J_{\pi_{t-1}}(\theta))
\end{aligned}
\end{equation}
When $J_{\pi^{*}}(s,\theta) \geq J_{\pi_{t-1}}(\theta)$, we could deduce that $\log\frac{J_{\pi_{t}}(\theta)}{J_{\pi^{*}}(s,\theta)} \leq 1$. So the posterior weight $W_{t}$ is less than 1. This result holds vice versa. The first term $1-W_{t} \leq 0$ when $J_{\pi^{*}}(s,\theta) \leq J_{\pi_{t-1}}(\theta)$. Therefore, we could prove that:
\begin{equation}
J_{\pi_{t}}(\theta) - J_{\pi_{t-1}}(\theta) = (1-W_{t})(J_{\pi^{*}}(s,\theta)-J_{\pi_{t-1}}(\theta)) \geq 0
\end{equation}
The sequence $J_{\pi_{t}}(\theta)$ is monotonically increased with time step $t$.
\end{proof}
\end{lemma}
\begin{lemma}
\label{lemma10}
Suppose Assumption \ref{assumption1} and Assumption \ref{assumption2} hold for some stochastic games $M$. Let $v_{i}$ be the state value in iteration $i$. Define $N$ as the maximum iteration number of the algorithm. Then $\pi_{t_{k}}$ is $\epsilon$-optimal after $N$ iterations.
\begin{proof}
Define $D = \min_{s}\{\mu_{i+1}(\pi)-\mu_{\pi}\}$ and $U = \max_{s}\{\mu_{i+1}(\pi)-\mu_{i}(\pi)\}$. Then we could deduce:
\begin{equation}
\begin{aligned}
D + \mu_{N}(\pi) &\leq \mu_{N+1}\\
&\leq W_{i}\mu_{N} + (1-W_{i})\pi_{i}^{*}(s,\theta)\\
&\leq W_{i}\mu_{N} + (1-W_{i})(r_{N} + \theta_{} v_{N})
\end{aligned}
\end{equation}
Since $0 \textless W_{i} \textless 1$, the upper equation could be turned to:
\begin{equation}
D \leq (1-W_{i})J_{\pi_{i}}(\theta)
\end{equation}
Let $\pi^{*}$ be the optimal policy under all states that satisfies $\pi^{*} := \sum_{s \in S}\pi_{i}^{*}(s,\theta)$. Then
\begin{equation}
D \leq (1-W_{i})J_{\pi_{i}}(\theta) \leq (1-W_{i})J_{\pi^{*}}(\theta)
\end{equation}
In a similar way, we could also prove $U \geq (1-W_{i})J_{\pi^{*}}(\theta)$. From the definition of the stopping criterion of the Policy Iteration algorithm, we could assume $U - D \leq (1-W_{i})\epsilon$. Therefore, we have
\begin{equation}
\begin{aligned}
U &\leq D + (1-W_{i})\gamma \\
U &\leq (1-W_{i})(J_{\pi_{i}}(\theta)+\epsilon)\\
(1-W_{i})J_{\pi^{*}} &\leq (1-W_{i})(J_{\pi_{i}}(\theta)+\epsilon)\\
J_{\pi^{*}} &\leq J_{\pi_{i}}(\theta)+\epsilon
\end{aligned}
\end{equation}
We could deduce that stationary policy $\pi$ is $\epsilon$-optimal after $N$ iterations.
\end{proof}
\end{lemma}
\subsection{Regret Bound Analysis}
After proving the astringency of the PPI method. We then move the proof of the regret bound. Which is the most popular metric for online learning method. Inside each episode, the regret could be separated into three parts. We could know the regret in time step $T$ would be represented as:
\begin{equation}
\begin{aligned}
Reg_{T} &= TJ_{\pi_{\hat{k}}}(\hat \theta) - \sum_{t = 1}^{T}r_{\pi_{t}}(s_{t},a_{t})\\
&= Reg_{T}^{1} + Reg_{T}^{2} + Reg_{T}^{3}
\end{aligned}
\end{equation}
We could define the following regret as:
\begin{equation}
\label{equation17}
\begin{array}{l}
Reg_{T}^{1} = TJ_{\pi_{\hat{k}}}(\hat \theta) - \sum_{t = 1}^{T}J_{\pi_{t}}(\hat \theta) \\ \\
Reg_{T}^{2}=\sum_{t = 1}^{T}J_{\pi_{t}}(\hat \theta) - \sum_{t = 1}^{T}J_{\pi_{t}}(\theta_{t}) \\ \\
Reg_{T}^{3}= \sum_{t = 1}^{T}J_{\pi_{t}}(\theta_{t}) - \sum_{t = 1}^{T}r_{\pi_{t}}(s_{t},a_{t})
\end{array}
\end{equation}
Where $J_{\pi_{\hat{k}}}(\hat{\theta})$ is the terminal average reward under terminal policy $\pi_{\hat{k}}$ and transition probability $\hat{\theta}$. Note that this value is a virtual value and only exists in theoretical analysis. $Reg_{T}^{1}$ represents the posterior difference between the total episodic reward and the total virtual instantaneous reward. We could assume such difference is calculated under the same transition probability since the transition probability is generated from the same priors. Since for any measurable function f and any $h_{t_{k}}$-measurable random variable $X$,$\mathbb{E}\left[f\left(\theta_{*}, X\right) \mid h_{t_{k}}\right]=\mathbb{E}\left[f\left(\theta_{k}, X\right) \mid h_{t_{k}}\right]$. This has been proved in previous studies.~\cite{osband2013more}
In order to bound the terminal regret $Reg_{T}^{1}$, we first bound the ratio between the expected optimal average discounted reward and the instantaneous discounted reward. Based on Assumption \ref{assumption1} and Assumption \ref{assumption4}, the expected optimal reward that an agent could achieve in the finite discounted MDP could be bounded by parameter $\Gamma$ and $\epsilon$.
\begin{lemma}
$$
\log \frac{J_{\pi_{*}}(\theta)}{J_{\pi_{t}}(\theta)} \leq \frac{\epsilon}{\Gamma}
$$
\begin{proof}
First, we could multiply $J_{\pi_{t}}(\theta)$ in order to construct the inequality. Let $J_{\pi_{t}}(\theta) = n$, $\epsilon = x$
\begin{equation}
\begin{aligned}
\lim _{n \rightarrow+\infty}\left(1+\frac{x}{n}\right)^{n} &= \lim _{n \rightarrow+\infty} e^{n \ln \left(1+\frac{x}{n}\right)} \\
&=e^{\lim _{n \rightarrow+\infty} \frac{\ln \left(1+\frac{x}{n}\right)}{\frac{1}{n}}}
\end{aligned}
\end{equation}
Apply the L'Hopital's Rule:
\begin{equation}
\begin{aligned}
\lim _{n \rightarrow+\infty}\left(1+\frac{x}{n}\right)^{n}&= e^{\lim _{n \rightarrow+\infty} \frac{\left(\frac{-x}{n^{2}}\right) \frac{1}{1+\frac{x}{n}}}{-\frac{1}{n^{2}}}}\\
&= e^{\lim _{n \rightarrow+\infty} \frac{x}{1+\frac{x}{n}}}=e^{x}
\end{aligned}
\end{equation}
Then, we could prove that $\left(1+\frac{x}{n}\right)^{n}$ is monotonically increased with $n$:
\begin{equation}
\begin{aligned}
(1+\frac{x}{n})^{2} &= 1 \cdot\underbrace{ \left(1+\frac{x}{n}\right) \cdot\left(1+\frac{x}{n}\right) \cdots \cdots\left(1+\frac{x}{n}\right)}_{n}\\
& \leq \left[\frac{1+(1+\frac{x}{n})+ \cdots +(1+\frac{x}{n})}{n+1}\right]^{n+1}\\
& = \left[\frac{1+n(1+\frac{x}{n})}{n+1}\right]^{n+1}\\
& = \left[1+\frac{x}{n(n+1)}\right]^{n+1}\\
& \leq \left[1+\frac{x}{n+1}\right]^{n+1}
\end{aligned}
\end{equation}
The first inequality holds for the arithmetic mean equality. We could deduce that $(1+\frac{x}{n})^{n} \leq e^{x}$. Therefore, we have:
\begin{equation}
J_{\pi_{t}}(\theta) \log \frac{J_{\pi_{*}}(\theta)}{J_{\pi_{t}}(\theta)} \leq \epsilon
\end{equation}
Based on Assumption \ref{assumption4}, we could deduce the upper bound of average discounted reward. Then the lemma could be proved.
\end{proof}
\end{lemma}
After bounding the log ratio between the expected optimal average reward and the instantaneous reward. We could then move to the bound of the instantaneous posterior weight $W_{t_{k}}$. Which is a crucial factor in the following proving process. At each time step, the posterior weight will be updated based on the previous policy and the observed experiment process. First we define the counter function $N_{\pi}(t):=\sum_{t=0}^{t-1} \sum_{\pi \in \Pi} \mathbbm{1}\left\{\pi_{e(t)}=\pi\right\}$ as the total number of the time instants during the period of $t$ when policy $\pi$ was conducted. When Assumption \ref{assumption3} holds, we could bound the posterior weight based on the count function in $k$ epoch and the average transition time $\tilde \tau$.
\begin{lemma}
\label{lemma3}
Under Assumption \ref{assumption3}, for each stationary near-optimal policy $\pi$ and epoch counter $k \geq 1$. The following upper bound holds for negative log-density.
$$
-\log W_{t_{k}}(\pi) \leq \frac{\epsilon}{\Gamma}|S|^{2}( \rho(k_{\pi}) \sqrt{k_{\pi}} + k_{\pi} \tilde{\tau}_{t_{k},k_{\pi}})
$$
\begin{proof}
When $W_{t_{k}} \leq 1$, we could have:
\begin{equation}
W_{t_{k}}(\theta):=\exp \sum_{\pi, s_{1}, s_{2}} \mathcal{H}\left(N_{\pi}(k), \pi\right) \log \frac{J_{\pi_{t}}(\theta)}{J_{\pi_{*}}(\theta)}
\end{equation}
Based on the definition of the counter $\mathcal{H}$, we could deduce the value of the posterior weight in a single epoch:
\begin{equation}
\begin{aligned}
&W_{t_{k}}(\theta)\\
&= \exp \left(\sum_{t=1}^{\infty} \mathbbm{1}\left\{\pi_{e(t)}=\pi,\left(S_{t}, S_{t+1}\right)=\left(s_{1}, s_{2}\right)\right\}\log \frac{J_{\pi_{t}}(\theta)}{J_{\pi_{*}}(\theta)}\right)\\
& = \exp \left(\sum_{\pi \in \Pi} \sum_{\left(s_{1}, s_{2}\right) \in \mathcal{S}^{2}} \sum_{t=1}^{T} \mathbbm{1}\left\{\pi_{e(i)}=\pi,\left(S_{t}, S_{t+1}\right)=\left(s_{1}, s_{2}\right)\right\}\right.\\
&\left.\quad \log \frac{J_{\pi_{t}}(\theta)}{J_{\pi_{*}}(\theta)}\right)\\
& = \exp \left(N_{\pi}(t) \sum_{\left(s_{1}, s_{2}\right) \in \mathcal{S}^{2}} \sum_{t=0}^{t-1} \frac{\mathbbm{1}\left\{\pi_{e(t)}=\pi,\left(S_{t}, S_{t+1}\right)=\left(s_{1}, s_{2}\right)\right\}}{N_{\pi}(t)}\right.\\
&\left.\quad \log \frac{J_{\pi_{t}}(\theta)}{J_{\pi_{*}}(\theta)}\right)\\
\end{aligned}
\end{equation}
Where $N_{\pi}(t):=\sum_{t=0}^{t-1}\sum_{\pi \in \Pi} \mathbbm{1}\left\{\pi_{e(t)}=\pi\right\}$ represents the total number of the time instants during the period of $t$ when policy $\pi$ was conducted.
When Assumption \ref{assumption3} holds, we could know that $N_{\pi}(t) = \tilde{\tau}_{\pi_{t_{k}},N_{\pi}(k)}$, where $N_{\pi}(k) :=\sum_{k=0}^{K}\sum_{\pi \in \Pi} \mathbbm{1}\left\{\pi_{e(k)}=\pi\right\} $ holds for the number of the epochs where policy $\pi$ was chosen(The notation of $\tau$ will be represented as $N_{\pi}(k) = k_{\pi}$, $\tilde{\tau}_{\pi_{t_{k}},N_{\pi}(k)} = \tilde{\tau}_{t_{k},k_{\pi}}$). Therefore, we could have:
\begin{equation}
\begin{aligned}
& -\log W_{t_{k}}(\pi)\\
& = -N_{\pi}(t) \sum_{\left(s_{1}, s_{2}\right) \in \mathcal{S}^{2}} \sum_{t=0}^{t-1} \frac{\mathbbm{1}\left\{\pi_{e(t)}=\pi,\left(S_{t}, S_{t+1}\right)=\left(s_{1}, s_{2}\right)\right\}}{N_{\pi}(t)}\log \frac{J_{\pi_{t}}(\theta)}{J_{\pi_{*}}(\theta)}\\
& = -\sum_{\left(s_{1}, s_{2}\right) \in \mathcal{S}^{2}} \tilde{\tau}_{t_{k},k_{\pi}} \mathcal{H}_{\left(s_{1}, s_{2}\right)}\left(\tilde{\tau}_{t_{k},k_{\pi}}, \pi\right) \log \frac{J_{\pi_{t}}(\theta)}{J_{\pi_{*}}(\theta)}\\
& = \sum_{\left(s_{1}, s_{2}\right) \in \mathcal{S}^{2}} \left[\tilde{\tau}_{t_{k},k_{\pi}} \mathcal{H}_{\left(s_{1}, s_{2}\right)}\left(\tilde{\tau}_{t_{k},k_{\pi}}, \pi\right) - k_{\pi}\tilde{\tau}_{t_{k},k_{\pi}}\theta_{\pi}(s_{1}|s_{2})\right]\log \frac{J_{\pi_{*}}(\theta)}{J_{\pi_{t}}(\theta)}\\
& + \sum_{\left(s_{1}, s_{2}\right) \in \mathcal{S}^{2}} k_{\pi}\tilde{\tau}_{t_{k},k_{\pi}}\theta(s_{1}|s_{2}) \log \frac{J_{\pi_{*}}(\theta)}{J_{\pi_{t}}(\theta)}\\
\end{aligned}
\end{equation}
The last equation is based on the logarithmic property $\log \frac{A}{B} = - \log \frac{B}{A}$. Based on the Assumption \ref{assumption3}, define $\rho(x) := O(\sqrt{\log \log(x)})$.
\begin{equation}
\begin{aligned}
- \log W_{t_{k}}(\pi)& \leq \sum_{\left(s_{1}, s_{2}\right) \in \mathcal{S}^{2}} \rho(k_{\pi}) \sqrt{k_{\pi}} \log \frac{J_{\pi_{*}}(\theta)}{J_{\pi_{t}}(\theta)}\\
&+ k_{\pi}\tilde{\tau}_{t_{k},k_{\pi}} \sum_{\left(s_{1}, s_{2}\right) \in \mathcal{S}^{2}}\theta(s_{1}|s_{2}) \log \frac{J_{\pi_{*}}(\theta)}{J_{\pi_{t}}(\theta)}\\
& \leq \frac{\epsilon}{\Gamma}|S|^{2}( \rho(k_{\pi}) \sqrt{k_{\pi}} + k_{\pi} \tilde{\tau}_{t_{k},k_{\pi}})
\end{aligned}
\end{equation}
\end{proof}
\end{lemma}
A proper optimization method should lead to promised margin between the expected discounted reward and the real reward. In order to achieve such results, numerous amount of iteration will be conducted. Therefore, from the astringency proof we proposed in section \ref{section4.1}, we could deduce the bound of the expected convergence time during the optimization process. In Lemma \ref{lemma3.3}, we give the bound the instantaneous difference between the real reward and the expected reward with $\sqrt{T}$. This bound is inversely proportional to $\sqrt{T}$ since our update method updates towards optimal direction(Lemma \ref{lemma0}). For the sake of brevity, the full proof will be shown in Appendix \ref{lemma3.3replacement}.
\begin{lemma}
\label{lemma3.3}
The difference between the local optimal average reward and the instantaneous average reward could be bounded by:
$$
|J_{\pi_{t}}-J^{*}| \leq \tilde{\mathcal{O}}(\frac{1}{\sqrt{T}})
$$
\begin{proof}
We could know that the current policy probability distribution is updated based on the previous distribution and the current optimal policy distribution:
\begin{equation}
\mu_{t}(\pi) = W_{t}\mu_{t}(\pi) + (1-W_{t})\pi_{t}^{*}(s,\theta_{t_{k}}
\end{equation}
We could extend this result to reward function:
\begin{equation}
\label{equation500}
\begin{aligned}
J_{\pi_{t}}&=W_{t} J_{\pi_{t-1}}+\left(1-W_{t}\right) J^{*}\left(\theta_{t}\right)\\
J_{\pi_{t}}^{2} &=W_{t}^{2} J_{\pi{t-1}}^{2}+\left(1-W_{t}\right)^{2} J^{*^{2}}+2 W_{t}\left(1-W_{t}\right) J^{*} J_{\pi_{t-1}} \\
& \leq W_{t}^{2} J_{\pi t}^{2}+\left(1-W_{t}\right)^{2} J^{* 2}+2 W_{t}\left(1-W_{t}\right) J^{*} J_{\pi_{t}}
\end{aligned}
\end{equation}
The inequality is based on the monotonicity of the algorithm. We could simplify Equation \ref{equation500}:
\begin{equation}
\begin{aligned}
\left(1-W_{t}^{2}\right) J_{\pi_{t}}^{2} &\leq \left(1-W_{t}\right)^{2} J^{*^{2}}+2 W_{t}\left(1-W_{t}\right) J^{*} J_{\pi_{t}} \\
\left(1+W_{t}\right) J_{\pi t}^{2} &\leq \left(1-W_{t}\right) J^{*^{2}}+2 W_{t} J^{*} J_{\pi_{t}} \\
J_{\pi_{t}}^{2}+W_{t} J_{\pi_{t}}^{2} &\leq J^{*^{2}}-W_{t} J^{*^{2}}+2 W_{t} J^{*} J_{\pi_{t}}\\
W_{t}\left(J_{\pi_{t}}^{2}+J^{*^{2}}\right) &\leq J^{*^{2}}-J_{\pi_{t}}^{2}+2 W_{t} J^{*} J_{\pi_{t}}\\
J_{\pi_{t}}^{2}+J^{*^{2}} &\leq \frac{1}{W_{t}}\left(J^{*^{2}}-J_{\pi_{t}}^{2}\right)+2 J^{*} J_{\pi_{t}}
\end{aligned}
\end{equation}
Based on the definition of the regret of each time step, we could deduce the bound of the instantaneous regret:
\begin{equation}
\begin{aligned}
\left(J_{\pi t}-J^{*}\right)^{2} &=J_{\pi_{t}}^{2}+J^{*^{2}}-2 J_{\pi t} J^{*} \\
& \leq \frac{1}{w_{t}}\left(J^{*^{2}}-J_{\pi_{t}}^{2}\right)+2 J^{*} J_{\pi_{t}}-2 J_{\pi_{t}} J^{*} \\
&=\frac{1}{w_{t}}\left(J^{*^{2}}-J_{\pi_{t}}^{2}\right) \\
&=\frac{1}{w_{t}}\left(J^{*}-J_{\pi_{t}}\right)\left(J^{*}+J_{\pi_{t}}\right)
\end{aligned}
\end{equation}
\begin{equation}
|J_{\pi_{t}}-J^{*}| \leq \frac{1}{W_{t}}|J_{\pi_{t}}+J^{*}|
\end{equation}
From Lemma \ref{lemma3}, we could know that $-\log W_{t_{k}}(\pi)$ is bounded by $B$, with $B = \frac{\epsilon}{\Gamma}|S|^{2}(\rho(k_{\pi}) \sqrt{k_{\pi}} + k_{\pi} \tilde{\tau}_{t_{k},k_{\pi}})$. Therefore, we could construct the following inequalities.
\begin{equation}
\begin{aligned}
W_{t_{k}}-1 &\geq \log W_{t_{k}} \geq -B\\
\frac{1}{W_{t_{k}}} &\leq \frac{1}{1-B}
\end{aligned}
\end{equation}
Factor $B$ is proportional to parameter $k_{\pi}$ which could be bounded by the total number of episode of under total time $T$. Therefore, we could bound $\frac{1}{W_{t}}$ by $T$(Ignoring the constants):
\begin{equation}
\begin{aligned}
\frac{1}{W_{t}} &\leq \frac{1}{1-\frac{\epsilon}{\Gamma}|S|^{2}(\rho(k_{\pi}) \sqrt{k_{\pi}} + k_{\pi} \tilde{\tau}_{t_{k},k_{\pi}})}\\
&\leq \frac{1}{1-\sqrt{\sqrt{T}}-\sqrt{T}}
\end{aligned}
\end{equation}
Based on Assumption \ref{assumption4}, the average discounted reward function is bounded by $\Gamma$. So the difference between the local optimal average reward and the instantaneous average reward could be bounded by:
\begin{equation}
\begin{aligned}
|J_{\pi_{t}}-J^{*}| &\leq \frac{1}{W_{t}}|J^{*}+J_{\pi_{t}|}\\
&\leq \frac{2}{1-\frac{\epsilon}{\Gamma}|S|^{2\Gamma}(\rho(k_{\pi}) \sqrt{k_{\pi}} + k_{\pi} \tilde{\tau}_{t_{k},k_{\pi}})}\\
&\leq \tilde{\mathcal{O}} (\frac{2\Gamma^{2}}{S^{2}\sqrt{T}})
\end{aligned}
\end{equation}
\end{proof}
\end{lemma}
Therefore, we could combine the previous Lemmas together to get the final regret bound of $Reg_{T}^{1}$.
\begin{theorem}
\label{theorem4}
The first part of the regret in time step $T$ is bounded by:
$$
Reg_{T}^{1} \leq \tilde{\mathcal{O}}(\frac{\sqrt{T}}{S^{2}})
$$
\begin{proof}
From the definition before, we could know that $Reg_{T}^{1}$ could be represented as:
\begin{equation}
Reg_{T}^{1} = TJ_{\pi_{\hat{k}}}(\hat \theta) - \sum_{t = 1}^{T}J_{\pi_{t}}(\hat \theta)
\end{equation}
Since this theorem won't involve the transformation of the transition probability. So let $J_{\pi}(\theta) = J_{\pi}$. Based on the update rule of the posterior distribution $\mu_{t+1}(\pi)$ of policy $\pi$. We could divide the average discounted return into several parts:\\
At time step $t = T$, we could assume the instantaneous regret equals to zero:
\begin{equation}
Reg_{t_{T}}^{1} = J_{\pi_{\hat k}}- J_{\pi_{T}} = 0
\end{equation}
At time step $t = T-1$, define the local optimal average discounted return as $J_{\pi}^{*}$. Note that this local optimal value is virtual. The instantaneous regret could be represented as:
\begin{equation}
\begin{aligned}
Reg_{t_{T-1}}^{1} &= J_{\pi_{\hat k}} - J_{\pi_{T-1}}\\
&= W_{t-1}J_{\pi_{T-1}} + (1-W_{t-1})J_{\pi}^{*} - J_{\pi_{T-1}}\\
&= (W_{t-1}-1)J_{\pi_{T-1}} + (1-W_{t-1})J_{\pi}^{*}\\
&= (1-W_{t-1})(J_{\pi}^{*} - J_{\pi_{T-1}})
\end{aligned}
\end{equation}
In a similar fashion, at time step $t = T-2$, the instantaneous regret could be represented as:
\begin{equation}
\begin{aligned}
Reg_{t_{T-2}}^{1} &= J_{\pi_{\hat k}} - J_{\pi_{T-2}}\\
&= J_{\pi_{\hat k}} - J_{\pi_{T-1}} + J_{\pi_{T-1}} - J_{\pi_{T-2}}\\
&= (1-W_{t-1})(J_{\pi}^{*} - J_{\pi_{T-1}}) + (1-W_{t-2})(J_{\pi}^{*} - J_{\pi_{T-1}})
\end{aligned}
\end{equation}
Based on Lemma \ref{lemma3.3}, the difference between the local optimal value and the current average return could be bounded by:
\begin{equation}
|J_{\pi}^{*} - J_{\pi_{t}}| \leq \tilde{\mathcal{O}}(\frac{1}{\sqrt{T}})
\end{equation}
The sub-optimal models are sampled when their posterior probability is larger than $\frac{1}{T}$. This ensures the time complexity of the Thompson sampling process is no more than $O(1)$. So we could deduce the total regret in time step $T$.
\begin{equation}
\begin{aligned}
Reg_{T}^{1} &= \frac{1}{T} (Reg_{t_{T-1}}^{1} + Reg_{t_{T-2}}^{1} + \cdots + Reg_{t_{1}}^{1})\\
&\leq \tilde{\mathcal{O}}(\frac{2\Gamma^{2}}{S^{2}\sqrt{T}})(\frac{T-1}{T} + \frac{T-2}{T} + \cdots \frac{1}{T})\\
&\leq \tilde{\mathcal{O}}(\frac{\Gamma^{2}\sqrt{T}}{S^{2}})
\end{aligned}
\end{equation}
\end{proof}
\end{theorem}
After giving the first part of the total regret bound, we then move to the proof of the second and third part. Based on the definition in Equation \ref{equation17}, the second regret bound is mainly related to the difference between transition probability. So we could denote the average discounted return as $J_{\pi_{t}}(\hat \theta) = J(\hat \theta)$, $J_{\pi_{t}}(\theta_{t}) = J(\theta_{t})$. Note that the $\sum_{t=1}^{T}J(\hat \theta)$ is a virtual value which represents the average discounted return in time step $T$ with transition probability $\hat \theta$. From the previous definition of the Bellman iterator of the average discounted return, we could deduct the bound of $Reg_{T}^{2}$(The full proof will be shown in the appendix for the sake of brevity):
\begin{theorem}
\label{theorem5}
The second part of the regret in time step $T$ is bounded by:
$$
Reg_{T}^{2} \leq \tilde{\mathcal{O}}( D\sqrt{SAT} )
$$
\end{theorem}
Finally, for the last part of the regret bound. $Reg_{T}^{3}$ is calculated by the difference between the instantaneous virtual average reward and the real reward. In Theorem \ref {theorem6}, we decompose the regret into two parts $Y_{t}^{1}$ and $Y_{t}^{2}$. We then use the Azuma-Hoeffding's inequality to bound $Y_{t}^{1}$ and $Y_{t}^{2}$ respectively. Therefore, we could get the regret bound of $Reg_{T}^{3}$(The full proof will be shown in the appendix):
\begin{theorem}
\label{theorem6}
The third part of the regret in time step $T$ is bounded by:
$$
Reg_{T}^{3} \leq \tilde{\mathcal{O}}(D\sqrt{ST})
$$
\end{theorem}
\section{Conclusion}
In this paper, we propose a policy-based posterior optimization method that achieves the best total regret bound $\tilde{\mathcal{O}}(\Gamma^{2}\sqrt{T}/S^{2})$ in finite-horizon stochastic game. This algorithm provides a new vision on the trade-off problem between exploration and exploitation by solving a posterior update problem. The posterior update problem could be solved by balancing between long-term policy and current greedy policy. Our research results shows that this posterior sampling method outperforms other optimization algorithms both theoretically and empirically.
In the future work, we aim to extend the application scope of our algorithm to continuous space. Sampling method had been proved to be efficient in discrete environment. But it still occurs many obstacles in this area. Our approach solves the discrete problems with count-based posterior weight. Such idea could be transplanted to continuous environment as well. We could represent the difference between the state of the continuous spaces with specific metric. Then adopt our method in such environment.
| 15,156 |
\section{Introduction}
As usual, we will say that $K\subset {\mathbb R}^d$ is a convex body if $K$ is a convex, compact subset of ${\mathbb R}^d$ equal to the closure of its interior. We say that $K$ is origin-symmetric if $K=-K$, where $\lambda K=\{\lambda {\boldsymbol x}: {\boldsymbol x} \in K\},$ for $\lambda \in {\mathbb R}$. For a set $K$
we denote by dim$(K)$ its dimension, that is, the dimension of the affine hull of $K$. We define $K+L=\{{\boldsymbol x}+{\boldsymbol y}: {\boldsymbol x}\in K, {\boldsymbol y} \in L\}$ to be the Minkowski sum of $K, L \subset {\mathbb R}^d$. We will also denote by $\mbox{\rm vol}_d$ the $d$-dimensional Hausdorff measure, and if the body $K$ is $d$-dimensional we will call $\mbox{\rm vol}_d(K)$ the volume of $K$. Finally, let us denote by ${\boldsymbol \xi}^\perp$ a hyperplane perpendicular to a unit vector ${\boldsymbol \xi}$, i.e.
$$
{\boldsymbol \xi}^\perp=\{{\boldsymbol x}\in {\mathbb R}^d: {\boldsymbol x}\cdot {\boldsymbol \xi} =0\}.
$$
We refer to \cite{Ga, K3, BGVV, RZ, Sch} for general definitions and properties of convex bodies.
The slicing problem of Bourgain \cite{Bo1, Bo2} is, undoubtedly, one of the major open problems in convex geometry asking if a convex, origin-symmetric body of volume one must have a large (in volume) hyperplane section. More precisely, it asks whether there exists an absolute constant ${\mathcal L}_1$ so that for any origin-symmetric convex body $K$ in ${\mathbb R}^d$
\begin{equation}\label{eq:kold}
\mbox{\rm vol}_d(K)^{\frac {d-1}{d}} \le {\mathcal L}_1 \max_{{\boldsymbol \xi} \in {\mathbb S}^{d-1}} \mbox{\rm vol}_{d-1}(K\cap {\boldsymbol \xi}^\bot).
\end{equation}
The problem is still open, with the best-to-date estimate of ${\mathcal L}_1\le O(d^{1/4})$ established
by Klartag \cite{Kl}, who improved the previous estimate of Bourgain \cite{Bo2}, we refer to \cite{MP} and \cite{BGVV} for detailed information and history of the problem. Recently, Alexander Koldobsky proposed an interesting generalization of the slicing problem \cite{K1,K2,K4,K5,K6}:
Does there exists an absolute constant ${\mathcal L}_2$ so that for every even measure $\mu$
on ${\mathbb R}^d$, with a positive density, and for every origin-symmetric convex body $K$ in ${\mathbb R}^d$ such that
\begin{equation}
\label{eq:cont}\mu(K)\le {\mathcal L}_2\max_{{\boldsymbol \xi}\in {\mathbb S}^{d-1}} \mu(K\cap {\boldsymbol \xi}^\bot) \mbox{\rm vol}_d(K)^{\frac 1d}?
\end{equation}
Koldobsky was able to solve the above question for a number of special cases of the body $K$ and provide a general estimate of $O(\sqrt{d})$. The most amazing fact here is that the constant ${\mathcal L}_2$ in (\ref{eq:cont}) can be chosen independent of the measure $\mu$ under the assumption that $\mu$ has even positive density. In addition, Koldobsky and the second named author were able to prove in \cite{KoZ} that ${\mathcal L}_2$ is of order $O\left(d^{1/4}\right)$ if one assumes that the measure $\mu$ is $s$-concave. We note that the assumption of positive density is essential for the above results and (\ref{eq:cont}) is simply not true if this condition is dropped. Indeed, to create a counterexample consider an even measure $\mu$ on ${\mathbb R}^2$ uniformly distributed over $2N$ points on the unit circle, then the constant ${\mathcal L}_2$ in (\ref{eq:cont}) will depend on $N$.
During the 2013 AIM workshop on ``Sections of convex bodies'' Koldobsky asked if it is possible to provide a discrete analog of inequality (\ref{eq:cont}): Let ${\mathbb Z}^d$ be the standard integer lattice in ${\mathbb R}^d$, $K$ be a convex, origin-symmetric body, define $\#K=\mbox{card}(K\cap{\mathbb Z}^d)$, the number of points of ${\mathbb Z}^d$ in $K$.
\vskip 1em
\noindent{\bf Question:} {\it Does there exist a constant ${\mathcal L}_3$ such that}
$$
\#K\leq {\mathcal L}_3 \max_{{\boldsymbol \xi}\in {\mathbb S}^{d-1}} \left( \#(K\cap {\boldsymbol \xi}^\perp)\right) \mbox{\rm vol}_d(K)^{\frac{1}{d}},
$$
{\it for all convex origin-symmetric bodies $K\subset {\mathbb R}^d$
containing $d$ linearly independent lattice points?}
\vskip 1em
We note here that we require that $K$ contains $d$ linearly
independent lattice points, i.e., $\dim(K\cap{\mathbb Z}^d)=d$, in order to eliminate the
degenerate case of a body (for example, take a box $[-1/n,
1/n]^{d-1}\times [-20, 20]$) whose maximal section contains all
lattice points in the body, but whose volume may be taken to 0 by eliminating a dimension.
Koldobsky's question is yet another example of an attempt to translate questions and facts from classical Convexity to more general settings including Discrete Geometry. The properties of sections of convex bodies with respect to the integer lattice were extensively studied in Discrete Tomography \cite{GGroZ, GGr1, GGr2, GGro}, where many interesting new properties were proved and a series of exciting open questions were proposed. It is interesting to note that after translation many questions become quite non-trivial and counterintuitive, and the answer may be quite different from the continuous case. In addition, finding the relation between the geometry of a convex set and the number of integer points contained in the set is always a non-trivial task. One can see this, for example, from the history of Khinchin's flatness theorem \cite{Ba1, Ba2, BLPS, KL}.
The main goal of this paper is to study Koldobsky's question.
In Section 2 we will show the solution for the $2$-dimensional
case. The solution is based on the classical Minkowski's First and
Pick's theorems from the Geometry of Numbers and gives a general idea
of the approach to be used in Sections 3 and 4. In Section 3 we apply
a discrete version of the theorem of F. John due to T. Tao and V. Vu
\cite{TV1} to give a partial answer to Koldobsky's question and show
that the constant ${\mathcal L}_3$ can be chosen independent of the body $K$ and
as small as $O(d)^{7d/2}$. We start Section 4 with a case of
unconditional bodies and present a simple proof that in this case
${\mathcal L}_3$ can be chosen of order $O(d)$ which is best possible. After, we prove the
discrete analog of Brunn's theorem and use it to show that the constant
${\mathcal L}_3$, for the general case, can be chosen as small as $O(1)^d$.
In fact, we prove the slightly more general result that
$$
\# K \leq O(1)^d d^{d-m} \max\left(\# (K\cap H)\right)\,\, \mbox{\rm vol}_d(K)^{\frac{d-m}{d}},
$$
where the maximum is taken over all $m$-dimensional linear subspaces $H \subset {\mathbb R}^d$.
Finally, we also provide a short observation that ${\mathcal L}_1 \le {\mathcal L}_3$.
\smallskip
\noindent {\bf Acknowledgment}: We are indebted to Alexander Koldobsky and Fedor Nazarov for
valuable discussions.
\section{Solution in ${\mathbb Z}^2$}
Let us start with recalling two classical statements in the Geometry
of Numbers
(see \cite{TV2}, Theorem 3.28 pg 134 and \cite{BR}, Theorem 2.8 pg 38):
\begin{theorem}\label{th:M1} (Minkowski's First Theorem)
Let $K \subset {\mathbb R}^d$ be an origin-symmetric convex body such that $\mbox{\rm vol}_d(K)\ge 2^d$ then $K$ contains at least one non-zero element of ${\mathbb Z}^d$.
\end{theorem}
\begin{theorem} (Pick's Theorem)
Let $P$ be an integral $2$-dimensional convex polygon, then $A=I+\frac{1}{2}B-1$ where $A=\mbox{\rm vol}_2(P)$ is the area of the polygon, $I$ is the number of lattice points in the interior of $P$, and $B$ is the number of lattice points on the boundary.
\end{theorem}
\noindent Here a polygon is called integral if it can be described as the
convex hull of lattice points.
Now we will use the above theorems to show that the constant ${\mathcal L}_3$ in Koldobsky's question can be chosen independently of a convex, origin-symmetric body in ${\mathbb R}^2$.
\begin{theorem} Let $K$ be a convex origin-symmetric body in ${\mathbb R}^2$,
$\dim(K\cap{\mathbb Z}^2)=2$, then
$$
\#K\leq 4 \max_{{\boldsymbol \xi}\in {\mathbb S}^{1}} \#(K\cap {\boldsymbol \xi}^\perp)\,\mbox{\rm vol}_2(K)^{\frac{1}{2}}.
$$
\end{theorem}
\noindent{\bf Proof: } Let $s=\sqrt{\mbox{\rm vol}_2(K)/4}$, then by Minkowski's theorem, since $\mbox{\rm vol}_2(\frac{1}{s}K)=4$, there exists a non-zero vector ${\boldsymbol u}\in {\mathbb Z}^2 \cap \frac{1}{s} K$.
Then $s{\boldsymbol u} \in K$ and
$$\# \left(L_{\boldsymbol u} \cap K\right) \geq 2 \lfloor s \rfloor + 1
,$$ where $\lfloor s \rfloor$ is the integer part of $s$,
and $L_{\boldsymbol u}$ is the line containing ${\boldsymbol u}$ and the origin.
Next, consider $P=\hbox{\rm conv}(K\cap {\mathbb Z}^2)$, i.e., the convex hull of the
integral points inside $K$. $P$ is an integral $2$-dimensional convex polytope, and so by Pick's theorem we get that
$$
\mbox{\rm vol}_2(P)=I+\frac{1}{2}B-1\geq \frac{I+B}{2}-\frac{1}{2},
$$
using that $I \ge 1$. Thus
$$
\# P=I+B \leq 2\mbox{\rm vol}_2(P)+1\leq \frac{5}{2}\mbox{\rm vol}_2(P),$$
since the minmal volume of an origin-symmetric
integral convex polygon is at least 2.
We now have that
\begin{equation*}
\begin{split}
\# K & = \# P \leq \frac{5}{2}\mbox{\rm vol}_2(P) \leq \frac{5}{2}\mbox{\rm vol}_2(K)\\
&\leq \frac{5}{2} \,(2\,s)\, \mbox{\rm vol}_2(K)^\frac{1}{2} < 4\,( 2 \lfloor s \rfloor + 1)\,
\mbox{\rm vol}(K)_2^\frac{1}{2}\\
&\leq 4 \max_{{\boldsymbol \xi}\in {\mathbb S}^{1}} \#(K\cap {\boldsymbol \xi}^\perp)\,\mbox{\rm vol}_2(K)^{\frac{1}{2}}.
\end{split}
\end{equation*}
\begin{flushright
\section{Approach via Discrete F. John Theorem}
It is a standard technique to get a first estimate in slicing inequalities, i.e. ${\mathcal L}_1 \le O(\sqrt{d})$, by using the classical F. John theorem, \cite{J}, \cite{MS}, or \cite{BGVV}, which claims that for every convex origin-symmetric body $K\subset {\mathbb R}^d$ there exists an Ellipsoid $E$ such that
$ E\subset K \subset \sqrt{d} E$.
In this section we will use a recent discrete version of F. John's theorem, proved by T. Tao and V. Vu (see \cite{TV1, TV2}) to prove that the constant ${\mathcal L}_3$ in Koldobsky's question can be chosen independent of the origin-symmetric convex body $K\subset {\mathbb R}^d$. We first recall the definition of a generalized arithmetic progression (see \cite{TV1, TV2} for more details):
\begin{definition}
Let $G$ be an additive group, $N=(N_1, \ldots, N_d)$ an $d$-tuple of non-negative integers and ${\boldsymbol v}=({\boldsymbol v}_1, \ldots, {\boldsymbol v}_d)\in G^d$.
Then a generalized symmetric arithmetic progression ${\bf P}$ is a triplet $(N, {\boldsymbol v}, d)$. In addition, define
$$\text{\rm Image}({\bf P})= [-N, N] \cdot {\boldsymbol v} = \left\{n_1 {\boldsymbol v}_1 + \ldots + n_d {\boldsymbol v}_d : n_j \in [-N_j, N_j] \cap {\mathbb Z} \text{ for all } 1\leq j \leq d \right\}.$$
The progression is called proper if the map ${\boldsymbol n} \mapsto {\boldsymbol n}\cdot {\boldsymbol v}$
is injective, ${\boldsymbol v}=({\boldsymbol v}_1, \ldots, {\boldsymbol v}_d)$ is called its basis
vectors, and $d$ its rank.
\label{def:progression}
\end{definition}
Below is a version for ${\mathbb Z}^d$ of the Discrete John theorem from \cite{TV1} (Theorem 1.6 there):
\begin{theorem}\label{th:dj}
Let $K$ be a convex origin-symmetric body in ${\mathbb R}^d$. Then there exists a symmetric, proper, generalized arithmetic progression ${\bf P} \subset {\mathbb Z}^d$, such that $\mbox{rank}({\bf P}) \le d$
and
\begin{equation}\label{eq:incl}
(O(d)^{-3d/2} K) \cap {\mathbb Z}^d \subset \text{\rm Image}({\bf P}) \subset K \cap {\mathbb Z}^d,
\end{equation}
in addition
\begin{equation}\label{eq:size}
O(d)^{-7d/2} \#K \le \#{\bf P}.
\end{equation}
\end{theorem}
Now we are ready to state and prove our first estimate in Koldobsky's question and prove that
for any origin-symmetric convex body $K \subset {\mathbb R}^d$,
$\dim(K\cap{\mathbb Z}^d)=d$,
\begin{equation}\label{eq:viajohn}
\#K\leq O(d)^{7d/2} \max_{{\boldsymbol \xi}\in {\mathbb S}^{d-1}} \left( \#(K\cap {\boldsymbol \xi}^\perp)\right) \mbox{\rm vol}_d(K)^{\frac{1}{d}}.
\end{equation}
To prove (\ref{eq:viajohn}) we apply the discrete John's theorem to get a symmetric, proper, generalized arithmetic progression ${\bf P}=(N, {\boldsymbol v}, d)$ as in Definition \ref{def:progression}. We note that if $\mbox{rank}({\bf P}) < d$, then
there exists a hyperplane ${\boldsymbol \xi}^\perp$ such that ${\bf P} \subset {\boldsymbol \xi}^\perp$ and using (\ref{eq:size}) we get
$$
O(d)^{-7d/2} \#K \le \#({\bf P}) \leq \#(K\cap {\boldsymbol \xi}^\perp).
$$
By our assumption $\dim (K\cap{\mathbb Z}^d)=d$ we have $\mbox{\rm vol}_d(K)\geq 2^d/d!$
and so $\mbox{\rm vol}_d(K)^{\frac{1}{d}}> 2/d$. Thus
$$
\#K \leq O(d)^{7d/2}\,\#(K\cap {\boldsymbol \xi}^\perp)\,\mbox{\rm vol}_d(K)^{\frac{1}{d}}.
$$
Next we consider the case $\mbox{rank}({\bf P})=d$.
Without loss of generality, take $N_1\geq N_2\geq \ldots \geq N_d \ge 1$, then define ${\boldsymbol \xi}^\perp=\hbox{\rm span}\,\{{\boldsymbol v}_1, \dots, {\boldsymbol v}_{d-1}\}$.
Application of (\ref{eq:size}) gives
\begin{align*}
\#K \leq & O(d)^{7d/2} \#({\bf P}) \\
\leq &O(d)^{7d/2} \prod_{i=1}^{d} (2N_i+1)\\
= &O(d)^{7d/2} (2N_d+1) \prod_{i=1}^{d-1} (2N_i+1)\\
\leq &O(d)^{7d/2} \left(\prod_{i=1}^{d} (2N_i+1)\right)^{\frac{1}{d}} \#(K\cap {\boldsymbol \xi}^\perp).
\end{align*}
Where the last inequality follows from the minimality of $N_d$ and we use (\ref{eq:incl}) to claim that
$$\#(K\cap {\boldsymbol \xi}^\perp) \geq \prod_{i=1}^{d-1} (2N_i+1).$$
Now we consider the volume covered by our progression. Take a fundamental parallelepiped
$$\Pi= \left\{a_1 {\boldsymbol v}_1 + \ldots +a_d {\boldsymbol v}_d, \mbox{ where } a_i \in [0,1), \mbox{ for all } i=1, \dots, n \right\}.$$
Let $X=[-N,N-1]\cdot {\boldsymbol v}$, we notice that
$$
K\supset \bigcup\limits_{{\boldsymbol x}\in X} ({\boldsymbol x}+ \Pi),
$$
indeed from $\text{\rm Image}({\bf P})\subset K\cap {\mathbb Z}^d$ we get that the vertices of ${\boldsymbol x}+\Pi$ belong to $K \cap {\mathbb Z}^d$ for all ${\boldsymbol x}\in X$ and thus, by convexity, ${\boldsymbol x}+\Pi\subset K$ for all ${\boldsymbol x}\in X$. Next
$$
\mbox{\rm vol}_d(K) \geq \left(\prod_{k=1}^d 2 N_k \right) \det({\boldsymbol v}_1,\ldots, {\boldsymbol v}_d) \geq \prod_{k=1}^d 2 N_k,
$$
where the last inequality follows from the fact that ${\boldsymbol v}_1,\ldots, {\boldsymbol v}_d$ are independent vectors in ${\mathbb Z}^d$ and thus
$\det({\boldsymbol v}_1,\ldots, {\boldsymbol v}_d) \geq \det({\mathbb Z}^d)=1$.
Finally,
\begin{align*}\#K \leq & O(d)^{7d/2} \left(\prod_{i=1}^{d} (2N_i+1)\right)^{\frac{1}{d}} \#(K\cap {\boldsymbol \xi}^\perp) \\ \leq & O(d)^{7d/2} \left(\prod_{i=1}^{d} (2N_i)\right)^{\frac{1}{d}} \#(K\cap {\boldsymbol \xi}^\perp)
\\\leq & O(d)^{7d/2} \#(K\cap {\boldsymbol \xi}^\perp) \mbox{\rm vol}_d(K)^{\frac{1}{d}}.
\end{align*}
\section{The case of co-dimensional slices and improved bound on $C(d)$}
The goal of this section is to improve the estimate provided in Section 3. We will need to consider counting points intersecting a body with a general lattice, and so we will adapt our notation slightly. We refer to \cite{TV2}, \cite{BR} and \cite{HW} for the general facts and introduction on the properties of the cardinality of intersections of convex bodies and a lattice. Given a lattice $\Gamma$ we will take $\#(K\cap \Gamma)= \text{card}(K\cap \Gamma)$ and, as before, if the lattice is omitted we will take the lattice to be the standard integer lattice of appropriate dimension. We begin with the statement of Minkowski's Second Theorem which is an extension of Minkowski's First Theorem (Theorem \ref{th:M1} above) and can be found, for example, in \cite{HW} (Theorem 1.2) or \cite{TV2} (Theorem 3.30 pg 135). First we recall the definition of Successive Minima.
\begin{definition}
Let $\Gamma$ be a lattice in ${\mathbb R}^d$ of rank $k$, and let $K$ be an
origin-symmetric
convex body in ${\mathbb R}^d$. For $1\leq j\leq k$ define the successive minima to be
$$\lambda_j=\lambda_j(K, \Gamma)= \min\left\{\lambda>0 : \lambda \cdot
K \text{ contains } j \text{ linearly independent elements of }
\Gamma \right\}.$$
\end{definition}
Note, that it follows directly from the definition that $\lambda_k \ge
\lambda_{k-1}\ge...\ge\lambda_1$. In addition, the assumption that $K$ contains $d$ linearly independent lattice points of $\Gamma$ implies that $\Gamma$ has rank $d$ and that $\lambda_d\leq 1$. Moreover, according to the definition of the
successive minima there exists a set of linearly independent vectors
from $\Gamma$, ${\boldsymbol v}_1,\ldots,{\boldsymbol v}_k$, such that ${\boldsymbol v}_i$ lies on the
boundary of $\lambda_i \cdot K$ but the interior of $\lambda_i \cdot
K$ does not contain any lattice vectors outside the span of
${\boldsymbol v}_1,\ldots, {\boldsymbol v}_{i-1}$. The vectors ${\boldsymbol v}_1,\dots,{\boldsymbol v}_k$ are called a directional
basis, and we note that they may not necessarily form a basis of $\Gamma$.
\begin{theorem}\label{th:M2} (Minkowski's Second Theorem)
Let $\Gamma$ be a lattice in ${\mathbb R}^d$ of rank $d$, $K$ be an
origin-symmetric convex body with successive minima $\lambda_i$. Then,
$$
\frac{1}{d!} \prod_{i=1}^{d} \frac{2}{\lambda_i} \leq \frac{\mbox{\rm vol}(K)}{\det(\Gamma)} \leq \prod_{i=1}^{d} \frac{2}{\lambda_i}.
$$
\end{theorem}
Next we will study the behavior of constant ${\mathcal L}_3$ in the case of unconditional convex bodies. A set $K\subset {\mathbb R}^d$ is said to be unconditional if it is symmetric with respect to any coordinate hyperplane, i.e., $(\pm x_1, \pm x_2, \dots, \pm x_d) \in K$, for any ${\boldsymbol x} \in K$ and any choice of $\pm$ signs.
\begin{theorem}\label{th:unconditional}
Let $K\subset{\mathbb R}^d$ be an unconditional convex body with
$\dim(K\cap{\mathbb Z}^d)=d$. Then
$$
\#K\leq O(d) \max_{i=1,\dots,d} \left( \#(K\cap {\boldsymbol e}_i^\perp)\right)\,\mbox{\rm vol}_d(K)^{\frac{1}{d}},
$$
where ${\boldsymbol e}_1, \dots {\boldsymbol e}_d$ are the standard basis vectors in
${\mathbb R}^d$. Moreover, this bound is the best possible.
\end{theorem}
\noindent{\bf Proof: }
This result follows from the simple observation that the section of $K$ by a coordinate hyperplane
${{\boldsymbol e}_i}^\perp$ is maximal in cardinality among all parallel sections of $K$, i.e.
\begin{equation}\label{eq:uncon}
\#(K \cap ({\boldsymbol e}_i^\perp + t {\boldsymbol e}_i))\le \#(K \cap {\boldsymbol e}_i^\perp), \mbox{ for all } t\in {\mathbb R}, \mbox{ and } i=1,\dots,d.
\end{equation}
We can see this by considering a point ${\boldsymbol x} \in K \cap ({\boldsymbol e}_i^\perp + t {\boldsymbol e}_i)\cap {\mathbb Z}^d$.
Let $\bar{{\boldsymbol x}}$ be the reflection of ${\boldsymbol x}$ over ${{\boldsymbol e}_i}^\perp$, i.e., $\bar{{\boldsymbol x}}=(x_1, \dots, -x_i, \dots, x_d)$. Using unconditionality of $K$, we get that $\bar{{\boldsymbol x}}\in K$
and convexity gives us $({\boldsymbol x}+\bar{{\boldsymbol x}})/2 \in K \cap {\boldsymbol e}_i^\perp$.
Hence, the projection of a point in $K \cap ({\boldsymbol e}_i^\perp + t {\boldsymbol e}_i)$ is associated to a point in $K \cap {\boldsymbol e}_i^\perp$, which explains (\ref{eq:uncon}).
Let $\{\lambda_i\}_{i=1}^d$ be the successive minima of $K$ with
respect to ${\mathbb Z}^d$.
Using an argument similar to the one above one can show that that the vectors ${\boldsymbol v}_1, \dots, {\boldsymbol v}_d \in {\mathbb Z}^d$ associated with $\{\lambda_i\}_{i=1}^d$ may be taken as a rearrangement of ${\boldsymbol e}_1,\dots, {\boldsymbol e}_d$.
We may assume without loss of generality that $\lambda_d$ corresponds to ${\boldsymbol e}_d$. So ${\boldsymbol e}_d \in \lambda_d K$ and
$\frac{1}{\lambda_d} {\boldsymbol e}_d \in K$. Thus $\#(K\cap L_{{\boldsymbol e}_d}) \le 2 \lfloor \frac{1}{\lambda_d}\rfloor +1$, where, as before, $L_{{\boldsymbol e}_d}$ is a line containing ${\boldsymbol e}_d$ and the origin. Using (\ref{eq:uncon}), we get
$$
\#K\leq \left( 2 \left\lfloor \frac{1}{\lambda_d}\right\rfloor +1 \right) \#(K\cap {{\boldsymbol e}_d}^\perp).
$$
By assumption we have $\lambda_d \le 1$ and, using $\lambda_d \ge \lambda_i$, for all $i=1, \dots, d$, we get
$$
2 \left\lfloor \frac{1}{\lambda_d}\right\rfloor +1 \le \frac{3}{\lambda_d} \le O(d) \left(\frac{1}{d!} \prod_{i=1}^{d} \frac{2}{\lambda_i} \right)^{1/d}.
$$
Finally we use Theorem \ref{th:M2} to finish the proof:
$$
\#K\leq O(d) \#(K\cap {{\boldsymbol e}_d}^\perp) \,\mbox{\rm vol}_d(K)^{\frac{1}{d}}.
$$
The cross-polytope $B_1^d =\hbox{\rm conv}\{\pm{\boldsymbol e}_1,\dots,\pm{\boldsymbol e}_d\}$ of
$\mbox{\rm vol}(B_1^d)=2^d/d!$ shows that the bound is optimal up to
multiplication with constants.
\begin{flushright
The idea of the proof of the above theorem follows from the classical Brunn's theorem: the central hyperplane section of a convex origin-symmetric body is maximal in volume among all parallel sections (see \cite{Ga}, \cite{K3}, \cite{RZ}). One may notice that, in general, it may not be the case that the maximal hyperplane in cardinality for an origin-symmetric convex body passes through the origin. Indeed, see Figure 1 below, or consider an example of a cross-polytope $B_1^d=\{{\boldsymbol x}\in {\mathbb R}^d: \sum|x_i|\le 1\}$, then $\#(B_1^{d} \cap (1/\sqrt{d},\dots, 1/\sqrt{d})^\perp)=1$ but a face of $B_1^d$ contains $d$ integer points.
\begin{figure}[h!]\label{fig:BrunnCounterexample}
\includegraphics[scale=0.4]{Mat_ex}
\caption{Central Section may have less integer points.}
\end{figure}
So we see that there is no equivalent of Brunn's theorem in this setting.
Still, we propose the following analog of Brunn's theorem in the discrete setting:
\begin{theorem}\label{th:brunn} Consider a convex, origin-symmetric
body $K\subset {\mathbb R}^d$ and a lattice $\Gamma \subset {\mathbb R}^d$ of rank $d$, then
$$
\#(K \cap {\boldsymbol \xi}^\perp \cap \Gamma) \ge 9^{-(d-1)} \#(K \cap ({\boldsymbol \xi}^\perp+t {\boldsymbol \xi})\cap \Gamma), \mbox{ for all } t \in {\mathbb R}. $$
\end{theorem}
\noindent Before proving Theorem \ref{th:brunn} we need to recall two nice packing estimates (see Lemma 3.21, \cite{TV2}):
\begin{lemma}\label{lm:pack}
Let $\Lambda$ be a lattice in ${\mathbb R}^d$. If $A \subset {\mathbb R}^d$ is an arbitrary bounded set and
$P \subset {\mathbb R}^d$ is a finite non-empty set, then
\begin{equation}\label{eq:3_9}
\#\left(A\cap(\Lambda+P)\right) \le \# \left((A-A)\cap(\Lambda+P-P)\right).
\end{equation}
If $B\subset {\mathbb R}^d$ is a origin-symmetric convex body, then
\begin{equation}\label{eq:3_10}
(kB) \cap \Lambda\mbox{ can be covered by } (4k+1)^d \mbox{ translates of }B\cap\Lambda.
\end{equation}
\end{lemma}
\noindent{\bf Proof of Theorem \ref{th:brunn}:}
We first recall a standard observation, that the convexity of $K$ gives us
$$
K \cap {\boldsymbol \xi}^\perp \supset \frac{1}{2}(K \cap ({\boldsymbol \xi}^\perp+t {\boldsymbol \xi}))+ \frac{1}{2}(K \cap ({\boldsymbol \xi}^\perp-t {\boldsymbol \xi})).
$$
Let $\Gamma'=\Gamma \cap {\boldsymbol \xi}^\perp$ and assume that $\Gamma \cap ({\boldsymbol \xi}^\perp+t {\boldsymbol \xi}) \not =\emptyset$ (the statement of the theorem is trivial in the other case). Consider a point ${\boldsymbol \gamma}\in \Gamma \cap ({\boldsymbol \xi}^\perp+t {\boldsymbol \xi})$ then
$$
\Gamma \cap ({\boldsymbol \xi}^\perp+t {\boldsymbol \xi}) = {\boldsymbol \gamma}+\Gamma' \mbox{ and } \Gamma \cap ({\boldsymbol \xi}^\perp-t {\boldsymbol \xi}) = -{\boldsymbol \gamma}+\Gamma'.
$$
Moreover,
$$
K \cap {\boldsymbol \xi}^\perp \supset \frac{1}{2}\left(\left[K \cap ({\boldsymbol \xi}^\perp+t {\boldsymbol \xi})\right]-{\boldsymbol \gamma}\right)+ \frac{1}{2}\left(\left[K \cap ({\boldsymbol \xi}^\perp-t {\boldsymbol \xi})\right]+{\boldsymbol \gamma}\right).
$$
Thus
$$
\left(K \cap {\boldsymbol \xi}^\perp \right) \cap \Gamma' \supset \left[\frac{1}{2}\left(\left[K \cap ({\boldsymbol \xi}^\perp+t {\boldsymbol \xi})\right]-{\boldsymbol \gamma}\right)+ \frac{1}{2}\left(\left[K \cap ({\boldsymbol \xi}^\perp-t {\boldsymbol \xi})\right]+{\boldsymbol \gamma}\right) \right] \cap \Gamma'.
$$
Our goal is to estimate the number of lattice points on the right hand side of the above inclusion. Let
$$
B=\frac{1}{2}\left(\left[K \cap ({\boldsymbol \xi}^\perp+t {\boldsymbol \xi})\right]-{\boldsymbol \gamma}\right)
$$
then, using the symmetry of $K$, we get
$$
-B=\frac{1}{2}\left(\left[K \cap ({\boldsymbol \xi}^\perp-t {\boldsymbol \xi})\right]+{\boldsymbol \gamma}\right).
$$
Thus $B-B$ is an origin-symmetric convex body in ${\boldsymbol \xi}^\perp$. Next we use (\ref{eq:3_10}) from Lemma \ref{lm:pack} to claim that
$$
\#(2(B-B) \cap \Gamma') \le 9^{d-1} \#((B-B)\cap \Gamma').
$$
Notice that $2(B-B)=2B-2B$ thus we may use (\ref{eq:3_9}) from Lemma \ref{lm:pack} with $P=\{{\boldsymbol 0}\}$, ${\boldsymbol \xi}^\perp$ associated with ${\mathbb R}^{d-1}$, and $\Lambda=\Gamma^\prime$ to claim that
$$
\#(2(B-B)\cap \Gamma') =\#((2B-2B)\cap \Gamma' )\ge \# (2B\cap \Gamma')=\#(2B\cap \Gamma).
$$
Thus we proved that
\begin{align*}
\#\bigg(\bigg[\frac{1}{2}(K \cap & ({\boldsymbol \xi}^\perp+t {\boldsymbol \xi}) -{\boldsymbol \gamma})+ \frac{1}{2}(K \cap ({\boldsymbol \xi}^\perp-t {\boldsymbol \xi})+{\boldsymbol \gamma}) \bigg] \cap \Gamma \bigg) \\ \ge & 9^{-(d-1)} \# \left( \left[K \cap ({\boldsymbol \xi}^\perp+t {\boldsymbol \xi})-{\boldsymbol \gamma} \right]\cap \Gamma \right)
\end{align*}
but
$$
\#\left(\left[K \cap ({\boldsymbol \xi}^\perp+t {\boldsymbol \xi})-{\boldsymbol \gamma} \right] \cap \Gamma \right)=\#\left(\left[K \cap ({\boldsymbol \xi}^\perp+t {\boldsymbol \xi}) \right]\cap\Gamma \right).
$$
\begin{flushright
\begin{corollary}\label{cor:Brunn}
Consider a convex, origin-symmetric body $M\subset {\mathbb R}^n$, lattice
$\Lambda \subset {\mathbb R}^n$ and $m$-dimensional lattice subspace $H$,
i.e., it contains $m$ linearly independent points of $\Lambda$, then
$$
\#(M \cap H \cap \Lambda) \ge 9^{-m} \#(M \cap (H+{\boldsymbol z})\cap \Lambda), \mbox{ for all } {\boldsymbol z} \in {\mathbb R}^n. $$
\end{corollary}
\noindent{\bf Proof: } Let ${\boldsymbol z}\in{\mathbb R}^n$. Then we may assume ${\boldsymbol z} \in \Gamma\setminus\{H\cap\Gamma\}$ and let $U$ be the linear space spanned by $H$ and ${\boldsymbol z}$. Then $\dim(U)=m+1$ and the corollary follows from Theorem \ref{th:brunn} with $U$ associated with ${\mathbb R}^{m+1}$, $K= M \cap U$, and $\Gamma =\Lambda \cap U$.
\begin{flushright
Let ${\rm G}_{{\mathbb Z}}(i,d)$ be the set of all $i$-dimensional linear
subspaces containing $i$-linearly independent lattice vectors of ${\mathbb Z}^d$, i.e., the
set of all $i$-dimensional lattice hyperplanes. The next theorem gives a general bound on the number of integer points in co-dimensional slices.
\begin{theorem}
Let $K \subset {\mathbb R}^d$ be an origin-symmetric convex body with $\dim
(K\cap{\mathbb Z}^d)=d$. Then
\begin{equation}
\# K \leq O(1)^d\, d^{d-m} \max\{\# (K\cap H): H\in
{\rm G}_{{\mathbb Z}}(m,d)\}\,\, \mbox{\rm vol}_d(K)^{\frac{d-m}{d}}.
\end{equation}
\label{tm:main}
\end{theorem}
Obviously, for $m=d-1$ we obtain the estimate for hyperplane slices
\begin{equation}
\#K\leq O(1)^d \max_{{\boldsymbol \xi}\in {\mathbb S}^{d-1}} \left( \#(K\cap
{\boldsymbol \xi}^\perp)\right)\,\mbox{\rm vol}_d(K)^{\frac{1}{d}}.
\label{eq:slice_bound}
\end{equation}
\medskip
\noindent{\bf Proof: }
Let $\{\lambda_i^*\}_{i=1}^d$ be the successive minima of the polar
body $$K^*=\{{\boldsymbol y}\in{\mathbb R}^d: {\boldsymbol y}\cdot,{\boldsymbol x}\leq 1, \text{ for all }{\boldsymbol x}\in
K\}$$ with respect to ${\mathbb Z}^d$ and let ${\boldsymbol v}_1, \dots {\boldsymbol v}_d \in {\mathbb Z}^d$ be
the associated directional basis. These vectors are linearly independent and
${\boldsymbol v}_i\in\lambda_i^*\,K^*$ for all $i$. Thus we have
\begin{equation}\label{eq:polar}
K\subseteq \{{\boldsymbol x}\in{\mathbb R}^d : |{\boldsymbol v}_i \cdot {\boldsymbol x}|\leq \lambda_i^*, \,
1\leq i\leq d\}.
\end{equation}
Let $U=\hbox{\rm span}\,\{{\boldsymbol v}_1,\dots,{\boldsymbol v}_{d-m}\}$ and let $\overline H=U^\perp$
be the orthogonal complement of $U$. Observe that $\overline H\in
{\rm G}_{{\mathbb Z}}(m,d)$. Since for ${\boldsymbol z}\in{\mathbb Z}^d$ we have ${\boldsymbol v}_i \cdot
{\boldsymbol z}\in{\mathbb Z}$, $1\leq i\leq d$, we also have ${\boldsymbol v}_i \cdot ({\boldsymbol z}\big
|U)\in{\mathbb Z}$, $1\leq i\leq d-m$, where ${\boldsymbol z}\big |U$ is the orthogonal
projection onto $U$. In view of \eqref{eq:polar} we obtain
\begin{equation}
\label{eq:bound_pro}
(K\cap{\mathbb Z}^d)\big|U \subset \{{\boldsymbol y}\in U : {\boldsymbol v}_i \cdot {\boldsymbol y}\in{\mathbb Z}\text{ and }| {\boldsymbol v}_i\cdot {\boldsymbol y}|\leq \lambda_i^*, \,
1\leq i\leq d-m\},
\end{equation}
and thus
\begin{equation}
\#((K\cap{\mathbb Z}^d)\big|U)\leq \prod_{i=1}^{d-m}\left(2\,\lfloor\lambda_i^*\rfloor +1\right).
\end{equation}
Due to our assumption that $K$ contains $d$-linearly independent lattice points we have that
$\lambda_1^*\geq 1$; otherwise \eqref{eq:polar} implies ${\boldsymbol v}_1 \cdot
{\boldsymbol z}=0$ for all ${\boldsymbol z}\in K\cap{\mathbb Z}^d$. So we conclude by \eqref{eq:bound_pro}
\begin{equation}
\begin{split}
\# K & \leq \#((K\cap{\mathbb Z}^d)|U)\,\max\{\#(K\cap ({\boldsymbol z}+\overline H)): {\boldsymbol z}\in{\mathbb Z}^d \} \\
& \leq \max\{\#(K\cap ({\boldsymbol z}+\overline H)): {\boldsymbol z}\in{\mathbb Z}^d \}\,3^{d-m}\prod_{i=1}^{d-m}\lambda_i^* \\
&\leq 3^{d-m}\,O(1)^m \#(K\cap \overline H) \,\prod_{i=1}^{d-m}\lambda_i^*
\leq O(1)^d\,\#(K\cap \overline H) \,\prod_{i=1}^{d-m}\lambda_i^* .
\end{split}
\label{eq:bound1}
\end{equation}
Here the last step follows from Corollary \ref{cor:Brunn}, the co-dimensional version of the discrete Brunn's theorem.
Next Minkowski's Second theorem (Theorem \ref{th:M2}) gives the upper bound
\begin{equation}
\lambda_1^*\cdot\ldots\cdot\lambda_d^* \mbox{\rm vol}_d(K^*)\leq 2^d
\end{equation}
and so we find
\begin{equation}
\left(\prod_{i=1}^{d-m}\lambda_i^*\right)^d\mbox{\rm vol}_d(K^*)^{d-m}\leq \left(\prod_{i=1}^{d}\lambda_i^*\right)^{d-m}\mbox{\rm vol}_d(K^*)^{d-m}\leq 2^{d(d-m)}.
\end{equation}
Hence
\begin{equation}
\prod_{i=1}^{d-m}\lambda_i^* \leq 2^{d-m}\, \mbox{\rm vol}_d(K^*)^{\frac{m-d}{d}}.
\label{eq:mink}
\end{equation}
By the Bourgain-Milman inequality (isomorphic version of reverse
Santal\'{o} inequality see \cite{BM, GPV, Na, Ku} or \cite{RZ}), there
exists an absolute constant $c>0$ with
$$
c^d \frac{4^d}{d!} \leq \mbox{\rm vol}_d(K)\mbox{\rm vol}_d(K^*)
$$
and so we get
\begin{equation}
\mbox{\rm vol}_d(K^*)^{\frac{m-d}{d}} \leq O(d)^{d-m}\mbox{\rm vol}_d(K)^{\frac{d-m}{d}}.
\end{equation}
Thus together with \eqref{eq:mink} and \eqref{eq:bound1} we obtain
\begin{equation}
\# K \leq O(1)^d\,d^{d-m}\, \max\{\# (K\cap H): H\in
{\rm G}_{{\mathbb Z}}(m,d)\}\, \mbox{\rm vol}_d(K)^{\frac{d-m}{d}}.
\end{equation}
\begin{flushright
\begin{remark} We notice that the methods used in Section 3,
i.e. computation via discrete version of the John theorem (Theorem
\ref{th:dj} from above), can also be used to provide a bound for
general co-dimensional sections. But such computation gives the estimate of order $O(d)^{7d/2}$ which is worse than the one in the above theorem.
\end{remark}
\begin{remark} Observe that Theorem \ref{tm:main} can be restated for an arbitrary $d$-dimensional lattice $\Lambda$: Let $\Lambda$ be a lattice in ${\mathbb R}^d$ and $K \subset {\mathbb R}^d$ be an origin-symmetric convex body with $\dim
(K\cap \Lambda)=d$. Then
\begin{equation}
\# K \leq O(1)^d\, d^{d-m} \max\{\# (K\cap H): H\in
{\rm G}_{\Lambda}(m,d)\}\,\, \left(\frac{\mbox{\rm vol}_d(K)}{\det(\Lambda)}\right)^{\frac{d-m}{d}}.
\end{equation}
\end{remark}
We also notice that the methods used in proofs of the Theorem \ref{th:unconditional} and Theorem \ref{tm:main} can be used to provide an estimate for the co-dimensional slices of an unconditional convex body:
\begin{theorem}
Let $K \subset {\mathbb R}^d$ be an unconditional convex body with $\dim
(K\cap{\mathbb Z}^d)=d$. Then
\begin{equation}
\# K \leq O(d)^{d-m} \max\{\# (K\cap H): H\in
{\rm G}_{{\mathbb Z}}(m,d)\}\,\, \mbox{\rm vol}_d(K)^{\frac{d-m}{d}}.
\end{equation}
\label{tm:unconmain}
\end{theorem}
\noindent{\bf Proof: } First we notice that if $K$ is an unconditional body and $H$ is a coordinate subspace of dimension $m$ (i.e. it is spanned by $m$ coordinate vectors) with $
K \cap (H + {\boldsymbol z}) \not = \emptyset,
$
then $
K \cap (H + {\boldsymbol z})
$ must be an unconditional convex body in $(H + {\boldsymbol z})$. Thus, using this property together with the proof of Theorem \ref{th:unconditional} we get that for any unconditional body $K$ and for any coordinate subspace $H$
$$
\#(K \cap H \cap {\mathbb Z}^d) \ge \#(K \cap (H+{\boldsymbol z})\cap {\mathbb Z}^d), \mbox{ for all } {\boldsymbol z} \in {\mathbb R}^d.
$$
Next we follow the steps of the proof of Theorem \ref{tm:main} and similarly to (\ref{eq:bound1})
get
$$
\# K \leq 3^{d-m} \#(K\cap \overline H) \,\prod_{i=1}^{d-m}\lambda_i^*.
$$
Finally, we finish the proof using Minkoswki's Second theorem and the Bourgain-Milman inequality.
\begin{flushright
\begin{remark} We also would like to test our estimates against two classical examples
\begin{enumerate}
\item[{\bf(A)}] For the cube $B_\infty^d = \{{\boldsymbol x}\in{\mathbb R}^d : |{\boldsymbol x}|_\infty\leq 1\}$ we have
$\# B_\infty^d =3^d$,\\
$\max\{\# (B_\infty^d\cap H): H\in {\rm G}_{{\mathbb Z}}(m,d)\}=3^m $, and $\mbox{\rm vol}_d(B_\infty^d)^{\frac{d-m}{d}} = 2^{d-m}$.
\item[{\bf (B)}] For the cross polytope $B_1^d = \{{\boldsymbol x}\in{\mathbb R}^d : |{\boldsymbol x}|_1\leq 1\}$ we have
$\# B_1^d =2\,d+1$,\\
$\max\{\# (B_1^d\cap H): H\in {\rm G}_{{\mathbb Z}}(m,d)\}=2\,m+1$, and $\mbox{\rm vol}_d(B_1^d)^{\frac{d-m}{d}} \sim \frac{c^{d-m}}{d^{d-m}}$.
\end{enumerate}
These examples show that we expect our constant to grow exponentially in the case of higher co-dimensional slices, though we do not expect our current estimates to be sharp.
\end{remark}
We finish this section with a remark about the relationship between the constant in the original slicing inequality, ${\mathcal L}_1$, and the constant in the discrete version, ${\mathcal L}_3$. Using the general idea from \cite{GGroZ} and Gauss's Lemma on the intersection of a large convex body with a lattice we will show that ${\mathcal L}_1 \le {\mathcal L}_3$.
Consider a convex symmetric body $K$ and let ${\mathcal L}_1(K)>0$ be such that
$$\mbox{\rm vol}_d(K)^{\frac {d-1}{d}} = {\mathcal L}_1(K) \max_{{\boldsymbol \xi} \in {\mathbb S}^{d-1}} \mbox{\rm vol}_{d-1}(K\cap {\boldsymbol \xi}^\bot).$$
Thus $${\mathcal L}_1=\max \{{\mathcal L}_1(K): K \subset {\mathbb R}^d, K \mbox{ is convex, origin-symmetric body, } d \ge 1\}.$$ Then $$\mbox{\rm vol}_d(K)^{\frac {d-1}{d}} \geq {\mathcal L}_1(K) \mbox{\rm vol}_{d-1}(K\cap {\boldsymbol \xi}^\bot), \,\, \forall {\boldsymbol \xi}\in {\mathbb S}^{d-1}.$$
Our goal is to study a section of $K$ with a maximal number of points from ${\mathbb Z}^d$, if $K\cap {\boldsymbol \xi}^\perp$ is such a section, then, without loss of generality, we may assume that ${\mathbb Z}^d \cap {\boldsymbol \xi}^\perp$ is a lattice of a full rank $d-1$. Indeed, if ${\mathbb Z}^d \cap {\boldsymbol \xi}^\perp$ has a rank less then $d-1$ we may rotate ${\boldsymbol \xi}$ to catch $d-1$ linearly independent vectors in ${\boldsymbol \xi}^\perp$, without decreasing the number of integer points in $K\cap {\boldsymbol \xi}^\perp$. Now, we may use Gauss's Lemma (see for example Lemma 3.22 in \cite{TV2}) to claim that for $r$ large enough we have
$$\#(rK) = r^d \mbox{\rm vol}_d(K) + O\left(r^{d-1}\right) \text{ and}$$
$$\#(rK \cap {\boldsymbol \xi}^\perp)=\frac{r^{d-1}\mbox{\rm vol}_{d-1}(K)}{\det\left({\mathbb Z}^d \cap {\boldsymbol \xi}^\perp\right)} + O\left(r^{d-2}\right).$$
Which we can rearrange to get the following two equations
$$\mbox{\rm vol}_{d}\left(K\right) = \frac{1}{r^{d}} \#\left(rK \right) + O\left(\frac{1}{r}\right) \text{ and}$$
$$\mbox{\rm vol}_{d-1}\left(K\cap {\boldsymbol \xi}^\perp\right) = \frac{1}{r^{d-1}} \#\left(rK \cap {\boldsymbol \xi}^\perp\right) \det\left({\mathbb Z}^d\cap {\boldsymbol \xi}^\perp\right) + O\left(\frac{1}{r}\right).$$
Next, using that $\det\left({\mathbb Z}^d \cap {\boldsymbol \xi}^\perp \right)\geq 1$ and $\mbox{\rm vol}_d(K) \geq {\mathcal L}_1(K) \mbox{\rm vol}\left(K \cap {\boldsymbol \xi}^\perp\right) \mbox{\rm vol}_d^{\frac{1}{d}}(K)$ we get
\begin{align*}
\frac{1}{r^{d}} \#\left(rK \right) \geq& {\mathcal L}_1(K) \left(\frac{1}{r^{d-1}}\right) \#\left(rK \cap {\boldsymbol \xi}^\perp\right) \det\left({\mathbb Z}^d\cap {\boldsymbol \xi}^\perp\right)\mbox{\rm vol}_d^{\frac{1}{d}}(K) + O\left(\frac{1}{r}\right)\\
\geq& {\mathcal L}_1(K) \left(\frac{1}{r^{d}}\right)\#\left(rK \cap {\boldsymbol \xi}^\perp\right)\mbox{\rm vol}_d^{\frac{1}{d}}(rK)+ O\left(\frac{1}{r}\right).
\end{align*}
Then for $\epsilon>0$ there is a sufficiently large $r_0$ such that for all $r>r_0$
$$\#(rK)\geq \left({\mathcal L}_1(K)-\epsilon\right)\#\left(rK \cap {\boldsymbol \xi}^\perp\right)\mbox{\rm vol}_d^{\frac{1}{d}}(rK).$$
So then if $\#(rK) \leq {\mathcal L}_3 \max_{{\boldsymbol \xi}\in {\mathbb S}^{d-1}} \# \left(rK \cap {\boldsymbol \xi}^\perp\right) \mbox{\rm vol}_d^{\frac{1}{d}}(rK)$ we have that ${\mathcal L}_1(K)-\epsilon \leq {\mathcal L}_3$ for all $d$, $\epsilon$, and bodies $K$.
Which leads us to conclude that ${\mathcal L}_1\leq {\mathcal L}_3$.
| 15,411 |
\section{Introduction}
The Marchenko-Pastur (MP) theorem \cite{MP} is one of the key results in the random matrix theory. It states that, with probability one, the empirical spectral distribution of a {\it sample covariance matrix}
\begin{equation}
\label{e8}
\wh \Sigma_n=\frac{1}{n}\sum_{k=1}^n \x_{pk}\x_{pk}^\top
\end{equation}
weakly converges to the MP law with parameter $\rho>0$ as $n\to\infty$ if $p=p(n)$ satisfies $p/n\to \rho$ and, for each $p,$ $\{\x_{pk}\}_{k=1}^n$ are i.i.d. copies of an isotropic $\bR^p$-valued random vector $\x_p$ satisfying certain assumptions (for definitions, see Section 2).
In the classical case, entries of $\x_p$ are assumed to be i.i.d. copies of some random variable with mean zero and unit variance (e.g., see Theorem 3.6 in \cite{BS}). In general, entries of $\x_p=(X_{p1},\ldots,X_{pp}) $ can be any independent random variables that have mean zero and unit variance and satisfy the Lindeberg condition
\begin{equation}\label{lin}
L_p(\varepsilon)=\frac{1}{p}\sum_{k=1}^p\e X_{pk}^2 I(|X_{pk}|>\varepsilon \sqrt{p})\to 0\quad\text{for all }\varepsilon>0\text{ as }p\to\infty
\end{equation}
(see \cite{P}). The independence assumption can be relaxed in a number of ways. E.g., in \cite{PP}, the MP theorem is proven for isotropic random vectors $\x_p$ with a centred log-concave distribution.
All mentioned assumptions imply that quadratic forms $\x_p^\top A_p\x_p$ concentrate near their expectations up to an error term $o( p)$ with probability $1-o(1)$ when $A_p$ is any $p\times p$ real matrix with the spectral norm $\|A_p\|=O(1)$. In fact, this condition is sufficient for the Marchenko-Pastur theorem (see \cite{BZ}, \cite{G}, \cite{PP}, Theorem 19.1.8 in \cite{PS}, and \cite{Y}).
Recently, it was also proved in \cite{CT} that extreme eigenvalues of $\wh\Sigma_n$ converge in probability to the edges of the support of the limiting Marchenko-Pastur law if a form of the concentration property for quadratic forms holds (for details, see \cite{CT}).
However, as noted in \cite{A}, the above concentration property for quadratic forms is {\it not} necessary in general. Namely, take $p=2q$ for $q=q(n)$ and consider \[\x_p=\sqrt{2}(\z_q \xi, \z_q (1-\xi)),\]
where $\z_q$ is a standard normal vector in $\bR^q$, $\xi$ is a random variable independent of $\z_q$, and $\p(\xi=0)=\p(\xi=1)=1/2$. Then $\wh\Sigma_n$ is a 2-block-diagonal matrix such that each block satisfies the MP theorem. It can be directly checked that $(\wh\Sigma_n)_{n=1}^\infty$ also satisfy the MP theorem and each $\x_p$ is an isotropic random vector for which the above concentration property doesn't hold.
There are many results related to the MP theorem, where some other dependence assumptions are considered. E.g., see \cite{A}, \cite{BM}, \cite{mp}, \cite{GNT}, \cite{PM}, \cite{MPP}, \cite{OR}, \cite{PSc}, \cite{Y}.
In the present paper we consider the case when not only $\x_p$ but also $C_q\x_p$ satisfies the MP theorem for each sequence $(C_q)_{p=1}^\infty$, where $q=q(p)\leqslant p$ and $C_q$ is a $q\times p$ matrix with $C_qC_q^\top =I_q$ for the $q\times q$ identity matrix $I_q$. We prove that the weak concentration property for quadratic forms is a necessary and sufficient assumption in this case. In addition, we derive this property under quite general assumptions recently studied in \cite{PM}. We also show that this property implies some other results in the random matrix theory beyond the MP theorem.
The paper is structured as follows. Section 2 contains some preliminaries, main assumptions, and notation. Main results are presented in Section 3. Section 4 deals with proofs. Some additional results are given in Appendices.
\section{Preliminaries and notation}
We now introduce assumptions and notation that will be used throughout the paper.
For each $p\geqslant 1,$ let $\x_p$ be a random vector in $\bR^p$. We call $\x_p$ isotropic if $\e\x_p\x_p^\top=I_p$ for the $p\times p$ identity matrix $I_p$. Let also $ \X_{pn}$ be a $p\times n$ matrix with columns $\{\x_{pk}\}_{k=1}^n$ that are i.i.d. copies of $\x_p$, unless otherwise stated. Then, for $\Sigma_n$ given in \eqref{e8},
\[\wh\Sigma_n=\frac{1}{n}\X_{pn}\X_{pn}^\top.\]
Define also the MP law $\mu_\rho$ with parameter $\rho>0$ by
\[d\mu_\rho=\max\{1-1/\rho,0\}\,d\delta_0+\frac{\sqrt{(b-x)(x-a)}}{2\pi x\rho }I(x\in [a,b])\,dx, \]
where $\delta_c$ is a Dirac function with mass at $c$, $a=(1-\sqrt{\rho})^2,$ and $b=(1+\sqrt{\rho})^2.$ Set also $\mathbb C^+=\{z\in\mathbb C:\,\Im(z)>0\}$.
For a real symmetric $p\times p$ matrix $A$ with eigenvalues $\lambda_1,\ldots,\lambda_p$, its empirical spectral distribution is defined by \[\mu_A=\frac{1}{p}\sum_{k=1}^p\delta_{\lambda_k}.\]
If, in addition, $A$ is positive semi-definite, then $A^{1/2}$ will be the principal square root of $A$. If $A$ is a complex rectangular matrix, then $\|A\|$ will be the spectral norm of $A$, i.e. $\|A\|=(\lambda_{\max}(A^*A))^{1/2}$, where $A^*=\overline{A}^\top$ and $\lambda_{\max}$ denotes the maximal eigenvalue. All matrices below will be real, unless otherwise stated. Let also $\|v\|$ be the Euclidean norm of $v=(v_1,\ldots,v_p)\in\mathbb C^p$, i.e. $\|v\|=\big(\sum_{i=1}^p|v_i|^2\big)^{1/2}.$
Consider the following assumptions.
$(\mathrm A0)$ $p=p(n)$ satisfies $p/n\to \rho$ for some $\rho>0$ as $n\to\infty$.
$(\mathrm A1)$ $(\x_p^\top A_p\x_p-\tr( A_p))/p\pto 0$ as $p\to\infty$ for all sequences of real symmetric positive semi-definite $p\times p$ matrices $A_p$ with uniformly bounded spectral norms $\|A_p\|$.
Assumption $(\mathrm A1)$ is a form of the weak law of large numbers for quadratic forms. Stronger forms of $(\mathrm A1)$ (with convergence in $L_2$ instead of convergence in probability) studied in the papers \cite{BZ}, \cite{PP}, and in the book of \cite{PS} (see Chapter 19). In the special case of isotropic $\x_p$, $\e (\x_p^\top A_p\x_p)=\tr( A_p)$ and $(\mathrm A1)$ states that $\x_p^\top A_p\x_p$ concentrates near its expectation up to a term $o(p)$ with probability $1-o(1)$ when $\|A_p\|=O(1)$. It is proved in \cite{Y1} (see also Lemma \ref{l2} in Section 4) that
\begin{align} \label{equiv}
&\text{if each $\x_p$ has independent entries with mean zero and unit variance,}\nonumber\\
&\text{then $(\mathrm A1)$ is equivalent to the Lindeberg condition \eqref{lin}.}
\end{align} For general isotropic $\x_p$, we can equivalently reformulate $(\mathrm A1)$ as follows (for a proof, see Appendix A).
\begin{proposition}\label{p2}
If $\x_p,$ $p\geqslant 1,$ are isotropic, then $(\mathrm A1 )$ holds iff $(\mathrm A1^*)$ holds, where
\begin{quote}
$(\mathrm A1^*)$ $(\x_p^\top \Pi_p\x_p-\tr( \Pi_p))/p\pto 0$ as $p\to\infty$ for all sequences of $p\times p$ orthogonal projection matrices matrices $\Pi_p$.
\end{quote}
\end{proposition}
We will also need a more general form of $(\text{A1})$ designed for non-isotropic $\x_p$ (namely, when $\e( \x_p\x_p^\top)=\Sigma_p\neq I_p$). For each $p\geqslant 1,$ let $\Sigma_p$ be a $p\times p$ symmetric positive semi-definite matrix $\Sigma_p$. Consider the following assumptions.
$(\text{A2})$ $(\x_p^\top A_p\x_p-\tr(\Sigma_p A_p))/p\pto 0$ as $p\to\infty$ for all sequences of real symmetric positive semi-definite $p\times p$ matrices $A_p$ with uniformly bounded spectral norms $\|A_p\|$.
$(\text{A3})$ $\tr(\Sigma_p^2)/p^2\to 0$ as $p\to\infty$.
\\
The next proposition shows that $(\text{A2})$ and $(\text{A3})$ are equivalent in the Gaussian case (for a proof, see Appendix A).
\begin{proposition}\label{p1}
For each $p\geqslant 1,$ let $\x_p$ be a Gaussian vector with mean zero and variance $\Sigma_p$. Then $(\mathrm A2)$ holds if and only if $(\mathrm A3)$ holds.
\end{proposition}
We now give a particular and quite general example of $\x_p,$ $p\geqslant 1,$ satisfying $(\mathrm A2)$.
\begin{proposition}\label{p3}
For each $p\geqslant 1,$ let $\x_p=(X_{p1},\ldots,X_{pp})$ be a random vector with mean zero and variance $\Sigma_p$. Suppose there is a nonincreasing sequence $\{\Gamma_j\}_{j=0}^\infty$ such that $\Gamma_j\to0, $ $j\to\infty$,
\[\e|\e(X_{pk}|\cF_{k-j}^p)|^2\leqslant \Gamma_j\quad \text{and}\quad \e|\e(X_{pk}X_{pl}|\cF_{k-j}^p)-\e(X_{pk}X_{pl})|\leqslant \Gamma_j\]
for all $1\leqslant k\leqslant l\leqslant p$ and $j=0,\ldots,k$, where $\cF_l^p=\sigma(X_{pk},k\leqslant l),$ $l\geqslant 1,$ and $\cF_0^p$ is the trivial $\sigma$-algebra. If \eqref{lin} holds, then $(\mathrm A2)$ holds.
\end{proposition}
Proposition \ref{p3} is closely related to Theorem 5 in \cite{PM}, where the same dependence conditions are considered, but another result is proven. We will prove Proposition \ref{p3} using Lindeberg's method and Bernstein's block technique as in the proof of Theorem 5 in \cite{PM} (see Appendix A).
Let us also give other versions of $(\text{A2})$ and $(\text{A3})$ allowing some dependence and heterogeneity in $\x_{pk}$ over $k$. For $\bR^p$-valued random vectors $\{\x_{pk}\}_{k=1}^n$, let $\cF_0^p$ be the trivial $\sigma$-algebra and $\cF_{k}^p=\sigma(\x_{pl},1\leqslant l\leqslant k)$. For given symmetric positive semi-definite $p\times p$ matrices $\{\Sigma_{pk}\}_{k=1}^n$, introduce the following assumption.
$(\mathrm A2^*)$ For all $\varepsilon>0$ and every stochastic array $\{A_{pk},p\geqslant k\geqslant 1\}$ with symmetric positive semi-definite symmetric $\cF_{k-1}^p$-measurable random $p\times p$ matrices $A_{pk}$ having $\|A_{pk}\|\leqslant 1$ a.s.,
\[\frac{1}{n}\sum_{k=1}^n\p\big(|\x_{pk}^\top A_{pk}\x_{pk}-\tr(\Sigma_{pk}A_{pk})|>\varepsilon p\big)\to0.\]
$(\mathrm A3^*)$ $(np^2)^{-1}\sum_{k=1}^n\tr(\Sigma_{pk}^2)\to 0$ as $p,n\to\infty$.
In fact, $(\mathrm A2^*)$ and $(\mathrm A3^*)$ are {\it average} versions of $(\mathrm A2)$ and $(\mathrm A3)$, respectively. In particular, if $\x_{pk}$ has independent centred entries and variance $\Sigma_{pk}$ for all $p,k$, then $(\mathrm A2^*)$ follows from $(\mathrm A3^*)$ and the (second) Lindeberg condition
\begin{equation*}\label{lin2}
\frac{1}{np}\sum_{k=1}^n\sum_{j=1}^p\e X_{kj}^2 I(|X_{kj}|>\varepsilon \sqrt{p})\to 0\quad\text{for all\quad}\varepsilon>0
\end{equation*}
as $p,n\to\infty$, where $X_{kj}=X_{kj}(p),$ $j=1,\ldots,p$, are entries of $\x_{pk}$. The latter can be checked directly as in the proof of Proposition 2.1 in \cite{Y1}.
Finally, we introduce the key limit property for a sequence of random matrices. Let $p=p(n)$ be such that $p/n\to\rho>0 $ and, for each $n,$ let $M_n$ be a symmetric positive semi-definite $p\times p$ matrix. We say that $(M_n)_{n=1}^\infty$ satisfies $(\mathrm {MP})$ if,
\begin{quote} with probability one, $\mu_{C_qM_nC_q^\top }$ weakly converges to $\mu_{\rho_1}$
as $n\to\infty$ for all $\rho_1\in(0,\rho]$ when $q=q(n)\leqslant p(n)$ satisfies $q/n\to \rho_1$ and $(C_q)_{n=1}^\infty$ is any sequence of matrices such that the size of $C_q$ is $q\times p$ and $C_q C_q^\top=I_q$.
\end{quote}
\section{Main results}
First, we derive necessary and sufficient conditions in the classical setting.
\begin{theorem}\label{t3} For each $p\geqslant 1,$ let $\x_p$ be isotropic and have centred independent entries. If $p=p(n)$ satisfies $(\mathrm A0)$, then $ \mu_{\wh\Sigma_n} $ weakly converges to $\mu_\rho$ almost surely as $n\to\infty$ iff \eqref{lin} holds for given $p=p(n)$.
\end{theorem}
This result is not new. As far as we know, it was initially established by Girko via a different and less transparent method (e.g., see Theorem 4.1 in \cite{G1}). In our proof of Theorem \ref{t3}, the necessity part follows from the lemma below.
\begin{lemma}\label{ml1}
Let $(\mathrm A0)$ hold and $\x_p$ be isotropic for all $p=p(n)$. If, with probability one, $\mu_{\wh \Sigma_n}$ weakly converges to $\mu_\rho$ as $n\to\infty$, then, for given $p=p(n)$,
\begin{equation}\label{e9}
\frac{ \x_p^\top \x_p-1}{p}\pto 0,\quad n\to\infty.
\end{equation}
\end{lemma}
The classical independence setting differs a lot from the general case of isotropic distributions. Namely, when entries of each $\x_p$ are independent and orthonormal, \eqref{e9} is equivalent to $(\mathrm A1)$ (see the proof of Theorem \ref{t3}). In general, \eqref{e9} doesn't imply $(\mathrm A1)$ as in the counterexample from the Introduction. To get $(\mathrm A1)$, we need more than convergence of $\mu_{\wh\Sigma_n}$, e.g., $(\mathrm{MP})$.
We now state the main result of this paper.
\begin{theorem}\label{t2} Let $(\mathrm A0)$ hold and $\wh\Sigma_n,$ $n\geqslant 1,$ be as in \eqref{e8}. If $(\mathrm A1)$ holds, then $(\wh\Sigma_n)_{n=1}^\infty$ satisfies $(\mathrm{MP})$. Conversely, if the latter holds and $\x_p$ is isotropic for all $p=p(n)$, then $(\mathrm A1)$ holds for given $p=p(n)$.
\end{theorem}
The proof of the necessity part in Theorem \ref{t2} follows from Lemma \ref{ml1}. The sufficiency part can be proved directly as in \cite{Y}, but we prefer to derive it from a more general result.
\begin{lemma}\label{ml}
Let $(\mathrm A2)$ and $(\mathrm A3)$ hold for some $\Sigma_p,$ $p\geqslant 1$. If $(A0)$ holds for some $p=p(n)$, then, for all $z\in\mathbb C^+$,
\begin{equation}\label{st}
\lim_{n\to\infty}\frac{1}{p}\Big[\tr\big(n^{-1}\wh\X_{pn}\wh\X_{pn}^\top+B_p-zI_p\big)^{-1}- \tr\big(n^{-1}\wh\Z_{pn}\wh\Z_{pn}^\top+B_p-zI_p\big)^{-1}\Big]=0
\end{equation} with probability one, where, for each $n\geqslant 1,$ \begin{itemize}
\item[] $\wh\X_{pn}=\X_{pn}+C_{pn}$ and $ \wh\Z_{pn}=\Z_{pn}+C_{pn},$
\item[] $\Z_{pn}$ is a $p\times n$ matrix with i.i.d. centred Gaussian columns having variance $\Sigma_p$,
\item[] $B_p$ is a $p\times p$ nonrandom matrix with $\|B_p\|=O(1),$
\item[] $C_{pn}$ is a $p\times n$ nonrandom matrix such that $\|n^{-1}C_{pn}C_{pn}^\top\|=O(1).$
\end{itemize}
\end{lemma}
\begin{remark}\label{r2}{\normalfont By the definition of $\mu_{A}$,
\[\int_{-\infty}^\infty\frac{ \mu_{A}(d\lambda)}{\lambda -z}=\frac{1}{p} \tr\big(A-zI_p)^{-1},\quad z\in\mathbb C^+,\] where $A$ is a symmetric $p\times p$ matrix. Therefore, by the Stieltjes continuity theorem (e.g., see Exercise 2.4.10 in \cite{Tao}), \eqref{st} implies that if $\mu_{\wh\Sigma_n}$ with $\wh\Sigma_n=n^{-1}\wh\Z_{pn}\wh\Z_{pn}^\top+B_p$ weakly converges to some measure $\mu $ a.s., then $\mu_{\Sigma_n}$ with $\Sigma_n=n^{-1}\wh\X_{pn}\wh\X_{pn}^\top+B_p$ weakly converges to the same measure $\mu $ a.s.}
\end{remark}
We now consider a generalization of Lemma \ref{ml} allowing some dependence and heterogeneity in $\x_{pk}$ over $k$. Let $\X_{pn}$ be a $p\times n$ random matrix with columns $\{\x_{pk}\}_{k=1}^n$. Let also $\{\Sigma_{pk}\}_{k=1}^n$ be symmetric positive semi-definite $p\times p$ matrices and
$\Z_{pn}$ be a $p\times n$ Gaussian random matrix whose $k$-th column $\z_{pk}$, $k=1,\ldots,n$, has mean zero and variance $\Sigma_{pk}$. Assume also that $\Z_{pn}$ and $\X_{pn}$ are independent.
\begin{lemma}\label{ml2}
Let $(\mathrm A2^*)$, $(\mathrm A3^*)$, and $(\mathrm A0)$ hold for some $p=p(n)$. Then, for all $z\in\mathbb C^+$, \eqref{st} with convergence in probability holds, where $B_p$, $C_{pn}$, $\wh\X_{pn}$, and $\wh \Z_{pn}$ are as in Lemma \ref{ml}.
\end{lemma}
\section{Proofs}
\noindent{\bf Proof of Theorem \ref{t3}.} By \eqref{equiv}, the Lindeberg method \eqref{lin} is equivalent to $(\mathrm A1)$ (see also Lemma \ref{l2} below). However, the proof of this proposition is still valid without the assumption that $\x_p$ has mean zero. Now, if $(\mathrm A1)$ holds, then $\mu_{\wh\Sigma_n}$ weakly converges to $\mu_\rho$ almost surely as $n\to\infty$ by Theorem \ref{t2}.
Suppose $\mu_{\wh\Sigma_n}$ weakly converges to $\mu_\rho$ a.s. We need to prove \eqref{lin}. By Lemma \ref{ml1}, \eqref{e9} holds. By the Gnedenko-Kolmogorov necessary and sufficient conditions for relative stability (see (A) and (B) in \cite{H}), \eqref{lin} is equivalent to \eqref{e9}. This finishes the proof of the theorem. Q.e.d.
\noindent{\bf Proof of Lemma \ref{ml1}.} We will proceed as in \cite{Y1}. Let $\x_p=\x_{p,n+1}$ be independent of the matrix $\X_{pn}$ and distributed as its columns $\{\x_{pk}\}_{k=1}^n$. Define also \[A_n=n\wh\Sigma_n=\X_{pn}\X_{pn}^\top=\sum_{k=1}^n\x_{pk}\x_{pk}^\top\quad \text{and }\quad B_n=A_n+\x_{p}\x_{p}^\top=\sum_{k=1}^{n+1}\x_{pk}\x_{pk}^\top.\]
Fix $\varepsilon>0.$ The matrix $B_n +\varepsilon nI_p$ is non-degenerate and
\[p=\tr\big((B_n +\varepsilon nI_p)(B_n +\varepsilon nI_p)^{-1}\big)=\sum_{k=1}^{n+1}\x_{pk}^\top(B_n +\varepsilon nI_p)^{-1}\x_{pk}+\varepsilon n\,\tr(B_n +\varepsilon nI_p)^{-1}.\]
Taking expectations and using the exchangeability of $\{\x_{pk}\}_{k=1}^{n+1}$,
\begin{align}\label{e1}
p=&(n+1)\e \x_p^\top(B_n +\varepsilon nI_p)^{-1}\x_p+\varepsilon n\,\e \tr(B_n +\varepsilon nI_p)^{-1}.
\end{align}
Define $f_n(\varepsilon)=\tr(A_n+\varepsilon nI_p)^{-1}$. By Facts 5 and 7 in Appendix C,
\[\e\tr(B_n+\varepsilon nI_p)^{-1}=\e f_n(\varepsilon)+o(1)\quad\text{and}\quad \e \x_p^\top(B_n +\varepsilon nI_p)^{-1}\x_p=O(1).\]
Thus,
\begin{equation}
\label{e3}
p/n=\e \x_p^\top(B_n +\varepsilon nI_p)^{-1}\x_p+\varepsilon \e \tr(A_n +\varepsilon nI_p)^{-1}+o(1).
\end{equation}
Let now $\Z_{pn}$ be a $p\times n$ matrix with i.i.d. $\mathcal N(0,1)$ entries. By the MP theorem, $\mu_{\widetilde\Sigma_n}$ weakly converges to $\mu_\rho$ a.s., where $\widetilde\Sigma_n=n^{-1}C_n$ and $C_n=\Z_{pn}\Z_{pn}^\top$. Therefore, by the Stieltjes continuity theorem (see Theorem B.9 in \cite{BS}),
\[\p(S_n(z)-s_n(z)\to 0 \text{\;\;for all }z\in\mathbb C^+)=1,\]
where $S_n(z)$ and $s_n(z)$ are the Stieltjes transforms defined by
\[S_n(z)=\int_{-\infty}^\infty\frac{ \mu_{\wh\Sigma_n}(d\lambda)}{\lambda -z}=p^{-1}\tr\big(\wh\Sigma_n-zI_p\big)^{-1}=p^{-1}\tr\big(n^{-1}A_n-zI_p\big)^{-1},\]
\[s_n(z)=\int_{-\infty}^\infty\frac{ \mu_{\wt\Sigma_n}(d\lambda)}{\lambda -z}=p^{-1}\tr\big(\wt \Sigma_n-zI_p\big)^{-1}=p^{-1}\tr\big(n^{-1}C_n-zI_p\big)^{-1}.\]
By Fact 8 in Appendix C and $p/n\to \rho >0,$ the latter implies that
\begin{equation}
\label{e101}
\p(\tr(A_n +\varepsilon nI_p)^{-1}-\tr(C_n +\varepsilon nI_p)^{-1}\to 0\text{\;\;for all }\varepsilon>0)=1.\end{equation}
Therefore, by the dominated convergence theorem,
\[
\e \tr(A_n +\varepsilon nI_p)^{-1}=\e \tr(C_n +\varepsilon nI_p)^{-1}+o(1)\text{\;\;for all }\varepsilon>0.
\]
In addition, arguing as above, we derive
\begin{equation}
\label{e4}
p/n=\e \z_p^\top(D_n +\varepsilon nI_p)^{-1}\z_p+\varepsilon \e \tr(C_n +\varepsilon nI_p)^{-1}+o(1),
\end{equation}
where $D_n=C_n+\z_p\z_p^\top$ and $\z_p$ is a $\bR^p$-valued standard normal vector independent of $C_n$. Subtracting \eqref{e3} from \eqref{e4}, we get
\[\e \x_p^\top(B_n +\varepsilon nI_p)^{-1}\x_p=\e \z_p^\top(D_n +\varepsilon nI_p)^{-1}\z_p+o(1)\quad\text{for all }\varepsilon>0.\]
By Fact 7 in Appendix C and \eqref{e5} (see Appendix B),
\[\e \z_p^\top(D_n +\varepsilon nI_p)^{-1}\z_p=\e \frac{Z_n}{1+Z_n}=\e \frac{\e (Z_n|C_n)}{1+\e (Z_n|C_n) }-R_n,\]
where $Z_n=\z_p^\top(C_n +\varepsilon nI_p)^{-1}\z_p$ and
\begin{align*}
R_n=\e\frac{(Z_n-\e (Z_n|C_n))^2}{(1+Z_n)(1+\e (Z_n|C_n))^2}&\leqslant \e (Z_n-\e( Z_n|C_n))^2=\e \var(Z_n|C_n)
\end{align*}
Since $\z_p$ and $C_n$ are independent, $\e (Z_n|C_n)=\tr(C_n +\varepsilon nI_p)^{-1},$ and \[\|(C_n +\varepsilon nI_p)^{-1}\|\leqslant (\varepsilon n)^{-1},\]
we have $\e \var(Z_n|C_n)=o(1)$ by \eqref{v1} (see Appendix A). Hence, $R_n\to 0.$
By \eqref{e101} and the dominated convergence theorem,
\begin{align*}
\e\frac{\e(Z_n|C_n)}{1+\e(Z_n|C_n)}&=\e\frac{ \tr(A_n +\varepsilon nI_p)^{-1}}{1+ \tr(A_n +\varepsilon nI_p)^{-1}}+o(1).
\end{align*}
Combining the above relations with Fact 7 in Appendix C yields
\begin{align*}\e \x_p^\top(B_n +\varepsilon nI_p)^{-1}\x_p&=\e\frac{ \x_p^\top(A_n +\varepsilon nI_p)^{-1}\x_p}{1+\x_p^\top(A_n +\varepsilon nI_p)^{-1}\x_p}\\
&=\e\frac{ \tr(A_n +\varepsilon nI_p)^{-1}}{1+ \tr(A_n +\varepsilon nI_p)^{-1}}+o(1).
\end{align*}
Additionally, $\e(\x_p^\top (A_n +\varepsilon nI_p)^{-1}\x_p |A_n)=\tr(A_n +\varepsilon nI_p)^{-1}$ a.s. by the independence of $\x_p$ and $A_n.$
\begin{lemma} \label{l4}
For each $n\geqslant 1,$ let $Z_n$ be a random variable such that $Z_n\geqslant 0$ a.s. and $\e Z_n$ is bounded over $n$. If $Y_n,$ $n\geqslant 1,$ are such random elements that
\[\e\frac{Z_n}{1+Z_n}-\e\frac{\e (Z_n|Y_n) }{1+\e (Z_n|Y_n)}\to 0,\quad n\to\infty,\]
then $Z_n-\e(Z_n|Y_n) \pto0.$
\end{lemma}
The proof of Lemma \ref{l4} is given in Appendix B. Using Lemma \ref{l4}, we conclude that
$\x_p^\top(A_n +\varepsilon nI_p)^{-1}\x_p-\tr(A_n +\varepsilon nI_p)^{-1}\pto 0.$
Multiplying by $\varepsilon$ and $n/p,$ we finally arrive at
\[p^{-1}(\x_p^\top(\varepsilon^{-1}\wh\Sigma_n + I_p)^{-1}\x_p-\tr(\varepsilon^{-1}\wh\Sigma_n +I_p)^{-1})\pto 0\quad\text{for all $\varepsilon>0,$}\]
where $\wh\Sigma_n=A_n/n$ is independent of $\x_p$.
Thus, we can find $\varepsilon_n$ that slowly tend to infinity and are such that
\[J_n=p^{-1}(\x_p^\top(\varepsilon_n^{-1}\wh\Sigma_n + I_p)^{-1}\x_p-\tr(\varepsilon_n^{-1}\wh\Sigma_n +I_p)^{-1})\pto 0.\]
We know that $\mu_{\wh\Sigma_n}$ weakly converges to $\mu_\rho$ a.s. The support of $\mu_\rho$ is bounded. Hence,
writing $\varepsilon_n^{-1}\wh\Sigma_n=\sum_{k=1}^p\lambda_ke_ke_k^\top$ for some $\lambda_k=\lambda_k(n)\geqslant0$ and orthonormal vectors $e_k=e_k(n)\in\bR^p$, $k=1,\ldots,p,$ we conclude that
\[\frac{1}{p}\sum_{k=1}^pI(\lambda_k>\delta_n)\pto0\]
when $\delta_n=K\varepsilon_n^{-1}\to0$ and $K>0$ is large enough. In addition,
\[J_n-\frac{\x_p^\top \x_p-1}{p}=U_n+V_n,\]
where
\[U_n=\frac{1}{p}\sum_{k:\,\lambda_k\leqslant\delta_n}((\x_p,e_k)^2-1)\bigg(\frac{1}{\lambda_k+1}-1\bigg),\]
\[V_n=\frac{1}{p}\sum_{k:\,\lambda_k>\delta_n}((\x_p,e_k)^2-1)\bigg(\frac{1}{\lambda_k+1}-1\bigg).\]
We finish the proof by showing that $U_n\pto 0$ and $V_n\pto0.$
By the independence of $\wh\Sigma_n$ and $\x_p,$ we have $\e((\x_p,e_k)^2|\wh\Sigma_n)=e_k^\top e_k=1$. In addition, \[\e |U_n|=\e[ \e(|U_n||\wh\Sigma_n)]\leqslant \frac{2}p\,\e\sum_{k:\,\lambda_k\leqslant\delta_n}\frac
{\lambda_k}{\lambda_k+1}\leqslant 2\delta_n=o(1),\]
\[\e |V_n|=\e[ \e(|V_n||\wh\Sigma_n)]\leqslant \frac{2}p\e\sum_{k=1}^p I(\lambda_k>\delta_n)=o(1).\]
Finally, we conclude that $( \x_p^\top \x_p-1)/p=J_n-(U_n+V_n)\pto0.$ Q.e.d.
\noindent{\bf Proof of Theorem \ref{t2}.} Let $(\mathrm A1)$ hold.
Fix some $q=q(n)$ such that $q\leqslant p$ and $q/n\to\rho_1>0.$ For each $q,$ let $C_q$ be a $q\times p$ matrix with $C_q C_q^\top=I_q.$ For such $C_q,$ $(\mathrm A1)$ implies that $(\mathrm A2)$ holds for $(\x_p,\Sigma_p,p)$ replaced by $(C_q\x_p,I_q,q)$. By Lemma \ref{ml} (without $B_p$ and $C_{pn}$) and Remark \ref{r2}, $\mu_{C_q \wh\Sigma_n C_q^\top}$ and $\mu_{\widetilde\Sigma_n}$ weakly converge to the same limit a.s., where $\widetilde\Sigma_n= n^{-1}\Z_{qn}\Z_{qn}^\top$ and $\Z_{qn}$ is a $q\times n$ random matrix with i.i.d. $\mathcal N(0,1)$ entries. By the MP theorem, $\mu_{\widetilde \Sigma_n}$ weakly converges to $\mu_{\rho_1}$ a.s. Thus, $(\wh\Sigma_n)_{n=1}^\infty$ satisfies $(\mathrm{MP})$.
Let now $(\wh\Sigma_n)_{n=1}^\infty$ satisfies $(\mathrm {MP})$ and let $\x_p$ be isotropic for each $p=p(n)$. We need to show that $(\mathrm A1)$ holds for given $p=p(n)$. By Proposition \ref{p2}, $(\mathrm A1)$ is equivalent to $(\mathrm A1^*)$. Let us verify $(\mathrm A1^*)$. Fix any sequence of orthogonal projectors $\Pi_p, $ $p=p(n),$ such that the size of $\Pi_p$ is $p\times p$. We need to show that
\[\frac{1}{p}\big(\x_p^\top \Pi_p \x_p-\tr(\Pi_p)\big)\pto0.\]
The latter is equivalent to the fact that
\[\frac{1}{p_k}\big(\x_{p_k}^\top \Pi_{p_k} \x_{p_k}-\tr(\Pi_{p_k})\big)\pto0\]
for any subsequence $\Pi_{p_k}$ such that $\tr(\Pi_{p_k})/p_k $ has a limit as $k\to\infty$. If this limit is zero, then, obviously, the above convergence holds.
Assume further w.l.o.g. that $q/n$ (or, equivalently, $q/p$) has a positive limit, where $q=\tr(\Pi_p)$. Namely, let $q/n\to \rho_1>0 $ (and $q/p\to\rho_1/\rho$). Write $\Pi_p=C_p^\top D_pC_p ,$ where $C_p$ is a $p\times p$ orthogonal matrix and $D_p$ is a diagonal matrix whose diagonal entries are $1,\ldots,1,0,\ldots,0$ with $q$ ($=\tr(\Pi_p)$) ones. Let $C_{qp}$ is the $q\times p$ upper block of $C_p$. Then
$\x_p^\top \Pi_p \x_p=(C_{qp}\x_p)^\top C_{qp}\x_p$.
By $(\mathrm{MP})$, $\mu_{C_{qp}\Sigma_n C_{qp}^\top}$ weakly converges to $\mu_{\rho_1}$ a.s. In addition, $C_{qp}\x_p$ is isotropic, since $\e C_{qp}\x_p(C_{qp}\x_p)^\top=C_{qp}C_{qp}^\top=I_q.$ Hence, by Lemma \ref{ml1}, we get
\[\frac{1}{q}\big((C_{qp}\x_p)^\top C_{qp}\x_p-q\big)=(\rho/ \rho_1+o(1))\frac{1}{p}\big(\x_p^\top \Pi_p \x_p-\tr(\Pi_p)\big)\pto0.\]
This proves that $(\mathrm A1^*)$ holds. Q.e.d.
{\bf Proof of Lemma \ref{ml}.} Fix $z\in\mathbb C^+.$
Assume w.l.o.g. that $B_p$ is positive semi-definite (we can always replace $B_p$ by $B_p+mI_p$ for $m=\sup_{p}\|B_p\|$ and change $z$ to $z+m$). First we proceed as in Step 1 of the proof of Theorem 1.1 in \cite{BZ} (see also the proof of (4.5.6) on page 83 in \cite{BS}) to show that $S_n(z)-\e S_n(z)\to 0$ a.s. as $n\to\infty,$ where
\[S_n(z)=p^{-1}\tr\big(n^{-1}\wh\X_{pn}\wh\X_{pn}^\top+B_p-zI_p\big)^{-1}.\] Similar arguments yield that $s_n(z)-\e s_n(z)\to 0$ a.s. as $n\to\infty,$ where \[s_n(z)=p^{-1}\tr\big(n^{-1}\wh\Z_{pn}\wh\Z_{pn}^\top+B_p-zI_p\big)^{-1}.\] Hence, we only need to show that $\e S_n(z)-\e s_n(z)\to 0$.
First, we consider the case when all entries of $C_{pn}$ are zeros, i.e. $\wh \X_{pn}=\X_{pn}$.
Let $\X_{pn}$ and $\Z_{pn}$ be independent. We will use Lindeberg's method as in the proof of Theorem 6.1 in \cite{G}.
Recall that
\[\X_{pn}\X_{pn}^\top=\sum_{k=1}^n\x_{pk}\x_{pk}^\top\quad\text{and}\quad
\Z_{pn}\Z_{pn}^\top=\sum_{k=1}^n\z_{pk}\z_{pk}^\top,\]
where $\{\x_{pk}\}_{k=1}^n$ and $\{\z_{pk}\}_{k=1}^n$ are columns of $\X_{pn}$ and $\Z_{pn}$, respectively. If $\z_p$ is a centred Gaussian vector with variance $\Sigma_p$ that is independent of $\x_p,$ then $\{(\x_k,\z_k)\}_{k=1}^n$ are i.i.d. copies of $(\x_p,\z_p)$. In what follows, we omit the index $p$ and, for example, write $(\x_k,\z_k)$ instead of $(\x_{pk},\z_{pk})$.
Let us now prove that
\begin{equation}\label{e102}
\e|S_n(z)-s_n(z)|\to 0.
\end{equation}
Using this representation, we derive that
\begin{align*}
S_n(z)&=\frac{1}{p}\tr\Big(n^{-1}\sum_{k=1}^n\x_{k}\x_{k}^\top+B_p-z I_p\Big)^{-1},\\
s_n(z)&=\frac{1}{p}\tr\Big(n^{-1}\sum_{k=1}^n\z_{k}\z_{k}^\top+B_p-z I_p\Big)^{-1},
\end{align*}
and $|S_n(z)-s_n(z)|\leqslant \sum_{k=1}^n|\Delta_{k}|/p,$ where
\begin{align*}
\Delta_{k} =\tr\Big(C_{k}+\frac{\x_{k}\x_{k}^\top }n+B_p-z I_p\Big)^{-1}-\tr\Big(C_{k}+ \frac{\z_{k}\z_{k}^\top }{n}+B_p-z I_p\Big)^{-1}
\end{align*}
for
$C_{1}=\sum_{i=2}^n \z_{i}\z_{i}^\top /n,$ $C_{n}=\sum_{i=1}^{n-1} \x_{i}\x_{i}^\top /n,$ and
\begin{equation}\label{e103}
C_{k}=\frac{1}{n}\sum_{i=1}^{k-1} \x_{i}\x_{i}^\top+\frac{1}{n}\sum_{i=k+1}^n \z_{i}\z_{i}^\top ,\quad 1<k<n.
\end{equation}
By Fact 7 in Appendix C,
\[
\tr(C+ww^\top-zI_p)^{-1}-\tr(C-zI_p)^{-1}=-\frac{w^\top (C-zI_p)^{-2}w}{1+w^\top (C-zI_p)^{-1}w}
\] for any real $p\times p$ matrix $C$ and $w\in\bR^p$. Adding and subtracting $\tr(C_{k}+B_p-z I_p)^{-1}$ yield
\[\Delta_{k} =-\frac{ \x_{k}^\top A_{k}^{2}\x_{k}/n}{1+ \x_{k}^\top A_{k}\x_{k}/n}+\frac{ \z_{k}^\top A_{k}^2\z_{k}/n}{1+\z_{k}^\top A_{k}\z_{k}/n},\] where we set $A_{k}=A_{k}(z)=(C_{k}+B_p-zI_p)^{-1}$, $1\leqslant k\leqslant n.$
Let us show that \[\frac1p\sum_{k=1}^n\e|\Delta_{k}|\to 0.\]
The latter implies that $\e |S_n(z)- s_n(z)|\to 0$.
Fix some $k\in\{1,\ldots,n\}$. For notational simplicity, we will further write \[\x_p,\z_p,A,\Delta,C\quad \text{instead of}\quad \x_{k},\z_k,A_{k},\Delta_{k},C_{k}\] and use the following properties: $C+B_p$ is a real symmetric positive semi-definite $p\times p$ random matrix, $(\x_p,\z_p)$ is independent of $A=(C+B_p-zI_p)^{-1}$.
Fix any $\varepsilon>0$ and let further $v=\Im(z)$ ($ >0$). Take
\[D=\bigcap_{j=1}^2\{|\x_p^\top A^j \x_p-\z_p^\top A^j \z_p|\leqslant \varepsilon p\}\]
and derive that $\e |\Delta|\leqslant \e|\Delta|I(D)+2\p\big(\overline{D}\big)/v,$ where $|\Delta|\leqslant 2/v$ by Fact 5 in Appendix C.
By the law of iterated mathematical expectations and Fact 4 in Appendix C,
\[\p\big(\overline{D}\big)=\e\big[\p\big(\overline{D}|A\big)\big]\leqslant 2\sup_{A_p}\p(|\x_p^\top A_p \x_p-\z_p^\top A_p \z_p|>\varepsilon p),\]
where $ A_p$ is any $p\times p$ complex symmetric matrix with $\|A_p\|\leqslant M:=\max\{v^{-1},v^{-2}\}$.
To estimate $\e|\Delta|I(D)$ we need the following technical lemma that is proved in Appendix B.
\begin{lemma} \label{l3} Let $z_1,z_2,w_1,w_2\in\mathbb C$. If $|z_1-z_2|\leqslant \gamma,$ $|w_1-w_2|\leqslant \gamma,$
\[\frac{|z_1|}{|1+w_1|}\leqslant M,\] and $|1+w_2|\geqslant\delta$ for some $\delta,M>0$ and $\gamma\in(0,\delta/2)$, then,
for some $C=C(\delta,M)>0,$
\[\Big|\frac{z_1}{1+w_1}-\frac{z_2}{1+w_2}\Big|\leqslant C\gamma.\]
\end{lemma}
Since $\z_p^\top A\z_p/n=\tr((\z_p\z_p^\top/n) A)$, Fact 6 in Appendix C implies that
\begin{equation}\label{1w}
|1+\z_p^\top A\z_p/n|\geqslant \delta:=\frac{v}{|z|}.
\end{equation}
Take $\gamma=\varepsilon p/n,$
\[(z_1,w_1)=(\x_p^\top A^2 \x_p, \x_p^\top A \x_p)/n,\qquad
(z_2,w_2)=(\z_p^\top A^2 \z_p, \z_p^\top A \z_p)/n\]
in Lemma \ref{l3}. By Fact 5 in Appendix C, $ |z_1|/|1+w_1| \leqslant 1/ v .$
By \eqref{1w},
\[|1+w_2|\geqslant \delta>\frac{2\varepsilon p}n=2\gamma\]
for small enough $\varepsilon>0$ and large enough $p$ (since $p/n\to \rho>0$).
Using Lemma \ref{l3}, we derive
\[\e|\Delta| I(D)\leqslant C(\delta,1/v)\,\frac{\varepsilon p}{n}.\]
Combining all above estimates together yields
\[ \e |\Delta_{k}| \leqslant C(\delta,1/v)\frac{\varepsilon p}{n}+ \frac 4v\,\sup_{ A_p}\p(|\x_p^\top A_p \x_p-\z_p^\top A_p \z_p|>\varepsilon p)
\]
for each $k=1,\ldots,n$ and
\begin{align}\label{las} \frac1p\sum_{k=1}^n\e |\Delta_{k}| &\leqslant C(\delta,1/v)\varepsilon + \frac {4n}{vp}\,\sup_{ A_p}\p(|\x_p^\top A_p \x_p-\z_p^\top A_p \z_p|>\varepsilon p).\end{align}
Note also that
\begin{align}\label{e106}\p(|\x_p^\top A_p \x_p-\z_p^\top A_p \z_p|>\varepsilon p)\leqslant& \p(|\x_p^\top A_p \x_p-\tr(\Sigma_pA_p)|>\varepsilon p/2)+\nonumber\\&\quad+\p(|\z_p^\top A_p \z_p -\tr(\Sigma_pA_p)|>\varepsilon p/2).\end{align}
Taking $\varepsilon$ small enough and then $p$ large enough, we can make the right-hand side of \eqref{las} arbitrarily small by the following lemma (that also holds for $\z_p$ instead of $\x_p$ by $(\mathrm A3)$ and Proposition \ref{p1}).
\begin{lemma} \label{l2} Let $(\mathrm A2)$ holds. Then, for each $\varepsilon,M>0$,
\begin{equation}\label{limsup}
\lim_{p\to\infty}\sup_{A_p}\p(|\x_p^\top A_p \x_p-\tr(\Sigma_pA_p)|>\varepsilon p)=0,\end{equation}
where the supremum is taken over all complex $p\times p$ matrices $A_p$ with $\|A_p\|\leqslant M.$
\end{lemma}
The proof of Lemma \ref{l2} can be found in Appendix B. Finally, we conclude that
\[\lim_{n\to\infty}\frac1p\sum_{k=1}^n\e |\Delta_{k}| =0.\]
This finishes the proof in the case when all entries of $C_{pn}$ are zeros.
Consider now the case with nonzero $C_{pn}$. Let $ c_k=c_k(n),$ $1\leqslant k\leqslant n,$ be columns of $C_{pn}$. Since $\|n^{-1}C_{pn}C_{pn}^\top\|=\|n^{-1}C_{pn}^\top C_{pn}\|=O(1)$ and $p/n\to\rho>0,$ we have $\max_{1\leqslant k\leqslant n}(c_k^\top c_k)=O(p)$, where $c_k^\top c_k$ are diagonal entries of $n^{-1}C_{pn}^\top C_{pn}$.
We also have
\[\wh\X_{pn}\wh\X_{pn}^\top=\sum_{k=1}^n\wh\x_{k}\wh\x_{k}^\top\quad\text{and}\quad
\wh\Z_{pn}\wh\Z_{pn}^\top=\sum_{k=1}^n\wh\z_{k}\wh\z_{k}^\top,\]
where $\wh \x_k=\x_k+c_k$ and $\wh \z_k=\z_k+c_k$ (here $\{(\x_k,\z_k)\}_{k=1}^p$ are i.i.d. copies of $(\x_p,\z_p)$). Arguing as above, we conclude that \eqref{las} holds with
\[\sup_{ A_p}\p(|\x_p^\top A_p \x_p-\z_p^\top A_p \z_p|>\varepsilon p)\] replaced by \[\frac{1}{n}\sum_{k=1}^n\sup_{ A_p}\p(|(\x_p+c_k)^\top A_p (\x_p+c_k)-(\z_p+c_k)^\top A_p (\z_p+c_k)|>\varepsilon p).\]
Recalling that $A_p$ is symmetric, we derive
\[|(\x_p+c_k)^\top A_p (\x_p+c_k)-(\z_p+c_k)^\top A_p (\z_p+c_k)|\leqslant\]\[\leqslant |\x_p^\top A_p \x_p- \z_p^\top A_p \z_p|+2|c_k^\top A_p \x_p|+2|c_k^\top A_p \z_p|.\]
In addition,
\[|c_k^\top A_p \x_p|^2/p^2\leqslant |\x_p^\top M_k\x_p-\tr(M_k)|/p +|\tr(M_k)|/p \] and the same inequality holds for $\z_p$ instead of $\x_p,$ where $M_k=p^{-1} A_p^* c_k c_k^\top A_p.$ Note also that, uniformly in $A_p$ with $\|A_p\|\leqslant M,$
\[\|M_k\|\leqslant \frac{\|A_p\|^2}p\max_{1\leqslant k\leqslant n}(c_k^\top c_k) =O(1)\] as well as \[\frac{|\tr(M_k)|}p=\frac{| c_k^\top A_pA_p^* c_k| }{p^2 }\leqslant \frac{\|A_p\|^2}{p^2}\max_{1\leqslant k\leqslant n} (c_k^\top c_k) =o(1).\]
Combining these estimates, Lemma \ref{l2}, $(\mathrm A2)$, $(\mathrm A3)$, and Proposition \ref{p1}, we get
\[\frac{1}{n}\sum_{k=1}^n\sup_{ A_p}\p(|(\x_p+c_k)^\top A_p (\x_p+c_k)-(\z_p+c_k)^\top A_p (\z_p+c_k)|>\varepsilon p) \to 0\]
for all fixed $\varepsilon>0$. Now, we can finish the proof as in the case with zero $C_{pn}.$ Q.e.d.
\noindent{\bf Proof of Lemma \ref{ml2}.} As in the proof of Lemma \ref{ml}, we can assume that $B_p$ is positive definite and, first, consider the case with null $C_{pn}$. Write further $\x_k$ and $\z_k$ instead of $\x_{pk}$ and $\z_{pk}$ and note that $(\mathrm A2^*)$ implies that, for all $\varepsilon,M>0,$
\begin{equation}
\label{e105}
\frac{1}{n}\sum_{k=1}^n\e\sup_{A_{pk}}\p(|\x_k^\top A_{pk} \x_k-\tr(\Sigma_{pk}A_{pk} )|>\varepsilon p|\cF_{k-1}^p)\to0,
\end{equation}
where the $k$-th supremum is taken over all $\cF_{k-1}^p$-measurable symmetric positive semi-definite random $p\times p$ matrices $A_{pk}$ with $\|A_{pk}\|\leqslant M $ a.s. Recall also that $\cF_{k-1}^p=\sigma(\x_l,l\leqslant k-1)$. Arguing in the same way as in the proof of Lemma \ref{l2}, it can be shown that \eqref{e105} holds even when the $k$-th supremum is taken over all $\cF_{k-1}^p$-measurable complex random $p\times p$ matrices $A_{pk}$ with $\|A_{pk}\|\leqslant M $ a.s.
Due to $(\mathrm A3^*)$ and \eqref{inequality} (see Appendix A), the same relation holds for $\x_k$ replaced by $\z_k$ (here $\z_k$ is independent of $\cF_{k-1}^p$).
Now, using the same arguments as in the proof of Lemma \ref{ml} after \eqref{e102}, we conclude that \eqref{las} holds with
\[\sup_{ A_p}\p(|\x_p^\top A_p \x_p-\z_p^\top A_p \z_p|>\varepsilon p)\] replaced by
\[V_n(\varepsilon):=\frac{1}{n}\sum_{k=1}^n[\p(|\x_k^\top A_k \x_k-\z_k^\top A_k \z_k|>\varepsilon p)+\p(|\x_k^\top A_k^2 \x_k-\z_k^\top A_k^2 \z_k|>\varepsilon p)],\]
where $A_k=(C_k+B_p-zI_p)^{-1}$ has $\|A_k\|\leqslant M=\max\{1/\Im(z),1/|\Im(z)|^2\}$ a.s. and $C_k$ is defined \eqref{e103}.
Each $C_k$ can be written as the sum of two matrices $C_{k1} $ and $C_{k2}$ such that $C_{k1}$ is $\cF_{k-1}^p$-measurable and $C_{k2}$
is a function of $\z_l, l\geqslant k+1$. Since $\{\z_k\}_{k=1}^n$ are mutually independent and independent from everything else,
\begin{equation}
\label{e104}
V_n(\varepsilon)\leqslant \frac{2}{n}\sum_{k=1}^n\e\sup_{A_{pk}}\p(|\x_k^\top A_{pk} \x_k-\z_k^\top A_{pk} \z_k|>\varepsilon p|\cF_{k-1}^p),\end{equation}
where the $k$-th supremum is taken over all $\cF_{k-1}^p$-measurable complex $p\times p$ random matrices $A_{pk}$ having $\|A_{pk}\|\leqslant M $ a.s. Thus, it follows from \eqref{e106}, \eqref{e105}, and the same relation with $\x_k$ replaced by $\z_k$ that
\[V_n(\varepsilon)\to0\quad \text{for all}\quad\varepsilon>0.\]
Now, we can finish the proof as in the proof of Lemma \ref{ml} after \eqref{e106}. Q.e.d.
\section*{Appendix A}
\noindent{\bf Proof of Proposition \ref{p2}.} By definition, $(\mathrm A1^*)$ follows from $(\mathrm A1)$. Suppose $(\mathrm A1^*)$ holds. Let us show that $(\mathrm A1)$ holds. Note that if $\Pi_p,$ $p\geqslant1,$ are orthogonal projectors, then
$Z_p\pto 1$ and $\e Z_p=1$ as $p\to\infty,$ where
$Z_p=1+(\x_p^\top \Pi_p\x_p-\tr(\Pi_p))/p\geqslant 0$ a.s. Therefore, $Z_p\to 1$ in $L_1$ for any sequence of orthogonal projectors $\Pi_p,$ $p\geqslant 1$. Thus,
\[\sup_{\Pi_p}\e|\x_p^\top \Pi_p\x_p-\tr(\Pi_p)|=o(p),\]
where the supremum is taken over all $p\times
p $ orthogonal projectors $\Pi_p.$ Any $p\times p$ diagonal matrix $D $ with diagonal entries $\lambda_1\geqslant \ldots\geqslant \lambda_p\geqslant 0$ ($\lambda_1>0$) can be written as \[\frac{D}{\lambda_1}=\sum_{k=1}^pw_k D_k,\]
where $\lambda_{p+1}=0,$ $w_k=(\lambda_k-\lambda_{k+1})/\lambda_1\geqslant 0$ are such that $\sum_{k=1}^pw_k=1$, and each $D_k$ is a diagonal matrix with diagonal entries in $1,\ldots,1,0,\ldots,0$ ($k$ ones). Hence, any symmetric positive semi-definite matrix $A_p$ with $\|A_p\|=1$ can be written as a convex combination of some orthogonal projectors. As a result, by the convexity of the $L_1$-norm,
\[\sup_{ A_p}\e|\x_p^\top A_p\x_p-\tr( A_p)|\leqslant \sup_{ \Pi_p}\e|\x_p^\top \Pi_p\x_p-\tr(\Pi_p)| =o(p),\]
where $A_p$ as above. We conclude that $(\mathrm A1)$ holds. Q.e.d.
\noindent{\bf Proof of Proposition \ref{p1}.} Suppose $(\mathrm A3)$ holds. Let us show that $(\mathrm A2)$ holds. The latter will follow from the inequality
\begin{equation}\label{inequality}
I:=\p(|\x_p^\top A_p \x_p-\tr(\Sigma_pA_p)|>\varepsilon p)\leqslant
\frac{2\|A_p\|^2\tr(\Sigma_p^2)}{(\varepsilon p)^{2}}
\end{equation}
valid for any $\varepsilon>0$ and any $p\times p$ symmetric positive semi-definite matrix $A_p$. By Chebyshev's inequality, $I\leqslant \var(\x_p^\top A_p \x_p)(\varepsilon p)^{-2},$ since $\e(\x_p^\top A_p \x_p)=\tr(\Sigma_pA_p)$. As a result, we need to verify that
\begin{equation}
\label{v1}
\var(\x_p^\top A_{p} \x_p) \leqslant 2\|A_p\|^2\tr(\Sigma_p^2). \end{equation}
We have $\x_p^\top A_p \x_p=\z_p^\top D_p\z_p,$ where $\z_p$ is a standard normal vector and $D_p$ is a diagonal matrix whose diagonal entries $\{\lambda_k\}_{k=1}^p$ are eigenvalues of $B_p=\Sigma_p^{1/2}A_p\Sigma_p^{1/2}$. By a direct calculation, $\var(\z_p^\top D_{p} \z_p)=2\,\tr(D_p^2)=2\,\tr(B_p^2)$. By Fact 1 in Appendix C and the identity $\tr( AB)=\tr(BA)$, $\tr(B_p^2)=\tr( A_p\Sigma_p A_p\Sigma_p )\leqslant \|A_p\|^2\tr(\Sigma_p^2)$. Thus, we get \eqref{inequality}.
Assume now that $(\mathrm A2)$ holds. Let us show that $(\mathrm A3)$ holds. Take $A_p=I_p$. Hence,
\[\frac{\x_p^\top \x_p-\tr(\Sigma_p)}{p}=\frac{\z_p^\top D_p \z_p-\tr(D_p)}{p}\pto 0,\quad p\to\infty,\]
where $\z_p$ is as above and $D_p$ is a diagonal matrix whose diagonal entries $\{\lambda_{k}\}_{k=1}^p$ are $\Sigma_p$'s eigenvalues arranged in descending order, i.e. $\|\Sigma_p\|=\lambda_{1}\geqslant\ldots\geqslant \lambda_{p}\geqslant0$ and $\lambda_k=\lambda_k(p)$, $1\leqslant k\leqslant p$.
Let $\y_p=(Y_1,\ldots,Y_p)$ be an independent copy of $\z_p=(Z_1,\ldots,Z_p)$. Therefore,
$(\z_p^\top D_p \z_p-\y_p^\top D_p \y_p)/p\pto 0$
and
\begin{align*}\e\exp\{i(\z_p^\top D_p \z_p- \y_p^\top D_p \y_p)/p\}=&\prod_{k=1}^p\e\exp\{i \lambda_{k}(Z_k^2-Y_k^2)/p\}=\prod_{k=1}^p|\varphi(\lambda_{k}/p)|^2\to 1
\end{align*}
as $p\to\infty$, where $\varphi(t)=\e\exp\{i tZ_1^2\}$, $t\in\bR$. Hence, \[|\varphi(\|\Sigma_p\|/p)|^2=\frac{1}{|1-2i\|\Sigma_p\|/p|}=
\frac{1}{\sqrt{1+4\|\Sigma_p\|^2/p^2}}\to 1\] and $\|\Sigma_p\|/p\to0 $. As a result,
\begin{align*}\prod_{k=1}^p|\varphi(\lambda_{k}/p)|^2
=&\prod_{k=1}^p\frac1{\sqrt{1+4\lambda_{k}^2/p^2}}
=\exp\Big\{(-2+\varepsilon_p)\sum_{k=1}^p\lambda_{k}^2/p^2\Big\}\to 1\end{align*} for some $\varepsilon_p=o(1)$.
Thus, $\tr(\Sigma_p^2)/p^2\to0,$ i.e. $(\mathrm A3)$ holds. Q.e.d.
\noindent{\bf Proof of Proposition \ref{p3}.} For each $p\geqslant 1$, let $A_p$ be a positive semi-definite symmetric $p\times p$ matrix with $\|A_p\|=O(1)$ as $p\to\infty$. We need to show that
\begin{equation}\label{e11}
\frac{1}{p}(\x_p^\top A_p\x_p-\tr(\Sigma_pA_p))\pto0,
\end{equation}
Let further $\y_p=(Y_{p1},\ldots,Y_{pp})$ and $\z_p=(Z_{p1},\ldots,Z_{pp})$ be centred Gaussian random vectors in $\bR^p$ with variances $A_p$ and $\Sigma_p$, respectively. Suppose also $\x_p,$ $\y_p,$ and $\z_p$ are mutually independent.
We have $|\e X_{pk}X_{pl}|\leqslant |\e X_{pk}\e[X_{pl}|\cF_k^p]|\leqslant \sqrt{\Gamma_0\Gamma_{l-k}}$ for $k\leqslant l$ and
\[\frac{\tr(\Sigma_p^2)}{p^2}=\frac{1}{p^2}\sum_{k,l=1}^p|\e X_{pk}X_{pl}|^2\leqslant
\frac{1}{p^2}\sum_{k,l=1}^{p}\sqrt{\Gamma_0\Gamma_{|l-k|}}\leqslant\frac2p\sum_{j=0}^p\sqrt{\Gamma_0\Gamma_j}=o(1)\]
(recall that $\Gamma_j\to 0,$ $j\to\infty$). Thus, $(\mathrm A3)$ holds and, by Proposition \ref{p1}, \begin{equation}\label{e10}
\frac{1}{p}(\z_p^\top A_p\z_p-\tr(\Sigma_pA_p))\pto0 .
\end{equation}
We need a technical lemma proved in Appendix B.
\begin{lemma}\label{l5} For each $n\geqslant 1,$ let $Z_n$ be a random variable such that $Z_n\geqslant 0$ a.s. If $\e Z_n=O(1)$ as $n\to\infty,$ then $Z_n-\e Z_n\pto0$ iff $\e \exp\{-Z_n\}-\exp\{-\e Z_n\}\to0 $.
\end{lemma}
Note that $\tr(\Sigma_p)/p\leqslant \Gamma_0$ and, by Fact 1 in Appendix C, $\tr(\Sigma_pA_p)/p=O(1)$. Thus, by $\tr(\Sigma_pA_p)=\e(\x_p^\top A_p\x_p)=\e(\z_p^\top A_p\z_p)$, Lemma \ref{l5}, and \eqref{e10}, we can prove \eqref{e11} by showing that
\begin{align*}
\Delta_p=\e\exp\{-\x_p^\top A_p\x_p/(2p)\}- \e\exp\{-\z_p^\top A_p\z_p/(2p)\}\to0.
\end{align*}
It follows from the independence $\y_p$ and $(\x_p,\z_p)$ that
\[\Delta_p=\e\exp\{i(\x_p^\top\y_p)/\sqrt{p}\}- \e\exp\{i(\z_p^\top\y_p)/\sqrt{p}\},\]
where we have used that $\e((\x_p^\top\y_p)^2|\x_p)=\x_p^\top A_p\x_p$ and $\e((\z_p^\top\y_p)^2|\z_p)=\z_p^\top A_p\z_p$ a.s.
Now, we will proceed in the same way as in the proof of Theorem 5 in \cite{PM}. Fix some $q,j\in \bN$ and assume w.l.o.g. that $m=p/(q+j)$ is integer (we can always add no more than $q+j$ zeros to $\x_p,\y_p,\z_p$). Let
\[\wt\x_p=(\wt X_{1},{\bf 0}_j,\wt X_{2},{\bf 0}_j,\ldots,\wt X_{m},{\bf 0}_j)\]
for the null vector ${\bf 0}_j$ in $\bR^j,$ where entries of $\wt X_{r}$ ($1\leqslant r\leqslant m$) are $X_{pl}$ for \[l=l_{r-1}+1,\ldots,l_{r-1} +q\quad\text{and}\quad l_{r-1}=(r-1)(q+j).\] Put also $\Delta \x_p=({\bf 0}_q,\Delta X_{1},\ldots,{\bf 0}_q,\Delta X_{m})$, where entries of $\Delta X_{r}$ are $X_{pl} $ for \[l=l_{r-1}+q+1,\ldots,l_r.\]
Define $\wt\y_p,$ $\wt\z_p$, $\Delta\y_p$, and $\Delta\z_p$ (with $\wt Y_r,\wt Z_r,\Delta Y_r,\Delta Z_r$) similarly.
Since $\x_p^\top\y_p=\wt \x_p^\top\wt \y_p+(\Delta \x_p)^\top \Delta \y_p$ and $|\exp\{ia\}-\exp\{ib\}|\leqslant |b-a|$ for $a,b\in\bR,$ we have
\begin{align*}
|\e \exp\{ i(\x_p^\top\y_p)/\sqrt{p} \}- \e \exp\{i(\wt \x_p^\top\wt \y_p)/\sqrt{p} \} |\leqslant
\frac{\e|(\Delta \x_p)^\top \Delta \y_p|}{\sqrt{p}}\leqslant \frac{\sqrt{\e|(\Delta \x_p)^\top \Delta \y_p|^2}}{\sqrt{p}}
\end{align*}
and, by the independence of $\Delta \x_p$ and $\Delta \y_p$,
\begin{align}\label{e16}
\e|(\Delta \x_p)^\top \Delta \y_p|^2\leqslant \e ( (\Delta \x_p)^\top V_p (\Delta \x_p ) )\leqslant \|V_p\| \e (\Delta \x_p)^\top (\Delta \x_p) \leqslant \|A_p\| mj,
\end{align}
where $V_p=\var(\Delta\y_p)$ and, obviously, $\|V_p\|\leqslant \|\var(\y_p)\|=\|A_p\|$.
We can bound $\e \exp\{i(\z_p^\top\y_p)/\sqrt{p} \}- \e \exp\{i(\wt \z_p^\top\wt \y_p)/\sqrt{p}\}$ in the same way. Combining these estimates with $m/p\leqslant 1/q$, we arrive at
\[\Delta_p= \e \exp\{i(\wt \x_p^\top\wt \y_p)/\sqrt{p}\}- \e \exp\{i(\wt \z_p^\top\wt \y_p)/\sqrt{p}\}+O(1)\sqrt{j/q}.\]
Fix some $\varepsilon>0$ and set $\wh \x_p=(\wh X_1,{\bf 0}_j,\ldots,\wh X_{m},{\bf 0}_j)$ for $\wh X_r$ having entries \[\wh X_{pl}=X_{pl}I(|X_{pl}|\leqslant \varepsilon\sqrt{p})-\e_{l_{r-1}-j}( X_{pl}I(|X_{pl}|\leqslant \varepsilon\sqrt{p})) ,\quad l=l_{r-1}+1,\ldots,l_{r-1}+q,\]
hereinafter $\e_l=\e\big(\cdot|\cF_{\max\{l,0\}}^p\big),$ $l\leqslant p$.
Analogously, let $\wh \z_p=(\wh Z_1,{\bf 0}_j,\ldots,\wh Z_{m},{\bf 0}_j)$, where entries of $\wh Z_r$ are \[\wh Z_{pl}=Z_{pl}-\e^*_{l_{r-1}-j} Z_{pl},\quad l=l_{r-1}+1,\ldots,l_{r-1}+q.\]
Here $\e_l^*=\e_l^*(\cdot|\cG_{\max\{l,0\}}^p)$ for $\cG^p_{l}=\sigma(Z_{pk},1\leqslant k\leqslant l),$ $l\geqslant 1,$ and the trivial $\sigma$-algebra $\cG_0^p$. Obviously, $\e_l^*Z_{pk}$ is a linear function in $ Z_{ps},$ $s\leqslant l$ (because $\z_p$ is a Gaussian vector) and, as a result, $\wh\z_p$ is a Gaussian vector. Also,
\[\e \wh Z_{ps}\wh Z_{pt}=\e(\e^*_{l_{r-1}-j} \wh Z_{ps}\wh Z_{pt})= \e Z_{ps} Z_{pt}-\e(\e^*_{l_{r-1}-j} Z_{ps})(\e^*_{l_{r-1}-j} Z_{pt})\]
for $s,t=l_{r-1}+1,\ldots,l_{r-1}+q, $ and, as a result,
\begin{equation}
\label{e13}
\var(\wt Z_r)-\var(\wh Z_r)=\e E_rE_r^\top,\quad r=1,\ldots,m,
\end{equation}
with $E_r$ is a vector with entries $\e_{l_{r-1}-j}^*Z_{pl},$ $l=l_{r-1}+1,\ldots,l_{r-1}+q.$
We have
\begin{align*}
|\e &\exp\{i(\wt \x_p^\top\wt \y_p)/\sqrt{p}\}- \e \exp\{i(\wt \z_p^\top\wt \y_p)/\sqrt{p}\}|\leqslant\\
&\leqslant |\e \exp\{i(\wt \x_p^\top\wt \y_p)/\sqrt{p}\}- \e \exp\{i(\wh \x_p^\top\wt \y_p)/\sqrt{p}\}|+\\
&\quad +|\e \exp\{i(\wt \z_p^\top\wt \y_p)/\sqrt{p}\}- \e \exp\{i(\wh \z_p^\top\wt \y_p)/\sqrt{p}\}|\\
&\quad +|\e \exp\{i(\wh \x_p^\top\wt \y_p)/\sqrt{p}\}- \e \exp\{i(\wh \z_p^\top\wt \y_p)/\sqrt{p}\}|.
\end{align*}
Let us estimate the first term in the right-hand side of the last inequality. Arguing as in \eqref{e16}, we infer that, for $U_p=\var(\wt \y_p)$ (having $\|U_p\|\leqslant \|\var(\y_p)\|=\|A_p\|$), \begin{align*}
|\e &\exp\{i(\wt \x_p^\top\wt \y_p)/\sqrt{p}\}- \e \exp\{i(\wh \x_p^\top\wt \y_p)/\sqrt{p}\}|\leqslant \frac{1}{\sqrt{p}}\e|(\wt \x_p-\wh\x_p)^\top\wt \y_p|\leqslant \\
& \leqslant \frac{1}{\sqrt{p}}\big(\e (\wt \x_p-\wh\x_p)^\top U_p (\wt \x_p-\wh\x_p)\big)^{1/2}\leqslant \frac{\sqrt{\|A_p\|}}{\sqrt{p}} (\e \|\wt \x_p-\wh\x_p\|^2)^{1/2}.
\end{align*}
Additionally, by $(a+b+c)^2\leqslant 3(a^2+b^2+c^2),$ $a,b,c\in\bR,$
\begin{align*}
\frac{1}{p}\e \|\wt \x_p-\wh\x_p\|^2\leqslant 3(\delta_1+\delta_2+\delta_3),
\end{align*}
where, by \eqref{lin}, $\e X_{pl}=0 $ for all $(p,l),$
\[X_{pl}-\wh X_{pl}=X_{pl}I(|X_{pl}|> \varepsilon\sqrt{p})+\e_{l_{r-1}-j}X_{pl}-\e_{l_{r-1}-j}( X_{pl}I(|X_{pl}|>\varepsilon\sqrt{p})),\]
and Jensen's inequality, we have
\begin{align*}
&\delta_1= p^{-1}\sum \e X_{pl}^2I(|X_{pl}|> \varepsilon\sqrt{p}) \leqslant L_p(\varepsilon) =o(1),\\
&\delta_2= p^{-1}\sum\e\big|\e_{l_{r-1}-j} (X_{pl}I(|X_{pl}|> \varepsilon\sqrt{p}) )\big|^2 \leqslant \delta_1=o(1),\\
&\delta_3= p^{-1}\sum\e\big|\e_{l_{r-1}-j} X_{pl} \big|^2 \leqslant \Gamma_j
\end{align*}
and \begin{equation}
\label{e15}
\text{the sum }\sum=\sum_{r=1}^m\sum_{l=l_{r-1}+1}^{l_{r-1}+q}\text{ has no more than $p$ terms.}
\end{equation}
Hence,
\begin{align} \label{e17}
\frac{1}{p}\e \|\wt \x_p-\wh\x_p\|^2\leqslant 3 \Gamma_j +o(1).
\end{align}
and $\e \exp\{i(\wt \x_p^\top\wt \y_p)/\sqrt{p}\}- \e \exp\{i(\wh \x_p^\top\wt \y_p)/\sqrt{p}\}=O(1)\sqrt{\Gamma_j}+o(1)$.
Using similar arguments, we see that
\begin{align*}
|\e \exp\{i(\wt \z_p^\top\wt \y_p)/\sqrt{p}\}- \e \exp\{i(\wh \z_p^\top\wt \y_p)/\sqrt{p}\}|^2&\leqslant \frac{\|A_p\|}{p}\e \|\wt \z_p-\wh\z_p\|^2\\
&\quad =\frac{\|A_p\|}{p}\sum\e\big|\e_{l_{r-1}-j}^* Z_{pl}\big|^2
\end{align*}
with $\sum$ as in \eqref{e15}. Since $\x_p$ and $\z_p$ have the same covariance structure and $\z_p$ is Gaussian, then \begin{equation}
\label{e14}
\e|\e_{l_{r-1}-j}^* Z_{pl} \big|^2\leqslant \e\big|\e_{l_{r-1}-j} X_{pl}\big|^2\leqslant \Gamma_j
\end{equation} (see Lemma 14 in \cite{PM}). Thus,
$p^{-1}\sum\e\big|\e^*_{l_{r-1}-j} Z_{pl} \big|^2\leqslant \Gamma_j$ for
$\sum$ from \eqref{e15}.
Combining these estimates, we deduce that
\[|\Delta_p|\leqslant |\e \exp\{i(\wh \x_p^\top\wt \y_p)/\sqrt{p}\}- \e \exp\{i(\wh \z_p^\top\wt \y_p)/\sqrt{p}\}|+O(1)(\sqrt{j/q}+\sqrt{\Gamma_j}).\]
Now,
\[|\e \exp\{i(\wh \x_p^\top\wt \y_p)/\sqrt{p}\}- \e \exp\{i(\wh \z_p^\top\wt \y_p)/\sqrt{p}\}|\leqslant \]
\[\leqslant\sum_{r=1}^m|\e \exp\{i(C_r+\wh X_r^\top \wt Y_r/\sqrt{p})\}- \e \exp\{i(C_r+\wh Z_r^\top \wt Y_r/\sqrt{p})\}|,\]
where, for $ r=1,\ldots, m$, \[C_r=p^{-1/2}\sum_{j<r} \wh X_j^\top \wt Y_j+p^{-1/2}\sum_{j>r} \wh Z_j^\top \wt Y_j\]
and the sum over the empty set is zero.
Expanding $\exp\{i(C_r+x)\}$ in Taylor's series around $x=0$, we derive
\[\exp\{i(C_r+x)\}=\exp\{i C_r\}(1+ix-x^2/2+\theta(x))\quad\text{for}\quad|\theta(x)|\leqslant |x|^3/6\]
and $|\e \exp\{i(C_r+\wh X_r^\top \wt Y_r/\sqrt{p})\}- \e \exp\{i(C_r+\wh Z_r^\top \wt Y_r/\sqrt{p})\}|\leqslant |R_{r1}|+|R_{r2}|+|R_{r3}|,$
where
\[R_{r1}=\frac{1}{\sqrt{p}}\e \exp\{i C_r\}( \wh X_r^\top \wt Y_r-\wh Z_r^\top \wt Y_r), \]
\[R_{r2}=\frac{1}{ p}\e \exp\{i C_r\}( (\wh X_r^\top \wt Y_r)^2-(\wh Z_r^\top \wt Y_r)^2), \]
\[R_{r3}=\frac{1}{ p\sqrt{p}}\e (| \wh X_r^\top \wt Y_r|^3+|\wh Z_r^\top \wt Y_r|^3).\]
Putting $R_k=\sum_{r=1}^m R_{rk}$, $k=1,2,3,$ we see that
\begin{equation}\label{e20}
|\Delta_p|\leqslant R_1+R_2+R_3+O(1)(\sqrt{j/q}+\sqrt{\Gamma_j}).
\end{equation}
First, we will show that $R_1=0.$ Note that, by the definition of $\wh \x_p$, $\e \wh X_1={\bf 0}_q $ and, for $r>1$, $\e[\wh X_r|\wh X_{r-1},\ldots,\wh X_1]={\bf 0}_q$
a.s. In addition, by the definition of $\wh \z_p,$ $\wh Z_{r }$ and $\wh Z_{s }$ are uncorrelated when $r \neq s.$ Thus, $\wh Z_1,\ldots,\wh Z_m$ are mutually independent Gaussian vectors with mean zero. Since $ \wh \x_p,\wt\y_p,\wh \z_p $ are also mutually independent, then, with probability one,
\[\e[\wh X_r|\wt\y_p,(\wh X_{l})_{l< r},(\wh Z_{l})_{l> r}]=\e[\wh X_r|(\wh X_{l})_{l< r}]={\bf 0}_q, \]
\[\e[\wh Z_r|\wt\y_p,(\wh X_{l})_{l< r}, (\wh Z_{l})_{l> r}]=\e \wh Z_r ={\bf 0}_q ,\]
where $r=1,\ldots,m$ and $\wh Z_{m+1}=\wh X_0={\bf 0}_q$. As a result, $R_1= 0.$
Let us prove that $R_3=o(1)+O(1)\varepsilon\sqrt{q}.$ Since $\wh X_r^\top \wt Y_r|\wh X_r\sim\mathcal N(0,\wh X_r^\top \wt V_r\wh X_r) $ for $\wt V_r=\var(\wt Y_r)$ (obviously, $\|\wt V_r\|\leqslant\|\var(\y_p)\|=\|A_p\|$), we have
\[\e |\wh X_r^\top \wt Y_r|^3= C_0\e|\wh X_r^\top \wt V_r\wh X_r|^{3/2}\leqslant
C_0\|A_p\|^{3/2}\e\|\wh X_r\|^{3 }
\]
for an absolute constant $C_0>0$ and each $r=1,\ldots,m$. Using the fact that entries of $\wh X_r$ are bounded by $2\varepsilon\sqrt{p}$ and the estimate
\begin{equation}
\label{e19}
\e\|\wh X_r\|^2\leqslant \sum_{l=l_{r-1}+1}^{l_{r-1}+q}\e X_{pl}^2I(|X_{pl}|\leqslant \varepsilon\sqrt{p}) \leqslant q\Gamma_0,
\end{equation}
and recalling that $mq\leqslant p$, we obtain
\[\frac{1}{p\sqrt{p}}\sum_{r=1}^m\e |\wh X_r^\top \wt Y_r|^3\leqslant C_0\|A_p\|^{3/2}\frac{m (q\Gamma_0) ( 4\varepsilon^2 pq )^{1/2}}{p\sqrt{p}}\leqslant 2C_0\Gamma_0\|A_p\|^{3/2} \varepsilon\sqrt{q }.\]
In addition, as $\wh Z_r$ is a centred Gaussian vector whose entries have variance not greater than $\Gamma_0$ and $\big(\sum_{i=1}^q|a_i|/q\big)^{3/2}\leqslant \sum_{i=1}^q|a_i|^{3/2}/q $ for any $a_i\in\bR$,
\[\e\|\wh Z_r\|^{3}=\e\Big|\sum_{l=l_{r-1}+1}^{l_{r-1}+q}\wh Z_{pl}^2\Big|^{3/2}\leqslant q^{1/2}\sum_{l=l_{r-1}+1}^{l_{r-1}+q}\e|\wh Z_{pl}|^{3 }=C_0q^{3/2}\Gamma_0^{3/2}
\] for $C_0>0 $ as above. Hence, arguing as above, we infer
\[\frac{1}{p\sqrt{p}}\sum_{r=1}^m\e |\wh Z_r^\top \wt Y_r|^3\leqslant C_0\|A_p\|^{3/2}\frac{m(C_0q^{3/2}\Gamma_0^{3/2})}{p\sqrt{p}}\leqslant C_0^2\Gamma_0^{3/2}\|A_p\|^{3/2} \sqrt{q/p}=o(1
).\]
This proves that $R_3=o(1)+O(1)\varepsilon\sqrt{q}.$
To finish the proof, we need a good bound on $R_2=\sum_{r=1}^mR_{r2}$. First, note that, by the independence of $\wh Z_r$ from everything else,
\[R_{r2}=\frac{1}{p}\e \exp\{i C_r\}( (\wh X_r^\top \wt Y_r)^2- \wt Y_r^\top \var(\wh Z_r)\wt Y_r). \]
In addition, by \eqref{e13}, Fact 1 in Appendix C, and \eqref{e14},
\[|\e \exp\{i C_r\}( \wt Y_r^\top \var(\wt Z_r)\wt Y_r- \wt Y_r^\top \var(\wh Z_r)\wt Y_r)|\leqslant \e ( \wt Y_r^\top (\e E_rE_r^\top ) \wt Y_r)\leqslant\]
\[\leqslant \tr( \var(\wt Y_r) \e E_rE_r^\top)\leqslant \|\var(\wt Y_r)\|\tr(\e E_rE_r^\top)\leqslant \|A_p\|\e\|E_r\|^2=\|A_p\| q\Gamma_j.\]
Hence, recalling that $mq\leqslant p $ and $\var(\wt Z_r)=\var(\wt X_r)$, we get
\[R_2=\frac{1}{p}\sum_{r=1}^m \e \exp\{i C_r\}( (\wh X_r^\top \wt Y_r)^2- \wt Y_r^\top \var(\wt X_r)\wt Y_r)+O(1)\Gamma_j.\]
Fix some integer $a>1$ and let
\[C_r^{-a}=p^{-1/2}\sum_{j\leqslant r-a} \wh X_j^\top \wt Y_j+p^{-1/2}\sum_{j>r} \wh Z_j^\top \wt Y_j,\quad r=1,\ldots,m,\] where the sum over the empty set is zero. Since all entries of $\wh X_j$, $j\geqslant 1,$ are bounded by $2\varepsilon \sqrt{p}$, we have
\begin{align} \label{e21}
|\e (\exp\{i C_r\}&-\exp\{iC_r^{-a}\})( (\wh X_r^\top \wt Y_r)^2- \wt Y_r^\top \var(\wt X_r)\wt Y_r)|\leqslant \nonumber\\
&\leqslant \frac{1}{\sqrt{p}} \sum_{r-a<j<r}\e |\wh X_j^\top \wt Y_j|\, |(\wh X_r^\top \wt Y_r)^2- \wt Y_r^\top \var(\wt X_r)\wt Y_r)|\nonumber\\
&\leqslant \frac{2\varepsilon \sqrt{p}}{\sqrt{p}} \sum_{r-a<j<r}\e \| \wt Y_j\|_1 |(\wh X_r^\top \wt Y_r)^2- \wt Y_r^\top \var(\wt X_r)\wt Y_r)|\nonumber\\
&\leqslant
2\varepsilon aq^3 \sup (\e|\wh X_{ps} \wh X_{pt}|+\e|X_{ps}X_{pt}|)\e|Y_{pk}Y_{ps}Y_{pt}|\nonumber\\
&\leqslant
4\varepsilon aq^3 \Gamma_0 \sup \e|Y_{pk}|^3 \nonumber\\
&\leqslant C_1\varepsilon aq^3 \Gamma_0\|A_p\|^{3/2},
\,
\end{align}
where $C_1>0$ is an absolute constant, $\|x\|_1=|x_1|+\ldots+|x_q|$ for $x=(x_1,\ldots,x_q),$ and the supremum is taken over all $1\leqslant k,s,t\leqslant p$.
In addition, we have
\[
|\e \exp\{i C_r^{-a}\}( (\wh X_r^\top \wt Y_r)^2- (\wt X_r^\top \wt Y_r)^2)|\leqslant \e|(\wh X_r^\top \wt Y_r)^2- (\wt X_r^\top \wt Y_r)^2)|\]
and, by the Cauchy-Schwartz inequality and the independence of $(\wh X_r,\wt X_r)$ and $\wt Y_r$,
\begin{align*}
\e|(\wh X_r^\top \wt Y_r)^2- (\wt X_r^\top \wt Y_r)^2)|
&\leqslant \big(\e ((\wh X_r-\wt X_r)^\top \wt Y_r)^2\big)^{1/2}\big(\e ((\wt X_r+\wh X_r)^\top \wt Y_r)^2 \big)^{1/2}\\
&\quad=\big(\e\| \wt V_r^{1/2}(\wh X_r-\wt X_r)\|^2 \big)^{1/2}\big(\e\|\wt V_r^{1/2}(\wt X_r+\wh X_r)^\top\|^2\big)^{1/2}\\
&\leqslant \|A_p\|\big(\e\|\wh X_r-\wt X_r\|^2 \big)^{1/2}\big(2\e\|\wt X_r\|^2+2\e\|\wh X_r\|^2\big)^{1/2} \end{align*}
where $\wt V_r=\var(\wt Y_r)$ has $\|\wt V_r\|\leqslant \|\var(\y_p)\|\leqslant \|A_p\|.$ Summing over $r$ and using \eqref{e17} (as well as \eqref{e19}), we arrive at
\begin{align*}
\frac{1}{p}\sum_{r=1}^m&\big(\e\|\wh X_r-\wt X_r\|^2 \big)^{1/2}\big(2\e\|\wt X_r\|^2+2\e\|\wh X_r\|^2\big)^{1/2}\leqslant \\
&\leqslant \Big(\frac{1}{p}\sum_{r=1}^m \e\|\wh X_r-\wt X_r\|^2 \Big)^{1/2}\Big( \frac{2}{p}\sum_{r=1}^m( \e\|\wt X_r\|^2+ \e\|\wh X_r\|^2)\Big)^{1/2}\\
&\quad =\Big(\frac{1}{p}\e\|\wh \x_p-\wt \x_p\|^2 \Big)^{1/2}\Big( \frac{2}{p} ( \e\|\wt \x_p\|^2+ \e\|\wh \x_p\|^2)\Big)^{1/2}\\
&\leqslant \Big(3\Gamma_j+o(1)\Big)^{1/2} (2\Gamma_0)^{1/2}.
\end{align*}
Combining bounds after \eqref{e21} (and using $mq\leqslant p$), we conclude that
\begin{align*}
|R_2|&\leqslant \frac{1}{p}\sum_{r=1}^m |D_r|+O(1)(\Gamma_j+\varepsilon aq^2)+O(1)\sqrt{\Gamma_j+o(1)}
\end{align*}
for $D_r=\e \exp\{i C_r^{-a}\}( (\wt X_r^\top \wt Y_r)^2- \wt Y_r^\top \var(\wt X_r)\wt Y_r).$
Again, the mutual independence of $\x_p,\wt \y_p,\wh \z_p$ implies that
\[\e\big((\wt X_r^\top \wt Y_r)^2\big| \wt\y_p,(X_{pl})_{l\leqslant (l_{r-a}-j)^+}, (\wh Z_j)_{j>r}\big)=\wt Y_r^\top \e_{l_{r-a}-j}(\wt X_r \wt X_r^\top) \wt Y_r\quad\text{a.s.}\]
and $D_r=\e e^{i C_r^{-a}}( \wt Y_r^\top [\e_{l_{r-a}-j}(\wt X_r \wt X_r^\top)- \var(\wt X_r)]\wt Y_r)$. Hence, as \[l_{r-1}+1-(l_{r-a}-j)=j+1+(a-1)(q+j)>aj,\]
we conclude that
\begin{align*}
|D_r|&\leqslant \e | \wt Y_r^\top (\e_{l_{r-a}-j}(\wt X_r \wt X_r^\top)- \e (\wt X_r \wt X_r^\top) )\wt Y_r)|\\
&\leqslant \sum_{s,t=l_{r-1}+1}^{l_{r-1}+q} \e | Y_{ps}Y_{pt}|\e|\e_{l_{r-a}-j} ( X_{ps} X_{qt} ) - \e ( X_{ps} X_{pt} ) |\\&\leqslant q^2\|A_p\| \Gamma_{aj}.
\end{align*}
The latter (with $mq\leqslant p$) implies that
\[|R_2|\leqslant O(1)\sqrt{ o(1)+\Gamma_j}+O(1)(\Gamma_j+\varepsilon aq^2) +q\Gamma_{aj}. \]
It follows follows from \eqref{e20} and obtained bounds on $R_1,R_2,R_3$ that, for a sufficiently large constant $C>0,$
\[\varlimsup_{p\to \infty} \Delta_p\leqslant C\big(\sqrt{j/q}+\sqrt{ \Gamma_j}+\Gamma_j+\varepsilon aq^2 +q\Gamma_{aj}+\varepsilon\sqrt{q}\big)=\Delta.\]
Tending $\varepsilon\to 0$, we see that $\Delta\to \Delta'= C\big(\sqrt{j/q}+\sqrt{ \Gamma_j}+\Gamma_j+q\Gamma_{aj}\big)$. Then, taking $a\to\infty$ yields $\Delta'\to\Delta''= C\big(\sqrt{j/q}+\sqrt{ \Gamma_j}\big)$. Finally, taking $q,j\to\infty$ and $j/q\to 0$, we get
$\Delta''\to0$. This finishes the proof of the proposition. Q.e.d.
\section*{Appendix B}
\noindent{\bf Proof of Lemma \ref{l4}.} We have
\begin{align}
\label{e5}
\e \frac{Z_n}{1+Z_n}-\e\frac{\e (Z_n|Y_n)}{1+\e (Z_n|Y_n)}&=\e \frac{Z_n-\e( Z_n|Y_n)}{(1+Z_n)(1+\e (Z_n|Y_n))}\nonumber\\
&=\e \frac{Z_n-\e(Z_n|Y_n)}{(1+\e(Z_n|Y_n))^2}-\e \frac{(Z_n-\e( Z_n|Y_n))^2}{(1+Z_n)(1+\e (Z_n|Y_n))^2}\nonumber\\
&=-\e \frac{(Z_n-\e( Z_n|Y_n))^2}{(1+Z_n)(1+\e (Z_n|Y_n))^2}.
\end{align}
As a result, we see that
\[\frac{(Z_n-\e (Z_n|Y_n))^2}{(1+Z_n)(1+\e (Z_n|Y_n))^2}\pto0.\]
Since $\e Z_n$ is bounded and $Z_n\geqslant 0$ a.s., we conclude that $Z_n$ and $\e(Z_n|Y_n)$ are bounded asymptotically in probability and $Z_n-\e(Z_n|Y_n)\pto0.$ Q.e.d.
\noindent{\bf Proof of Lemma \ref{l3}.} We have
\begin{align*} I=\frac{z_1}{1+w_1}-\frac{z_2}{1+w_2}&=\frac{z_1(1+w_2)-z_2(1+w_1)-z_1w_1+w_1z_1}{(1+w_1)(1+w_2)}\\&=
\frac{(z_1-z_2)+z_1(w_2-w_1)+w_1(z_1-z_2)}{(1+w_1)(1+w_2)}.\end{align*}
It follows from $|z_1-z_2|\leqslant \gamma$, $|w_1-w_2|\leqslant \gamma,$ and $|z_1|/|1+w_1|\leqslant M$ that
\[|I|\leqslant \frac{\gamma(1+|z_1|+|w_1|)}{|1+w_1||1+w_2|}\leqslant \frac{\gamma}{|1+w_2||1+w_1|}+\frac{\gamma M}{|1+w_2|}+\frac{\gamma}{|1+w_2|}\,\frac{|w_1|}{|1+w_1|}.\]
In addition, we have $|1+w_2| \geqslant \delta,$
\[|1+w_1|=|1+w_2+(w_1-w_2)|\geqslant \delta-\gamma\geqslant \delta /2,\]
\[
\frac{|w_1|}{|1+w_1|}= \dfrac{2}{|1+w_1|} I(|w_1|\leqslant 2)+
\dfrac{|w_1|}{|w_1|-1}\,I(|w_1|>2)\leqslant\begin{cases}
4/\delta,& |w_1|\leqslant 2,\\
2,& |w_1|>2.\end{cases}\]
Finally, we conclude that $|I|\leqslant \gamma(2/\delta^2+M/\delta +4/\min\{\delta^2,2\delta\}).$ Q.e.d.
\noindent{\bf Proof of Lemma \ref{l2}.} For any given $\varepsilon,M>0$, set
\begin{equation}\label{IM}
I_0(\varepsilon,M)=\varlimsup_{p\to\infty}\sup_{A_p}\p(|\y_p^\top A_p \y_p-\tr(\Sigma_p A_p)|>\varepsilon p),
\end{equation}
where the supremum is taken over all real symmetric $p\times p$ matrices $A_p$ with $\|A_p\|\leqslant M.$ By this definition, there are $p_j\to\infty$ and $A_{p_j}$ with $\|A_{p_j}\|\leqslant M$ such that
\[I_0(\varepsilon,M)=\lim_{j\to\infty}\p(|\y_{p_j}^\top A_{p_j} \y_{p_j}-\tr(\Sigma_{p_j}A_{p_j})|>\varepsilon p_j).\]
Every real symmetric matrix $A_p$ can be written as $A_p=A_{p1}-A_{p2}$ for real symmetric positive semi-definite $p\times p$ matrices $A_{pk},$ $k=1,2$, with $\|A_{pk}\|\leqslant \|A_p\|$. Moreover,
for any $\varepsilon>0$ and $p\geqslant 1$,
\begin{align} \label{ineq}
\p(|\y_p^\top A_p \y_p-\tr(\Sigma_p A_p)|>\varepsilon p)\leqslant &\sum_{k=1}^2 \p(|\y_p^\top A_{pk} \y_p-\tr(\Sigma_p A_{pk})|>\varepsilon p/2) .\end{align}
Hence, it follows from $(\mathrm A2)$ that $I_0(\varepsilon,M)=0$ for any $\varepsilon,M>0.$
If $A_p$ is any real $p\times p$ matrix and $B_p=( A_p^\top +A_p)/2$, then $\y_p^\top A_p\y_p= \y_p^\top B_p\y_p$ and, by Fact 2 in Appendix C, $\|B_p\|\leqslant \|A_p\|.$
In addition, $\tr(\Sigma_pA_p)=\tr( A_p\Sigma_p)=\tr(( A_p\Sigma_p)^\top)=\tr(\Sigma_pA_p^\top )=\tr(\Sigma_pB_p).$
Thus, if $I_1(\varepsilon,M)$ is defined as $I_0(\varepsilon,M)$ in \eqref{IM} with the supremum taken over all real $p\times p$ matrices $A_p$ with $\|A_p\|\leqslant M$, then
\begin{equation}\label{I1M}
I_1(\varepsilon,M)=I_0(\varepsilon,M)=0\quad\text{ for any $\varepsilon,M>0.$}
\end{equation}
Define now $I_2(\varepsilon,M)$ similarly to $I_0(\varepsilon,M)$ in \eqref{IM} with the supremum taken over all complex $p\times p$ matrices $A_p$ with $\|A_p\|\leqslant M$. Every such $A_p$ can be written as $A_p=A_{p1}+iA_{p2}$ for real $p\times p$ matrices $A_{pk},$ $k=1,2.$ By Fact 3 in Appendix C, $
\|A_{pk}\|\leqslant \|A_p\|$, $k=1,2$. Thus, \eqref{ineq} and \eqref{I1M} yield $I_2(\varepsilon,M)=0$ for any $\varepsilon,M>0.$ Q.e.d.
\noindent{\bf Proof of Lemma \ref{l5}.} If $Z_n-\e Z_n\pto 0$ as $n\to\infty$, then $e^{\e Z_n-Z_n}-1\to 0$. Since $\e Z_n\geqslant 0$, $e^{-\e Z_n }\leqslant 1$ and $e^{-\e Z_n}(e^{\e Z_n-Z_n}-1)= e^{-Z_n}-e^{-\e Z_n}\to 0$. By the dominated convergence theorem, $\e (e^{-Z_n}-e^{-\e Z_n})\to 0$.
Suppose now $\e e^{-Z_n}-e^{-\e Z_n}\to 0$. By Jensen's inequality,
\[ e^{-\e Z_n}+o(1) = \e e^{-Z_n}\geqslant (\e e^{-Z_n/2})^2\geqslant (e^{-\e Z_n/2})^2= e^{-\e Z_n}.\]
Note also that $\e Z_n$ is bounded. Therefore, there is $c>0$ such that $e^{-\e Z_n/2}\geqslant c+o(1)$, $n\to\infty.$
Hence, $\var(e^{-Z_n/2})\to0$ as $n\to\infty$ and $\e e^{-Z_n/2}-e^{-\e Z_n/2}\to0.$ Combing these facts yields $e^{-Z_n/2}-e^{-\e Z_n/2}=e^{-\e Z_n/2}(e^{(\e Z_n-Z_n)/2}-1)\pto0$. This shows that $e^{(\e Z_n-Z_n)/2}\pto 1$ or $Z_n-\e Z_n\pto0$. Q.e.d.
\section*{Appendix C}
In this section we list a few useful well-known facts from linear algebra. Write $A_1\succ A_2$ for real $p\times p$ matrices $A_1,A_2$ if $A_1-A_2$ is positive semi-definite.
Let $B$ and $C$ be real $p\times p$ matrices, $A$ be a complex $p\times p$ matrix, and $z\in \mathbb C^+$.
\\
{\bf Fact 1.} $\tr(BC)\leqslant \|B\|\tr(C)$ and $\tr((BC)^2)\leqslant \|B\|^2\tr(C^2)$ if $B,C$ are symmetric and positive semi-definite.
\\
{\bf Fact 2.} If $C=(B^\top+B)/2$, then $\|C\|\leqslant \|B\|$.
\\
{\bf Fact 3.} If $A=B+iC,$ then $\|B\|\leqslant \|A\|$ and $\|C\|\leqslant \|A\|.$
\\
{\bf Fact 4.} If $C$ is symmetric, then $\|(C-zI_p)^{-1}\|\leqslant 1/\Im(z) $.
\\
{\bf Fact 5.} If $w\in\bR^p$ and $C$ is symmetric, then
\[ \frac{|w^\top (C-zI_p)^{-2} w|}{|1+w^\top (C-zI_p)^{-1} w|}\leqslant \frac{1}{\Im(z)}.\]
If, in addition, $C$ is positive definite, then
\[0\leqslant \frac{w^\top C^{-2} w}{1+w^\top C^{-1} w}\leqslant \|C^{-1}\| \frac{w^\top C^{-1} w}{1+w^\top C^{-1} w}\leqslant \|C^{-1}\|.\]
{\bf Fact 6.} If $B$ and $C$ are symmetric and positive semi-definite, then
\[|1+\tr(B(C-zI_p)^{-1})|\geqslant \frac{\Im(z)}{|z|}.\]
{\bf Fact 7.} If $A$ is invertible and $w\in\mathbb C^p$ satisfy $1+w^\top A^{-1}w\neq 0$, then
\[\tr(A+ww^\top )^{-1}=\tr (A^{-1})-\frac{ w^\top A^{-2}w}{1+ w^\top A^{-1}w}\quad\text{and}\quad w^\top (A+ww^\top)^{-1}w=\frac{ w^\top A^{-1}w}{1+ w^\top A^{-1}w}.\]
{\bf Fact 8.} If $C$ is symmetric and positive semi-definite and $\varepsilon,v>0$, then
\[|\tr(C-(-\varepsilon+iv)I_p)^{-1}-\tr(C+\varepsilon I_p)^{-1}|\leqslant \frac{pv}{\varepsilon^2}.\]
\noindent{\bf Proof of Fact 1.} Since $\|B\|I_p\succ B$, \[\|B\|C=C^{1/2}(\|B\|I_p)C^{1/2}\succ C^{1/2}BC^{1/2}\]
and $\|B\|^2C^2\succ (C^{1/2}BC^{1/2})^2$. Thus, $\tr(BC)=\tr(C^{1/2}BC^{1/2})\leqslant \|B\|\tr(C)$ and \[\tr(BCBC)=\tr((C^{1/2}BC^{1/2})^2)\leqslant \|B\|^2\tr(C^2).\]
\noindent{\bf Proof of Fact 2.} We have \[
\|B\|=\sqrt{\lambda_{\max}(B^\top B)}=\sqrt{\lambda_{\max}(BB^\top)}=\|B^\top\|\quad\text{and}\quad\|C\|\leqslant\frac{\|B\|+\|B^\top\|}{2}=\|B\|.\]
\noindent{\bf Proof of Fact 3.} If $A=B+iC$, then \[
\|A\|=\sup_{y\in\mathbb C^p:\,\|y\|=1}\|Ay\|\geqslant \sup_{x\in\mathbb R^p:\,\|x\|=1}\|Ax\|\geqslant
\sup_{x\in\mathbb R^p:\,\|x\|=1}\|Bx\|=\|B\|, \]
where we have used the fact that $\|Ax\|^2=\|Bx\|^2+\|Cx\|^2,$ $x\in\bR^p,$ and, for some nonzero $x_0\in\bR^p$,
$B^\top Bx_0=\|B\|^2x_0$ and $\|Bx_0\|=\|B\|\|x_0\|$. Similarly, we get that $\|A\|\geqslant \|C\|$. Q.e.d.
\noindent{\bf Proof of Fact 4.} The spectral norm of $A=(C-zI_p)^{-1}$ is the square root of $\lambda_{\max} ( A^*A),$ where $A^*=\overline{A^\top}=(C-\overline{z}I_p)^{-1}$. If $z=u+iv$ for $u\in \bR$ and $v=\Im(z)>0,$ then
\[A^*A=(C-\overline{z}I_p)^{-1}(C-zI_p)^{-1}=((C-uI_p)^2+v^2 I_p)^{-1}.\]
Hence, $\lambda_{\max}(A^*A)\leqslant 1/v^2$ and $\|A\|\leqslant 1/v.$ Q.e.d.
\noindent{\bf Proof of Fact 5.} Write $C=\sum_{k=1}^p\lambda_ke_ke_k^\top$ for some $\lambda_k\in\bR$ and orthonormal vectors $e_k\in\bR^p,$ $1\leqslant k\leqslant p.$ Then the result follows from the inequalities
\begin{align*} |1+&w^\top (C-zI_p)^{-1} w|\geqslant\Im(w^\top (C-zI_p)^{-1} w)=
\Im\Big(\sum_{k=1}^p \frac{(w^\top e_k)^2}{\lambda_k-z}\Big)=\\
&=\Im (z) \sum_{k=1}^p\frac{(w^\top e_k)^2}{|\lambda_k-z|^2}\geqslant
\Im (z) \Big|\sum_{k=1}^p\frac{(w^\top e_k)^2}{(\lambda_k-z)^2}\Big|=\Im (z)|w^\top (C-zI_p)^{-2} w|.
\end{align*}
Now, if $C$ is positive definite, then $C^{-k},$ $k=1,2,$ are also positive definite and \[w^\top C^{-2} w=(C^{-1/2}w)^\top C^{-1} (C^{-1/2}w)\leqslant \|C^{-1}\|\|C^{-1/2}w\|^2=\|C^{-1}\|(w^\top C^{-1}w).\]
The latter implies desired bounds. Q.e.d.
\noindent{\bf Proof of Fact 6.} Write $z=u+iv$ for $u\in \bR$ and $v>0$. We need to prove that
\[|z||1+\tr(B(C-zI_p)^{-1})|=|z+\tr(B(C/z-I_p)^{-1})|\geqslant v.\]
Since
\[|z+\tr(B(C/z-I_p)^{-1})|\geqslant \Im(z+\tr(B(C/z-I_p)^{-1}))=v+\Im(\tr(B(C/z-I_p)^{-1})),\]
we only need to check that $\Im\big(\tr(B(C/z-I_p)^{-1})\big)\geqslant0.$
Let
\begin{equation} \label{B}S=\Big(\frac{u}{|z|^2}C-I_p\Big)^2+\frac{v^2}{|z|^4}C^2.\end{equation}
Such $S$ is invertible, symmetric, and positive definite, since \[S=(C/z-I_p)(C/z-I_p)^*=(C/z-I_p)^*(C/z-I_p)=\]\[=
(C/\overline{z}-I_p)(C/z-I_p)=\frac{1}{|z|^2} C^2-\frac{2u}{|z|^2} C+I_p\] and $C/z-I_p=(C-zI_p)/z$ is invertible, where $A^*=\overline{A^\top}$.
Additionally,
\begin{align*}
(C/z-I_p)^{-1}=(C/\overline{z}-I_p)S^{-1}=\Big(\frac{u}{|z|^2}C-I_p+\frac{iv}{|z|^2} C\Big)S^{-1}.
\end{align*}
Therefore,
\[\Im\big(\tr(B(C/z-I_p)^{-1})\big)=\frac{v}{|z|^2}\tr(B CS^{-1}).\]
By the definition of $S$, $C^{1/2}$ and $S^{-1}$ commute, $CS^{-1}=C^{1/2}S^{-1}C^{1/2}$, and
\[\tr(B CS^{-1})=\tr(B C^{1/2}S^{-1}C^{1/2})=\tr(B^{1/2} C^{1/2}S^{-1}C^{1/2}B^{1/2}).\]
As it is shown above, $S$ is symmetric and positive definite. Hence, $S^{-1}$ is symmetric positive definite and
$QS^{-1}Q^\top$ is symmetric positive semi-definite for any $p\times p$ matrix $Q$. Taking $Q=B^{1/2} C^{1/2}$ yields
\[\tr(B CS^{-1})=\tr(QS^{-1}Q^\top)\geqslant 0.\]
This proves the desired bound. Q.e.d.
\noindent{\bf Proof of Fact 7.} The Sherman-Morrison formula states that
\[(A+ww^\top )^{-1}= A^{-1}-\frac{ A^{-1}ww^\top A^{-1}}{1+ w^\top A^{-1}w}\quad\text{when}\quad 1+ w^\top A^{-1}w\neq 0.\]
Taking traces we get the first identity. Multiplying by $w$ and $w^\top,$ we arrive at
\[w^\top(A+ww^\top )^{-1}w= w^\top A^{-1}w-\frac{(w^\top A^{-1}w)^2}{1+ w^\top A^{-1}w}=\frac{ w^\top A^{-1}w}{1+ w^\top A^{-1}w}.\]
Q.e.d.
\noindent{\bf Proof of Fact 8.} We have
\[ (C-(-\varepsilon+iv)I_p)^{-1}-(C+\varepsilon I_p)^{-1}=iv(C-(-\varepsilon+iv)I_p)^{-1}(C+\varepsilon I_p)^{-1}.\]
Arguing as in the proof of Fact 4, we see that
\[\|(C-(-\varepsilon+iv)I_p)^{-1}(C+\varepsilon I_p)^{-1}\|\leqslant
\|(C-(-\varepsilon+iv)I_p)^{-1}\|\,\|(C+\varepsilon I_p)^{-1}\|\leqslant \varepsilon^{-2}.\]
Combining these relations and using the inequality $|\tr(A)|\leqslant p\|A\|$, we get the desired bound. Q.e.d.
| 34,944 |
\section{Introduction}
Throughout this paper, $\mathbb{N}$ is the set of positive integers, $m,n,r,\lambda\in \mathbb{N}$, and $[n]:=\{1,\dots,n\}$. In a mathematics workshop with $mn$ mathematicians from $n$ different areas, each area consisting of $m$ mathematicians, we want to create a collaboration network. For this purpose, we would like to schedule daily meetings between groups of size three, so that (i) two people of the same area meet one person of another area, (ii) each person has exactly $r$ meeting(s) each day, and (iii) each pair of people of the same area have exactly $\lambda$ meeting(s) with each person of another area by the end of the workshop. Using hypergraph amalgamation-detachment, we prove a more general theorem. In particular we show that above meetings can be scheduled if: $3\divides rm$, $2\divides rnm$ and $r\divides 3\lambda(n-1)\binom{m}{2}$.
A {\it hypergraph} $\mathcal G$ is a pair $(V,E)$ where $V$ is a finite set called the vertex set, $E$ is the edge multiset, where every edge is itself a multi-subset of $V$. This means that not only can an edge occur multiple times in $E$, but also each vertex can have multiple occurrences within an edge.
The total number of occurrences of a vertex $v$ among all edges of $E$ is called the {\it degree}, $d_{\mathcal G}(v)$ of $v$ in $\mathcal G$. For $h\in \mathbb{N}$, $\mathcal G$ is said to be $h$-{\it uniform} if $|e|=h$ for each $e\in E$.
For $r, r_1,\dots,r_k\in \mathbb{N}$, an $r$-factor in a hypergraph $\mathcal G$ is a spanning $r$-regular sub-hypergraph, and an {\it $(r_1,\dots,r_k)$-factorization} is a partition of the edge set of $\mathcal G$ into $F_1,\dots, F_k$ where $F_i$ is an $r_i$-factor for $i\in [k]$. We abbreviate $(r,\dots,r)$-factorization to $r$-factorization.
The hypergraph $K_n^h:=(V,\binom{V}{h})$ with $|V|=n$ (by $\binom{V}{h}$ we mean the collection of all $h$-subsets of $V$) is called a {\it complete} $h$-uniform hypergraph. In connection with Kirkman's schoolgirl problem \cite{Kirk1847}, Sylvester conjectured that $K_n^h$ is 1-factorable if and only if $h\divides n$. This conjecture was finally settled by Baranyai \cite{Baran75}. Let $\scr K_{n\times m}^3$ denote the 3-uniform hypergraph with vertex partition $\{V_i:i\in [n]\}$, so that $V_i=\{x_{ij}: j\in [m]\}$ for $i \in [n]$, and with edge set $E=\{\{x_{ij}, x_{ij'}, x_{kl}\} : i,k\in [n], j,j',l\in [m], j\neq j', i\neq k\}$.
One may notice that finding an $r$-factorization for $ \scr K_{n\times m}^3$ is equivalent to scheduling the meetings between mathematicians with the above restrictions for the case $\lambda=1$.
If we replace every edge $e$ of $\mathcal G$ by $\lambda$ copies of $e$, then we denote the new hypergraph by $\lambda \mathcal G$. In this paper, the main result is the following theorem which is obtained by proving a more general result (see Theorem \ref{moregenres}) using amalgamation-detachment techniques.
\begin{theorem}\label{type2suff}
$\lambda \scr K_{m\times n}^{3}$ is $(r_1,\ldots,r_k)$-factorable if
\begin{enumerate
\item [(S1)] $3\divides r_im$ for $i\in [k]$,
\item [(S2)] $2\divides r_imn$ for $i\in [k]$, and
\item [(S3)] $\sum_{i=1}^k r_i= 3\lambda(n-1)\binom{m}{2}$.
\end{enumerate}
\end{theorem}
In particular, by letting $r=r_1=\dots=r_k$ in Theorem \ref{type2suff}, we solve the Mathematicians Collaboration Problem in the following case.
\begin{corollary}
$\lambda \scr K_{m\times n}^{3}$ is $r$-factorable if
\begin{enumerate}
\item [\textup {(i)}] $3\divides rm$,
\item [\textup {(ii)}] $2\divides rnm$, and
\item [\textup {(iii)}] $r\divides 3\lambda(n-1)\binom{m}{2}$.
\end{enumerate}
\end{corollary}
The two results above can be seen as analogues of Baranyai's theorem for complete 3-uniform ``multipartite" hypergraphs. We note that in fact, Baranyai \cite{Baran79} solved the problem of factorization of complete uniform multipartite hypergraphs, but here we aim to solve this problem under a different notion of ``multipartite". In Baranyai's definition, an edge can have at most one vertex from each part, but here we allow an edge to have two vertices from each part (see the definition of $\scr K_{m\times n}^{3}$ above). More precise definitions together with preliminaries are given in Section \ref{term}, the main result is proved in Section \ref{factorizationcor}, and related open problems are discussed in the last section.
Amalgamation-detachment technique was first introduced by Hilton \cite{H2} (who found a new proof for decompositions of complete graphs into Hamiltonian cycles), and was more developed by Hilton and Rodger \cite{HR}.
Hilton's method was later genealized to arbitrary graphs \cite{bahrodjgtold}, and later to hypergraphs \cite{bahCPC1,Bahhyp1,bahrodjgt13, bahnew16} leading to various extensions of Baranyai's theorem (see for example \cite{bahCPC1, bahCCA1}). The results of the present paper, mainly relies on those from \cite{bahCPC1} and \cite{Nash87}. For the sake of completeness, here we give a self contained exposition.
\section{More Terminology and Preliminaries} \label{term}
Recall that an edge can have multiple copies of the same vertex. For the purpose of this paper, all hypergraphs (except when we use the term graph) are 3-uniform, so an edge is always of one of the forms $\{u,u,u\}, \{u,u,v\}$, and $\{u,v,w\}$ which we will abbreviate to $\{u^3\}, \{u^2,v\}$, and $\{u,v,w\}$, respectively. In a hypergraph $\mathcal G$, $\mult_{\mathcal G}(.)$ denotes the multiplicity; for example $\mult_{\mathcal G}(u^3)$ is the multiplicity of an edge of the form $\{u^3\}$. Similarly, for a graph $G$, $\mult(u,v)$ is the multiplicity of the edge $\{u,v\}$. A \textit{k-edge-coloring} of a hypergraph $\mathcal G$ is a mapping $K:E(\mathcal G)\rightarrow [k]$, and the sub-hypergraph of $\mathcal G$ induced by color $i$ is denoted by $\mathcal G(i)$. Whenever it is not ambiguous, we drop the subscripts, and also we abbreviate $d_{\mathcal{G}(i)}(u)$ to $d_i(u)$, $\mult_{\mathcal G(i)}(u^3)$ to $\mult_{i}(u^3)$, etc..
Factorizations of the complete graph, $K_n$, is studied in a very general form in \cite{MJohnson07, Johnstone00}, however for the purpose of this paper, a $\lambda$-fold version is needed:
\begin{theorem}\textup{(Bahmanian, Rodger \cite[Theorem 2.3]{bahrodsurvey1})}\label{bahrodsurvey1thmfac}
$\lambda K_n$ is $(r_1,\dots,r_k)$-factorable if and only if $r_i n$ is even for $i\in[ k]$ and $\sum_{i=1}^k r_i=\lambda (n-1)$.
\end{theorem}
Let $K_{n}^{*}$ denote the 3-uniform hypergraph with $n$ vertices in which $\mult(u^2, v)=1$, and $\mult(u^3)=\mult(u, v, w)=0$ for distinct vertices $u,v,w$. A (3-uniform) hypergraph $\mathcal G=(V,E)$ is {\it $n$-partite}, if there exists a partition $\{V_1,\dots,V_n\}$ of $V$ such that for every $e\in E$, $|e\cap V_i|=1, |e\cap V_j|=2$ for some $i,j\in [n]$ with $i\neq j$. For example, both $K_n^*$ and $\scr K_{m\times n}^{3}$ are $n$-partite. We need another simple but crucial lemma:
\begin{lemma} If $r_in$ is even for $i\in [k]$, and $\sum_{i=1}^k r_i= \lambda (n-1)$, then $\lambda K_{n}^{*}$ is $(3r_1,\ldots,3r_k)$-factorable. \label{3rfaclemma}
\end{lemma}
\begin{proof}
Let $G=\lambda K_n$ with vertex set $V$. By Theorem \ref{bahrodsurvey1thmfac}, $G$ is $(r_1,\ldots,r_k)$-factorable. Using this factorization, we obtain a $k$-edge-coloring for $G$ such that $d_{G(i)}(v)=r_i$ for every $v\in V$ and every color $i\in [k]$. Now we form a $k$-edge-colored hypergraph $\mathcal H$ with vertex set $V$ such that $\mult_{\mathcal H(i)}(u^2,v)=\mult_{G(i)}(u,v)$ for every pair of distinct vertices $u,v\in V$, and each color $i\in [k]$. It is easy to see that $\mathcal H\cong\lambda K_{n}^{*}$ and $d_{\mathcal H(i)}(v)=3r_i$ for every $v\in V$ and every color $i\in [k]$. Thus we obtain a $(3r_1,\ldots,3r_k)$-factorization for $\lambda K_{n}^{*}$.
\end{proof}
If the multiplicity of a vertex $\alpha$ in an edge $e$ is $p$, we say that $\alpha$ is {\it incident} with $p$ distinct {\it hinges}, say $h_1(\alpha,e),\dots,h_p(\alpha,e)$, and we also say that $e$ is {\it incident} with $h_1(\alpha,e),\dots,h_p(\alpha,e)$. The set of all hinges in $\mathcal G$ incident with $\alpha$ is denoted by $H_{\mathcal G}(\alpha)$; so $|H_{\mathcal G}(\alpha)|$ is in fact the degree of $\alpha$.
Intuitively speaking, an {\it $\alpha$-detachment} of a hypergraph $\mathcal G$ is a hypergraph obtained by splitting a vertex $\alpha$ into one or more vertices and sharing the incident hinges and edges among the subvertices. That is, in an $\alpha$-detachment $\mathcal G'$ of $\mathcal G$ in which we split $\alpha$ into $\alpha$ and $\beta$, an edge of the form $\{\alpha^p,u_1,\dots,u_z\}$ in $\mathcal G$ will be of the form $\{\alpha^{p-i},\beta^{i},u_1,\dots,u_z\}$ in $\mathcal G'$ for some $i$, $0\leq i\leq p$. Note that a hypergraph and its detachments have the same hinges. Whenever it is not ambiguous, we use $d'$, $\mult'$, etc. for degree, multiplicity and other hypergraph parameters in $\mathcal G'$.
Let us fix a vertex $\alpha$ of a $k$-edge-colored hypergraph $\mathcal G=(V,E)$. For $i\in [k]$, let $H_i(\alpha)$ be the set of hinges each of which is incident with both $\alpha$ and an edge of color $i$ (so $d_i(\alpha)=|H_i(\alpha)|$). For any edge $e\in E$, let $H^e(\alpha)$ be the collection of hinges incident with both $\alpha$ and $e$. Clearly, if $e$ is of color $i$, then $H^e(\alpha)\subset H_i(\alpha)$.
A family $\scr A$ of sets is \textit{laminar} if, for every pair $A, B$ of sets belonging to $\scr A$, either $A\subset B$, or $B\subset A$, or $A\cap B=\varnothing$. We shall present two lemmas, both of which follow immediately from definitions.
\begin{lemma}\label{lamAlem}
Let $\scr A = \{H_{1}(\alpha),\ldots,H_{k}(\alpha)\} \cup \{ H^e(\alpha) : e \in E\}$. Then $\scr A$ is a laminar family of subsets of $H(\alpha)$.
\end{lemma}
For each $p\in \{1,2\}$, and each $U \subset V\backslash\{\alpha\}$, let $H(\alpha^p, U)$ be the set of hinges each of which is incident with both $\alpha$ and an edge of the form $\{\alpha^p\}\cup U$ in $\mathcal G$ (so $|H(\alpha^p, U)|=p\mult(\{\alpha^p,U\}$).
\begin{lemma}\label{lamBlem}
Let $\scr B=\{H(\alpha^p, U): p\in \{1,2\}, U \subset V\backslash\{\alpha\}\}$. Then $\scr B$ is a laminar family of disjoint subsets of $H(\alpha)$.
\end{lemma}
If $x, y$ are real numbers, then $\lfloor x \rfloor$ and $\lceil x \rceil$ denote the integers such that $x-1<\lfloor x \rfloor \leq x \leq \lceil x \rceil < x+1$, and $x\approx y$ means $\lfloor y \rfloor \leq x\leq \lceil y \rceil$. We need the following powerful lemma:
\begin{lemma}\textup{(Nash-Williams \cite[Lemma 2]{Nash87})}\label{laminarlem}
If $\scr A, \scr B$ are two laminar families of subsets of a finite set $S$, and $n\in \mathbb{N}$, then there exist a subset $A$ of $S$ such that
\begin{eqnarray*}
|A\cap P|\approx |P|/n \mbox { for every } P\in \scr A \cup \scr B.
\end{eqnarray*}
\end{lemma}
\section{Proofs}\label{factorizationcor}
Notice that $\lambda\scr K_{m\times n}^{3}$ is a $3\lambda(n-1)\binom{m}{2}$-regular hypergraph with $nm$ vertices and $2\lambda m\binom{n}{2}\binom{m}{2}$ edges. To prove Theorem \ref{type2suff}, we prove the following seemingly stronger result.
\begin{theorem} \label{moregenres}
Let $3\divides r_im$ and $2\divides r_imn$ for $i\in [k]$, and $\sum_{i=1}^k r_i= 3\lambda(n-1)\binom{m}{2}$. Then for all $\ell=n,n+1,\dots,mn$ there exists a $k$-edge-colored $\ell$-vertex $n$-partite hypergraph $\mathcal G=(V,E)$ and a function $g:V\rightarrow \mathbb N$ such that the following conditions are satisfied:
\begin{enumerate}
\item [(C1)] $\sum_{v\in W} g(v)=m$ for each part $W$ of $\mathcal G$;
\item [(C2)] $\mult (u^2,v)=\lambda \binom{g(u)}{2} g(v)$ for each pair of vertices $u,v$ from different parts of $\mathcal G$;
\item [(C3)] $\mult (u,v,w)=\lambda g(u) g(v)g(w)$ for each pair of distinct vertices $u,w$ from the same part, and $v$ from a different part of $\mathcal G$;
\item [(C4)] $d_{i}(u)=r_i g(u)$ for each color $i\in [k]$ and each $u\in V$.
\end{enumerate}
\end{theorem}
\begin{remark}\textup{ It is implicitly understood that every other type of edge in $\mathcal G$ is of multiplicity 0.
}\end{remark}
Before we prove Theorem \ref{moregenres}, we show how Theorem \ref{type2suff} is implied by Theorem \ref{moregenres}.
\\
\noindent {\it \bf Proof of Theorem \ref{type2suff}.}
It is enough to take $\ell=mn$ in Theorem \ref{moregenres}. Then there exists an $n$-partite hypergraph $\mathcal G=(V,E)$ of order $mn$ and a function $g:V\rightarrow \mathbb N$ such that by (C1) $\sum_{v\in W} g(v)=m$ for each part $W$ of $\mathcal G$. This implies that $g(v)=1$ for each $v\in V$ and that each part of $\mathcal G$ has $m$ vertices. By (C2), $\mult_{\mathcal G} (u^2,v)=\lambda \binom{1}{2}(1)=0$ for each pair of vertices $u,v$ from different parts of $\mathcal G$, and by (C3), $\mult_{\mathcal G} (u,v,w)=\lambda$ for each pair of vertices $u,v$ from the same part and $w$ from a different part of $\mathcal G$. This implies that $\mathcal G \cong \lambda \scr K_{m\times n}^{3}$.
Finally, by (C4), $\mathcal G$ admits a $k$-edge-coloring such that $d_{\mathcal G(i)}(v)=r_i$ for each color $i\in [ k]$. This completes the proof.
\qed
The idea of the proof of Theorem \ref{moregenres} is that each vertex $\alpha$ will be split into $g(\alpha)$ vertices and that this will be done by ``splitting off" single vertices one at a time.
\\
\noindent {\it \bf Proof of Theorem \ref{moregenres}.} We prove the theorem by induction on $\ell$.
First we prove the basis of induction, case $\ell=n$. Let $\mathcal G=(V,E)$ be $\lambda m\binom{m}{2}K_{n}^{*}$ and let $g(v)=m$ for all $v\in V$. Since $\mathcal G$ has $n$ vertices, it is $n$-partite (each vertex being a partite set). Obviously, $\sum_{v\in W} g(v)=g(v)=m$ for each part $W$ of $\mathcal G$. Also, $\mult(u^2,v)=\lambda m\binom{m}{2}=\lambda \binom{g(u)}{2} g(v)$ for each pair of vertices $u,v$ from distinct parts of $\mathcal G$, so (C2) is satisfied. Since there is only one vertex in each part, (C3) is trivially satisfied.
Since for $i\in [k]$, $2\divides \frac{r_imn}{3}$ and $\sum_{i=1}^k \frac{r_im}{3}= \lambda m(n-1)\binom{m}{2}$, by Lemma \ref{3rfaclemma}, $\mathcal G$ is $(mr_1,\ldots,mr_k)$-factorable. Thus, we can find a $k$-edge-coloring for $\mathcal G$ such that $d_{\mathcal G(j)}(v)=mr_i=r_ig(v)$ for $i\in [k]$, and therefore (C4) is satisfied.
Suppose now that for some $\ell \in \{ n, n+1, \ldots, mn-1 \}$, there exists a $k$-edge-colored $n$-partite hypergraph $\mathcal G=(V,E)$ of order $\ell$ and a function $g:V \rightarrow {\mathbb N}$ satisfying properties (C1)--(C4) from the statement of the theorem. We shall now construct an $n$-partite hypergraph $\mathcal G'$ of order $\ell+1$ and a function $g':V(\mathcal G') \rightarrow {\mathbb N}$ satisfying (C1)--(C4).
Since $\ell<mn$, $\mathcal G$ is $n$-partite and (C1) holds for $\mathcal G$, there exists a vertex $\alpha$ of $\mathcal G$ with $g(\alpha)>1$. The graph $\mathcal G'$ will be constructed as an $\alpha$-detachment of $\mathcal G$ with the help of laminar families $$\scr A:= \{H_{1}(\alpha),\ldots,H_{k}(\alpha)\} \cup \{ H^e(\alpha) : e \in E\}$$ and $$
\scr B:= \{H(\alpha^p, U): p\in \{1,2\}, U \subset V\backslash\{\alpha\}\}.$$
By Lemma \ref{laminarlem}, there exists a subset $Z$ of $H(\alpha)$ such that
\begin{equation}\label{lamapp1'} |Z\cap P|\approx |P|/g(\alpha), \mbox{ for every } P\in \mathscr A \cup \scr B. \end{equation}
Let $\mathcal G'=(V',E')$ with $V'=V\cup\{\beta\}$ be the hypergraph obtained from $\mathcal G$ by splitting $\alpha$ into two vertices $\alpha$ and $\beta$ in such a way that hinges which were incident with $\alpha$ in $ \mathcal G$ become incident in $\mathcal G'$ with $\alpha$ or $\beta$ according to whether they do not or do belong to $Z$, respectively. More precisely,
\begin{equation}\label{hinge1'}
H'(\beta)=Z, \quad H'(\alpha)=H( \alpha)\backslash Z.
\end{equation}
So $\mathcal G'$ is an $\alpha$-detachment of $\mathcal{G}$ and the colors of the edges are preserved. Let $g':V' \rightarrow {\mathbb N}$ so that $g'(\alpha)=g(\alpha)-1, g'(\beta)=1$, and $g'(u)=g(u)$ for each $u\in V'\backslash \{\alpha,\beta\}$.
It is obvious that $\mathcal G'$ is of order $\ell+1$, $n$-partite, and $\sum_{v\in W} g'(v)=m$ for each part $W$ of $\mathcal G'$ (the new vertex $\beta$ belongs to the same part of $\mathcal{G'}$ as $\alpha$ belongs to). Moreover, it is clear that $\mathcal G'$ satisfies (C2)--(C4) if $\{\alpha,\beta\} \cap \{u,v,w\}=\emptyset$. For the rest of the argument, we will repeatedly use the definitions of $\mathscr A, \mathscr B$, (\ref{lamapp1'}), and (\ref{hinge1'}).
For $i\in [k]$ we have
\begin{eqnarray*}
d'_i(\beta) &=& |Z\cap H_i(\alpha)|\approx |H_i(\alpha)|/g(\alpha)
= d_i(\alpha)/g(\alpha)=r_i=r_ig'(\beta),\\
d'_i(\alpha) &=& d_i(\alpha)-d'_i(\beta)=r_ig(\alpha)-r_i=r_i(g(\alpha)-1)=r_ig'(\alpha),
\end{eqnarray*}
so $\mathcal{G'}$ satisfies (C4).
Let $u\in V'$ so that $u$ and $\alpha$ (or $\beta$) belong to different parts of $\mathcal{G'}$. We have
\begin{eqnarray*}
\mult'(\beta,u^2)&=&|Z\cap H(\alpha,\{u^2\})|\approx |H(\alpha,\{u^2\})|/g(\alpha)=\mult(\alpha,u^2)/g(\alpha)\\
&=&\lambda \binom{g(u)}{2}=\lambda \binom{g'(u)}{2} g'(\beta),\\
\mult' (\alpha,u^2)&=&\mult(\alpha,u^2)-\mult' (\beta,u^2)=\lambda \binom{g(u)}{2} g(\alpha)-\lambda \binom{g(u)}{2}
=\lambda \binom{g'(u)}{2} g'(\alpha).
\end{eqnarray*}
Recall that $g(\alpha)\geq 2$, and for every $e\in E$ and $i\in[k]$, $|H^e(\alpha)|\leq 2$, and thus $|Z\cap H^e(\alpha)|\approx |H^e(\alpha)|/g(\alpha)\leq 1$. This implies that
\begin{eqnarray*}
\mult'(\beta^2,u)=0=\lambda \binom{g'(\beta)}{2} g'(u),
\end{eqnarray*}
and so $\mult(\alpha^2,u)=\mult' (\alpha^2,u)+\mult' (\alpha,\beta,u)$. Now we have
\begin{eqnarray*}
\mult' (\alpha,\beta,u)&=&|Z\cap H(\alpha^2,\{u\})| \approx |H(\alpha^2,\{u\})|/g(\alpha)\\
&=&2\mult(\alpha^2,u)/g(\alpha)=\lambda (g(\alpha)-1)g(u)=\lambda g'(\alpha)g'(\beta)g'(u),\\
\mult' (\alpha^2,u)&=&\mult(\alpha^2,u)-\mult' (\alpha,\beta,u)=\lambda \binom{g(\alpha)}{2}g(u) -\lambda (g(\alpha)-1)g(u)\\
&=&\lambda \binom{g(\alpha)-1}{2} g(u)=\lambda \binom{g'(\alpha)}{2} g'(u).
\end{eqnarray*}
Therefore $\mathcal{G'}$ satisfies (C2).
Let $u,v\in V'$ so that $u,v$ belong to different parts of $\mathcal{G'}$, $u,\alpha$ belong to the same part of $\mathcal{G'}$, and $u\notin\{\alpha,\beta\}$. We have
\begin{eqnarray*}
\mult'(\beta, u, v)&=&|Z\cap H(\alpha,\{u,v\})|\approx |H(\alpha,\{u,v\})|/g(\alpha)=\mult(\alpha,u,v)/g(\alpha)\\
&=&\lambda g(u)g(v)=\lambda g'(\beta)g'(u)g'(v),\\
\mult' (\alpha,u,v)&=&\mult(\alpha,u,v)-\mult'(\beta,u,v)=\lambda (g(\alpha)-1)g(u)g(v)= \lambda g'(\alpha)g'(u)g'(v).
\end{eqnarray*}
Finally, let $u,v\in V'$ so that $u,v$ belong to the same part of $\mathcal{G'}$, and $u,\alpha$ belong to different parts of $\mathcal{G'}$, and $u\notin\{\alpha,\beta\}$. By an argument very similar to the one above, we have
\begin{eqnarray*}
\mult'(u, v,\beta)&=&\lambda g'(u)g'(v)g'(\beta),\\
\mult' (u,v,\alpha)&=& \lambda g'(u)g'(v)g'(\alpha).
\end{eqnarray*}
Therefore $\mathcal{G'}$ satisfies (C3), and the proof is complete.
\qed
\section{Final Remarks}\label{remarks}
We define $\scr K_{m_1,\ldots,m_n}^{3}$ similar to $\scr K_{m\times n}^{3}$ with the difference that in $\scr K_{m_1,\ldots,m_n}^{3}$ we allow different parts to have different sizes. It seems reasonable to conjecture that
\begin{conjecture} \label{type2suffconj}
$\lambda \scr K_{m_1,\ldots,m_n}^{3}$ is $(r_1,\ldots,r_k)$-factorable if and only if
\begin{enumerate
\item [\textup {(i)}] $m_i=m_j:=m$ for $i,j\in [n]$,
\item [\textup {(ii)}] $3\divides r_imn$ for $i\in [k]$, and
\item [\textup {(iii)}] $\sum_{i=1}^{k} r_i= 3\lambda(n-1)\binom{m}{2}$.
\end{enumerate}
\end{conjecture}
We prove the necessity as follows. Since $\lambda \scr K_{m\times n}^{3}$ is factorable, it must be regular. Let $u$ and $v$ be two vertices from two different parts, say $p^{th}$ and $q^{th}$ parts respectively. Then we have the following sequence of equivalences:
\begin{align*}
d(u)&=d(v) &\iff \\
\sum\nolimits_{\scriptstyle 1 \leq i \leq n \hfill \atop \scriptstyle i \ne p \hfill} \binom{m_i}{2}+(m_p-1)\sum\nolimits_{\scriptstyle 1 \leq i \leq n \hfill \atop \scriptstyle i \ne p \hfill}m_i&=\\
\sum\nolimits_{\scriptstyle 1 \leq i \leq n \hfill \atop \scriptstyle i \ne q \hfill} \binom{m_i}{2}+(m_q-1)\sum\nolimits_{\scriptstyle 1 \leq i \leq n \hfill \atop \scriptstyle i \ne q \hfill}m_i &&\iff \\
\binom{m_q}{2}+\sum\nolimits_{\scriptstyle 1 \leq i \leq n \hfill \atop \scriptstyle i \ne p,q \hfill} \binom{m_i}{2}+(m_p-1)(m_q+\sum\nolimits_{\scriptstyle 1 \leq i \leq n \hfill \atop \scriptstyle i \ne p,q \hfill}m_i)&= \\
\binom{m_p}{2}+\sum\nolimits_{\scriptstyle 1 \leq i \leq n \hfill \atop \scriptstyle i \ne p,q \hfill} \binom{m_i}{2}+(m_q-1)(m_p+\sum\nolimits_{\scriptstyle 1 \leq i \leq n \hfill \atop \scriptstyle i \ne p,q \hfill}m_i) &&\iff& \\
\binom{m_p}{2}-\binom{m_q}{2}+m_pm_q-m_p-m_pm_q+m_q+(m_p-m_q)\sum\nolimits_{\scriptstyle 1 \leq i \leq n \hfill \atop \scriptstyle i \ne p,q \hfill}m_i)&=0 &\iff \\
m_p^2-m_q^2-3m_p+3m_q+2(m_p-m_q)\sum\nolimits_{\scriptstyle 1 \leq i \leq n \hfill \atop \scriptstyle i \ne p,q \hfill}m_i)&=0 &\iff \\
(m_p-m_q)(m_p+m_q-3+2\sum\nolimits_{\scriptstyle 1 \leq i \leq n \hfill \atop \scriptstyle i \ne p,q \hfill}m_i)&=0 & \iff \\%\mathop\iff \limits^{n\geq 3} \\
m_p&=m_q:=m.
\end{align*}
This proves (i). The existence of an $r_i$-factor implies that $3\divides r_imn$ for $i\in [k]$. Since each $r_i$-factor is an $r_i$-regular spanning sub-hypergraph and $\lambda \scr K_{m\times n}^{3}$ is $3\lambda(n-1)\binom{m}{2}$-regular, we must have $\sum_{i=1}^{k} r_i=3\lambda(n-1)\binom{m}{2}$.
In Theorem \ref{type2suff}, we made partial progress toward settling Conjecture \ref{type2suffconj}, however at this point, it is not clear to us whether our approach will work for the remaining cases.
\section{Acknowledgement}
The author is deeply grateful to Professors Chris Rodger, Mateja \v Sajna, and the anonymous referee for their constructive comments.
| 9,000 |
\section{Introduction}
The numerical value of the Fermi constant $G_F$ is conventionally defined via the muon lifetime within the SM. Even though this measurement is extremely precise~\cite{Webber:2010zf,Tishchenko:2012ie,Gorringe:2015cma}
\begin{equation}
\label{Mulan}
G_F^\mu=1.1663787(6) \times 10^{-5} \,\text{GeV}^{-2},
\end{equation}
at the level of $0.5\,\text{ppm}$, its determination of the Fermi constant is not necessarily free of BSM contributions. In fact, one can only conclude the presence or absence of BSM effects by comparing $G_F^\mu$ to another independent determination. This idea was first introduced by Marciano in Ref.~\cite{Marciano:1999ih}, concentrating on $Z$-pole observables and the fine-structure constant $\alpha$. In addition to a lot of new data that has become available since 1999, another option already mentioned in Ref.~\cite{Marciano:1999ih}---the determination of $G_F$ from $\beta$ and kaon decays using CKM unitarity---has become of particular interest due to recent hints for the (apparent) violation of first-row CKM unitarity. These developments motivate a fresh look at the Fermi constant, in particular on its extraction from a global EW fit and via CKM unitarity, as to be discussed in the first part of this Letter.
The comparison of the resulting values for $G_F$ shows that with modern input these two extractions are close in precision, yet still lagging behind muon decay by almost three orders of magnitude. Since the different $G_F$ determinations turn out to display some disagreement beyond their quoted uncertainties, the second part of this Letter is devoted to a systematic analysis of possible BSM contributions in SM effective field theory (SMEFT)~\cite{Buchmuller:1985jz,Grzadkowski:2010es} to see which scenarios could account for these tensions without being excluded by other constraints. This is important to identify BSM scenarios that could be responsible for the tensions, which
will be scrutinized with forthcoming data in the next years.
\section{Determinations of $\boldsymbol{G_F}$}
Within the SM, the Fermi constant $G_F$ is defined by, and is most precisely determined from, the muon lifetime~\cite{Tishchenko:2012ie}
\begin{equation}
\frac{1}{\tau_{\mu}}=\frac{(G_F^\mu)^2m_{\mu}^5}{192\pi^3}(1+\Delta q),
\end{equation}
where $\Delta q$ includes the phase space, QED, and hadronic radiative corrections. The resulting numerical value in Eq.~\eqref{Mulan} is so precise that its error can be ignored in the following. To address the question whether $G_F^\mu$ subsumes BSM contributions, however, alternative independent determinations of $G_F$ are indispensable, and their precision limits the extent to which BSM contamination in $G_F^\mu$ can be detected.
In Ref.~\cite{Marciano:1999ih}, the two best independent determinations were found as
\begin{align}
\label{Marciano}
G_F^{Z\ell^+\ell^-}&=1.1650(14)\big({}^{+11}_{-6}\big)\times 10^{-5}\,\text{GeV}^{-2},\notag\\
G_F^{(3)}&=1.1672(8)\big({}^{+18}_{-7}\big)\times 10^{-5}\,\text{GeV}^{-2},
\end{align}
where the first variant uses the width for $Z\to\ell^+\ell^-(\gamma)$, while the second employs $\alpha$ and $\sin^2\theta_W$, together with the appropriate radiative corrections. Since the present uncertainty in $\Gamma[Z\to\ell^+\ell^-(\gamma)]=83.984(86)\,\text{MeV}$~\cite{Zyla:2020zbs} is only marginally improved compared to the one available in 1999, the update
\begin{equation}
G_F^{Z\ell^+\ell^-}=1.1661(16)\times 10^{-5}\,\text{GeV}^{-2}
\end{equation}
does not lead to a gain in precision, but the shift in the central value improves agreement with $G_F^\mu$. The second variant, $G_F^{(3)}$, is more interesting, as here the main limitation arose from the radiative corrections, which have seen significant improvements regarding the input values for the masses of the top quark, $m_t$, the Higgs boson, $M_H$, the strong coupling, $\alpha_s$, and the hadronic running of $\alpha$. In fact, with all EW parameters determined, it now makes sense to use the global EW fit, for which $G_F^\mu$ is usually a key input quantity, instead as a tool to determine $G_F$ in a completely independent way.
The EW observables included in our fit ($W$ mass, $\sin^2\theta_W$, and
$Z$-pole observables~\cite{Schael:2013ita,ALEPH:2005ab}) are given in Table~\ref{ObsEW}, with the other input parameters summarized in Table~\ref{ParamEW}. Here, the hadronic running $\Delta\alpha_\text{had}$ is taken from $e^+e^-$ data, which is insensitive to the changes in $e^+e^-\to\text{hadrons}$ cross sections~\cite{Aoyama:2020ynm,Davier:2017zfy,Keshavarzi:2018mgv,Colangelo:2018mtw,Hoferichter:2019mqg,Davier:2019can,Keshavarzi:2019abf,Hoid:2020xjs} recently suggested by lattice QCD~\cite{Borsanyi:2020mff}, as long as these changes are concentrated at low energies~\cite{Crivellin:2020zul,Keshavarzi:2020bfy,Malaescu:2020zuc,Colangelo:2020lcg}.
We perform the global EW fit (without using experimental input for $G_F$) in a Bayesian framework using the publicly available \texttt{HEPfit} package~\cite{deBlas:2019okz}, whose Markov Chain Monte Carlo (MCMC) determination of posteriors is powered by the Bayesian Analysis Toolkit (\texttt{BAT})~\cite{Caldwell:2008fw}. As a result, we find
\begin{equation}
\label{GFEW}
G_F^\text{EW}\Big|_\text{full}=1.16716(39)\times 10^{-5}\,\text{GeV}^{-2},
\end{equation}
a gain in precision over $G_F^{(3)}$ in Eq.~\eqref{Marciano} by a factor $5$. As depicted in Fig.~\ref{GFplot}, this value lies above $G_F^\mu$ by $2\sigma$, reflecting the known tensions within the EW fit~\cite{Baak:2014ora,deBlas:2016ojx}. For comparison, we also considered a closer analog of $G_F^{(3)}$, by only keeping $\sin^2\theta_W$ from Table~\ref{ObsEW} in the fit, which gives
\begin{equation}
G_F^\text{EW}\Big|_\text{minimal}=1.16728(83)\times 10^{-5}\,\text{GeV}^{-2},
\end{equation}
consistent with Eq.~\eqref{GFEW}, but with a larger uncertainty. The pull of $G_F^\text{EW}$ away from $G_F^\mu$ is mainly driven by $M_W$, $\sin^2\theta_W$ from the hadron colliders, $A_\ell$, and $A_{\rm FB}^{0, \ell}$.
\begin{table}[t!]
\centering
\begin{tabular}{c c}
\toprule
\begin{tabular}{l r r }
$M_W\,[\text{GeV}]$ & ~\cite{Zyla:2020zbs} & $80.379(12)$ \\
$\Gamma_W\,[\text{GeV}]$ & ~\cite{Zyla:2020zbs} & $2.085(42)$ \\
$\text{BR}(W\to \text{had})$ & ~\cite{Zyla:2020zbs} & $0.6741(27)$ \\
$\text{BR}(W\to \text{lep})$ & ~\cite{Zyla:2020zbs} & $0.1086(9)$ \\
$\text{sin}^2\theta_{\rm eff(QFB)}$ & ~\cite{Zyla:2020zbs} & $0.2324(12)$ \\
$\text{sin}^2\theta_{\rm eff(Tevatron)}$ & ~\cite{Aaltonen:2018dxj} & $0.23148(33)$ \\
$\text{sin}^2\theta_{\rm eff(LHC)}$ & ~\cite{Aaij:2015lka,Aad:2015uau,ATLAS:2018gqq,Sirunyan:2018swq} & $0.23129(33)$ \\
$\Gamma_Z\,[\text{GeV}]$ &~\cite{ALEPH:2005ab} &$2.4952(23)$ \\
$\sigma_h^{0}\,[\text{nb}]$ &~\cite{ALEPH:2005ab} &$41.541(37)$ \\
$P_{\tau}^{\rm pol}$ &~\cite{ALEPH:2005ab} &$0.1465(33)$ \\
\end{tabular} &
\begin{tabular}{l r r}
$A_{\ell}$ &~\cite{ALEPH:2005ab} &$0.1513(21)$ \\
$R^0_{\ell}$ &~\cite{ALEPH:2005ab} &$20.767(25)$ \\
$A_{\rm FB}^{0, \ell}$&~\cite{ALEPH:2005ab} &$0.0171(10)$ \\
$R_{b}^{0}$ &~\cite{ALEPH:2005ab} &$0.21629(66)$\\
$R_{c}^{0}$ &~\cite{ALEPH:2005ab} &$0.1721(30)$ \\
$A_{\rm FB}^{0,b}$ &~\cite{ALEPH:2005ab} &$0.0992(16)$\\
$A_{\rm FB}^{0,c}$ &~\cite{ALEPH:2005ab} &$0.0707(35)$ \\
$A_{b}$ &~\cite{ALEPH:2005ab} &$0.923(20)$ \\
$A_{c}$ &~\cite{ALEPH:2005ab} &$0.670(27)$ \\
\end{tabular}\\
\botrule
\end{tabular}
\caption{EW observables included in our global fit together with their current experimental values.\label{ObsEW}}
\end{table}
\begin{table}[t!]
\centering
\begin{tabular}{l r}
\hline\hline
Parameter & Prior \\
\hline
$\alpha\times 10^3$~\cite{Zyla:2020zbs} & $7.2973525664(17)$ \\
$\Delta\alpha_\text{had}\times 10^4$~\cite{Davier:2019can,Keshavarzi:2019abf} & $276.1(1.1)$ \\
$\alpha_s(M_Z)$~\cite{Zyla:2020zbs,Aoki:2019cca} & $0.1179(10)$\\
$M_Z\,\,[{\rm GeV}]$~\cite{Zyla:2020zbs,Barate:1999ce,Abbiendi:2000hu,Abreu:2000mh,Acciarri:2000ai} & $91.1876(21)$\\
$M_H\,\,[{\rm GeV}]$~\cite{Zyla:2020zbs,Aad:2015zhl,Aaboud:2018wps,Sirunyan:2017exp} & $125.10(14)$ \\
$m_{t}\,\,[{\rm GeV}]$~\cite{Zyla:2020zbs,Khachatryan:2015hba,TevatronElectroweakWorkingGroup:2016lid,Aaboud:2018zbu,Sirunyan:2018mlv}& $172.76(30)$ \\
\hline\hline
\end{tabular}
\caption{Parameters of the EW fit together with their (Gaussian) priors. \label{ParamEW}}
\end{table}
\begin{figure}[t]
\includegraphics[width=1\linewidth]{GF_Plot.pdf}
\caption{Values of $G_F$ from the different determinations.\label{GFplot} }
\end{figure}
As a third possibility, one can determine the Fermi constant from superallowed $\beta$ decays, taking $V_{us}$ from kaon or $\tau$ decays and assuming CKM unitarity ($|V_{ub}|$ is also needed, but the impact of its uncertainty is negligible). This is particularly interesting given recent hints for the apparent violation of first-row CKM unitarity, known as the Cabibbo angle anomaly (CAA). The significance of the tension crucially depends on the radiative corrections applied to $\beta$ decays~\cite{Marciano:2005ec,Seng:2018yzq,Seng:2018qru,Gorchtein:2018fxl,Czarnecki:2019mwq,Seng:2020wjq,Hayen:2020cxh,Hardy:2020qwl}, but also on the treatment of tensions between $K_{\ell 2}$ and $K_{\ell 3}$
decays~\cite{Moulson:2017ive} and the constraints from $\tau$ decays~\cite{Amhis:2019ckw}, see Ref.~\cite{Crivellin:2020lzu} for more details.
In the end, quoting a significance around $3\sigma$ should give a realistic representation of the current situation, and for definiteness we will thus use the estimate of first-row CKM unitarity from Ref.~\cite{Zyla:2020zbs}
\begin{align}
\big|V_{ud}\big|^2+\big|V_{us}\big|^2+\big|V_{ub}\big|^2
= 0.9985(5).
\label{1throw}
\end{align}
In addition, we remark that there is also a deficit in the first-column CKM unitarity relation~\cite{Zyla:2020zbs}
\begin{equation}
\big|V_{ud}\big|^2+\big|V_{cd}\big|^2+\big|V_{td}\big|^2 = 0.9970(18),
\end{equation}
less significant than Eq.~\eqref{1throw}, but suggesting that if the deficits were due to BSM effects, they would likely be related to $\beta$ decays. For the numerical analysis, we will continue to use Eq.~\eqref{1throw} given the higher precision.
The deficit in Eq.~\eqref{1throw} translates to
\begin{equation}
\label{GFCKM}
G_F^\text{CKM} = 0.99925(25)\times G_F^\mu =1.16550(29)\times 10^{-5}\,\text{GeV}^{-2}.
\end{equation}
Comparing the three independent determinations of $G_F$ in Fig.~\ref{GFplot}, one finds the situation that $G_F^\text{EW}$ lies above $G_F^\mu$ by $2\sigma$, $G_F^\text{CKM}$ below $G_F^\mu$ by $3\sigma$, and the tension between $G_F^\text{EW}$ and $G_F^\text{CKM}$ amounts to $3.4\sigma$. To bring all three determinations into agreement within $1\sigma$, an effect in at least two of the underlying processes is thus necessary. This leads us to study BSM contributions to
\begin{enumerate}
\vspace{-1mm}
\item $\mu\to e\nu\nu$ transitions,
\vspace{-1mm}
\item $Z\to \ell\ell,\nu\nu$, $\alpha_2/\alpha$, $M_Z/M_W$,
\vspace{-1mm}
\item superallowed $\beta$ decays,
\end{enumerate}
\vspace{-1mm}
where the second point gives the main observables in the EW fit, with $\alpha_2/\alpha$ a proxy for the ratio of $SU(2)_L$ and $U(1)_Y$ couplings. We do not consider the possibility of BSM effects in kaon, $\tau$, or $D$ decays, as this would require a correlated effect with a relating symmetry. Furthermore, as shown in Ref.~\cite{Crivellin:2020lzu}, the sensitivity to a BSM effect in superallowed $\beta$ decays is enhanced by a factor $|V_{ud}|^2/|V_{us}|^2$ compared to kaon, $\tau$, or $D$ decays. This can also be seen from Eq.~\eqref{1throw} as $|V_{ud}|$ gives the dominant contribution.
BSM explanations of the discrepancies between these determinations of $G_F$ have been studied in the literature in the context of the CAA~\cite{Belfatto:2019swo,Coutinho:2019aiy,Crivellin:2020lzu,Capdevila:2020rrl,Endo:2020tkb,Crivellin:2020ebi,Kirk:2020wdk,Alok:2020jod,Crivellin:2020oup,Crivellin:2020klg,Crivellin:2021egp}. In this Letter, we will analyze possible BSM effects
in all three $G_F$ determinations using an EFT approach with gauge-invariant dimension 6 operators~\cite{Buchmuller:1985jz,Grzadkowski:2010es}.
\section{SMEFT analysis}
Dimension-$6$ operators that can explain the differences among the determinations of $G_F$ can be grouped into the following classes
\begin{enumerate}[A.]
\vspace{-1mm}
\item four-fermion operators in $\mu\to e\nu\nu$,
\vspace{-1mm}
\item four-fermion operators in $u\to d e\nu$,
\vspace{-1mm}
\item modified $W$--$u$--$d$ couplings,
\vspace{-1mm}
\item modified $W$--$\ell$--$\nu$ couplings,
\vspace{-1mm}
\item other operators affecting the EW fit.
\vspace{-1mm}
\end{enumerate}
Global fits to a similar set of effective operators have been considered in Refs.~\cite{Han:2004az,Falkowski:2014tna,Falkowski:2015krw,Ellis:2018gqa,Skiba:2020msb}, here, we will concentrate directly on the impact on $G_F$ determinations, following the conventions of Ref.~\cite{Grzadkowski:2010es}.
\subsection{Four-fermion operators in $\boldsymbol{\mu\to e\nu\nu}$}
Not counting flavor indices, there are only two operators
that can generate a neutral current involving four leptons:
\begin{align}
Q_{\ell \ell }^{ijkl} = {{\bar \ell }_i}{\gamma ^\mu }{\ell _j}{{\bar \ell }_k}{\gamma ^\mu }{\ell _l},\qquad
Q_{\ell e}^{ijkl} &= {{\bar \ell }_i}{\gamma ^\mu }{\ell _j}{{\bar e}_k}{\gamma ^\mu }{e_l}.
\end{align}
Not all flavor combinations are independent, e.g.,
$Q_{\ell \ell }^{ijkl}=Q_{\ell \ell }^{klij}=Q_{\ell \ell }^{ilkj}=Q_{\ell \ell }^{kjil}$ due to Fierz identities and $Q_{\ell \ell(e) }^{jilk}=Q_{\ell \ell(e) }^{ijkl*}$ due to Hermiticity.
Instead of summing over flavor indices, it is easiest to absorb these terms into a redefinition of the operators whose latter two indices are $12$, which contribute directly to $\mu\to e\nu\nu$. Therefore, we have to consider 9 different flavor combinations for both operators:
{\setlength{\leftmargini}{12pt}
\begin{enumerate}
\vspace{-1mm}
\item $Q_{\ell \ell }^{2112}$
contributes to the SM amplitude (its coefficient is real by Fierz identities and Hermiticity). Therefore, it can give a constructive or destructive effect in the muon lifetime and does not affect the Michel parameters~\cite{Michel:1949qe,Michel:1954eua,Kinoshita:1957zz,Scheck:1977yg,Fetscher:1986uj,Danneberg:2005xv,MacDonald:2008xf,Bayes:2011zza}. In order to bring $G_F^\text{CKM}$ and $G_F^\mu$ into agreement at $1\sigma$ we need
\begin{equation}
C_{\ell \ell }^{2112}\approx -1.4\times 10^{-3} G_F.
\end{equation}
This Wilson coefficient is constrained by LEP searches for $e^+e^-\to \mu^+\mu^-$~\cite{Schael:2013ita}
\begin{equation}
- \frac{4\pi}{(9.8\,\text{TeV})^2} < C_{\ell \ell }^{1221} < \frac{4\pi}{(12.2\,\text{TeV})^2},
\end{equation}
a factor $8$ weaker than preferred by the CAA, but within reach of future $e^+e^-$ colliders.
\vspace{-1mm}
\item Even though $Q_{\ell e}^{2112}$ has a vectorial Dirac structure, it leads to a scalar amplitude after applying Fierz identities. Its interference with the SM amplitude is usually expressed in terms of the Michel parameter $\eta=\Re C_{\ell e }^{2112}/(2\sqrt{2}G_F)$, leading to a correction $1-2\eta m_e/m_\mu$. In the absence of right-handed neutrinos the restricted analysis from Ref.~\cite{Danneberg:2005xv} applies, constraining the shift in $G_F^\mu$ to $0.68\times 10^{-4}$, well below the required effect to obtain $1\sigma$ agreement with $G_F^\text{CKM}$ or $G_F^\text{EW}$.
\vspace{-1mm}
\item The operators $Q_{\ell \ell (e) }^{1212}$ could contribute to muon decay as long as the neutrino flavors are not detected. To reconcile $G_F^\text{CKM}$ and $G_F^\mu$ within $1\sigma$ we need $|C_{\ell \ell }^{1212}|\approx 0.045\,G_F$ or $|C_{\ell e }^{1212}|\approx 0.09\,G_F$. Both solutions are excluded by muonium--anti-muonium oscillations ($M=\mu^+ e^-$)~\cite{Willmann:1998gd}
\begin{equation}
{\cal P}(\bar{M}\text{--}M)<8.3\times 10^{-11}/S_B,
\label{exp:mmbar}
\end{equation}
with correction factor $S_B=0.35$ ($C_{\ell \ell }^{1212}$) and $S_B=0.78$ ($C_{\ell e }^{1212}$) for the extrapolation to zero magnetic field.
Comparing to the prediction for the rate~\cite{Feinberg:1961zza,Hou:1995np,Horikawa:1995ae}
\begin{align}
{\cal P}(\bar{M}\text{--}M) &=
\frac{8(\alpha \mu_{\mu e})^6\tau_\mu^2 G_F^2}{\pi^2}
\, \left|{C_{\ell \ell (e) }^{1212}}/{G_F}\right|^2\notag\\
&=3.21\times 10^{-6} \left|{C_{\ell \ell (e) }^{1212}}/{G_F}\right|^2,
\end{align}
with reduced mass $\mu_{\mu e}=m_\mu m_e/(m_\mu+m_e)$, the limits become
$|C_{\ell \ell (e) }^{1212}|<8.6 (5.8)\times 10^{-3}G_F$.
\vspace{-1mm}
\item For $Q_{\ell \ell (e) }^{1112}$ again numerical values of $
|C_{\ell \ell (e) }^{1112}|\approx 0.09\, G_F$ are
preferred (as for all the remaining Wilson coefficients in this list). Both operators give tree-level effects in $\mu\to 3 e$, e.g.,
\begin{equation}
\text{Br}\left[ {\mu \to 3e} \right] = \frac{{m_\mu ^5\tau_\mu}}{{768{\pi ^3}}}{\left| {C_{\ell \ell }^{1112}} \right|^2}=0.25\bigg|\frac{C_{\ell \ell }^{1112}}{G_F}\bigg|^2,
\end{equation}
which exceeds the experimental limit on the branching ratio of $1.0\times 10^{-12}$~\cite{Bellgardt:1987du} by many orders of magnitude (the result for $C_{\ell e }^{1112}$ is smaller by a factor $1/2$).
\vspace{-1mm}
\item The operators $Q_{\ell \ell (e) }^{2212}$ and $Q_{\ell \ell (e) }^{3312}$ contribute at the one-loop level to $\mu\to e$ conversion and $\mu\to 3e$ and at the two-loop level to $\mu\to e\gamma$~\cite{Crivellin:2017rmk}. Here the current best bounds come from $\mu\to e$ conversion. Using Table~3 in Ref.~\cite{Crivellin:2017rmk} we have
\begin{align}
\left| {C_{\ell \ell }^{3312}} \right| &<6.4\times 10^{-5} G_F,\notag\\
\left| {C_{\ell \ell }^{2212}} \right| &<2.8\times 10^{-5} G_F,
\end{align}
excluding again a sizable BSM effect, and similarly for $Q_{\ell e }^{3312}$ and $Q_{\ell e }^{2212}$.
\vspace{-1mm}
\item $Q_{\ell \ell (e) }^{2312}$, $Q_{\ell \ell (e) }^{3212}$, $Q_{\ell \ell (e) }^{1312}$, and $Q_{\ell \ell (e) }^{3112}$ contribute to $\tau\to \mu \mu e$ and $\tau\to \mu ee$, respectively,
which excludes a sizable effect in analogy to $\mu\to 3e$
above~\cite{Hayasaka:2010np,Lees:2010ez,Amhis:2019ckw}.
\vspace{-1mm}
\end{enumerate} }
Other four-quark operators can only contribute via loop effects, which leads us to conclude that the only viable mechanism proceeds via a modification of the SM operator.
\subsection{Four-fermion operators in $\boldsymbol{d\to u e\nu}$}
First of all, the operators $Q_{\ell equ}^{\left( 1 \right)1111}$ and $Q_{\ell equ}^{\left( 3 \right)1111}$ give rise to $d\to u e\nu$ scalar amplitudes. Such amplitudes lead to enhanced effects in $\pi\to\mu\nu/\pi\to e\nu$ with respect to $\beta$ decays and therefore can only have a negligible impact on the latter once the stringent experimental bounds~\cite{Aguilar-Arevalo:2015cdf,Zyla:2020zbs} are taken into account. Furthermore, the tensor amplitude generated by $Q_{\ell equ}^{\left( 3 \right)ijkl}$ has a suppressed matrix element in $\beta$ decays.
Therefore, we are left with $Q_{\ell q}^{\left( 3 \right)1111}$, for which we only consider the flavor combination that leads to interference with the SM. The CAA prefers
$C_{\ell q}^{\left( 3 \right)1111}\approx 0.7\times 10^{-3}G_F$.
Via $SU(2)_L$ invariance, this operator generates effects in neutral-current (NC) interactions
\begin{equation}
{{\cal L}_{{\rm{NC}}}} = C_{\ell q}^{\left( 3 \right)1111}\left( {\bar d{\gamma ^\mu }{P_L}d - \bar u{\gamma ^\mu }{P_L}u} \right)\bar e{\gamma _\mu }{P_L}e.
\end{equation}
Note that since the SM amplitude for $\bar uu (\bar dd)\to e^+e^-$, at high energies, has negative (positive) sign, we have constructive interference in both amplitudes. Therefore, the latest nonresonant dilepton searches by ATLAS~\cite{Aad:2020otl} lead to
\begin{equation}
C_{\ell q}^{\left( 3 \right)1111} \lesssim 0.8\times10^{-3}G_F.
\end{equation}
Hence, four-fermion operators affecting $d\to u e\nu$ transitions can bring $G_F^\text{CKM}$ into $1\sigma$ agreement with $G_F^\mu$, but are at the border of the LHC constraints.
\subsection{Modified $\boldsymbol{W}$--$\boldsymbol{u}$--$\boldsymbol{d}$ couplings}
There are only two operators modifying the $W$ couplings to quarks
\begin{align}
Q_{\phi q}^{\left( 3 \right)ij} &= {\phi ^\dag }i
\overset{\leftrightarrow}{D}^I_\mu
\phi {{\bar q}_i}{\gamma ^\mu }{\tau ^I}{q_j},\notag\\
Q_{\phi ud}^{ij} &= {\phi ^\dag }i\overset{\leftrightarrow}{D}_\mu \phi {{\bar u}_i}{\gamma ^\mu }{d_j}.
\end{align}
First of all, $Q_{\phi ud}^{ij}$ generates right-handed $W$--quark couplings, which can only slightly alleviate the CAA, but not solve it~\cite{Grossman:2019bzp}. $Q_{\phi q}^{\left( 3 \right)ij}$ generates modifications of the left-handed $W$--quark couplings and data prefer
\begin{equation}
C_{\phi q}^{\left( 3 \right)11} \approx 0.7\times 10^{-3}G_F.
\end{equation}
Due to $SU(2)_L$ invariance, in general effects in $D^0$--$\bar D^0$ and $K^0$--$\bar K^0$ mixing are generated. However, in case of alignment with the down-sector, the effect in $D^0$--$\bar D^0$ is smaller than the experimental value and thus not constraining as the SM prediction cannot be reliably calculated. Furthermore, the effects in $D^0$--$\bar D^0$ and $K^0$--$\bar K^0$ mixing could be suppressed by assuming that $C_{\phi q}^{\left( 3 \right)ij}$ respects a global $U(2)^2$ symmetry.
\subsection{Modified $\boldsymbol{W}$--$\boldsymbol{\ell}$--$\boldsymbol{\nu}$ couplings}
Only the operator
\begin{align}
Q_{\phi \ell }^{(3)ij}={\phi ^\dag }i
\overset{\leftrightarrow}{D}^I_\mu
\phi {{\bar \ell}_i}{\gamma ^\mu }{\tau ^I}{\ell_j}
\end{align}
generates modified $W$--$\ell$--$\nu$ couplings at tree level. In order to avoid the stringent bounds from charged lepton flavor violation, the off-diagonal Wilson coefficients, in particular $C_{\phi \ell }^{(3)12}$, must be very small. Since they also do not generate amplitudes interfering with the SM ones, their effect can be neglected. While $C_{\phi \ell }^{(3)11}$ affects $G_F^\mu$ and $G_F^\text{CKM}$ in the same way, $C_{\phi \ell }^{(3)22}$ only enters in muon decay. Therefore, agreement between $G_F^\mu$ and $G_F^\text{CKM}$ can be achieved by $C_{\phi \ell }^{(3)11}<0$, $C_{\phi \ell }^{(3)22}>0$, and $|C_{\phi \ell }^{(3)22}|<|C_{\phi \ell }^{(3)11}|$ without violating lepton flavor universality tests such as $\pi(K)\to\mu\nu/\pi(K)\to e\nu$ or $\tau\to\mu\nu\nu/\tau(\mu)\to e\nu\nu$~\cite{Coutinho:2019aiy,Crivellin:2020lzu,Pich:2013lsa}. However, $C_{\phi \ell }^{(3)ij}$ also affects $Z$ couplings to leptons and neutrinos, which enter the global EW fit.
\subsection{Electroweak fit}
The impact of modified gauge-boson--lepton couplings on the global EW fit, generated by $Q_{\phi \ell}^{\left( 3 \right)ij}$ and
\begin{align}
Q_{\phi \ell}^{\left( 1 \right)ij} &= {\phi ^\dag }i\overset{\leftrightarrow}{D}_\mu \phi {{\bar \ell}_i}{\gamma ^\mu }{\ell_j},
\end{align}
can be minimized by only affecting $Z\nu\nu$ but not $Z\ell\ell$, by imposing $C_{\phi \ell }^{(1)ij}=-C_{\phi \ell }^{(3)ij}$. In this way, in addition to the Fermi constant, only the $Z$ width to neutrino changes and the fit improves significantly compared to the SM~\cite{Coutinho:2019aiy}, see Fig.~\ref{FinalPlot} for the preferred parameter space. One can even further improve the fit by assuming $C_{\phi \ell }^{(1)11}=-C_{\phi \ell }^{(3)11}$, $C_{\phi \ell }^{(1)22}=-3C_{\phi \ell }^{(3)22}$, which leads to a better description of $Z\to\mu\mu$ data. Furthermore, the part of the tension between $G_F^\text{EW}$ and $G_F^{\mu}$ driven by the $W$ mass can be alleviated by the operator $Q_{\phi W B}=\phi^\dagger \tau^I\phi W^I_{\mu\nu}B^{\mu\nu}$.
\begin{figure}[t]
\includegraphics[width=0.9\linewidth]{FinalPlot.pdf}
\caption{Example of the complementarity between the $G_F$ determinations from muon decay ($G_F^\mu$), CKM unitarity ($G_F^\text{CKM}$), and the global EW fit ($G_F^\text{EM}$) in case of $C_{\phi \ell }^{(3)ii}=-C_{\phi \ell }^{(1)ii}$, corresponding to modifications of neutrino couplings to gauge bosons (the EW fit also includes $\tau\to\mu\nu\nu/\tau(\mu)\to e\nu\nu$~\cite{Amhis:2019ckw,Zyla:2020zbs,Pich:2013lsa}). Here, we show the preferred $1\sigma$ regions obtained by requiring that two or all three $G_F$ determinations agree. The contour lines show the value of the Fermi constant extracted from muon decay once BSM effects are taking into account.} \label{FinalPlot}
\end{figure}
\section{Conclusions and outlook}
Even though the Fermi constant is determined extremely precisely by the muon lifetime, Eq.~\eqref{Mulan},
its constraining power on BSM effects is limited by the precision of the second-best determination. In this Letter we derived in a first step two alternative independent determinations, from the EW fit, Eq.~\eqref{GFEW}, and superallowed $\beta$ decays using CKM unitarity, Eq.~\eqref{GFCKM}. The latter determination
is more precise than the one from the EW fit, even though the precision of $G_F^\text{EW}$ increased by a factor 5 compared to Ref.~\cite{Marciano:1999ih}. Furthermore, as shown in Fig.~\ref{GFplot}, both determinations display a tension of $2\sigma$ and $3\sigma$ compared to $G_F^\mu$, respectively.
In a second step, we investigated how these hints of BSM physics can be explained within the SMEFT framework. For BSM physics in $G_F^\mu$ we were able to rule out all four-fermion operators, except for $Q_{\ell\ell}^{2112}$, which generates a SM-like amplitude, and modified $W$--$\ell$--$\nu$ couplings, from $Q_{\phi \ell }^{(3)ij}$. Therefore, both constructive and destructive interference is possible, which would bring $G_F^\mu$ into agreement with $G_F^\text{CKM}$ or $G_F^\text{EW}$, respectively, at the expense of increasing the tension with the other determination. To achieve a better agreement among the three different values of $G_F$, also BSM effects in $G_F^\text{CKM}$ and/or $G_F^\text{EW}$ are necessary. In the case of $G_F^\text{CKM}$, only a single four-fermion operator, $Q_{\ell q}^{\left( 3 \right)1111}$, and $Q_{\phi \ell }^{(3)ij}$ remain. Finally, modified gauge-boson--lepton couplings, via $Q_{\phi \ell}^{\left( 3 \right)ij}$ and $Q_{\phi \ell}^{\left( 1 \right)ij}$, can not only change $G_F^\text{CKM}$ and $G_F^{\mu}$, but also affect the EW fit via the $Z$-pole observables, which can further improve the global agreement with data, see Fig.~\ref{FinalPlot}. This figure also demonstrates the advantage of interpreting the tensions in terms of $G_F$, defining a transparent benchmark for comparison both in SMEFT and concrete BSM scenarios, and allows one to constrain the amount of BSM contributions to muon decay.
Our study highlights the importance of improving the precision of the alternative independent determinations of $G_F^\text{CKM}$ and $G_F^\text{EW}$ in order to confirm or refute BSM contributions to the Fermi constant. Concerning $G_F^\text{CKM}$, improvements in the determination of $|V_{ud}|$ should arise from advances in nuclear-structure~\cite{Cirgiliano:2019nyn,Martin:2021bud} and EW radiative corrections~\cite{Feng:2020zdc}, while experimental developments~\cite{Fry:2018kvq,Soldner:2018ycf,Wang:2019pts,Serebrov:2019puy,Gaisbauer:2016kan,Ezhov:2018cta,Callahan:2018iud} could make the determination from neutron decay~\cite{Pattie:2017vsj,Markisch:2018ndu,Czarnecki:2018okw} competitive and, in combination with $K_{\ell 3}$ decays, add another complementary
constraint on $|V_{ud}|/|V_{us}|$ via pion $\beta$ decay~\cite{Pocanic:2003pf,Czarnecki:2019iwz}.
Further, improved measurements of
$|V_{cd}|$ from $D$ decays~\cite{Ablikim:2019hff} could bring the precision of the first-column CKM unitarity relation close to the first-row one, which, in turn, could be corroborated via improved
$|V_{us}|$ determinations from $K_{\ell3}$ decays~\cite{Yushchenko:2017fzv,Junior:2018odx,Babusci:2019gqu}.
The precision of $G_F^\text{EW}$ will profit in the near future from LHC measurements of $m_t$ and $M_W$, in the mid-term future from the Belle-II EW precision program~\cite{Kou:2018nap}, and in the long-term from future $e^+e^-$ colliders such as the FCC-ee~\cite{Abada:2019zxq}, ILC~\cite{Baer:2013cma}, CEPC~\cite{An:2018dwb}, or CLIC~\cite{Aicheler:2012bya}, which could achieve a precision at the level of $10^{-5}$.
\begin{acknowledgments}
We thank David Hertzog and Klaus Kirch for valuable discussions,
and the ATLAS collaboration, in particular Noam Tal Hod, for clarifications concerning the analysis of Ref.~\cite{Aad:2020otl}.
Support by the Swiss National Science Foundation, under Project Nos.\ PP00P21\_76884 (A.C., C.A.M) and PCEFP2\_181117 (M.H.) is gratefully acknowledged.
\end{acknowledgments}
| 11,013 |
\section{Introduction}
Modern Penning-trap experiments based on the continuous Stern-Gerlach effect provide very precise
measurements of the Zeeman splitting of energy levels in one- and few-electron ions
\cite{sturm:11,koehler:16,arapoglou:19}. The linear Zeeman splitting is usually parameterized in
terms of the $g$ factor of the atomic system. The fractional accuracy of the recent measurements
of the $g$ factors of H-like and Li-like ions has reached few parts in $10^{-11}$
\cite{sturm:14,glazov:19}. Combined with dedicated theoretical calculations, these measurements
provided the determination of the electron mass \cite{mohr:16:codata} and one of the best tests
of the bound-state quantum electrodynamics (QED) \cite{sturm:13:Si}. Extension of these tests
towards heavier ions are anticipated in the future \cite{vogel:19}, which might open new ways for
determination of the fine-structure constant $\alpha$ \cite{shabaev:06:prl,yerokhin:16:gfact:prl}
and searches for physics beyond the Standard Model \cite{debierre:20}.
In view of the very high accuracy of the measurements, theoretical investigations of atomic $g$
factors often need to be carried out without any expansion in the nuclear binding strength
parameter $\Za$ (where $Z$ is the nuclear charge number). In such calculations, the
electron-electron interaction has to be treated by perturbation theory. The starting point of the
perturbation expansion is the hydrogenic approximation, i.e., the approximation of
non-interacting electrons. The electron-correlation corrections come from the exchange of virtual
photons between the electrons. An exchange by each photon leads to the suppression of the
corresponding correction by a parameter of $1/Z$. The first-order perturbation correction
$\sim\!1/Z^1$ is due to the one-photon exchange. This correction is relatively simple and was
calculated for Li-like ions first in Ref.~\cite{shabaev:02:li} and later reproduced in
Refs.~\cite{yerokhin:16:gfact:pra,cakir:20}.
The QED calculation of the two-photon exchange correction $\sim\!1/Z^2$ is a difficult task.
First calculations of this correction were accomplished in Refs.~\cite{volotka:12,volotka:14}. In
these studies, results were reported for just four ions and their numerical uncertainty was
significant on the level of the current experimental precision. In the present work we will
perform an independent calculation of the two-photon exchange correction for the ground state of
Li-like ions. Our goals will be to cross-check the previous calculations, to improve the
numerical accuracy, and to study the $Z$-dependence of the two-photon correction in the low-$Z$
region, checking the consistency of the applied method with the $Z\alpha$-expansion calculations
performed recently in Ref.~\cite{yerokhin:17:gfact}.
The relativistic units ($\hbar=c=m=1$) and the Heaviside charge units ($ \alpha = e^2/4\pi$,
$e<0$) will be used throughout this paper.
\section{Electronic structure corrections to the $\boldsymbol{g}$ factor}
In the present work we assume the nucleus to be spinless and the electron configuration to be a
valence electron $v$ beyond a closed shell of core electrons denoted by $c$. Contributions to the
$g$ factor can be formally obtained as corrections induced by the effective magnetic interaction
\cite{yerokhin:10:sehfs}
\begin{align}
V_g(\bfr) = \frac{1}{\mu_v}\, (\bfr \times\balpha)_z\,,
\end{align}
where $\balpha$ is the vector of Dirac matrices and $\mu_v$ is the angular momentum projection of
the valence electron.
To the zeroth order in the electron-electron interaction, the $g$ factor of the ground state of a
Li-like ion is given by the expectation value of the magnetic potential $V_g$ on the hydrogenic
Dirac wave function of the valence $2s$ state. For the point nucleus, the result is known in the
closed form
\begin{align}
g_{\rm Dirac} = \lbr 2s | V_g| 2s \rbr = \frac23 \bigg( \Big[2\sqrt{1-(\Za)^2}+2\Big]^{1/2}+1\bigg)\,.
\end{align}
Corrections to the $g$ factor of a Li-like ion due to the presence of core electrons are
evaluated by perturbation theory in the electron-electron interaction, with the expansion
parameter $1/Z$. The leading correction of order $1/Z^1$ is induced by the one-photon exchange
between the valence and core electrons. The corresponding correction was calculated in
Ref.~\cite{shabaev:02:li} (see also Ref.~\cite{yerokhin:16:gfact:pra}) and can be written as
\begin{align}\label{eq0:4}
\Delta g_{\rm 1ph} = &\ 2 \sum_{\mu_{c}} \Big[ \Lambda_{\rm 1ph}(vcvc)+\Lambda_{\rm 1ph}(cvcv)
\nonumber \\ &
-\Lambda_{\rm 1ph}(cvvc)-\Lambda_{\rm 1ph}(vccv) \Big]
\,,
\end{align}
where the summation runs over the angular-momentum projection of the core electron $\mu_c$ and
\begin{align}
\Lambda_{\rm 1ph}(abcd) = &\ \sum_{n\ne a} \frac{\lbr a | V_g | n\rbr \lbr nb| I(\Delta_{db}) | cd\rbr}{\vare_a-\vare_n}
\nonumber \\ &
+ \frac14\, \lbr ab| I'(\Delta_{db})|cd\rbr\,\Big( \lbr d|V_g|d\rbr - \lbr b|V_g|b\rbr\Big)
\,.
\end{align}
Here, $\Delta_{ab} = \vare_a-\vare_b$, $I(\omega)$ is the operator of the electron-electron
interaction, and $I'(\omega) =
\partial I(\omega)/(\partial \omega)$.
The electron-electron interaction operator $I(\omega)$ is defined as
\begin{equation}\label{a1}
I(\omega,\bfr_{1},\bfr_{2}) = e^2\, \alpha_{1}^{\mu} \alpha_{2}^{\nu}\, D_{\mu\nu}(\omega,\bfr_{1},\bfr_2)\,,
\end{equation}
where $\alpha^{\mu}_{a} = (1,\balpha_{a})$ is the four-vector of Dirac matrices acting on
$\bfr_a$, $D_{\mu\nu}$ is the photon propagator, and $\omega$ is the photon energy. In the
present work we use the photon propagator in the Feynman and Coulomb gauges. In the Feynman
gauge, the electron-electron interaction takes the simplest form,
\begin{equation}
I_{\rm Feyn}(\omega) = \alpha\,\big(1 - \balpha_1\cdot\balpha_2\big)\,
\frac{e^{i|\omega|r_{12}}}{r_{12}}\,,
\end{equation}
where $r_{12} = |\bfr_{12}| = |\bfr_{1} - \bfr_{2}|$, and $|\omega|$ should be understood as
$|\omega| = \sqrt{\omega^2+i\epsilon}$, where $\epsilon$ is a positive infinitesimal addition.
The electron-electron interaction operator in the Coulomb gauge reads
\begin{align}\label{eq:I:coul}
I_{\rm Coul}(\omega) = &\ \alpha\Biggl[
\frac1{r_{12}} - \balpha_1\cdot\balpha_2\, \frac{e^{i|\omega|r_{12}}}{r_{12}}
\nonumber \\ &
+ \frac{\left( \balpha_1\cdot\bnabla_1\right)\left(\balpha_2\cdot\bnabla_2\right)}{\omega^2}
\frac{e^{i|\omega|r_{12}}-1}{r_{12}} \Biggr]\,.
\end{align}
It can be easily seen that the one-photon exchange correction $\Delta g_{\rm 1ph}$ given by
Eq.~(\ref{eq0:4}) can be obtained from the corresponding correction to the Lamb shift,
\begin{align}
\Delta E_{\rm 1ph}(vc) = \sum_{\mu_c} \Big[
\lbr cv| I(0) | cv\rbr - \lbr vc| I(\Delta_{vc}) | cv\rbr\Big]\,,
\end{align}
by perturbing this expression with the magnetic potential $V_g$. Specifically, one perturbs the
one-electron wave functions,
\begin{align}\label{eq0:9}
|a\rbr \to |a\rbr + |\delta a\rbr\,, \ \ |\delta a\rbr = \sum_{n \neq a} \frac{|n\rbr \lbr n| V_g|a\rbr}{\vare_a - \vare_n}\,,
\end{align}
and energies,
\begin{align}
\vare_a \to \vare_a + \delta \vare_a\,, \ \ \delta\vare_a = \lbr a | V_g| a\rbr\,.
\end{align}
In this work, we use this approach in order to obtain formulas for the two-photon exchange
corrections to the $g$ factor. We start with the two-photon exchange correction for the Lamb
shift, graphically represented in Fig.~\ref{fig:Feyn:Lamb}. The Feynman diagrams for the $g$
factor in Fig.~\ref{fig:Feyn:gfact} are obtained from the Lamb-shift diagrams by inserting the
magnetic interaction $V_g$ in all possible ways. The corresponding formulas for the $g$ factor
are obtained by using formulas for the two-photon exchange correction for the Lamb shift derived
in Refs.~\cite{shabaev:94:ttg1,yerokhin:01:2ph} and perturbing them with the magnetic potential
$V_g$.
The two-photon exchange correction to the $g$ factor is conveniently represented as a sum of the
direct (``dir''), exchange (``ex''), and the three-electron (``3el'') contributions, obtained as
perturbations of the corresponding Lamb-shift corrections. Furthermore, each of the three
contributions is sub-divided into the irreducible (``ir'') and reducible (``red'') parts. The
reducible parts are induced by the intermediate states degenerate in energy with the energy of
the reference state of the ion. We thus represent the total two-photon exchange correction to the
$g$ factor as the sum of three irreducible and three reducible contributions,
\begin{align} \label{eq0:11}
\Delta g_{\rm 2ph} = &\ \Delta g_{\rm ir, dir} + \Delta g_{\rm ir, ex} + \Delta g_{\rm 3el, ir}
\nonumber \\ &
+ \Delta g_{\rm red, dir} + \Delta g_{\rm red, ex} + \Delta g_{\rm 3el, red}\,.
\end{align}
We now examine each of these terms one by one.
\begin{figure}
\centerline{
\resizebox{0.5\textwidth}{!}
\includegraphics{2phot1.eps}
}
}
\caption{Feynman diagrams representing the two-photon exchange correction to the Lamb shift.
The three graphs are referred to, from left to right, as the ladder, the crossed, and the
three-electron diagrams, respectively. The double line denotes the electron propagating in
the field of the nucleus, the wavy line denotes the virtual photon.
\label{fig:Feyn:Lamb}}
\end{figure}
\begin{figure}
\centerline{
\resizebox{0.245\textwidth}{!}
\includegraphics{gfact-2phot-ladder.eps}
}
\hspace*{0.05cm}
\resizebox{0.245\textwidth}{!}
\includegraphics{gfact-2phot-crossed.eps}
}
}\vspace*{0.1cm}
\centerline{
\resizebox{0.5\textwidth}{!}
\includegraphics{gfact-2phot-3body.eps}
}
}
\caption{Feynman diagrams representing the two-photon exchange corrections to the $g$ factor. The
wavy line terminated by a cross denotes the magnetic interaction.
\label{fig:Feyn:gfact}}
\end{figure}
\subsection{Direct irreducible part}
The direct irreducible contribution comes from the ladder (``lad'') and crossed (``cr'')
diagrams. For the Lamb shift, this contribution is given by Eq.~(32) of
Ref.~\cite{yerokhin:01:2ph}. Changing the variable $\omega \to -\omega$ in the ladder part and
using the property $I(\omega) = I(-\omega)$, we write the expression as
\begin{align}\label{eq:1}
\Delta E_{\rm ir, dir} &\ =
\frac{i}{2\pi} \int_{-\infty}^{\infty} d\omega\,
\Bigg[ \sum_{n_1n_2 \ne cv,vc}\!
\frac{F_{\rm lad, dir}(\omega, n_1n_2)}
{(\tilde{\Delta}_{cn_1}+\omega)(\tilde{\Delta}_{vn_2}-\omega)}
\nonumber \\ & \
+ \sum_{n_1n_2 \ne cv}
\frac{F_{\rm cr, dir}(\omega, n_1n_2)}
{(\tilde{\Delta}_{cn_1}-\omega)(\tilde{\Delta}_{vn_2}-\omega)}
\Bigg]\,,
\end{align}
where $\tilde{\Delta}_{an} \equiv \vare_a - \vare_n(1-i0)$. Furthermore,
\begin{align}
F_{\rm lad, dir}(\omega, n_1n_2) &\ =
\sum_{\mu_c \mu_1\mu_2}
\lbr cv| I(\omega) |n_1n_2\rbr\, \lbr n_1n_2| I(\omega)| cv\rbr
\,, \\
F_{\rm cr, dir}(\omega, n_1n_2) &\ =
\sum_{\mu_c \mu_1\mu_2}
\lbr cn_2| I(\omega) |n_1v\rbr\, \lbr n_1v| I(\omega)| cn_2\rbr
\,,
\end{align}
where $\mu_1$ and $\mu_2$ denote the angular-momentum projections of the states $n_1$ and $n_2$,
respectively. The summations over $n$'s run over the complete spectrum of the Dirac equation,
implying the sum over the corresponding relativistic angular quantum numbers $\kappa_n$ and the
principal quantum numbers of the discrete spectrum and the integration over the continuum part of
the spectrum. The terms excluded from the summation over $n_{1}$ and $n_2$ in Eq.~(\ref{eq:1})
will be accounted for by the reducible part.
Formulas for the $g$ factor are obtained by perturbing the above expressions with the magnetic
potential $V_g$. One perturbs the initial-state and intermediate-state wave functions and
energies in the denominators. Perturbations of wave functions lead to the corrections $\delta
F$'s,
\begin{align} \label{eq:6}
\delta F_{\rm lad, dir}(\omega, n_1n_2) = &\
2\, \sum_{\mu_c \mu_1\mu_2}
\Big[
\lbr cv| I(\omega) |n_1n_2\rbr\, \lbr n_1n_2| I(\omega)| \delta cv\rbr
\nonumber \\
&
+\lbr cv| I(\omega) |n_1n_2\rbr\, \lbr n_1n_2| I(\omega)| c \delta v\rbr
\nonumber \\
&
+\lbr cv| I(\omega) |n_1n_2\rbr\, \lbr \delta n_1n_2| I(\omega)| c v\rbr
\nonumber \\
&
+\lbr cv| I(\omega) |n_1n_2\rbr\, \lbr n_1 \delta n_2| I(\omega)| c v\rbr
\Big]\,,
\end{align}
\begin{align}
\delta F_{\rm cr, dir}(\omega, n_1n_2) = &\
2\,\sum_{\mu_c \mu_1\mu_2}
\Big[
\lbr cn_2| I(\omega) |n_1v\rbr\, \lbr n_1v| I(\omega)| \delta cn_2\rbr
\nonumber \\
&
+ \lbr cn_2| I(\omega) |n_1 \delta v\rbr\, \lbr n_1v| I(\omega)| cn_2\rbr
\nonumber \\
&
+ \lbr cn_2| I(\omega) |n_1 v\rbr\, \lbr \delta n_1v| I(\omega)| cn_2\rbr
\nonumber \\
&
+ \lbr c \delta n_2| I(\omega) |n_1 v\rbr\, \lbr n_1v| I(\omega)| cn_2\rbr
\Big]\,,
\end{align}
with perturbed wave functions $|\delta a\rbr$ defined by Eq.~(\ref{eq0:9}). Perturbations of
energies in the denominators leads to corrections $\delta_1F$ and $\delta_2F$,
\begin{widetext}
\begin{align}
\delta_1 F_{\rm lad, dir}(\omega, n_1n_2) = &\
\sum_{\mu_c \mu_1\mu_2}
\big(V_{n_1n_1}-V_{cc}\big)\,
\lbr cv| I(\omega) |n_1n_2\rbr\, \lbr n_1n_2| I(\omega)| cv\rbr
\,, \\
\delta_2 F_{\rm lad, dir}(\omega, n_1n_2) = &\
\sum_{\mu_c \mu_1\mu_2}
\big(V_{n_2n_2}-V_{vv}\big)\,
\lbr cv| I(\omega) |n_1n_2\rbr\, \lbr n_1n_2| I(\omega)| cv\rbr
\,, \\
\delta_1 F_{\rm cr, dir}(\omega, n_1n_2) = &\
\sum_{\mu_c \mu_1\mu_2}
\big(V_{n_1n_1}-V_{cc}\big)
\lbr cn_2| I(\omega) |n_1v\rbr\, \lbr n_1v| I(\omega)| cn_2\rbr
\,, \\
\delta_2 F_{\rm cr, dir}(\omega, n_1n_2) = &\
\sum_{\mu_c \mu_1\mu_2}
\big(V_{n_2n_2}-V_{vv}\big)
\lbr cn_2| I(\omega) |n_1v\rbr\, \lbr n_1v| I(\omega)| cn_2\rbr
\,,
\end{align}
where $V_{ab} = \lbr a|V_g|b\rbr$. Finally, the correction to the $g$ factor is
\begin{align}\label{eq:4}
\Delta g_{\rm ir, dir} = &\
\frac{i}{2\pi} \int_{-\infty}^{\infty} d\omega\,
\Bigg\{
\sum_{n_1n_2 \ne cv,vc}
\bigg[
\frac{\delta F_{\rm lad, dir}(\omega,n_1n_2)}
{(\tilde{\Delta}_{cn_1}+\omega)(\tilde{\Delta}_{vn_2}-\omega)}
+ \frac{\delta_1 F_{\rm lad, dir}(\omega,n_1n_2)}
{(\tilde{\Delta}_{cn_1}+\omega)^2(\tilde{\Delta}_{vn_2}-\omega)}
+ \frac{\delta_2 F_{\rm lad, dir}(\omega,n_1n_2)}
{(\tilde{\Delta}_{cn_1}+\omega)(\tilde{\Delta}_{vn_2}-\omega)^2}
\bigg]
\nonumber \\ &
+ \sum_{n_1n_2 \ne cv}
\bigg[
\frac{\delta F_{\rm cr, dir}(\omega,n_1n_2)}
{(\tilde{\Delta}_{cn_1}-\omega)(\tilde{\Delta}_{vn_2}-\omega)}
+ \frac{\delta_1 F_{\rm cr, dir}(\omega,n_1n_2)}
{(\tilde{\Delta}_{cn_1}-\omega)^2(\tilde{\Delta}_{vn_2}-\omega)}
+ \frac{\delta_2 F_{\rm cr, dir}(\omega,n_1n_2)}
{(\tilde{\Delta}_{cn_1}-\omega)(\tilde{\Delta}_{vn_2}-\omega)^2}
\bigg]
\Bigg\}\,.
\end{align}
\subsection{Exchange irreducible part}
The exchange irreducible contribution for the Lamb shift is given by (see Eq.~(32) of
Ref.~\cite{yerokhin:01:2ph})
\begin{align}\label{eq:11}
\Delta E_{\rm ir, ex} = &\
\frac{i}{2\pi} \int_{-\infty}^{\infty} d\omega\,
\bigg[ \sum_{n_1n_2 \ne cv,vc}
\frac{F_{\rm lad, ex}(\omega,n_1n_2)}
{(\tilde{\Delta}_{vn_1}-\omega)(\tilde{\Delta}_{cn_2}+\omega)}
+
\sum_{n_1n_2 \ne cc,vv}
\frac{F_{\rm cr, ex}(\omega,n_1n_2)}
{(\tilde{\Delta}_{vn_1}-\omega)(\tilde{\Delta}_{vn_2}-\omega)}
\bigg]
\,,
\end{align}
where the functions $F$ are given by
\begin{align}
F_{\rm lad, ex}(\omega,n_1n_2) &\ =
(-1)\sum_{\mu_c \mu_{1}\mu_{2}}
\lbr vc| I(\omega) |n_1n_2\rbr\, \lbr n_1n_2| I(\widetilde{\omega})| cv\rbr
\,,
\\
F_{\rm cr, ex}(\omega,n_1n_2) &\ =
(-1)\sum_{\mu_c \mu_{1}\mu_{2}}
\lbr vn_2| I(\omega) |n_1v\rbr\, \lbr n_1c| I(\widetilde{\omega})| cn_2\rbr
\,,
\end{align}
with $\widetilde{\omega} \equiv \omega - \Delta$ and $\Delta \equiv \vare_v -\vare_c$. Again, the
terms excluded from the summation over $n_1$ and $n_2$ will be accounted for by the corresponding
reducible part. It should be mentioned that our present definition of the irreducible part
differs slightly from that of Ref.~\cite{yerokhin:01:2ph}. Specifically, we here do {\em not}
exclude from the summation over $n_1$ and $n_2$ the state separated by the finite nuclear size
effect from the reference state ({\em i.e.}, the $2p_{1/2}$ state for $v = 2s$ state), since it
leads to unnecessary complications in the case of the $g$ factor.
The formulas for the $g$ factor are obtained similarly to the direct contribution, by perturbing
expressions for the Lamb shift with the potential $V_g$,
\begin{align}\label{eq:14}
\Delta g_{\rm ir, ex} = &\
\frac{i}{2\pi} \int_{-\infty}^{\infty} d\omega\,
\Bigg\{
\sum_{n_1n_2 \ne cv,vc}
\bigg[
\frac{\delta F_{\rm lad, ex}(\omega,n_1n_2)}
{(\tilde{\Delta}_{vn_1}-\omega)(\tilde{\Delta}_{cn_2}+\omega)}
+ \frac{\delta_1 F_{\rm lad, ex}(\omega,n_1n_2)}
{(\tilde{\Delta}_{vn_1}-\omega)^2(\tilde{\Delta}_{cn_2}+\omega)}
+ \frac{\delta_2 F_{\rm lad, ex}(\omega,n_1n_2)}
{(\tilde{\Delta}_{vn_1}-\omega)(\tilde{\Delta}_{cn_2}+\omega)^2}
\bigg]
\nonumber \\ &
+ \sum_{n_1n_2 \ne cc,vv}
\bigg[
\frac{\delta F_{\rm cr, ex}(\omega,n_1n_2)}
{(\tilde{\Delta}_{vn_1}-\omega)(\tilde{\Delta}_{vn_2}-\omega)}
+ \frac{\delta_1 F_{\rm cr, ex}(\omega,n_1n_2)}
{(\tilde{\Delta}_{vn_1}-\omega)^2(\tilde{\Delta}_{vn_2}-\omega)}
+ \frac{\delta_2 F_{\rm cr, ex}(\omega,n_1n_2)}
{(\tilde{\Delta}_{vn_1}-\omega)(\tilde{\Delta}_{vn_2}-\omega)^2}
\bigg]
\Bigg\}\,.
\end{align}
The perturbations of the $F$ functions by the magnetic potential $V_g$ are defined as follows
\begin{align}
\delta F_{\rm lad, ex}(\omega,n_1n_2) = &\
(-2)\sum_{\mu_c \mu_1 \mu_2}
\Big[
\lbr vc| I(\omega) |n_1n_2\rbr\, \lbr n_1n_2| I(\tilde{\omega})| \delta cv\rbr
+ \lbr vc| I(\omega) |n_1n_2\rbr\, \lbr n_1n_2| I(\tilde{\omega})| c \delta v\rbr
\nonumber \\ &
+ \lbr vc| I(\omega) |n_1n_2\rbr\, \lbr \delta n_1n_2| I(\tilde{\omega})| c v\rbr
+ \lbr vc| I(\omega) |n_1n_2\rbr\, \lbr n_1 \delta n_2| I(\tilde{\omega})| c v\rbr
\nonumber \\ &
- \frac12 \big(V_{vv}-V_{cc}\big)\,
\lbr vc| I(\omega) |n_1n_2\rbr\, \lbr n_1n_2| I'(\tilde{\omega})| cv\rbr
\Big]\,,
\\
\delta F_{\rm cr, ex}(\omega,n_1n_2) = &\
(-2)\sum_{\mu_c \mu_1 \mu_2}
\Big[
\lbr vn_2| I(\omega) |n_1v\rbr\, \lbr n_1c| I(\tilde{\omega})| \delta cn_2\rbr
+ \lbr vn_2| I(\omega) |n_1\delta v\rbr\, \lbr n_1c| I(\tilde{\omega})| cn_2\rbr
\nonumber \\ &
+ \lbr vn_2| I(\omega) |n_1v\rbr\, \lbr \delta n_1c| I(\tilde{\omega})| cn_2\rbr
+ \lbr v\delta n_2| I(\omega) |n_1v\rbr\, \lbr n_1c| I(\tilde{\omega})| cn_2\rbr
\nonumber \\ &
- \frac12 \big(V_{vv}-V_{cc}\big)\,
\lbr vn_2| I(\omega) |n_1v\rbr\, \lbr n_1c| I'(\tilde{\omega})| cn_2\rbr
\Big]\,,
\end{align}
and
\begin{align}
\delta_1 F_{\rm lad, ex}(\omega,n_1n_2) = &\
\sum_{\mu_c \mu_1 \mu_2}
\big(V_{vv} -V_{n_1n_1}\big)\,
\lbr vc| I(\omega) |n_1n_2\rbr\, \lbr n_1n_2| I(\tilde{\omega})| cv\rbr\,,
\\
\delta_2 F_{\rm lad, ex}(\omega,n_1n_2) = &\
\sum_{\mu_c \mu_1 \mu_2}
\big(V_{cc} - V_{n_2n_2}\big)\,
\lbr vc| I(\omega) |n_1n_2\rbr\, \lbr n_1n_2| I(\tilde{\omega})| cv\rbr\,,
\\
\delta_1 F_{\rm cr, ex}(\omega,n_1n_2) = &\
\sum_{\mu_c \mu_1 \mu_2}
\big(V_{vv} - V_{n_1n_1}\big)\,
\lbr vn_2| I(\omega) |n_1v\rbr\, \lbr n_1c| I(\tilde{\omega})| cn_2\rbr \,,
\\
\delta_2 F_{\rm cr, ex}(\omega,n_1n_2) = &\
\sum_{\mu_c \mu_1 \mu_2}
\big(V_{vv} - V_{n_2n_2}\big)\,
\lbr vn_2| I(\omega) |n_1v\rbr\, \lbr n_1c| I(\tilde{\omega})| cn_2\rbr\,.
\end{align}
\subsection{Direct reducible part}
The reducible part of the two-electron diagrams for the Lamb shift is given by Eq.~(41) of
Ref.~\cite{yerokhin:01:2ph}. Separating the direct contribution, we write it as
\begin{align}\label{eq2:31}
\Delta E_{\rm red, dir} &\ =
\frac{-i}{4\pi}\int_{-\infty}^{\infty}
d\omega\,
\bigg\{
\bigg[ \frac{1}{(\omega+i0)^2}+\frac{1}{(\omega-i0)^2}\bigg]\,F_{\rm lad, dir}(\omega,cv)
\nonumber \\ &
+
\bigg[ \frac{1}{(\omega+\Delta+i0)^2}+\frac{1}{(\omega+\Delta-i0)^2}\bigg]\,
F_{\rm lad, dir}(\omega,vc)\,
-
\frac{2}{(\omega-i0)^2}\, F_{\rm cr, dir}(\omega,cv)\,
\bigg\}\,.
\end{align}
\end{widetext}
We note that the crossed $(cv)$ term in the above expression exactly coincides with the one
excluded from the summation in Eq.~(\ref{eq:1}). The ladder $(cv)$ and $(vc)$ terms in the above
expression are very similar to those excluded from the summation in Eq.~(\ref{eq:1}) but differ
by signs of $i0$. Specifically, the terms excluded from the summation over $n_{1,2}$ in
Eq.~(\ref{eq:1}) contained poles both at $\omega = i0$ and $\omega = -i0$ (or at $\omega =
-\Delta + i0$ and $\omega = -\Delta - i0$), thus ``squeezing'' the integration contour between
the two poles, causing singularities. By contrast, the ladder terms in Eq.~(\ref{eq2:31}) have
double poles, from one side of the integration contour. Therefore, the integration contour can be
``moved away'' from the pole (assuming a finite photon mass in the case of $\omega = 0$), so
there is no real singularities in Eq.~(\ref{eq2:31}).
Taking into account that the ladder $(cv)$ term and the crossed $(cv)$ term cancel each other (as
proven in Ref.~\cite{shabaev:94:ttg1}), the expression in simplified further to yield
\begin{align}\label{eq2:33}
\Delta E_{\rm red, dir} =
-\frac{i}{2\pi}{\cal P}\intinf d\omega\,
\frac{1}{\omega+\Delta}\, F_{\rm lad, dir}^{\prime}(\omega,vc)
\,,
\end{align}
where ${\cal P}$ denotes the principal value of the integral and $F^{\prime}(\omega) =
\partial F(\omega)/(\partial\omega)$.
Corrections to the $g$ factor arise through perturbations of the Lamb-shift formulas with the
magnetic potential $V_g$. We divide them into two parts,
\begin{align}
\Delta g_{\rm red, dir} = \Delta g_{\rm red, dir, wf} + \Delta g_{\rm red, dir, en}\,,
\end{align}
where the first term is induced by perturbations of the wave functions and the second, by
perturbations of the energies. The perturbed-wave-function part is immediately obtained from
Eq.~(\ref{eq2:33}) as
\begin{align}\label{eq2:35}
\Delta g_{\rm red, dir, wf} =&\ -\frac{i}{2\pi} {\cal P} \intinf d\omega\,
\frac1{\omega+\Delta}\, \delta F_{\rm lad, dir}^{\prime}(\omega,vc)\,,
\end{align}
where $\delta F_{\rm lad, dir}$ is defined by Eq.~(\ref{eq:6}). The derivation of the
energy-perturbed reducible part is more difficult and is carried out by perturbing the formulas
given by Eq.~(47) of Ref.~\cite{shabaev:94:ttg1}. This derivation is potentially problematic,
because vanishing contributions to the Lamb shift may induce nonzero magnetic perturbations. For
example, the energy difference $\Delta_{an} = \vare_a - \vare_n$ induces a perturbation $\lbr
a|V_g|a\rbr - \lbr n|V_g|n\rbr$. If $\vare_n = \vare_a$ and $\mu_n \neq \mu_a$, the energy
difference vanishes but the magnetic perturbation survives. In order to avoid potential
ambiguities, we fix the reducible part by requirement of the gauge invariance of the total
correction to the $g$ factor. The result for the direct energy-perturbation reducible part is
\begin{align}\label{eq2:36}
\Delta g_{\rm red, dir, en} &\ = \frac14 \Big[
-\delta_1 F_{\rm lad, dir}^{\prime\prime}(0,cv)
-\delta_2 F_{\rm lad, dir}^{\prime\prime}(0,cv)
\nonumber \\ &\
-\delta_1 F_{\rm lad, dir}^{\prime\prime}(\Delta,vc)
-\delta_2 F_{\rm lad, dir}^{\prime\prime}(\Delta,vc)
\nonumber \\ &\
+ \delta_1 F^{\prime\prime}_{\rm cr, dir}(0,cv)
+\delta_2 F^{\prime\prime}_{\rm cr, dir}(0,cv)
\Big]
\nonumber \\ &\
+ \frac{i}{4\pi} {\cal P} \intinf
d\omega\,
\frac{\delta_1 F_{\rm lad, dir}^{\prime\prime}(\omega,vc)
-\delta_2 F_{\rm lad, dir}^{\prime\prime}(\omega,vc)}{\omega+\Delta}
\,,
\end{align}
where $F^{\prime\prime}(\omega) =\partial^2 F(\omega)/(\partial\omega)^2$.
It should be pointed out that the second integration by parts, leading to the second derivative
of the photon exchange operator $I''(\omega)$ in Eq.~(\ref{eq2:36}) is potentially troublesome.
The reason is that the imaginary part of the first derivative $I'(\omega)$ is discontinuous at
$\omega = 0$. Specifically, ${\rm Im}\big[I'(0_+)\big] = -{\rm Im}\big[I'(0_-)\big] \ne 0$. This
discontinuity leads, in principle, to appearance of additional off-integral terms in
Eq.~(\ref{eq2:36}). We found, however, that their numerical contributions are completely
negligible for the case under consideration in the present paper. The same holds for the exchange
reducible part.
\begin{widetext}
\subsection{Exchange reducible part}
The reducible exchange correction for the Lamb shift is given by Eq.~(41) of
Ref.~\cite{yerokhin:01:2ph}. We write this correction as
\begin{align}
\Delta E_{\rm red, ex} = \frac{i}{2\pi}\int_{-\infty}^{\infty}
d\omega\,
\Bigg\{ & \
-\frac12\, F_{\rm lad, ex}(\omega,cv)\,
\bigg[ \frac{1}{(\Delta-\omega-i0)^2}+\frac{1}{(\Delta-\omega+i0)^2}\bigg]
\nonumber \\ &
-\frac12\, F_{\rm lad, ex}(\omega,vc)\,
\bigg[ \frac{1}{(-\omega-i0)^2}+\frac{1}{(-\omega+i0)^2}\bigg]
\nonumber \\ &
+\, F_{\rm cr, ex}(\omega,cc)\,
\frac{1}{(\Delta-\omega+i0)^2}
+\, F_{\rm cr, ex}(\omega,vv)\,
\frac{1}{(-\omega+i0)^2}
\Bigg\}\,.
\end{align}
We observe that the crossed $(cc)$ and $(vv)$ terms in the above expression exactly coincide with
the two terms excluded from the summation in Eq.~(\ref{eq:11}). The ladder $(cv)$ and $(vc)$
terms in the above expression are similar to those excluded from the summation in
Eq.~(\ref{eq:11}) but differ from them by the signs of $i0$. We evaluate the above expression by
integrating by parts and taking the principal value of the integral, separating the pole
contribution. The result is
\begin{align}
\Delta E_{\rm red, ex} = &\ -\frac12 \Big[F^{\prime}_{\rm cr, ex}(\Delta,cc)
+ F^{\prime}_{\rm cr, ex}(0,vv)\Big]
\nonumber \\ &
+ \frac{i}{2\pi} {\cal P} \intinf
d\omega\,
\Bigg[
\frac{F_{\rm lad, ex}^{\prime}(\omega,cv)}{\Delta-\omega}\,
+ \frac{F_{\rm lad, ex}^{\prime}(\omega,vc)}{-\omega}\,
- \frac{F_{\rm cr, ex}^{\prime}(\omega,cc)}{\Delta-\omega}\,
- \frac{F_{\rm cr, ex}^{\prime}(\omega,vv)}{-\omega}\,
\Bigg]\,.
\end{align}
It should be mentioned that the individual terms in the brackets under the integral in the above
formula contain singularities at $\omega = 0$ and $\omega = \Delta$. When the ladder and exchange
terms are combined together, however, the singularities disappear and the principal value of the
resulting integral becomes well defined and can be calculated numerically.
The reducible exchange contribution for the $g$ factor is the sum of perturbations of the wave
functions and perturbations of the energies, $\Delta g_{\rm red, ex} = \Delta g_{\rm red, ex, wf
}+ \Delta g_{\rm red, ex, en}$, where
\begin{align}
\Delta g_{\rm red, ex, wf} = &\ -\frac12 \Big[\delta F^{\prime}_{\rm cr, ex}(\Delta,cc)
+ \delta F^{\prime}_{\rm cr, ex}(0,vv)\Big]
\nonumber \\ &
+ \frac{i}{2\pi} {\cal P} \intinf
d\omega\,
\Bigg[
\frac{\delta F_{\rm lad, ex}^{\prime}(\omega,cv)}{\Delta-\omega}\,
+ \frac{\delta F_{\rm lad, ex}^{\prime}(\omega,vc)}{-\omega}\,
- \frac{\delta F_{\rm cr, ex}^{\prime}(\omega,cc)}{\Delta-\omega}\,
- \frac{\delta F_{\rm cr, ex}^{\prime}(\omega,vv)}{-\omega}\,
\Bigg]\,,
\end{align}
\begin{align}
\Delta g_{\rm red, ex, en} = &\
\frac14 \bigg[-\delta_1 F_{\rm lad, ex}^{\prime\prime}(\omega,cv)
-\delta_2 F_{\rm lad, ex}^{\prime\prime}(\omega,cv)
-\delta_1 F_{\rm lad, ex}^{\prime\prime}(\omega,vc)
-\delta_2 F_{\rm lad, ex}^{\prime\prime}(\omega,vc)
\nonumber \\ &
\ \ \ \ \ \
+\delta_1 F^{\prime\prime}_{\rm cr, ex}(\Delta,cc)
+\delta_2 F^{\prime\prime}_{\rm cr, ex}(\Delta,cc)
+\delta_1 F^{\prime\prime}_{\rm cr, ex}(0,vv)
+\delta_2 F^{\prime\prime}_{\rm cr, ex}(0,vv)\bigg]
\nonumber \\ &
+ \frac{i}{4\pi} {\cal P} \intinf
d\omega\,
\Bigg[
\frac{\delta_1 F_{\rm lad, ex}^{\prime\prime}(\omega,cv)
-\delta_2 F_{\rm lad, ex}^{\prime\prime}(\omega,cv)}{\omega-\Delta}
+ \frac{\delta_1 F_{\rm lad, ex}^{\prime\prime}(\omega,vc)
-\delta_2 F_{\rm lad, ex}^{\prime\prime}(\omega,vc)}{\omega}
\nonumber \\ &
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
- \frac{ \delta_1 F_{\rm cr, ex}^{\prime\prime}(\omega,cc)
+\delta_2 F_{\rm cr, ex}^{\prime\prime}(\omega,cc)}{\omega-\Delta}
- \frac{ \delta_1 F_{\rm cr, ex}^{\prime\prime}(\omega,vv)
+\delta_2 F_{\rm cr, ex}^{\prime\prime}(\omega,vv)}{\omega}
\Bigg]\,.
\end{align}
\subsection{Three-electron part}
The three-electron irreducible contribution to the Lamb shift is given by Eq. (14) of
Ref.~\cite{yerokhin:01:2ph},
\begin{align} \label{eq0a}
\Delta E_{\rm 3el, ir} = \sum_{PQ} (-1)^{P+Q} {\sum_{n}}^{\prime} \
\frac{I_{P2\,P3\,n\,Q3}(\Delta_{P3Q3})\,
I_{P1\,n\,Q1\,Q2}(\Delta_{Q1P1})}
{\vare_{Q1}+\vare_{Q2}-\vare_{P1}-\vare_n} \,.
\end{align}
Here, ``1'', ``2'', and ``3'' label the three electrons of the ions (in arbitrary order), the
operators $P$ and $Q$ permute the initial-state and the final-state electrons, $(-1)^P$ and
$(-1)^Q$ are the sign of permutations, and the prime on the sum means that the terms with the
vanishing denominator are excluded from the summation. Furthermore, $\Delta_{ab} \equiv \vare_a -
\vare_b$, and $I_{abcd}(\Delta) \equiv \lbr ab|I(\Delta)|cd\rbr$.
The corrections to the $g$ factor are obtained as first-order perturbations of Eq.~(\ref{eq0a})
by $V_g$. It is convenient to split the whole contribution into the perturbations of the external
wave functions (``pwf''), external energies (``en''), and the propagator (``ver''),
\begin{align}\label{eq0b0}
\Delta g_{\rm 3el, ir} = \Delta g^{\rm 3el}_{\rm ir, pwf} + \Delta g^{\rm 3el}_{\rm ir, en} + \Delta g^{\rm 3el}_{\rm ir, ver}\,,
\end{align}
\begin{align} \label{eq0b}
\Delta g^{\rm 3el}_{\rm ir, pwf} = &\ 2 \sum_{PQ} (-1)^{P+Q} {\sum_{n}}^{\prime} \
\frac1{\vare_{Q1}+\vare_{Q2}-\vare_{P1}-\vare_n}
\biggl[
I_{P2\,P3\,n\,\delta Q3}(\Delta_{P3Q3})\,I_{P1\,n\,Q1\,Q2}(\Delta_{Q1P1})
\nonumber \\ &
+ I_{P2P3nQ3}(\Delta_{P3Q3})\,I_{P1\,n\,\delta Q1\,Q2}(\Delta_{Q1P1})
+ I_{P2P3nQ3}(\Delta_{P3Q3})\,I_{P1\,n\,Q1\,\delta Q2}(\Delta_{Q1P1})
\biggr]
\ ,
\end{align}
\begin{align} \label{eq0d}
&\ \Delta g^{\rm 3el}_{\rm ir, en} =\sum_{PQ} (-1)^{P+Q} {\sum_{n}}^{\prime} \
\biggl[
- \left( V_{Q1Q1} + V_{Q2Q2} - V_{P1P1} \right)
\frac{I_{P2\,P3\,n\,Q3}(\Delta_{P3Q3})\,
I_{P1\,n\,Q1\,Q2}(\Delta_{Q1P1})} {(\vare_{Q1}+\vare_{Q2}-\vare_{P1}-\vare_n)^2}
\nonumber \\ &
+
\frac{\left( V_{P3P3} - V_{Q3Q3} \right)\, I'_{P2P3nQ3}(\Delta_{P3Q3})\,
I_{P1\,n\,Q1\,Q2}(\Delta_{Q1P1})
+
\left( V_{Q1Q1} - V_{P1P1} \right)
I_{P2\,P3\,n\,Q3}(\Delta_{P3Q3})\,
I'_{P1\,n\,Q1\,Q2}(\Delta_{Q1P1})
} {\vare_{Q1}+\vare_{Q2}-\vare_{P1}-\vare_n}
\biggr]\,,
\end{align}
\begin{align} \label{eq0c}
\Delta g^{\rm 3el}_{\rm ir, ver} = \sum_{PQ} (-1)^{P+Q} {\sum_{n_1n_2}} \,\Xi\,\,
\frac{I_{P2\,P3\,n_1\,Q3}(\Delta_{P3Q3}) \,V_{n_1n_2}\,
I_{P1\,n_2\,Q1\,Q2}(\Delta_{Q1P1})}
{(\vare_{Q1}+\vare_{Q2}-\vare_{P1}-\vare_{n_1})(\vare_{Q1}+\vare_{Q2}-\vare_{P1}-\vare_{n_2})} \ ,
\end{align}
where the operator $\Xi$ acts on energy denominators $\Delta_1$, $\Delta_2$ as follows:
\begin{eqnarray} \label{eqII17}
\Xi \, \frac{X}{\Delta_1\,\Delta_2} = \left\{
\begin{array}{cl}
\displaystyle \frac{ X}{ \Delta_1\Delta_2}\,,
& \mbox{if}\ \Delta_1\ne0 \ \mbox{and}\ \Delta_2\ne0\,,\\[0.35cm]
\displaystyle -\frac{ X}{\Delta_1^2}\,, &\mbox{if}\
\Delta_1\ne0 \ \mbox{and}\ \Delta_2=0\,,\\[0.35cm]
\displaystyle -\frac{X}{\Delta_2^2}\,, &\mbox{if}\
\Delta_1=0 \ \mbox{and}\ \Delta_2\ne0\,,\\[0.35cm]
0\,, &\mbox{if}\
\Delta_1=0 \ \mbox{and}\ \Delta_2=0\,.\\
\end{array}
\right.
\end{eqnarray}
The three-electron reducible correction to the Lamb shift is given by Eq.~(19) of
Ref.~\cite{yerokhin:01:2ph},
\begin{align} \label{eq2a}
\Delta {E}_{\rm 3el, red} =&\ \frac12 \sum_{PQ} (-1)^{P+Q} \sum_{{\mu_n}\atop{\vare_n = \vare_{Q1}+\vare_{Q2}-\vare_{P1}}}\!\!\!\!\!\!
\bigg[
I_{P2\,P3\,n\,Q3}\pr(\Delta_{P3Q3})\,I_{P1\,n\,Q1\,Q2}(\Delta_{Q1P1})
+I_{P2\,P3\,n\,Q3}(\Delta_{P3Q3})\,I_{P1\,n\,Q1\,Q2}\pr(\Delta_{Q1P1})
\bigg] \,.
\end{align}
The corresponding corrections to the $g$ factor arise as perturbations of the wave functions and
energies,
\begin{align} \label{eq2b}
\Delta {g}_{\rm 3el, red} =&\ \sum_{PQ} (-1)^{P+Q} \sum_{{\mu_n}\atop{\vare_n = \vare_{Q1}+\vare_{Q2}-\vare_{P1}}}\!\!\!\!\!\!
\biggl\{
I_{P2\,P3\,n\,Q3}\pr(\Delta_{P3Q3})\,I_{P1\,n\,\delta Q1\,Q2}(\Delta_{Q1P1}) + I_{P2\,P3\,n\,Q3}(\Delta_{P3Q3})\,I_{P1\,n\,\delta Q1\,Q2}\pr(\Delta_{Q1P1})
\nonumber \\ &
+ I_{P2\,P3\,n\,Q3}\pr(\Delta_{P3Q3})\,I_{P1\,n\,Q1\,\delta Q2}(\Delta_{Q1P1}) + I_{P2\,P3\,n\,Q3}(\Delta_{P3Q3})\,I_{P1\,n\,Q1\,\delta Q2}\pr(\Delta_{Q1P1})
\nonumber \\ &
+ I_{P2\,P3\,n\,\delta Q3}\pr(\Delta_{P3Q3})\,I_{P1\,n\,Q1\,Q2}(\Delta_{Q1P1}) + I_{P2\,P3\,n\,\delta Q3}(\Delta_{P3Q3})\,I_{P1\,n\,Q1\,Q2}\pr(\Delta_{Q1P1})
\nonumber \\ &
+ I_{P2\,P3\,n\,Q3}\pr(\Delta_{P3Q3})\,I_{P1\,\delta n\,Q1\,Q2}(\Delta_{Q1P1}) + I_{P2\,P3\,n\,Q3}(\Delta_{P3Q3})\,I_{P1\,\delta n\,Q1\,Q2}\pr(\Delta_{Q1P1})
\nonumber \\ &
+ (V_{P3P3}-V_{Q3Q3})
\,\Bigl[ I\ppr_{P2\,P3\,n\,Q3}(\Delta_{P3Q3})I_{P1\,n\, Q1\,Q2}(\Delta_{Q1P1})
+ I\pr_{P2\,P3\,n\,Q3}(\Delta_{P3Q3})I_{P1\,n\, Q1\,Q2}\pr(\Delta_{Q1P1})\Big]
\nonumber \\ &
+
(V_{Q1Q1}-V_{P1P1}) \,
\Big[ I\pr_{P2\,P3\,n\,Q3}(\Delta_{P3Q3})I\pr_{P1\,n\, Q1\,Q2}(\Delta_{Q1P1})
+ I_{P2\,P3\,n\,Q3}(\Delta_{P3Q3})I\ppr_{P1\,n\, Q1\,Q2}(\Delta_{Q1P1})\Big]
\biggr\}
\, .
\end{align}
\section{MBPT approximation}
In this section we obtain formulas for the two-photon exchange correction to the $g$ factor in
the approximation of the relativistic many-body perturbation theory (MBPT). The corresponding
formulas can be obtained from the QED expressions by (i) using the Coulomb gauge in the photon
propagators and neglecting the energy dependence, $I(\omega) \to I_{\rm Coul}(0)$, and (ii)
restricting the summations over the Dirac spectrum to the positive-energy part. Under these
assumptions, all reducible contributions vanish and the integrals over $\omega$ can be performed
by the Cauchy theorem. The $\omega$ integral for the crossed contribution vanishes, so the total
two-electron correction comes only from the ladder irreducible part. Performing the $\omega$
integrations in Eqs.~(\ref{eq:4}) and (\ref{eq:14}), we obtain the two-electron contribution in
the MBPT approximation as
\begin{align} \label{eq3a}
\Delta g_{\rm 2el}^{\rm MBPT}
= &\ \left.\sum_{\underset{\vare_{n_1},\vare_{n_2} > 0}{n_1n_2}}\right.^{\!\!\!\!\!\!\prime}\,\,\,
\Bigg[ \frac{\delta F_{\rm lad, dir}(0,n_1n_2)+ \delta F_{\rm lad, ex}(0,n_1n_2)}{\vare_v+\vare_c-\vare_{n_1}-\vare_{n_2}}
\nonumber \\ &
+
\frac{\delta_1 F_{\rm lad, dir}(0,n_1n_2)+ \delta_2 F_{\rm lad, dir}(0,n_1n_2)
+ \delta_1 F_{\rm lad, ex}(0,n_1n_2)+\delta_2 F_{\rm lad, ex}(0,n_1n_2)}{(\vare_v+\vare_c-\vare_{n_1}-\vare_{n_2})^2}
\Bigg]\,.
\end{align}
\end{widetext}
Here, the prime on the sum means that the terms with the vanishing denominator should be omitted,
and the summation over $n_1$ and $n_2$ is performed over the positive-energy part of the Dirac
spectrum. We note that Eq.~(\ref{eq3a}) can be also obtained directly by perturbing the
two-photon MBPT correction for the Lamb shift, given by Eq.~(43) of Ref.~\cite{yerokhin:01:2ph}.
The three-electron MBPT correction is immediately obtained from Eqs.~(\ref{eq0b0})-(\ref{eq0c}),
after the substitution $I(\omega) \to I_{\rm Coul}(0)$ and the restriction of the summations to
the positive-energy part of the spectrum.
We note that the standard formulation of MBPT assumes the restriction of all summations over the
Dirac spectrum to the positive-energy part. The consistent treatment of the negative-energy
spectrum is possible only within the QED theory. However, it can be easily observed that one can
include some negative-energy contributions already in the MBPT formulas, namely, in those cases
when it does not lead to the so-called continuum dissolution, i.e., vanishing energy
denominators. Specifically, one can include the negative-energy spectrum in the three-electron
contributions, Eqs.~(\ref{eq0b0})-(\ref{eq0c}), and in the magnetic perturbations of the wave
functions, Eq.~(\ref{eq0:9}). We will refer to this variant of the MBPT as ``MBPT-neg''. We will
demonstrate that such partial inclusion of the negative-energy spectrum within MBPT is crucially
important to approximately reproduce the QED results in the region of small nuclear charges,
whereas the standard MBPT yields a very much different result, even in the limit of $Z\to 0$.
Previously the same conclusion was drawn by the St.~Petersburg group
\cite{volotka:priv,wagner:13}.
The connection between the QED and MBPT formulas was extensively used in this work for checking
the numerical procedure for the $\omega$ integrations. Specifically, after neglecting the energy
dependence of the photon propagators in the Coulomb gauge, we checked that the numerical $\omega$
integration yields the same result as the analytical integration by the Cauchy theorem.
\section{Numerical evaluation}
We now turn to the numerical evaluation of the two-photon exchange corrections. Since the
calculation of the three-electron contributions is relatively straightforward, we concentrate
mainly on the two-electron terms. The direct and exchange irreducible contributions given by
Eqs.~(\ref{eq:4}) and (\ref{eq:14}) represent the main computational difficulty. It is
advantageous to deform the contour of the $\omega$ integration in them, in order to escape strong
oscillations of the photon propagators $\propto e^{i|\omega|r_{12}}$ for large real values of
$\omega$. Deforming the contour, one needs to take into account the branch cuts of the photon
propagators and the pole structure of the Dirac propagators. The analytical structure of the
integrand as a function of complex $\omega$ is shown in Fig.~\ref{fig:CD} for the direct part and
in Fig.~\ref{fig:CX} for the exchange part, respectively.
For the evaluation of the direct irreducible contribution, we use two different choices of the
$\omega$-integration contour. The first choice is the standard Wick rotation, $\omega \to
i\omega$, which splits the correction into the pole contribution and the integral along the
imaginary $\omega$ axis. This contour was used in the previous Lamb-shift calculations
\cite{yerokhin:01:2ph,mohr:00:pra}. The advantage of this choice is that the analysis of the pole
terms is the simplest. There are, however, also some difficulties. The first problem is the
rapidly-varying structure of the integrand in the vicinity of $\omega = 0$, due to poles of the
electron propagators lying near the imaginary axis. The second difficulty is that the
contributions with $n_2 = v$ in Eq.~(\ref{eq:4}) contain singular terms $\sim 1/\omega^2$, which
need to be integrated by parts before the numerical evaluation.
In order to achieve a more regular behaviour of the integrand for small $\omega$, we adopted the
contour $C_D$, shown in Fig.~\ref{fig:CD}. This contour is convenient for the numerical
evaluation, especially for low values of $Z$. Its disadvantage is the presence of a pole on the
low-energy part of the integration contour and thus the need to evaluate the principal value of
the integral. We find, however, that the contour $C_D$ is very similar to the contour $C_X$ used
in the evaluation of the exchange contribution and discussed in detail below. Because of this
similarity, we were able to use essentially the same numerical procedure both for the direct and
the exchange part. We checked that our numerical evaluation of the integral along the contour
$C_D$ leads to the same results as the integration along the Wick-rotated contour.
\begin{figure*}
\centerline{
\resizebox{0.45\textwidth}{!}
\includegraphics{CDLAD3.eps}
}
\hspace*{0.5cm}
\resizebox{0.45\textwidth}{!}
\includegraphics{CDCR3.eps}
}
}
\caption{The singularities of the integrand in the complex $\omega$ plane and the integration contour $C_D$, for
the ladder direct contribution (left panel) and the crossed direct contribution (right panel).
\label{fig:CD}}
\end{figure*}
\begin{figure*}
\centerline{
\resizebox{0.45\textwidth}{!}
\includegraphics{CXLAD3.eps}
}
\hspace*{0.5cm}
\resizebox{0.45\textwidth}{!}
\includegraphics{CXCR3.eps}
}
}
\caption{The contour $C_X$ and singularities of the integrand in the complex $\omega$-plane, for
the ladder exchange contribution (left panel) and the crossed exchange contribution (right panel).
\label{fig:CX}}
\end{figure*}
For the numerical evaluation of the exchange irreducible part, we use the contour $C_X$ depicted
in Fig.~\ref{fig:CX}. This contour was suggested in Ref.~\cite{mohr:00:pra} for the Lamb shift
and later used for the $g$ factor and hyperfine structure in
Refs.~\cite{volotka:12,kosheleva:20}. As can be seen from Fig.~\ref{fig:CX}, the deformation of
the contour from $(-\infty,\infty)$ to $C_X$ leads to appearance of pole terms at $\omega = 0$
and $\omega = \Delta$. In the case of the Lamb shift, the pole terms are identified as follows,
for the ladder contribution,
\begin{align}\label{eq:29}
&\frac{i}{2\pi} \int_{-\infty}^{\infty} d\omega\, \frac{F(\omega)}{(\tilde{\Delta}_1-\omega)(\tilde{\Delta}_2+\omega)}
\underset{\scriptscriptstyle \Delta_1\ne -\Delta_2}{=} \frac{F(\Delta_1)}{\Delta_1+\Delta_2}\,\delta(\Delta_1-\Delta)
\nonumber \\ &
+ \frac{F(-\Delta_2)}{\Delta_1+\Delta_2}\,\delta(\Delta_2)
+ \frac{i}{2\pi} \int_{C_X} d\omega\, \frac{F(\omega)}{(\tilde{\Delta}_1-\omega)(\tilde{\Delta}_2+\omega)}
\,,
\end{align}
where $\tilde{\Delta}_i$ denotes $\Delta_i$ with the infinitesimal imaginary addition according
to Eqs.~(\ref{eq:4}) and (\ref{eq:14}). For the crossed contribution, the corresponding equation
reads
\begin{align}
&\frac{i}{2\pi} \int_{-\infty}^{\infty} d\omega\, \frac{F(\omega)}{(\tilde{\Delta}_1-\omega)(\tilde{\Delta}_2-\omega)}
\underset{\scriptscriptstyle \Delta_1\ne \Delta_2}{=} \ \frac{F(\Delta_1)}{\Delta_2-\Delta_1}\,\delta(\Delta_1-\Delta)
\nonumber \\ &
+ \frac{F(\Delta_2)}{\Delta_1-\Delta_2}\,\delta(\Delta_2-\Delta)
+ \frac{i}{2\pi} \int_{C_X} d\omega\, \frac{F(\omega)}{(\tilde{\Delta}_1-\omega)(\tilde{\Delta}_2-\omega)}
\,. \label{eq:30}
\end{align}
For the $g$ factor, formulas with squared energy denominators are required, which can be obtained
by a formal differentiation of the above formulas over $\Delta_1$ and $\Delta_2$.
For a numerical evaluation, the integral over $C_X$ is represented as a sum of three pieces,
\begin{align}
\frac{i}{2\pi}\int_{C_X}d\omega &\ \frac{I(\omega)\,I(\omega-\Delta)}{(\tilde{\Delta}_1-\omega)(\tilde{\Delta}_2\pm\omega)}
=
\nonumber \\ &
-\frac1{\pi} \int_0^{\delta}d\omega\,
\frac{{\rm Im}\big[I(\omega)\big]\,I(\Delta-\omega)}{(\tilde{\Delta}_1-\omega)(\tilde{\Delta}_2\pm\omega)}
\nonumber \\ &
-\frac1{\pi} \int_{\delta}^{\Delta} d\omega\,
\frac{I(\omega)\,{\rm Im}\big[I(\Delta-\omega)\big]}{(\tilde{\Delta}_1-\omega)(\tilde{\Delta}_2\pm\omega)}
\nonumber \\ &
-\frac1{\pi} \, {\rm Re}\int_{0}^{\infty} d\omega\,
\frac{I(\delta + i\omega)\,I(\delta + i\omega-\Delta)}{(\Delta_1-\delta - i\omega)(\Delta_2\pm (\delta + i\omega))}\,,
\end{align}
where $\delta$ is a free parameter $0 < \delta < \Delta$. A typical value of $\delta = \Delta/2$
was used. An advantage of the contour $C_X$ is that the integrand has a more regular behaviour at
the end points of the intervals, $\omega = 0$ and $\omega = \Delta$, because ${\rm
Im}\,I(\omega)\propto\omega$ as $\omega \to 0$. There are, however, singularities inside the
intervals along the real axis, and thus the infinitesimal imaginary terms $i0$ should be retained
for them. For the $g$ factor, we encounter single, double, and even triple poles on the
integration contour. Specifically, for the $v = 2s$ reference state, the singularities arise from
the intermediate states $n_{1}$ and/or $n_2 = 2p_{1/2}$, whose energy $\vare_{2p_{1/2}} <
\vare_{2s}$ is separated from the reference-state energy by the finite nuclear size effect. To
deal with these singularities, we introduce subtractions obtained by expanding the integrand in
the Taylor series in the vicinity of the poles. The subtractions remove singularities and make
the integrand a regular and smooth function suitable for the numerical integration. The
subtracted terms are then re-added, with the principal-value integrals calculated analytically.
The corresponding formulas are summarized in Appendix~\ref{sec:subtractions}.
In order to check our numerical procedure, we performed calculations also by using a different
integration contour, namely, the contour $C_{\rm irr}$ suggested in Ref.~\cite{yerokhin:01:2ph}
(shown in Fig.~5 of that work). A very good agreement of numerical results obtained with two
different contours was used as a confirmation of the internal consistency of the numerical
procedure.
For the numerical evaluation of the reducible direct and exchange contributions, we used the
$\omega$-integration contour consisting of three sections: $(-\delta -i \infty,-\delta)$, $(-
\delta,\delta)$, $(\delta,\delta + i\infty)$. The parameter $\delta$ of the contour was taken to
be $\delta > \Delta$, which allowed us to evaluate the principal value of the integrals at points
$\omega = 0$ and $\omega = \pm \Delta$.
The summations over the Dirac intermediate states were performed by using the dual kinetic
balance basis-set method \cite{shabaev:04:DKB}, with the basis set constructed with the
B-splines. The standard two-parameter Fermi model was used to represent the nuclear charge
distribution, with the nuclear radii taken from Ref.~\cite{angeli:13}. The infinite partial-wave
summation over the relativistic angular momentum quantum number $\kappa$ was performed up to
$|\kappa_{\rm max}| = 25$, with the remaining tail estimated by the polynomial fitting of the
expansion terms in $1/|\kappa|$. The largest numerical uncertainty was typically induced by
convergence in the number of basis functions. Our final values were typically obtained by
performing calculations with $N = 85$ and $N = 105$ $B$-splines and extrapolating the results to
$N\to \infty$ as $\delta g = \delta g_{N = 105} + 0.93\,(\delta g_{N = 105}-\delta g_{N = 85}$),
where the numerical coefficient was obtained by analysing the convergence pattern of our
numerical results. An example of the convergence study with respect to $N$ is presented in
Table~\ref{tab:convergence}.
In the present work we perform calculations in the Feynman and the Coulomb gauge. The expressions
for the matrix elements of the electron-electron interaction in the Coulomb gauge are summarized
in Appendix~\ref{app:coul}. We note that in the present work (unlike, e.g., in
Ref.~\cite{yerokhin:01:2ph}) we use the expression for the Coulomb-gauge matrix element
[Eq.~(\ref{eq:Rcoul})] that does not rely on commutator relations for the wave functions. This
expression is valid for the general case when the wave functions in the matrix element are not
eigenfunctions of the Dirac Hamiltonian, in particular, when they are the magnetic perturbations
of the Dirac wave functions. Another advantage of this expression is that it allows a numerical
evaluation of the Coulomb-gauge radial integrals and their derivatives for very small but
nonvanishing photon energies $\omega$. The region of small but nonzero $\omega$ is usually
numerically unstable for the expressions based on the commutator relations, especially for the
second derivative of the photon propagator $I''(\omega)$.
\begin{table}
\caption{Convergence study for the direct irreducible contribution $\Delta g_{\rm ir, dir}$ for $Z = 14$
as a function of the number of
$B$-splines in the basis set $N$, in Feynman gauge, in units of $10^{-6}$.
\label{tab:convergence}}
\begin{ruledtabular}
\begin{tabular}{lw{4.6}w{4.6}}
$N$ & \multicolumn{1}{c}{$\Delta g_{\rm ir, dir}$}
& \multicolumn{1}{c}{Increment}
\\ \hline\\[-5pt]
55 & -9.442\,31 & \\
70 & -9.442\,86 & -0.000\,55 \\
85 & -9.443\,08 & -0.000\,22 \\
105 & -9.443\,20 & -0.000\,12 \\
130 & -9.443\,26 & -0.000\,06 \\
Extrap. & -9.443\,31 & -0.000\,06 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}[t]
\centerline{
\resizebox{1.1\columnwidth}{!}
\includegraphics{results1c.eps}
} }
\vskip0.5cm
\caption{The two-photon exchange correction to the $g$ factor of the ground state
of Li-like ions in different approaches (QED, MBPT with the negative continuum contribution, NRQED).
The dotted line is a polynomial fit to the numerical data, to guide the eye.
\label{fig:2ph}}
\end{figure}
\begin{table*}
\caption{Individual two-photon exchange contributions to the $g$ factor of the ground state
of Li-like ions, in Feynman and Coulomb gauge. Units are $10^{-6}$.
\label{tab:2ph:Si}}
\begin{ruledtabular}
\begin{tabular}{llw{4.6}w{4.6}w{4.6}w{4.6}w{4.6}w{4.6}w{4.6}}
$Z$ & \multicolumn{1}{c}{Gauge}
& \multicolumn{2}{c}{Direct}
& \multicolumn{2}{c}{Exchange}
& \multicolumn{2}{c}{3-electron}
& \multicolumn{1}{c}{Total}
\\
& & \multicolumn{1}{c}{Irred}& \multicolumn{1}{c}{Red}
& \multicolumn{1}{c}{Irred}& \multicolumn{1}{c}{Red}
& \multicolumn{1}{c}{Irred}& \multicolumn{1}{c}{Red}\\
\hline\\[-5pt]
14 &Feynman& -9.4433 & 0.0015 & -0.0739 & 0.0302 & 2.6352 & -0.0285 & -6.8787\,(1) \\
&Coulomb& -9.4417 & 0.0000 & -0.0457 & 0.0020 & 2.6320 & -0.0253 & -6.8787\,(2) \\[3pt]
83 &Feynman& -16.6698 & -0.0053 & -3.4213 & 1.0489 & 11.0343 & -1.0481 & -9.0613\,(6) \\
&Coulomb& -16.6175 & -0.0631 & -2.8158 & 0.4425 & 10.8548 & -0.8621 & -9.0612\,(6) \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\begin{table*}
\caption{Numerical results for the two-photon exchange correction to the $g$ factor of the ground state
of Li-like ions, in units of $10^{-6}$. QED results are obtained in the Feynman gauge. ``MBPT'' labels results obtained within
the standard relativistic many-body perturbation theory. ``MBPT-neg'' labels results obtained within
MBPT supplemented by the correction from the negative-continuum part of the Dirac spectrum.
\label{tab:2ph}}
\begin{ruledtabular}
\begin{tabular}{rw{4.6}w{4.6}w{4.6}w{4.8}w{4.6}w{4.4}}
\multicolumn{1}{c}{Z}
& \multicolumn{1}{c}{Direct}
& \multicolumn{1}{c}{Exchange}
& \multicolumn{1}{c}{3-electron}
& \multicolumn{1}{c}{Total QED}
& \multicolumn{1}{c}{MBPT-neg}
& \multicolumn{1}{c}{MBPT} \\
\hline\\[-5pt]
6 & -9.3333 & -0.0068 & 2.5038 & -6.8363\,(3) & -6.8262 & -22.243 \\
8 & -9.3524 & -0.0128 & 2.5216 & -6.8436\,(2) & -6.8257 & -22.250 \\
10 & -9.3769 & -0.0209 & 2.5447 & -6.8531\,(3) & -6.8251 & -22.258 \\
12 & -9.4066 & -0.0312 & 2.5730 & -6.8648\,(2) & -6.8245 & -22.268 \\
14 & -9.4418 & -0.0437 & 2.6067 & -6.8787\,(1) & -6.8239 & -22.280 \\
& & & & -6.876\,^a \\
18 & -9.5287 & -0.0755 & 2.6909 & -6.9133\,(2) & -6.8229 & -22.309 \\
20 & -9.5808 & -0.0949 & 2.7417 & -6.9341\,(3) & -6.8227 & -22.327 \\
24 & -9.7031 & -0.1408 & 2.8615 & -6.9824\,(3) & -6.8231 & -22.367 \\
28 & -9.8512 & -0.1962 & 3.0073 & -7.0401\,(3) & -6.8252 & -22.414 \\
32 & -10.0273 & -0.2612 & 3.1813 & -7.1072\,(3) & -6.8299 & -22.468 \\
40 & -10.4743 & -0.4213 & 3.6250 & -7.2706\,(4) & -6.8508 & -22.594 \\
54 & -11.6370 & -0.8099 & 4.7900 & -7.6569\,(5) & -6.9540 & -22.856 \\
70 & -13.8228 & -1.4897 & 7.0179 & -8.2945\,(5) & -7.2907 & -23.193 \\
82 & -16.4083 & -2.2884 & 9.7062 & -8.9905\,(6) & -7.8666 & -23.490 \\
83 & -16.6750 & -2.3725 & 9.9863 & -9.0613\,(6) & -7.9348 & -23.519 \\
92 & -19.5455 & -3.3111 & 13.0311 & -9.8254\,(7) & -8.7545 & -23.837 \\
\end{tabular}
\end{ruledtabular}
$^a\,$ Volotka {\em et al.} 2014 \cite{volotka:14}.
\end{table*}
\begin{table*}[t]
\caption{Binding corrections to the $g$-factor of the ground state of Li-like ions, in $10^{-6}$.
The sum of all binding contributions is the difference of the atomic $g$ factor and the
free-electron $g$ factor, $g_e = 2.002\,319\,304\,361\,5\,(6)$
\cite{hanneke:08}.
\label{tab:gtotal}}
\begin{ruledtabular}
\begin{tabular}{llw{4.8}w{4.8}w{4.8}w{4.8}}
\multicolumn{1}{l}{Effect}
& \multicolumn{1}{c}{Contribution}
& \multicolumn{1}{c}{$^{ 12}$C$^{ 3+}$}
& \multicolumn{1}{c}{$^{ 16}$O$^{ 5+}$}
& \multicolumn{1}{c}{$^{ 28}$Si$^{ 11+}$}
& \multicolumn{1}{c}{$^{ 40}$Ca$^{ 17+}$}
\\
\hline\\[-5pt]
\multicolumn{1}{l}{Electronic structure}
&$1/Z^0$ & -319.6997 & -568.6205 & -1745.2493 & -3573.9891 \\
&$1/Z^1$ & 137.4194 & 183.3202 & 321.5908 & 461.1500 \\
&$1/Z^2$ & -6.8363\,(3) & -6.8436\,(2) & -6.8787\,(1) & -6.9341\,(3) \\
&$1/Z^{3+}$ & 0.1478\,(6) & 0.1377\,(8) & 0.0942\,(15) & 0.0695\,(25) \\[3pt]
\multicolumn{1}{l}{One-loop QED}
&$1/Z^0$ & 0.1978 & 0.3629 & 1.2244 & 2.7349 \\
&$1/Z^1$ & -0.0974\,(7) & -0.1329\,(5) & -0.2460\,(6) & -0.3675\,(6) \\
&$1/Z^{2+}$ & 0.0091\,(1) & 0.0092\,(2) & 0.0096\,(6) & 0.0100\,(11) \\[3pt]
\multicolumn{1}{l}{Recoil}
&$1/Z^0$ & 0.0219 & 0.0293 & 0.0515 & 0.0742 \\
&$1/Z^{1}$ & -0.0075 & -0.0076 & -0.0076 & -0.0076 \\
&$1/Z^{2+}$ & -0.0005 & -0.0004 & -0.0003 & -0.0002 \\[3pt]
\multicolumn{1}{l}{Two-loop QED}
&$1/Z^0$ & -0.0003 & -0.0005 & -0.0017 & -0.0044\,(3) \\
&$1/Z^1$ & 0.0001 & 0.0002 & 0.0003\,(1) & 0.0005\,(4) \\[3pt]
\multicolumn{1}{l}{Finite nuclear size}
&$1/Z^0$ & 0.0001 & 0.0002 & 0.0026 & 0.0144 \\
&$1/Z^1$ & & -0.0001 & -0.0005 & -0.0020\,(1) \\[3pt]
\multicolumn{1}{l}{Radiative recoil}
&$1/Z^0$ & & & & -0.0001 \\[3pt]
\multicolumn{1}{l}{Total theory}
&$g-g_e$ & -188.8455\,(10) & -391.7458\,(10) & -1429.4107\,(17) & -3117.2515\,(27) \\[1pt]
&$g$ & 2\,002\,130.4588\,(10) & 2\,001\,927.5585\,(10) & 2\,000\,889.8937\,(17) & 1\,999\,202.0529\,(27) \\[1pt]
\multicolumn{1}{l}{Previous theory}
&$g$ & 2\,002\,130.457\,(5)^a & 2\,001\,927.558\,(10)^a& 2\,000\,889.8944\,(34)^b & 1\,999\,202.041\,(13)^c \\[1pt]
\multicolumn{1}{l}{Experiment}
& $g$ & & & 2\,000\,889.88845\,(14)^b & 1\,999\,202.0406\,(11)^d\\[1pt]
& & & & 2\,000\,889.8884\,(19)^e
\end{tabular}
\end{ruledtabular}
$^a$ Yerokhin {\em et al.} 2017 \cite{yerokhin:17:gfact}; $^b$ Glazov {\em et al.} 2019
\cite{glazov:19}; $^c$ Volotka {\em et al.} 2014 \cite{volotka:14}; $^d$ K\"ohler {\em et al.}
2016 \cite{koehler:16}; $^e$ Wagner {\em et al.} 2013 \cite{wagner:13}.
\end{table*}
\section{Results and discussion}
Numerical results of our calculations of the two-photon exchange corrections for the ground state
of Li-like ions are presented in Tables~\ref{tab:2ph:Si} and \ref{tab:2ph}.
Table~\ref{tab:2ph:Si} contains a breakdown of our calculations in two gauges for $Z = 14$ and $Z
= 83$ and demonstrates the gauge invariance of our numerical results. Table~\ref{tab:2ph}
presents the final results of our calculations for $Z = 6$ -- $ 92$. It also compares results of
the QED calculation with those obtained within the standard MBPT and the MBPT with the partial
inclusion of the negative-energy spectrum (``MBPT-neg''). We observe that the standard MBPT
yields the two-photon exchange correction by about three times larger than the complete QED
results. The disagreement is evidently present even in the limit of $Z \to 0$. On the contrary,
the MBPT-neg approach closely reproduces the QED treatment in the region of low values of $Z$.
Previously the same conclusion was reached by the St.~Petersburg group
\cite{volotka:priv,wagner:13}.
For low values of the nuclear charge $Z$, results of our QED and MBPT-neg calculations can be
compared with the prediction of the nonrelativistic QED (NRQED) theory based on the explicitly
correlated three-electron wave function \cite{yerokhin:17:gfact}. According to
Ref.~\cite{yerokhin:17:gfact}, the two-photon exchange correction in the limit $Z\to 0$ is given
by (see Eq. (16) of that work)
\begin{align}
\Delta g_{\rm 2ph}(Z = 0) = -0.128\,204\,(9)\,\alpha^2 = -6.8270\,(5)\times 10^{-6}\,.
\end{align}
Fig.~\ref{fig:2ph} shows the comparison of the QED, MBPT-neg, and NRQED results. We observe that
all three methods yield results converging to each other in the limit $Z\to 0$. The difference
between the results by different methods scales as $\propto (\Za)^2$ as expected. It is
interesting that the MBPT-neg approach does not yield any significant improvement over the NRQED
treatment for low- and medium-$Z$ ions.
The QED calculation of the two-photon exchange correction for the $g$ factor of Li-like ions was
previously carried out in Refs.~\cite{volotka:12,volotka:14}. Unfortunately, the numerical
results were presented only for four ions and mostly in the form of the total electron-electron
interaction correction. The only ion for which the calculations are directly comparable is
silicon, $Z = 14$, for which we find some tension. Our calculation yields $-6.8787\,(1) \times
10^{-6}$, whereas Ref.~\cite{volotka:14} reported $-6.876 \times 10^{-6}$. As an additional
cross-check, we performed calculations for the two-photon exchange correction to the ground-state
hyperfine splitting of Li-like bismuth (which is another example of a magnetic perturbation
potential) and found agreement with results listed in Table~I of Ref.~\cite{volotka:12}.
Having obtained results for the two-photon exchange correction, we are now in a position to
update the theoretical predictions for the ground-state $g$ factor of Li-like ions.
Table~\ref{tab:gtotal} presents a compilation of all known binding corrections to the $g$ factor
of the ground state of four Li-like ions, C$^{3+}$, O$^{5+}$, Si$^{11+}$, and Ca$^{ 17+}$. As
compared to the analogous compilation in our previous investigation \cite{yerokhin:17:gfact}, we
introduced several improvements. The two-photon exchange correction (i.e., the
electronic-structure contribution of relative order $1/Z^2$) is computed in the present work.
Beside this, we included the one-loop $1/Z$ and $1/Z^{2+}$ QED effects from our recent work
\cite{yerokhin:20:gfact} and the nuclear recoil corrections of relative orders $1/Z^0$, $1/Z^1$,
and $1/Z^{2+}$ calculated by Shabaev {\em et al.}~\cite{shabaev:17:prl}. Furthermore, we added
the two-loop $(\Za)^5$ effects calculated recently by Czarnecki and co-workers
\cite{czarnecki:18,czarnecki:20} and by us \cite{yerokhin:13:twoloopg}. The two-loop results in
those studies were reported for the $1s$ hydrogenic state. We here convert them to the $2s$ state
by assuming the $1/n^3$ scaling. Having in mind that the result for the nonlogarithmic $(\Za)^5$
term is not complete, we ascribe the uncertainty of 20\% to the two-loop $(\Za)^5$ correction.
The uncertainty due to higher-order two-loop effects was evaluated on the basis of available
one-loop results, with the extension factor of 2.
One of the largest uncertainties of the theoretical predictions comes from the higher-order
electronic-structure correction $\sim\!1/Z^{3+}$. The values in Table~\ref{tab:gtotal} for this
correction are obtained within NRQED in Ref.~\cite{yerokhin:17:gfact}. Theoretical estimates for
their uncertainties are obtained by taking the relative deviation of the NRQED and full QED
results for the $1/Z^2$ correction and multiplying it by the extension factor of 2.
The comparison presented in Table~\ref{tab:gtotal} shows agreement of the present theoretical
$g$-factor values with previous theoretical predictions. In particular, for C$^{3+}$ and
O$^{5+}$, our results are in excellent agreement with, but 5-10 times more precise than our
earlier results in Ref.~\cite{yerokhin:17:gfact}. For Si$^{11+}$ and Ca$^{17+}$, our total
$g$-factor values are in agreement with the previous theoretical results of the St.~Petersburg
group \cite{volotka:14,glazov:19}. We note, however, that some tension exists between the
calculations on the level of individual contributions. Specifically, the total electron-electron
interaction correction for silicon in Ref.~\cite{glazov:19} is reported as $314.812\,(3)\times
10^{-6}$, whereas our calculation yields $314.806\,(2)\times 10^{-6}$. This deviation disappears
when the electron-structure correction is combined with the $1/Z$ QED contribution.
Table~\ref{tab:gtotal} also compares the obtained theoretical predictions with experimental
results available for two Li-like ions, Si$^{11+}$ and Ca$^{ 17+}$. In both cases theoretical
values deviate from the experimentally observed $g$ factors, by 3.1$\,\sigma$ for silicon and
4.2$\,\sigma$ for calcium. The discrepancy grows with the increase of $Z$. Such effect could be
caused by some unknown contribution missing in theoretical calculations.
The largest uncertainty in the theoretical predictions for Li-like silicon and calcium presently
comes from the higher-order electron correction $\sim\!1/Z^{3}$. This contribution cannot be
calculated rigorously to all orders in $\Za$ but needs to be treated by approximate methods or
within the $\Za$ expansion. It should be pointed out that our calculations and those by the
St.~Petersburg group \cite{volotka:14,glazov:19} use different approaches for handling this
correction. Our result is based on the $\Za$ expansion, whereas the St.~Petersburg group used
screening potentials in the two-photon-exchange calculations and explicitly computed the
three-photon-exchange contribution within the Breit approximation \cite{glazov:19}. The
$\Za$-expansion approach is most suitable in the low-$Z$ region, whereas the screening-potential
method is advantageous for high-$Z$ ions. For light ions, the $\Za$-expansion results can be
improved further, by performing the NRQED calculation of the next-order term in the $\alpha$
expansion.
Summarizing, we performed calculations of the two-photon exchange corrections to the $g$ factor
of the ground state of Li-like ions without an expansion in the nuclear binding strength
parameter $Z\alpha$. The calculations were carried out in two gauges, the Feynman and the Coulomb
ones, thus allowing an explicit test of the gauge invariance. In the low-$Z$ region, the obtained
results were checked against those delivered by two different and independent methods, namely,
the relativistic many-body perturbation theory with a partial inclusion of the negative-energy
continuum and the nonrelativistic quantum electrodynamics. It was demonstrated that all three
methods yield consistent results in the limit of small nuclear charges.
Our calculation improves the overall accuracy of theoretical predictions of the $g$ factor of
Li-like ions, especially in the low-$Z$ region. An agreement with previous theoretical
calculations is found. However, the theoretical predictions are shown to systematically deviate
from the experimental results for Li-like silicon and calcium, by approx.~3 and 4 standard
deviations, respectively. The reason for these discrepancies is not known at present, but is
likely to be on the theoretical side. We conclude that further work is needed in order to find
the reasons behind the observed discrepancies.
\begin{acknowledgments}
Work of V.A.Y. is supported by the Russian Science Foundation (Grant No. 20-62-46006). Z.H. and
C.H.K. are supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
Project-ID 273811115 – SFB 1225.
\end{acknowledgments}
| 28,716 |
\section{Introduction} \label{sec:introduction}
Numerical simulations of physical systems have been used for decades (for a review of numerical
methods in statistical physics, see Refs. \cite{landau} and \cite{newman}). In recent
years, however, this method has received a renewed interest, due to the increase
in computational power and, more important, due to the introduction of new algorithms.
The developments in the algorithms have the goal to allow for more efficient simulations,
in many different directions: the calculation of the density of states using flat histograms,
which allows for obtaining information
at any temperature from one single simulation \cite{wang};
the use of bitwise operations and storage, which increases by a great amount the speed of
the simulation and saves memory \cite{penna1990}; and the
introduction of cluster algorithms, which updates collections of spins, decreasing
the autocorrelation time and reducing critical slowing down \cite{landau,newman,
swendsen, wolff}.
The Metropolis algorithm \cite{metropolis}, for example, which had been the main choice of
algorithm for a long time, suffers from a severe critical slowing-down effect near critical points.
This is in part due to the fact that it updates one spin each time (therefore, being in the
general category of single-spin algorithms). Critical slowing down is measured through the
dynamic critical exponent $z$, defined from the dependence of the autocorrelation time $\tau$
on the linear size of the simulated lattice, $L$, at the critical temperature $T_c$. This dependence
is assumed to be in the form $\tau \sim L^z$. Therefore, smaller values of $z$ lead to smaller
autocorrelation times and more efficient algorithms, since more configurations can be used to
calculate the necessary averages. The Metropolis algorithm, for example, when applied to the Ising model in two dimensions,
presents $z \sim 2.16$ \cite{nightingale}.
One possible way to overcome the difficulty of critical slowing down is to design algorithms
which update clusters of spins (the so-called cluster algorithms), that may have a
much lower value of $z$: this is the case for the Swendsen-Wang \cite{swendsen}
and Wolff \cite{wolff} algorithms, for which $z$ may be zero
for the two-dimensional Ising model \cite{baillie,coddington}. See also Ref. \cite{girardi} for another
possible dependence of the autocorrelation time on the lattice size $L$.
An alternative (and generalization) to these last two cluster algorithms, the
Niedermayer algorithm, was introduced some time ago \cite{niedermayer} but,
to the best of our knowledge, has only had his dynamic behavior
studied in detail for the Ising and $XY$ models \cite{girardi}. Our goal in this work is to
study in a systematic way the dynamic behavior of the Niedermayer algorithm, applied to Potts models in two dimensions,
for number of states $q=2$, $3$, and $4$,
for some values of the free parameter $E_0$ (see below), in
order to determine the best choice of this parameter for these models. Note also that the critical temperature
for two-dimensional Potts models are exactly known \cite{wu},
which allows for a more precise determination of $z$.
The Potts model is defined by the Hamiltonian \cite{wu}
\begin{equation}
{\cal H} = -J \sum_{<i,j>} \delta_{S_i,S_j}, \label{hamiltonianapotts}
\end{equation}
where $S_i=1, 2, \ldots, q$, $\forall i$, the sum is over nearest neighbors on a lattice (in our case, a square lattice)
and $\delta_{S_i,S_j}$ is the Kronecker delta ($\delta_{S_i,S_j}=1$, if $S_i=S_j$, and $\delta_{S_i,S_j}=0$,
if $S_i \neq S_j$). We treat the ferromagnetic case in this work, i.e, $J>0$. In two dimensions, the phase transition
for this model is a continuous one for $q \leq 4$. In our study, we restrict ourselves to this interval.
This work is organized as follows: in the next section we present
the Niedermayer algorithm for the Potts model and its relation to Metropolis' and Wolff's. In Section
\ref{sec:timeandexponent} we review some features connected to the autocorrelation
time and the dynamic exponent $z$, in Section \ref{sec:results} we present
and discuss our results, and in the last section we summarize the results.
\section{The Niedermayer algorithm} \label{sec:algorithm}
The Niedermayer algorithm was introduced some time ago as a cluster algorithm,
motivated by the possibility to diminish the value of the dynamic
critical exponent. It may be seen as a generalization of the Wolff or the Swendsen-Wang algorithms.
The main idea is to build a cluster of spins and accept its updating
as a single entity, with a parameter $E_0$ which controls the nature and size of the cluster and its
acceptance ratio, as explained below. In this work,
we have chosen to build the clusters according to the Wolff algorithm (they
can be constructed according to the Swendsen-Wang rule but
the results will not differ qualitatively in two
dimensions and in higher dimensions Wolff algorithm is superior
to Swendsen-Wang's).
In the Niedermayer algorithm, a spin in the lattice, which we will call the {\it seed} spin, is randomly chosen, being the
first spin of the cluster. First-neighbors of this spin may be
considered part of the cluster, with a probability
\begin{equation}
P_{add} (E_{ij}) = \left\{ \begin{array}{lcl}
1-e^{K \left( E_{ij}-E_0 \right) } & , & \mbox{if} \;\; E_{ij}<E_0 \\
0 & , & \mbox{otherwise}
\end{array},
\right. \label{eq:Padd}
\end{equation}
where $K = J/kT$, $T$ is the temperature, $J$ is the exchange constant,
and $E_{ij}$ is the energy between
nearest-neighbor spins in unities of $J$ (i.e, $E_{ij} =- \delta_{ s_i s_j}$). First-neighbors of added spins may be
added to the cluster, according to the probability given above.
Each spin has more than one chance to be part of the cluster, since it
may have more than one first-neighbor in it. When no
more spins can be added, all spins in the cluster have their state changed to a new state
with an acceptance ratio $A$. Assuming that, at the frontier of the cluster
there are $m$ bonds linking spins in the same state \textit{in the old configuration}
and $n$ bonds linking spins in the same state
\textit{in the new configuration} (there are also $p$ bonds in the border of the cluster which
are in different states in the old and in the new configurations, but they cancel
out in the expression below), $A$ satisfies:
\begin{equation}
\frac{A(a\rightarrow b)}{A(b\rightarrow a)} = \left[ e^{ K}
\left( \frac{1-P_{add}(-J)}{1-P_{add}(0)} \right) \right]^{n-m}, \label{eq:acceptanceratio}
\end{equation}
where $a\rightarrow b$ represents the possible updating process,
from the ``old'' ($a$) to the ``new'' ($b$) state, which differ from the
flipping of all spins in the cluster, and $b\rightarrow a$ represents the
opposite move. This expression ensures that detailed balance is satisfied \cite{newman}.
One has to consider three different intervals for $E_0$:
\begin{itemize}
\item [(i)] for $-1 \leq E_0 < 0$, only spins in the same state as the seed
may be added to the cluster, with probability $P_{add} = 1 - e^{-K(1+E_0)}$.
The new state of the cluster is randomly chosen between the remaining $q-1$ states.
The acceptance ratio (Eq. \ref{eq:acceptanceratio}) cannot be chosen to be one
always and is given by $A = e^{-K E_0 (m-n)}$, if $n<m$ (i.e, if the energy
increases when the spins in the cluster are changed), or by $A=1$, if
$n>m$ (i.e, if the energy
decreases when the spins in the cluster are changed). If $E_0=-1$, we obtain
the Metropolis algorithm, since only one-spin clusters are allowed
(according to Eq. (\ref{eq:Padd}), $P_{add}=0$ for $E_0=-1$) and the
acceptance ratio is $A = e^{-K \Delta E}$ for positive $\Delta E$ and $1$
otherwise, where $\Delta E = (m-n)$ is the difference in energy when the spin is
changed, in units of $J$;
\item [(ii)] for $E_0 = 0$, again only spins in the same state can take part of
the cluster, with probability $P_{add} = 1 - e^{-K}$. Now, the acceptance
ratio can be chosen to be $1$, i.e, the cluster of like spins is always
changed. Again, the new state of the cluster is randomly chosen between the remaining $q-1$ states.
This is the celebrated Wolff algorithm;
\item [(iii)] for $E_0 > 0$, spins in different states
may be part of the cluster. Consider a spin already in the cluster: the probability of adding one
of its first-neighbors to the cluster is $P_{add} = 1 - e^{-K(1+E_0)}$ if they are in the
same state or $P_{add} = 1 - e^{-K E_0}$ otherwise.
The acceptance ratio is again always $1$. To change the state, for each cluster, we randomly choose a $\Delta q$ between $1$ and $q-1$ and perform a cyclic sum. Note
that for $E_0 \gg 0$ nearly all spins will be in the cluster and
the algorithm will be clearly inefficient (in fact, it will not be ergodic for
$E_0 \rightarrow \infty$). Therefore, we expect that, if the optimal choice of
$E_0$ is greater than $0$, it will not be much greater than this value.
\end{itemize}
After constructing a cluster, possible updating it and calculating the relevant thermodynamic functions
for the new configuration, the whole process is repeated with a new seed spin. In this way, the Markov
chain of configurations is generated.
\section{Autocorrelation time and dynamic exponent} \label{sec:timeandexponent}
To calculate the relevant averages from a numerical simulation, one has to
build a Markov chain of spin configurations and use data from uncorrelated configurations along this chain.
Therefore, one important quantity is the autocorrelation time for a given quantity, say $\Phi(t)$,
obtained from the autocorrelation function $\rho(t)$:
\begin{eqnarray}
\rho(t) & = & \int \left[ \Phi(t') - <\Phi> \right] \left[ \Phi(t'+t) - <\Phi> \right] dt' \nonumber \\
& = & \int \left[ \Phi(t') \Phi(t'+t) - <\Phi>^2 \right] dt', \label{eq:autocorrelation}
\end{eqnarray}
Since time is a discrete quantity on Monte Carlo simulations, one has to discretize the above equation \cite{newman}:
\begin{eqnarray}
\rho(t) & = & \frac{1}{t_{max}-t} \sum_{t'=0}^{t_{max}-t} \left[ \Phi(t') \Phi(t'+t) \right] - \nonumber \\
& & \frac{1}{(t_{max}-t)^2} \sum_{t'=0}^{t_{max}-t} \Phi(t') \times \sum_{t'=0}^{t_{max}-t} \Phi(t'+t)
\label{eq:discrete}
\end{eqnarray}
It is usually assumed that the autocorrelation function behaves, in its simplest form, as \cite{newman}:
\begin{equation}
\rho(t) = A e^{-t/\tau}, \label{eq:functionoftime}
\end{equation}
This hypothesis has to be corroborated by data and, in some cases, more than one
exponential term is required \cite{wansleben}. One point worth mentioning is that the
autocorrelation function is not well behaved
for long times, due to bad statistics (this is evident from Eq. \ref{eq:discrete},
since few ``measurements'' are available for long times). Then, one has to
choose the region where the
straight line will be adjusted very carefully and it turns out that the
value of $\tau$ so obtained is strongly dependent on this choice. One other
possible way to measure $\tau$ is to integrate $\rho(t)$, assuming
a single exponential dependence on (past and forward) time, and obtain:
\begin{equation}
\tau = \frac{1}{2} \int_{-\infty}^{\infty} \frac{\rho(t)}{\rho(0)} dt, \label{eq:tau}
\end{equation}
with:
\begin{equation}
\rho(t) \equiv e^{-|t|/\tau}.
\end{equation}
Eq. \ref{eq:tau}, when discretized, leads to \cite{salas}:
\begin{equation}
\tau = \frac{1}{2} + \sum_{t=1}^{\infty} \frac{\rho(t)}{\rho(0)}. \label{eq:somadetau}
\end{equation}
The sum in the previous equation cannot be carried out for large values of $t$,
since bad statistics would lead to unreliable results. In order to truncate the sum at some
point, we use a cutoff (see Ref.\cite{salas} and references
therein), defined as the value in time where the noise in the data is clearly greater
than the signal itself. We then obtain a first estimate of $\tau$ using Eq. \ref{eq:somadetau} and
then make the integral of $\rho(t)/\rho(0)$
from the value of the cutoff to infinity. A criterion to accept the cutoff is that the value of this
integral is smaller than the statistical uncertainty in calculating $\tau$.
A note on the definition of ``time'' is worth stressing here. In the Metropolis algorithm,
time is measured in Monte Carlo steps ($MCS$); one $MCS$ is defined as
the attempt to flip $N$ spins, where $N$ is the number of spins in
the (finite) lattice being simulated (in our case, $N=L^2$, where
$L$ is the linear size of the lattice). For cluster algorithms, one unity of time is defined as
the ``time'' taken to build and possibly change a cluster, $t$. A rescaling of the quantity is necessary, to
be able to compare the results for different values of $E_0$, namely:
\begin{equation}
t_{MCS} = t \frac{<n>}{N},
\end{equation}
where $t_{MCS}$ is the time measured in $MCS$ and $<n>$ is the mean cluster
size. Note that, for Metropolis, $<n>=1$ and $1$ $MCS$
is the ``time'' taken to try to flip $N$ spins, as usual. In this work, this rescaling has been
done and all times are expressed in $MCS$.
Our first attempt was to fit the autocorrelation time to the expected behavior, namely
$\tau \sim L^z$, in the critical region, where $z$ is the dynamic exponent. We have taken as
the chosen quantity to calculate $\tau$ the one which better adjusts to Eq. \ref{eq:functionoftime}. Usually,
different quantities lead to autocorrelation functions which behave differently as a function
of time. A typical example
is shown in Fig. \ref{fig:magnatizacaoeenergia}, where both the magnetization
and the energy autocorrelation functions for the $q=3$ Potts model are depicted as functions of time, for the
Niedermayer
algorithm with $E_0=-0.25$ and linear sizes $L=16$ (main graph) and $L=128$ (inset).
The magnetization autocorrelation function presents an abrupt drop for small times and $L=16$.
This shows that this function is not properly described
by a single exponential. On the other hand, the energy autocorrelation time
follows a straight line even for the smallest times. Therefore, we should calculate $\tau$ from the
latter, for $L=16$, using Eq. \ref{eq:somadetau}.
Note, however, that, when $L$ is increased, the picture changes and now the magnetization
autocorrelation function is well described by a single exponential (for small and intermediate values of
time), as depicted in the inset of Fig. \ref{fig:magnatizacaoeenergia}. Whenever a
crossover like this is present, we measure the dynamic exponent from the behavior
for large values of $L$ and for the function which is well described by a single exponential for this
range of $L$, using Eq. \ref{eq:somadetau}. The fact that, for intermediate values of $t$,
the slopes of both curves in Fig. \ref{fig:magnatizacaoeenergia} (main graph and inset) appear to be the same, is
an indication that the autocorrelation times for both are the same.
However, we have already
commented on the drawback of calculating $\tau$ from the slope of the autocorrelation function on a semi-log graph.
As final note, we would like to mention that we used helical boundary conditions and 20 independent runs (each with a
different seed for the pseudo random number generator) were made for each $E_0$ and $L$. For each seed, at least
$4 \times 10^6$ trial changes were made, in order to calculate the autocorrelation functions and their respective autocorrelation
times. The values we quote are the average of the values obtained for each seed of the pseudo random number generator and
the uncertainty in $\tau$ is the standard deviation of these $20$ values.
\section{Results and Discussion} \label{sec:results}
We have simulated the case $E_0=-1$ (Metropolis algorithm) as a test to our code.
The result is presented in Fig. \ref{fig:metropolis} for the magnetization autocorrelation time and,
as we can see, all three cases have the same qualitative behavior. For $q=2$ and $3$ the
dynamic exponent $z$ takes approximately the same value
($z\simeq2.16$), while for $q=4$ it assumes a higher value, namely $z=2.21\pm0.02$. These values
are consistent with those in the literature \cite{binder}.
From now on we will not comment on $q=2$, since this case has been treated in Ref.
\cite{girardi}. Our simulation for $E_0 = -0.75$ is presented in Fig. \ref{fig:tao-0,75}, where the magnetization and
energy autocorrelation times are depicted as functions of $L$. For $q=3$ and $L \geq 32$ and for $q=4$ and
$L \geq 64$, the dynamic behavior (i.e., $z$) is the same as for the Metropolis algorithm.
In Figs. \ref{fig:tao-0,75}$b$ ($q=3$) and \ref{fig:tao-0,75}$d$
($q=4$) we present the
behavior of the average cluster size $<n>$ versus lattice size $L$: in both cases, for $L \geq \tilde{L} = 64$,
$<n>$ is constant. As
already discussed, the autocorrelation function which is well described by only one exponential presents the
greatest autocorrelation time and for, $E_0=-0.75$, this happens for the magnetization's autocorrelation function.
The dynamic exponent is calculated for $L \geq \tilde{L}$ and we obtain $z=2.16 \pm 0.05$ and
$z= 2.18 \pm 0.09$ for $q=3$ and $q=4$, respectively. These results agree, within error bars, with the values
quoted in the literature \cite{binder} and with our values obtained for $E_0=-1$.
For $E_0=-0.25$, our result is depicted in Fig. \ref{fig:tao-0,25} for $q=3$ ( the result $q=4$ is qualitatively the same);
the behavior follows the same overall trend observed for $E_0=-0.75$, with a
different value of $\tilde{L}$. For $L<128$ the autocorrelation time for the energy is greater than for the magnetization and the average cluster size $<n>$ increases with $L$. For $L \geq \tilde{L} = 128$, the picture changes and the autocorrelation time for the magnetization is the greater one and $<n>$ is constant. The value of $z$ is consistent with the one for the Metropolis algorithm (again, $z$ is calculated for $L\geq \tilde{L}$).
For other values of $-1\leq E_0 < 0 $ we observe the same qualitative behavior. There is always a $\tilde{L}$, such that,
for $L<\tilde{L}$, $<n>$ increases with $L$ and, for $L>\tilde{L}$, $<n>$ is constant and the magnetization's autocorrelation
time is the one well described by a single exponential. The exponent $z$, always calculated for $L \geq \tilde{L}$, is the
same as for the Metropolis algorithm. This can be linked to the constancy of $<n>$ in this interval: since $L$ increases and $<n>$
remains the same, the fraction $<n>/L^2$ decreases and the behavior is the same as a single-spin algorithm.
We simulated the Wolff algorithm to perform another check of our algorithm and to compare with our results
for $E_0 \neq 0$. In Fig. \ref{fig:wolff} we present the autocorrelation time and $<n>$ versus $L$ for $q=3$ and $q=4$. To
calculate $z$ we used a different approach here \cite{girardi1,picco}.
We perform a power-law fitting using three consecutive lattice size
(e.g $L=512$, $1024$, and $2048$) and call $L_{min}$ the smallest size. We then plot $z$ versus $1/L_{min}$,
as seen in Fig. \ref{fig:picco} \cite{girardi1,picco}. The power-law fitting for $q=3$ is a good fit for all lattice sizes
and we obtain $z=0.55\pm0.02$, which
agrees with a previous estimate \cite{coddington2}. For $q=4$, only for $L \geq 128$ we obtain a good fit, with $z=1.00\pm0.02$.
This value is somewhat above a previous calculation \cite{li} for the Swendsen-Wang algorithm. If we restrict our data to
$L \leq 256$, as in the previous reference, we obtain $z=0.94 \pm 0.01$, which just overlaps with the result
of Ref. \cite{li} (namely, $z = 0.89 \pm 0.05$).
We plot the data for $q=2$ for comparison with our results for $q=3$ and $4$:
it seems clear that, for that value of $q$, the asymptotic regime has not yet been reached, for the lattice sizes
we simulated (in fact, it is not clear if there is an asymptotic regime) \cite{girardi}.
We would like to note that a good indication of the result quality is the relation $<n> \propto L^{\gamma/\nu}$.
As we can see in Figs.
\ref{fig:wolff}$b$ and \ref{fig:wolff}$d$, our estimates of $\gamma/\nu$ from a log-log plot of
$<n>$ vs. $L$ (namely, $\gamma/\nu = 1.734 \pm 0.001$ and $1.7498 \pm 0.0009$ for $q=3$ and $4$,
respectively)
are, within error bars, the same of the (conjectured) exact results (namely, $\gamma/\nu=26/15=1.73333....$
for $q=3$ \cite{wu} and $\gamma/\nu=7/4=1.75$ for $q=4$ \cite{wu,creswick}).
In order to compare our results for different values of $E_0$, we have plotted $z$ vs. $E_0$ for both $q=3$ and
$q=4$ (see Fig. \ref{fig:zxE0}): we see that $z$ is approximately constant for any $E_0 \neq 0$ and strongly decreases
for $E_0=0$. In fact, the subtle decrease of $z$ as $E_0$ approaches $0$ may be a crossover effect: the closer
we get to the Wolff algorithm, the greater the value of $\tilde{L}$ and one has to go to larger and larger lattices
to obtain the correct dynamic behavior.
Therefore, we have shown that the Wolff algorithm is still the most efficient procedure, when compared to
the generalizations for $E_0<0$. But we still have to check the dynamic behavior for $E_0>0$.
We know that for $E_0 \rightarrow \infty$
the algorithm is not ergodic. So we expect that, if a better value for $E_0$ exists, when compared to $E_0=0$,
it is not much greater than this last value. To address this
question we simulate the cases $E_0=0.05$ and $0.1$.
In Fig. \ref{fig:grande} we show the results for $q=3$ (a)
and $q=4$ (b), both calculated from the energy autocorrelation function.
As we can see, for $q=3$ the autocorrelation time $\tau$ for $E_0=0.05$ is much greater than for
$E_0=0$ and grows faster than for the latter. For $q=4$, $\tau$ is slightly
greater for $E_0=0.05$ than for $E_0=0$, although our result is consistent with the same value of $z$ for both cases. However,
one has to consider that the implementation of the algorithm for $E_0=0.05$ is more complex than for $E_0=0$.
\section{Conclusion} \label{sec:summary}
In this work we studied the Niedermayer algorithm applied to the two-dimensional Potts model with 2, 3, and
4 states. Our goal was
to determine which value of $E_0$ leads to the optimal algorithm, i.e.,
to the smallest value of the dynamic exponent $z$. We observe that for $-1 \leq E_0 < 0$ there is a lattice size $\tilde{L}$,
such that, for $L \geq \tilde{L}$, the average size of updates clusters is constant and the dynamic behavior of the algorithm is the same as
for Metropolis'. The value of $\tilde{L}$ increases with $E_0$ and diverges for
$E_0=0$ (Wolff algorithm). When we look to the auto-correlation function, we notice that for $L < \tilde{L}$ the auto-correlation
time of the energy is greater than for the magnetization and the opposite happens for $L \geq \tilde{L}$. As we show in Fig. \ref{fig:magnatizacaoeenergia}, the quantity with greater autocorrelation time have the auto-correlation function
well described by a single exponential. For $E_0=0$ we regain the Wolff algorithm.
In Table 1 we summarize our findings,
which show that the Wolff algorithm, $E_0=0$, is more efficient than its generalization for $E_0<0$.
There is still the possibility that some value of $E_0> 0$ may present a lower value of $z$, when
compared to $E_0=0$. We show that, if this value exists, it is lower than $E_0<0.05$ and the complexity
of the algorithm is greater than the improvement in the dynamic behavior.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|} \hline
q$\;$ & $-1\leq E_0 <0$ &Wolff ($E_0=0$)\\ \hline
2 & 2.16(1) & undefined\\ \hline
3 & 2.162(7) & 0.55(2) \\\hline
4 & 2.21(2) & 1.00(2) \\\hline
\end{tabular}
\end{center}
\label{tab:resultado}
\caption{ Values for the dynamic exponent $z$ for the three models studied here and for $-1 \le E_0 \leq 0$.}
\end{table}
\section*{Acknowledgments}
The authors would like to thank the Brazilian agencies
FAPESC, CNPq, and CAPES for partial financial support.
\bibliographystyle{elsarticle-num}
| 7,697 |
\section{Introduction}
Exchanging data objects with untrusted code is a delicate matter because
of the risk of creating a data space that is accessible by an attacker.
Consequently, secure programming guidelines for Java such as those proposed by
Sun \cite{SunGuidelines:2010} and CERT \cite{CertGuidelines:2010} stress the
importance of using defensive \emph{copying} or \emph{cloning} before
accepting or handing out references to an internal mutable object. There are
two aspects of the problem:
\begin{enumerate}[(1)]
\item If the result of a method is a reference to an internal mutable object,
then the receiving code may modify the internal state. Therefore, it is
recommended to make copies of mutable objects that are returned as
results, unless the intention is to share state.
\item If an argument to a method is a reference to an object coming
from hostile code, a local copy of the object should be
created. Otherwise, the hostile code may be able to modify the internal
state of the object.
\end{enumerate}
\noindent A common way for a class to provide facilities for copying objects is
to implement a \ttt{clone()} method that overrides the cloning
method provided by \ttt{java.lang.Object}.
The following code snippet, taken from Sun's Secure Coding
Guidelines for Java, demonstrates how a \ttt{date} object is cloned
before being returned to a caller:
\begin{lstlisting}
public class CopyOutput {
private final java.util.Date date;
...
public java.util.Date getDate() {
return (java.util.Date)date.clone(); }
}
\end{lstlisting}
\noindent However, relying on calling a polymorphic \ttt{clone} method
to ensure secure copying of objects may prove insufficient, for two
reasons. First, the implementation of the \ttt{clone()} method is
entirely left to the programmer and there is no way to enforce that an
untrusted implementation provides a sufficiently \emph{deep} copy of
the object. It is free to leave references to parts of the original
object being copied in the new object. Second,
even if the current \ttt{clone()} method works properly, sub-classes may
override the \ttt{clone()} method and replace it with a method that does not
create a sufficiently deep clone. For the above example to behave correctly,
an additional class invariant is required, ensuring that the \ttt{date} field
always contains an object that is of class \ttt{Date} and not one of its
sub-classes. To quote from the CERT guidelines for secure Java programming:
\emph{``Do not carry out defensive copying using the clone() method in
constructors, when the (non-system) class can be subclassed by untrusted code.
This will limit the malicious code from returning a crafted object when the
object's clone() method is invoked.''} Clearly, we are faced with a situation
where basic object-oriented software engineering principles (sub-classing and
overriding) are at odds with security concerns.
To reconcile these two aspects in a manner that provides semantically
well-founded guarantees of the resulting code, this paper proposes a formalism
for defining \emph{cloning policies} by annotating classes and specific copy
methods, and a static enforcement mechanism that will guarantee that all
classes of an application adhere to the copy policy. Intuitively,
policies impose non-sharing constraints between the structure
referenced by a field of an object and the structure returned by the
cloning method. Notice, that we do not enforce that
a copy method will always return a target object that is functionally
equivalent to its source. Nor does our method prevent a sub-class from
making a copy of a structure using new fields that are not governed by
the declared policy. For a more detailed example of these limitations,
see Section~\ref{sec:pol:limitations}.
\subsection{Cloning of Objects}
\label{sec:intro:cloning}
For objects in Java to be cloneable, their class must implement the
empty interface \ttt{Cloneable}. A default \ttt{clone} method is
provided by the class \ttt{Object}: when invoked on an object of a
class, \ttt{Object.clone} will create a new object of that class and
copy the content of each field of the original object into the new
object. The object and its clone share all sub-structures of the
object; such a copy is called \textit{shallow}.
It is common for cloneable classes to override the default clone
method and provide their own implementation. For a generic
\ttt{List} class, this could be done as follows:
\begin{lstlisting}
public class List<V> implements Cloneable
{
public V value;
public List<V> next;
public List(V val, List<V> next) {
this.value = val;
this.next = next; }
public List<V> clone() {
return new List(value,(next==null)?null:next.clone()); }
}
\end{lstlisting}
Notice that this cloning method performs a shallow copy of the list,
duplicating the spine but sharing all the elements between the list and its
clone. Because this amount of sharing may not be desirable (for the reasons
mentioned above), the programmer is free to implement other versions of
\ttt{clone()}. For example, another way of cloning a list is by copying both
the list spine and its elements\footnote{To be type-checked by the Java
compiler it is necessary to add a cast before calling \ttt{clone()} on
\ttt{value}. A cast to a sub interface of \ttt{Cloneable} that declares a
\ttt{clone()} method is necessary.}, creating what is known as a \textit{deep}
copy.
\begin{lstlisting}
public List<V> deepClone() {
return new List((V) value.clone(),
(next==null ? null : next.deepClone())); }
\end{lstlisting}
\noindent A general programming pattern for methods that clone objects works by first
creating a shallow copy of the object by calling the \ttt{super.clone()}
method, and then modifying certain fields to reference new copies of the original
content. This is illustrated in the following snippet, taken from the class
\ttt{LinkedList} in Fig.~\ref{fig:linkedlist}:
\begin{lstlisting}
public Object clone() { ...
clone = super.clone(); ...
clone.header = new Entry<E>(null, null, null); ...
return clone;}
\end{lstlisting}
\noindent
There are two observations to be made about the analysis of such methods.
First, an analysis that tracks the depth of the clone being returned will have
to be flow-sensitive, as the method starts out with a shallow copy that is
gradually being made deeper. This makes the analysis more costly. Second,
there is no need to track precisely modifications made to parts of the memory
that are not local to the clone method, as clone methods are primarily
concerned with manipulating memory that they allocate themselves. This
will have a strong impact on the design choices of our analysis.
\subsection{Copy Policies}
\label{sec:intro:pol}
The first contribution of the paper is a proposal for a set of semantically
well-defined program annotations, whose purpose is to enable the expression of
policies for secure copying of objects. Introducing a copy policy language
enables class developers to state explicitly the intended behaviour of copy
methods.
In the basic form of the copy policy formalism, fields of classes are
annotated with \ttt{@Shallow} and \ttt{@Deep}. Intuitively, the annotation
\ttt{@Shallow} indicates that the field is referencing an object, parts of which
may be referenced from elsewhere. The annotation \ttt{@Deep}(\ttt{X}) on a
field \ttt{f} means
that
\begin{inparaenum}[\itshape a\upshape)]
\item upon return from \ttt{clone()}, the object referenced by this field \ttt{f} is not referenced
from elsewhere, and
\item the field \ttt{f} is copied according to the copy policy identified by \ttt{X}.
\end{inparaenum}
Here, \ttt{X} is either the name of a specific policy or if omitted,
it designates the default policy of the class of the field.
For example, the following annotations:
\begin{lstlisting}
class List { @Shallow V value; @Deep List next; ...}
\end{lstlisting}
specifies a default policy for the class \ttt{List} where the \ttt{next} field
points to a list object that also respects the default copy policy for
lists. Any method in the \ttt{List} class, labelled with the \ttt{@Copy}
annotation, is meant to respect this default policy.
In addition it is possible to define other copy policies and annotate specific
\emph{copy methods} (identified by the annotation \ttt{@Copy(...)}) with the name
of these policies.
For example, the annotation\footnote{Our implementation uses a sightly
different policy declaration syntax because of the limitations imposed by the
Java annotation language.}
\begin{lstlisting}
DL: { @Deep V value; @Deep(DL) List next;};
@Copy(DL) List<V> deepClone() {
return new List((V) value.clone(),
(next==null ? null : next.deepClone())); }
\end{lstlisting}
can be used to specify a list-copying method that also ensures that
the \ttt{value} fields of a list of objects are copied according to
the copy policy of their class (which is a stronger policy than
that imposed by the annotations of the class \ttt{List}). We give a
formal definition of the policy annotation language in
Section~\ref{section-annotations}.
The annotations are meant to ensure a certain degree of non-sharing between
the original object being copied and its clone. We want to state
explicitly that the parts of the clone that can be accessed via fields marked \ttt{@Deep}
are unreachable from any part of the heap that was accessible before the call
to \ttt{clone()}. To make this intention precise, we provide a formal
semantics of a simple programming language extended with policy annotations
and define what it means for a program to respect a policy
(Section~\ref{sec:policysemantics}).
\subsection{Enforcement}
\label{sec:intro:typ}
The second major contribution of this work is to make the developer's intent,
expressed by copy policies, statically enforceable using a type system. We
formalize this enforcement mechanism by giving an interpretation of the policy
language in which annotations are translated into graph-shaped type
structures. For example, the default annotations of the \ttt{List} class defined above
will be translated into the graph that is depicted to the right in
Fig.~\ref{fig:listab} (\ttt{res} is the name given to
the result of the copy method). The left part shows the concrete heap structure.
Unlike general purpose shape analysis, we take into account the programming
methodologies and practice for copy methods, and design a type system
specifically tailored to the enforcement of copy policies. This means that
the underlying analysis must be able to track precisely all modifications to
objects that the copy method allocates itself (directly or indirectly) in a
flow-sensitive manner. Conversely, as copy methods should not modify
non-local objects, the analysis will be designed to be more approximate when
tracking objects external to the method under analysis, and the type system
will accordingly refuse methods that attempt such non-local modifications. As
a further design choice, the annotations are required to be verifiable
modularly on a class-by-class basis without having to perform an analysis of
the entire code base, and at a reasonable cost.
\begin{figure}
\centering
\includegraphics[width=.8\linewidth]{listalpha}
\caption{A linked structure (left part) and its abstraction (right part).}
\label{fig:listab}
\end{figure}
As depicted in Fig.~\ref{fig:listab}, concrete memory cells
are either abstracted as
\begin{inparaenum}[\itshape a\upshape)]
\item $\topout$ when they are not allocated in the copy method itself (or its callee);
\item $\top$ when they are just marked as \emph{maybe-shared}; and
\item circle nodes of a deterministic
graph when they are locally allocated and not shared. A single
circle furthermore expresses a singleton concretization.
\end{inparaenum}
In this example, the abstract heap representation matches the graph
interpretation of annotations, which means that the instruction set that
produced this heap state satisfies the specified copy policy.
Technically, the intra-procedural component of our analysis corresponds to
heap shape analysis with the particular type of graphs that we have defined.
Operations involving non-local parts of the heap are rapidly discarded.
Inter-procedural analysis uses the signatures of copy methods provided by the
programmer. Inheritance is dealt with by stipulating that inherited fields
retain their ``shallow/deep'' annotations. Redefinition of a method must
respect the same copy policy and other copy methods can be added to a
sub-class.
The detailed definition of the analysis, presented as a set of type inference
rules, is given in Section~\ref{sec:type-system}.
This article is an extended version of a paper presented at ESOP'11~\cite{JensenKP:Esop11}.
We have taken advantage of the extra space to provide improved and more detailed
explanations, in particular of the inference mechanism and of what is
exactly is being enforced by our copy policies. We have also added
details of the proof of correctness of the enforcement
mechanism. The formalism of copy policies and the correctness theorem
for the core language defined in Section~\ref{section-annotations} have been
implemented and verified mechanically in Coq~\cite{clone-webpage}.
The added details about the proofs should especially facilitate the
understanding of this Coq development
\section{Language and Copy Policies}\label{section-annotations}
\begin{figure}
\centering
\begin{small}
\begin{frameit}
\[
\var{x},\var{y} \in \Var \qquad
\var{f} \in \Field \qquad
\var{m} \in \Meth \qquad
\var{cn} \in \ClassName \qquad
X \in \Polid
\]
\[
\begin{array}{rrrrl}\\
p&\in&\prog & \dasig & \overline{cl} \\
\mathit{cl}&\in&\Class & \dasig & \classkeyword~cn~[\extendskeyword~cn]~\{ \overline{\cp}~ \overline{\md} \} \\
\cp &\in& \CopySignature &\dasig& X : \{\tau\} \\
\tau&\in&\Pol & \dasig & \overline{(X,f)} \\%(X,f) ~; ~\ldots~ \\%
\md&\in&\MethDecl & \dasig & \Copy(X)~\HAssign{m(x)}{}c \\
c&\in&\comm & \dasig &\HAssign{x}{y}
\mid \HAssign{x}{y.f}
\mid \HAssign{x.f}{y}
\mid \HAssign{x}{\snull}\\
&&& & \mid \HNew{x}{cn}
\mid \HCall{x}{\var{cn}:X}{y} \mid \UnkownCall{x}{y} \mid \HReturn{x} \\
&&& & \mid c;c
\mid \HIf{c}{c}
\mid \HWhile{c}\\
\end{array}
\]
\\[2ex]
\begin{minipage}[t]{.9965\linewidth}
\textbf{Notations:} We write $\preceq$ for the reflexive transitive closure
of the subclass relation induced by a (well-formed) program that is fixed in
the rest of the paper. We write $\overline{x}$ a sequence of syntactic
elements of form $x$.
\end{minipage}
\end{frameit}
\end{small}
\caption{Language Syntax.}
\label{fig:syntax}
\end{figure}
The formalism is developed for a small, imperative language extended with
basic, class-based object-oriented features for object allocation, field
access and assignment, and method invocation. A program is a collection of
classes, organized into a tree-structured class hierarchy via the
$\extendskeyword$ relation. A class consists of a series of copy method
declarations with each its own policy $X$, its name $m$, its formal parameter $x$
and commands $c$ to execute. A sub-class inherits the copy methods of its
super-class and can re-define a copy method defined in one of its
super-classes. We only consider copy methods. Private methods (or static
methods of the current class) are inlined by the type checker. Other method
calls (to virtual methods) are modeled by a special instruction
$\UnkownCall{x}{y}$ that assigns an arbitrary value to $\var{x}$ and possibly
modifies all heap cells reachable from $\var{y}$ (except itself).
The other commands are standard.
The copy method call $\HCall{x}{\var{cn}:X}{y}$ is a virtual call. The method
to be called is the copy method of name $m$ defined or inherited by the
(dynamic) class of the object stored in variable $y$. The subscript annotation
$\var{cn{:}X}$ is used as a static constraint. It is supposed that the type
of $y$ is guaranteed to be a sub-class of class $cn$ and that $cn$ defines a
method $m$ with a copy policy $X$. This is ensured by standard bytecode
verification and method resolution.
We suppose given a set of policy identifiers $\Polid$, ranged over by $X$. A
copy policy declaration
has the form $ X : \{\tau\}$ where $X$ is the identifier of the policy
signature and $\tau$ is a policy. The policy $\tau$ consists of a set of
field annotations $(X,f) ~; ~\ldots$ where $f$ is a \emph{deep} field that
should reference an object which can only be accessed via the returned pointer
of the copy method and which respects the copy policy identified by $X$.
The use of policy identifiers makes it possible to write recursive definitions
of copy policies, necessary for describing copy properties of recursive
structures.
Any other field is implicitly \emph{shallow}, meaning that no copy properties are
guaranteed for the object referenced by the field. No further copy
properties are given for the sub-structure starting at \emph{shallow} fields.
For instance, the default copy policy declaration of the class \ttt{List} presented in
Sec.~\ref{sec:intro:pol} writes: $\mathtt{List.default}:~\{ (\mathtt{List.default},\mathtt{next}) \}$.
We assume that for a given program, all copy policies have been grouped
together in a finite map $\Pi_p:\Polid\to\Pol$. In the rest of the paper, we
assume this map is complete, \emph{i.e.} each policy name $X$ that appears in
an annotation is bound to a unique policy in the program $p$.
The semantic model of the language defined here is store-based: $$
\begin{small} \begin{array}{rrlcl} l &\in& \Loc \\ v &\in& \Val &=& \Loc \cup
\{ \vnull \} \\ \rho &\in& \Env &=& \Var \to \Val \\ o &\in& \Object &=&
\Field \to \Val \\ h &\in& \Heap &=& \Loc \ptofin \left( \ClassName \times
\Object\right) \\ \st{\rho,h,A}&\in& \State &=&
\Env\times\Heap\times\Power(\Loc) \end{array} \end{small} $$
A program state consists of an environment $\rho$ of local variables,
a store $h$ of locations mapping\footnote{We note $\ptofin$ for
partial functions on finite domains.} to objects in a heap and a set
$A$ of \emph{locally allocated locations}, \emph{i.e.}, the locations
that have been allocated by the current method invocation or by one of its
callees. This last component does not influence the semantic
transitions: it is used to express the type system interpretation
defined in Sec.~\ref{sec:type-system}, but is not used in the final
soundness theorem. Each object is modeled in turn as a pair composed
with its dynamic class and a finite function from field names to values
(references or the specific \vnull\ reference for null values). We do
not deal with base values such as integers because their immutable
values are irrelevant here.
\begin{figure*}
{\centering\scriptsize
$$
\begin{array}{c}
\inferrule
{~~}
{\left(\HAssign{x}{y},\stb{\rho,h,A}\right) \leadsto \str{\rho[x\mapsto\rho(y)],h,A}}
\qquad
\inferrule
{~~}
{\left(\HAssign{x}{\snull},\stb{\rho,h,A}\right) \leadsto \str{\rho[x\mapsto\vnull],h,A}}
\\[3ex]
\inferrule
{\rho(y) \in \dom(h)}
{\left(\HAssign{x}{y.f},\stb{\rho,h,A}\right) \leadsto \str{\rho[x\mapsto h(\rho(y),f)],h,A}}
\qquad
\inferrule
{\rho(x) \in \dom(h)}
{\left(\HAssign{x.f}{y},\stb{\rho,h,A}\right) \leadsto \str{\rho,h[(\rho(x),f)\mapsto \rho(y)],A}}
\\[3ex]
\inferrule
{l \not\in \dom(h)}
{\left(\HNew{x}{\var{cn}},\stb{\rho,h,A}\right) \leadsto \str{\rho[x\mapsto l],h[l \mapsto (\var{cn},o_{\vnull})],A\cup\{l\}}}
\\[1ex]
\inferrule
{~
{\left(\HReturn{x},\stb{\rho,h,A}\right) \leadsto \str{\rho[\var{ret}\mapsto\rho(x)],h,A}}
\\[3ex]
\inferrule
{\begin{array}{c}
h(\rho(y))=(\var{cn}_y,\underscore) \quad \lookup(\var{cn}_y,m) = \left(\Copy(X')~\HAssign{m(a)}c\right)
\quad \var{cn}_y \preceq \var{cn} \cr
(c,\stb{\rho_{\vnull}[a\mapsto\rho(y)],h,\emptyset}) \leadsto \str{\rho',h',A'} \cr
\end{array}}
{\left(\HCall{x}{\var{cn}:X}{y},\stb{\rho,h,A}\right) \leadsto \str{\rho[x\mapsto \rho'(\ret)],h',A\cup A'}}
\\[3ex]
\inferrule
{
\begin{array}[c]{c}
\dom(h)\subseteq\dom(h')\quad \forall l\in\dom(h)\setminus\Reach{h}{\rho(y)},~ h(l)=h'(l) \cr
\forall l\in\dom(h)\setminus\Reach{h}{\rho(y)},~
\forall l'\in\dom(h'), l\in\Reach{h'}{l'} \Rightarrow l'\in\dom(h)\setminus\Reach{h}{\rho(y)} \cr
v \in \{\vnull\}+\Reach{h}{\rho(y)}\cup(\dom(h')\setminus\dom(h))
\end{array}
}
{\left(\UnkownCall{x}{y},\stb{\rho,h,A}\right) \leadsto \str{\rho[x\mapsto v],h',A \backslash \Reachp{h}{\rho(y)}}}\\[3ex]
\inferrule
{{\left(c_1,\stb{\rho,h,A}\right) \leadsto \str{\rho_1,h_1,A_1}}\quad
{\left(c_2,\stb{\rho_1,h_1,A_1}\right) \leadsto \str{\rho_2,h_2,A_2}}}
{\left(c_1; c_2,\stb{\rho,h,A}\right) \leadsto \str{\rho_2,h_2,A_2}}\\[3ex]
\inferrule
{\left(c_1,\stb{\rho,h,A}\right) \leadsto \str{\rho_1,h_1,A_1}}
{\left(\HIf{c_1}{c_2},\stb{\rho,h,A}\right) \leadsto \str{\rho_1,h_1,A_1}}\qquad
\inferrule
{\left(c_2,\stb{\rho,h,A}\right) \leadsto \str{\rho_2,h_2,A_2}}
{\left(\HIf{c_1}{c_2},\stb{\rho,h,A}\right) \leadsto \str{\rho_2,h_2,A_2}}\\[3ex]
\inferrule{~~}
{\left(\HWhile{c},\stb{\rho,h,A}\right) \leadsto \str{\rho,h,A}}\qquad
\inferrule
{\left(c; \HWhile{c},\stb{\rho,h,A}\right) \leadsto \str{\rho',h',A'}}
{\left(\HWhile{c},\stb{\rho,h,A}\right) \leadsto \str{\rho',h',A'}}
\end{array}
$$}\vspace{3 pt}
\begin{minipage}[t]{.9965\linewidth}\footnotesize
\textbf{Notations:} We write $h(l,f)$ for the value $o(f)$ such that
$l\in\dom(h)$ and $h(l)=o$. We write $h[(l,f)\mapsto v]$ for the heap $h'$
that is equal to $h$ except that the $f$ field of the object at location $l$
now has value $v$. Similarly, $\rho[x\mapsto v]$ is the environment $\rho$
modified so that $x$ now maps to $v$. The object $o_\vnull$ is the object
satisfying $o_\vnull(f)=\vnull$ for all field $f$, and $\rho_\vnull$ is the
environment such that $\rho_\vnull(x)=\vnull$ for all variables $x$. We
consider methods with only one parameter and name it $p$. $\lookup$
designates the dynamic lookup procedure that, given a class name $\var{cn}$
and a method name $m$, find the first implementation of $m$ in the class
hierarchy starting from the class of name $\var{cn}$ and scanning the
hierarchy bottom-up. It returns the corresponding method declaration.
\var{ret} is a specific local variable name that is used to store the result
of each method. $\Reach{h}{l}$ (resp. $\Reachp{h}{l}$) denotes the set of
values that are reachable from any sequence (resp. any non-empty sequence) of
fields in $h$.
\end{minipage}
\caption{Semantic Rules.}\label{fig:semrules}
\end{figure*}
The operational semantics of the language is defined (Fig.~\ref{fig:semrules})
by the evaluation relation $\leadsto$ between configurations $\comm \times
\State$ and resulting states $\State$. The set of locally allocated locations
is updated by both the $\HNew{x}{\var{cn}}$ and the $\HCall{x}{\var{cn}:X}{y}$
statements. The execution of an unknown method call $\UnkownCall{x}{y}$
results in a new heap $h'$ that keeps all the previous objects that were not
reachable from $\rho(y)$. It assigns the variable $x$ a reference that was
either reachable from $\rho(y)$ in $h$ or that has been allocated during this
call and hence not present in $h$.
\subsection{Policies and Inheritance}
We impose restrictions on the way that inheritance can interact with
copy policies. A method being re-defined in a sub-class can impose
further constraints on how fields of the objects returned as
result should be copied. A field already annotated \emph{deep} with policy $X$ must have
the same annotation in the policy governing the re-defined
method but a field annotated as \emph{shallow} can be annotated \emph{deep}
for a re-defined method.
\begin{defi}[Overriding Copy Policies]
A program $p$ is well-formed with respect to overriding copy policies if and
only if for any method declaration
$\Copy(X')~\HAssign{m(x)}{}\ldots$
that overrides (\emph{i.e.} is declared with this signature in a subclass
of a class $\mathit{cl}$) another method declaration
$\Copy(X)~\HAssign{m(x)}{}\ldots$ declared in $\mathit{cl}$,
we have
\[\Pi_p(X) \subseteq \Pi_p(X').\]
\end{defi}
\noindent
Intuitively, this definition imposes that the overriding copy policy is stronger than the
policy that it overrides.
Lemma~\ref{lem:mono} below states this formally.
\begin{exa}
The \ttt{java.lang.Object} class provides a \ttt{clone()} method of policy
$\{\}$ (because its native \ttt{clone()} method is \emph{shallow} on all
fields). A class $\mathtt{A}$ declaring two fields $\mathtt{f}$ and
$\mathtt{g}$ can hence override the \ttt{clone()} method and give it a
policy $\{(X,\mathtt{g})\}$. If a class $\mathtt{B}$ extends $\mathtt{A}$ and
overrides \ttt{clone()}, it must assign it a policy of the form
$\{(X,\mathtt{g});~ \ldots ~\}$ and could declare the field $\mathtt{f}$ as
\emph{deep}. In our implementation, we let the programmer leave the policy
part that concerns fields declared in superclasses implicit, as it is
systematically inherited. \end{exa}
\subsection{Semantics of Copy Policies}
\label{sec:policysemantics}
The informal semantics of the copy policy annotation of a method is:
\begin{quote}
A copy method satisfies a copy policy $X$ if and only if no memory cell that is
reachable from the result of this method following only fields with
\emph{deep} annotations in $X$, is reachable from another local variable of the
caller.
\end{quote}
We formalize this by giving, in Fig.~\ref{fig:annot:semantics}, a
semantics to copy policies based on access paths. An access path
consists of a variable $x$ followed by a sequence of field names $f_i$
separated by a dot. An access path $\pi$ can be evaluated to a value
$v$ in a context $\st{\rho,h}$ with a judgement
$\evalexpr{\rho}{h}{\pi}{v}$. Each path $\pi$ has a root variable
$\proot{\pi}\in\Var$. A judgement $\vdash \pi:\tau$ holds when a path
$\pi$ follows only deep fields in the policy $\tau$. The rule
defining the semantics of copy policies can be paraphrased as follows:
For any path
$\pi$ starting in $x$ and leading to location $l$ only following deep
fields in policy $\tau$, there cannot be another path leading to the
same location $l$ which does not start in $x$.
\begin{figure}
\scriptsize\centering
\begin{minipage}[t]{.9965\linewidth}
\bf Access path syntax
\end{minipage}
$\begin{array}{rrrrl}
\pi & \in &\AccessPath & \dasig & x \mid \pi.f
\end{array}
$
\begin{minipage}[t]{.9965\linewidth}
\bf Access path evaluation
\end{minipage}
\inferrule{}{\evalexpr{\rho}{h}{x}{\rho(x)}}
\hspace*{5ex}\inferrule
{\evalexpr{\rho}{h}{\pi}{l} \quad h(l) = o}
{\evalexpr{\rho}{h}{\pi.f}{o(f)}}
$
\begin{minipage}[t]{.9965\linewidth}
\bf Access path root
\end{minipage}
$\proot{x}=x \hspace*{5ex} \proot{\pi.f} = \proot{\pi}$
\begin{minipage}[t]{.9965\linewidth}
{\bf Access path satisfying a policy}\\
We suppose given $\Pi_p:\Polid\to\Pol$ the set of copy policies of the
considered program $p$.
\end{minipage}
$\inferrule{~}{\vdash x:\tau}\hspace*{5ex}
\inferrule
{(X_1~f_1)\in\tau, (X_2~f_2)\in\Pi_p(X_1), \cdots, (X_n~f_n)\in\Pi_p(X_{n-1})
}
{\vdash x.f_1.\ldots.f_n : \tau}
$
\begin{minipage}[t]{.9965\linewidth}
{\bf Policy semantics}
\end{minipage}
$
\inferrule
{\left.\begin{array}{rl}
\forall \pi, \pi'\in\AccessPath, \forall l,l'\in\Loc, &x=\proot{\pi},\quad \proot{\pi'}\not=x,\cr
&\evalexpr{\rho}{h}{\pi}{l}~,\quad \evalexpr{\rho}{h}{\pi'}{l'}, \cr
&\vdash \pi : \tau
\end{array}\right\}
\text{implies}~ l\not= l'
}
{\rho,h,x \models \tau}
$
\caption{Copy Policy Semantics}\label{fig:annot:semantics}
\end{figure}
\begin{defi}[Secure Copy Method]
\label{def:secure-copy-method}
A method $m$ is said \emph{secure} wrt. a copy signature $\Copy(X)\{\tau\}$ if
and only if for all heaps $h_1,h_2\in\Heap$, local environments
$\rho_1,\rho_2\in\Env$, locally allocated locations $A_1,A_2\in\Power(\Loc) $, and variables $x,y\in\Var$,
$$
(\HCall{x}{\var{cn}:X}{y}, \st{\rho_1,h_1,A_1}) \leadsto \st{\rho_2,h_2,A_2}
~~\text{implies}~~\rho_2,h_2,x \models \tau$$
\end{defi}
\noindent Note that because of virtual dispatch, the method executed
by such a call may not be the method found in $\var{cn}$ but an
overridden version of it. The security policy requires that all
overriding implementations still satisfy the policy $\tau$.
\begin{lem}[Monotonicity of Copy Policies wrt. Overriding]\label{lem:mono}
$$
\tau_1\subseteq\tau_2 ~\text{implies}~
\forall h,\rho,x,~~ \rho,h,x \models \tau_2 ~\Rightarrow \rho,h,x \models \tau_1
$$
\end{lem}
\begin{proof}
[See Coq proof \texttt{Overriding.copy\_policy\_monotony}~\cite{clone-webpage}]
Under these hypotheses, for all access paths $\pi$, $\vdash\pi:\tau_1$ implies
$\vdash\pi:\tau_2$. Thus the result holds by definition of $\models$.
\end{proof}
Thanks to this lemma, it is sufficient to prove that each method is secure wrt.~its own copy
signature to ensure that all potential overridings will be also secure
wrt.~that copy signature.
\subsection{Limitations of Copy Policies}
\label{sec:pol:limitations}
The enforcement of our copy policies will ensure that certain sharing
constraints are satisfied between fields of an object and its
clone. However, in the current formalism we restrict the policy to
talk about fields that are actually present in a class. The policy
does not ensure properties about fields that are added in
sub-classes. This means that an attacker could copy \emph{e.g.}, a
list by using a new field to build the list, as in the following
example.
\begin{lstlisting}
public class EvilList<V> extends List<V>
{
@Shallow public List<V> evilNext;
public EvilList(V val, List<V> next) {
super(val,null);
this.evilNext = next; }
public List<V> clone() {
return new EvilList(value,evilNext); }
// redefinition of all other methods to use the evilNext field
// instead of next
}
\end{lstlisting}
\noindent
The enforcement mechanism described in this article will determine
that the \ttt{clone()} method of class \ttt{EvilList} respects the copy
policy declared for the \ttt{List} class in
Section~\ref{sec:intro:pol} because this policy only speaks about the
\ttt{next} field which is set to \ttt{null}. It will fail to discover
that the class \ttt{EvilList} creates a shallow copy of lists through
the \ttt{evilNext} field. In order to prevent this attack, the policy
language must be extended, \emph{e.g.}, by adding a facility for
specifying that all fields \emph{except} certain, specifically named
fields must be copied deeply. The enforcement of such policies will likely
be able to reuse the analysis technique described below.
\section{Type and Effect System}
\label{sec:type-system}
The annotations defined in the previous section are convenient for
expressing a copy policy but are not sufficiently expressive for
reasoning about the data structures being copied. The static
enforcement of a copy policy hence relies on a translation of
policies into a graph-based structure (that we shall call types)
describing parts of the environment of local variables and the heap
manipulated by a program. In particular, the types can express useful
alias information between variables and heap cells. In this section,
we define the set of types, an approximation (sub-typing) relation
$\sqsubseteq$ on types, and an inference system for assigning types to
each statement and to the final result of a method.
The set of types is defined using the following symbols:
\begin{align*}
n &\in\Node
&t &\in\BaseType = \Node + \{ \bot, \topout, \top \} \\
\Gamma &\in \Var \to \BaseType
&\Delta &\in \LSG = \Node \ptofin \Field \to \BaseType \\
\Theta &\in \Power(\Node)
&T &\in \Type = (\Var \to \BaseType)\times\LSG\times\Power(\Node)
\end{align*
We assume given a set $\Node$ of nodes. A value can be given a \emph{base
type} $t$ in $\Node + \{ \bot, \topout, \top \}$. A node $n$ means the
value has been locally allocated and is not shared.
The symbol $\bot$ means that the value is equal to the null reference
$\vnull$. The symbol $\topout$ means that the value contains a location that
cannot reach a locally allocated object. The symbol $\top$ is the specific
``no-information'' base type.
As is standard in analysis of memory structures, we distinguish
between nodes that represent exactly one memory cell and nodes that
may represent several cells. If a node representing only one cell
has an edge to another node, then this edge can be forgotten and \emph{replaced} when we
assign a new value to the node---this is called a \emph{strong}
update. If the node represents several cells, then the assignment may
not concern all these cells and edges cannot be forgotten. We can only
\emph{add} extra out-going edges to the node---this is termed a
\emph{weak} update. In the graphical representations of types, we use
singly-circled nodes to designate "weak" nodes and doubly-circled
nodes to represent "strong" nodes.
A type is a triplet
$T=(\Gamma,\Delta,\Theta)\in\Type$ where
\begin{description}
\item[$\Gamma$] is a typing environment that maps (local) variables to base
types.
\item[$\Delta$] is a graph whose nodes are elements of $\Node$. The edges of
the graphs are labeled with field names. The successors of a node is a base
type. Edges over-approximate the concrete points-to relation.
\item[$\Theta$] is a set of nodes that represents necessarily only one
concrete cell each. Nodes in $\Theta$ are eligible to strong update while
others (weak nodes) can only be weakly updated.
\end{description}
\begin{TR}
\begin{exa}
\label{ex:type:graph}
The default \ttt{List} policy of Sec.~\ref{sec:intro:pol} translates into the
type
\begin{align*}
\Gamma &= [\mathtt{res}\mapsto n_1, \mathtt{this}\mapsto \topout] \\
\Delta &=
[ (n_1,\mathtt{next})\mapsto n_2,(n_2,\mathtt{next})\mapsto n_2,(n_1,\mathtt{value})\mapsto \top,(n_2,\mathtt{value})\mapsto \top ] \\
\Theta &= \{n_1\}.
\end{align*}
As mentioned in Sec~\ref{sec:intro:typ}, this type enjoys a graphic representation
corresponding to the right-hand side of Fig.~\ref{fig:listab}.
\end{exa}
\end{TR}
In order to link types to the heap structures they represent, we
will need to state reachability predicates in the abstract domain. Therefore,
the path evaluation relation is extended to types using the following
inference rules:
\begin{small}
\begin{equation*}
\inferrule{~
{\tevalexpr{\Gamma}{\Delta}{x}{\Gamma(x)}}
\quad
\inferrule{\tevalexpr{\Gamma}{\Delta}{\pi}{n}}
{\tevalexpr{\Gamma}{\Delta}{\pi.f}{\Delta[n,f]}}
\quad
\inferrule{\tevalexpr{\Gamma}{\Delta}{\pi}{\top}}
{\tevalexpr{\Gamma}{\Delta}{\pi.f}{\top}}
\quad
\inferrule{\tevalexpr{\Gamma}{\Delta}{\pi}{\topout}}
{\tevalexpr{\Gamma}{\Delta}{\pi.f}{\topout}}
\end{equation*}
\end{small}
Notice both $\topout$ and $\top$ are considered as sink nodes for path
evaluation purposes
\footnote{The sink nodes status of $\top$ (resp.~$\topout$) can be
understood as a way to state the following invariant enforced by our type
system: when a cell points to an unspecified (resp.~foreign) part of the
heap, all successors of this cell are also unspecified (resp.~foreign).}.
\subsection{From Annotation to Type}
The set of all copy policies $\Pi_p\subseteq\CopySignature$ can be
translated into a graph $\Delta_p$ as described hereafter. We assume a
naming process that associates to each policy name $X\in\Polid$ of a program a
unique node $n'_X\in\Node$.
\begin{equation*}
\Delta_p = \displaystyle\bigcup_{X:\{ (X_1,f_1);\ldots ;(X_k,f_k)\} \in \Pi_p}
\left[ (n'_X,f_1) \mapsto n'_{X_1}, \cdots, (n'_X,f_k) \mapsto n'_{X_k} \right]
\end{equation*}
Given this graph, a policy $\tau=\{(X_1,f_1);\ldots ;(X_k,f_k)\}$ that is declared
in a class $\var{cl}$ is translated into a triplet:
$$
\Phi(\tau)=\left(n_\tau,
\Delta_p\cup\left[ (n_\tau,f_1) \mapsto n'_{X_1}, \cdots, (n_\tau,f_k) \mapsto n'_{X_k} \right],
\{n_\tau\}\right)
$$
Note that we \emph{unfold} the possibly cyclic graph $\Delta_p$ with an extra
node $n_\tau$ in order to be able to catch an alias information between this
node and the result of a method, and hence declare $n_\tau$ as strong. Take
for instance the type in Fig.~\ref{fig:listab}: were it not for this
unfolding step, the type would have consisted only in a weak node and a $\top$
node, with the variable \ttt{res} mapping directly to the former. Note also
that it is not necessary to keep (and even to build) the full graph $\Delta_p$
in $\Phi(\tau)$ but only the part that is reachable from $n_\tau$.
\subsection{Type Interpretation}
The semantic interpretation of types is given in
Fig.~\ref{fig:type:interpret}, in the form of a relation
$${\InterpM{\rho}{h}{A}{\Gamma}{\Delta}{\Theta}}$$ that states when a local
allocation history $A$, a heap $h$ and an environment $\rho$ are coherent with
a type $(\Gamma,\Delta,\Theta)$.
The interpretation judgement amounts to checking that
\begin{inparaenum}[(i)]
\item for every path $\pi$ that leads to a value $v$ in the concrete
memory and to a base type $t$ in the
graph, $t$ is a correct description of $v$, as formalized by the auxiliary type interpretation
${\Interpret{\rho}{h}{A}{\Gamma}{\Delta}{v}{t}}$;
\item every strong node in $\Theta$ represents a uniquely reachable value in
the concrete memory.
\end{inparaenum}
The auxiliary judgement ${\Interpret{\rho}{h}{A}{\Gamma}{\Delta}{v}{t}}$ is
defined by case on $t$. The null value is represented by any type. The
symbol $\top$ represents any value and $\topout$ those values that do not
allow to reach a locally allocated location. A node $n$ represents a locally
allocated memory location $l$ such that every concrete path $\pi$ that leads
to $l$ in $\st{\rho,h}$ leads to node $n$ in $(\Gamma,\Delta)$.
\begin{figure}
{\centering
\begin{minipage}[t]{.9965\linewidth}
\bf Auxiliary type interpretatio
\end{minipage}
\scriptsize$$
\begin{array}{c}
\inferrule{~~}
{\Interpret{\rho}{h}{A}{\Gamma}{\Delta}{\vnull}{t}}
~~~~
\inferrule{~~}
{\Interpret{\rho}{h}{A}{\Gamma}{\Delta}{v}{\top}}
~~~~
\inferrule{
\Reach{h}{l}\cap A = \emptyset
}
{\Interpret{\rho}{h}{A}{\Gamma}{\Delta}{l}{\topout}}
\\[5ex]
\inferrule{
l \in A \qquad
n\in\dom(\Delta) \qquad
\forall \pi,~ \evalexpr{\rho}{h}{\pi}{l} ~\Rightarrow
\evalexpr{\Gamma}{\Delta}{\pi}{n} }
{\Interpret{\rho}{h}{A}{\Gamma}{\Delta}{l}{n}}
\end{array}
$$}
\begin{minipage}[t]{.9965\linewidth}
\bf Main type interpretation
\end{minipage}
{\scriptsize$$\inferrule{
\begin{array}[t]{l}
\forall \pi, \forall t, \forall v, \cr
~~~\left.\begin{array}[c]{l}
\tevalexpr{\Gamma}{\Delta}{\pi}{t} \cr
\evalexpr{\rho}{h}{\pi}{v}
\end{array}\right\rbrace \Rightarrow
{\Interpret{\rho}{h}{A}{\Gamma}{\Delta}{v}{t}}
\end{array}
\\
\begin{array}[t]{l}
\forall n\in\Theta,~\forall \pi, \forall \pi', \forall l,\forall l',\cr
~~~
\left.\begin{array}[c]{l}
\tevalexpr{\Gamma}{\Delta}{\pi}{n} ~\land~
\tevalexpr{\Gamma}{\Delta}{\pi'}{n} \cr
\evalexpr{\rho}{h}{\pi}{l} ~\land ~
\evalexpr{\rho}{h}{\pi'}{l'}
\end{array}\right\rbrace
\Rightarrow l = l'
\end{array}
}
{\InterpM{\rho}{h}{A}{\Gamma}{\Delta}{\Theta}}
$$}
\caption{Type Interpretation}\label{fig:type:interpret}
\end{figure}
We now establish a semantic link between policy semantics and type
interpretation. We show that if the final state of a copy method can be given a type
of the form $\Phi(\tau)$ then this is a secure method wrt. the policy $\tau$.
\begin{thm}
\label{th:policy-type}
Let $\Phi(\tau)=(n_\tau,\Delta_\tau,\Theta_\tau)$, $\rho\in\Env,
A\in\Power(\Loc)$, and $x\in\Var$. Assume that, for all $y\in\Var$ such that $y$ is
distinct from $x$, $A$ is not reachable from $\rho(y)$ in a
given heap $h$, \emph{i.e.} $ \Reach{h}{\rho(y)}\cap A = \emptyset$. If there
exists a state of the form $\st{\rho',h,A}$, a return variable $\res$ and a
local variable type $\Gamma'$ such that $\rho'(\res)=\rho(x)$,
$\Gamma'(\res)=n_\tau$ and
$\InterpM{\rho'}{h}{A}{\Gamma'}{\Delta_\tau}{\Theta_\tau}$, then $\rho,h,x
\models \tau$ holds.
\end{thm}
\begin{proof}
[See Coq proof \texttt{InterpAnnot.sound_annotation_to_type}~\cite{clone-webpage}]
We consider two paths $\pi'$ and $x.\pi$ such that
$\proot{\pi'}\not=x$, $\evalexpr{\rho}{h}{\pi'}{l}$, $\vdash x.\pi:\tau$,
$\evalexpr{\rho}{h}{x.\pi}{l}$ and look for a contradiction.
Since $\vdash x.\pi:\tau$ and $\Gamma'(\res)=n_\tau$, there exists a
node $n\in\Delta_\tau$ such that
$\tevalexpr{\Gamma'}{\Delta_\tau}{ \res.\pi}{n}$. Furthermore
$\evalexpr{\rho'}{h}{\res.\pi}{l}$ so we can deduce that $l\in A$. Thus we
obtain a contradiction with $\evalexpr{\rho}{h}{\pi'}{l}$ because any
path that starts from a variable other than $x$ cannot reach the
elements in $A$.
\end{proof}
\subsection{Sub-typing}
\begin{figure}
\centering\scriptsize
\begin{minipage}[t]{.9965\linewidth}
\bf Value sub-typing judgment
\end{minipage}\\
$
\inferrule{t\in \BaseType}{\bot \leq_\sigma t} \quad
\inferrule{t\in \BaseType \backslash \Node}{t \leq_\sigma \top} \quad
\inferrule{~~}{\topout \leq_\sigma \topout} \quad
\inferrule{n\in\Node}{n \leq_\sigma \sigma(n)}
$
\begin{minipage}[t]{.9965\linewidth}
\bf Main sub-typing judgment
\end{minipage}\\[-0.9cm] ~
\begin{prooftree}
\AxiomC{
\begin{minipage}[t]{0.75\linewidth}
\renewcommand{\theequation}{$\text{ST}_\arabic{equation}$}
\begin{align}
&\sigma\in\dom (\Delta_1) \rightarrow \dom (\Delta_2) + \{\top\}
\label{eq:st1}\\
\begin{split}
&\forall t_1\in\BaseType, \forall \pi\in\AccessPath, \tevalexpr{\Gamma_1}{\Delta_1}{\pi}{t_1}
\Rightarrow \exists t_2\in \BaseType, t_1 \leq_\sigma t_2 \wedge \tevalexpr{\Gamma_2}{\Delta_2}{\pi}{t_2}
\end{split}
\label{eq:st2} \\
&\forall n_2\in\Theta_2, ~ \exists n_1\in\Theta_1,~ \sigma^{-1}(n_2) = \{n_1\} \label{eq:st3}
\end{align}
\end{minipage}}
\UnaryInfC{$(\Gamma_1,\Delta_1,\Theta_1)\sqsubseteq (\Gamma_2,\Delta_2,\Theta_2)$}
\end{prooftree}
\caption{Sub-typing}\label{fig:subtype:rules}
\end{figure}
To manage control flow merge points we rely on a sub-typing relation
$\sqsubseteq$ described in Fig.~\ref{fig:subtype:rules}.
A sub-type relation ${(\Gamma_1,\Delta_1,\Theta_1)\sqsubseteq
(\Gamma_2,\Delta_2,\Theta_2)}$ holds if and only if \eqref{eq:st1} there
exists a fusion function $\sigma$ from $\dom(\Delta_1)$ to $\dom(\Delta_2)+\{\top\}$.
$\sigma$ is a mapping that merges nodes and edges in $\Delta_1$ such that
\eqref{eq:st2} every element $t_1$ of $\Delta_1$ accessible from a path $\pi$
is mapped to an element $t_2$ of $\Delta_2$ accessible from the same path,
such that $t_1 \leq_\sigma t_2$. In particular, this means that all successors of
$t_1$ are mapped to successors of $t_2$. Incidentally, because $\top$ acts as
a sink on paths, if $t_1$ is mapped to $\top$, then all its successors are
mapped to $\top$ too.
Finally, when a strong node in $\Delta_1$ maps to a strong node in
$\Delta_2$, this image node cannot be the image of any other node in
$\Delta_1$---in other terms, $\sigma$ is injective on strong nodes \eqref{eq:st3}.
Intuitively, it is possible to go up in the type partial order either by
merging, or by forgetting nodes in the initial graph.
The following example shows three ordered types and their corresponding fusion functions.
On the left, we forget the node pointed to by $\mathtt{y}$ and hence forget
all of its successors (see~\eqref{eq:st2}).
On the right we fusion two strong nodes to obtain a weak node.
\begin{center}
\includegraphics[width=.8\textwidth]{sqsubseteq.pdf}
\end{center}
\noindent
The logical soundness of this sub-typing relation is formally proved with
two intermediate lemmas.
The first one states that paths are preserved between subtypes, and that
they evaluate into basetypes that are related by the subtyping function.
\begin{lem}[Pathing in subtypes]
\label{lem:subtype}
Assume $\InterpMMM{\rho}{h}{A}{T_1}$, and let $\sigma$ be the fusion map
defined by the assertion $T_1\sqsubseteq T_2$. For any $\pi,r$ such that
$\evalexpr{\rho}{h}{\pi}{r}$, for any $t_2$ such that
$\tevalexpr{\Gamma_2}{\Delta_2}{\pi}{t_2}$:
\begin{align*}
\exists t_1 \leq_\sigma t_2,\quad
\tevalexpr{\Gamma_1}{\Delta_1}{\pi}{t_1}.
\end{align*}
\end{lem}
\proof
[See Coq proof \texttt{Misc.Access_Path_Eval_subtyp}~\cite{clone-webpage}]
The proof follows directly from the definition of type interpretation and subtyping.\qed
The second lemma gives a local view on logical soundness of subtyping.
\begin{lem}[Local logical soundness of subtyping]
\label{lem:subtype2}
Assume $(\Gamma_1,\Delta_1,\Theta_1)\sqsubseteq
(\Gamma_2,\Delta_2,\Theta_2)$, and let $v$ be a value
and $t_1,t_2$ some types.
$$
{\InterpretS{\rho}{h}{A}{(\Gamma_1,\Delta_1)}{v}{t_1}}
~~
\text{and}
~~t_1 \sqsubseteq_\sigma t_2~~
\text{implies}
~~
{\InterpretS{\rho}{h}{A}{(\Gamma_2,\Delta_2)}{v}{t_2}}.
$$
\end{lem}
\proof
[See Coq proof \texttt{Misc.Interp_monotone}~\cite{clone-webpage}]
We make a case for each rules of $t_1 \sqsubseteq_\sigma t_2$. The only non-trivial case
is for $v=l\in\Loc$, $t_1 = n \in\Node$ and $t_2 = \sigma(n) \in \dom(\Delta_2)$.
In this case we have to prove
$\forall \pi,~ \evalexpr{\rho}{h}{\pi}{l} ~\Rightarrow
\tevalexpr{\Gamma_2}{\Delta_2}{\pi}{\sigma(n)} $.
Given such a path $\pi$, the hypothesis
${\InterpretS{\rho}{h}{A}{(\Gamma_1,\Delta_1)}{l}{n}}$ gives
us $\tevalexpr{\Gamma_1}{\Delta_1}{\pi}{n}$. Then subtyping hypothesis $ST_2$
gives us a base type $t'_2$ such that
$n \sqsubseteq_\sigma t'_2$ and $\tevalexpr{\Gamma_2}{\Delta_2}{\pi}{t'_2}$.
But necessarily $t_2=t'_2$ so we are done. \qed
The logical soundness of this sub-typing relation is then formally proved with the
following theorem.
\begin{thm} For any type $T_1,T_2\in\Type$ and $\st{\rho,h,A}\in \State$,
${T_1\sqsubseteq T_2}$
and
$\InterpMMM{\rho}{h}{A}{T_1}$
imply
$\InterpMMM{\rho}{h}{A}{T_2}$.
\end{thm}
\begin{proof}
\begin{CA}
See~\cite{JensenKP10} and the companion Coq development.
\end{CA}
\begin{TR}[See Coq proof \texttt{InterpMonotony.Interpretation_monotone}~\cite{clone-webpage}]
We suppose $T_1$ is of the form $(\Gamma_1,\Delta_1,\Theta_1)$
and $T_2$ of the form $(\Gamma_2,\Delta_2,\Theta_2)$.
From the definition of the main type interpretation (Fig~\ref{fig:type:interpret}), we reduce
the proof to proving the following two subgoals.
First, given a path $\pi$, a base type $t_2$ and a value $v$ such that
$\evalexpr{\Gamma_2}{\Delta_2}{\pi}{t_2}$
and
$\evalexpr{\rho}{h}{\pi}{v}$, we must prove that
${\InterpretS{\rho}{h}{A}{(\Gamma_2,\Delta_2)}{v}{t_2}}$ holds.
Since $(\Gamma_1,\Delta_1,\Theta_1)\sqsubseteq (\Gamma_2,\Delta_2,\Theta_2)$,
there exists, by Lemma~\ref{lem:subtype}, a base type $t_1$ such that
$\tevalexpr{\Gamma_1}{\Delta_1}{\pi}{t_1}$ and
$t_1 \sqsubseteq_\sigma t_2$. Since $\InterpMMM{\rho}{h}{A}{T_1}$ holds we
can argue that ${\InterpretS{\rho}{h}{A}{(\Gamma_1,\Delta_1)}{v}{t_1}}$ holds
too and conclude with Lemma~\ref{lem:subtype2}.
Second, given a strong node $n_2\in\Theta_2$, two paths $\pi$ and $\pi'$ and two locations
$l$ and $l'$ such that
$\tevalexpr{\Gamma_2}{\Delta_2}{\pi}{n_2}$,
$\tevalexpr{\Gamma_2}{\Delta_2}{\pi'}{n_2}$,
$\evalexpr{\rho}{h}{\pi}{l}$ and
$\evalexpr{\rho}{h}{\pi}{l'}$, we must prove that $l=l'$.
As previously, there exists by Lemma~\ref{lem:subtype}, $t_1$ and $t'_1$ such that
$\tevalexpr{\Gamma_1}{\Delta_1}{\pi}{t_1}$, $t_1 \sqsubseteq_\sigma n_2$,
$\tevalexpr{\Gamma_1}{\Delta_1}{\pi'}{t'_1}$ and
$t'_1 \sqsubseteq_\sigma n_2$. But then, by $(ST_3$), there exists some strong node
$n_1$ such that $t_1 = t'_1 = n_1$ and we can obtain the desired
equality from the hypothesis
$\InterpMMM{\rho}{h}{A}{\Gamma_1,\Delta_1,\Theta_1}$.
\end{TR}
\end{proof}
\subsection{Type and Effect System}
\label{sec:typesystem}
The type system verifies, statically and class by class, that a program
respects the copy policy annotations relative to a declared copy policy.
The core of the type system concerns the typability of commands, which is
defined through the following judgment:
$$
\Gamma,\Delta,\Theta \vdash c : \Gamma',\Delta',\Theta'.
$$
The judgment is valid if the execution of command $c$ in a state satisfying
type $(\Gamma,\Delta,\Theta)$ will result in a state satisfying
$(\Gamma',\Delta',\Theta')$ or will diverge.
\begin{figure}
\centering\scriptsize
\begin{minipage}[t]{.9965\linewidth}
\bf Command typing rules
\end{minipage}\\
$
\begin{array}{c}
\inferrule{~~}
{\tstb{\Gamma,\Delta,\Theta} \vdash \HAssign{x}{y} : \tstr{\Gamma[\var{x}\mapsto\Gamma(\var{y})],\Delta,\Theta}}
\quad
\inferrule{n~\text{fresh in}~\Delta}
{\tstb{\Gamma,\Delta,\Theta} \vdash \HNew{x}{\var{cn}} : \tstr{\Gamma[\var{x}\mapsto n],\Delta[(n,\_)\mapsto\bot],\Theta\cup\{n\}}}
\\[3ex]
\inferrule{\Gamma(\var{y})=t\qquad t\in\{\topout,\top\}}
{\tst{\Gamma,\Delta,\Theta} \vdash \HAssign{x}{y.f} : \tstr{\Gamma[\var{x}\mapsto t],\Delta,\Theta}}
\quad
\inferrule{\Gamma(\var{y})=n}
{\tstb{\Gamma,\Delta,\Theta} \vdash \HAssign{x}{y.f} : \tstr{\Gamma[\var{x}\mapsto \Delta[n,f]],\Delta,\Theta}}
\\[3ex]
\inferrule{\Gamma(\var{x})=n\quad n\in\Theta}
{\tstb{\Gamma,\Delta,\Theta} \vdash \HAssign{x.f}{y} : \tstr{\Gamma,\Delta[n,f \mapsto \Gamma(y)],\Theta}}
\\[3ex]
\inferrule{\Gamma(\var{x})=n\quad n\not\in\Theta
\quad (\Gamma,\Delta[n,f \mapsto \Gamma(y)],\Theta) \sqsubseteq (\Gamma',\Delta',\Theta')
\quad (\Gamma,\Delta,\Theta) \sqsubseteq (\Gamma',\Delta',\Theta')
}
{\tstb{\Gamma,\Delta,\Theta} \vdash \HAssign{x.f}{y} : \tstr{\Gamma',\Delta',\Theta'}}
\\[3ex]
\inferrule{
\begin{array}[c]{c}
\tstb{\Gamma,\Delta,\Theta} \vdash c_1 : \tstr{\Gamma_1,\Delta_1,\Theta_1} \quad
(\Gamma_1,\Delta_1,\Theta_1)\sqsubseteq (\Gamma',\Delta',\Theta') \cr
\tstb{\Gamma,\Delta,\Theta} \vdash c_2 : \tstr{\Gamma_2,\Delta_2,\Theta_2} \quad
(\Gamma_2,\Delta_2,\Theta_2)\sqsubseteq (\Gamma',\Delta',\Theta')
\end{array}}
{\tstb{\Gamma,\Delta,\Theta} \vdash \HIf{c_1}{c_2} : \tstr{\Gamma',\Delta',\Theta'}}
\\[3ex]
\inferrule{\tstb{\Gamma',\Delta',\Theta'} \vdash c : \tstr{\Gamma_0,\Delta_0,\Theta_0} \quad (\Gamma,\Delta,\Theta)\sqsubseteq (\Gamma',\Delta',\Theta') \quad (\Gamma_0,\Delta_0,\Theta_0)\sqsubseteq (\Gamma',\Delta',\Theta')}
{\tstb{\Gamma,\Delta,\Theta} \vdash \HWhile{c} : \tstr{\Gamma',\Delta',\Theta'}}
\\[3ex]
\inferrule{\tstb{\Gamma,\Delta,\Theta} \vdash c_1 : \tstr{\Gamma_1,\Delta_1,\Theta_1} \quad \tstb{\Gamma_1,\Delta_1,\Theta_1} \vdash c_2 : \tstr{\Gamma_2,\Delta_2,\Theta_2}}
{\tstb{\Gamma,\Delta,\Theta} \vdash c_1; c_2 : \tstr{\Gamma_2,\Delta_2,\Theta_2}}
\\[4ex]
\inferrule{
\Pi_p(X) = \tau \quad \Phi(\tau) = (n_\tau,\Delta_\tau) \quad
\nodes(\Delta)\cap \nodes(\Delta_\tau) = \emptyset \quad
(\Gamma(y) = \bot) \vee (\Gamma(y)=\topout)}
{\tstb{\Gamma,\Delta,\Theta} \vdash \HCall{x}{\var{cn}:X}{y} : \tstr{\Gamma[x\mapsto n_\tau],\Delta\cup\Delta_\tau,\Theta\cup\{n_\tau\}}}
\\[4ex]
\inferrule{
\begin{array}[c]{c}
\Pi_p(X) = \tau \quad \Phi(\tau) = (n_\tau,\Delta_\tau) \quad
\nodes(\Delta)\cap \nodes(\Delta_\tau) = \emptyset \cr
\mathit{KillSucc}_{n}(\Gamma,\Delta,\Theta) = (\Gamma',\Delta',\Theta')\quad
\Gamma(y) = n
\end{array}}
{\tstb{\Gamma,\Delta,\Theta} \vdash \HCall{x}{\var{cn}:X}{y} : \tstr{\Gamma'[x\mapsto n_\tau],\Delta'\cup\Delta_\tau,\Theta'\cup\{n_\tau\}}}
\\[4ex]
\inferrule{(\Gamma(y) = \bot) \vee (\Gamma(y)=\topout)}
{\tstb{\Gamma,\Delta,\Theta} \vdash \UnkownCall{x}{y} : \tstr{\Gamma[x\mapsto \topout],\Delta,\Theta}}
\quad
\inferrule{
\mathit{KillSucc}_{n}(\Gamma,\Delta,\Theta) = (\Gamma',\Delta',\Theta')\quad
\Gamma(y) = n}
{\tstb{\Gamma,\Delta,\Theta} \vdash \UnkownCall{x}{y} : \tstr{\Gamma'[x\mapsto \topout],\Delta',\Theta'}}
\\%[1ex]
\inferrule{~~}
{\tstb{\Gamma,\Delta,\Theta} \vdash \HReturn{x} : \tstr{\Gamma[\var{ret}\mapsto\Gamma(x)],\Delta,\Theta}}
\end{array}
$\\
\begin{minipage}[t]{.9965\linewidth}
\bf Method typing rule
\end{minipage}\\
$
\inferrule{
\begin{array}[c]{c}
[~\cdot\mapsto\bot][x\mapsto \topout], \emptyset, \emptyset \vdash c : \Gamma,\Delta,\Theta \cr
\Pi_p(X) = \tau \quad \Phi(\tau) = (n_\tau,\Delta_\tau) \quad
(\Gamma, \Delta, \Theta) \sqsubseteq (\Gamma',\Delta_\tau,\{n_\tau\})
\quad \Gamma'(\var{ret}) = n_\tau
\end{array}}
{ \vdash \Copy(X)~\HAssign{m(x)}{}c}
$\\
\begin{minipage}[t]{.9965\linewidth}
\bf Program typing rule
\end{minipage}
$
\inferrule{\forall \var{cl}\in p, ~ \forall\md\in\var{cl},~~ \vdash \md }
{\vdash p}
$
\vspace{10 pt}
\begin{minipage}[t]{.9965\linewidth}\footnotesize
\textbf{Notations:} We write
$\Delta[(n,\_)\mapsto\bot]$ for the update of $\Delta$ with a new node $n$ for
which all successors are equal to $\bot$.
We write
$\mathit{KillSucc}_{n}$ for the function that removes all
nodes reachable from $n$ (with at least one step) and sets all its successors
equal to $\top$.
\end{minipage}
\caption{Type System}\label{fig:type:rules}
\end{figure}
Typing rules are given in Fig.~\ref{fig:type:rules}. We explain a selection of
rules below. The rules for $\HIf{}{}$, $\HWhile{}$, sequential
composition and most of the assignment rules are standard for flow-sensitive
type systems. The rule for $\HNew{x}{}$ ``allocates'' a fresh node $n$ with no
edges in the graph $\Delta$ and let $\Gamma(x)$ references this node.
There are two rules concerning the instruction $\HAssign{x.f}{y}$ for
assigning values to fields. Assume that the variable $x$ is
represented by node $n$ (\emph{i.e.}, $\Gamma(x) = n$). In the first
case (strong update), the node is strong and we update destructively (or add) the edge
in the graph $\Delta$ from node $n$ labeled $f$ to point to the value
of $\Gamma(y)$. The previous edge (if any) is lost because
$n\in\Theta$ ensures that all concrete cells represented by $n$ are
affected by this field update. In the second case (weak update), the node is
weak. In order to be conservative, we must merge the previous shape
with its updated version since the content of $x.f$ is updated but an
other cell mays exist and be represented by $n$ without being
affected by this field update.
As for method calls $m(\var{y})$, two cases arise depending on whether the
method $m$ is copy-annotated or not. In each case, we also reason
differently depending on the type of the argument $\var{y}$. If a
method is associated with a copy policy $\tau$, we compute the
corresponding type $(n_\tau,\Delta_\tau)$ and type the result of
$\HCall{x}{\var{cn}:X}{y}$ starting in $(\Gamma,\Delta,\Theta)$ with
the result type consisting of the environment $\Gamma$ where $x$ now
points to $n_\tau$, the heap described by the disjoint union of
$\Delta$ and $\Delta_\tau$. In addition, the set of strong nodes is
augmented with $n_\tau$ since a copy method is guaranteed to return a
freshly allocated node. The method call may potentially modify the
memory referenced by its argument $\var{y}$, but the analysis has no
means of tracking this. Accordingly, if $\var{y}$ is a locally allocated memory
location of type $n$, we must remove all nodes reachable from $n$, and
set all the successors of $n$ to $\top$. The other case to consider
is when the method is
not associated with a copy policy (written $ \UnkownCall{x}{y}$). If the parameter $\var{y}$ is null
or not locally allocated, then there is no way for the method call to
access locally allocated memory and we know that $\var{x}$ points to a
non-locally allocated object. Otherwise, $\var{y}$ is a locally allocated
memory location of type $n$, and we must kill all its successors in
the abstract heap.
Finally, the rule for method definition verifies the coherence of the
result of analysing the body of a method $m$
with its copy annotation $\Phi(\tau)$. Type checking extends trivially to
all methods of the program.
Note the absence of a rule for typing an instruction $\HAssign{x.f}{y}$ when
$\Gamma(\var{x})=\top$ or $\topout$. In a first attempt, a sound rule would
have been
\begin{small}
\begin{equation*}
\inferrule
{\Gamma(\var{x})=\top}
{\Gamma,\Delta \vdash \HAssign{x.f}{y} : \Gamma,\Delta[~\cdot,f \mapsto \top]}
\end{equation*}
\end{small}%
Because $\var{x}$ may point to any part of the local shape we must
conservatively forget all knowledge about the field
$\var{f}$. Moreover we should also warn the caller of the current
method that a field $\var{f}$ of his own local shape may have been
updated. We choose to simply reject copy methods with such patterns.
Such a policy is strong but has the merit to be easily
understandable to the programmer: a copy method should only modify
locally allocated objects to be typable in our type system. For
similar reasons, we reject methods that attempt to make a method call on
a reference of type $\top$ because we can not track side effect
modifications of such methods without losing the modularity of the verification mechanism.
\begin{figure}
\centering
\begin{minipage}{.5\linewidth}
\lstset{basicstyle=\ttfamily\scriptsize,numbers=left,numberstyle=\scriptsize,numbersep=2pt}
\begin{lstlisting}
class LinkedList<E> implements Cloneable {
private @Deep Entry<E> header;
private static class Entry<E> {
@Shallow E element;
@Deep Entry<E> next;
@Deep Entry<E> previous;
}
@Copy public Object clone() {
LinkedList<E> clone = null;
clone = (LinkedList<E>) super.clone();
clone.header = new Entry<E>;
clone.header.next = clone.header;
clone.header.previous = clone.header;
Entry<E> e = this.header.next;
while (e != this.header) {
Entry<E> n = new Entry<E>;
n.element = e.element;
n.next = clone.header;
n.previous = clone.header.previous;
n.previous.next = n;
n.next.previous = n;
e = e.next;
}
return clone;
}
}
\end{lstlisting}
\end{minipage}
\begin{minipage}{.49\linewidth}
\includegraphics[width=\linewidth]{L1}
\includegraphics[width=\linewidth]{L2}
\includegraphics[width=\linewidth]{L2p}
\includegraphics[width=\linewidth]{L3}
\includegraphics[width=\linewidth]{L4}
\includegraphics[width=\linewidth]{L5}
\includegraphics[width=\linewidth]{L6}
\includegraphics[width=\linewidth]{L7}
\end{minipage}
\caption{Intermediate Types for \ttt{java.util.LinkedList.clone()}}
\label{fig:linkedlist}
\end{figure}
\begin{exa}[Case Study: \ttt{java.util.LinkedList}]
In this example, we demonstrate the use of the type system on a challenging
example taken from the standard Java library.
The companion
web page provides a more detailed explanation of this example~\cite{clone-webpage}.
The class
\ttt{java.util.LinkedList} provides an implementation of doubly-linked lists.
A list is composed of a first cell that points through a field
\ttt{header} to a collection of doubly-linked cells. Each cell has a
link to the previous and the next cell and also to an element of
(parameterized) type \ttt{E}. The clone method provided in
\ttt{java.lang} library implements a ``semi-shallow'' copy where only cells of
type \ttt{E} may be shared between the source and the result of the
copy.
In Fig.~\ref{fig:linkedlist} we present a modified version of the
original source code: we have inlined all method calls, except those
to copy methods and removed exception handling that leads to an
abnormal return from the method\footnote{Inlining is automatically
performed by our tool and exception control flow graph is managed as
standard control flow but omitted here for simplicity.}. Note that
there was one method call in the original code that was virtual and
hence prevented inlining. It has been necessary to make a private
version of this method. This makes sense because such a virtual call
actually constitutes a potentially dangerous hook in a cloning method,
as a re-defined implementation could be called when cloning a subclass
of \ttt{Linkedlist}.
In Fig.~\ref{fig:linkedlist} we provide several intermediate types
that are necessary for typing this method ($T_i$ is the type before
executing the instruction at line $i$). The call
to \ttt{super.clone} at line 12 creates a shallow copy of the header
cell of the list, which contains a reference to the original list. The
original list is thus shared, a fact which is
represented by an edge to $\topout$ in type $T_{13}$.
The copy method then progressively constructs a deep copy of the list,
by allocating a new node (see type $T_{14}$) and setting all paths
\ttt{clone.header}, \ttt{clone.header.next} and
\ttt{clone.header.previous} to point to this node. This is reflected
in the analysis by a \emph{strong update} to the node representing
path \ttt{clone.header} to obtain the type $T_{16}$ that precisely
models the alias between the three paths \ttt{clone.header},
\ttt{clone.header.next} and \ttt{clone.header.previous} (the Java
syntax used here hides the temporary variable that is introduced to be
assigned the value of \ttt{clone.header} and then be updated).
This type $T_{17}$ is the loop invariant necessary for type checking
the whole loop. It is a super-type of $T_{16}$ (updated with
$e\mapsto\topout$) and of $T_{24}$ which represents the memory at the
end of the loop body. The body of the loop allocates a new list cell
(pointed to by variable \ttt{n}) (see type $T_{19}$) and inserts it
into the doubly-linked list. The assignment in line 22 updates the
weak node pointed to by path \ttt{n.previous} and hence merges the
strong node pointed to by \ttt{n} with the weak node pointed to by
\ttt{clone.header}, representing the spine of the list. The
assignment at line 23 does not modify the type $T_{23}$.
Notice that the types used in this example show that a flow-insensitive
version of the analysis could not have found this information. A
flow-insensitive analysis would force the merge of the types at all program
points, and the call to \ttt{super.clone} return a type that is less precise
than the types needed for the analysis of the rest of the method.
\end{exa}
\subsection{Type soundness}
\label{sec:soudness}
The rest of this section is devoted to the soundness proof of the type system.
We consider the types
$T=(\Gamma,\Delta,\Theta),
T_1=(\Gamma_1,\Delta_1,\Theta_1),T_2=(\Gamma_2,\Delta_2,\Theta_2)\in\Type$, a
program $c\in\prog$, as well as the configurations $\st{\rho,h,A},
\st{\rho_1,h_1,A_1}, \st{\rho_2,h_2,A_2}\in\State$.
Assignments that modify the heap space can also modify the reachability
properties of locations. This following lemma indicates how to reconstruct a path to $l_f$ in the \emph{initial} heap from a path
$\pi'$ to a given location $l_f$ in the \emph{assigned} heap.
\begin{lem}[Path decomposition on assigned states]
\label{lem:pathdec}
Assume given a path $\pi$, field $f$, and locations $l,l'$ such that
$\evalexpr{\rho}{h}{\pi}{l'}$.
and assume that for any path $\pi'$ and a location $l_f$ we have
$\evalexpr{\rho}{h[l,f\mapsto l']}{\pi'}{l_f}$. Then, either
\begin{equation*}
\evalexpr{\rho}{h}{\pi'}{l_f
\end{equation*}
or $\exists \pi_z, \pi_1, \ldots, \pi_n, \pi_f$ such that:
\begin{equation*}
\left\{
\begin{aligned}
&\pi'=\pi_z.f.\pi_1.f.\ldots.f.\pi_n.f.\pi_f \\
&\evalexpr{\rho}{h}{\pi_z}{l}\\%\ \wedge\ \pathdiff{\rho}{h}{\pi_z}{l,f} \\
&\evalexpr{\rho}{h}{\pi.\pi_f}{l_f}\\%\ \wedge\ \pathdiff{\rho}{h}{\pi.\pi_f}{l,f} \\
&\forall \pi_1,\ldots,\pi_n, \evalexpr{\rho}{h}{\pi.\pi_i}{l
\end{aligned}
\right.
\end{equation*}
The second case of the conclusion of this lemma is illustrated in
Fig.~\ref{fig:pathdec}.
\end{lem}
\proof
[See Coq proof \texttt{Misc.Access_Path_Eval_putfield_case}~\cite{clone-webpage}]
The proof is done by induction on $\pi$. \qed
\begin{figure}%
\centering
\subfloat[in $\st{\rho,h}$]{\includegraphics[width=.35\textwidth]{pathdec1}}\qquad
\subfloat[in $\st{\rho,h[l,f\mapsto l']}$]{\includegraphics[width=.35\textwidth]{pathdec2}}
\caption{An Illustration of Path Decomposition on Assigned States.}
\label{fig:pathdec}
\end{figure}
We extend the previous lemma to paths in both the concrete heap and the graph types.
\begin{lem}[Pathing through strong field assignment]
\label{lem:strongassign}
Assume $\InterpMMM{\rho}{h}{A}{\Gamma,\Delta,\Theta}$ with $\rho(x)=l_x\in A$,
$\Gamma(x)=n_x\in \Theta$, $\rho(y)=l_y$, and $\Gamma(y)=t_y$. Additionally,
suppose that for some path $\pi$, value $v$, and type $t$:
\begin{gather*}
\evalexpr{\rho}{h[l_x,f\mapsto l_y]}{\pi}{v} \\
\tevalexpr{\Gamma}{\Delta[n_x,f\mapsto t_y]}{\pi}{t}.
\end{gather*}
Then at least one of the following four statements hold:
\begin{align*}
&\left(\evalexpr{\rho}{h}{\pi}{v} \wedge \tevalexpr{\Gamma}{\Delta}{\pi}{t}\right) \tag{1}\\
&\left(\exists\pi', \evalexpr{\rho}{h}{y.\pi'}{v} \wedge \tevalexpr{\Gamma}{\Delta}{y.\pi'}{t}\right) \tag{2}\\
&t=\top \tag{3} \\
&v=\vnull \tag{4}
\end{align*}
\end{lem}
\proof
[See Coq proof \texttt{Misc.strong_subst_prop}~\cite{clone-webpage}]
The non-trivial part of the lemma concerns the situation when $t \neq
\top$ and $v \neq \vnull$). In that case,
the proof relies on Lemma~\ref{lem:pathdec}. The two parts of the disjunction
in Lemma~\ref{lem:pathdec} are used to prove one of the two first
statements. If the first
part of the disjunction holds, we can assume that $\evalexpr{\rho}{h}{\pi}{v}$. Then, since
$\InterpMMM{\rho}{h}{A}{\Gamma,\Delta,\Theta}$, we also have
$\tevalexpr{\Gamma}{\Delta}{\pi}{t}$. This implies the first main statement.
If the second part of the disjunction holds, then, by observing that
$\evalexpr{\rho}{h}{y}{l_y}$, we can derive the sub-statement $\exists \pi_f,
\evalexpr{\rho}{h}{y.\pi_f}{v}$. As previously, by assumption we also have
$\tevalexpr{\Gamma}{\Delta}{y.\pi_f}{t}$, which implies our second main
statement.
\qed
We first establish a standard subject reduction theorem and then prove type
soundness. We assume that all methods of the considered program are
well-typed.
\begin{thm}[Subject Reduction]\label{theo2}
Assume ${T_1 \vdash c: T_2}$ and
$\InterpMM{\rho_1}{h_1}{A_1}{T_1}$.\\
If
$(c,\st{\rho_1,h_1,A_1}) \leadsto \st{\rho_2,h_2,A_2}$
then
$\InterpMM{\rho_2}{h_2}{A_2}{T_2}$.
\end{thm}
\proof
[See Coq proof \texttt{Soundness.subject_reduction}~\cite{clone-webpage}]
The proof proceeds by structural induction on the instruction $c$. For
each reduction rule concerning $c$ (Fig.~\ref{fig:semrules}), we prove
that the resulting state is in relation to the type $T_2$, as defined
by the main type interpretation rule in
Fig.~\ref{fig:type:interpret}. This amounts to verifying that the two
premises of the type interpretation rule are satisfied. One premise
checks that all access paths lead to related (value,node) pairs. The
other checks that all nodes that are designated as ``strong'' in the
type interpretation indeed only represent unique locations.
We here present the most intricate part of the proof, which concerns
graph nodes different from $\top$, $\bot$, or $\topout$, and focus here on variable and field assignment.
The entire proof has been checked using the Coq proof management system.
\underline{If $c\equiv \HAssign{x}{y}$}
then $\st{\rho_2,h_2,A_2} = \st{\rho_1[x\mapsto\rho_1(y)],h_1,A_1}$.
In $(\rho_2,h_2,\Gamma_2,\Delta_2)$ take a path $x.\pi$: since
$\rho_2(x)=\rho_1(y)$ and $\Gamma_2(x)=\Gamma_1(y)$, $y.\pi$ is also a path
in $(\rho_1,h_1,\Gamma_1,\Delta_1)$. Given that
$\InterpMMM{\rho_1}{h_1}{A_1}{\Gamma_1,\Delta_1,\Theta_1}$, we know that
$y.\pi$ will lead to a value $v$ (formally,
$\evalexpr{\rho_1}{h_1}{y.\pi}{v}$) and a node $n$ (formally,
$\tevalexpr{\Gamma_1}{\Delta_1}{y.\pi}{n}$) such that
${\Interpret{\rho_1}{h_1}{A_1}{\Gamma_1}{\Delta_1}{v}{n}}$. The two paths
being identical save for their prefix, this property also holds for $x.\pi$
(formally, ${\Interpret{\rho_1[x\mapsto\rho_1(y)]}{h_1}{A_1}{\Gamma_1[x\mapsto\Gamma_1(y)]}{\Delta_1}{v}{n}}$,
$\evalexpr{\rho_1[x\mapsto\rho_1(y)]}{h_1}{x.\pi}{v}$,
and $\tevalexpr{\Gamma_1[x\mapsto\Gamma_1(y)]}{\Delta_1}{x.\pi}{n}$).
Paths not beginning with $x$ are not affected by the assignment, and so we
can conclude that the first premise is satisfied.
For the second premise, let $n \in \Theta_2$ and assume that $n$ can
be reached by two paths $\pi$ and $\pi'$ in $\Delta_2$. If none or
both of the paths begin with $x$ then, by assumption, the two paths
will lead to the same location in $h_2=h_1$. Otherwise, suppose
that, say, $\pi$ begins with $x$ and $\pi'$ with a different
variable $z$ and that they lead to $l$ and $l'$ respectively. Since
$\Gamma_2(x)=\Gamma_1(y)$ and $\Delta_2=\Delta_1$, then by assumption there is a path $\pi''$ in
$(\rho_1,h_1,\Delta_1,\Gamma_1)$ that starts with $y$, and such that
$\evalexpr{\rho_1}{h_1}{\pi''}{l}$. As $z$ is not affected by the assignment,
we also have that $\evalexpr{\rho_1}{h_1}{\pi'}{l'}$. Therefore, as $n \in
\Theta_1$ and $\InterpMMM{\rho_1}{h_1}{A_1}{\Gamma_1,\Delta_1,\Theta_1}$, we
can conclude that $l = l'$. This proves that
$\InterpMMM{\rho_1[x\mapsto\rho_1(y)]}{h_1}{A_1}{\Gamma_1[\var{x}\mapsto\Gamma_1(\var{y})],\Delta_1,\Theta_1}$.
\underline{If $c\equiv \HAssign{x.f}{y}$}
then $\st{\rho_2,h_2,A_2} = \st{\rho_1,h_1[(\rho_1(x),f)\mapsto \rho_1(y)],A_1}$.
Two cases arise, depending on whether $n = \Gamma(x)$ is a strong or a weak node.
\underline{If $c\equiv \HAssign{x.f}{y}$ and $n=\Gamma(x)\in\Theta$}
the node $n$ represents a unique concrete cell in $h$. To check the first
premise, we make use of Lemma~\ref{lem:strongassign} on a given path
$\pi$, a node $n$ and a location $l$ such that
$\tevalexpr{\Gamma_2}{\Delta_2}{\pi}{n}$ and
$\evalexpr{\rho_2}{h_2}{\pi}{l}$. This yields one of two main hypotheses.
In the first case $\pi$ is not modified by the $f$-redirection (formally,
$\evalexpr{\rho_1}{h_1}{\pi}{l} \wedge
\tevalexpr{\Gamma_1}{\Delta_1}{\pi}{n}$), and by assumption
$\Interpret{\rho_1}{h_1}{A_1}{\Gamma_1}{\Delta_1}{l}{n}$. Now consider any
path $\pi_0$ such that $\evalexpr{\rho_2}{h_2}{\pi_0}{l}$: there is a
node $n_0$ such that $\tevalexpr{\rho_2}{h_2}{\pi_0}{n_0}$, and we can
reapply Lemma~\ref{lem:strongassign} to find that
$\Interpret{\rho_1}{h_1}{A_1}{\Gamma_1}{\Delta_1}{l}{n_0}$. Hence $n_0=n$,
and since nodes in $\Delta_1$ and $\Delta_2$ are untouched by the field
assignment typing rule, we can conclude that the first premise is
satisfied. In the second case ($\pi$ is modified by the $f$-redirection)
there is a path $\pi'$ in $(\rho_1,h_1,\Gamma_1,\Delta_1)$ such that
$y.\pi'$ leads respectively to $l$ and $n$ (formally
$\evalexpr{\rho_1}{h_1}{y.\pi'}{l} \wedge
\tevalexpr{\Gamma_1}{\Delta_1}{y.\pi'}{n}$).
As in the previous case, for
any path $\pi_0$ such that $\evalexpr{\rho_2}{h_2}{\pi_0}{l}$ and
$\tevalexpr{\rho_2}{h_2}{\pi_0}{n_0}$, we have
$\Interpret{\rho_1}{h_1}{A_1}{\Gamma_1}{\Delta_1}{l}{n_0}$, which implies
that $n=n_0$, and thus we can conclude
that $\pi_0$ leads to the same node as $\pi$, as required.
For the second premise, let $n \in \Theta_2$ and assume that $n$ can be
reached by two paths $\pi$ and $\pi'$ in $\Delta_2$. The application of
Lemma~\ref{lem:strongassign} to both of these paths yields the following
combination of cases:
\begin{desCription}
\item\noindent{\hskip-12 pt\bf neither path is modified by the $f$-redirection:}\ formally,
$\evalexpr{\rho_1}{h_1}{\pi}{l} \wedge \tevalexpr{\Gamma_1}{\Delta_1}{\pi}{n} \wedge
\evalexpr{\rho_1}{h_1}{\pi'}{l'} \wedge \tevalexpr{\Gamma_1}{\Delta_1}{\pi'}{n}$.
By assumption, $l=l'$.
%
\item\noindent{\hskip-12 pt\bf one of the paths is modified by the $f$-redirection:}\ without loss
of generality, assume $\evalexpr{\rho_1}{h_1}{\pi'}{l'} \wedge
\tevalexpr{\Gamma_1}{\Delta_1}{\pi}{n}$, and there is a path $\pi_\star$ in
$(\rho_1,h_1,\Gamma_1,\Delta_1)$ such that $y.\pi_\star$ leads to $l$
in the heap, and $n$ in the graph (formally,
$\evalexpr{\rho_1}{h_1}{y.\pi_\star}{l} \wedge
\tevalexpr{\Gamma_1}{\Delta_1}{y.\pi_\star}{n}$). By assumption, $l=l'$.
%
\item\noindent{\hskip-12 pt\bf both paths are modified by the $f$-redirection:}\ we can find two
paths $\pi_\star$ and $\pi'_\star$ such that $y.\pi_\star$ leads to $l$
in the heap and $n$ in the graph, and $y.\pi'_\star$ leads to $l'$ in
the heap and $n$ in the graph (formally, $\evalexpr{\rho_1}{h_1}{y.\pi_\star}{l} \wedge
\tevalexpr{\Gamma_1}{\Delta_1}{y.\pi_\star}{n} \wedge
\evalexpr{\rho_1}{h_1}{y.\pi'_\star}{l'} \wedge
\tevalexpr{\Gamma_1}{\Delta_1}{y.\pi'_\star}{n}$). By assumption,
$l=l'$.
\end{desCription}
In all combinations, $l=l'$ in $h_1$. Since the rule for field assignment in
the operational semantics preserves the locations, then $l=l'$ in $h_2$.
This concludes the proof of $\InterpMMM{\rho_1}{h_1[(\rho_1(x),f)\mapsto
\rho_1(y)]}{A_1}{\Gamma_1,\Delta_1[n,f \mapsto \Gamma(y)],\Theta_1}$ when
$n\in\Theta$.
\underline{If $c\equiv \HAssign{x.f}{y}$ and $n=\Gamma(x)\notin\Theta$}
here $n$ may represent multiple concrete cells in $h$. Let $\sigma_1$ and
$\sigma_2$ be the mappings defined, respectively, by the hypothesis
$(\Gamma_1,\Delta_1,\Theta_1)\sqsubseteq(\Gamma_2,\Delta_2,\Theta_2)$
and
$(\Gamma_1,\Delta_1[n,f\mapsto
\Gamma_1(y)],\Theta_1)\sqsubseteq(\Gamma_2,\Delta_2,\Theta_2)$.
The first premise of the proof is proved by examining a fixed path $\pi$
in $(\rho_2,h_2,\Gamma_2,\Delta_2)$ that ends in $l_0$ in the concrete
heap, and $n_0'$ in the abstract graph. Applying Lemma~\ref{lem:pathdec}
to this path (formally, instantiating $l$ by $\rho_1(x)$, $l'$ by
$\rho_1(y)$, $l_f$ by $l_0$, $\pi$ by $y$, and $\pi'$ by $\pi$) yields two
possibilities. The \emph{first alternative} is when $\pi$ is not modified by the
$f$-redirection (formally, $\evalexpr{\rho_1}{h_1}{\pi}{l_0}$).
Lemma~\ref{lem:subtype} then asserts the existence of a node $n_0$ that
$\pi$ evaluates to in $(\rho_1,h_1,\Gamma_1,\Delta_1)$ (formally,
$\tevalexpr{\Gamma_1}{\Delta_1}{\pi}{n_0}$ with
$n'_0=\sigma_1(n_0)$).
Moreover, by assumption $n_0$ and $l_0$ are in correspondence (formally,
$\Interpret{\rho_1}{h_1}{A_1}{\Gamma_1}{\Delta_1}{l_0}{n_0}$). To prove
that $l_0$ and $n'_0$ are in correspondence, we refer to the auxiliary
type interpretation rule in Fig.~\ref{fig:type:interpret}, and prove that
given a path $\pi_0$ that verifies $\evalexpr{\rho_2}{h_2}{\pi_0}{l_0}$,
the proposition $\tevalexpr{\Gamma_2}{\Delta_2}{\pi_0}{n'_0}$ holds.
Using Lemma~\ref{lem:pathdec} on $\pi_0$ (formally, instantiating $l$ by
$\rho_1(x)$, $l'$ by $\rho_1(y)$, $l_f$ by $l_0$, $\pi$ by $y$, and $\pi'$
by $\pi_0$), the only non-immediate case is when $\pi_0$ goes through
$f$. In this case, $\pi_0 = \pi_z.f.\pi_1.f.\ldots.f.\pi_n.f.\pi_f$, and
we can reconstruct this as a path to $n'_0$ in $(\Gamma_2,\Delta_2)$ by
assuming there are two nodes $n_x'$ and $n_y'$ such that
$\sigma_1(\Gamma_1(x)) = n_x'$ and $\sigma_1(\Gamma_1(y)) = n_y'$, and
observing:
\begin{enumerate}[$\bullet$]
\item $\evalexpr{\rho_1}{h_1}{\pi_z}{\rho_1(x)}$ thus
$\tevalexpr{\Gamma_1}{\Delta_1}{\pi_z}{\Gamma_1(x)}$ by assumption.
Because access path evaluation is monotonic wrt. mappings (a direct
consequence of clause~\eqref{eq:st2} in the definition of sub-typing), we
can derive $\tevalexpr{\Gamma_2}{\Delta_2}{\pi_z}{n_x'}$;
%
\item for $i\in [1,n]$, $\evalexpr{\rho_1}{h_1}{y.\pi_i}{\rho_1(x)}$ thus
by assumption $\tevalexpr{\Gamma_1}{\Delta_1}{y.\pi_i}{\Gamma_1(x)}$. By
again using the monotony of access path evaluation, we can derive
$\tevalexpr{\Gamma_2}{\Delta_2}{y.\pi_i}{n_x'}$;
%
\item $\evalexpr{\rho_1}{h_1}{y.\pi_f}{l_0}$ thus by assumption
$\tevalexpr{\Gamma_1}{\Delta_1}{y.\pi_f}{n_0}$. Hence
$\tevalexpr{\Gamma_2}{\Delta_2}{y.\pi_f}{n'_0}$ by $n'_0 = \sigma_1(n_0)$
and due to monotonicity.
%
\item in $\Delta_1[n,f \mapsto \Gamma_1(y)]$, $n = \Gamma_1(x)$ points to
$\Gamma_1(y)$ by $f$. Note that because the types
$\Gamma_1,\Delta_1,\Theta_1$ and $\Gamma_1,\Delta_1[n,f\mapsto
\Gamma_1(y)],\Theta_1$ share the environment $\Gamma_1$, we have
$\sigma_2(\Gamma_1(x)) = \sigma_1(\Gamma_1(x)) = n_x'$ and
$\sigma_2(\Gamma_1(y)) = \sigma_1(\Gamma_1(y)) = n_y'$.
Hence in $\Delta_2$, $n_x'$ points to $n_y'$ by $f$, thanks to the
monotonicity of $\sigma_2$.
\end{enumerate}
This concludes the proof of $\tevalexpr{\Gamma_2}{\Delta_2}{\pi_0}{n'_0}$.
The cases when $n_x'$ and $n_y'$ do not exist are easily dismissed;
we refer the reader to the Coq development for more details.
We now go back to our first application of Lemma~\ref{lem:pathdec} and
tackle the \emph{second alternative} -- when $\pi$ is indeed modified by
the $f$-redirection. Here $\pi$ can be decomposed into
$\pi_z.f.\pi_1.f.\ldots.f.\pi_n.f.\pi_f$. We first find the node in
$\Delta_1$ that corresponds to the location $l_0$: for this we need to
find a path in $(\rho_1,h_1,\Gamma_2,\Delta_2)$ that leads to both
$l_0$ and $n'_0$ (in order to apply Lemma~\ref{lem:subtype}, we can no
longer assume, as in the first alternative, that $\pi$ leads to $l_0$ in
$\st{\rho_1,h_1}$). Since $\evalexpr{\rho_1}{h_1}{y.\pi_f}{l_0}$, the path
$y.\pi_f$ might be a good candidate: we prove that it points to $n'_0$
in $\Delta_2$. Assume the existence of two nodes $n_x'$ and $n_y'$ in
$\Delta_2$ such that $n_x' = \sigma_1(\Gamma_1(x)) =
\sigma_2(\Gamma_1(x))$ and $n_y' = \sigma_1(\Gamma_1(y)) =
\sigma_2(\Gamma_1(y))$ (the occurrences of non-existence are dismissed by
reducing them, respectively, to the contradictory case when
$\Gamma_2(x)=\top$, and to the trivial case when $n'_0=\top$; the equality
between results of $\sigma_1$ and $\sigma_2$ stems from the same reasons
as in bullet $4$ previously). From the first part of the decomposition of
$\pi$ one can derive, by assumption, that $\pi_z$ leads to $\Gamma_1(x)$
in $\Delta_1$, and, by monotonicity, to $n_x'$ in $\Delta_2$ (formally,
$\evalexpr{\rho_1}{h_1}{\pi_z}{\rho_1(x)}$ entails
$\tevalexpr{\Gamma_1}{\Delta_1}{\pi_z}{\Gamma_1(x)}$ implies
$\tevalexpr{\Gamma_2}{\Delta_2}{\pi_z}{n_x'}$). Similarly, we can derive
that for any $i\in[1,n]$,
$\tevalexpr{\Gamma_1}{\Delta_1}{y.\pi_i}{\Gamma_1(x)}$.
We use monotonicity both to infer that in $\Delta_2$, $n_x'$ points to $n_y'$
by $f$, and that for any $i\in[1,n]$
$\tevalexpr{\Gamma_2}{\Delta_2}{y.\pi_i}{n_x'}$. From these three
statements on $\Delta_2$, we have
$\tevalexpr{\Gamma_2}{\Delta_2}{\pi_z.f.\pi_1.f.\ldots.f.\pi_n}{n_x'}$,
and by path decomposition $\tevalexpr{\Gamma_2}{\Delta_2}{y.\pi_f}{n_0'}$.
This allows us to proceed and apply Lemma~\ref{lem:subtype}, asserting the
existence of a node $n_0$ in ($\rho_1,h_1,\Delta_1,\Gamma_1$) that
$y.\pi_f$ evaluates to (formally,
$\tevalexpr{\Gamma_1}{\Delta_1}{y.\pi_f}{n_0}$ with $n_0' =
\sigma_1(n_0)$). Moreover, by assumption $n_0$ and $l_0$ are in
correspondence (formally,
$\Interpret{\rho_1}{h_1}{A_1}{\Gamma_1}{\Delta_1}{l_0}{n_0}$). The proof
schema from here on is quite similar to what was done for the first
alternative. As previously, we demonstrate that $l_0$ and $n'_0$ are in
correspondence by taking a path $\pi_0$ such that
$\evalexpr{\rho_2}{h_2}{\pi_0}{l_0}$ and proving
$\tevalexpr{\Gamma_2}{\Delta_2}{\pi_0}{n'_0}$. As previously, using
Lemma~\ref{lem:pathdec} on $\pi_0$ (formally, instantiating $l$ by
$\rho_1(x)$, $l'$ by $\rho_1(y)$, $l_f$ by $l_0$, $\pi$ by $y$, and $\pi'$
by $\pi_0$), the only non-immediate case is when $\pi_0$ goes through
$f$. In this case, $\pi_0 = \pi'_z.f.\pi'_1.f.\ldots.f.\pi'_n.f.\pi'_f$,
and we can reconstruct this as a path to $n'_0$ in $(\Gamma_2,\Delta_2)$
by observing:
\begin{enumerate}[$\bullet$]
\item $\evalexpr{\rho_1}{h_1}{\pi'_z}{\rho_1(x)}$ thus
$\tevalexpr{\Gamma_1}{\Delta_1}{\pi'_z}{\Gamma_1(x)}$ by assumption.
Monotonicity yields $\tevalexpr{\Gamma_2}{\Delta_2}{\pi'_z}{n_x'}$;
%
\item for $i\in [1,n]$, $\evalexpr{\rho_1}{h_1}{y.\pi'_i}{\rho_1(x)}$ thus
by assumption $\tevalexpr{\Gamma_1}{\Delta_1}{y.\pi'_i}{\Gamma_1(x)}$, and
by monotonicity we can derive $\tevalexpr{\Gamma_2}{\Delta_2}{y.\pi'_i}{n_x'}$;
%
\item $\evalexpr{\rho_1}{h_1}{y.\pi_f}{l_0}$ thus since $l_0$ and $n_0$
are in correspondence, $\tevalexpr{\Gamma_1}{\Delta_1}{y.\pi_f}{n_0}$.
Hence $\tevalexpr{\Gamma_2}{\Delta_2}{y.\pi_f}{n'_0}$ by $n'_0 =
\sigma_1(n_0)$ and by monotonicity.
\end{enumerate}
This concludes the proof of $\tevalexpr{\Gamma_2}{\Delta_2}{\pi_0}{n'_0}$,
hence the demonstration of the first premise.
We are now left with the second premise, stating the unicity of strong
node representation. Assume $n\in\Theta_2$ can be reached by two paths
$\pi$ and $\pi'$ in $\rho_2,h_2,\Gamma_2,\Delta_2$, we use
Lemma~\ref{lem:pathdec} to decompose both paths. The following cases
arise:
\begin {desCription}
\item\noindent{\hskip-12 pt\bf neither path is modified by the $f$-redirection:}\ formally,
$\evalexpr{\rho_1}{h_1}{\pi}{l} \wedge
\evalexpr{\rho_1}{h_1}{\pi'}{l'}$. Lemma~\ref{lem:subtype} ensures the
existence of $n'$ such that $n = \sigma_1(n')$ and
$\tevalexpr{\Gamma_1}{\Delta_1}{\pi}{n'} \wedge
\tevalexpr{\Gamma_1}{\Delta_1}{\pi'}{n'}$ Thus by assumption, $l=l'$.
%
\item\noindent{\hskip-12 pt\bf one of the paths is modified by the $f$-redirection:}\ without loss
of generality, assume $\evalexpr{\rho_1}{h_1}{\pi}{l}$, and
$\pi'=\pi_0.f.\pi_1.f.\ldots.f.\pi_n.f.\pi_f$ with
$\evalexpr{\rho_1}{h_1}{y.\pi_f}{l'}$. From the first concrete path
expression, Lemma~\ref{lem:subtype} ensures that there is $n'$ such that
$n=\sigma_1(n')$, and $\tevalexpr{\Gamma_1}{\Delta_1}{\pi}{n'}$.
Moreover, by assumption there exists a strong node $n''$ in $\Delta_1$
that $y.\pi_f$ leads to, and that is mapped to $n$ by $\sigma_1$
(formally, $n=\sigma_1(n'')$ and
$\tevalexpr{\Gamma_1}{\Delta_1}{y.\pi_f}{n''}$). Clause~\eqref{eq:st3}
of the definition of subtyping states the uniqueness of strong source
nodes: thus $n'=n''$, and we conclude by assumption that $l=l'$.
%
\item\noindent{\hskip-12 pt\bf both paths are modified by the $f$-redirection:}\
Lemma~\ref{lem:pathdec} provides two paths $\pi_f$ and $\pi'_f$ such
that $y.\pi_f$ leads to $l$, and $y.\pi'_f$ leads to $l'$ in the heap
(formally, $\evalexpr{\rho_1}{h_1}{y.\pi_\star}{l} \wedge
\evalexpr{\rho_1}{h_1}{y.\pi'_\star}{l'}$). As in the previous case, we
use the assumption and clause~\eqref{eq:st3} of main sub-typing
definition to exhibit the common node $n'$ in $\Delta_1$ that is
pointed to by both paths. By assumption, $l=l'$.
\end {desCription}
This concludes the demonstration of the second premise, and of the
$c\equiv \HAssign{x.f}{y}$ case in our global induction.
\qed
%
\begin{thm}[Type soundness]
If ${ \vdash p}$ then
all methods $m$ declared in the program $p$
are secure, \emph{i.e.}, respect their copy policy.
\end{thm}
\begin{CA}
For the proofs see~\cite{JensenKP10} and the companion Coq development.
\end{CA}
\begin{TR}
\begin{proof}
[See Coq proof \texttt{Soundness.soundness}~\cite{clone-webpage}]
Given a method $m$ and a copy signature $\Copy(X)\{\tau\}$ attached to it, we show that
for all heaps $h_1,h_2\in\Heap$, local environments
$\rho_1,\rho_2\in\Env$, locally allocated locations $A_1,A_2\in\Power(\Loc) $, and variables $x,y\in\Var$,
$$
(\HCall{x}{\var{cn}:X}{y}, \st{\rho_1,h_1,A_1}) \leadsto \st{\rho_2,h_2,A_2}
~~\text{implies}~~\rho_2,h_2,x \models \tau.$$
Following the rule defining the semantics of calls to copy methods, we
consider the situation where there exists a potential overriding
of $m$,
$\Copy(X')~\HAssign{m(a)}c$ such that
$$(c,\st{\rho_{\vnull}[a\mapsto\rho_1(y)],h_1,\emptyset}) \leadsto \st{\rho',h_2,A'}.$$
Writing $\tau'$ for the policy attached to $X'$, we know that $\tau \subseteq \tau'$.
By Lemma~\ref{lem:mono}, it is then sufficient to prove that
$\rho_2,h_2,x \models \tau'$ holds.
By typability of $\Copy(X')~\HAssign{m(a)}c$, there exist $\Gamma',\Gamma,\Delta,\Theta$ such that
$\Gamma'(\var{ret}) = n_\tau$,
$(\Gamma, \Delta, \Theta) \sqsubseteq (\Gamma',\Delta_\tau,\{n_\tau\})$ and
$[~\cdot\mapsto\bot][x\mapsto \topout], \emptyset, \emptyset \vdash c : \Gamma,\Delta,\Theta$.
Using Theorem~\ref{theo2}, we know that
$\InterpMMM{\rho'}{h_2}{A'}{\Gamma,\Delta,\Theta}$ holds and
by subtyping between
$(\Gamma, \Delta, \Theta)$ and $(\Gamma',\Delta_\tau,\{n_\tau\})$ we
obtain that $\InterpMMM{\rho'}{h_2}{A'}{\Gamma',\Delta_\tau,\{n_\tau\}}$ holds.
Theorem~\ref{th:policy-type} then immediately yields that $\rho_2,h_2,x \models \tau$.
\end{proof}
\end{TR}
\section{Inference}\label{sec:inference}
In order to type-check a method with the previous type system, it is necessary
to infer intermediate types at each loop header, conditional junction
points and weak field assignment point. A standard approach consists in turning the previous typing problem
into a fixpoint problem in a suitable sup-semi-lattice structure. This section
presents the lattice that we put on $(\Type,\sqsubseteq)$. Proofs are
generally omitted by lack of space but can be found in the companion report.
Typability is then checked by computing a suitable least-fixpoint in this
lattice. We end this section by proposing a widening operator that is
necessary to prevent infinite iterations.
\begin{TR}
\begin{lem}
The binary relation $\sqsubseteq$ is a preorder on $\Type$.
\end{lem}
\begin{proof}
The relation is reflexive because for all type $T\in\Type$, $T\sqsubseteq_\textit{id} T$.
The relation is transitive because if there exists types $T_1, T_2, T_3\in\Type$
such that $T_1\sqsubseteq_{\sigma_1}T_2$ and $T_2\sqsubseteq_{\sigma_2}T_3$
for some fusion maps $\sigma_1$ and $\sigma_2$ then $T_1\sqsubseteq_{\sigma_3}T_3$
for the fusion map $\sigma_3$ define by $\sigma_3(n) = \sigma_2(\sigma_1(n))$
if $\sigma_1(n)\in\Node$ or $\top$ otherwise.
\end{proof}
\end{TR}
We write $\equiv$ for the equivalence relation defined by $T_1\equiv
T_2$ if and only if $T_1\sqsubseteq T_2$ and $T_2\sqsubseteq
T_1$. Although this entails that $\sqsubseteq$ is a partial order structure on top of
$(\Type,\equiv)$, equality and order testing remains
difficult using only this definition. Instead of considering the
quotient of $\Type$ with $\equiv$, we define a notion of
\emph{well-formed} types on which $\sqsubseteq$ is antisymmetric. To
do this, we assume that the set of nodes, variable names and field
names are countable sets and we write $n_i$ (resp. $x_i$ and $f_i$)
for the $i$th node (resp. variable and field). A type
$(\Gamma,\Delta,\Theta)$ is \emph{well-formed} if every node in
$\Delta$ is reachable from a node in $\Gamma$ and the nodes in
$\Delta$ follow a canonical numbering based on a breadth-first
traversal of the graph.
Any type can be \emph{garbage-collected} into a canonical well-formed
type by removing all unreachable nodes from variables
and renaming all remaining nodes using a fixed
strategy based on a total ordering on variable names and field names
and a breadth-first traversal. We call this transformation $\gc$. The
following example shows the effect of $\gc$ using a canonical numbering.
\begin{center}
\includegraphics[width=.7\textwidth]{exgc}
\end{center}
\noindent
Since by definition, $\sqsubseteq$ only deals with reachable nodes, the $\gc$
function is a $\equiv$-morphism and respects type interpretation.
This means that an inference engine can at any time replace a type by a garbage-collected
version. This is useful to perform an equivalence test in order to
check whether a fixpoint iteration has ended.
\begin{lem}
For all well-formed types $T_1,T_2\in\Type$,
$T_1\equiv T_2~~\text{iff}~~ T_1=T_2$.
\end{lem}
\begin{defi}
Let $\sqcup$ be an operator that merges two types according to the algorithm
in Fig.~\ref{fig:join:code}.
\end{defi}
\begin{figure}
\begin{minipage}[c]{0.48\linewidth}
\lstset{basicstyle=\ttfamily\scriptsize\color{black},float,emph={fusion,succ,updatemap,lift,ground,elementOf},emphstyle=\underbar,mathescape=true}
\begin{lstlisting}
// Initialization.
// $\alpha$-nodes are sets in $\BaseType$.
// $\alpha$-transitions can be
// non-deterministic.
$\alpha$ = lift($\Gamma_1$,$\Gamma_2$,$\Delta_1 \cup \Delta_2$)
// Determinize $\alpha$-transitions:
// start with the entry points.
for $\{(x,t);(x,t')\} \subseteq (\Gamma_1\times\Gamma_2)$ {
fusion($\{t,t'\}$)
}
// Determinize $\alpha$-transitions:
// propagate inside the graph.
while $\exists u\in\alpha,\exists f\in\Field, |succ(u,f)| > 1$ {
fusion(succ($u$,$f$))
}
// $\alpha$ is now fully determinized:
// convert it back into a type.
$(\Gamma,\Delta,\Theta)$ = ground($\Gamma_1$,$\Gamma_2$,$\alpha$)
\end{lstlisting}
\end{minipage}
%
\begin{minipage}[c]{0.48\linewidth}
\lstset{basicstyle=\ttfamily\scriptsize\color{black},float,emph={fusion,succ,updatemap,lift,ground,elementOf},emphstyle=\underbar,mathescape=true}
\begin{lstlisting}
// $S$ is a set of $t\in\BaseType$.
// $\llparenthesis S\rrparenthesis$ denotes a node labelled by $S$.
void fusion ($S$) {
// Create a new $\alpha$-node.
NDG node n = $\llparenthesis S\rrparenthesis$
$\alpha \gets \alpha + n$
// Recreate all edges from the fused
// nodes on the new node.
for $t \in S$ {
for $f \in \Field$ {
if $\exists u, \alpha(t,f)=u$ {
// Outbound edges.
$\alpha \gets \alpha$ with $(n,f) \mapsto u$
}
if $\exists n', \alpha(n',f)=t$ {
// Inbound edges.
$\alpha \gets \alpha$ with $(n',f) \mapsto n$
}}
// Delete the fused node.
$\alpha \gets \alpha - t$
}}
\end{lstlisting}
\end{minipage}
\caption{Join Algorithm}
\label{fig:join:code}
\end{figure}
The procedure has $T_1=(\Gamma_1,\Delta_1,\Theta_1)$ and
$T_2=(\Gamma_2,\Delta_2,\Theta_2)$ as input, then takes the following steps.
\begin{enumerate}[(1)]
\item \label{enum:cell} It first makes the disjunct union of $\Delta_1$ and
$\Delta_2$ into a non-deterministic graph (\NDG) $\alpha$, where nodes are
labelled by sets of elements in $\BaseType$. This operation is performed by the
\ttt{lift} function, that maps nodes to singleton nodes, and fields to
transitions.
\item \label{enum:init} It joins together the nodes in $\alpha$ referenced
by $\Gamma_i$ using the \ttt{fusion} algorithm\footnote{Remark that
$\Gamma_i$-bindings are not represented in $\alpha$, but that node set
fusions are trivially traceable. This allows us to safely ignore $\Gamma_i$
during the following step and still perform a correct graph
reconstruction.}.
\item \label{enum:fus} Then it scans the \NDG and merges all
nondeterministic successors of nodes.
\item \label{enum:decell} Finally it uses the \ttt{ground} function to
recreate a graph $\Delta$ from the now-\emph{deterministic} graph $\alpha$.
This function operates by pushing a node set to a node labelled by the
$\leq_\sigma$-sup of the set. The result environment $\Gamma$ is derived
from $\Gamma_i$ and $\alpha$ before the $\Delta$-reconstruction.
\end{enumerate}
All state fusions are recorded in a map $\sigma$ which binds nodes in
$\Delta_1\cup\Delta_2$ to nodes in $\Delta$.
\begin{TR}
Figure~\ref{fig:auxfun} contains the auxiliary functions used in the above
procedure. Figure~\ref{fig:join:example} unfolds the algorithm on a small
example.
\end{TR}
\begin{TR}
\begin{figure}
\begin{minipage}{0.48\linewidth}
\lstset{basicstyle=\ttfamily\scriptsize\color{black},float,emph={fusion,succ,updatemap,lift,ground,elementOf},emphstyle=\underbar,mathescape=true}
\begin{lstlisting}
// Convert a type to an NDG
NDG lift ($\Gamma_1$,$\Gamma_2$,$\Delta$) {
NDG $\alpha$ = undef
for $x \in \text{Var}$ {
$\alpha \gets \alpha + \llparenthesis\{\Gamma_1(x)\}\rrparenthesis + \llparenthesis\{\Gamma_2(x)\}\rrparenthesis$
}
for $n \in \Delta$ {
if $\exists f\in\text{Fields}, \exists b\in\text{BaseType}, \Delta[n,f]=b$ {
$\alpha \gets \alpha + \llparenthesis\{n\}\rrparenthesis + \llparenthesis\{b\}\rrparenthesis$
$\alpha \gets \alpha$ with $(\llparenthesis \{n\}\rrparenthesis,f) \mapsto \llparenthesis \{b\}\rrparenthesis$
}}
return $\alpha$
}
\end{lstlisting}
\end{minipage}
\begin{minipage}{0.48\linewidth}
\lstset{basicstyle=\ttfamily\scriptsize\color{black},float,emph={fusion,succ,updatemap,lift,ground,elementOf},emphstyle=\underbar,mathescape=true}
\begin{lstlisting}
// Convert an NDG to a type
$\Type$ ground ($\Gamma_1$,$\Gamma_2$,$\alpha$) {
($\Var\to\BaseType$) $\Gamma$ = $\lambda x.\bot$
$\LSG$ $\Delta$ = undef
for $N\in \alpha$ {
for $x\in \Var, \Gamma_1(x)\in N \vee \Gamma_2(x)\in N$ {
$\Gamma \gets \Gamma$ with $x \mapsto \nmlz{N}$
}}
for $N,N' \in \alpha$ {
if $\alpha(N,f)=N'$ {
$\Delta \gets \Delta$ with $(\nmlz{N},f) \mapsto \nmlz{N'}$
}}
return $(\Gamma,\Delta)$
}
\end{lstlisting}
\end{minipage}
%
\begin{minipage}[c]{0.40\linewidth}
\lstset{basicstyle=\ttfamily\scriptsize\color{black},float,emph={fusion,succ,updatemap,lift,ground,elementOf},emphstyle=\underbar,mathescape=true}
\begin{lstlisting}
// $\leq_\sigma$-sup function
BaseType $\downarrow$ (N) {
if $\top\in N$ return $\top$
else if $\forall c\in N, c=\bot$ return $\bot$
else return freshNode()
}
\end{lstlisting}
\end{minipage}
\caption{Auxiliary join functions}
\label{fig:auxfun}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.6\linewidth]{algojoinexample1}
\caption{An example of join}
\label{fig:join:example}
\end{figure}
\end{TR}
\begin{thm}
The operator $\sqcup$ defines a sup-semi-lattice on types.
\end{thm}
\proof
\begin{CA}
See~\cite{JensenKP10}.
\end{CA}
\begin{TR}
First note that $\sigma_1$ and $\sigma_2$, the functions associated with the
two respective sub-typing relations, can easily be reconstructed from
$\sigma$.
\begin{desCription}
\item\noindent{\hskip-12 pt\bf Upper bound:}\
Let $(T,\sigma)= T_1\sqcup T_2$: we prove that $T_1 \sqsubseteq T$ and $T_2
\sqsubseteq T$.
%
Hypothesis~\eqref{eq:st2} is discharged by case analysis, on $t_1$ and on
$t_2$. The general argument used is that the join algorithm does not
delete any edges, thus preserving all paths in the initial graphs.
\item\noindent{\hskip-12 pt\bf Least of the upper bounds:}\
Let $(T,\sigma)= T_1 \sqcup T_2$. Assume there exists
$T'$ such that $T_1 \sqsubseteq T'$ and $T_2 \sqsubseteq T'$. Then we prove
that $T \sqsubseteq T'$.
%
The proof consists in checking that the join algorithm produces, at each
step, an intermediary pseudo-type $T$ such that $T\sqsubseteq T'$. The
concrete nature of the algorithm drives the following, more detailed
decomposition.
\newcommand{\sqsubseteq_\tau}{\sqsubseteq_\tau}
\newcommand{\sqsubseteq_{\tau^{*}}}{\sqsubseteq_{\tau^{*}}}
\begin{enumerate}[(1)]
\item \label{pr:autorel}
Given a function $\sigma$,
define a state mapping function $\leq_\tau$, and a
$\sqsubseteq_\sigma$-like relation $\sqsubseteq_\tau$ on non-deterministic graphs.
The aim with this relation is to emulate the properties of
$\sqsubseteq_\sigma$, lifting the partial order $\leq_\sigma$ on nodes to
sets of nodes (cells).
Lift $T'$ into an \NDG $\alpha'$.
\item \label{pr:unioncel}
Using the subtyping relations between $T_1$, $T_2$, and $T'$, establish
that the disjunct union and join steps produce an intermediary
\NDG $\beta$ such that $\beta \sqsubseteq_\tau \alpha'$.
\item \label{pr:fusion}
Ensure that the fusions operated by the join algorithm in $\beta$
produce an \NDG $\gamma$ such that $\gamma \sqsubseteq_\tau \alpha'$. This
is done by case analysis, and depending on whether the fusion takes place
during the first entry point processing phase, or if it occurs later.
\item \label{pr:decel}
Using the $\BaseType$-lattice, show that the ground operation on the
\NDG $\gamma$ produces a type $T$ that is a sub-type of $T'$.\qed
\end{enumerate}
\end{desCription}
\end{TR}
\noindent
The poset structure does have infinite ascending chains, as shown by
the following example.
\begin{center}
\includegraphics[width=.9\textwidth]{ascendingchain}
\end{center}
Fixpoint iterations may potentially result in such an infinite chain
so we have then to rely on a widening~\cite{CousotCousot77} operator to enforce
termination of fixpoint computations. Here we follow a pragmatic approach
and define a widening operator $\nabla\in\Type\times\Type\to\Type$ that takes
the result of $\sqcup$ and that collapses together (with the operator
\ttt{fusion} defined above) any node $n$ and its predecessors such that the
minimal path reaching $n$ and starting from a local variable is of length at
least 2.
\begin{TR}
This is illustrated by the following example.
\begin{center}
\includegraphics[width=.9\textwidth]{widen}
\end{center}
This ensures the termination of the fixpoint iteration because the number of
nodes is then bounded by $2N$ with $N$ the number of local variables
in the program.
\end{TR}
\section{Experiments}
The policy language and its enforcement mechanism have been implemented in the
form of a security tool for Java bytecode. Standard Java \ttt{@interface}
declarations are used to specify native annotations, which enable development
environments such as Eclipse to parse, identify and auto-complete
\ttt{@Shallow}, \ttt{@Deep}, and \ttt{@Copy} tags. Source code annotations
are being made accessible to bytecode analysis frameworks. Both the policy
extraction and enforcement components are implemented using the Javalib/Sawja
static analysis libraries\footnote{\url{http://sawja.inria.fr}} to derive
annotations and intermediate code representations, and to facilitate the
creation of an experimental Eclipse plugin.
In its standard mode, the tool performs a modular verification of
annotated classes. We have run experiments on several
classes of the standard library (specially in the package \ttt{java.util})
and have successfully checked realistic copy signatures for them.
These experiments have also
confirmed that the policy enforcement mechanism facilitates
re-engineering into more compact implementations of cloning methods in
classes with complex dependencies, such as those forming the
\ttt{gnu.xml.transform} package. For example, in the \ttt{Stylesheet}
class an inlined implementation of multiple deep copy methods for half
a dozen fields can be rewritten to dispatch these functionalities to
the relevant classes, while retaining the expected copy policy. This
is made possible by the modularity of our enforcement mechanism, which
validates calls to external cloning methods as long as their
respective policies have been verified. As expected, some cloning methods are
beyond the reach of the analysis. We have identified
one such method in GNU Classpath's \ttt{TreeMap} class, where the
merging of information at control flow merge points destroys too much
of the inferred type graph. A disjunctive form of abstraction seems
necessary to verify a deep copy annotation on such programs.
The analysis is also capable of processing un-annotated methods, albeit with
less precision than when copy policies are available---this is because it
cannot rely on annotations to infer external copy method types.
Nevertheless, this capability was used to test the tool on two large
code bases. The 17000 classes in Sun's \ttt{rt.jar} and the 7000
others in the GNU Classpath have passed our scanner
un-annotated. Among the 459 \ttt{clone()} methods we found in these
classes, only 15 have been rejected because of an assignment or method
call on non-local memory, as explained in
Section~\ref{sec:typesystem}. %
Assignment on non-local memory means here that the copying method is
updating fields of other objects than the result of the copy
itself. For such examples, our shape analysis seems too coarse to track
the dependencies between sources and copy targets.
For 78 methods we were unable to infer
the minimal, shallow signature $\{\}$ (the same signature as
\ttt{java.lang.Object.clone()}). In some cases, for instance in the
\ttt{DomAttr} class, this will happen when the copy method returns the result
of another, unannotated method call, and can be mitigated with additional copy
annotations. In other cases, merges between abstract values result in
precision losses: this is, for instance, the case for the \ttt{clone} method
of the TreeMap class, as explained above.
Our prototype confirms the efficiency of the enforcement technique:
these verifications took about 25s to run on stock hardware.
The prototype, the Coq formalization and proofs, as well as examples of annotated
classes can be found at \url{http://www.irisa.fr/celtique/ext/clones}.
\section{Related Work}
\label{sec:relatedwork}
Several proposals for programmer-oriented annotations of Java programs
have been published following Bloch's initial proposal of an annotation
framework for the Java language \cite{Bloch04}.
These proposals define the syntax of the annotations but often leave
their exact semantics unspecified. A notable exception is the set of
annotations concerning non-null annotations \cite{Fahndrich03} for which a precise
semantic characterization has emerged~\cite{hubert08}.
Concerning security, the GlassFish environment in Java offers program annotations of
members of a class (such
as \ttt{@DenyAll} or \ttt{@RolesAllowed}) for implementing role-based access control to
methods.
To the best of our knowledge, the current paper is the first to
propose a formal, semantically founded framework for secure cloning through program
annotation and static enforcement. The closest work in
this area is that of Anderson \emph{et al.}~\cite{AndersonGayNaik:PLDI09}
who have designed an annotation system for C data structures in order
to control sharing between threads. Annotation policies are enforced by
a mix of static and run-time verification. On the run-time verification
side, their approach requires an operator that can dynamically
``cast'' a cell to an unshared structure.
In contrast, our approach offers a completely static mechanism with
statically guaranteed alias properties.
Aiken \emph{et al.} proposes an analysis for checking and inferring
local non-aliasing of data~\cite{Aiken:03}. They propose to annotate C function parameters with
the keyword \ttt{restrict} to ensure that no other aliases to the data
referenced by the parameter are used during the execution of the
method. A type and effect system is defined for enforcing this
discipline statically. This analysis differs from ours in that it
allows aliases to exist as long as they are not used whereas we aim at
providing guarantees that certain parts of memory are without
aliases.
The properties tracked by our type system are close to
escape analysis~\cite{Blanchet99,ChoiGSSM99} but the analyses differ
in their purpose. While escape
analysis tracks locally allocated objects and tries to detect those
that do not escape after the end of a method execution, we are
specifically interested in tracking
locally allocated objects that escape from the result of a method, as
well as analyse their dependencies with respect to parameters.
Our static enforcement technique falls within the large area of static
verification of heap properties. A substantial amount of research has been
conducted here, the most prominent being region calculus~\cite{TofteTalpin97},
separation logic~\cite{OHearnYR04} and shape analysis~\cite{SagivRW02}. Of
these three approaches, shape analysis comes closest in its use of shape
graphs. Shape analysis is a large framework that allows to infer complex
properties on heap allocated data-structures like absence of dangling pointers
in C or non-cyclicity invariants. In this approach, heap cells are abstracted
by shape graphs with flexible object abstractions. Graph nodes can either
represent a single cell, hence allowing strong updates, or several cells
(summary nodes). \emph{Materialization} allows to split a summary node during
cell access in order to obtain a node pointing to a single cell.
The shape graphs that we use are not intended to do full shape analysis but
are rather specialized for tracking sharing in locally allocated objects. We
use a different naming strategy for graph nodes and discard all information
concerning non-locally allocated references. This leads to an analysis which
is more scalable than full shape analysis, yet still powerful enough
for verifying complex copy policies as demonstrated in the concrete case study
\ttt{java.util.LinkedList}.
Noble \emph{et al.}~\cite{Noble:98:Flexible} propose a prescriptive
technique for characterizing the aliasing, and more generally, the
topology of the object heap in object-oriented programs. This
technique is based on alias modes which have evolved into the notion
of ownership types \cite{Clarke:98:Ownership}. In this setting, the
annotation \ttt{@Repr} is used to specify that an object is
\emph{owned} by a specific object. It is called a
\emph{representation} of its owner. After such a declaration, the
programmer must manipulate the representation in order to ensure that
any access path to this object should pass trough its owner. Such a
property ensures that a \ttt{@Repr} field must be a \ttt{@Deep}
field in any copying method. Still, a \ttt{@Deep} field is not
necessarily a \ttt{@Repr} field since a copying method may want to
deeply clone this field without further interest in the global alias
around it. Cloning seems not to have been studied further in the
ownership community and ownership type checking is generally not
adapted to flow-sensitive verification, as required by the programming
pattern exhibited in existing code. In this example, if we annotate
the field \ttt{next} and \ttt{previous} with \ttt{@Repr}, the
\ttt{clone} local variable will not be able to keep the same
ownership type at line 12 and at line 26. Such a an example would
require ownership type systems to track the update of a reference in
order to catch that any path to a representation has been erased in
the final result of the method.
We have aimed at annotations that together with static
analysis allows to verify existing cloning methods.
Complementary to our approach,
Drossopoulou and Noble~\cite{pubsdoc:clonesPre} propose a system
that generate cloning methods from annotation inpired by ownership types.
\section{Conclusions and Perspectives}
Cloning of objects is an important aspect of exchanging data with
untrusted code. Current language technology for cloning does not
provide adequate means for defining and enforcing a secure copy policy
statically; a task which is made more difficult by important
object-oriented features such as inheritance and re-definition of
cloning methods. We have presented a flow-sensitive type system for
statically enforcing copy policies defined by the software developer
through simple program annotations. The annotation formalism deals
with dynamic method dispatch and addresses some of the problems posed
by redefinition of cloning methods in inheritance-based object
oriented programming language (but see
Section~\ref{sec:pol:limitations} for a discussion of current
limitations). The verification technique is designed to enable modular
verification of individual classes. By specifically targeting the
verification of copy methods, we consider a problem for which it is
possible to deploy a localized version of shape analysis that avoids
the complexity of a full shape analysis framework. This means that our
method can form part of an extended, security-enhancing Java byte code
verifier which of course would have to address, in addition to secure cloning, a wealth of other
security policies and security guidelines as \emph{e.g.}, listed on
the CERT web site for secure Java
programming~\cite{CertGuidelines:2010}.
The present paper constitutes the formal foundations for a secure
cloning framework. All theorems except those of
Section~\ref{sec:inference} have been mechanized in the Coq proof
assistant. Mechanization was particularly challenging because
of the storeless nature of our type interpretation but in the end
proved to be of great help to get the
soundness arguments right.
Several issues merit further investigations in order
to develop a full-fledged software security verification tool.
The extension of the policy language to be able to impose policies on
fields defined in sub-classes should be developed
(\emph{cf.}~discussion in Section~\ref{sec:pol:limitations}). We
believe that the
analysis defined in this article can be used to enforce such policies
but their precise semantics remains to be defined.
In the current approach, virtual methods without copy
policy annotations are considered as black boxes that may modify any object reachable
from its arguments. An extension of our copy annotations to virtual
calls should be worked out if we want to enhance our enforcement
technique and accept more secure copying methods. More advanced verifications
will be possible if we
develop a richer form of type signatures for methods where the formal
parameters may occur in copy policies, in order to express a relation
between copy properties of returning objects and parameter fields.
The challenge here is to provide sufficiently expressive signatures
which at the same time remain humanly readable software contracts.
The current formalisation has been developed for a sequential model of
Java. We conjecture that the extension to interleaving multi-threading
semantics is feasible and that it can be done without making major changes to the type system
because we only manipulate thread-local pointers.
An other line of work could be to consider the correctness of
\ttt{equals()} methods with respect to copying methods, since we
generally expect \ttt{x.clone().equals(x)} to be \ttt{true}. The
annotation system is already in good shape for such a work but a
static enforcement may require a major improvement of our specifically
tailored shape analysis.
\section{Acknowledgement}
We wish to thank the ESOP’11 and LMCS anonymous reviewers for their
helpful comments on this article. We specially thank the anonymous reviewer who suggested the
\ttt{EvilList} example presented in Section~\ref{sec:pol:limitations}.
\bibliographystyle{plain}
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\section{Introduction}
\label{sec:introduction}
Let $X$ be a complex projective $K3$ surface and let $T\colon X\to X$ be an automorphism with positive topological entropy $h>0$.
Thanks to a foundational results of Cantat \cite{Cantat}, there are closed positive currents $\eta_{\pm}$ which satisfy
\begin{equation*}
T^*\eta_{\pm}=e^{\pm h}\eta_{\pm},
\end{equation*}
and are normalized so that their cohomology classes satisfy $[\eta_+]\cdot[\eta_-]=1$. The classes $[\eta_{\pm}]$ belong to the boundary of the ample cone of $X$ and have vanishing self-intersection.
These eigencurrents have H\"older continuous local potentials \cite{DS}, and their wedge product $\mu=\eta_+\wedge\eta_-$ is well-defined by Bedford--Taylor theory, and is the unique $T$-invariant probability measure with maximal entropy.
When $(X,T)$ is not a Kummer example, it was shown by Cantat-Dupont \cite{CD} (with a new proof by the authors \cite{FT2} that also covers the nonprojective case) that $\mu$ is singular with respect to the Lebesgue measure $\dVol$; therefore there exists a Borel set of zero Lebesgue measure carrying the entire mass of $\mu$.
The authors conjectured (see \cite[Conjecture 7.3]{Tosatti_survey}) that the topological support $\supp \mu$ should nonetheless be equal to all of $X$, see also Cantat's \cite[Question 3.4]{Cantat2018_Automorphisms-and-dynamics:-a-list-of-open-problems}.
If this were true, it would also imply the same for each of the currents: $\supp \eta_{\pm}=X$.
In \cite{FilipTosatti2021_Canonical-currents-and-heights-for-K3-surfaces} the authors showed that, under mild assumptions on $X$, the eigencurrents $\eta_{\pm}$ fit into a continuous family of closed positive currents with continuous local potentials whose cohomology classes sweep out the boundary of the ample cone, perhaps after blowing up the boundary at the rational rays.
We called the corresponding closed positive currents the \emph{canonical currents}.
It is then natural to wonder whether \emph{all} such canonical currents are fully supported on $X$.
In this note we show that this is in fact not the case.
Namely, we show in \autoref{thm:gaps_in_the_support_of_canonical_currents} below:
\begin{theoremintro}[Gaps in the support]
\label{thm_intro:gaps_in_the_support}
There exists a projective K3 surface $X$ of type $(2,2,2)$, and an uncountable dense $F_{\sigma}$ set of rays $F\subset \partial \Amp(X)$ in the boundary of its ample cone, such that for every $f\in F$ the topological support of the unique canonical current $\eta_f$ is not all of $X$.
\end{theoremintro}
\noindent Note that because the rank of the Picard group of a very general K3 surface of type $(2,2,2)$ is $3$, there is no need to blow up the rational directions on the boundary.
Moreover, the canonical currents in the rational directions (i.e. those where the ray spanned by $f$ intersects $H^2(X,\mathbb{Q})$ nontrivially) have full support, see \autoref{rmk:avoidance_of_parabolic_points}.
The above result can be strengthened to show that there exist K3 surfaces defined over $\bR$, such that the supports of certain canonical currents are disjoint from the (nonempty) real locus (see \autoref{thm:full_gaps_in_the_real_locus}):
\begin{theoremintro}[Full gaps in the real locus]
\label{thm_intro:full_gaps_real_locus}
There exists a projective K3 surface $X$ of type $(2,2,2)$ defied over $\bR$ with $X(\bR)\neq \emptyset$, and an uncountable dense $F_{\sigma}$ set of rays $F\subset \partial \Amp(X)$ in the boundary of its ample cone, such that for every $f\in F$ the topological support of the unique canonical current $\eta_f$ is disjoint from $X(\bR)$.
\end{theoremintro}
In the examples we construct, $X(\bR)$ is homeomorphic to a $2$-sphere.
McMullen \cite[Thm.~1.1]{McMullen2002_Dynamics-on-K3-surfaces:-Salem-numbers-and-Siegel-disks} constructed \emph{nonprojective} K3 surfaces with automorphisms whose eigencurrents $\eta_{\pm}$ are not fully supported.
In fact, his examples have a Siegel disc: an invariant neighborhood of a fixed point on which the dynamics is holomorphically conjugate to a rotation, and where $\eta_{\pm}$ thus vanish.
Let us also note that Moncet \cite[Thm.~A]{Moncet2013_Sur-la-dynamique-des-diffeomorphismes-birationnels-des-surfaces-algebriques-reelles:-ensemble} constructed a birational automorphism of a rational surface $X$ defined over $\bR$, with positive dynamical degree and Fatou set containing $X(\bR)$.
Despite our \autoref{thm_intro:gaps_in_the_support} above, we do maintain hope that on projective K3 surfaces, the measure of maximal entropy (and therefore also $\eta_{\pm}$) is fully supported.
\subsubsection*{Acknowledgments}
\label{sssec:acknowledgements}
We are grateful to Roland Roeder for conversations on his work with Rebelo \cite{RebeloRoeder2021_Dynamics-of-groups-of-birational-automorphisms-of-cubic-surfaces-and-Fatou/Julia} that inspired this note, and to Serge Cantat for detailed feedback that improved our exposition.
We are also grateful to Serge Cantat for suggesting to combine our methods with an example of Moncet that led to \autoref{thm_intro:full_gaps_real_locus}.
This research was partially conducted during the period the first-named author served as a Clay Research Fellow, and was partially supported by the National Science Foundation under Grant No. DMS-2005470. The second-named author was partially supported by NSF grant DMS-2231783, and part of this work was conducted during his visit to the Center for Mathematical Sciences and Applications at Harvard University, which he would like to thank for the hospitality. This note is dedicated to the memory of Nessim Sibony, a dear colleague and friend, whose contributions to holomorphic dynamics and several complex variables remain an inspiration to us. He is greatly missed.
\section{Gaps in the support of canonical currents}
\label{sec:gaps_in_the_support_of_canonical_currents}
\paragraph{Outline}
We recall some constructions and estimates, originally based on an idea of Ghys \cite{Ghys1993_Sur-les-groupes-engendres-par-des-diffeomorphismes-proches-de-lidentite}, itself inspired by the Zassenhaus lemma on commutators of small elements in Lie groups.
In brief, the idea is that if two germs of holomorphic maps near the origin are close to the identity, then their commutator is even closer, and the estimates are strong enough to allow for an iteration argument.
The precise estimates that we need for \autoref{thm_intro:gaps_in_the_support} are contained in \autoref{prop:fixed_points_with_small_derivative}, and we follow Rebelo--Roeder \cite{RebeloRoeder2021_Dynamics-of-groups-of-birational-automorphisms-of-cubic-surfaces-and-Fatou/Julia} to establish the needed bounds.
We then recall some basic facts concerning the geometry of K3 surfaces in \autoref{ssec:2_2_2_surfaces_and_canonical_currents}, and establish the existence of gaps in the support of some of their canonical currents in \autoref{ssec:an_example_with_slow_commutators}.
\subsection{Commutator estimates}
\label{ssec:commutator_estimates}
In this section we introduce notation and collect some results, stated and proved by Rebelo--Roeder \cite{RebeloRoeder2021_Dynamics-of-groups-of-birational-automorphisms-of-cubic-surfaces-and-Fatou/Julia} but which have also been known and used in earlier contexts, e.g. by Ghys \cite{Ghys1993_Sur-les-groupes-engendres-par-des-diffeomorphismes-proches-de-lidentite}.
The results are concerned with commutators of germs of holomorphic maps in a neighborhood of $0\in \bC^d$.
\subsubsection{Derived series and commutators}
\label{sssec:derived_series}
Fix a set $S$, whose elements we regard as formal symbols which can be juxtaposed to form words.
Assume that $S$ is equipped with a fixed-point-free involution $s\mapsto s^{-1}$, i.e. any element has a unique corresponding ``inverse'' in the set.
Define the ``derived series'' of sets by
\[
S^{(0)}:= S \quad S^{(n+1)}:= \left[S^{(n)},S^{(n)}\right]
\]
where $[A,B]$ denotes the set of commutators $[a,b]:=aba^{-1}b^{-1}$ with $a\in A,b\in B$, and we omit the trivial commutators $[a,a^{-1}]$.
Denote the disjoint union by $S^{\bullet}:=\coprod_{n\geq 0} S^{(n)}$.
We will use the same notation in the case of a pseudogroup.
We also collect the next elementary result:
\begin{proposition}[Fast ramification]
\label{prop:fast_ramification}
Let $F_k$ denote the free group on $k$ generators $a_1,\ldots, a_k$.
Set $S^{(0)}:=\{a_1,\ldots, a_k,a_1^{-1},\ldots, a_k^{-1}\}$.
Then the ${\binom{k}{2}}$ elements $[a_i,a_j]\in S^{(1)}$ with $i<j$, generate a free subgroup of rank $\binom{k}{2}$ inside $F_k$.
\end{proposition}
\begin{proof}
Observe that $[a_i,a_j]^{-1}=[a_j,a_i]$.
Therefore, it suffices to check that any word of the form
\[
[a_{i_1},a_{j_1}]\cdots [a_{i_l},a_{j_l}] \cdots [a_{i_N},a_{j_N}]
\]
is never trivial, subject to the condition that for any $l$ the consecutive pairs don't obviously cancel out.
Equivalently, we assume that for any $l$ either $a_{i_l}\neq a_{j_{l+1}}$ or $a_{j_l}\neq a_{i_{l+1}}$.
But this can be immediately verified by writing out the expression in the generators $a_{\bullet}$, and observing that the only cancellations can occur if $a_{j_l}=a_{i_{l+1}}$.
However, the next cancellation is excluded by assumption so the reduced word has at least $4N-2(N-1)=2N+2$ letters and is nonempty.
\end{proof}
Later on, we will apply iteratively this proposition, starting with $k\geq 4$, an inequality which is preserved by $k\mapsto {\binom{k}{2}}$.
\subsubsection{Pseudogroup of transformations}
\label{sssec:pseudo_groups_of_transformations}
Let $B_0(\ve)\subset \bC^d$ denote the ball of radius $\ve>0$ centered at the origin in $\bC^d$.
Let $\gamma_1,\ldots, \gamma_k$ be injective holomorphic maps $\gamma_i\colon B_0(\ve)\to \bC^d$, which are thus biholomorphisms onto their ranges $\cR_{\gamma_i}:=\gamma_i(B_0(\ve))$.
Let $S$ denote the set with $2k$ symbols $\gamma_1,\ldots,\gamma_k,\gamma_1^{-1},\ldots,\gamma_k^{-1}$.
With $S^{\bullet}$ as in \autoref{sssec:derived_series}, assign to any element $\gamma\in S^{\bullet}$, whenever possible, the holomorphic map also denoted by $\gamma\colon \cD_{\gamma}\to \cR_{\gamma}$ with open sets $\cD_{\gamma},\cR_{\gamma}\subset \bC^d$ by expressing $\gamma$ in reduced form in the letters from $S$, and shrinking the domains/ranges according to the word.
For certain elements $\gamma$, these might well be empty sets.
Denote by $\id$ the identity transformation and by $\norm{f}_{C^0(K)}$ the supremum norm of a function or map $f$ on a set $K$.
\begin{theorem}[Common domain of definition]
\label{thm:common_domain_of_definition}
For any given $0<\ve\leq 1$, if
\[
\norm{\gamma_i^{\pm 1}-\id}_{C^0(B_{0}(\ve))} \leq \frac{\ve}{32},
\text{ for }i=1,\ldots,k
\]
then for every $n\geq 0$ and every $\gamma\in S^{(n)}$, its domain $\cD_{\gamma}$ contains $B_0(\ve/2)$ and furthermore it satisfies
\[
\norm{\gamma-\id}_{C^0(B_0(\ve/2))} \leq \frac{\ve}{2^n\cdot 32}.
\]
\end{theorem}
\noindent This result is proved as in \cite[Prop.~7.1]{RebeloRoeder2021_Dynamics-of-groups-of-birational-automorphisms-of-cubic-surfaces-and-Fatou/Julia} or \cite[Prop.~3.1]{Rebelo_Reis}, which state it for $k=2$.
Indeed, the estimates in the proof only involve the estimates on the ``seed'' transformations $\gamma_i$, and not their cardinality. We include the proof for the reader's convenience.
\begin{proof}We will show by induction on $n\geq 0$ that for every $\gamma\in S^{(n)}$ its domain $\cD_{\gamma}$ contains $B_0(\ve_n)$ where
$$\ve_n:=\ve-\frac{\ve}{4}\sum_{j=0}^{n-1}2^{-j}\geq \frac{\ve}{2},$$
and that
$$\norm{\gamma-\id}_{C^0(B_0(\ve_n))} \leq \frac{\ve}{2^n\cdot 32}.$$
The base case $n=0$ is obvious, and for the induction step the key result that we need is the following improvement \cite[Lemma 3.0]{Loray_Rebelo} of a result of Ghys \cite[Prop.~2.1]{Ghys1993_Sur-les-groupes-engendres-par-des-diffeomorphismes-proches-de-lidentite}: given constants $0<r,\delta,\tau<1$ with $4\delta+\tau<r$, if $f,g:B_0(r)\to \mathbb{C}^d$ are two injective holomorphic maps which satisfy
\begin{equation}\label{a}
\|f-\id\|_{C^0(B_{0}(r))}\leq \delta, \quad \|g-\id\|_{C^0(B_{0}(r))}\leq \delta,
\end{equation}
then their commutator $[f,g]$ is defined on $B_0(r-4\delta-\tau)$ and satisfies
\begin{equation}\label{b}
\|[f,g]-\id\|_{C^0(B_{0}(r-4\delta-\tau))}\leq \frac{2}{\tau}\|f-\id\|_{C^0(B_{0}(r))}\|g-\id\|_{C^0(B_{0}(r))}.
\end{equation}
We use this to prove the case $n+1$ of the induction by taking $$r:=\ve_n,\quad \delta:=\frac{\ve}{2^n\cdot 32},\quad \tau:=\frac{\ve}{2^n\cdot 8},$$
and applying it to two arbitrary $f,g\in S^{(n)}$. These satisfy \eqref{a} by induction hypothesis, and so $[f,g]$ is defined on the ball centered at the origin of radius
$$\ve_n-4\frac{\ve}{2^n\cdot 32}-\frac{\ve}{2^n\cdot 8}=\ve_{n+1},$$
and by \eqref{b} it satisfies
$$\|[f,g]-\id\|_{C^0(B_{0}(\ve_{n+1}))}\leq\frac{2}{\tau}\delta^2=\frac{\delta}{2}=\frac{\ve}{2^{n+1}\cdot 32},$$
as desired.
\end{proof}
The next result, appearing in \cite[Lemma~7.2]{RebeloRoeder2021_Dynamics-of-groups-of-birational-automorphisms-of-cubic-surfaces-and-Fatou/Julia}, will be useful in exhibiting explicit examples satisfying the assumptions of \autoref{thm:common_domain_of_definition}.
We will denote by $\id$ both the identity map and the identity matrix acting on $\bC^d$, and by $\norm{-}_{\rm Mat}$ the matrix norm on $n\times n$ matrices.
\begin{proposition}[Fixed points with small derivative]
\label{prop:fixed_points_with_small_derivative}
For any $0<\ve_0\leq 1$ and holomorphic map $\gamma:B_0(\ve_0)\to\bC^d$ satisfying
\[
\gamma(0)=0 \text{ and }\norm{D\gamma(0)-\id}_{\rm Mat} \leq \frac{1}{64},
\]
there exists $\ve_1$, depending on $\gamma$, with the following property.
For any $\ve\in (0,\ve_1)$, the map restricted to $B_0(\ve)$ satisfies
\begin{equation}\label{s}
\norm{\gamma -\id}_{C^0(B_0(\ve))}\leq \frac{\ve}{32}.
\end{equation}
\end{proposition}
\begin{proof}
For $0<\ve<\ve_1$ (where $\ve_1$ is to be determined), let $\Lambda_\ve(z_1,\dots,z_n)=(\ve z_1,\dots,\ve z_n)$ be the scaling map, and let $\gamma_\ve:=\Lambda_\ve^{-1}\circ\gamma\circ\Lambda_\ve$. This is a holomorphic map on $B_0(1)$ that satisfies
\[
\gamma_\ve(0)=0 \text{ and }\norm{D\gamma_\ve(0)-\id}_{\rm Mat} \leq \frac{1}{64}.
\]
An application of the Taylor formula gives
$$\norm{\gamma_\ve -\id}_{C^0(B_0(1))}\leq \norm{D\gamma_\ve(0)-\id}_{\rm Mat}+C_\gamma \ve\leq\frac{1}{64}+C_\gamma \ve,$$
for some constant $C_\gamma$ that depends on the size of the Hessian of $\gamma$. Thus, it suffices to choose $\ve_1=\frac{1}{64 C_\gamma},$ and we have
$$\norm{\gamma_\ve -\id}_{C^0(B_0(1))}\leq\frac{1}{32},$$
which is equivalent to \eqref{s}.
\end{proof}
\subsection{\texorpdfstring{$(2,2,2)$}{(2,2,2)}-surfaces and canonical currents}
\label{ssec:2_2_2_surfaces_and_canonical_currents}
For basic background on K3 surfaces, see \cite{BeauvilleBourguignonDemazure1985_Geometrie-des-surfaces-K3:-modules-et-periodes,Huybrechts2016_Lectures-on-K3-surfaces} and, for an introduction to complex automorphisms of K3 surfaces see \cite{Filip_notes_K3}.
Our main examples, the $(2,2,2)$-surfaces, were first noted by Wehler \cite{Wehler1988_K3-surfaces-with-Picard-number-2}.
\subsubsection{Setup}
\label{sssec:setup_2_2_2_surfaces_and_canonical_currents}
We work over the complex numbers.
Consider the $3$-fold $(\bP^1)^3$, with its family of smooth anticanonical divisors given by degree $(2,2,2)$-surfaces, i.e. let $\cU\subset \bC^{27}$ denote the parameter space of coefficients of an equation
\[
\sum_{0\leq i,j,k\leq 2}c_{ijk}x^iy^j z^k = 0 \quad \text{ in }(\bA^1)^3
\]
that yield smooth surfaces when compactified in $(\bP^1)^{3}$.
We will call these $(2,2,2)$-surfaces.
We consider for simplicity the full set of equations, without identifying surfaces equivalent under the action of $(\PGL_2)^{3}$.
\begin{definition}[Strict $(2,2,2)$ example]
\label{def:strict_222_example}
We will say that a $(2,2,2)$-surface is \emph{strict} if the rank of its \Neron--Severi group (over $\bC$) is the minimal possible, i.e. $3$.
\end{definition}
Note that a countable dense union of codimension-one subsets in $\cU$ consists of non-strict $(2,2,2)$-surfaces.
For strict $(2,2,2)$-surfaces, the \Neron--Severi group equipped with its intersection form is isometric to $\bR^{1,2}$ (after extension of scalars to $\bR$).
\subsubsection{Some recollections from topology}
\label{sssec:some_recollections_from_topology}
Recall that $F_\sigma$-sets are countable unions of closed sets, while $G_\delta$-sets are countable intersections of open ones.
It follows from standard results in the moduli theory of K3 surfaces that strict $(2,2,2)$-surfaces form a dense $G_\delta$-set in $\cU$, which in fact has full Lebesgue measure.
Indeed, parameters giving strict $(2,2,2)$-surfaces are the complement of countably many divisors in the full parameter space, see e.g. \cite{oguiso}.
\subsubsection{Involutions}
\label{sssec:involutions}
For any $u\in \cU$, denote the associated surface by $X_u\subset (\bP^1)^3$.
The projection onto one of the coordinate planes $X_u\to (\bP^1)^2$ is two-to-one and so $X_u$ admits an involution exchanging the two sheets.
Denote by $\sigma_x,\sigma_y,\sigma_z$ the three involutions obtained in this manner.
\subsubsection{Canonical currents}
\label{sssec:canonical_currents}
We can apply \cite[Thm.~1]{FilipTosatti2021_Canonical-currents-and-heights-for-K3-surfaces} to any strict $(2,2,2)$-surface $X_u$ with $u\in \cU$.
In that theorem a certain space $\partial^{\circ}\Amp_c(X_u)$ appears, which on strict $(2,2,2)$-surfaces reduces to the boundary of the ample cone $\partial\Amp(X_u)$, so it consists of nef cohomology classes $[\eta]\in \NS_\bR(X_u)\subset H^{1,1}(X_u)$ satisfying $[\eta]^2=0$.
Since $\NS_\bR(X_u)$ equipped with the intersection pairing is isometric to $\bR^{1,2}$, the space $\partial\Amp(X_u)$ is isomorphic to one component of the null-cone in this Minkowski space.
Note that in the general form of the result, one needs to replace the rational rays in $\partial\Amp(X_u)$ by their blowups; since in the case of a rank $3$ \Neron--Severi group it would mean blowing up rays on a surface, no extra points need to be added.
Next, \cite[Thm.~1]{FilipTosatti2021_Canonical-currents-and-heights-for-K3-surfaces} shows that each cohomology class $[\eta]\in \partial \Amp(X_u)$ has a canonical positive representative $\eta$, which additionally has $C^0$ potentials.
The representative is unique when the class is irrational, and a preferred representative in the rational (also called parabolic) classes exists that makes the entire family of currents continuous in the $C^0$-topology of the potentials for the currents.
We will show in \autoref{thm:gaps_in_the_support_of_canonical_currents} below that some of the canonical representatives do not have full support in $X_u$.
Specifically, we will show that there exists an open set $\cU_0\subset \cU$ and a dense $G_\delta$ set of $u\in \cU_0$ for which some of the canonical currents $\eta$ do not have full support in $X_u$.
But first, we will show that the set of cohomology classes $[\eta]$ for which the gaps in the support are constructed contain, after projectivization, a closed uncountable set.
\subsubsection{Free subgroups of automorphisms}
\label{sssec:free_subgroups_of_automorphisms}
We will consider subgroups of automorphisms of $X_u$ freely generated by five elements.
Specifically, $\sigma_x,\sigma_y,\sigma_z$ generate a group $\Gamma_{\sigma}\subseteq\Aut(X_u)$ isomorphic to $\left(\bZ/2\right)*\left(\bZ/2\right)*\left(\bZ/2\right)$, in other words there are no relations between them except that $\sigma_{i}^2=\id$ for $i=x,y,z$.
This can be verified by considering the action on the hyperbolic space inside the \Neron--Severi group of $X_u$ (see for instance \cite[Prop.6.1]{Filip2019_Tropical-dynamics-of-area-preserving-maps} for the explicit matrices corresponding to the action in the upper half-space model).
\begin{proposition}[Free group on five generators]
\label{prop:free_group_on_five_generators}
Consider the surjective homomorphism $\Gamma_{\sigma}\onto (\bZ/2)^{\oplus 3}$ sending $\sigma_x,\sigma_y,\sigma_z$ to $(1,0,0),(0,1,0),(0,0,1)$ respectively.
Then its kernel $K_{\sigma}$ is a free group on five generators.
\end{proposition}
The above homomorphism corresponds to evaluating the derivatives of the transformations at the common fixed of the transformations described in \autoref{sssec:fixed_point_and_derivatives}.
\begin{proof}
We will divide our analysis by looking at the homomorphisms $\Gamma_{\sigma}\onto (\bZ/2)^{\oplus 3}\onto \bZ/2$ where the last map sends each generator of a summand to the unique nonzero element.
Now the kernel of $\Gamma_{\sigma}\onto \bZ/2$ sending each $\sigma_i$ to $1\in \bZ/2$ is the free group on two letters, generated by $a:=\sigma_x\sigma_y$ and $b:=\sigma_y\sigma_z$.
Indeed this kernel is the fundamental group of the Riemann sphere with $3$ points removed.
Now $K_{\sigma}$ is contained with finite index in the free group on $a,b$, and is visibly given as the kernel of the surjection onto $(\bZ/2)^{\oplus 2}$ sending $a\mapsto (1,0)$ and $b\mapsto (0,1)$.
One can then work out the associated covering space and rank of free group, using the techniques in \cite[\S1.A]{Hatcher2002_Algebraic-topology}, and determine that $K_{\sigma}$ is a free group on $5$ generators.
Alternatively, the corresponding $(\bZ/2)^{\oplus 2}$-covering space of the triply punctured Riemann sphere can be visualized as a square-shaped ``pillowcase'' with four punctures at the corners, and two additional punctures in the center of the two faces.
The involutive automorphisms are rotations by $180^{\circ}$ through an axis that goes across two opposite punctures.
\end{proof}
\subsubsection{Largeness of the set currents with gaps}
\label{sssec:largeness_of_the_set_currents_with_gaps}
To continue, we select $\gamma_1,\gamma_2,\gamma_3,\gamma_4,\gamma_5\subset K_{\sigma}$ to be five elements freely generating the group.
Next, the construction of \autoref{sssec:derived_series} applies with $S:=\{\gamma_1,\ldots,\gamma_5,\gamma_1^{-1},\ldots\gamma_5^{-1}\}$ and yields a subset $S^{\bullet}\subset \Aut(X_u)$ consisting of iterated commutators.
Fix a \Kahler metric $\omega_0$ on $X_u$, with volume normalized to $[\omega_0]^2=1$, and let $\bH^2(X_u)$ denote the hyperbolic plane of all nef cohomology classes satisfying $[\omega]^2=1$.
\begin{proposition}[Uncountably many currents with gaps]
\label{prop:uncountably_many_currents_with_gaps}
The intersection of the closure of the set $S^{\bullet}\cdot [\omega_0]\subset \bH^2(X_u)$ with the boundary $\partial \bH^2(X_u)$ is an uncountable closed set.
\end{proposition}
\begin{proof}
That the set is closed follows from its definition.
To show that the set is uncountable, we will argue on the boundary of the free group on the five initial generators, and use that the natural map from the boundary of the free group to the hyperbolic space is injective, except perhaps at the countably many parabolic points.
For this, let $\cT$ denote the Cayley graph of the free group on five generators; it is a $10$-valent infinite regular tree.
Define the sequence of finite subtrees $\cT_k$, where $\cT_0$ consists of the identity vertex, and $\cT_{k+1}$ is obtained from $\cT_k$ by connecting the leaves of $\cT_k$ with the elements in $S^{(k+1)}$.
From \autoref{prop:fast_ramification} it follows that the number of new edges added to the leaves at each step is at least $3$.
Therefore, the number of infinite paths starting at the origin in $\cT_{\infty}:=\cup_{k\geq 0}\cT_k$ is uncountable, and the claim follows.
\end{proof}
\subsection{An example with slow commutators}
\label{ssec:an_example_with_slow_commutators}
\subsubsection{Setup}
\label{sssec:setup_an_example_with_slow_commutators}
To show that the assumptions of \autoref{prop:fixed_points_with_small_derivative} are satisfied in practice, we start with an explicit equation:
\begin{align}
\label{eqn:simplest_example}
(1+x^2)(1+y^2)(1+z^2) + xyz = 1
\end{align}
Let us note that \autoref{eqn:simplest_example} determines a \emph{singular} $(2,2,2)$-surface, with a singularity at the origin $0\in \bC^3$.
We will construct an open set $\cU_0$ of smooth $(2,2,2)$-surfaces by taking perturbations of the above equation.
\subsubsection{Automorphisms of ambient space}
\label{sssec:automorphisms_of_ambient_space}
Let $u_0\in \bC^{27}$ denote the point corresponding to the choice of parameters as in \autoref{eqn:simplest_example}, it lies outside $\cU$ but any analytic neighborhood of $u_0$ in $\bC^{27}$ intersects $\cU$ in a nonempty open set.
We have three explicit involutions $\sigma_{u_0,x},\sigma_{u_0,y},\sigma_{u_0,z}$:
\[
\sigma_{u_0,x}(x,y,z) = \left(\frac{-yz}{(1+y^2)(1+z^2)} - x, y, z\right)
\]
and similarly for $\sigma_{u_0,y},\sigma_{u_0,z}$, which we view as holomorphic maps defined in a neighborhood of $0\in \bC^3$.
\subsubsection{Fixed point and Derivatives}
\label{sssec:fixed_point_and_derivatives}
It is immediate from the explicit formulas that all three involutions preserve the point $0\in \bC^3$.
Furthermore, their derivatives at that point are matrices of order two:
\[
D\sigma_{u_0,x}(0,0,0) = \begin{bmatrix}
-1 & & \\
& 1 & \\
& & 1
\end{bmatrix}
\text{ and analogously for }\sigma_{u_0,y},\sigma_{u_0,z}.
\]
We now consider $\sigma_{u,x},\sigma_{u,y},\sigma_{u,z}$ for $u\in \bC^{27}$ in a sufficiently small neighborhood of $u_0$.
Then we can regard the $\sigma$'s as holomorphic maps defined in a neighborhood of $0\in \bC^3$, preserving the intersection of $X_u$ with the fixed neighborhood.
We can now use these observations to establish:
\begin{theorem}[Gaps in the support of canonical currents]
\label{thm:gaps_in_the_support_of_canonical_currents}
There exists a nonempty open set $\cU_0$ in the analytic topology of smooth $(2,2,2)$-surfaces with the following property.
For each strict K3 surface $X_u$ with $u\in \cU_0$, there exists a dense $F_\sigma$-set of rays $F$ on the boundary of the ample cone of $X_u$ such that for any $[\eta]\in F$, the canonical current $\eta$ provided by \cite[Thm.~1]{FilipTosatti2021_Canonical-currents-and-heights-for-K3-surfaces} is supported on a proper closed subset of $X_u$.
Furthermore $F$ determines an uncountable set of rays.
\end{theorem}
\noindent By a ``ray'' we mean one orbit of the $\bR_{>0}$-action by scaling, so that the ``set of rays'' is the projectivization of $\partial \Amp(X_u)$.
It is implicit in the the statement above that the set $F$ is disjoint from the countably many parabolic rays.
This is justified by \autoref{rmk:avoidance_of_parabolic_points} below.
\begin{proof}
We keep the notation as before the statement of the theorem and will consider $u\in \cU$ in a sufficiently small neighborhood of $u_0$.
Consider the subgroup $K_{\sigma}\subset \Aut(X)$ obtained by applying \autoref{prop:free_group_on_five_generators} to the group generated by the three involutions.
At the parameter $u_0$ all elements in $K_{\sigma}$ preserve the point with coordinate $(0,0,0)$ and have derivative equal to the identity there, see \autoref{sssec:fixed_point_and_derivatives}.
Fix now the five free generators $\gamma_{u,i} \in K_{\sigma}$ with $i=1,\ldots,5$, as per \autoref{prop:free_group_on_five_generators}.
Let $S^{(n)}$ denote the set of iterated commutators, as per \autoref{sssec:derived_series}.
\autoref{prop:uncountably_many_currents_with_gaps} yields for any strict $X_u$ an uncountable closed set $F_0\subset \partial \Amp(X_u)$ with the following property. Fixing $\omega_0$ a reference K\"ahler metric on $X_u$, for any $f\in F_0$ there exists a sequence $\{s_n\}$ of automorphisms of $X_u$, with $s_n\in S^{(n)}$, and a sequence of positive scalars $\lambda_n\to+\infty$ such that
\[ f = \lim_{n\to +\infty} \frac{1}{\lambda_n}(s_n)_{*}[\omega_0].\]
Note that $\lambda_n\to +\infty$ since the self-intersection of $(s_n)_*[\omega_0]$ is $1$, while the self-intersection of $f$ is zero.
Applying \cite[Thm.~4.2.2, pts. 4,5]{FilipTosatti2021_Canonical-currents-and-heights-for-K3-surfaces} then shows that in the weak sense of currents we have
\[
\eta_f = \lim_{n\to +\infty} \frac{1}{\lambda_n}(s_n)_{*}\omega_0,
\]
where $\eta_f$ is a canonical positive representative of the cohomology class $f$.
Furthermore, at this stage of the argument the cohomology class $f$ might be rational, but its canonical representative is in fact unique since we consider strict $(2,2,2)$-surfaces.
Nonetheless, see \autoref{rmk:avoidance_of_parabolic_points} below for why, in fact, this case does not occur.
\autoref{prop:fixed_points_with_small_derivative} applies to the finitely many generators $\gamma_{u_0,i}$, so \autoref{thm:common_domain_of_definition} applies to them as well on a fixed ball $B_0(\ve)$ around $0\in \bC^3$.
However, the assumptions of \autoref{thm:common_domain_of_definition} are stable under a small perturbation, so they hold for $\gamma_{u,i}$ for $u$ in a sufficiently small neighborhood of $u_0$.
Therefore, by \autoref{thm:common_domain_of_definition} \emph{all} the maps $s_n$ are within a bounded distance of the identity when restricted to $B_0(\ve/2)$.
However, the maps preserve the intersection of $X_u$ with $B_0(\ve/2)$ so the weak limit of $\frac{1}{\lambda_n}(s_n)_* \omega_0$ vanishes in $B_0(\ve/2)\cap X_u$.
We conclude that the support of $\eta_f$ avoids $B_0(\ve)$.
Finally, the action of $\Aut(X_u)$ on the (projectivized) boundary of the ample cone is minimal, i.e. every orbit is dense, and clearly the property of having a gap in the support is invariant under applying one automorphism.
It follows that the set $F:=\Aut(X_u)\cdot F_0$ is a dense $F_\sigma$-set with the required properties.
\end{proof}
\begin{remark}[Avoidance of parabolic points]
\label{rmk:avoidance_of_parabolic_points}
The set $F$ provided by \autoref{thm:gaps_in_the_support_of_canonical_currents} is disjoint from the countably many parabolic points.
The reason is that the canonical currents at the parabolic points have full support, since they are obtained as the pullback of currents from the base $\bP^1(\bC)$ of an elliptic fibration, but the corresponding currents on $\bP^1(\bC)$ have real-analytic potentials away from the finitely many points under the singular fibers.
The last assertion can be seen from following through the proof of \cite[Thm.~3.2.14]{FilipTosatti2021_Canonical-currents-and-heights-for-K3-surfaces} with real-analytic data.
\end{remark}
\begin{remark}[Zassenhausian points]
\label{rmk:zassenhausian_points}
Recall that relative to a lattice $\Gamma\subset \Isom(\bH^n)$ of isometries of a hyperbolic space, the boundary points in $\partial \bH^n$ can be called ``Liouvillian'' or ``Diophantine''.
Specifically, a Liouvillian point is one for which the geodesic ray with the point as its limit on the boundary makes very long excursions into the cusps of $\Gamma\backslash \bH^n$, while Diophantine points are ones for which the excursions into the cusps are controlled.
Both situations involve quantitative bounds.
The boundary points constructed using iterated commutators as in \autoref{sssec:derived_series}, with group elements lying deeper and deeper in the derived series of $\Gamma$, could then be called ``Zassenhausian''.
Note that in principle, geodesics with Zassenhausian boundary points will have good recurrence properties and will also be Diophantine.
It would be interesting to see if canonical currents corresponding to Liouvillian boundary points have full support or not.
\end{remark}
\subsection{An example with no support on the real locus}
\label{ssec:an_example_with_no_support_on_the_real_locus}
The above methods can be strengthened to construct an example of a current with no support on the real locus of a real projective K3 surface.
The starting point is a construction due to Moncet \cite[\S9.3]{Moncet2012_Geometrie-et-dynamique-sur-les-surfaces-algebriques-reelles}, who constructed real K3 surfaces with arbitrarily small entropy on the real locus.
We use some minor modifications for notational convenience, and emphasize that many different choices are possible for the initial singular real K3 surface.
Let us also note that these examples have a ``tropical'' analogue given by PL actions on the sphere, and the analogue of the finite-order action at the singular parameter corresponds to a finite order action by reflections on the cube, see \cite[\S6.2]{Filip2019_Tropical-dynamics-of-area-preserving-maps}.
\subsubsection{Setup}
\label{sssec:setup_an_example_with_no_support_on_the_real_locus}
Let $X_0$ denote the (singular) surface
\[
x^2 + y^2 + z^2 = 1
\]
compactified in $(\bP^1)^3$.
Its real locus $X_0(\bR)$ is a real $2$-dimensional sphere.
As before let $\cU\subset \bR^{27}$ be the subset of smooth $(2,2,2)$-surfaces, parametrized by the possible coefficients, and normalized such that the parameter $0\in \bR^{27}$ corresponds to $X_0$.
Note that $0\notin \cU$.
Let next $\cU'\subset \cU$ denote the subset of strict $(2,2,2)$-surfaces.
By the discussion in \autoref{sssec:some_recollections_from_topology} the set $\cU'$ is the complement of countably many divisors in $\cU$, and thus forms a dense $G_{\delta}$ set.
\begin{theorem}[Full gaps in the real locus]
\label{thm:full_gaps_in_the_real_locus}
There exists a nonempty open set $\cU_0\subset \cU\subset \bR^n$ in the analytic topology of smooth real $(2,2,2)$-surfaces with the following property.
For each strict K3 surface $X_u$ with $u\in \cU_0$, there exists a dense $F_\sigma$-set of rays $F$ on the boundary of the ample cone of $X_u$ such that for any $[\eta]\in F$, the support of the canonical current $\eta$ provided by \cite[Thm.~1]{FilipTosatti2021_Canonical-currents-and-heights-for-K3-surfaces} is disjoint from the real locus $X_u(\bR)$.
Furthermore $F$ determines an uncountable set of rays.
\end{theorem}
\subsubsection{Subgroup of slow automorphisms}
\label{sssec:subgroup_of_slow_automorphisms}
Let us first observe that the involution $\sigma_x$ acting on the surface $X_0$ in \autoref{sssec:setup_an_example_with_no_support_on_the_real_locus} is given by $\sigma_x(x,y,z)=(-x,y,z)$, and analogously for $\sigma_y,\sigma_z$.
Therefore, let $K_{\sigma}\subset \Gamma_{\sigma}$ be the group from \autoref{prop:free_group_on_five_generators} obtained as the kernel of this action; it is a free group on five generators $\gamma_i$ and acts nontrivially on any smooth and strict $(2,2,2)$-surface.
Even for smooth surfaces $X_u\subset (\bP^1)^{3}$, we will be interested only in their intersection with the affine chart $\bC^{3}$, and specifically a neighborhood of $X_0(\bR)$.
We will thus restrict to the locus where no additional real components arise.
\subsubsection{Good cover}
\label{sssec:good_cover}
Choose a finite cover of $X_0(\bR)\subset \bR^3$ by open sets $V_i\subset \bC^3$ such that we have biholomorphisms $\phi_i\colon V_i \to B_0(1)\subset \bC^3$ to a ball of radius $1$ around $0$, and the preimages of the smaller balls $V_i':=\phi_i^{-1}\left(B_0(\tfrac 14)\right)$ still cover $X_0(\bR)$.
Choose now a sufficiently small open neighborhood of the origin $\cU_0\subset \bR^{27}$ such that the following property is satisfied: For each of the five generators $\gamma_j$ of $K_{\sigma}$ and their inverses, we have for every chart $V_i$ that $\gamma_{ij}':=\phi_i \circ \gamma_j \circ \phi^{-1}_i$ satisfies:
\[
\gamma_{ij}'\colon B_0\left(\tfrac 12\right) \to B_0(1) \text{ is well-defined and }
\norm{\gamma_{ij}'-\id}_{B_{0}\left(\tfrac 12\right)}\leq \tfrac{1}{64}.
\]
Require also that for any $u\in \cU_0$ that $X_u(\bR)$ is nonempty and still covered by the sets $\{V_i'\}$.
\begin{proof}[Proof of \autoref{thm:full_gaps_in_the_real_locus}]
By \autoref{thm:common_domain_of_definition} all the commutators in the set $S^{(n)}$ as defined in \autoref{sssec:derived_series} are well-defined when conjugated to any of the charts $\phi_i$, and furthermore their distance to the identity transformation goes to zero as $n\to +\infty$.
As in the proof of \autoref{thm:gaps_in_the_support_of_canonical_currents}, let $s_n\in S^{(n)}$ be any sequence of such commutators such that the cohomology class $\tfrac{1}{\lambda_n}(s_n)_*[\omega_0]$ converges to some class $f$.
Then the canonical current $\eta_f$ has no support in the neighborhoods $V_i'$.
Since these still cover $X_u(\bR)$ for $u\in \cU_0$, the result follows.
\end{proof}
\bibliographystyle{sfilip_bibstyle}
| 13,435 |
\section{Introduction}
Named Entity Recognition (NER) predicts which word tokens refer to location, people, organization, time, and other entities from a word sequence. Deep neural network models have successfully achieved the state-of-the-art performance in NER tasks \cite{cohenmulti, chiu2016named, lample2016neural, shen2017deep} using monolingual corpus. However, learning from code-switching tweets data is very challenging due to several reasons: (1) words may have different semantics in different context and language, for instance, the word ``cola'' can be associated with product or ``queue'' in Spanish (2) data from social media are noisy, with many inconsistencies such as spelling mistakes, repetitions, and informalities which eventually points to Out-of-Vocabulary (OOV) words issue (3) entities may appear in different language other than the matrix language. For example ``todos los Domingos en Westland Mall" where ``Westland Mall" is an English named entity.
Our contributions are two-fold: (1) bilingual character bidirectional RNN is used to capture character-level information and tackle OOV words issue (2) we apply transfer learning from monolingual pre-trained word vectors to adapt the model with different domains in a bilingual setting. In our model, we use LSTM to capture long-range dependencies of the word sequence and character sequence in bilingual character RNN. In our experiments, we show the efficiency of our model in handling OOV words and bilingual word context.
\section{Related Work}
Convolutional Neural Network (CNN) was used in NER task as word decoder by \citet{collobert2011natural} and a few years later, \citet{huang2015bidirectional} introduced Bidirectional Long-Short Term Memory (BiLSTM) \cite{sundermeyer2012lstm}. Character-level features were explored by using neural architecture and replaced hand-crafted features \cite{dyer2015transition, lample2016neural, chiu2016named, limsopatham2016bidirectional}. \citet{lample2016neural} also showed Conditional Random Field (CRF) \cite{lafferty2001conditional} decoders to improve the results and used Stack memory-based LSTMs for their work in sequence chunking. \citet{aguilar2017multi} proposed multi-task learning by combining Part-of-Speech tagging task with NER and using gazetteers to provide language-specific knowledge. Character-level embeddings were used to handle the OOV words problem in NLP tasks such as NER \cite{lample2016neural}, POS tagging, and language modeling \cite{ling2015finding}.
\section{Methodology}
\begin{table*}[!htb]
\centering
\caption{OOV words rates on ENG-SPA dataset before and after preprocessing}
\label{eng-spa-oov-statistics}
\begin{tabular}{@{}lccccc@{}}
\hline
\multicolumn{1}{c}{\multirow{2}{*}{\textbf{}}} & \multicolumn{2}{|c|}{\textbf{Train}} & \multicolumn{2}{c}{\textbf{Dev}} & \multicolumn{1}{|c}{\textbf{Test}} \\ \cline{2-6}
\multicolumn{1}{c}{} & \multicolumn{1}{|c|}{All} & Entity & \multicolumn{1}{|c|}{All} & Entity & \multicolumn{1}{|c}{All} \\ \hline
Corpus & \multicolumn{1}{|c|}{-} & - & \multicolumn{1}{|c|}{18.91\%} & 31.84\% & \multicolumn{1}{|c}{49.39\%} \\ \hline
FastText (eng) \cite{mikolov2018advances} & \multicolumn{1}{|c|}{62.62\%} & 16.76\% & \multicolumn{1}{|c|}{19.12\%} & 3.91\% & \multicolumn{1}{|c}{54.59\%} \\ \hline
+ FastText (spa) \cite{grave2018learning} & \multicolumn{1}{|c|}{49.76\%} & 12.38\% & \multicolumn{1}{|c|}{11.98\%} & 3.91\% & \multicolumn{1}{|c}{39.45\%} \\ \hline
+ token replacement & \multicolumn{1}{|c|}{12.43\%} & 12.35\% & \multicolumn{1}{|c|}{7.18\%} & 3.91\% & \multicolumn{1}{|c}{9.60\%} \\ \hline
\textbf{+ token normalization} & \multicolumn{1}{|c|}{\textbf{7.94\%}} & \textbf{8.38\%} & \multicolumn{1}{|c|}{\textbf{5.01\%}} & \textbf{1.67\%} & \multicolumn{1}{|c}{\textbf{6.08\%}} \\ \hline
\end{tabular}
\end{table*}
\subsection{Dataset}
For our experiment, we use English-Spanish (ENG-SPA) Tweets data from Twitter provided by \citet{calcs2018shtask}. There are nine different named-entity labels. The labels use IOB format (Inside, Outside, Beginning) where every token is labeled as \textnormal{\tt B-label} in the beginning and follows with \textnormal{\tt I-label} if it is inside a named entity, or \textnormal{\tt O} otherwise. For example ``Kendrick Lamar'' is represented as \textnormal{\tt B-PER I-PER}. Table \ref{data-statistics-eng-spa} and Table \ref{data-statistics-eng-spa-named-entities} show the statistics of the dataset.
\begin{table}[!htb]
\centering
\caption{Data Statistics for ENG-SPA Tweets}
\label{data-statistics-eng-spa}
\begin{tabular}{@{}llll@{}}
\hline
& \multicolumn{1}{|c|}{\textbf{Train}} & \multicolumn{1}{c}{\textbf{Dev}} & \multicolumn{1}{|c}{\textbf{Test}} \\ \hline
\# Words & \multicolumn{1}{|c|}{616,069} & 9,583 & \multicolumn{1}{|c}{183,011} \\ \hline
\end{tabular}
\end{table}
\begin{table}[!htb]
\centering
\caption{Entity Statistics for ENG-SPA Tweets}
\label{data-statistics-eng-spa-named-entities}
\begin{tabular}{@{}rcc@{}}
\hline
\multicolumn{1}{r}{\textbf{Entities}} & \multicolumn{1}{|c|}{\textbf{Train}} & \multicolumn{1}{c}{\textbf{Dev}} \\ \hline
\# Person & \multicolumn{1}{|c|}{4701} & 75 \\ \hline
\# Location & \multicolumn{1}{|c|}{2810} & 10 \\ \hline
\# Product & \multicolumn{1}{|c|}{1369} & 16 \\ \hline
\# Title & \multicolumn{1}{|c|}{824} & 22 \\ \hline
\# Organization & \multicolumn{1}{|c|}{811} & 9 \\ \hline
\# Group & \multicolumn{1}{|c|}{718} & 4 \\ \hline
\# Time & \multicolumn{1}{|c|}{577} & 6 \\ \hline
\# Event & \multicolumn{1}{|c|}{232} & 4 \\ \hline
\# Other & \multicolumn{1}{|c|}{324} & 6 \\ \hline
\end{tabular}
\end{table}
``Person'', ``Location'', and ``Product'' are the most frequent entities in the dataset, and the least common ones are ``Time", ``Event", and ``Other'' categories. `Other'' category is the least trivial among all because it is not well clustered like others.
\subsection{Feature Representation}
In this section, we describe word-level and character-level features used in our model.
\textbf{Word Representation: } Words are encoded into continuous representation. The vocabulary is built from training data. The Twitter data are very noisy, there are many spelling mistakes, irregular ways to use a word and repeating characters. We apply several strategies to overcome the issue. We use 300-dimensional English \cite{mikolov2018advances} and Spanish \cite{grave2018learning} FastText pre-trained word vectors which comprise two million words vocabulary each and they are trained using Common Crawl and Wikipedia. To create the shared vocabulary, we concatenate English and Spanish word vectors.
For preprocessing, we propose the following steps:
\begin{enumerate}
\item \textbf{Token replacement: } Replace user hashtags (\#user) and mentions (@user) with ``USR", and URL (https://domain.com) with ``URL".
\item \textbf{Token normalization: } Concatenate Spanish and English FastText word vector vocabulary. Normalize OOV words by using one out of these heuristics and check if the word exists in the vocabulary sequentially
\begin{enumerate}
\item Capitalize the first character
\item Lowercase the word
\item Step (b) and remove repeating characters, such as \textit{``hellooooo"} into \textit{``hello"} or \textit{``lolololol"} into \textit{``lol"}
\item Step (a) and (c) altogether
\end{enumerate}
\end{enumerate}
Then, the effectiveness of the preprocessing and transfer learning in handling OOV words are analyzed. The statistics is showed in Table \ref{eng-spa-oov-statistics}. It is clear that using FastText word vectors reduce the OOV words rate especially when we concatenate the vocabulary of both languages. Furthermore, the preprocessing strategies dramatically decrease the number of unknown words.
\textbf{Character Representation: } We concatenate all possible characters for English and Spanish, including numbers and special characters. English and Spanish have most of the characters in common, but, with some additional unique Spanish characters. All cases are kept as they are.
\subsection{Model Description}
In this section, we describe our model architecture and hyper-parameters setting.
\textbf{Bilingual Char-RNN: } This is one of the approaches to learn character-level embeddings without needing of any lexical hand-crafted features. We use an RNN for representing the word with character-level information \cite{lample2016neural}. Figure \ref{fig:char-rnn} shows the model architecture. The inputs are characters extracted from a word and every character is embedded with $d$ dimension vector. Then, we use it as the input for a Bidirectional LSTM as character encoder, wherein every time step, a character is input to the network. Consider $a_t$ as the hidden states for word $t$.
\[ a_t = (a_1^1, a_t^2, ..., a_t^\textnormal{V}) \]
where $\textnormal{V}$ is the character length. The representation of the word is obtained by taking $a_t^\textnormal{V}$ which is the last hidden state.
\begin{figure}[!htb]
\centering
\includegraphics[width=1.0\linewidth]{img/char-rnn.pdf}
\caption{Bilingual Char-RNN architecture}
\label{fig:char-rnn}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=1.0\linewidth]{img/overall.pdf}
\caption{Main architecture}
\label{fig:overall}
\end{figure}
\textbf{Main Architecture: } Figure \ref{fig:overall} presents the overall architecture of the system. The input layers receive word and character-level representations from English and Spanish pre-trained FastText word vectors and Bilingual Char-RNN. Consider $\textbf{X}$ as the input sequence:
\[ \textbf{X} = {(x_1, x_2, ..., x_\textnormal{N})} \]
where N is the length of the sequence. We fix the word embedding parameters. Then, we concatenate both vectors to get a richer word representation $u_t$. Afterwards, we pass the vectors to bidirectional LSTM.
\[ u_t = x_t \oplus a_t \]
\[ \overrightarrow{h_t} = \overrightarrow{\textnormal{LSTM}}(u_t, \overrightarrow{h_{t-1}})\textnormal{, } \overleftarrow{h_t} = \overleftarrow{\textnormal{LSTM}}(u_t, \overleftarrow{h_{t-1}}) \]
\[ c_t = \overrightarrow{h_t} \oplus \overleftarrow{h_t} \]
where $\oplus$ denotes the concatenation operator. Dropout is applied to the recurrent layer. At each time step we make a prediction for the entity of the current token. A softmax function is used to calculate the probability distribution of all possible named-entity tags.
\[ y_t = \frac{e^{c_t}}{\sum_{j=1}^T e^{c_j}} \textnormal{, where } j = 1 \textnormal{, .., T} \]
where $\textnormal{y}_t$ is the probability distribution of tags at word $t$ and $\textnormal{T}$ is the maximum time step. Since there is a variable number of sequence length, we padded the sequence and applied mask when calculating cross-entropy loss function. Our model does not use any gazetteer and knowledge-based information, and it can be easily adapted to another language pair.
\begin{table*}[!htb]
\centering
\caption{Results on ENG-SPA Dataset ($\ddagger$ result(s) from the shared task organizer \cite{calcs2018shtask} \hspace{1mm} $\dagger$ without token normalization) }
\label{results-eng-spa}
\begin{tabular}{@{}llcc@{}}
\hline
\multicolumn{1}{c}{\textbf{Model}} & \multicolumn{1}{|c|}{\textbf{Features}} & \textbf{\begin{tabular}[c]{@{}c@{}}F1\\ Dev\end{tabular}} & \multicolumn{1}{|c}{\textbf{\begin{tabular}[c]{@{}c@{}}F1\\ Test\end{tabular}}} \\ \hline
Baseline$^\ddagger$ & \multicolumn{1}{|c|}{Word} & - & \multicolumn{1}{|c}{53.2802\%} \\ \hline
BiLSTM$^\dagger$ & \multicolumn{1}{|c|}{Word + Char-RNN} & 46.9643\% & \multicolumn{1}{|c}{53.4759\%} \\ \hline
BiLSTM & \multicolumn{1}{|c|}{FastText (eng)} & 57.7174\% & \multicolumn{1}{|c}{59.9098\%} \\ \hline
BiLSTM & \multicolumn{1}{|c|}{FastText (eng-spa)} & 57.4177\% & \multicolumn{1}{|c}{60.2426\%} \\ \hline
BiLSTM & \multicolumn{1}{|c|}{+ Char-RNN} & 65.2217\% & \multicolumn{1}{|c}{61.9621\%} \\ \hline
+ post & \multicolumn{1}{|c|}{} & \textbf{65.3865\%} & \multicolumn{1}{|c}{\textbf{62.7608\%}} \\ \hline \hline
\multicolumn{4}{l}{\textbf{Competitors$^\ddagger$}} \\ \hline
IIT BHU ($1^{st}$ place) & \multicolumn{1}{|c|}{-} & - & \multicolumn{1}{|c}{63.7628\% (+1.0020\%)} \\ \hline
FAIR \hspace{4.4mm} ($3^{rd}$ place) & \multicolumn{1}{|c|}{-} & - & \multicolumn{1}{|c}{62.6671\% (- 0.0937\%)} \\ \hline
\end{tabular}
\end{table*}
\subsection{Post-processing}
We found an issue during the prediction where some words are labeled with \textnormal{\tt O}, in between \textnormal{\tt B-label} and \textnormal{\tt I-label} tags. Our solution is to insert \textnormal{\tt I-label} tag if the tag is surrounded by \textnormal{\tt B-label} and \textnormal{\tt I-label} tags with the same entity category. Another problem we found that many \textnormal{\tt I-label} tags are paired with \textnormal{\tt B-label} in different categories. So, we replace \textnormal{\tt B-label} category tag with corresponding \textnormal{\tt I-label} category tag. This step improves the result of the prediction on the development set. Figure \ref{fig:post1} shows the examples.
\begin{figure}[!htb]
\centering
\includegraphics[width=1.0\linewidth]{img/post1.pdf}
\caption{Post-processing examples}
\label{fig:post1}
\end{figure}
\subsection{Experimental Setup}
We trained our LSTM models with a hidden size of 200. We used batch size equals to 64. The sentences were sorted by length in descending order. Our embedding size is 300 for word and 150 for characters. Dropout \cite{srivastava2014dropout} of 0.4 was applied to all LSTMs. Adam Optimizer was chosen with an initial learning rate of 0.01. We applied time-based decay of $\sqrt{2}$ decay rate and stop after two consecutive epochs without improvement. We tuned our model with the development set and evaluated our best model with the test set using harmonic mean F1-score metric with the script provided by \citet{calcs2018shtask}.
\section{Results}
Table \ref{results-eng-spa} shows the results for ENG-SPA tweets. Adding pre-trained word vectors and character-level features improved the performance. Interestingly, our initial attempts at adding character-level features did not improve the overall performance, until we apply dropout to the Char-RNN. The performance of the model improves significantly after transfer learning with FastText word vectors while it also reduces the number of OOV words in the development and test set. The margin between ours and first place model is small, approximately 1\%.
We try to use sub-words representation from Spanish FastText \cite{grave2018learning}, however, it does not improve the result since the OOV words consist of many special characters, for example, \textit{``/IAtrevido/Provocativo", ``Twets/wek"}, and possibly create noisy vectors and most of them are not entity words.
\section{Conclusion}
This paper presents a bidirectional LSTM-based model with hierarchical architecture using bilingual character RNN to address the OOV words issue. Moreover, token replacement, token normalization, and transfer learning reduce OOV words rate even further and significantly improves the performance. The model achieved 62.76\% F1-score for English-Spanish language pair without using any gazetteer and knowledge-based information.
\section*{Acknowledgments}
This work is partially funded by ITS/319/16FP of the Innovation Technology Commission, HKUST 16214415 \& 16248016 of Hong Kong Research Grants Council, and RDC 1718050-0 of EMOS.AI.
| 6,712 |
\section{Introduction}
\label{sec:introduction}
Deep neural networks trained with backpropagation have commonly attained superhuman performance in applications of computer vision \cite{krizhevsky2012imagenet} and many others \cite{schmidhuber2015deep} and are thus receiving an unprecedented research interest. Despite the rapid growth of the list of successful applications with these gradient-based methods, our theoretical understanding, however, is progressing at a more modest pace.
One of the salient features of deep networks today is that they often have far more model parameters than the number of training samples that they are trained on, but meanwhile some of the models still exhibit remarkably good generalization performance when applied to unseen data of similar nature, while others generalize poorly in exactly the same setting. A satisfying explanation of this phenomenon would be the key to more powerful and reliable network structures.
To answer such a question, statistical learning theory has proposed interpretations from the viewpoint of system complexity \cite{vapnik2013nature,bartlett2002rademacher,poggio2004general}. In the case of large numbers of parameters, it is suggested to apply some form of regularization to ensure good generalization performance. Regularizations can be explicit, such as the dropout technique \cite{srivastava2014dropout} or the $l_2$-penalization (weight decay) as reported in \cite{krizhevsky2012imagenet}; or implicit, as in the case of the early stopping strategy \cite{yao2007early} or the stochastic gradient descent algorithm itself \cite{zhang2016understanding}.
Inspired by the recent line of works \cite{saxe2013exact,advani2017high}, in this article we introduce a random matrix framework to analyze the training and, more importantly, the generalization performance of neural networks, trained by gradient descent. Preliminary results established from a toy model of two-class classification on a single-layer linear network are presented, which, despite their simplicity, shed new light on the understanding of many important aspects in training neural nets. In particular, we demonstrate how early stopping can naturally protect the network against overfitting, which becomes more severe as the number of training sample approaches the dimension of the data. We also provide a strict lower bound on the training sample size for a given classification task in this simple setting. A byproduct of our analysis implies that random initialization, although commonly used in practice in training deep networks \cite{glorot2010understanding,krizhevsky2012imagenet}, may lead to a degradation of the network performance.
From a more theoretical point of view, our analyses allow one to evaluate any functional of the eigenvalues of the sample covariance matrix of the data (or of the data representation learned from previous layers in a deep model), which is at the core of understanding many experimental observations in today's deep networks \cite{glorot2010understanding,ioffe2015batch}. Our results are envisioned to generalize to more elaborate settings, notably to deeper models that are trained with the stochastic gradient descent algorithm, which is of more practical interest today due to the tremendous size of the data.
\emph{Notations}: Boldface lowercase (uppercase) characters stand for vectors (matrices), and non-boldface for scalars respectively. $\mathbf{0}_p$ is the column vector of zeros of size $p$, and $\mathbf{I}_p$ the $p \times p$ identity matrix. The notation $(\cdot)^{\sf T}$ denotes the transpose operator. The norm $\| \cdot \| $ is the Euclidean norm for vectors and the operator norm for matrices. $\Im(\cdot)$ denotes the imaginary part of a complex number. For $x \in \mathbb{R}$, we denote for simplicity $(x)^+ \equiv \max(x,0)$.
In the remainder of the article, we introduce the problem of interest and recall the results of \cite{saxe2013exact} in Section~\ref{sec:problem}. After a brief overview of basic concepts and methods to be used throughout the article in Section~\ref{sec:preliminaries}, our main results on the training and generalization performance of the network are presented in Section~\ref{sec:performance}, followed by a thorough discussion in Section~\ref{sec:discuss} and experiments on the popular MNIST database \cite{lecun1998mnist} in Section~\ref{sec:validations}. Section~\ref{sec:conclusion} concludes the article by summarizing the main results and outlining future research directions.
\section{Problem Statement}
\label{sec:problem}
Let the training data $\mathbf{x}_1, \ldots, \mathbf{x}_n \in \mathbb{R}^p$ be independent vectors drawn from two distribution classes $\mathcal{C}_1$ and $\mathcal{C}_2$ of cardinality $n_1$ and $n_2$ (thus $n_1 + n_2 = n$), respectively. We assume that the data vector $\mathbf{x}_i$ of class $\mathcal{C}_a$ can be written as
\[
\mathbf{x}_i = (-1)^a \boldsymbol{\mu} + \mathbf{z}_i
\]
for $a = \{1,2\}$, with $\boldsymbol{\mu} \in \mathbb{R}^p$ and $\mathbf{z}_i$ a Gaussian random vector $\mathbf{z}_i \sim \mathcal{N}(\mathbf{0}_p, \mathbf{I}_p)$. In the context of a binary classification problem, one takes the label $y_i = -1$ for $\mathbf{x}_i \in \mathcal{C}_1$ and $y_j = 1$ for $\mathbf{x}_j \in \mathcal{C}_2$ to distinguish the two classes.
We denote the training data matrix $\mathbf{X} = \begin{bmatrix} \mathbf{x}_1, \ldots, \mathbf{x}_n \end{bmatrix} \in \mathbb{R}^{p \times n}$ by cascading all $\mathbf{x}_i$'s as column vectors and associated label vector $\mathbf{y} \in \mathbb{R}^n$. With the pair $\{\mathbf{X}, \mathbf{y}\}$, a classifier is trained using ``full-batch'' gradient descent to minimize the loss function $L(\mathbf{w})$ given by
\[
L(\mathbf{w}) = \frac1{2n} \| \mathbf{y}^{\sf T} - \mathbf{w}^{\sf T} \mathbf{X} \|^2
\]
so that for a new datum $\hat \mathbf{x}$, the output of the classifier is $\hat y = \mathbf{w}^{\sf T} \hat \mathbf{x}$, the sign of which is then used to decide the class of $\hat \mathbf{x}$. The derivative of $L$ with respective to $\mathbf{w}$ is given by
\[
\frac{\partial L(\mathbf{w})}{\partial \mathbf{w}} = - \frac1{n} \mathbf{X} (\mathbf{y} - \mathbf{X}^{\sf T} \mathbf{w}).
\]
The gradient descent algorithm \cite{boyd2004convex} takes small steps of size $\alpha$ along the \emph{opposite direction} of the associated gradient, i.e., $\mathbf{w}_{t+1} = \mathbf{w}_t - \alpha \frac{\partial L(\mathbf{w})}{\partial \mathbf{w}} \big|_{\mathbf{w} = \mathbf{w}_t}$.
Following the previous works of \cite{saxe2013exact,advani2017high}, when the learning rate $\alpha$ is small, $\mathbf{w}_{t+1}$ and $\mathbf{w}_t$ are close to each other so that by performing a continuous-time approximation, one obtains the following differential equation
\[
\frac{\partial \mathbf{w}(t)}{\partial t} = - \alpha \frac{\partial L(\mathbf{w})}{\partial \mathbf{w}} = \frac{\alpha}{n} \mathbf{X} \left(\mathbf{y} - \mathbf{X}^{\sf T} \mathbf{w}(t) \right)
\]
the solution of which is given explicitly by
\begin{equation}
\mathbf{w}(t) = e^{- \frac{\alpha t}n \mathbf{X} \mathbf{X}^{\sf T} } \mathbf{w}_0 + \left(\mathbf{I}_p - e^{- \frac{\alpha t}n \mathbf{X}\X^{\sf T} } \right) ( \mathbf{X}\X^{\sf T} )^{-1} \mathbf{X}\mathbf{y}
\label{eq:solution-de}
\end{equation}
if one assumes that $\mathbf{X} \mathbf{X}^{\sf T}$ is invertible (only possible in the case $p < n$), with $\mathbf{w}_0 \equiv \mathbf{w}(t=0)$ the initialization of the weight vector; we recall the definition of the exponential of a matrix $\frac1n \mathbf{X} \mathbf{X}^{\sf T}$ given by the power series $e^{\frac1n \mathbf{X} \mathbf{X}^{\sf T}} = \sum_{k=0}^\infty \frac1{k!} (\frac1n \mathbf{X} \mathbf{X}^{\sf T})^k = \mathbf{V} e^{\boldsymbol{\Lambda}} \mathbf{V}^{\sf T}$, with the eigendecomposition of $\frac1n \mathbf{X} \mathbf{X}^{\sf T} = \mathbf{V} \boldsymbol{\Lambda} \mathbf{V}^{\sf T}$ and $e^{\boldsymbol{\Lambda}}$ is a diagonal matrix with elements equal to the exponential of the elements of $\boldsymbol{\Lambda}$. As $t\to \infty$ the network ``forgets'' the initialization $\mathbf{w}_0$ and results in the least-square solution $\mathbf{w}_{LS} \equiv ( \mathbf{X}\X^{\sf T} )^{-1} \mathbf{X}\mathbf{y}$.
When $p>n$, $\mathbf{X}\X^{\sf T}$ is no longer invertible. Assuming $\mathbf{X}^{\sf T} \mathbf{X}$ is invertible and writing $\mathbf{X}\mathbf{y} = \left(\mathbf{X}\X^{\sf T}\right)\mathbf{X}\left(\mathbf{X}^{\sf T}\mathbf{X}\right)^{-1}\mathbf{y}$, the solution is similarly given by
\[
\mathbf{w}(t) = e^{- \frac{\alpha t}n \mathbf{X} \mathbf{X}^{\sf T} } \mathbf{w}_0 + \mathbf{X} \left(\mathbf{I}_n - e^{- \frac{\alpha t}n \mathbf{X}^{\sf T} \mathbf{X} } \right) ( \mathbf{X}^{\sf T} \mathbf{X} )^{-1} \mathbf{y}
\]
with the least-square solution $\mathbf{w}_{LS} \equiv \mathbf{X}( \mathbf{X}^{\sf T}\mathbf{X} )^{-1} \mathbf{y}$.
In the work of \cite{advani2017high} it is assumed that $\mathbf{X}$ has i.i.d.\@ entries and that there is no linking structure between the data and associated targets in such a way that the ``true'' weight vector $\bar \mathbf{w}$ to be learned is independent of $\mathbf{X}$ so as to simplify the analysis. In the present work we aim instead at exploring the capacity of the network to retrieve the (mixture modeled) data structure and position ourselves in a more realistic setting where $\mathbf{w}$ captures the different statistical structures (between classes) of the pair $(\mathbf{X},\mathbf{y})$. Our results are thus of more guiding significance for practical interests.
From \eqref{eq:solution-de} note that both $e^{-\frac{\alpha t}{n} \mathbf{X} \mathbf{X}^{\sf T}}$ and $\mathbf{I}_p - e^{-\frac{\alpha t}{n} \mathbf{X} \mathbf{X}^{\sf T}}$ share the same eigenvectors with the \emph{sample covariance matrix} $\frac1n \mathbf{X} \mathbf{X}^{\sf T}$, which thus plays a pivotal role in the network learning dynamics. More concretely, the projections of $\mathbf{w}_0$ and $\mathbf{w}_{LS}$ onto the eigenspace of $\frac1n \mathbf{X} \mathbf{X}^{\sf T}$, weighted by functions ($\exp(-\alpha t \lambda_i)$ or $1-\exp(-\alpha t \lambda_i)$) of the associated eigenvalue $\lambda_i$, give the temporal evolution of $\mathbf{w}(t)$ and consequently the training and generalization performance of the network. The core of our study therefore consists in deeply understanding of the eigenpairs of this sample covariance matrix, which has been largely investigated in the random matrix literature \cite{bai2010spectral}.
\section{Preliminaries}
\label{sec:preliminaries}
Throughout this paper, we will be relying on some basic yet powerful concepts and methods from random matrix theory, which shall be briefly highlighted in this section.
\subsection{Resolvent and deterministic equivalents}
\label{subsec:resolvent-and-its-D-E}
Consider an $n \times n$ Hermitian random matrix $\mathbf{M}$. We define its \emph{resolvent} $\mathbf{Q}_{\mathbf{M}}(z)$, for $z \in \mathbb{C}$ not an eigenvalue of $\mathbf{M}$, as
\[
\mathbf{Q}_{\mathbf{M}}(z) = \left( \mathbf{M} - z \mathbf{I}_n \right)^{-1}.
\]
Through the Cauchy integral formula discussed in the following subsection, as well as its central importance in random matrix theory, $\mathbf{Q}_{\frac1n \mathbf{X}\X^{\sf T}}(z)$ is the key object investigated in this article.
For certain simple distributions of $\mathbf{M}$, one may define a so-called \emph{deterministic equivalent} \cite{hachem2007deterministic,couillet2011random} $\bar\mathbf{Q}_{\mathbf{M}}$ for $\mathbf{Q}_{\mathbf{M}}$, which is a deterministic matrix such that for all $\mathbf{A}\in \mathbb{R}^{n \times n}$ and all $\mathbf{a,b} \in \mathbb{R}^n$ of bounded (spectral and Euclidean, respectively) norms, $\frac1n \tr \left( \mathbf{A} \mathbf{Q}_{\mathbf{M}} \right) - \frac1n \tr \left( \mathbf{A} \bar \mathbf{Q}_{\mathbf{M}} \right) \to 0$ and $\mathbf{a}^{\sf T} \left( \mathbf{Q}_{\mathbf{M}} - \bar \mathbf{Q}_{\mathbf{M}} \right) \mathbf{b} \to 0$ almost surely as $n \to \infty$. As such, deterministic equivalents allow to transfer random spectral properties of $\mathbf{M}$ in the form of deterministic limiting quantities and thus allows for a more detailed investigation.
\subsection{Cauchy's integral formula}
\label{subsec:cauchy-integral-and-residue}
First note that the resolvent $\mathbf{Q}_{\mathbf{M}}(z)$ has the same eigenspace as $\mathbf{M}$, with associated eigenvalue $\lambda_i$ replaced by $\frac1{\lambda_i - z}$. As discussed at the end of Section~\ref{sec:problem}, our objective is to evaluate functions of these eigenvalues, which reminds us of the fundamental Cauchy's integral formula, stating that for any function $f$ holomorphic on an open subset $U$ of the complex plane, one can compute $f(\lambda)$ by contour integration. More concretely, for a closed positively (counter-clockwise) oriented path $\gamma$ in $U$ with winding number one (i.e., describing a $360^\circ$ rotation), one has, for $\lambda$ contained in the surface described by $\gamma$, $\frac1{2\pi i} \oint_{\gamma} \frac{f(z)}{z - \lambda} dz = f(\lambda)$ and $\frac1{2\pi i} \oint_{\gamma} \frac{f(z)}{z - \lambda} dz = 0$ if $\lambda$ lies outside the contour of $\gamma$.
With Cauchy's integral formula, one is able to evaluate more sophisticated functionals of the random matrix $\mathbf{M}$. For example, for $f(\mathbf{M}) \equiv \mathbf{a}^{\sf T} e^{\mathbf{M}} \mathbf{b}$ one has
\[
f(\mathbf{M}) = -\frac1{2 \pi i} \oint_{\gamma} \exp(z) \mathbf{a}^{\sf T} \mathbf{Q}_{\mathbf{M}}(z) \mathbf{b}\ dz
\]
with $\gamma$ a positively oriented path circling around \emph{all} the eigenvalues of $\mathbf{M}$. Moreover, from the previous subsection one knows that the bilinear form $\mathbf{a}^{\sf T} \mathbf{Q}_{\mathbf{M}}(z) \mathbf{b}$ is asymptotically close to a non-random quantity $\mathbf{a}^{\sf T} \bar \mathbf{Q}_{\mathbf{M}}(z) \mathbf{b}$. One thus deduces that the functional $\mathbf{a}^{\sf T} e^{\mathbf{M}} \mathbf{b}$ has an asymptotically deterministic behavior that can be expressed as $-\frac1{2 \pi i} \oint_{\gamma} \exp(z) \mathbf{a}^{\sf T} \bar \mathbf{Q}_{\mathbf{M}}(z) \mathbf{b}\ dz$.
This observation serves in the present article as the foundation for the performance analysis of the gradient-based classifier, as described in the following section.
\section{Temporal Evolution of Training and Generalization Performance}
\label{sec:performance}
With the explicit expression of $\mathbf{w}(t)$ in \eqref{eq:solution-de}, we now turn our attention to the training and generalization performances of the classifier as a function of the training time $t$. To this end, we shall be working under the following assumptions.
\begin{Assumption}[Growth Rate]
As $n \to \infty$,
\begin{enumerate}
\item $\frac{p}{n} \to c \in (0,\infty)$.
\item For $a= \{1,2\}$, $\frac{n_a}{n} \to c_a \in (0,1)$.
\item $\| \boldsymbol{\mu} \| = O(1)$.
\end{enumerate}
\label{ass:growth-rate}
\end{Assumption}
The above assumption ensures that the matrix $ \frac1n \mathbf{X} \mathbf{X}^{\sf T}$ is of bounded operator norm for all large $n,p$ with probability one \cite{bai1998no}.
\begin{Assumption}[Random Initialization]
Let $\mathbf{w}_0 \equiv \mathbf{w}(t=0)$ be a random vector with i.i.d.\@ entries of zero mean, variance $\sigma^2/p$ for some $\sigma>0$ and finite fourth moment.
\label{ass:initialization}
\end{Assumption}
We first focus on the generalization performance, i.e., the average performance of the trained classifier taking as input an unseen new datum $\hat \mathbf{x}$ drawn from class $\mathcal{C}_1$ or $\mathcal{C}_2$.
\subsection{Generalization Performance}
\label{subsec:generalization-perf}
To evaluate the generalization performance of the classifier, we are interested in two types of misclassification rates, for a new datum $\hat \mathbf{x}$ drawn from class $\mathcal{C}_1$ or $\mathcal{C}_2$, as
\[
{\rm P}( \mathbf{w}(t)^{\sf T} \hat \mathbf{x} > 0~|~\hat \mathbf{x} \in \mathcal{C}_1), \quad {\rm P}( \mathbf{w}(t)^{\sf T} \hat \mathbf{x} < 0~|~\hat \mathbf{x} \in \mathcal{C}_2).
\]
Since the new datum $\hat \mathbf{x}$ is independent of $\mathbf{w}(t)$, $\mathbf{w}(t)^{\sf T} \hat \mathbf{x}$ is a Gaussian random variable of mean $\pm \mathbf{w}(t)^{\sf T} \boldsymbol{\mu}$ and variance $ \| \mathbf{w}(t) \|^2 $. The above probabilities can therefore be given via the $Q$-function: $Q(x) \equiv \frac1{\sqrt{2\pi}} \int_x^{\infty} \exp\left( -\frac{u^2}2 \right) du$. We thus resort to the computation of $\mathbf{w}(t)^{\sf T} \boldsymbol{\mu}$ as well as $ \mathbf{w}(t)^{\sf T} \mathbf{w}(t) $ to evaluate the aforementioned classification error.
For $\boldsymbol{\mu}^{\sf T} \mathbf{w}(t)$, with Cauchy's integral formula we have
\begin{align*}
&\boldsymbol{\mu}^{\sf T} \mathbf{w}(t) = \boldsymbol{\mu}^{\sf T} e^{- \frac{\alpha t}n \mathbf{X} \mathbf{X}^{\sf T} } \mathbf{w}_0 + \boldsymbol{\mu}^{\sf T} \left(\mathbf{I}_p - e^{- \frac{\alpha t}n \mathbf{X}\X^{\sf T} } \right) \mathbf{w}_{LS}\\
&= -\frac1{2\pi i} \oint_{\gamma} f_t(z) \boldsymbol{\mu}^{\sf T} \left( \frac1n \mathbf{X} \mathbf{X}^{\sf T} - z \mathbf{I}_p \right)^{-1} \mathbf{w}_0 \ dz \\
& -\frac1{2\pi i} \oint_{\gamma} \frac{1-f_t(z)}{z} \boldsymbol{\mu}^{\sf T} \left( \frac1n \mathbf{X} \mathbf{X}^{\sf T} - z \mathbf{I}_p \right)^{-1} \frac1n \mathbf{X}\mathbf{y} \ dz
\end{align*}
with $f_t(z) \equiv \exp(-\alpha t z)$, for a positive closed path $\gamma$ circling around all eigenvalues of $\frac1n \mathbf{X} \mathbf{X}^{\sf T}$. Note that the data matrix $\mathbf{X}$ can be rewritten as
\[
\mathbf{X} = -\boldsymbol{\mu} \j_1^{\sf T} + \boldsymbol{\mu} \j_2^{\sf T} + \mathbf{Z} = \boldsymbol{\mu} \mathbf{y}^{\sf T} + \mathbf{Z}
\]
with $\mathbf{Z} \equiv \begin{bmatrix} \mathbf{z}_1, \ldots, \mathbf{z}_n \end{bmatrix} \in \mathbb{R}^{p \times n}$ of i.i.d.\@ $\mathcal{N}(0,1)$ entries and $\j_a \in \mathbb{R}^n$ the canonical vectors of class $\mathcal{C}_a$ such that $(\j_a)_i = \delta_{\mathbf{x}_i \in \mathcal{C}_a}$. To isolate the deterministic vectors $\boldsymbol{\mu}$ and $\j_a$'s from the random $\mathbf{Z}$ in the expression of $\boldsymbol{\mu}^{\sf T} \mathbf{w}(t)$, we exploit Woodbury's identity to obtain
\begin{align*}
&\left( \frac1n \mathbf{X} \mathbf{X}^{\sf T} - z \mathbf{I}_p \right)^{-1} = \mathbf{Q}(z) - \mathbf{Q}(z) \begin{bmatrix} \boldsymbol{\mu} & \frac1n \mathbf{Z}\mathbf{y} \end{bmatrix} \\
&\begin{bmatrix} \boldsymbol{\mu}^{\sf T} \mathbf{Q}(z) \boldsymbol{\mu} & 1+\frac1n \boldsymbol{\mu}^{\sf T} \mathbf{Q}(z) \mathbf{Z}\mathbf{y} \\ * & -1 + \frac1n \mathbf{y}^{\sf T} \mathbf{Z}^{\sf T} \mathbf{Q}(z) \frac1n \mathbf{Z} \mathbf{y} \end{bmatrix}^{-1} \begin{bmatrix} \boldsymbol{\mu}^{\sf T} \\ \frac1n \mathbf{y}^{\sf T} \mathbf{Z}^{\sf T} \end{bmatrix} \mathbf{Q}(z)
\end{align*}
where we denote the resolvent $\mathbf{Q}(z) \equiv \left( \frac1n \mathbf{Z}\Z^{\sf T} - z \mathbf{I}_p \right)^{-1}$, a deterministic equivalent of which is given by
\[
\mathbf{Q}(z) \leftrightarrow \bar \mathbf{Q}(z) \equiv m(z) \mathbf{I}_p
\]
with $m(z)$ determined by the popular Marčenko–Pastur equation \cite{marvcenko1967distribution}
\begin{equation}
m(z) = \frac{1-c-z}{2cz} + \frac{\sqrt{(1-c-z)^2 - 4cz}}{2cz}
\label{eq:MP-equation}
\end{equation}
where the branch of the square root is selected in such a way that $\Im(z) \cdot \Im m(z) >0$, i.e., for a given $z$ there exists a \emph{unique} corresponding $m(z)$.
Substituting $\mathbf{Q}(z)$ by the simple form deterministic equivalent $m(z) \mathbf{I}_p$, we are able to estimate the random variable $\boldsymbol{\mu}^{\sf T} \mathbf{w}(t)$ with a contour integral of some deterministic quantities as $n,p \to \infty$. Similar arguments also hold for $\mathbf{w}(t)^{\sf T} \mathbf{w}(t)$, together leading to the following theorem.
\begin{Theorem}[Generalization Performance]
Let Assumptions~\ref{ass:growth-rate} and~\ref{ass:initialization} hold. As $n \to \infty$, with probability one
\begin{align*}
&{\rm P}( \mathbf{w}(t)^{\sf T} \hat \mathbf{x} > 0~|~\hat \mathbf{x} \in \mathcal{C}_1) - Q\left( \frac{E }{ \sqrt{V} } \right) \to 0 \\
&{\rm P}( \mathbf{w}(t)^{\sf T} \hat \mathbf{x} < 0~|~\hat \mathbf{x} \in \mathcal{C}_2) - Q\left( \frac{E }{ \sqrt{V} } \right)\to 0
\end{align*}
where
\begin{align*}
E &\equiv -\frac1{2\pi i} \oint_{\gamma} \frac{1-f_t(z)}{z} \frac{ \| \boldsymbol{\mu} \|^2 m(z) \ dz}{ \left( \| \boldsymbol{\mu} \|^2 +c \right) m(z) +1 } \\
V &\equiv \frac1{2\pi i} \oint_{\gamma} \left[\frac{ \frac1{z^2} \left(1-f_t(z)\right)^2\ }{ \left( \| \boldsymbol{\mu} \|^2 +c \right) m(z) +1 } - \sigma^2 f_t^2(z) m(z) \right]dz
\end{align*}
with $\gamma$ a closed positively oriented path that contains all eigenvalues of $\frac1n \mathbf{X} \mathbf{X}^{\sf T}$ and the origin, $f_t(z) \equiv \exp(-\alpha t z)$ and $m(z)$ given by Equation~\eqref{eq:MP-equation}.
\label{theo:generalize-perf}
\end{Theorem}
Although derived from the case $p<n$, Theorem~\ref{theo:generalize-perf} also applies when $p>n$. To see this, note that with Cauchy's integral formula, for $z\neq0$ not an eigenvalue of $\frac1n \mathbf{X} \mathbf{X}^{\sf T}$ (thus not of $\frac1n \mathbf{X}^{\sf T} \mathbf{X}$), one has $\mathbf{X} \left( \frac1n \mathbf{X}^{\sf T} \mathbf{X} - z \mathbf{I}_n \right)^{-1}\mathbf{y} = \left( \frac1n \mathbf{X}\X^{\sf T} - z \mathbf{I}_p \right)^{-1} \mathbf{X} \mathbf{y}$, which further leads to the same expressions as in Theorem~\ref{theo:generalize-perf}. Since $\frac1n \mathbf{X}\X^{\sf T}$ and $\frac1n \mathbf{X}^{\sf T}\mathbf{X}$ have the same eigenvalues except for additional zero eigenvalues for the larger matrix, the path $\gamma$ remains unchanged (as we demand that $\gamma$ contains the origin) and hence Theorem~\ref{theo:generalize-perf} holds true for both $p<n$ and $p>n$. The case $p=n$ can be obtained by continuity arguments.
\subsection{Training performance}
\label{eq:training-perf}
To compare generalization versus training performance, we are now interested in the behavior of the classifier when applied to the training set $\mathbf{X}$. To this end, we consider the random vector $\mathbf{X}^{\sf T} \mathbf{w}(t)$ given by
\[
\mathbf{X}^{\sf T} \mathbf{w}(t) = \mathbf{X}^{\sf T} e^{- \frac{\alpha t}n \mathbf{X} \mathbf{X}^{\sf T} } \mathbf{w}_0 + \mathbf{X}^{\sf T} \left(\mathbf{I}_p - e^{- \frac{\alpha t}n \mathbf{X}\X^{\sf T} } \right) \mathbf{w}_{LS}.
\]
Note that the $i$-th entry of $\mathbf{X}^{\sf T} \mathbf{w}(t)$ is given by the bilinear form $\mathbf{e}_i^{\sf T} \mathbf{X}^{\sf T} \mathbf{w}(t)$, with $\mathbf{e}_i$ the canonical vector with unique non-zero entry $[\mathbf{e}_i]_i = 1$. With previous notations we have
\begin{align*}
&\mathbf{e}_i^{\sf T} \mathbf{X}^{\sf T} \mathbf{w}(t)\\
&= -\frac1{2\pi i} \oint_{\gamma} f_t(z, t) \mathbf{e}_i^{\sf T} \mathbf{X}^{\sf T} \left( \frac1n \mathbf{X} \mathbf{X}^{\sf T} - z \mathbf{I}_p \right)^{-1} \mathbf{w}_0 \ dz\\
&-\frac1{2\pi i} \oint_{\gamma} \frac{1-f_t(z)}{z} \mathbf{e}_i^{\sf T} \frac1n \mathbf{X}^{\sf T} \left( \frac1n \mathbf{X} \mathbf{X}^{\sf T} - z \mathbf{I}_p \right)^{-1} \mathbf{X} \mathbf{y} \ dz
\end{align*}
which yields the following results.
\begin{Theorem}[Training Performance]
Under the assumptions and notations of Theorem~\ref{theo:generalize-perf}, as $n \to \infty$,
\begin{align*}
&{\rm P}( \mathbf{w}(t)^{\sf T} \mathbf{x}_i > 0~|~\mathbf{x}_i \in \mathcal{C}_1) - Q\left( \frac{E_* }{ \sqrt{V_* - E_*^2 } } \right) \to 0 \\
&{\rm P}( \mathbf{w}(t)^{\sf T} \mathbf{x}_i < 0~|~\mathbf{x}_i \in \mathcal{C}_2) - Q\left( \frac{E_* }{ \sqrt{V_* - E_*^2} } \right)\to 0
\end{align*}
almost surely, with
\begin{align*}
E_* &\equiv \frac1{2\pi i} \oint_{\gamma} \frac{1-f_t(z)}{z} \frac{dz}{ \left( \| \boldsymbol{\mu} \|^2 +c \right) m(z) +1 } \\
V_* &\equiv \frac1{2\pi i} \oint_{\gamma} \left[\frac{ \frac1{z} \left(1-f_t(z)\right)^2\ }{ \left( \| \boldsymbol{\mu} \|^2 +c \right) m(z) +1 } - \sigma^2 f_t^2(z) z m(z) \right] dz.
\end{align*}
\label{theo:training-perf}
\end{Theorem}
\begin{figure}[htb]
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\caption{Training and generalization performance for $\boldsymbol{\mu} = [2;\mathbf{0}_{p-1}]$, $p=256$, $n=512$, $\sigma^2 =0.1$, $\alpha = 0.01$ and $c_1 = c_2 = 1/2$. Results obtained by averaging over $50$ runs.}
\label{fig:train-and-general-perf}
\end{center}
\vskip -0.1in
\end{figure}
In Figure~\ref{fig:train-and-general-perf} we compare finite dimensional simulations with theoretical results obtained from Theorem~\ref{theo:generalize-perf}~and~\ref{theo:training-perf} and observe a very close match, already for not too large $n,p$. As $t$ grows large, the generalization error first drops rapidly with the training error, then goes up, although slightly, while the training error continues to decrease to zero. This is because the classifier starts to over-fit the training data $\mathbf{X}$ and performs badly on unseen ones. To avoid over-fitting, one effectual approach is to apply regularization strategies \cite{bishop2007pattern}, for example, to ``early stop'' (at $t=100$ for instance in the setting of Figure~\ref{fig:train-and-general-perf}) in the training process. However, this introduces new hyperparameters such as the optimal stopping time $t_{opt}$ that is of crucial importance for the network performance and is often tuned through cross-validation in practice. Theorem~\ref{theo:generalize-perf} and~\ref{theo:training-perf} tell us that the training and generalization performances, although being random themselves, have asymptotically deterministic behaviors described by $(E_*, V_*)$ and $(E, V)$, respectively, which allows for a deeper understanding on the choice of $t_{opt}$, since $E, V$ are in fact functions of $t$ via $f_t(z) \equiv \exp(-\alpha t z)$.
Nonetheless, the expressions in Theorem~\ref{theo:generalize-perf} and~\ref{theo:training-perf} of contour integrations are not easily analyzable nor interpretable. To gain more insight, we shall rewrite $(E, V)$ and $(E_*, V_*)$ in a more readable way. First, note from Figure~\ref{fig:eigvenvalue-distribution-XX} that the matrix $\frac1n \mathbf{X}\X^{\sf T}$ has (possibly) two types of eigenvalues: those inside the \emph{main bulk} (between $\lambda_- \equiv (1-\sqrt{c})^2$ and $\lambda_+ \equiv (1+\sqrt{c})^2$) of the Marčenko–Pastur distribution
\begin{equation}\label{eq:MP-distribution}
\nu(dx) = \frac{ \sqrt{ (x- \lambda_-)^+ (\lambda_+ - x)^+ }}{2\pi c x} dx + \left( 1- \frac1c\right)^+ \delta(x)
\end{equation}
and a (possibly) isolated one\footnote{The existence (or absence) of outlying eigenvalues for the sample covariance matrix has been largely investigated in the random matrix literature and is related to the so-called ``spiked random matrix model''. We refer the reader to \cite{benaych2011eigenvalues} for an introduction. The information carried by these ``isolated'' eigenpairs also marks an important technical difference to \cite{advani2017high} in which $\mathbf{X}$ is only composed of noise terms.} lying away from $[\lambda_-,\lambda_+]$, that shall be treated separately. We rewrite the path $\gamma$ (that contains all eigenvalues of $\frac1n \mathbf{X}\X^{\sf T}$) as the sum of two paths $\gamma_b$ and $\gamma_s$, that circle around the main bulk and the isolated eigenvalue (if any), respectively. To handle the first integral of $\gamma_b$, we use the fact that for any nonzero $\lambda \in \mathbb{R}$, the limit $\lim_{z\in\mathbb{Z} \to \lambda} m(z) \equiv \check m(\lambda)$ exists \cite{silverstein1995analysis} and follow the idea in \cite{bai2008clt} by choosing the contour $\gamma_b$ to be a rectangle with sides parallel to the axes, intersecting the real axis at $0$ and $\lambda_+$ and the horizontal sides being a distance $\varepsilon \to 0$ away from the real axis, to split the contour integral into four single ones of $\check m(x)$. The second integral circling around $\gamma_s$ can be computed with the residue theorem. This together leads to the expressions of $(E, V)$ and $(E_*, V_*)$ as follows\footnote{We defer the readers to Section~\ref{sm:detailed-deduction} in Supplementary Material for a detailed exposition of Theorem~\ref{theo:generalize-perf}~and~\ref{theo:training-perf}, as well as \eqref{eq:E}-\eqref{eq:Var-star}.}
\begin{align}
E &= \int \frac{ 1-f_t(x) }{x} \mu(dx) \label{eq:E}\\
%
V &= \frac{\|\boldsymbol{\mu}\|^2 + c}{\|\boldsymbol{\mu}\|^2} \int \frac{ (1-f_t(x))^2 \mu(dx)}{x^2} + \sigma^2 \int f_t^2(x) \nu(dx) \label{eq:Var} \\
%
%
%
E_* &= \frac{\|\boldsymbol{\mu}\|^2 + c}{\|\boldsymbol{\mu}\|^2} \int \frac{ 1-f_t(x) }{x} \mu(dx) \label{eq:E-star} \\
%
V_* &= \frac{\|\boldsymbol{\mu}\|^2 + c}{\|\boldsymbol{\mu}\|^2} \int \frac{ (1-f_t(x))^2 \mu(dx)}{x} + \sigma^2 \int x f_t^2(x) \nu(dx) \label{eq:Var-star}
\end{align}
where we recall $f_t(x) = \exp(-\alpha t x)$, $\nu(x)$ given by \eqref{eq:MP-distribution} and denote the measure
\begin{equation}\label{eq:definition-measure}
\mu(dx) \equiv \frac{\sqrt{(x-\lambda_-)^+(\lambda_+ -x)^+}}{ 2\pi(\lambda_s - x) } dx + \frac{ (\|\boldsymbol{\mu}\|^4-c)^+}{\|\boldsymbol{\mu}\|^2} \delta_{\lambda_s}(x)
\end{equation}
as well as
\begin{equation}\label{eq:lambda_s}
\lambda_s = c+1 +\| \boldsymbol{\mu}\|^2 + \frac{c}{\| \boldsymbol{\mu}\|^2}\ge (\sqrt{c}+1)^2
\end{equation}
with equality if and only if $\| \boldsymbol{\mu}\|^2 = \sqrt{c}$.
A first remark on the expressions of \eqref{eq:E}-\eqref{eq:Var-star} is that $E_*$ differs from $E$ only by a factor of $\frac{\|\boldsymbol{\mu}\|^2+c}{\|\boldsymbol{\mu}\|^2}$. Also, both $V$ and $V_*$ are the sum of two parts: the first part that strongly depends on $\boldsymbol{\mu}$ and the second one that is independent of $\boldsymbol{\mu}$. One thus deduces for $\|\boldsymbol{\mu}\| \to 0$ that $E \to 0$ and
\[
V \to \int \frac{ (1-f_t(x))^2 }{x^2} \rho(dx) + \sigma^2 \int f_t^2(x) \nu(dx) > 0
\]
with $\rho(dx) \equiv \frac{\sqrt{(x-\lambda_-)^+(\lambda_+ -x)^+}}{ 2\pi (c+1) } dx$ and therefore the generalization performance goes to $Q(0) = 0.5$. On the other hand, for $\|\boldsymbol{\mu}\| \to \infty$, one has $ \frac{E}{\sqrt{V}} \to \infty $ and hence the classifier makes perfect predictions.
In a more general context (i.e., for Gaussian mixture models with generic means and covariances as investigated in \cite{benaych2016spectral}, and obviously for practical datasets), there may be more than one eigenvalue of $\frac1n \mathbf{X}\X^{\sf T}$ lying outside the main bulk, which may not be limited to the interval $[\lambda_-,\lambda_+]$. In this case, the expression of $m(z)$, instead of being explicitly given by \eqref{eq:MP-equation}, may be determined through more elaborate (often implicit) formulations. While handling more generic models is technically reachable within the present analysis scheme, the results are much less intuitive. Similar objectives cannot be achieved within the framework presented in \cite{advani2017high}; this conveys more practical interest to our results and the proposed analysis framework.
\begin{figure}[htb]
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\caption{Eigenvalue distribution of $\frac1n \mathbf{X}\X^{\sf T}$ for $\boldsymbol{\mu} = [1.5;\mathbf{0}_{p-1}]$, $p=512$, $n=1\,024$ and $c_1 = c_2 = 1/2$.}
\label{fig:eigvenvalue-distribution-XX}
\end{center}
\vskip -0.1in
\end{figure}
\section{Discussions}
\label{sec:discuss}
In this section, with a careful inspection of \eqref{eq:E} and~\eqref{eq:Var}, discussions will be made from several different aspects. First of all, recall that the generalization performance is simply given by $ Q\left( \frac{\boldsymbol{\mu}^{\sf T} \mathbf{w}(t)}{ \| \mathbf{w}(t) \| } \right)$, with the term $\frac{\boldsymbol{\mu}^{\sf T} \mathbf{w}(t)}{ \| \mathbf{w}(t) \| }$ describing the alignment between $\mathbf{w}(t)$ and $\boldsymbol{\mu}$, therefore the best possible generalization performance is simply $Q(\|\boldsymbol{\mu}\|)$. Nonetheless, this ``best'' performance can never be achieved as long as $p/n \to c >0$, as described in the following remark.
\begin{Remark}[Optimal Generalization Performance]
Note that, with Cauchy–Schwarz inequality and the fact that $\int \mu(dx) = \| \boldsymbol{\mu} \|^2$ from \eqref{eq:definition-measure}, one has
\[
E^2 \le \int \frac{(1-f_t(x))^2}{x^2} d\mu(x) \cdot \int d\mu(x) \le \frac{\|\boldsymbol{\mu}\|^4}{\|\boldsymbol{\mu}\|^2+c} V
\]
with equality in the right-most inequality if and only if the variance $\sigma^2 = 0$. One thus concludes that $E/\sqrt{V} \le \|\boldsymbol{\mu}\|^2/\sqrt{\|\boldsymbol{\mu}\|^2 + c}$ and the best generalization performance (lowest misclassification rate) is $Q (\|\boldsymbol{\mu}\|^2/\sqrt{\|\boldsymbol{\mu}\|^2 + c})$ and can be attained only when $\sigma^2 = 0$.
\label{rem:optimal-generalization-perf}
\end{Remark}
The above remark is of particular interest because, for a given task (thus $p, \boldsymbol{\mu}$ fixed) it allows one to compute the \emph{minimum} training data number $n$ to fulfill a certain request of classification accuracy.
As a side remark, note that in the expression of $E/\sqrt{V}$ the initialization variance $\sigma^2$ only appears in $V$, meaning that random initializations impair the generalization performance of the network. As such, one should initialize with $\sigma^2$ very close, but not equal, to zero, to obtain symmetry breaking between hidden units \cite{goodfellow2016deeplearning} as well as to mitigate the drop of performance due to large $\sigma^2$.
In Figure~\ref{fig:optimal-perf-and-time-vs-sigma2} we plot the optimal generalization performance with the corresponding optimal stopping time as functions of $\sigma^2$, showing that small initialization helps training in terms of both accuracy and efficiency.
\begin{figure}[tbh]
\vskip 0.1in
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\caption{ Optimal performance and corresponding stopping time as functions of $\sigma^2$, with $c = 1/2$, $\|\boldsymbol{\mu}\|^2=4$ and $\alpha=0.01$.}
\label{fig:optimal-perf-and-time-vs-sigma2}
\end{center}
\vskip -0.1in
\end{figure}
Although the integrals in \eqref{eq:E} and \eqref{eq:Var} do not have nice closed forms, note that, for $t$ close to $0$, with a Taylor expansion of $f_t(x) \equiv \exp(-\alpha tx)$ around $\alpha t x = 0$, one gets more interpretable forms of $E$ and $V$ without integrals, as presented in the following subsection.
\subsection{Approximation for $t$ close to $0$}
\label{subsec:t-close-to-0}
Taking $t = 0$, one has $f_t(x) = 1$ and therefore $E = 0$, $V= \sigma^2 \int \nu(dx)= \sigma^2$, with $\nu(dx)$ the Marčenko–Pastur distribution given in \eqref{eq:MP-distribution}. As a consequence, at the beginning stage of training, the generalization performance is $Q(0) = 0.5$ for $\sigma^2 \neq 0$ and the classifier makes random guesses.
For $t$ not equal but close to $0$, the Taylor expansion of $f_t(x) \equiv \exp(-\alpha tx)$ around $\alpha t x=0$ gives
\[
f_t(x) \equiv \exp(-\alpha t x) \approx 1 -\alpha t x + O(\alpha^2 t^2 x^2).
\]
Making the substitution $x = 1+c-2\sqrt{c} \cos\theta$ and with the fact that $\int_0^{\pi} \frac{ \sin^2\theta }{ p + q \cos\theta } d\theta = \frac{p \pi}{q^2} \left( 1 - \sqrt{1-q^2/p^2 } \right)$ (see for example 3.644-5 in \cite{gradshteyn2014table}), one gets $E = \tilde{E} + O(\alpha^2 t^2)$ and $V = \tilde{V}+ O(\alpha^2 t^2)$, where
\[
\tilde{E} \equiv \frac{\alpha t}{2} g(\boldsymbol{\mu},c) + \frac{ (\|\boldsymbol{\mu} \|^4 -c)^+ }{\| \boldsymbol{\mu} \|^2} \alpha t = \| \boldsymbol{\mu} \|^2 \alpha t
\]
\begin{align*}
\tilde{V} &\equiv \frac{\|\boldsymbol{\mu}\|^2 + c}{\|\boldsymbol{\mu}\|^2} \frac{ (\|\boldsymbol{\mu} \|^4 - c)^+}{\| \boldsymbol{\mu} \|^2} \alpha^2 t^2 + \frac{\|\boldsymbol{\mu}\|^2 + c}{\|\boldsymbol{\mu}\|^2} \frac{\alpha^2 t^2}2 g(\boldsymbol{\mu},c) \\
& + \sigma^2 (1+c) \alpha^2 t^2 - 2\sigma^2 \alpha t + \left(1-\frac1c\right)^+ \sigma^2 \\
&+ \frac{\sigma^2}{2c} \left( 1+c - (1+\sqrt{c}) |1-\sqrt{c}| \right) \\
&= (\| \boldsymbol{\mu}\|^2 + c + c\sigma^2) \alpha^2 t^2 + \sigma^2 ( \alpha t - 1 )^2
\end{align*}
with $g(\boldsymbol{\mu},c) \equiv \| \boldsymbol{\mu}\|^2 + \frac{c}{\| \boldsymbol{\mu}\|^2} - \left( \| \boldsymbol{\mu}\| + \frac{\sqrt{c}}{\| \boldsymbol{\mu}\|} \right) \left| \| \boldsymbol{\mu}\| - \frac{ \sqrt{c} }{\| \boldsymbol{\mu}\|} \right| $ and consequently $\frac12 g(\boldsymbol{\mu},c) + \frac{(\|\boldsymbol{\mu} \|^4 -c)^+}{\| \boldsymbol{\mu} \|^2} = \| \boldsymbol{\mu}\|^2$. It is interesting to note from the above calculation that, although $E$ and $V$ seem to have different behaviors\footnote{This phenomenon has been largely observed in random matrix theory and is referred to as ``phase transition''\cite{baik2005phase}.} for $\| \boldsymbol{\mu}\|^2 > \sqrt{c}$ or $c>1$, it is in fact not the case and the extra part of $\| \boldsymbol{\mu}\|^2 > \sqrt{c}$ (or $c>1$) compensates for the singularity of the integral, so that the generalization performance of the classifier is a smooth function of both $\| \boldsymbol{\mu}\|^2 $ and $c$.
Taking the derivative of $\frac{ \tilde{E}}{ \sqrt{ \tilde{V}} } $ with respect to $t$, one has
\[
\frac{\partial }{\partial t} \frac{ \tilde{E}}{ \sqrt{ \tilde{V}} } = \frac{ \alpha (1-\alpha t) \sigma^2 }{ {\tilde V}^{3/2} }
\]
which implies that the maximum of $\frac{ \tilde{E}} {\sqrt{ \tilde{V}}}$ is $ \frac{ \| \boldsymbol{\mu}\|^2 }{ \sqrt{ \| \boldsymbol{\mu}\|^2 +c+c\sigma^2} }$ and can be attained with $t = 1/\alpha$. Moreover, taking $t=0$ in the above equation one gets $\frac{\partial }{\partial t} \frac{ \tilde{E}}{ \sqrt{ \tilde{V}} } \big|_{t=0} = \frac{\alpha}{\sigma}$. Therefore, large $\sigma$ is harmful to the training efficiency, which coincides with the conclusion from Remark~\ref{rem:optimal-generalization-perf}.
The approximation error arising from Taylor expansion can be large for $t$ away from $0$, e.g., at $t = 1/\alpha$ the difference $E - \tilde{E}$ is of order $O(1)$ and thus cannot be neglected.
\subsection{As $t \to \infty$: least-squares solution}
As $t \to \infty$, one has $f_t(x) \to 0$ which results in the least-square solution $\mathbf{w}_{LS} = (\mathbf{X}\X^{\sf T})^{-1} \mathbf{X} \mathbf{y} $ or $\mathbf{w}_{LS} = \mathbf{X} (\mathbf{X}^{\sf T}\mathbf{X})^{-1} \mathbf{y} $ and consequently
\begin{equation}
\frac{ \boldsymbol{\mu}^{\sf T} \mathbf{w}_{LS} }{ \| \mathbf{w}_{LS}\| } = \frac{ \| \boldsymbol{\mu} \|^2 }{\sqrt{\| \boldsymbol{\mu} \|^2 + c}} \sqrt{
1-\min\left(c,\frac1c\right) }.
\label{eq:perf-w-LS}
\end{equation}
Comparing \eqref{eq:perf-w-LS} with the expression in Remark~\ref{rem:optimal-generalization-perf}, one observes that when $t \to \infty$ the network becomes ``over-trained'' and the performance drops by a factor of $\sqrt{1- \min (c, c^{-1} ) }$. This becomes even worse when $c$ gets close to $1$, as is consistent with the empirical findings in \cite{advani2017high}. However, the point $c=1$ is a singularity for \eqref{eq:perf-w-LS}, but not for $\frac{E}{ \sqrt{V} }$ as in \eqref{eq:E} and~\eqref{eq:Var}. One may thus expect to have a smooth and reliable behavior of the well-trained network for $c$ close to $1$, which is a noticeable advantage of gradient-based training compared to simple least-square method. This coincides with the conclusion of \cite{yao2007early} in which the asymptotic behavior of solely $n \to \infty$ is considered.
In Figure~\ref{fig:approximation-t-small} we plot the generalization performance from simulation (blue line), the approximation from Taylor expansion of $f_t(x)$ as described in Section~\ref{subsec:t-close-to-0} (red dashed line), together with the performance of $\mathbf{w}_{LS}$ (cyan dashed line). One observes a close match between the result from Taylor expansion and the true performance for $t$ small, with the former being optimal at $t=100$ and the latter slowly approaching the performance of $\mathbf{w}_{LS}$ as $t$ goes to infinity.
In Figure~\ref{fig:approximation-c=1} we underline the case $c=1$ by taking $p=n=512$ with all other parameters unchanged from Figure~\ref{fig:approximation-t-small}. One observes that the simulation curve (blue line) increases much faster compared to Figure~\ref{fig:approximation-t-small} and is supposed to end up at $0.5$, which is the performance of $\mathbf{w}_{LS}$ (cyan dashed line). This confirms a serious degradation of performance for $c$ close to $1$ of the classical least-squares solution.
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\caption{ Generalization performance for $\boldsymbol{\mu} = \begin{bmatrix}2;\mathbf{0}_{p-1}\end{bmatrix}$, $p=256$, $n=512$, $c_1 = c_2 = 1/2$, $\sigma^2 = 0.1$ and $\alpha = 0.01$. Simulation results obtained by averaging over $50$ runs.}
\label{fig:approximation-t-small}
\end{center}
\vskip -0.1in
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\begin{figure}[htb]
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\caption{ Generalization performance for $\boldsymbol{\mu} = \begin{bmatrix}2;\mathbf{0}_{p-1}\end{bmatrix}$, $p=512$, $n=512$, $c_1 = c_2 = 1/2$, $\sigma^2 = 0.1$ and $\alpha = 0.01$. Simulation results obtained by averaging over $50$ runs.}
\label{fig:approximation-c=1}
\end{center}
\vskip -0.1in
\end{figure}
\subsection{Special case for $c = 0$}
\label{subsec:c}
One major interest of random matrix analysis is that the ratio $c$ appears constantly in the analysis. Taking $c = 0$ signifies that we have far more training data than their dimension. This results in both $\lambda_-$, $\lambda_+ \to 1$, $\lambda_s \to 1 + \| \boldsymbol{\mu} \|^2$ and
\begin{align*}
E &\to \| \boldsymbol{\mu} \|^2 \frac{ 1-f_t(1 + \| \boldsymbol{\mu} \|^2 ) }{ 1+ \| \boldsymbol{\mu} \|^2 } \\
V &\to \| \boldsymbol{\mu} \|^2 \left( \frac{ 1-f_t(1 + \| \boldsymbol{\mu} \|^2 ) }{ 1+ \| \boldsymbol{\mu} \|^2 } \right)^2 + \sigma^2 f_t^2(1).
\end{align*}
As a consequence, $\frac{ E }{ \sqrt{ V} } \to \| \boldsymbol{\mu}\|$ if $\sigma^2 = 0$. This can be explained by the fact that with sufficient training data the classifier learns to align perfectly to $\boldsymbol{\mu}$ so that $\frac{ \boldsymbol{\mu}^{\sf T} \mathbf{w}(t) }{ \|\mathbf{w}(t) \| } = \| \boldsymbol{\mu} \|$. On the other hand, with initialization $\sigma^2 \neq 0$, one always has $\frac{ E }{ \sqrt{ V} } < \| \boldsymbol{\mu}\|$. But still, as $t$ goes large, the network forgets the initialization exponentially fast and converges to the optimal $\mathbf{w}(t)$ that aligns to $\boldsymbol{\mu}$.
In particular, for $\sigma^2 \neq 0$, we are interested in the optimal stopping time by taking the derivative with respect to $t$,
\[
\frac{\partial }{\partial t} \frac{ E }{ \sqrt{ V } } = \frac{\alpha \sigma^2 \|\boldsymbol{\mu}\|^2}{ V^{3/2} } \frac{ \|\boldsymbol{\mu}\|^2 f_t(1+\|\boldsymbol{\mu}\|^2) + 1 }{1+\|\boldsymbol{\mu}\|^2} f_t^2(1) > 0
\]
showing that when $c= 0$, the generalization performance continues to increase as $t$ grows and there is in fact no ``over-training'' in this case.
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\addlegendentry{{ Theory: generalization performance }};
\begin{scope}
\spy[black!50!white,size=1.6cm,circle,connect spies,magnification=2] on (0.6,0.4) in node [fill=none] at (4,1.5);
\end{scope}
\end{axis}
\end{tikzpicture}
\caption{Training and generalization performance for MNIST data (number $1$ and $7$) with $n=p=784$, $c_1 = c_2 = 1/2$, $\alpha=0.01$ and $\sigma^2 = 0.1$. Results obtained by averaging over $100$ runs.}
\label{fig:MNIST-simu}
\end{center}
\vskip -0.1in
\end{figure}
\section{Numerical Validations}
\label{sec:validations}
We close this article with experiments on the popular MNIST dataset \cite{lecun1998mnist} (number $1$ and $7$). We randomly select training sets of size $n=784$ vectorized images of dimension $p=784$ and add artificially a Gaussian white noise of $-10\si{\deci\bel}$ in order to be more compliant with our toy model setting. Empirical means and covariances of each class are estimated from the full set of $13\,007$ MNIST images ($6\,742$ images of number $1$ and $6\,265$ of number $7$). The image vectors in each class are whitened by pre-multiplying $\mathbf{C}_a^{-1/2}$ and re-centered to have means of $\pm \boldsymbol{\mu}$, with $\boldsymbol{\mu}$ half of the difference between means from the two classes. We observe an extremely close fit between our results and the empirical simulations, as shown in Figure~\ref{fig:MNIST-simu}.
\section{Conclusion}
\label{sec:conclusion}
In this article, we established a random matrix approach to the analysis of learning dynamics for gradient-based algorithms on data of simultaneously large dimension and size. With a toy model of Gaussian mixture data with $\pm \boldsymbol{\mu}$ means and identity covariance, we have shown that the training and generalization performances of the network have asymptotically deterministic behaviors that can be evaluated via so-called deterministic equivalents and computed with complex contour integrals (and even under the form of real integrals in the present setting). The article can be generalized in many ways: with more generic mixture models (with the Gaussian assumption relaxed), on more appropriate loss functions (logistic regression for example), and more advanced optimization methods.
In the present work, the analysis has been performed on the ``full-batch'' gradient descent system. However, the most popular method used today is in fact its ``stochastic'' version \cite{bottou2010large} where only a fixed-size ($n_{batch}$) randomly selected subset (called a \emph{mini-batch}) of the training data is used to compute the gradient and descend \emph{one} step along with the opposite direction of this gradient in each iteration. In this scenario, one of major concern in practice lies in determining the optimal size of the mini-batch and its influence on the generalization performance of the network \cite{keskar2016large}. This can be naturally linked to the ratio $n_{batch}/p$ in the random matrix analysis.
Deep networks that are of more practical interests, however, need more efforts. As mentioned in \cite{saxe2013exact,advani2017high}, in the case of multi-layer networks, the learning dynamics depend, instead of each eigenmode separately, on the coupling of different eigenmodes from different layers. To handle this difficulty, one may add extra assumptions of independence between layers as in \cite{choromanska2015loss} so as to study each layer separately and then reassemble to retrieve the results of the whole network.
\section*{Acknowledgments}
We thank the anonymous reviewers for their comments and constructive suggestions. We would like to acknowledge this work is supported by the ANR Project RMT4GRAPH (ANR-14-CE28-0006) and the Project DeepRMT of La Fondation Sup{\'e}lec.
| 32,720 |
\section{Introduction}
\label{sec:introduction}
The hierarchy problem and the dark matter are two main motivations for new physics near the electroweak (EW) scale. In the standard model (SM), the Higgs field receives large quadratically divergent contributions to its potential from the interactions with SM particles, in particular, the top quark and weak gauge bosons. For a natural EW symmetry breaking scale, new particles are expected to be close to the EW scale to cut off these quadratic contributions. On the other hand, a stable weakly interacting massive particle (WIMP) with a mass around the EW scale gives a right amount of thermal relic from the Hot Big Bang to account for the dark matter in the universe. It is called ``WIMP miracle.'' Such a dark matter particle candidate also often appears naturally in models which address the hierarchy problem. The most popular and most studied examples are the supersymmetric (SUSY) extensions of SM. With a conserved $R$-parity, the lightest neutralino is stable and represents a good dark matter candidate. The models that can explain both the hierarchy problem and dark matter are particularly attractive because they provide a link between the two mysterious problems.
The new particles related to the hierarchy problem and the WIMPs have been extensively searched at various experiments. So far none of them has been discovered. The LHC has put very strong bounds on new colored particles that can cancel the SM top loop contribution to the Higgs mass. Except for some special cases, the bounds on the masses of new colored particles generically exceed 1 TeV. This would imply a quite severe tuning of the Higgs mass if the top loop is not canceled below 1 TeV. Direct searches of DM also put strong bounds on the scattering cross sections of the DM particle with nucleons. A big fraction of the expected region of the WIMP parameter space from typical SUSY models is excluded, though there are still surviving scenarios. These null experimental results have prompted people to wonder that the standard pictures such as SUSY might not be realized at the electroweak scale in nature. Alternative solutions to the hierarchy problem and DM where the interactions between new particles and SM particles are stealthier should be taken more seriously.
For the hierarchy problem, the ``neutral naturalness'' models gained increasing attentions in recent years. In these models, the top quark partners which regularize the top loop contribution to the Higgs mass do not carry SM color quantum numbers, and hence are not subject to the strong bounds from the LHC. The mirror twin Higgs model~\cite{Chacko:2005pe} is the first example and is probably the stealthiest one. The twin sector particles are SM singlets but charged under their own $SU(3) \times SU(2) [\times U(1)]$ gauge group. They are related to the SM sector by a $Z_2$ symmetry. As a result, the mass terms of the Higgs fields of the SM and twin sectors exhibit an enhanced $SU(4)$ symmetry. The 125 GeV Higgs boson arises as a pseudo-Nambu-Goldstone boson (PNGB) of the spontaneously broken $SU(4)$ symmetry. The twin sector particles are difficult to produce at colliders because they do not couple to SM gauge fields. The main experimental constraints come from the mixing between the SM Higgs and the twin Higgs, which are rather weak. The model can still be natural without violating current experimental bounds.
The next question is whether the neutral naturalness models like the twin Higgs possess good dark matter candidates. In the fraternal version of the twin Higgs model~\cite{Craig:2015pha}, people have shown that there are several possible dark matter candidates~\cite{Garcia:2015loa,Craig:2015xla,Garcia:2015toa,Farina:2015uea}. The fraternal twin Higgs takes a minimal approach in addressing the hierarchy problem using the twin Higgs mechanism. In this model, the twin fermion sector only contains the twin partners of the third generation SM fermions, since only the top Yukawa coupling gives a large contribution to the Higgs mass that needs to be regularized below the TeV scale. The twin $U(1)$ gauge boson can be absent or can have a mass without affecting the naturalness. The fraternal twin Higgs model can avoid potential cosmological problems of an exact mirror twin Higgs model which contains many light or massless particles in the twin sector. Refs.~\cite{Garcia:2015loa,Craig:2015xla} showed that the twin tau can be a viable dark matter candidate. The correct relic density is obtained for a twin tau mass in the range of 50--150 GeV, depending on other model parameters. If twin hypercharge is gauged, then the preferred mass is lighter, in the 1--20 GeV range~\cite{Craig:2015xla}. Another possibility is asymmetric dark matter from the twin baryon made of twin $b$-quarks, where the relic density is set by the baryon asymmetry in the twin sector~\cite{Garcia:2015toa,Farina:2015uea}.
In this paper, we explore a new scenario where the dark matter is the twin neutrino, in the mass range $\sim$ 0.1--10~GeV. In previous studies of twin tau dark matter, its stability is protected by the twin $U(1)_\text{EM}$ symmetry, which is assumed to be a good symmetry, either gauged or global. Here we consider that the twin $U(1)_\text{EM}$ is broken so that the twin photon acquires a mass to avoid potential cosmological problems. In this case, the twin tau and the twin neutrino can mix so the twin tau can decay to the twin neutrino if the twin tau is heavier. The twin neutrino, on the other hand, being the lightest twin fermion, can be stable due to the conservation of the twin lepton number or twin lepton parity. An interesting scenario is that if the twin photon, twin neutrino, and twin tau all have masses of the same order around a few GeV or below, the right amount of dark matter relic density from twin neutrinos can be obtained. The relic density is controlled by the coannihilation and recently discovered coscattering processes~\cite{DAgnolo:2017dbv,Garny:2017rxs}.
The coscattering phase is considered as the fourth exception in the calculation of thermal relic abundances in addition to the three classical cases enumerated in Ref.~\cite{Griest:1990kh}. It is closely related to the coannihilation case as both require another state with mass not far from the dark matter mass so that the partner state can play an important role during decoupling. The difference is that in the coannihilation phase the relic density is controlled by the freeze-out of the annihilation processes of these particles, while in coscattering phase the relic density is controlled by the freeze-out of the inelastic scattering of a dark matter particle into the partner state. Because of the energy threshold of the upward scattering, the coscattering process has a strong momentum dependence. This makes the relic density calculation quite complicated. The standard DM calculation tools such as micrOMEGAs~\cite{Belanger:2018ccd}, DarkSUSY~\cite{Bringmann:2018lay}, and MadDM~\cite{Ambrogi:2018jqj} do not apply and one needs to solve the momentum-dependent Boltzmann equations. Also, because of the momentum dependence of the coscattering process, we find that there are parameter regions of mixed phase, i.e., the relic abundance is controlled partially by coannihilation and partially by coscattering. We investigate in detail the relevant parameter space and perform calculations of the relic abundance in different phases, including situations where it is controlled by coannihilation, by coscattering, or by both processes. The calculation in the mixed phase is more involved and we discuss a relatively simple method to obtain the DM abundance with good accuracies.
The twin neutrino DM does not couple to SM directly. Its couplings to matter through mixings of the photons or the Higgses between the SM sector and the twin sector are suppressed, so the direct detection experiments have limited sensitivities. Some of the main constraints come from indirect detections and searches of other associated particles. Its annihilation through twin photon is constrained by Cosmic Microwave Background (CMB), 21cm line absorption, and Fermi-LAT data. For associated particles, the light dark photon searches provide some important constraints and future probes.
The paper is organized as follows. In Sec.~\ref{sec:FTH} we first give a brief summary of the fraternal twin Higgs model and its possible DM candidates. Then we focus the discussion on the sector of our DM scenario, i.e., the twin neutrino as the DM, and its coannihilation/coscattering partners, the twin tau and the twin photon. In Sec.~\ref{sec:Kinematics} we enumerate the relevant processes and discuss their roles in controlling the DM abundance in different scenarios. Sec.~\ref{sec:calculation} describes how to evaluate the DM relic density in different phases, including the coannihilation phase, the coscattering phase, and the mixed phase. Some details about the calculations are collected in the Appendices. Our numerical results for some benchmark models are presented in Sec.~\ref{sec:numerical}. In Sec.~\ref{sec:constraints} we discuss various experimental constraints and future probes of this DM scenario. The conclusions are drawn in Sec.~\ref{sec:conclusions}.
\section{Fraternal Twin Higgs and Light DM}
\label{sec:FTH}
The twin Higgs model postulates a mirror (or twin) sector which is related to the SM sector by a $Z_2$ symmetry. The particles in the twin sector are completely neutral under the SM gauge group but charged under the twin $SU(3)\times SU(2) \times U(1)$ gauge group. Due to the $Z_2$ symmetry, the Higgs fields of the SM sector and the twin sector exhibit an approximate $U(4)$ (or $O(8)$) symmetry. The $U(4)$ symmetry is spontaneously broken down to $U(3)$ by the Higgs vacuum expectation values (VEVs). A phenomenological viable model requires that the twin Higgs VEV $f$ to be much larger than the SM Higgs VEV $v$, $f/v \gtrsim 3$, so that the light uneaten pseudo-Nambu-Goldstone boson (pNGB) is an SM-like Higgs boson. (The other six Nambu-Goldstone bosons are eaten and become the longitudinal modes of the $W,\, Z$ bosons of the SM sector and the twin sector.) This requires a small breaking of the $Z_2$ symmetry. The one-loop quadratically divergent contribution from the SM particles to the Higgs potential is cancelled by the twin sector particles, which are heavier than their SM counterparts by the factor of $f/v$. The model can be relatively natural for $f/v \sim 3-5$.
The twin sector particles are not charged under the SM gauge group so it is difficult to look for them at colliders. However, if there is an exact mirror content of the SM sector and the couplings respect the $Z_2$ symmetry, there will be light particles (photon, electron, neutrinos) in the twin sector. They can cause cosmological problems by giving a too big contribution to $N_\text{eff}$.\footnote{Some solutions within the mirror twin Higgs framework can be found in Refs.~\cite{Farina:2015uea,Chacko:2016hvu,Craig:2016lyx,Barbieri:2016zxn,Csaki:2017spo,Barbieri:2017opf}.} In addition, in general one expects a kinetic mixing between two $U(1)$ gauge fields. If the twin photon is massless, its kinetic mixing with the SM photon is strongly constrained. On the other hand, these light particles have small couplings to the Higgs and hence play no important roles in the hierarchy problem. One can take a minimal approach to avoid these light particles by only requiring the $Z_2$ symmetry on the parts which are most relevant for the hierarchy problem. This is the fraternal twin Higgs (FTH) model proposed in Ref.~\cite{Craig:2015pha}. The twin sector of the FTH model can be summarized below.
\begin{itemize}
\item The twin $SU(2)$ and $SU(3)$ gauge couplings should be approximately equal to the SM $SU(2)$ and $SU(3)$ gauge couplings. The twin hypercharge does not need to be gauged. If it is gauged, its coupling can be different from the SM hypercharge coupling, as long as it is not too big to significantly affect the Higgs potential. Also, the twin photon can be massive by spontaneously breaking the $U(1)$ gauge symmetry or simply writing down a Stueckelberg mass term.
\item There is a twin Higgs doublet. Together with the SM Higgs doublet there is an approximate $U(4)$-invariant potential. The twin Higgs doublet acquires a VEV $f\gg v$, giving masses to the twin weak gauge bosons and twin fermions.
\item The twin fermion sector contains the third generation fermions only. The twin top Yukawa coupling needs to be equal to the SM top Yukawa coupling to a very good approximation so that their contributions to the Higgs potential can cancel. The twin bottom and twin leptons are required for anomaly cancellation, but their Yukawa couplings do not need to be equal to the corresponding ones in the SM, as long as they are small enough to not generate a big contribution to the Higgs potential.
\end{itemize}
The collider phenomenology of the FTH model mainly relies on the mixing of the SM and twin Higgs fields. In typical range of the parameter space, one often expects displaced decays that constitute an interesting experimental signature. Here we focus on the DM. A natural candidate is the twin tau. Since its Yukawa coupling needs not to be related to the SM tau Yukawa coupling, its mass can be treated as a free parameter. It is found that a right amount of thermal relic abundance can be obtained for a twin tau mass in the range of 50--150 GeV if the twin hypercharge is not gauged~\cite{Garcia:2015loa,Craig:2015xla}. It corresponds to a twin tau Yukawa coupling much larger than the SM tau Yukawa coupling. The requirement that the twin tau Yukawa coupling does not reintroduce the hierarchy problem puts a upper limit $\sim 200$ GeV on the twin tau mass. A twin tau lighter than $\sim 50$ GeV would generate an overabundance which overcloses the universe. The relic density can be greatly reduced if a light twin photon also exists, because it provides additional annihilation channels for the twin tau. If the twin photon coupling strength is similar to the SM photon coupling, the annihilation will be too efficient and it will be difficult to obtain enough DM. For a twin photon coupling $\hat{\alpha} \sim 0.03\, \alpha_\text{EM}$, a right amount of relic density can be obtained for a twin tau mass in the range of 1--20 GeV~\cite{Craig:2015xla}.
In the twin tau DM discussion, its stability is assumed to be protected by the twin $U(1)_\text{EM}$ symmetry. However, if twin $U(1)_\text{EM}$ is broken and the twin photon has a mass, the twin tau may be unstable and could decay to the twin neutrino if the twin neutrino is lighter. This is because that the twin tau and the twin neutrino can mix due to the twin $U(1)_\text{EM}$ breaking effect. On the other hand, if the twin lepton number (or parity) is still a good symmetry, the lightest state that carries the twin lepton number (parity) will be stable. In this paper we will assume that it is the twin neutrino and consider its possibility of being the DM.
\subsection{Twin lepton mixings}
If the twin $U(1)_\text{EM}$ (or equivalently twin hypercharge) is broken, it is possible to write down various Dirac and Majorana masses between the left-handed and right-handed twin tau and twin neutrino fields. For simplicity, we consider the case where the twin lepton number remains a good symmetry, which can be responsible for the stability of DM. This forbids the Majorana mass terms.
The twin tau and twin neutrino receive the usual Dirac masses from the twin Higgs VEV,
\begin{eqnarray}
-\mathcal{L}& \supset & y_{{\tau}_B} \, {L}_B \, \widetilde{H}_B \,{\tau}_{B,R}^c + y_{{\nu}_B}\, {L}_B \, {H}_B\, {\nu}_{B,R}^c + \text{h.c.} \nonumber \\
&\supset & \frac{y_{{\tau}_B} f}{\sqrt{2}}\, {\tau}_{B,L} {\tau}_{B,R}^c + \frac{y_{{\nu}_B} f}{\sqrt{2}}\, {\nu}_{B,L} {\nu}_{B,R}^c + \text{h.c.},
\label{eq:diracmass}
\end{eqnarray}
where the subscript $B$ represents the twin sector fields, and $\widetilde{H}_B = i\sigma_2 H^\ast_B$ transforms as $(\bf{3}, \bf{2})_{-1/2}$ under the twin gauge group. The twin hypercharge breaking can be parameterized by a spurion field $S$ which is a singlet under $SU(3)_B \times SU(2)_B$ but carries $+1$ twin hypercharge (also $+1$ twin electric charge). It can come from a VEV of a scalar field which breaks $U(1)_B$ spontaneously. The radial component is assumed to be heavier than the relevant particles (twin photon, tau, and neutrino) here and plays no role in the following discussion. Using the spurion we can write down the following additional lepton-number conserving mass terms,
\begin{eqnarray}
-\mathcal{L}\supset\frac{ d_1}{\Lambda} S {L}_B \widetilde{H}_B {\nu}_{B,R}^c + \frac{d_2}{\Lambda} S^\dagger {L}_B {H}_B {\tau}_{B,R}^c \, .
\label{eq:mixingmass}
\end{eqnarray}
The mass matrix of the twin tau and twin neutrino is then given by
\begin{eqnarray}
(\begin{array}{cc}
{\tau}_{B,R}^c & {\nu}_{B,R}^c\\
\end{array})
\left(\begin{array}{cc}
m_{{\tau}_B} & \mu_{2}\\
\mu_{1} & m_{{\nu}_B}
\end{array}\right)
\left(\begin{array}{c}
{\tau}_{B,L} \\
{\nu}_{B,L}
\end{array}\right),
\label{eqn:mixing 1}
\end{eqnarray}
where
\begin{eqnarray}
m_{{\tau}_B} = \frac{y_{\hat{\tau}} f}{\sqrt{2}}, \quad m_{{\nu}_B} = \frac{y_{\hat{\nu}} f}{\sqrt{2}}, \quad \mu_1 = \frac{d_1 f S}{\sqrt{2}\Lambda}, \quad \mu_2 = \frac{d_2 f S }{\sqrt{2}\Lambda}.
\end{eqnarray}
The mass matrix can be diagonalized by the rotations
\begin{eqnarray}
\left(\begin{array}{c}
\hat{\tau}_R^{c}\\
\hat{\nu}_R^{c}
\end{array}\right)=\left(\begin{array}{cc}
\cos\theta_1 &\sin\theta_1\\
-\sin\theta_1 & \cos\theta_1
\end{array}\right)
\left(\begin{array}{c}
{\tau}_{B,R}^{c}\\
{\nu}_{B,R}^{c}
\end{array}\right),\quad
\left(\begin{array}{c}
\hat{\tau}_L^{} \\
\hat{\nu}_L^{}
\end{array}\right)=\left(\begin{array}{cc}
\cos\theta_2 &\sin\theta_2\\
-\sin\theta_2 & \cos\theta_2
\end{array}\right)
\left(\begin{array}{c}
{\tau}_{B,L}^{} \\
{\nu}_{B,L}^{}
\end{array}\right) \, ,\label{eqn:thetamatrix}
\end{eqnarray}
where the mass eigenstates in the twin sector are labelled with a hat (\,$\hat{}\,$).
We assume that the off-diagonal masses $|\mu_1|, |\mu_2| \ll m_{{\tau}_B} - m_{{\nu}_B}$ so that the mixing angles $\theta_1, \theta_2$ are small. This is reasonable given that $\mu_1, \mu_2$ arise from higher dimensional operators and require an insertion of the twin hypercharge breaking VEV, which is assumed to be small for a light twin photon. For our analysis, to reduce the number of independent parameters, we further assume that one of the off-diagonal masses dominates, i.e., $\mu_1 \gg \mu_2$, so that we can ignore $\mu_2$. In this case, we obtain two Dirac mass eigenstates $\hat{\tau}$ and $\hat{\nu}$ which are labeled by their dominant components. The two mass eigenvalues are
\begin{equation}
m^2_{\hat{\tau},\hat{\nu}}=\frac{1}{2} \left(\mu_{1}^2+m_{{\tau}_B}^2+m_{{\nu}_B}^2\pm\sqrt{\left(\mu_{1}^2+m_{{\tau}_B}^2+m_{{\nu}_B}^2\right)^2-4 m_{{\tau}_B}^2 m_{{\nu}_B}^2}\right),
\end{equation}
and the two mixing angles are given by
\begin{equation}
\sin\theta_1 = \frac{\mu_{1} m_{{\tau}_B}}{m_{{\tau}_B}^2 -m_{{\nu}_B}^2 +\mu_{1}^2}, \quad \sin\theta_2=\frac{m_{{\nu}_B}}{m_{{\tau}_B}}\sin\theta_1\, ,
\end{equation}
in the small mixing angle limit. There is no qualitative difference in our result if $\mu_1$ and $\mu_2$ are comparable except that the two mixing angles become independent.
We are interested in the region of parameter space where the twin tau $\hat{\tau}$, twin neutrino $\hat{\nu}$, and twin photon $\hat{\gamma}$ have masses of the same order in the range $\sim 0.1-10$~GeV, with $m_{\hat{\tau}} > m_{\hat{\nu}} > m_{\hat{\gamma}}$. Compared with Ref.~\cite{DAgnolo:2017dbv}, the twin neutrino $\hat{\nu}$ plays the role of $\chi$ which is the DM, $\hat{\tau}$ corresponds to $\psi$, the coannihilation/coscattering partner of the DM particle, and $\hat{\gamma}$ corresponds to the mediator $\phi$. Following Ref.~\cite{DAgnolo:2017dbv}, we define two dimensionless parameters,
\begin{equation}
r \equiv \frac{m_{\hat{\gamma}}}{m_{\hat{\nu}}}, \qquad \Delta \equiv \frac{m_{\hat{\tau}}-m_{\hat{\nu}}}{m_{\hat{\nu}}},
\end{equation}
which are convenient for our discussion. The region of interest has $r < 1$ and $0< \Delta \lesssim 1$.
The mass spectrum and mixing pattern would be more complicated if Majorana masses for the twin leptons are allowed. In addition to the standard Majorana mass for the right-handed twin neutrino, all other possible terms can arise from higher dimensional operators with insertions of the spurion field $S$ (and the twin Higgs field $H_B$), filling the $4\times 4$ mass matrix of $({\tau}_{B,L}, {\nu}_{B,L}, {\tau}_{B,R}^c, {\nu}_{B,R}^c)$. There are four mass eigenstates and many more mixing angles. The stability of the lightest eigenstate can be protected by the twin lepton parity in this case. Because the twin photon couples off-diagonally to Majorana fermions, the analysis of annihilation and scattering needs to include all four fermion eigenstates, which becomes quite complicated. Nevertheless, one can expect that there are regions of parameter space where the correct relic abundance can be obtained through coannihilation and/or coscattering processes in a similar way to the case studied in this work.
\section{Relevant Processes for the Thermal Dark Matter Abundance}
\label{sec:Kinematics}
At high temperature, the SM sector and the twin sector stay in thermal equilibrium through the interactions due to the Higgs mixing and the kinetic mixing of the $U(1)$ gauge fields. As the universe expands, the heavy species in the twin sector decouple from the thermal bath and only the light species including twin photon ($\hat{\gamma}$), twin tau ($\hat{\tau}$), and twin neutrino ($\hat{\nu}$) survive. These light species talk to SM mainly via the kinetic mixing term, $-(\epsilon/2) F_{\mu\nu}\hat{F}^{\mu\nu}$, between $\gamma$ and $\hat{\gamma}$. We assume that the kinetic mixing is big enough ($ \epsilon \gtrsim 10^{-9}$) to keep the twin photon in thermal equilibrium by scattering off light SM leptons~\cite{DAgnolo:2017dbv} during the DM freeze-out. The DM abundance is then controlled by several annihilation and scattering processes.
{\bf Annihilation processes}:
\begin{equation*}
\hat{\nu}\tn \to \hat{\gamma}\gd~(\hm{A}), \qquad
\hat{\tau}\hat{\nu}\to \hat{\gamma}\gd~(\hm{C_A}), \qquad
\hat{\tau}\tt\to \hat{\gamma}\gd~( \hm{C_S}).
\end{equation*}
The coupling of $\hat{\nu}$ to $\hat{\gamma}$ arises from mixing with the twin tau. As a result, for small mixings the usual annihilation process $\hm{A}$ for the $\hat{\nu}$ DM is suppressed by $\theta_1^4$, while the coannihilation process $\hm{C_A}$ (where the subscript $\hm{A}$ stands for asymmetric) is suppressed by $\theta_1^2$. There is no mixing angle suppression for the coannihilation process $\hm{C_S}$ (where the subscript $\hm{S}$ represents symmetric or sterile). On the other hand, the Boltzmann suppression goes the other way, the Boltzmann factors for the three processes are $\sim e^{-2m_{\hat{\nu}}/T}$, $e^{-(m_{\hat{\nu}}+m_{\hat{\tau}})/T}$, and $e^{-2m_{\hat{\tau}}/T}$ respectively.
{\bf (Co)scattering process}:
\begin{equation*}
\hat{\nu}\hat{\gamma}\to \hat{\tau}\hat{\gamma}~(\hm{S})
\end{equation*}
It is suppressed by $\theta_1^2$. Because $\hat{\tau}$ is assumed to be heavier than $\hat{\nu}$, the initial states particles $\hat{\nu}$ and $\hat{\gamma}$ must carry enough momenta for this process to happen. The coscattering process therefore has a strong momentum dependence. Ignoring the momentum dependence for a moment, the Boltzmann factor can be estimated to be $\sim e^{-(m_{\hat{\tau}}+m_{\hat{\gamma}})/T}$.
In addition, there is also the decay process
\begin{equation*}
\hat{\tau} \to \hat{\nu}\hat{\gamma}^{(\ast)}~(\hm{D})
\end{equation*}
On-shell decay only occurs if $m_{\hat{\tau}} > m_{\hat{\nu}} + m_{\hat{\gamma}}\; ( \Delta > r)$. In this case the inverse decay ($\hm{ID}$) plays a similar role as the coscattering process since both convert $\hat{\nu}$ to $\hat{\tau}$, but the rate is much larger. It turns out that if the (inverse) decay is open, the relic abundance is simply determined by the coannihilation because the inverse decay process decouples later.
The majority of the parameter space we focus on has $m_{\hat{\tau}}< m_{\hat{\nu}}+m_{\hat{\gamma}}\; (\Delta < r)$. In this case, the twin photon has to be off-shell then decays to SM fermions. It is further suppressed by $\epsilon^2$ so it can be ignored during the freeze-out. It is however responsible for converting the remaining $\hat{\tau}$ to $\hat{\nu}$ eventually after the freeze-out.
If the mixing is large and/or $\Delta$ is large so that $\theta_1^2 e^{-2m_{\hat{\nu}}/T} > e^{-(m_{\hat{\tau}}+m_{\hat{\nu}})/T}$ during freeze-out, then the annihilation process $\hm{A}$ will dominate and we will have the usual WIMP scenario. However, for such a WIMP DM lighter than 10 GeV, this has been ruled out by the CMB constraint (see discussion in Sec.~\ref{sec:constraints}). Therefore we focus on the opposite limit $\theta_1^2 e^{-2m_{\hat{\nu}}/T} < e^{-(m_{\hat{\tau}}+m_{\hat{\nu}})/T}$, i.e., small mixing and small $\Delta$. In this case we have $\hm{C_S} > \hm{C_A} >\hm{A}$ in terms of rates. The coscattering process $\hm{S}$ has the same $\theta_1$ dependence as $\hm{C_A}$, but is less Boltzmann suppressed because $m_{\hat{\gamma}} < m_{\hat{\nu}}$, so the coscattering can keep $\hat{\tau}$ and $\hat{\nu}$ in kinetic equilibrium after $\hm{C_A}$ freezes out. In this simple-minded picture, the DM relic density is then determined by freeze-out of $\hm{C_S}$ and $\hm{S}$. Denoting their freeze-out temperatures by $T_{\hm{C_S}}$ and $T_{\hm{S}}$, then there are two main scenarios.
\begin{enumerate}
\item $T_{\hm{C_S}} > T_{\hm{S}}$: This occurs if $\theta_1^2 e^{-(m_{\hat{\tau}}+m_{\hat{\gamma}})/T} \gg e^{-2m_{\hat{\tau}}/T}$ during freeze-out so that $\hm{C_S}$ freezes out earlier. After that the total number of $\hat{\nu}$ and $\hat{\tau}$ in a comoving volume is fixed. The coscattering process only re-distributes the densities between $\hat{\nu}$ and $\hat{\tau}$, but eventually all $\hat{\tau}$'s will decay down to $\hat{\nu}$'s. The DM relic density is determined by $\hm{C_S}$. This is the coannihilation phase. It is schematically depicted in the left panel of Fig.~\ref{fig:Cartoons1}.
\item $T_{\hm{C_S}} < T_{\hm{S}}$: In the opposite limit, $\hm{S}$ freezes out before $\hm{C_S}$, and hence stops converting $\hat{\nu}$ into $\hat{\tau}$. On the other hand, $\hm{C_S}$ is still active and will annihilate most of the leftover $\hat{\tau}$'s. The relic density in this case is determined by the coscattering process $\hm{S}$. (Remember that $\hm{A}$ has frozen out earlier.) This is the coscattering phase discovered in Ref.~\cite{DAgnolo:2017dbv}. It is illustrated in the right panel of Fig.~\ref{fig:Cartoons1}.
\end{enumerate}
\begin{figure}
\captionsetup{singlelinecheck = false, format= hang, justification=raggedright, font=footnotesize, labelsep=space}
\includegraphics[scale=0.4]{Cartoon_C1N}
\includegraphics[scale=0.4]{Cartoon_SN}
\caption{Schematic plots of different scenarios, displayed by the freeze-out temperature and different momentum of $\hat{\nu}$. {\bf Left:} Coannihilation phase where the DM relic density is dominantly determined by $\hm{C_S}$. {\bf Right:} Coscattering phase discussed in Ref.~\cite{DAgnolo:2017dbv} where DM relic density is determined by $\hm{S}$. }
\label{fig:Cartoons1}
\end{figure}
In the above discussion, we have associated each process with a single freeze-out temperature. This is a good approximation for the annihilation and coannihilation processes, but not for the coscattering process which has a strong momentum dependence. In the coscattering phase, the processses $\hat{\nu} \hat{\gamma} \to \hat{\nu}\hat{\gamma}$, $\hat{\nu}\tn \to \hat{\nu}\tn$ are suppressed by $\theta_1^4$ and hence are expected to freeze out earlier and cannot re-equilibrate the $\hat{\nu}$ momentum. Consequently, different momentum modes in the coscattering process freeze out at different time, with low momentum modes freeze out earlier. If $\theta_1^2 e^{-(m_{\hat{\tau}}+m_{\hat{\gamma}})/T} \sim e^{-2m_{\hat{\tau}}/T}$ during freeze-out, we can have a situation that coscattering of the low momentum modes freezes out earlier than $\hm{C_S}$ while the coscattering of the high momentum modes freezes out later than $\hm{C_S}$. This is illustrated in the left panel of Fig.~\ref{fig:Cartoons2}. In this case, the relic density of low momentum modes is determined by the coscattering process and the relic density of high momentum modes is determined by the coannihilation process $\hm{C_S}$. We have a mixed coscattering/coannihilation phase where the relic density is determined by both $\hm{S}$ and $\hm{C_S}$.
\begin{figure}
\captionsetup{singlelinecheck = false, format= hang, justification=raggedright, font=footnotesize, labelsep=space}
\includegraphics[scale=0.4]{Cartoon_SC1}
\includegraphics[scale=0.4]{Cartoon_SC2}
\caption{Schematic illustrations of the mixed coscattering/coannihilation phases. {\bf Left:} The $\hm{S}/\hm{C_S}$ mixed phase: for low (high) momentum modes $\hm{S}$ freezes out earlier (later) than $\hm{C_S}$. {\bf Right:} The $\hm{C_A}/\hm{S}$ mixed phase: for low (high) momentum modes $\hm{C_A}$ freezes out earlier (later) than $\hm{S}$.}
\label{fig:Cartoons2}
\end{figure}
If the masses of the twin photon and the twin neutrino are close, we have $\theta_1^2 e^{-(m_{\hat{\tau}}+m_{\hat{\gamma}})/T} \sim \theta_1^2 e^{-(m_{\hat{\tau}}+m_{\hat{\nu}})/T} <e^{-2m_{\hat{\tau}}/T}$ during freeze-out from our assumption. One expects that this belongs to the coscattering phase since $T_{\hm{C_S}} < T_{\hm{S}}$. However, the rates of $\hm{S}$ and $\hm{C_A}$ become comparable in this limit so we have $T_{\hm{S}} \sim T_{\hm{C_A}} > T_{\hm{C_S}}$. Due to the strong momentum dependence of $\hm{S}$, one can have a situation depicted in the right panel of Fig.~\ref{fig:Cartoons2}. The coscattering of low momentum modes freezes out early, but their comoving density is still reduced by the coannihilation $\hm{C_A}$ until $\hm{C_A}$ freezes out. The relic density of high momentum modes is determined by $\hm{S}$ as in the coscattering phase. In this case we have another mixed coscattering/coannihilation phase where the relic density is determined together by $\hm{C_A}$ and $\hm{S}$. Finally, if the mixing is not very small so that the rates of $\hm{C_A}$ and $\hm{C_S}$ are not far apart, the freeze-out time of the coscattering process can even cut through that of both $\hm{C_A}$ and $\hm{C_S}$, although it can only happen in some rare corner of the parameter space. The relic density is then determined by all three processes, with the low momentum modes controlled by $\hm{C_A}$, intermediate momentum modes governed by $\hm{S}$, and high momentum modes determined by $\hm{C_S}$.
Due to the momentum dependence of the coscattering process, the relic density calculations of the coscattering and mixed phases are more complicated. We describe the calculation for each case in the next section.
\section{Relic Abundance Calculations in Various Phases}
\label{sec:calculation}
In this section we describe the calculations of DM relic abundance in different phases.
\subsection{Coannihilation}
In the coannihilation phase, the coscattering process $\hm{S}$ decouples late enough to keep $\hat{\tau}$ and $\hat{\nu}$ in chemical equilibrium even after all annihilation processes freeze out, so we have
\begin{equation}
\frac{n_{\hat{\tau}}(T)}{n_{\hat{\nu}}(T)}=\frac{n_{\hat{\tau}}^{\rm{eq}}(T)}{n_{\hat{\nu}}^{\rm{eq}}(T)} ,
\label{eqn:CE}
\end{equation}
where $n$ ($n^{\rm{eq}}$) is the (equilibrium) number density.
We can simply write down the Boltzmann equation for the total DM number density $n_{\rm{tot}}(T)=n_{\hat{\tau}}(T)+n_{\hat{\nu}}(T)$~\cite{Griest:1990kh,Mizuta:1992qp}:
\begin{equation}
\dot{n}_{\rm{tot}}+3Hn_{\rm{tot}}=-\left\langle\sigma v\right\rangle_{C_S}\left(n_{\hat{\tau}}^2-(n_{\hat{\tau}}^{\text{eq}})^2\right)
-\left\langle\sigma v\right\rangle_{C_A}\left(n_{\hat{\tau}}n_{\hat{\nu}}-n_{\hat{\tau}}^{\text{eq}}n_{\hat{\nu}}^{\text{eq}}\right)
-\left\langle\sigma v\right\rangle_{A}\left(n_{\hat{\nu}}^2-(n_{\hat{\nu}}^{\text{eq}})^2\right).
\label{eqn:COA}
\end{equation}
It can be easily solved and is incorporated in the standard DM relic density calculation packages. In the parameter region that we are interested, the right-handed side is mostly dominated by the $\hm{C_S}$ term.
\subsection{Coscattering}
The DM density calculation in the coscattering phase is more involved and was discussed in detail in Ref.~\cite{DAgnolo:2017dbv}. There the authors provided an approximate solution based on the integrated Boltzmann equation:
\begin{equation}\label{eqn:boltzmannS}
\dot{n}_{\hat{\nu}}+3Hn_{\hat{\nu}}=-\left\langle\sigma v\right\rangle_S (n_{\hat{\nu}}-n_{\hat{\nu}}^{\rm eq})n_{\hat{\gamma}}^{\rm eq} .
\end{equation}
However, as mentioned earlier, the kinetic equilibrium of $\hat{\nu}$ will not be maintained during the freeze-out of the coscattering process and different momentum modes freeze out at different time. The simple estimate from Eq.~(\ref{eqn:boltzmannS}) is not always accurate. Here we reproduce the calculation from the unintegrated Boltzmann equation to keep track of the momentum dependence.
The unintegrated Boltzmann equation of the density distribution in the momentum space $f(p, t)$ for the coscattering process $\hat{\nu}(p) + \hat{\gamma}(k) \to \hat{\tau}(p') + \hat{\gamma}(k')$ is given by
\begin{equation}
\left(\partial_t-H p\partial_p\right) f_{\hat{\nu}}(p,t)=\frac{1}{E_p}C[f_{\hat{\nu}}](p,t),
\label{eqn:collisionop}
\end{equation}
where the collision operator is defined as \cite{DAgnolo:2017dbv}
\begin{equation}
C[f_{\hat{\nu}}](p,t)=\frac{1}{2}\int d\Omega_{\hm k} d\Omega_{\hm p'} d\Omega_{\hm k'} |\overline{\mathcal{M}}|^2 [f_{\hat{\tau}}(p',t)f_{\hat{\gamma}}(k',t)-f_{\hat{\nu}}(p,t)f_{\hat{\gamma}}(k,t)](2\pi)^4\delta^4(\sum p^\mu).
\label{eqn:collisionop2}
\end{equation}
In the above expression $d\Omega_{\hm p}=d^3p/[(2\pi)^3 2E_p]$ is the Lorentz-invariant integration measure and $|\overline{\mathcal{M}}|^2$ is the squared amplitude averaged over initial and summed over final state quantum numbers.
In this phase $\hat{\tau}$ and $\hat{\gamma}$ can be assumed to be in thermal equilibrium with the thermal bath, $f_{\hat{\tau}(\hat{\gamma})} = f^{\rm{eq}}_{\hat{\tau}(\hat{\gamma})}$, from the processes $\hat{\tau} \hat{\gamma} \leftrightarrow \hat{\tau} \hat{\gamma}$, $\hat{\tau} \hat{\tau} \leftrightarrow \hat{\gamma} \hat{\gamma}$ and $\hat{\gamma}$ interactions with SM fields. Using $f^{\rm{eq}}_{\hat{\tau}} (p', t) f^{\rm{eq}}_{\hat{\gamma}} (k', t) = f^{\rm{eq}}_{\hat{\nu}}(p, t) f^{\rm{eq}}_{\hat{\gamma}} (k,t)$,
the right-hand side of Eq.~(\ref{eqn:collisionop}) can be simplifed as
\begin{equation}\label{eqn:c3}
\frac{1}{E_p}C[f_{\hat{\nu}}](p,t)=[f_{\hat{\nu}}^{\text{eq}}(p,t)-f_{\hat{\nu}}(p,t)]\widetilde{C}(p,t),
\end{equation}
where the reduced collision operator $\widetilde{C}(p,t)$ takes the form,
\begin{align}
\widetilde{C}(p,t) &=\frac{1}{2E_p}\int d\Omega_{\hm k} f_{\hat{\gamma}}^{\text{eq}}(k,t)\int d\Omega_{\hm p'} d\Omega_{\hm k'}|\overline{\mathcal{M}}|^2 (2\pi)^4\delta^4({p^\mu}+{k^\mu}-{p^{\prime\mu}}-{k^{\prime\mu}})\\
&=\frac{1}{2E_p}\int d\Omega_{\hm k} f_{\hat{\gamma}}^{\text{eq}}(k,t) j(s)\sigma(s),
\label{eqn:coredefinition}
\end{align}
with the Lorentz-invariant flux factor $j(s) = 2E_p 2 E_k |\hm{v}_p - \hm{v}_k| $. The calculation of $\widetilde{C}$ from the coscattering is described in Appendix~\ref{sec:collision}.
The left-hand side of Eq.~(\ref{eqn:collisionop}) can be written as a single term by defining the comoving momentum $q\equiv p \,a$,
then Eq.~(\ref{eqn:collisionop}) becomes a first order differential equation of
the scale factor $a$ for each comoving momentum $q$:
\begin{equation}
Ha\partial_a f_{\hat{\nu}}(q,a) =[f_{\hat{\nu}}^{\text{eq}}(q,a)-f_{\hat{\nu}}(q,a)]\widetilde{C}(q,a),
\label{eqn:collisioncore}
\end{equation}
where we have written the distribution $f$ as a function of $q$ and $a$, instead of $p$ and $t$.
Taking the boundary condition $f_{\hat{\nu}} (q, a_0) = f^{\rm eq}_{\hat{\nu}} (q, a_0)$ at an early time $a_0$, the solution is given by
\begin{equation}
f_{\hat{\nu}}(q,a)=f_{\hat{\nu}}^{\text{eq}}(q,a)-\int_{a_0}^a da' \frac{d f_{\hat{\nu}}^{\text{eq}}(q,a')}{d a'} e^{-\int_{a'}^a \frac{\widetilde{C}(q,a^{\prime\prime})}{ H a^{\prime\prime}}da^{\prime\prime}}.
\label{eqn:coscattersolution}
\end{equation}
At early time and high temperature where $\widetilde{C}(q,a) \gg H(a)$, the second term on the right-hand side can be neglected and $\hat{\nu}$ density is given by the equilibrium density as expected. At late time and low temperature, $\widetilde{C}(q,a) \ll H(a)$, the $\hat{\nu}$ comoving density stops changing. For each comoving momentum $q$, one can find a time $a' = a_f(q)$ beyond which the exponent is small so that the exponential factor is approximately 1. Then the final $\hat{\nu}$ density is roughly given by $f^{\rm eq}_{\hat{\nu}}( q, a_f(q))$. The $a_f(q)$ can be viewed as the freeze-out scale factor for coscattering of the comoving momentum $q$ of $\hat{\nu}$. It is roughly determined by $\widetilde{C}(q,a_f(q))\simeq H(a_f(q))$.
As mentioned in the previous section, if $m_{\hat{\tau}} > m_{\hat{\nu}} +m_{\hat{\gamma}} \, (r<\Delta)$, the inverse decay $\hat{\nu} + \hat{\gamma} \to \hat{\tau}$ plays a similar role as the coscattering process. Its contribution should be added to the collision operator, which is calculated in Appendix~\ref{sec:ID}. It is larger than the coscattering contribution, as it requires less energy to produce the final state. In the parameter region that we consider, it always makes the second term in Eq.~(\ref{eqn:coscattersolution}) negligible before $\hm{C_S}$ freezes out. Therefore, it goes back to the coannihilation phase if this on-shell decay is allowed.
\subsection{Mixed Phases}
From the discussion in the previous section, mixed phases occur when $T_{\hm{S}} \sim T_{\hm{C_S}}$ or $T_{\hm{S}} \sim T_{\hm{C_A}}$. We first consider the case
$T_{\hm{ C_A}}\sim T_{\hm S} > T_{\hm{C_S}}$ ($\hm{C_A}/\hm{S}$ mixed phase). This happens when $m_{\hat{\gamma}} \simeq m_{\hat{\nu}}$. Because the coannihilation process $\hm{C_A}$, $\hat{\nu}(p) + \hat{\tau}(k) \to \hat{\gamma}(p') + \hat{\gamma}(k')$, is also important in this case, the contribution from $\hm{C_A}$ to the collision operator,
\begin{equation}
\widetilde{C}_{\hm{C_A}}(p,t) =\frac{1}{2E_p}\int d\Omega_{\hm k} f_{\hat{\tau}}^{\text{eq}}(k,t)\int d\Omega_{\hm p'} d\Omega_{\hm k'}|\overline{\mathcal{M}}_{\hm{C_A}}|^2 (2\pi)^4\delta^4({p}+{k}-{p'}-{k'}),
\end{equation}
should be included in Eq.~(\ref{eqn:collisionop})
in addition to the coscattering contribution of Eq.~(\ref{eqn:collisionop2}).
Since $T_{\hm{C_S}}$ is assumed to be small, $\hat{\tau}$ stays in thermal equilibrium during the decoupling of $\hm{ S}$ and $\hm{ C_A}$.
In contrast to $\hm S$, there is no kinematic threshold in $\hm{ C_A}$, so it has very weak momentum dependence. Hence we can treat it as a function of time/temperature only. In Eq.~(\ref{eqn:coscattersolution}) we can replace $\widetilde{C}$ by $\widetilde{C}_{\hm S}+\widetilde{C}_{\hm{C_A}}$ and conduct the computation in the same manner as in the coscattering phase. In this $\hm{C_A}/\hm{S}$ mixed phase, $\widetilde{C}_{\hm{C_A}}> \widetilde{C}_{\hm S}$ for low momentum modes so their freeze out temperature is determined by $\hm{C_A}$, while for high momentum modes $\widetilde{C}_{\hm S}> \widetilde{C}_{\hm{C_A}}$ and their contributions is given by the coscattering result, as illustrated in the right panel of Fig.~\ref{fig:Cartoons2}
For the other mixed phase ($\hm{S}/\hm{C_S}$) where $T_{\hm{ C_S}}\sim T_{\hm{ S}}< T_{\hm{C_A}}$, the calculation is more complicated.
In this case $\hat{\tau}$ is no longer in thermal equilibrium with the thermal bath, $f_{\hat{\tau}}$ itself is unknown, hence cannot be set to equal $f_{\hat{\tau}}^{\rm eq}$. As a result, Eq.~(\ref{eqn:c3}) no longer holds. Moreover, different from the usual coannihilation scenario, $\hat{\tau}$ and $\hat{\nu}$ are not in chemical equilibrium. A complete solution requires solving the two coupled Boltzmann equations for $\hat{\tau}$ and $\hat{\nu}$ in this case, which is numerically expensive. However, we can assume that $\hat{\tau}$ is still in kinetic equilibrium with the SM sector (i.e., has the canonical distribution up to an unknown overall factor) due to the elastic scattering with $\hat{\gamma}$. We can write
$f_{\hat{\tau}}/f_{\hat{\tau}}^{\rm eq}=Y_{\hat{\tau}}/Y_{\hat{\tau}}^{\rm eq}$ where $Y_{\hat{\tau}}= n_{\hat{\tau}}/s$ is the comoving number density. The term in the square bracket of Eq.~(\ref{eqn:collisionop2}) can be written as:
\begin{eqnarray}
& &f_{\hat{\tau}}(p',t)f_{\hat{\gamma}}(k',t)-f_{\hat{\nu}}(p,t)f_{\hat{\gamma}}(k,t) = f^{\text{eq}}_{\hat{\tau}}(p',t)\frac{Y_{\hat{\tau}}(t)}{Y_{\hat{\tau}}^{\text{eq}}(t)}f_{\hat{\gamma}}^{\text{eq}}(k',t)-f_{\hat{\nu}}(p,t)f_{\hat{\gamma}}^{\text{eq}}(k,t)\\
&=&
f_{\hat{\gamma}}^{\text{eq}}(k, t)[\frac{Y_{\hat{\tau}}(t)}{Y_{\hat{\tau}}^{\text{eq}}(t)}f^{\text{eq}}_{\hat{\nu}}(p,t)-f_{\hat{\nu}}(p,t)].\nonumber
\label{eqn:tauratio}
\end{eqnarray}
This corresponds to replacing $f_{\hat{\nu}}^{\rm eq}(q, a)$ by $(Y_{\hat{\tau}}(a)/ Y^{\rm eq}_{\hat{\tau}} (a)) f_{\hat{\nu}}^{\rm eq}(q, a)$ in Eq.~(\ref{eqn:collisioncore}). The solution Eq.~(\ref{eqn:coscattersolution}) is modified to~\cite{Garny:2017rxs}
\begin{equation}
f_{\hat{\nu}}(q,a)=\frac{Y_{\hat{\tau}}(a)}{Y_{\hat{\tau}}^{\text{eq}}(a)}f_{\hat{\nu}}^{\text{eq}}(q,a)+\int_{a_0}^a da' \frac{-d \frac{Y_{\hat{\tau}}(a')}{Y_{\hat{\tau}}^{\text{eq}}(a')}f_{\hat{\nu}}^{\text{eq}}(q,a')}{d a'} e^{-\int_{a'}^a \frac{\widetilde{C}(q,a'')}{ H a''}da''}.
\label{eqn:intersolution}
\end{equation}
Of course we do not know $Y_{\hat{\tau}}(a)$ in advance. One way to solve this problem is to employ the iterative method~\cite{Garny:2017rxs} which is described in Appendix~\ref{sec:iteration}. One starts with some initial guess of $Y_{\hat{\tau}} (a)$ to obtain the solution for Eq.~(\ref{eqn:intersolution}), then use that solution in the Boltzmann equation for $n_{\hat{\tau}}$ to obtain a new $Y_{\hat{\tau}}(a)$, and repeat the procedure until the result converges. We find that a good first guess is to simply take $Y_{\hat{\tau}}(a)$ to be the one obtained in the coannihilation calculation. In coannihilation, $\hat{\tau}$ and $\hat{\nu}$ are in chemical equilibrium:
\begin{equation}
\frac{Y_{\hat{\tau}}(a)}{Y_{\hat{\tau}}^{\text{eq}}(a)}f_{\hat{\nu}}^{\text{eq}}(q,a)\simeq \frac{Y_{\hat{\nu}}(a)}{Y_{\hat{\nu}}^{\text{eq}}(a)}f_{\hat{\nu}}^{\text{eq}}(q,a) = f^{\text{CA}}_{\hat{\nu}}(q,a)\simeq f^{\text{CA}}_{\text{tot}}(q,a),
\end{equation}
where the superscript CA indicates that the result is obtained from the coanihillation-only estimation (Eq.~(\ref{eqn:COA})).
Notice that in this case the first term in Eq.~(\ref{eqn:intersolution}) is simply the contribution from coannihilation. If $T_{\hm{S}} < T_{\hm{C_S}}$, $\hm{S}$ will still be active when $\hm{C_S}$ freezes out, $\widetilde{C}(q, a'')/(Ha'')$ will be large and the second term in Eq.~(\ref{eqn:intersolution}) will be suppressed. We obtain the correct coannihilation limit. On the other hand, if $T_{\hm{S}} >T_{\hm{C_S}}$, the coannihilation $\hm{C_S}$ is effective to keep ${Y_{\hat{\tau}}(a)}/{Y_{\hat{\tau}}^{\text{eq}}(a)}\approx1$ and Eq.~(\ref{eqn:intersolution}) returns to the coscattering result in Eq.~(\ref{eqn:coscattersolution}). Using $Y_{\hat{\tau}}(a)$ from the coannihalation calculation in Eq.~(\ref{eqn:intersolution}) gives the correct results in both the coannihilation and coscattering limits. The expression interpolates between these two limits in the mixed phase and it turns out to be an excellent approximation to the correct relic density even without performing the iterations. (See Appendix~\ref{sec:iteration}.)
\begin{figure}
\captionsetup{singlelinecheck = false, format= hang, justification=raggedright, font=footnotesize, labelsep=space}
\includegraphics[scale=0.28]{YRatio_C1}
\includegraphics[scale=0.28]{YRatio_C1_BM2}
\includegraphics[scale=0.28]{YRatio_C1_BM3}
\caption{Benchmark results for relic density calculation in the coscattering (left), mixed (central), and coannihilation (right) phases. The red curves are the contributions from the first term of Eq.~(\ref{eqn:intersolution}) which corresponds to a pure coannihilation calculation. The blue solid curves are due to the coscattering contributions from the second term of Eq.~(\ref{eqn:intersolution}). The purple curves are the total contributions. We can see that in the coscattering phase and the coannihilation phase the total contribution is dominated by one term, while in the mixed phase both terms give comparable contributions. The blue dashed curves are calculated from the coscattering formula of Eq.~(\ref{eqn:collisionop}). Only in the coscattering phase the blue dashed curve approximates the correct result. }
\label{fig:YratioBM123}
\end{figure}
For completeness, we can include the $\hm{C_A}$ contribution in Eq.~(\ref{eqn:intersolution}), then this result also applies to the case if $T_{\hm{S}}$ cuts through both $T_{\hm{C_A}}$ and $T_{\hm{C_S}}$. Fig.~\ref{fig:YratioBM123} shows the differential DM density as a function of the comoving momentum in different phases, all calculated from Eq.~(\ref{eqn:intersolution}) including all contributions. We will use this formula for numerical calculations of DM densities in the next section in all phases.
\section{Numerical results }
\label{sec:numerical}
In this section we present the numerical calculations of the dark matter relic abundance for various parameter choices of the model. We fix $\hat{e} =0.3$ for the twin electromagnetic gauge coupling and calculate the DM density in units of the observed $\Omega_{\text{obs}} h^2 = 0.12$. All calculations are performed using Eq.~(\ref{eqn:intersolution}) including contributions from all relevant processes.
\begin{figure}
\centering
\captionsetup{singlelinecheck = false, format= hang, justification=raggedright, font=footnotesize, labelsep=space}
\includegraphics[scale=0.4]{Contour_1G_BM010.pdf}
\includegraphics[scale=0.4]{Contour_1G_BM018.pdf}
\caption{Relic density for $m_{\hat{\nu}}=1$~GeV and $\Delta = 0.1$ and 0.18. Contours of fixed $\log_{10}(\Omega/\Omega_{\rm{OBS}})$ values are depicted. The shaded green region indicates the mixed phase where $T_{\hm S}\sim T_{\hm{C_S}}$. The region to the lower right of the green band is in the coscattering phase while the coannihilation phase is in the upper left corner. In the region to the right of the orange dashed line, $\hm{C_A}$ becomes important and it enters the $\hm{C_A}/\hm{S}$ mixed phase. The contours become slightly less sensitive to $m_{\hat{\gamma}}$ or $r$. The red dashed line indicates $r=\Delta$. To its left the $\hat{\tau} \leftrightarrow \hat{\gamma}+\hat{\nu}$ decay and inverse decay are open. Their large rates keep $\hat{\tau},\hat{\nu}$ in chemical equilibrium, making this region coannihilation-like.
}
\label{fig:Omega_1G}
\end{figure}
In Fig.~\ref{fig:Omega_1G}, we consider $m_{\hat{\nu}}= 1$~GeV and plot the DM density dependence on $r =m_{\hat{\gamma}}/m_{\hat{\nu}}$ and the mixing angle $\theta_1$, for $\Delta = (m_{\hat{\tau}}-m_{\hat{\nu}})/m_{\hat{\nu}} = 0.1$ and $0.18$. The contours are in $\log_{10} (\Omega_{\rm DM}/ \Omega_{\rm obs})$ and the ``0'' contour represents points which produce the observed DM density.
The coscattering phase sits in the lower-right region, as for small $\theta_1$ and large $r$ the coscattering process $\hm{S}$ is suppressed and freezes out earlier. The upper-left region, on the other hand, belongs to the coannihilation phase. The green band separating them corresponds to the $\hm{S}/\hm{C_S}$ mixed phase. The boundaries of the green band are determined by the condition $T_{\hm{S}} (q=p\,a=0) = T_{\hm{C_S}}$ and $T_{\hm{S}} (q=25) = T_{\hm{C_S}}$. The contribution to the relic density from modes with $q>25$ is small and is ignored in the coscattering calculation. The orange vertical dashed line corresponds to $T_{\hm{S}} (q=0) = T_{\hm{C_A}}$. To the right of it $\hm{C_A}$ becomes relevant and we enter the $\hm{C_A}/ \hm{S}$ mixed phase. The DM relic density is slightly reduced by $\hm{C_A}$ compared to a pure coscattering calculation. The red vertical dashed line indicates $r=\Delta$. To the left of it the on-shell decay and inverse decay $\hat{\tau} \leftrightarrow \hat{\nu} \hat{\gamma}$ are open so this whole region is in the coannihilation phase.
The relic density in the coannhilation phase is mostly independent of $\theta_1$ because it is mainly controlled by $\hm{C_S}$ which hardly depends on $\theta_1$. Only at larger $\theta_1$ values when $\hm{C_A}$ and $\hm{A}$ become relevant the DM relic density shows some $\theta_1$ dependence. The dependence on $r$ of the relic density in the coannihilation phase is also mild, as it mainly affects the phase space of the coannihilation process. In the coscattering phase, the DM relic density increases as $\Delta$ increases, which can be seen by comparing the two plots in Fig.~\ref{fig:Omega_1G}. This is because a larger gap between $m_{\hat{\tau}}$ and $m_{\hat{\nu}}$ requires a higher threshold momentum for $\hat{\gamma}$ to make $\hm{S}$ happen. Therefore the coscattering is suppressed, resulting in a larger relic density. For $m_{\hat{\nu}}= 1$~GeV, $\Delta=0.1$, the observed DM density is produced in the coscattering phase, while for $\Delta = 0.18$ it moves to the mixed phase or coannihilation phase.
In Figs.~\ref{fig:Omega_100M} and \ref{fig:Omega_10G} we show the results for $m_{\hat{\nu}}=100$~MeV and 10 GeV. A larger DM mass will give a larger relic density if all other parameters are fixed. Consequently for a correct relic density we need a smaller (larger) $\Delta$ for a larger (smaller) $m_{\hat{\nu}}$. The $\Delta$ values are chosen to be 0.2 and 0.26 for the two plots with $m_{\hat{\nu}}=100$~MeV, and 0.04 and 0.1 for the two plots with $m_{\hat{\nu}}= 10$~GeV. The behaviors of the contours are similar to the case of $m_{\hat{\nu}}=1$~GeV.
\begin{figure}[ht]
\centering
\captionsetup{singlelinecheck = false, format= hang, justification=raggedright, font=footnotesize, labelsep=space}
\includegraphics[scale=0.4]{Contour_100M_BM020.pdf}
\includegraphics[scale=0.4]{Contour_100M_BM026.pdf}
\caption{Similar to Fig.~\ref{fig:Omega_1G}, but for $m_{\hat{\nu}}=100$~MeV and $\Delta = 0.2$ and 0.26.}
\label{fig:Omega_100M}
\end{figure}
\begin{figure}[ht]
\centering
\captionsetup{singlelinecheck = false, format= hang, justification=raggedright, font=footnotesize, labelsep=space}
\includegraphics[scale=0.4]{Contour_10G_BM004.pdf}
\includegraphics[scale=0.4]{Contour_10G_BM010.pdf}
\caption{Similar to Fig.~\ref{fig:Omega_1G}, but for $m_{\hat{\nu}}=10$~GeV and $\Delta = 0.04$ and 0.1.}
\label{fig:Omega_10G}
\end{figure}
\section{Experimental Constraints and Tests}
\label{sec:constraints}
In this section we discuss the current experimental constraints and future experimental tests of this model. A comprehensive summary for this type of DM scenarios can be found in Ref.~\cite{DAgnolo:2018wcn}.
\subsection{Direct Detection}
The dark matter $\hat{\nu}$ interacts with SM particles through the Higgs portal or dark photon portal. The Higgs portal is suppressed by the Higgs mixing between the SM and twin sectors, and the twin neutrino Yukawa coupling. The dark photon portal is suppressed by the kinetic mixing $\epsilon$, and also the mixing angle $\theta_1$. Most current DM direct detection experiments are based on heavy nuclei recoiling when the nuclei scatter with the DM particles. They become ineffective for light DM less than a few GeV, but could potentially constrain the parameter space of heavier twin neutrino region, as the Yukawa coupling also becomes larger for a heavier twin neutrino. For $m_{\hat{\nu}}$=10~GeV and $f/v =3$, the $\hat{\nu}$-nucleon cross section is of $\mathcal{O}(10^{-47})$~cm$^2$, which is dominated by the Higgs portal. Such $\hat{\nu}$ DM is not yet constrained by recent liquid xenon DM detectors~\cite{daSilva:2017swg,Aprile:2017iyp,Cui:2017nnn}. A conservative estimate according to Ref.~\cite{Djouadi:2011aa} gives an upper limit of $\sim$22(60)~GeV on $m_{\hat{\nu}}$ for $f/v =3(5)$.
For even lighter DM, recent upgrades/proposals of detecting DM-electron scattering can largely increase the sensitivity for sub-GeV mass DM~\cite{Essig:2013lka,Alexander:2016aln}. In our model, the $\hat{\nu}$-$e$ couplings from both the dark photon portal and the Higgs portal are highly suppressed. The elastic cross section of $\hat{\nu}-e$ scattering from the dark photon portal is
\begin{eqnarray}
\sigma_{\nu e} &\simeq& \frac{g^2 \hat{e}^2 \epsilon^2 \theta_1^4}{\pi m_{\hat{\gamma}}^4}\bigg(\frac{m_e m_{\hat{\gamma}}}{m_e+m_{\hat{\gamma}}}\bigg)^2 \nonumber \\
&\simeq& 4.3\times10^{-38} \bigg(\frac{\hat{e}}{0.3}\bigg)^2 \bigg(\frac{\epsilon}{10^{-3}}\bigg)^2\bigg(\frac{\theta_1}{10^{-1}}\bigg)^4 \bigg(\frac{10~\text{MeV}}{m_{\hat{\nu}}}\bigg)^4 \bigg(\frac{0.5}{r}\bigg)^4 ~~\text{cm}^2,
\end{eqnarray}
where the reference values of mixing parameters $\epsilon$, $\theta_1$ have been chosen close to the upper bounds to maximize the cross section. This is not yet constrained by recent electron-scattering experiments, including SENSEI~\cite{Crisler:2018gci}, Xenon10~\cite{Essig:2012yx}, DarkSide-50~\cite{Agnes:2018oej}. Future upgrades will be able to probe part of the parameter space with large mixings.
It is also worth mentioning that there are also crystal experiments based on phonon signals coming from DM scattering off nuclei in the detector, such as CRESST-III~\cite{Petricca:2017zdp}. Thanks to the low energy threshold ($\mathcal{O}$(50~eV)), these experiments will also be sensitive to the sub-GeV DM mass region. For such low DM mass, the dark photon portal becomes important and can dominate over the Higgs portal interaction if the mixings are not too small. However, the current constraint still can not put any bounds on $m_{\hat{\nu}}$ even for the $f/v$=3 case. Significant progress in the future could be helpful to constrain the parameter space for a light $\hat{\nu}$.
\subsection{Indirect Constraints Induced from DM Annihilation}
Light DM is in general strongly constrained by indirect searches due to its high number density. WIMP models with annihilation cross section $\left\langle\sigma v\right\rangle \simeq 10^{-26}\text{cm}^3/\text{s}$ and $m_{\text{DM}} \lesssim10$~GeV have already been ruled out~\cite{Ade:2015xua}. In our model the DM relic density is not determined by the DM annihilation process, but by the coannihilation and coscattering processes. The DM annihilation is dominated by $\hat{\nu}\tn\to\hat{\gamma}\gd\to 4f$, which is suppressed by $\hat{e}^4\theta_1^4$. An upper bound on the DM annihilation cross section gives a constraint on the combination of the parameters $\hat{e} \theta_1$. Fermi-LAT data~\cite{Geringer-Sameth:2014qqa} has put an upper limit on the DM annihilation cross section for DM heavier than 6~GeV.
A stronger constraint comes from CMB observables, which restrict the net energy deposited from DM annihilation into visible particles during the reionization era~\cite{Liu:2016cnk}. The constraints from the Fermi-LAT and the Planck data are plotted in Fig.~\ref{fig:ind1}.
\begin{figure}
\captionsetup{singlelinecheck = false, format= hang, justification=raggedright, font=footnotesize, labelsep=space}
\includegraphics[scale=0.6]{Indirect_Limit.pdf}
\caption{The Fermi-LAT, CMB and EDGES bounds on DM annihilation rate in terms of $\hat{e}\theta_1$ as a function of $m_{\hat{\nu}}$. The different color curves correspond to different experiments, with the one inferred from the EDGES to be the strongest assuming that no other effect can enhance the 21cm line absorption. The solid curves represent the benchmark with $r=0.05$ and $\Delta=0.5$, where the constraints are strongest. The dashed and dotted curves are for different choices of $r$ and $\Delta$. All bounds are deduced from $ee$ final states.
}
\label{fig:ind1}
\end{figure}
Note that the annihilation of $\hat{\nu}\tn$ can produce $4e$ instead of $2e$. This may modify the bounds derived from the $2e$ final state. The total energy injection is the same, and the $4e$ final state will result in more electrons but with lower energies. Ref.~\cite{Elor:2015bho} performed a detailed study in comparing the constrains for cascade decays with different numbers of final sate particles. After convoluting with the energy dependence of the efficiency factor $f_{\text{eff}}$~\cite{Slatyer:2015jla}, it is found that the effects due to multi-step decay is rather mild. The constraints on the $4e$ and $2e$ final states from the Planck data are roughly the same. On the other hand, a higher-step cascade tends to soften the spectrum and thus slightly weakens the constraint from the Fermi-LAT result~\cite{Elor:2015bho}. The proposed ground-based CMB Stage-4 experiment~\cite{Abazajian:2016yjj} is expected to improve the constraint by a factor of 2 to 3 compared to Planck.
The DM annihilation after the CMB era will also heat the intergalactic hydrogen gas and erase the absorption features of 21cm spectrum around $z\simeq 17$. The recent measurement by the EDGES experiment instead observed an even stronger absorption than the standard astrophysical expectation~\cite{Bowman:2018yin}. If one interprets this result as a constraint that the DM annihilation should not significantly reduce the absorption, the observed brightness temperature then suggests an even stronger bound than the one from CMB~\cite{DAmico:2018sxd,Cheung:2018vww,Berlin:2018sjs,Barkana:2018qrx}. In Fig.~\ref{fig:ind1} we also plot the most conservative constraint in terms of $\hat{e}\times \theta_1$ according to Ref.~\cite{DAmico:2018sxd}, taking the efficiency factor to be 1. For $m_{\hat{\nu}} =1$~GeV, the upper limit for $\hat{e} \theta_1$ is between $10^{-2}$ and $10^{-3}$ depending on other model parameters.\footnote{
The $\hat{\nu}\tn$ annihilation cross section depends on the model parameters as
$$
\left\langle\sigma v\right\rangle \propto -\frac{\theta_1 ^4\hat{e}^4 \left(1-r^2\right)^{3/2} \left(\left(\Delta ^4+4 \Delta
^3-8 \Delta -4\right) r^2-2 \Delta ^2 (\Delta +2)^2\right)}{32 \pi
(\Delta +1)^4 r^2 m_{\hat{\nu}}^2 \left(\Delta ^2+2 \Delta
-r^2+2\right)^2}.
$$
For a fixed mixing angle, the annihilation rate reaches its maximum around $\Delta\sim 0.5$ for very small $r$, while for larger $r$, it decreases as $\Delta$ increases.}
The constraint gets more stringent for lighter $m_{\hat{\nu}}$.
\subsection{Constraints induced by the Light Twin Photon}
\label{sec:lightphoton}
In our scenario $\hat{\gamma}$ is the lightest twin sector particle. An on-shell $\hat{\gamma}$ decays through its kinetic mixing with the SM photon and there is no invisible decay mode to the twin sector. In this case, it is well described by two parameters: the twin photon mass $m_{\hat{\gamma}}= rm_{\hat{\nu}}$ and the kinematic mixing with the SM photon $\epsilon$. Experimental constraints on dark photon have been extensively studied. Summeries of current status can be found in Refs.~\cite{Curtin:2013fra,Curtin:2014cca,Alexander:2016aln,Ilten:2018crw}.
A lower bound on $m_{\hat{\gamma}}$ comes from the effective number of neutrinos ($N_{\rm eff}$)~\cite{Boehm:2013jpa}. A light $\hat{\gamma}$ can stay in thermal equilibrium with photons and electrons after the neutrinos decouple at $T\sim 2.3$~MeV. The entropy transferred from $\hat{\gamma}$ to the photon bath will change the neutrino-photon temperature ratio, and therefore modify $N_{\rm eff}$. Using the results in Ref.~\cite{Boehm:2013jpa} and the Planck data~\cite{Ade:2015xua}, we obtain a lower bound on $m_{\hat{\gamma}}$ around $11$~MeV. It may be further improved to $\sim 19$~MeV by the future CMB-S4 experiment~\cite{Abazajian:2016yjj}.
There are also many constraints on $\epsilon$ depending on $m_{\hat{\gamma}}$.
The current upper bounds for $\epsilon$ mostly come from colliders, fixed target experiments
and meson decay experiments, in searching for prompt decay products. (See Ref.~\cite{Ilten:2018crw} for a summary and an extended reference list.) These experiments constrain $\epsilon \lesssim 10^{-3}$ in the mass range that we consider (except for a few narrow gaps at the meson resonances). The lower bounds for $\epsilon$ come from $\hat{\gamma}$ displaced decays from various beam dump experiments~\cite{Bjorken:2009mm,Bergsma:1985is,
Konaka:1986cb,Riordan:1987aw,Bjorken:1988as,Bross:1989mp,
Davier:1989wz,Athanassopoulos:1997er,Astier:2001ck,
Essig:2010gu,Williams:2011qb,Blumlein:2011mv,Gninenko:2012eq,
Blumlein:2013cua}, and also from supernova SN1987A~\cite{Chang:2016ntp}. {The Big Bang Nucleosynthesis (BBN) would also constrain the lifetime of $\hat{\gamma}$. However, the decay of $\hat{\gamma}$ is only suppressed by $\epsilon$ and the BBN does not introduce extra constraints for $\epsilon \gtrsim 10^{-10}$~\cite{Chang:2016ntp}, which is required in this model to keep the DM sector in thermal contact with the SM.}
These bounds are summarized in Fig.~\ref{fig:darkphoton}.
\begin{figure}[h]
\captionsetup{singlelinecheck = false, format= hang, justification=raggedright, font=footnotesize, labelsep=space}
\includegraphics[scale=0.44]{DarK_G_New}
\caption{{ Constraints on the kinetic mixing parameter $\epsilon$ and the twin photon mass $m_{\hat{\gamma}}$. The green and cyan shaded regions are ruled out by lab experiments. The magenta shaded region is the constraint from SN1897A cooling. In the red shaded region $\epsilon$ is too small to keep $\hat{\gamma}$ in thermal equilibrium with the SM. We also plot 6 benchmark models which give the correct DM relic density from the numerical calculations in Sec.~\ref{sec:numerical}. For small enough $\epsilon$, models 1, 3, 4, and 6 are in the coscattering phase and model 2 is in the mixed phase. At large values of $\epsilon$ these model curves turn right because the three-body (inverse) decay rate becomes large and freezes out after the coannihilation process, driving the models into the coannihilation phase. Model 5 is in the coannihilation phase for all $\epsilon$ large enough to keep $\hat{\gamma}$ in thermal equilibrium. The ticks on each benchmark curve represent $\hat{\tau}$ lifetime, starting from $\tau(\hat{\tau})$=1~sec and increasing by $10^2$ each tick below. The dashed parts of the curves are ruled out by the BBN constraint.}}
\label{fig:darkphoton}
\end{figure}
\subsection{Constraints Induced from $\hat{\tau}$ decay}
As we require $m_{\hat{\nu}}>m_{\hat{\gamma}}$, the twin photon only decays to SM particles, thus $\hat{\tau}$ can only be pair produced in a lab via an off-shell twin photon or from the Higgs boson decay. The constraints from $\hat{\tau}$ pair production from off-shell $\hat{\gamma}$ are weaker compared to the ones from $\hat{\gamma}$ visible decay modes described in the previous subsection. Moreover, $\hat{\nu}$ pair production via $\hat{\gamma}^*$ will be further suppressed by $\theta_1^4$, leaving $h\to \hat{\nu}\tn/\hat{\tau}\tt$ to be the main production channel.
At the LHC, the $\hat{\tau}$ produced from $h$/$\hat{\gamma}^*$ will be long-lived in general, if the two-body decay $\hat{\tau} \to \hat{\nu}\hat{\gamma}$ is forbidden ($r>\Delta$), because the leading three-body decay is suppressed by $\theta_1^2\epsilon^2 \hat{e}^2 e^2$.
Assuming $\hat{\tau}$ and $\hat{\nu}$ are Dirac fermions
and taking $m_{\hat{\tau}} \simeq m_{\hat{\nu}}$, the current upper bound of the Higgs invisible decay branching ratio, Br($h \to$ invisible) $< 24\%$~\cite{Khachatryan:2016vau,Khachatryan:2016whc}, constrains $m_{\hat{\nu}}$ to be $\lesssim 19\,(52)$~GeV for $f/v=3\,(5)$. HL-LHC is expected to improve the Higgs invisible branching ratio measurement to 6-8\%~\cite{Dawson:2013bba}, which would translate to a bound $\lesssim 11\,(30)$~GeV for $m_{\hat{\nu}}$. Future $e^+e^-$ colliders can probe BR($h\to$ invisible) to the sub-percent level~\cite{Gomez-Ceballos:2013zzn,Fujii:2015jha,CEPC-SPPCStudyGroup:2015csa,Abramowicz:2016zbo}. A 0.3\% measurement can constrain $m_{\hat{\nu}}$ down to $\lesssim 2\,(6)$~GeV for $f/v=3\,(5)$.
The $\hat{\tau}$ decay width can be expressed analytically in the small $\Delta$ limit ($1\gtrsim r \gg \Delta$):
\begin{equation}
\Gamma_{\hat{\tau} \to \hat{\nu} e^+e^-} \simeq \frac{\theta_1^2 \Delta ^5 e^2 \hat{e}^2 \epsilon ^2}{60 \pi ^3 r^4} m_{\hat{\nu}},
\end{equation}
which strongly depends on $\Delta$ and $r$. Numerically the proper decay length is given by
\begin{equation}
c\tau (\hat{\tau}) \approx \frac{8.8\times10^6}{N_f} \left(\frac{0.2}{\Delta}\right)^5 \left(\frac{10^{-3}}{\theta_1}\right)^2 \left(\frac{0.3}{\hat{e}}\right)^2 \left(\frac{10^{-3}}{\epsilon}\right)^2\left(\frac{r}{0.5}\right)^4 \left(\frac{1~\rm{GeV}}{m_{\hat{\nu}}}\right)\rm{cm},
\end{equation}
where $N_f$ is the number of SM fermions that can appear in the final state.
The dark sector could be probed by searching for $\hat{\tau}$ displaced decays at the HL-LHC for $c\tau(\hat{\tau})\sim \mathcal{O}$(1)~m~\cite{Curtin:2017izq}. Longer decay lengths may be tested at future proposed experiments, such as SHiP~\cite{Alekhin:2015byh}, MATHUSLA~\cite{Curtin:2017izq}, CODEX-b~\cite{Gligorov:2017nwh}, and FASER~\cite{Feng:2017uoz}.
If $\tau(\hat{\tau})\gtrsim \mathcal{O}$(1)~second, the decays of $\hat{\tau}$ thermal relic will inject energy during the BBN era and the recombination era. However, the constraint is weakened by the fact that $\hat{\tau}$ only makes up a small fraction of $\Omega$ after it freezes out ($\Omega_{\hat{\tau}} h^2 \lesssim 10^{-3}$), also the fact that only a small fraction $\frac{m_{\hat{\tau}}-m_{\hat{\nu}}}{m_{\hat{\tau}}}=\frac{\Delta}{1+\Delta}$ of energy would be injected. The strongest bound comes from the electromagnetic decay products and depends on the model parameters. Typically, the lifetime of the long-lived $\hat{\tau}$ is only constrained by BBN to be shorter than $\sim 10^6$~seconds~\cite{Poulin:2016anj,Kawasaki:2017bqm}.
The extra energy injection from $\hat{\tau}$ decays could distort the CMB blackbody spectrum which can potentially be captured by the proposed PIXIE mission~\cite{Kogut:2011xw}. It may give a slightly stronger bound than BBN on $\tau(\hat{\tau})$~\cite{Poulin:2016anj}, also around $10^{6}$~sec for our typical benchmark models.
In Fig~\ref{fig:darkphoton}, we also plot several benchmark points with fixed $r$, $\Delta$, and $\theta_1$, that give rise to the correct DM relic density from our numerical results in Sec.~\ref{sec:numerical}. The vertical parts indicates that the thermal relic density is mostly independent of the gauge kinematic mixing parameter $\epsilon$, as long as it can keep the DM in thermal equilibrium with SM before freeze-out. For large values of $\epsilon$, the three-body (inverse) decay rate gets larger, and can even freeze out later than the coannihilation process $\hm{C_S}$. It drives the benchmark models to the coannihilation phase even if it was in the coscattering or mixed phase for smaller values of $\epsilon$. This occurs for all our benchmark models except for model 5 (red line) which is in the coannihilation phase for all $\epsilon$. The transition to the coannihilation phase reduces the DM number density, hence it needs a larger DM mass to compensate the effect. This explains the turn of the curves at lager $\epsilon$ values. However, except for model 2 (orange line), the turns occur in the region which has been ruled out by other experiments. For smaller $\epsilon$, the $\hat{\tau}$ lifetime becomes longer, which is constrained by the BBN bound. The region that violates the BBN constraint is indicated by dashed lines.
\section{Conclusions}
\label{sec:conclusions}
The necessity of DM in the universe is one of the strongest evidences of new physics beyond the SM. Experimental searches in various fronts so far have not revealed the nature of the DM. For the most popular WIMP DM scenario, recent advancements in experiments have covered significant fractions of the allowed parameter space, even though there are still viable parameter space left. People have taken more seriously the possibility that DM resides in a more hidden sector, and hence has escaped our intensive experimental searches. Even in this case, it would be more satisfactory if it is part of a bigger story, rather than just arises in an isolated sector for no particular reason. In this paper, we consider DM coming from a particle in the twin sector of the fraternal twin Higgs model, which itself is motivated by the naturalness problem of the SM EW symmetry breaking and non-discovery of the colored top partners at the colliders. Although the relevant particles for the DM relic density in our study, i.e., the twin neutrino, twin tau, and twin photon, have little effect on the naturalness of the EW scale, they are an integral part of the full theory that solves the naturalness problem, just like the neutralinos in a supersymmetric SM.
To obtain the correct DM relic density, the interplay of the twin neutrino, twin tau, and twin photon is important. The DM relic density is determined by the order of the freeze-out temperatures of various annihilation and scattering processes. It is in the coannihilation phase if the twin tau annihilation freezes out earlier than the twin neutrino to twin tau scattering. In the opposite limit, it realizes the recently discovered coscattering phase. There is also an intermediate regime where the DM relic density is determined by both coannihilation and coscattering processes due to the momentum dependence of the coscattering process. The calculation of the DM relic density in this mixed phase is more complicated and has not been done in the literature and we provide a reasonably simple way to evaluate it with very good accuracies.
There are many experimental constraints but none of them can cover the whole parameter space. Direct detection with nuclei recoiling can only constrain heavier DM, above a few GeV. The experiments based on electron scattering are not yet sensitive to this model. Future upgrades may be able to probe the region of the parameter space with large mixings between the twin tau and the twin neutrino. Indirect constraints from DM annihilation are more sensitive to smaller DM mass with large enough twin tau -- twin neutrino mixings. Other constraints rely on the coannihilation/coscattering partners of the DM. Twin photon is subject to various dark photon constraints. Twin tau typically has a long lifetime so it is constrained by BBN and CMB. Its displaced decays may be searched at colliders with dedicated detectors or strategies. A big chunk of the parameter space still survives all the constraints. A complete coverage of the parameter space directly is not easy in the foreseeable future. An indirect test may come from the test of the whole fraternal twin Higgs model at a future high energy collider, if other heavier particles in the model can be produced.
\section*{acknowledgments}
We would like to thank Zackaria Chacko, Rafael Lang, Natalia Toro, Yuhsin Tsai, and Po-Jen Wang for useful discussions. This work is supported in part by the US Department of Energy grant DE-SC-000999. H.-C.~C. was also supported by The Ambrose Monell Foundation at the Institute for Advanced Study, Princeton.
| 22,626 |
\section{Introduction}
Let $X_0: M^n\to \mathbb{H}^{n+1}$ be a smooth embedding such that $M_0=X_0(M)$ is a closed smooth hypersurface in the hyperbolic space $\mathbb{H}^{n+1}$. We consider a smooth family of immersions $X:M^n\times [0,T)\rightarrow \mathbb{H}^{n+1}$ satisfying
\begin{equation}\label{flow-VMCF-0}
\left\{\begin{aligned}
\frac{\partial}{\partial t}X(x,t)=&~(\phi(t)-\Psi(x,t))\nu(x,t),\\
X(\cdot,0)=&~X_0(\cdot),
\end{aligned}\right.
\end{equation}
where $\nu(x,t)$ is the unit outward normal of $M_t=X(M,t)$, $\Psi$ is a smooth curvature function evaluated at the point $(x,t)$ of $M_t$, the global term $\phi(t)$ is chosen to impose a constraint on the enclosed volume or quermassintegrals of $M_t$.
The volume preserving mean curvature flow in hyperbolic space was first studied by Cabezas-Rivas and Miquel \cite{Cab-Miq2007} in 2007. By imposing horospherically convexity (the condition that all principal curvatures exceed $1$, which will also be called h-convex) on the initial hypersurface, they proved that the solution exists for all time and converges smoothly to a geodesic sphere. Some other mixed volume preserving flows were considered in \cite{Mak2012,WX} with speed given by homogeneous degree one functions of the principal curvatures.
Recently Bertini and Pipoli \cite{Be-Pip2016} succeeded in treating flows by more general functions of mean curvature, including in particular any positive power of mean curvature. In a recent paper \cite{And-Wei2017-2}, the first and the third authors proved the smooth convergence of quermassintegral preserving flows with speed given by any positive power of a homogeneous degree one function $f$ of the principal curvatures for which the dual function $f_*(x_1,\cdots,x_n) = (f(x_1^{-1},\cdots,x_n^{-1}))^{-1}$ is concave and approaches zero on the boundary of the positive cone. This includes in particular the volume preserving flow by positive powers of $k$-th mean curvature for h-convex hypersurfaces. Note that in all the above mentioned work, the initial hypersurface is assumed to be h-convex.
One reason to consider constrained flows of the kind considered here is to prove geometric inequalities: In particular, the convergence of the volume-preserving mean curvature flow to a sphere implies that the area of the initial hypersurface is no less than that of a geodesic sphere with the same enclosed volume, since the area is non-increasing while the volume remains constant under the flow. The same motivation lies behind \cite{WX}, where inequalities between quermassintegrals were deduced from the convergence of certain flows.
In this paper, we make the following contributions:
\begin{itemize}
\item[(1)] In the first part of the paper, we weaken the horospherical convexity condition, allowing instead hypersurfaces for which the intrinsic sectional curvatures are positive. We
consider the flow \eqref{flow-VMCF-0} for hypersurfaces with positive sectional curvature and with speed $\Psi$ given by any positive power of a smooth, symmetric, strictly increasing and homogeneous of degree one function of the Weingarten matrix $\mathcal{W}$ of $M_t$. Here we say a hypersurface $M$ in hyperbolic space has positive sectional curvature if its sectional curvature $R_{ijij}^M>0$ for any $1\leq i< j\leq n$, which by Gauss equation is equivalent to the principal curvatures of $M$ satisfying $\kappa_i\kappa_j>1$ for $1\leq i\neq j\leq n$. This is a weaker condition than h-convexity. As a consequence we deduce inequalities between volume and other quermassintegrals for hypersurfaces with positive sectional curvature, extending inequalities previously known only for horospherically convex hypersurfaces.
\item[(2)] In the second part of this paper, we consider flows \eqref{flow-VMCF-0} for strictly h-convex hypersurfaces in which the speed $\Psi$ is homogeneous as a function of the shifted Weingarten matrix $\mathcal{W}-\mathrm{I}$ of $M_t$, rather than the Weingarten matrix itself. Using these flows we are able to prove a new class of integral inequalities for horospherically convex hypersurfaces.
\item[(3)] In order to understand these new functionals we introduce some new machinery for horospherically convex regions, including a horospherical Gauss map and a horospherical support function. We also develop an interesting connection (closely related to the results of \cite{EGM}) between flows of h-convex hypersurfaces in hyperbolic space by functions of principal curvatures, and conformal flows of conformally flat metrics on $S^n$ by functions of the eigenvalues of the Schouten tensor. This allows us to translate our results to convergence theorems for metric flows, and our isoperimetric inequalities to corresponding results for conformally flat metrics. We expect that these will prove useful in future work.
\end{itemize}
We will describe our results in more detail in the rest of this section:
\subsection{Volume preserving flow with positive sectional curvature}
Suppose that the initial hypersurface $M_0$ has positive sectional curvature. We consider the smooth family of immersions $X:M^n\times [0,T)\rightarrow \mathbb{H}^{n+1}$ satisfying
\begin{equation}\label{flow-VMCF}
\left\{\begin{aligned}
\frac{\partial}{\partial t}X(x,t)=&~(\phi(t)-F^{\alpha}(\mathcal{W}))\nu(x,t),\\
X(\cdot,0)=&~X_0(\cdot),
\end{aligned}\right.
\end{equation}
where $\alpha>0$, $\nu(x,t)$ is the unit outward normal of $M_t=X(M,t)$, $F$ is a smooth, symmetric, strictly increasing and homogeneous of degree one function of the Weingarten matrix $\mathcal{W}$ of $M_t$. The global term $\phi(t)$ in \eqref{flow-VMCF} is defined by
\begin{equation}\label{s1:phit}
\phi(t)=\frac{1}{|M_t|}\int_{M_t}F^{\alpha} d\mu_t
\end{equation}
such that the volume of $\Omega_t$ remains constant along the flow \eqref{flow-VMCF}, where $d\mu_t$ is the area measure on $M_t$ with respect to the induced metric.
Since $F(\mathcal{W})$ is symmetric with respect to the components of $\mathcal{W}$, by a theorem of Schwarz \cite{Scharz75} we can write $F(\mathcal{W})=f(\kappa)$ as a symmetric function of the eigenvalues of $\mathcal{W}$. We assume that $f$ satisfies the following assumption:
\begin{assump}\label{s1:Asum}
Suppose $f$ is a smooth symmetric function defined on the positive cone $\Gamma_+:=\{\kappa=(\kappa_1,\cdots,\kappa_n)\in \mathbb{R}^n: \kappa_i>0,~\forall~i=1,\cdots,n\}$, and satisfies
\begin{itemize}
\item[(i)] $f$ is positive, strictly increasing, homogeneous of degree one and is normalized such that $f(1,\cdots,1)=1$;
\item[(ii)] For any $i\neq j$,
\begin{equation}\label{s1:asum-1}
(\frac{\partial f}{\partial \kappa_i}\kappa_i-\frac{\partial f}{\partial \kappa_j}\kappa_j)(\kappa_i-\kappa_j)~\geq~0.
\end{equation}
\item[(iii)] For all $(y_1,\cdots,y_n)\in \mathbb{R}^n$,
\begin{equation}\label{s1:asum-2}
\sum_{i,j}\frac{\partial^2 \log f}{\partial\kappa_i\partial\kappa_j}y_iy_j+\sum_{i=1}^n\frac 1{\kappa_i}\frac{\partial\log f}{\partial \kappa_i}y_i^2~\geq~0.
\end{equation}
\end{itemize}
\end{assump}
\noindent Examples satisfying Assumption \ref{s1:Asum} include $f=n^{-1/k}S_k^{1/k}$ $(k>0)$ and $f=E_k^{1/k}$ (see, e.g., \cite{GaoLM17,GuanMa03}), where
\begin{equation*}
E_{k}={\binom{n}{k}}^{-1}\sigma_k(\kappa)={\binom{n}{k}}^{-1}\sum_{1\leq i_1<\cdots<i_k\leq n}\kappa_{i_1}\cdots \kappa_{i_k},\qquad k=1,\cdots,n.
\end{equation*}
is the (normalized) $k$-th mean curvature of $M_t$ and $ S_k(\kappa)=\sum_{i=1}^n\kappa_i^k$ is the $k$-th power sum of $\kappa$ for $k>0$. The inequalities \eqref{s1:asum-1} and \eqref{s1:asum-2} are equivalent to the statement that $\log F$ is a convex function of the components of $\log{\mathcal W}$, which is the map with the same eigenvectors as ${\mathcal W}$ and eigenvalues $\log\kappa_i$. In particular, if $f_1$ and $f_2$ are two symmetric functions satisfying \eqref{s1:asum-1} and \eqref{s1:asum-2}, then the function $f_1^\alpha$ with $\alpha>0$ and the product $f_1f_2$ also satisfy \eqref{s1:asum-1} and \eqref{s1:asum-2}. Note that the Cauchy-Schwarz inequality and \eqref{s1:asum-2} imply that any symmetric function $f$ satisfying \eqref{s1:asum-2} must be inverse concave, i.e., its dual function
$$f_*(z_1,\cdots,z_n)=f(z_1^{-1},\cdots,z_n^{-1})^{-1}$$
is concave with respect to its argument.
The first result of this paper is the following convergence result for the flow \eqref{flow-VMCF}:
\begin{thm}\label{thm1-1}
Let $X_0: M^n\to \mathbb{H}^{n+1}$ be a smooth embedding such that $M_0=X_0(M)$ is a closed hypersurface in $\mathbb{H}^{n+1}$ $(n\geq 2)$ with positive sectional curvature. Assume that $f$ satisfies Assumption \ref{s1:Asum}, and either
\begin{itemize}
\item[(i)] $f_*$ vanishes on the boundary of $\Gamma_+$, and
\begin{equation}\label{thm1-cond1}
\lim_{x\to 0+}f(x,\frac 1x,\cdots,\frac 1x)~=~+\infty,
\end{equation}
and $\alpha>0$, or
\item[(ii)] $n=2$, $f=(\kappa_1\kappa_2)^{1/2}$ and $\alpha\in [1/2,2]$.
\end{itemize}
Then the flow \eqref{flow-VMCF} with global term $\phi(t)$ given by \eqref{s1:phit} has a smooth solution $M_t$ for all time $t\in [0,\infty)$, and $M_t$ has positive sectional curvature for each $t>0$ and converges smoothly and exponentially to a geodesic sphere of radius $r_{\infty}$ determined by $\mathrm{Vol}(B(r_{\infty}))=\mathrm{Vol}(\Omega_0)$ as $t\to\infty$.
\end{thm}
\begin{rem}
Examples of function $f$ satisfying Assumption \ref{s1:Asum} and the condition (i) of Theorem \ref{thm1-1} include:
\begin{itemize}
\item[a).] $n\geq 2$, $f=n^{-1/k}S_k^{1/k}$ with $k>0$;
\item[b).] $n\geq 3$, $f= E_k^{1/k}$ with $k=1,\cdots,n$;
\item[c).] $n=2, f=(\kappa_1+\kappa_2)/2$.
\end{itemize}\end{rem}
\begin{rem}
We remark that the contracting curvature flows for surfaces with positive scalar curvature in hyperbolic 3-space $\mathbb{H}^3$ have been studied by the first two authors in a recent work \cite{And-chen2014}.
\end{rem}
As a key step in the proof of Theorem \ref{thm1-1}, we prove in \S \ref{sec:PSC} that the positivity of sectional curvatures of the evolving hypersurface $M_t$ is preserved along the flow \eqref{flow-VMCF} with any $f$ satisfying Assumption \ref{s1:Asum} and any $\alpha>0$. In order to show that the positivity of sectional curvatures are preserved, we consider the sectional curvature as a function on the frame bundle $O(M)$ over $M$, and apply a maximum principle. This requires a rather delicate computation, using inequalities on the Hessian on the total space of $O(M)$ to show the required inequality on the time derivative at a minimum point. The argument is related to that used by the first author to prove a generalised tensor maximum principle in \cite{And2007}*{Theorem 3.2}, but cannot be deduced directly from that result. The argument combines the ideas of the generalised tensor maximum principle with those of vector bundle maximum principles for reaction-diffusion equations \cites{AH,Ha1986}.
We remark that the flow \eqref{flow-VMCF} with
\begin{equation}\label{s1:f-quoti}
f~=~\left(\frac{E_k}{E_l}\right)^{\frac 1{k-l}},\quad 1\leq l<k\leq n
\end{equation}
and any power $\alpha>0$ does not preserve positive sectional curvatures: Counterexamples can be produced in the spirit of the constructions in \cite{Andrews-McCoy-Zheng}*{Sections 4--5}.
The remaining parts of the proof of Theorem \ref{thm1-1} will be given in \S \ref{sec:thm1-pf}. In \S \ref{sec:4-1}, we will derive a uniform estimate on the inner radius and outer radius of the evolving domains $\Omega_t$ along the flow \eqref{flow-VMCF}. Recall that the inner radius $\rho_-$ and outer radius $\rho_+$ of a bounded domain $\Omega$ are defined as
\begin{equation*}
\rho_-= ~\sup \bigcup_{p\in\Omega}\{\rho>0:\ B_{\rho}(p)\subset\Omega\} \qquad
\rho_+= ~\inf\bigcup_{p\in\Omega}\{\rho>0: \Omega\subset B_{\rho}(p)\},
\end{equation*}
where $B_{\rho}(p)$ denotes the geodesic ball of radius $\rho$ and centered at some point $p$ in the hyperbolic space. All the previous papers \cite{And-Wei2017-2,Be-Pip2016,Cab-Miq2007,Mak2012,WX} on constrained curvature flows in hyperbolic space focus on horospherically convex domains, which have the property that $\rho_+\leq c(\rho_-+\rho_-^{1/2})$, see e.g. \cite{Cab-Miq2007,Mak2012}. However, no such property is known for hypersurfaces with positive sectional curvature. Our idea to overcome this obstacle is to use an Alexandrov reflection argument to bound the diameter of the domain $\Omega_t$ enclosed by the flow hypersurface $M_t$. Then we project the domain $\Omega_t$ to the unit ball in Euclidean space $\mathbb{R}^{n+1}$ via the Klein model of the hyperbolic space. The upper bound on the diameter of $\Omega_t$ implies that this map has bounded distortion. This together with the preservation of the volume of $\Omega_t$ gives a uniform lower bound on the inner radius of $\Omega_t$.
Then in \S \ref{sec:4-2} we adapt Tso's technique \cite{Tso85} to derive an upper bound on the speed if $f$ satisfies Assumption \ref{s1:Asum}, where the positivity of sectional curvatures of $M_t$ will be used to estimate the zero order terms of the evolution equation of the auxiliary function. In \S \ref{sec:4-3}, we will complete the proof of Theorem \ref{thm1-1} by obtaining uniform bounds on the principal curvatures. In the case (i) of Theorem \ref{thm1-1}, the upper bound of $f$ together with the positivity of sectional curvatures imply the uniform two-sides positive bound of the principal curvatures of $M_t$. In the case (ii) of Theorem \ref{thm1-1}, the estimate $1\leq \kappa_1\kappa_1=f(\kappa)^2\leq C$ does not prevent $\kappa_2$ from going to infinity. Instead, we will obtain the estimate on the pinching ratio $\kappa_2/\kappa_1$ by applying the maximum principle to the evolution equation of
$G(\kappa_1,\kappa_2)=(\kappa_1\kappa_2)^{\alpha-2}(\kappa_2-\kappa_1)^2$ with $\alpha\in [1/2,2]$. This idea has been applied by the first two authors in \cite{And1999,And-chen2012} to prove the pinching estimate for the contracting flow by powers of Gauss curvature in $\mathbb{R}^3$. Once we have the uniform estimate on the principal curvatures of the evolving hypersurfaces, higher regularity estimates can be derived by a standard argument. A continuation argument then yields the long time existence of the flow and the Alexandrov reflection argument as in \cite[\S 6]{And-Wei2017-2} implies the smooth convergence of the flow to a geodesic sphere.
\subsection{Alexandrov-Fenchel inequalities}
The volume preserving curvature flow is a useful tool in the study of hypersurface geometry. We will illustrate an application of Theorem \ref{thm1-1} in the proof of Alexandrov-Fenchel type inequalities (involving the quermassintegrals) for hypersurfaces in hyperbolic space. Recall that for a convex domain $\Omega$ in hyperbolic space, the quermassintegral $W_k(\Omega)$ is defined as follows (see \cite{Sant2004,Sol2006}):\footnote{Note that the definition \eqref{S1:Wk-def1} is different with the definition given in \cite{Sol2006} by a constant multiple $\frac{n+1-k}{n+1}$.}
\begin{equation}\label{S1:Wk-def1}
W_k(\Omega)=~\frac{\omega_{k-1}\cdots\omega_0}{\omega_{n-1}\cdots\omega_{n-k}}\int_{\mathcal{L}_k}\chi(L_k\cap\Omega)dL_k,\quad k=1,\cdots,n,
\end{equation}
where $\mathcal{L}_k$ is the space of $k$-dimensional totally geodesic subspaces $L_k$ in $\mathbb{H}^{n+1}$ and $\omega_n$ denotes the area of $n$-dimensional unit sphere in Euclidean space. The function $\chi$ is defined to be $1$ if $L_k\cap\Omega\neq \emptyset$ and to be $0$ otherwise. Furthermore, we set
\begin{equation*}
W_0(\Omega)=|\Omega|,\quad W_{n+1}(\Omega)=|\mathbb{B}^{n+1}|=\frac{\omega_n}{n+1}.
\end{equation*}
If the boundary of $\Omega$ is smooth, we can define the principal curvatures $\kappa=(\kappa_1,\cdots,\kappa_n)$ and the curvature integrals
\begin{equation}\label{s1:CurInt}
V_{n-k}(\Omega)~=~\int_{\partial\Omega}E_k(\kappa)d\mu,\quad k=0,1,\cdots,n
\end{equation}
of the boundary $M=\partial\Omega$. The quermassintegrals and the curvature integrals of a smooth convex domain $\Omega$ in $\mathbb{H}^{n+1}$ are related by the following equations (see \cite{Sol2006}):
\begin{align}
V_{n-k}(\Omega) ~=&~ (n-k)W_{k+1}(\Omega)+ kW_{k-1}(\Omega),\quad k=1,\cdots,n\label{s1:quermass-1}\\
V_n(\Omega) ~=&~nW_1(\Omega)~=~|\partial\Omega|.\label{s1:quermass-2}
\end{align}
In \cite{WX}, Wang and Xia proved the Alexandrov-Fenchel inequalities for smooth h-convex domain $\Omega$ in $\mathbb{H}^{n+1}$, which states that there holds
\begin{equation}\label{s1:AF-WX}
W_k(\Omega)~\geq f_k\circ f_l^{-1}(W_l(\Omega))
\end{equation}
for any $0\leq l<k\leq n$, with equality if and only if $\Omega$ is a geodesic ball, where $f_k: \mathbb{R}_+\to \mathbb{R}_+$ is an increasing function defined by $f_k(r)=W_k(B(r))$, the $k$-th quermassintegral of the geodesic ball of radius $r$. The proof in \cite{WX} is by applying the quermassintegral preserving flow for smooth h-convex hypersurfaces with speed given by the quotient \eqref{s1:f-quoti} and $\alpha=1$, and is similar with the Euclidean analogue considered by McCoy \cite{McC2005}. The inequality \eqref{s1:AF-WX} can imply the following inequality
\begin{equation}\label{s1:AF-WX2}
\int_{\partial\Omega}E_kd\mu~\geq~|\partial\Omega|\left(1+\left(\frac{|\partial\Omega|}{\omega_n}\right)^{-2/n}\right)^{k/2}
\end{equation}
for smooth h-convex domains, which compares the curvature integral \eqref{s1:CurInt} and the boundary area. Note that the inequality \eqref{s1:AF-WX2} with $k=2$ was proved earlier by the third author with Li and Xiong \cite{LWX-2014} for star-shaped and $2$-convex domains using the inverse curvature flow in hyperbolic space. For the other even $k$, the inequality \eqref{s1:AF-WX2} was also proved for smooth h-convex domains using the inverse curvature flow by Ge, Wang and Wu \cite{GWW-2014JDG}. It's an interesting problem to prove the inequalities \eqref{s1:AF-WX} and \eqref{s1:AF-WX2} under an assumption that is weaker than h-convexity.
Applying the result in Theorem \ref{thm1-1}, we show that the \emph{h-convexity} assumption for the inequality \eqref{s1:AF-WX} can be replaced by the weaker assumption of \emph{positive sectional curvature} in the case $l=0$ and $1\leq k\leq n$.
\begin{cor}\label{s1:cor-1}
Let $M=\partial\Omega$ be a smooth closed hypersurface in $\mathbb{H}^{n+1}$ which has positive sectional curvature and encloses a smooth bounded domain $\Omega$. Then for any $n\geq 2$ and $k=1,\cdots,n$, we have
\begin{equation}\label{s1:AF-0}
W_k(\Omega)~\geq f_k\circ f_0^{-1}(W_0(\Omega)),
\end{equation}
where $f_k: \mathbb{R}_+\to \mathbb{R}_+$ is an increasing function defined by $f_k(r)=W_k(B(r))$, the $k$-th quermassintegral of the geodesic ball of radius $r$. Moreover, equality holds in \eqref{s1:AF-0} if and only if $\Omega$ is a geodesic ball.
\end{cor}
The quermassintegral $W_k(\Omega_t)$ of the evolving domain $\Omega_t$ along the flow \eqref{flow-VMCF} with $F=E_k^{1/k}$ satisfies (see Lemma \ref{s2:lem2})
\begin{equation*}
\frac d{dt}W_k(\Omega_t)~=~\int_{M_t}E_k(\phi(t)-E_k^{\alpha/k})d\mu_t,
\end{equation*}
which is non-positive for each $\alpha>0$ by the choice \eqref{s1:phit} of $\phi(t)$ and the H\"{o}lder inequality. This means that $W_k(\Omega_t)$ is monotone decreasing along the flow \eqref{flow-VMCF} with $F=E_k^{1/k}$ unless $E_k$ is constant on $M_t$ (which is equivalent to $M_t$ being a geodesic sphere). Then Corollary \ref{s1:cor-1} follows from the monotonicity of $W_k$ and the convergence result in Theorem \ref{thm1-1}.
\subsection{Volume preserving flow for horospherically convex hypersurfaces}
In the second part of this paper, we will consider the flow of h-convex hypersurfaces in hyperbolic space with speed given by functions of the shifted Weingarten matrix $\mathcal{W}-\mathrm{I}$ plus a global term chosen to preserve modified quermassintegrals of the evolving domains.
Let us first define the following modified quermassintegrals:
\begin{equation}\label{s1:Wk-td}
\widetilde{W}_k(\Omega)~:=~\sum_{i=0}^k(-1)^{k-i}\binom{k}{i}W_i(\Omega),\quad k=0,\cdots,n,
\end{equation}
for a h-convex domain $\Omega$ in hyperbolic space. Thus $\widetilde{W}_k$ is a linear combination of the quermassintegrals of $\Omega$. In particular, $\widetilde{W}_0(\Omega)=|\Omega|$ is the volume of $\Omega$. The modified quermassintegrals defined in \eqref{s1:Wk-td} satisfy the following property:
\begin{prop}\label{s1:wk-prop}
The modified quermassintegral ${\widetilde W}_k$ is monotone with respect to inclusion for h-convex domains: That is, if $\Omega_0$ and $\Omega_1$ are h-convex domains with $\Omega_0\subset\Omega_1$, then $\widetilde W_k(\Omega_0)\leq\widetilde W_k(\Omega_1)$.
\end{prop}
This property is not obvious from the definition \eqref{s1:Wk-td} and its proof will be given in \S \ref{sec:h-convex}. We will first investigate some of the properties of horospherically convex regions in hyperbolic space $\mathbb{H}^{n+1}$. In particular, for such regions we define a \emph{horospherical Gauss map}, which
is a map to the unit sphere, and we show that each horospherically convex region is completely described in terms of a scalar function $u$ on the sphere $\mathbb{S}^n$ which we call the \emph{horospherical
support function}. There are interesting formal similarities between this situation and that
of convex Euclidean bodies. We show that the h-convexity of a region $\Omega$ is equivalent to that the following matrix
\begin{equation*}
A_{ij} = \bar\nabla_j\bar\nabla_k\varphi-\frac{|\bar\nabla\varphi|^2}{2\varphi}\bar g_{ij}+\frac{\varphi-\varphi^{-1}}{2}\bar g_{ij}
\end{equation*}
on the sphere $\mathbb{S}^n$ is positive definite, where $\bar{g}_{ij}$ is the standard round metric on $\mathbb{S}^n$, $\varphi={\mathrm e}^u$ and $u$ is the horospherical support function of $\Omega$. The shifted Weingarten matrix $\mathcal{W}-\mathrm{I}$ is related to the matrix $A_{ij}$ by
\begin{equation}\label{eq:A-vs-W}
A_{ij} = \varphi^{-1}\left[\left({\mathcal W}-\mathrm{I}\right)^{-1}\right]_i^k\bar g_{kj}.
\end{equation}
Using this characterization of h-convex domains, for any two h-convex domains $\Omega_0$ and $\Omega_1$ with $\Omega_0\subset\Omega_1$ we can find a foliation of h-convex domains $\Omega_t$ which is expanding from $\Omega_0$ to $\Omega_1$. This can be used to prove Proposition \ref{s1:wk-prop} by computing the variation of $\widetilde{W}_k$. We expect that the description of horospherically convex regions which we develop here will be useful in further investigations beyond the scope of this paper.
The flow we will consider is the following:
\begin{equation}\label{flow-VMCF-2}
\left\{\begin{aligned}
\frac{\partial}{\partial t}X(x,t)=&~(\phi(t)-F(\mathcal{W}-\mathrm{I}))\nu(x,t),\\
X(\cdot,0)=&~X_0(\cdot)
\end{aligned}\right.
\end{equation}
for smooth and strictly h-convex hypersurface in hyperbolic space, where $F$ is a smooth, symmetric, homegeneous of degree one function of the shifted Weingarten matrix $\mathcal{W}-\mathrm{I}=(h_i^j-\delta_i^j)$. For simplicity, we denote $S_{ij}=h_i^j-\delta_i^j$. Note that the eigenvalues of $(S_{ij})$ are the shifted principal curvatures $\lambda=(\lambda_1,\cdots,\lambda_n)=(\kappa_1-1,\cdots,\kappa_n-1)$. Again by a theorem of Schwarz \cite{Scharz75}, $F(\mathcal{W}-\mathrm{I})=f(\lambda)$, where $f$ is a smooth symmetric function of $n$ variables $\lambda=(\lambda_1,\cdots,\lambda_n)$. We choose the global term $\phi(t)$ in \eqref{flow-VMCF-2} as
\begin{equation}\label{s1:phit-2}
\phi(t)=\left(\int_{M_t}E_l(\lambda)d\mu_t\right)^{-1}\int_{M_t}E_l(\lambda)Fd\mu_t,\quad l=0,\cdots,n
\end{equation}
such that $\widetilde{W}_l(\Omega_t)$ remains constant, where $\Omega_t$ is the domain enclosed by the evolving hypersurface $M_t$.
We will prove the following result for the flow \eqref{flow-VMCF-2} with $\phi(t)$ given in \eqref{s1:phit-2}.
\begin{thm}\label{thm1-5}
Let $n\geq 2$ and $X_0: M^n\to \mathbb{H}^{n+1}$ be a smooth embedding such that $M_0=X_0(M)$ is a smooth closed and strictly h-convex hypersurface in $\mathbb{H}^{n+1}$. If $f$ is a smooth, symmetric, increasing and homogeneous of degree one function, and either
\begin{itemize}
\item[(i)] $f$ is concave and f approaches zero on the boundary of the positive cone $\Gamma_+$, or
\item[(ii)] $f$ is concave and inverse concave, or
\item[(iii)] $f$ is inverse concave and its dual function $f_*$ approaches zero on the boundary of positive cone $\Gamma_+$, or
\item[(iv)] $n=2$,
\end{itemize}
then the flow \eqref{flow-VMCF-2} with the global term $\phi(t)$ given by \eqref{s1:phit-2} has a smooth solution $M_t$ for all time $t\in [0,\infty)$, and $M_t$ is strictly h-convex for any $t>0$ and converges smoothly and exponentially to a geodesic sphere of radius $r_{\infty}$ determined by $\widetilde{W}_l(B(r_{\infty}))=\widetilde{W}_l(\Omega_0)$ as $t\to\infty$.
\end{thm}
Constrained curvature flows in hyperbolic space by homogeneous of degree one, concave and inverse concave function of the principal curvatures were studied by Makowski \cite{Mak2012} and Wang and Xia in \cite{WX}. The quermassintegral preserving flow by any positive power of a homogeneous of degree one function of the principal curvatures, which is inverse concave and its dual function $f_*$ approaches zero on the boundary of positive cone $\Gamma_+$, was studied recently by the first and the third authors in \cite{And-Wei2017-2}. Note that the speed function $f$ of the flow \eqref{flow-VMCF-2} in Theorem \ref{thm1-5} is not a homogeneous function of the principal curvatures $\kappa_i$ and there are essential differences in the analysis compared with the previously mentioned work \cite{And-Wei2017-2,Mak2012,WX}.
The key step in the proof of Theorem \ref{thm1-5} is a pinching estimate for the shifted principal curvatures $\lambda_i$. That is, we will show that the ratio of the largest shifted principal curvature $\lambda_n$ to the smallest shifted principal curvature $\lambda_1$ is controlled by its initial value along the flow \eqref{flow-VMCF-2}. For the proof, we adapt methods from the proof of pinching estimates of the principal curvatures for contracting curvature flows \cite{And1994,And2007,And2010,Andrews-McCoy-Zheng} and the constrained curvature flows in Euclidean space \cite{McC2005,Mcc2017}. In particular, in the case (iii) we define the tensor $T_{ij}=S_{ij}-\varepsilon F\delta_i^j$ and show that the positivity of $T_{ij}$ is preserved by applying the tensor maximum principle (proved by the first author in \cite{And2007}). The inverse concavity is used to estimate the sign of the gradient terms. This case is similar with the pinching estimate for the contracting curvature flow in Euclidean case in \cite[Lemma 11]{Andrews-McCoy-Zheng}. Although the proof there is given in terms of the Gauss map parametrisation of the convex solutions of the flow in Euclidean space, which is not available in hyperbolic space, we can deal with the gradient terms directly using the inverse concavity property of $f$.
To prove Theorem \ref{thm1-5}, we next show that the inner radius and outer radius of the enclosed domain $\Omega_t$ of the evolving hypersurface $M_t$ satisfies a uniform estimate $0<C^{-1}<\rho_-(t)\leq \rho_+(t)\leq C$ for some positive constant $C$. This relies on the preservation of $\widetilde{W}_l(\Omega_t)$ and the monotonicity of $\widetilde{W}_l$ under the inclusion of h-convex domains stated in Proposition \ref{s1:wk-prop}. With the estimate on the inner radius and outer radius, the technique of Tso \cite{Tso85} yields the upper bound on $F$ and the Harnack inequality of Krylov and Safonov \cite{KS81} yields the lower bound on $F$. The pinching estimate then gives the estimate on the shifted principal curvatures $\lambda_i$. The long time existence and the convergence of the flow then follows by a standard argument.
The result in Theorem \ref{thm1-5} is useful in the study of the geometry of hypersurfaces. The first application of Theorem \ref{thm1-5} is the following rigidity result.
\begin{cor}\label{s1:cor-2}
Let $M$ be a smooth, closed and strictly h-convex hypersurface in $\mathbb{H}^{n+1}$ with principal curvatures $\kappa=(\kappa_1,\cdots,\kappa_n)$ satisfying $f(\lambda)=C$
for some constant $C>0$, where $\lambda=(\lambda_1,\cdots,\lambda_i)$ with $\lambda_i=\kappa_i-1$ and $f$ is a symmetric function satisfying the condition in Theorem \ref{thm1-5}. Then $M$ is a geodesic sphere.
\end{cor}
The second application of Theorem \ref{thm1-5} is a new class of Alexandrov-Fenchel type inequalities between quermassintegrals of h-convex hypersurface in hyperbolic space.
\begin{cor}\label{s1:cor-3}
Let $M=\partial\Omega$ be a smooth, closed and strictly h-convex hypersurface in $\mathbb{H}^{n+1}$. Then for any $0\leq l<k\leq n$, there holds
\begin{equation}\label{s1:AF}
\widetilde{W}_k(\Omega)~\geq~ \tilde{f}_k\circ \tilde{f}_l^{-1}(\widetilde{W}_l(\Omega)),
\end{equation}
with equality holding if and only if $\Omega$ is a geodesic ball. Here the function $\tilde{f}_k: \mathbb{R}_+\to \mathbb{R}_+$ is defined by $\tilde{f}_k(r)=\widetilde{W}_k(B(r))$, which is an increasing function by Proposition \ref{s1:wk-prop}. $\tilde{f}_l^{-1}$ is the inverse function of $\tilde{f}_l$.
\end{cor}
The inequality \eqref{s1:AF} can be obtained by applying Theorem \ref{thm1-5} with $f$ chosen as
\begin{equation}\label{s1:F}
f=\left(\frac{E_k(\lambda)}{E_l(\lambda)}\right)^{\frac 1{k-l}},\quad 0\leq l<k\leq n
\end{equation}
in the flow \eqref{flow-VMCF-2}. We see that along the flow \eqref{flow-VMCF-2} with such $f$, the modified quermassintegral $\widetilde{W}_l(\Omega_t)$ remains a constant and $\widetilde{W}_k(\Omega_t)$ is monotone decreasing in time by the H\"{o}lder inequality. In fact, by Lemma \ref{s2:lem3} the modified quermassintegral evolves by
\begin{align}\label{s1:eq2}
\frac d{dt}\widetilde{W}_k(\Omega_t)~=&~ \int_{M_t}E_k(\lambda)\left(\phi(t)-\left(\frac{E_k(\lambda)}{E_l(\lambda)}\right)^{\frac 1{k-l}}\right)d\mu_t.
\end{align}
Applying the H\"{o}lder inequality to the equation \eqref{s1:eq2} yields that $\widetilde{W}_k(\Omega_t)$ is monotone decreasing in time unless $E_k(\lambda)=C E_l(\lambda)$ on $M_t$ (which is equivalent to that $M_t$ is a geodesic sphere by Corollary \ref{s1:cor-2}). Since the flow exists for all time and converges to a geodesic sphere $B_r$, the inequality \eqref{s1:AF} follows from the monotonicity of $\widetilde{W}_k(\Omega_t)$ and the preservation of $\widetilde{W}_l(\Omega_t)$.
\begin{rem}
We remark that the inequalities \eqref{s1:AF} are new and can be viewed as an improvement of the inequalities \eqref{s1:AF-WX}. For example, the inequality \eqref{s1:AF} with $l=0$ implies that
\begin{equation}\label{s1:AF-rem1}
\sum_{i=0}^k(-1)^{k-i}\binom ki\biggl(W_i(\Omega)-f_i\circ f_0^{-1}(W_0(\Omega))\biggr)~\geq~0.
\end{equation}
By induction on $k$, \eqref{s1:AF-rem1} implies that each $W_i(\Omega)-f_i\circ f_0^{-1}(W_0(\Omega))$ is nonnegative for $h$-convex domains. Thus our inequalities \eqref{s1:AF} imply that the linear combinations of $W_i(\Omega)-f_i\circ f_0^{-1}(W_0(\Omega))$ as in \eqref{s1:AF-rem1} are also nonnegative for h-convex domains.
\end{rem}
\begin{ack}
The second author is grateful to the Mathematical Sciences Institute at the Australian National University for its hospitality during his visit, when some of this work was completed.
\end{ack}
\section{Preliminaries}\label{sec:pre}
In this section we collect some properties of smooth symmetric functions $f$ of $n$ variables, and recall the evolution equations of geometric quantities along the flows \eqref{flow-VMCF} and \eqref{flow-VMCF-2}.
\subsection{Properties of symmetric functions}
For a smooth symmetric function $F(A)=f(\kappa(A))$, where $A=(A_{ij})\in \mathrm{Sym}(n)$ is symmetric matrix and $\kappa(A)=(\kappa_1,\cdots,\kappa_n)$ give the eigenvalues of $A$, we denote by $\dot{F}^{ij}$ and $\ddot{F}^{ij,kl}$ the first and second derivatives of $F$ with respect to the components of its argument, so that
\begin{equation*}
\frac{\partial}{\partial s}F(A+sB)\bigg|_{s=0}=\dot{F}^{ij}(A)B_{ij}
\end{equation*}
and
\begin{equation*}
\frac{\partial^2}{\partial s^2}F(A+sB)\bigg|_{s=0}=\ddot{F}^{ij,kl}(A)B_{ij}B_{kl}
\end{equation*}
for any two symmetric matrixs $A,B$. We also use the notation
\begin{equation*}
\dot{f}^i(\kappa)=\frac{\partial f}{\partial \kappa_i}(\kappa),\quad \ddot{f}^{ij}(\kappa)=\frac{\partial^2 f}{\partial \kappa_i\partial \kappa_j}(\kappa).
\end{equation*}
for the first and the second derivatives of $f$ with respect to $\kappa$. At any diagonal $A$ with distinct eigenvalues $\kappa=\kappa(A)$, the first derivatives of $F$ satisfy
\begin{equation*}
\dot{F}^{ij}(A)~=~\dot{f}^i(\kappa)\delta_{ij}
\end{equation*}
and the second derivative of $F$ in direction $B\in \mathrm{Sym}(n)$ is given in terms of $\dot{f}$ and $\ddot{f}$ by (see \cite{And2007}):
\begin{equation}\label{s2:F-ddt}
\ddot{F}^{ij,kl}(A)B_{ij}B_{kl}=\sum_{i,j}\ddot{f}^{ij}(\kappa)B_{ii}B_{jj}+2\sum_{i>j}\frac{\dot{f}^i(\kappa)-\dot{f}^j(\kappa)}{\kappa_i-\kappa_j}B_{ij}^2.
\end{equation}
This formula makes sense as a limit in the case of any repeated values of $\kappa_i$.
From the equation \eqref{s2:F-ddt}, we have
\begin{lem}\label{s2:lem0}
Suppose $A$ has distinct eigenvalues $\kappa=\kappa(A)$. Then $F$ is concave at $A$ if and only if $f$ is concave at $\kappa$ and
\begin{equation}\label{s2:f-conc}
\left(\dot{f}^k-\dot{f}^l\right)(\kappa_k-\kappa_l)~\leq ~0,\quad \forall~k\neq l.
\end{equation}
\end{lem}
In this paper, we also need the inverse concavity property of $f$ in many cases. We include the properties of inverse concave function in the following lemma.
\begin{lem}[\cite{And2007,And-Wei2017-2}]\label{s2:lem1}
\begin{itemize}
\item[(i)]If $f$ is inverse concave, then
\begin{equation}\label{s2:f-invcon-1}
\sum_{k,l=1}^n\ddot{f}^{kl}y_ky_l+2\sum_{k=1}^n\frac {\dot{f}^k}{\kappa_k}y_k^2~\geq ~2f^{-1}(\sum_{k=1}^n\dot{f}^ky_k)^2
\end{equation}
for any $y=(y_1,\cdots,y_n)\in \mathbb{R}^n$, and
\begin{equation}\label{s2:f-invcon}
\frac{\dot{f}^k-\dot{f}^l}{\kappa_k-\kappa_l}+\frac{\dot{f}^k}{\kappa_l}+\frac{\dot{f}^l}{\kappa_k}\geq~0,\quad \forall~k\neq l.
\end{equation}
\item[(ii)] If $f=f(\kappa_1,\cdots,\kappa_n)$ is inverse concave, then
\begin{equation}\label{s2:f-invcon-0}
\sum_{i=1}^n\dot{f}^i\kappa_i^2~\geq~f^2.
\end{equation}
\end{itemize}
\end{lem}
\subsection{Evolution equations}
Along any smooth flow
\begin{equation}\label{s2:flow}
\frac{\partial}{\partial t}X(x,t)=~\varphi(x,t)\nu(x,t)
\end{equation}
of hypersurfaces in hyperbolic space $\mathbb{H}^{n+1}$, where $\varphi$ is a smooth function on the evolving hypersurfaces $M_t=X(M^n,t)$, we have the following evolution equations on the induced metric $g_{ij}$, the induced area element $d\mu_t$ and the Weingarten matrix $\mathcal{W}=(h_i^j)$ of $M_t$:
\begin{align
\frac{\partial}{\partial t}g_{ij} =&~ 2\varphi h_{ij} \label{evl-g}\\
\frac{\partial}{\partial t}d\mu_t=& ~nE_1 \varphi d\mu_t\label{evl-dmu}\\
\frac{\partial}{\partial t}h_i^j=&~-\nabla^j\nabla_i\varphi-\varphi (h_i^kh_k^j-\delta_i^j)\label{evl-h}
\end{align}
From the evolution equations \eqref{evl-dmu} and \eqref{evl-h}, we can derive the evolution equation of the curvature integral $V_{n-k}$:
\begin{align}\label{s2:evl-Vk}
\frac d{dt}V_{n-k}(\Omega_t)=&\frac d{dt}\int_{M_t}E_kd\mu_t\nonumber\\
=&\int_{M_t}\left(\frac{\partial}{\partial t} E_k+nE_1E_k\varphi\right)d\mu_t\nonumber\\
=& \int_{M_t}\left(-\frac{\partial E_k}{\partial h_i^j}\nabla^j\nabla_i\varphi -\varphi \frac{\partial E_k}{\partial h_i^j}(h_i^kh_k^j-\delta_i^j)+nE_1E_k\varphi\right)d\mu_t\displaybreak[0]\nonumber\\
= & \int_{M_t}\varphi \biggl((n-k)E_{k+1}+kE_{k-1}\biggr)d\mu_t,
\end{align}
where we used integration by part and the fact that $\frac{\partial E_k}{\partial h_i^j}$ is divergence free. Since the quermassintegrals are related to the curvature integrals by the equations \eqref{s1:quermass-1} and \eqref{s1:quermass-2}, applying an induction argument to the equation \eqref{s2:evl-Vk} yields that
\begin{lem}[cf.\cite{And-Wei2017-2,WX}]\label{s2:lem2}
Along the flow \eqref{s2:flow}, the quermassintegral $W_k$ of the evolving domain $\Omega_t$ satisfies
\begin{equation*
\frac d{dt}W_k(\Omega_t)~=~\int_{M_t}E_k(\kappa)\varphi d\mu_t,\quad k=0,\cdots,n.
\end{equation*}
\end{lem}
We can also derive the following evolution equation for the modified quermassintegrals.
\begin{lem}\label{s2:lem3}
Along the flow \eqref{s2:flow}, the modified quermassintegral $\widetilde{W}_k$ of the evolving domain $\Omega_t$ satisfies
\begin{equation*
\frac d{dt}\widetilde{W}_k(\Omega_t)~=~\int_{M_t}E_k(\lambda)\varphi d\mu_t,\quad k=0,\cdots,n,
\end{equation*}
where $\lambda=(\lambda_1,\cdots,\lambda_n)=(\kappa_1-1,\cdots,\kappa_n-1)$ are the shifted principal curvatures of $M_t$.
\end{lem}
\proof
Firstly, we derive the formula for $\sigma_{k}(\lambda)$ in terms of $\sigma_i(\kappa), i=0,\cdots,k$. By the definition of the elementary symmetric polynomial, we have
\begin{equation*
\prod_{i=1}^n(t+\lambda_i)~=\sum_{k=0}^n\sigma_k(\lambda)t^{n-k}.
\end{equation*}
On the other hand,
\begin{align*
\prod_{i=1}^n(t+\lambda_i) =&~\prod_{i=1}^n(t-1+\kappa_i) ~=~ \sum_{l=0}^n\sigma_l(\kappa)(t-1)^{n-l} \nonumber\\
= &~ \sum_{l=0}^n\sigma_l(\kappa)\sum_{i=0}^{n-l}\binom{n-l}it^i(-1)^{n-l-i} \nonumber\\
=&~\sum_{k=0}^n\left(\sum_{i=0}^k\binom{n-i}{k-i}(-1)^{k-i}\sigma_i(\kappa)\right)t^{n-k}.
\end{align*}
Comparing the coefficients of $t^{n-k}$, we have
\begin{align*}
\sigma_k(\lambda)~=&~\sum_{i=0}^k\binom{n-i}{k-i}(-1)^{k-i}\sigma_i(\kappa)\nonumber\\
=&~\sum_{i=0}^k\binom{n-i}{k-i}(-1)^{k-i}\binom{n}{i}E_i(\kappa)\nonumber\\
=&~\binom{n}k\sum_{i=0}^k(-1)^{k-i}\binom{k}{i}E_i(\kappa).
\end{align*}
Equivalently, we have
\begin{equation}\label{s2:sk-3}
E_k(\lambda)~=~{\binom{n}{k}}^{-1}\sigma_k(\lambda)~=~\sum_{i=0}^k(-1)^{k-i}\binom{k}{i}E_i(\kappa).
\end{equation}
Then by the definition \eqref{s1:Wk-td} of $\widetilde{W}_k$ and Lemma \ref{s2:lem2},
\begin{align*
\frac d{dt}\widetilde{W}_k(\Omega_t)~=&\sum_{i=0}^k(-1)^{k-i}\binom{k}{i}\frac d{dt}W_i(\Omega_t)\nonumber\displaybreak[0]\\
=&\sum_{i=0}^k(-1)^{k-i}\binom{k}{i}\int_{M_t}E_i(\kappa) \varphi d\mu_t ~=~ \int_{M_t}E_k(\lambda)\varphi d\mu_t.
\end{align*}
\endproof
If we consider the flow \eqref{flow-VMCF}, i.e., $\varphi=\phi(t)-\Psi(\mathcal{W})$, using \eqref{evl-h} and the Simons' identity we have the evolution equations for the curvature function $\Psi=\Psi(\mathcal{W})$ and the Weingarten matrix $\mathcal{W}=(h_i^j)$ of $M_t$ (see \cite{And-Wei2017-2}):
\begin{equation}\label{s2:evl-Psi}
\frac{\partial}{\partial t}\Psi=~\dot{\Psi}^{kl}\nabla_k\nabla_l\Psi+(\Psi-\phi(t))(\dot{\Psi}^{ij}h_i^kh_k^j-\dot{\Psi}^{ij}\delta_i^j),
\end{equation}
and
\begin{align}\label{s2:evl-h}
\frac{\partial}{\partial t}h_i^j =& \dot{\Psi}^{kl}\nabla_k\nabla_lh_i^j+\ddot{\Psi}^{kl,pq}\nabla_ih_{kl}\nabla^jh_{pq} +(\dot{\Psi}^{kl}h_k^rh_{rl}+\dot{\Psi}^{kl}g_{kl})h_i^j\nonumber\\
&\quad -\dot{\Psi}^{kl}h_{kl}(h_i^ph_{p}^j+\delta_i^j)+(\Psi-\phi(t))(h_i^kh_{k}^j-\delta_i^j),
\end{align}
where $\nabla$ denotes the Levi-Civita connection with respect to the induced metric $g_{ij}$ on $M_t$, and $\dot{\Psi}^{kl}, \ddot{\Psi}^{kl,pq}$ denote the derivatives of $\Psi$ with respect to the components of the Weingarten matrix $\mathcal{W}=(h_i^j)$.
If we consider the flow \eqref{flow-VMCF-2} of h-convex hypersurfaces, i.e., $\varphi=\phi(t)-F(\mathcal{W}-\mathrm{I})$, we have the similar evolution equation for the curvature function $F$
\begin{align}\label{s2:evl-F}
\frac{\partial}{\partial t}F=&~\dot{F}^{kl}\nabla_k\nabla_lF+(F-\phi(t))(\dot{F}^{ij}h_i^kh_k^j-\dot{F}^{ij}\delta_i^j),
\end{align}
and a parabolic type equation for the Weingarten matrix $\mathcal{W}=(h_i^j)$ of $M_t$:
\begin{align}\label{s2:evl-h2}
\frac{\partial}{\partial t}h_i^j =& \dot{F}^{kl}\nabla_k\nabla_lh_i^j+\ddot{F}^{kl,pq}\nabla_ih_{kl}\nabla^jh_{pq} +(\dot{F}^{kl}h_k^rh_{rl}+\dot{F}^{kl}g_{kl})h_i^j\nonumber\\
&\quad -\dot{F}^{kl}h_{kl}(h_i^ph_{p}^j+\delta_i^j)+(F-\phi(t))(h_i^kh_{k}^j-\delta_i^j).
\end{align}
However, we observe that $\dot{F}^{kl}, \ddot{F}^{kl,pq}$ in \eqref{s2:evl-F} and \eqref{s2:evl-h2} denote the derivatives of $F$ with respect to the components of shifted Weingarten matrix $\mathcal{W}-\mathrm{I}$, so the homogeneity of $F$ implies that $\dot{F}^{kl}(h_{k}^l-\delta_k^l)=F$. Denote the components of the shifted Weingarten matrix by $S_{ij}=h_i^j-\delta_i^j$. Then the equation \eqref{s2:evl-h2} implies that
\begin{align}\label{s2:evl-S}
\frac{\partial}{\partial t}S_{ij} =&~ \dot{F}^{kl}\nabla_k\nabla_lS_{ij}+\ddot{F}^{kl,pq}\nabla_ih_{kl}\nabla^jh_{pq} +(\dot{F}^{kl}S_{kr}S_{rl}+2F-2\phi(t))S_{ij}\nonumber\\
&\quad -(\phi(t)+\dot{F}^{kl}\delta_k^l) S_{ik}S_{kj}+\dot{F}^{kl}S_{kr}S_{rl}\delta_i^j.
\end{align}
\subsection{A generalised maximum principle}
In \S \ref{sec:Pinch}, we will use the tensor maximum principle to prove that the pinching estimate along the flow \eqref{flow-VMCF-2}. For the convenience of readers, we include here the statement of the tensor maximum principle, which was first proved by Hamilton \cite{Ha1982} and was generalized by the first author in \cite{And2007}.
\begin{thm}[\cite{And2007}]\label{s2:tensor-mp}
Let $S_{ij}$ be a smooth time-varying symmetric tensor field on a compact manifold $M$, satisfying
\begin{equation*}
\frac{\partial}{\partial t}S_{ij}=a^{kl}\nabla_k\nabla_lS_{ij}+u^k\nabla_kS_{ij}+N_{ij},
\end{equation*}
where $a^{kl}$ and $u$ are smooth, $\nabla$ is a (possibly time-dependent) smooth symmetric connection, and $a^{kl}$ is positive definite everywhere. Suppose that
\begin{equation}\label{s2:TM2}
N_{ij}v^iv^j+\sup_{\Lambda}2a^{kl}\left(2\Lambda_k^p\nabla_lS_{ip}v^i-\Lambda_k^p\Lambda_l^qS_{pq}\right)\geq 0
\end{equation}
whenever $S_{ij}\geq 0$ and $S_{ij}v^j=0$. If $S_{ij}$ is positive definite everywhere on $M$ at $t=0$ and on $\partial M$ for $0\leq t\leq T$, then it is positive on $M\times[0,T]$.
\end{thm}
\section{Preserving positive sectional curvature}\label{sec:PSC}
In this section, we will prove that the flow \eqref{flow-VMCF} preserves the positivity of sectional curvatures, if $\alpha>0$ and $f$ satisfies Assumption \ref{s1:Asum}.
\begin{thm}\label{s3:thm-2}
If the initial hypersurface $M_0$ has positive sectional curvature, then along the flow \eqref{flow-VMCF} in $\mathbb{H}^{n+1}$ with $f$ satisfying Assumption \ref{s1:Asum} and any power $\alpha>0$ the evolving hypersurface $M_t$ has positive sectional curvature for $t>0$.
\end{thm}
\proof
The sectional curvature defines a smooth function on the Grassmannian bundle of two-dimensional subspaces of $TM$. For convenience we lift this to a function on the orthonormal frame bundle $O(M)$ over $M$: Given
a point $x\in M$ and $t\geq 0$, and a frame ${\mathbb O} = \{e_1,\cdots,e_n\}$ for $T_xM$ which is orthonormal with respect to $g(x,t)$, we define
$$
G(x,t,{\mathbb O}) = h_{(x,t)}(e_1,e_1)h_{(x,t)}(e_2,e_2)-h_{(x,t)}(e_1,e_2)^2-1.
$$
We consider a point $(x_0,t_0)$ and a frame ${\mathbb O}_0 = \{\bar e_1,\cdots,\bar e_n\}$ at which a new minimum of the function $G$ is attained, so that we have $G(x,t,{\mathbb O})\geq G(x_0,t_0,{\mathbb O}_0)$ for all $x\in M$ and all $t\in[0,t_0]$, and all ${\mathbb O}\in F(M)_{(x,t)}$. The fact that ${\mathbb O}_0$ achieves the minimum of $G$ over the fibre $F(M)_{(x_0,t_0)}$ implies that $e_1$ and $e_2$ are eigenvectors of $h_{(x_0.t_0)}$ corresponding to $\kappa_1$ and $\kappa_2$, where $\kappa_1\leq\kappa_2\leq\cdots\leq\kappa_n$ are the principal curvatures at $(x_0,t_0)$. Since $G$ is invariant under rotation in the subspace orthogonal to $\bar e_1$ and $\bar e_2$, we can assume that $h(\bar e_i,\bar e_i)=\kappa_i$ and $h(\bar e_i,\bar e_j)=0$ for $i\neq j$.
The time derivative of $G$ at $(x_0,t_0,{\mathbb O}_0)$ is given by Equation \eqref{s2:evl-h}, noting that the frame ${\mathbb O}(t)$ for $T_xM$ defined by $\frac{d}{dt}e_i(t) = (F^\alpha-\phi){\mathcal W}(e_i)$ remains orthonormal with respect to $g(x,t)$ if $e_i(t_0)=\bar e_i$ for each $i$. This yields the following:
\begin{align}\label{eq:dGdt}
\frac{\partial}{\partial t}G|_{(x_0,t_0,{\mathbb O}_0)}
&= \kappa_1\frac{\partial}{\partial t}h_{2}^{2}+\kappa_2\frac{\partial}{\partial_t}h_{1}^{1}\notag\\
&=\kappa_1\dot\Psi^{kl}\nabla_k\nabla_lh_{22}+\kappa_2\dot\Psi^{kl}\nabla_k\nabla_lh_{11}+
\kappa_1\ddot\Psi(\nabla_2h,\nabla_2h)+\kappa_2\ddot\Psi(\nabla_1h,\nabla_1h)\notag\\
&\quad\null+2\left(\dot{\Psi}^{kl}h_k^rh_{rl}+\dot{\Psi}^{kl}g_{kl}\right)\kappa_1\kappa_2-(\alpha-1)\Psi \kappa_1\kappa_2(\kappa_1+\kappa_2)\notag\\
&\quad\null-(\alpha+1)\Psi(\kappa_1+\kappa_2)-\phi(t)(\kappa_1\kappa_2-1)(\kappa_1+\kappa_2).
\end{align}
Since $\Psi=f^{\alpha}$, we have the following:
\begin{align*}
2&\left(\dot{\Psi}^{kl}h_k^rh_{rl}+\dot{\Psi}^{kl}g_{kl}\right)\kappa_1\kappa_2-(\alpha-1)\Psi \kappa_1\kappa_2(\kappa_1+\kappa_2)-(\alpha+1)\Psi(\kappa_1+\kappa_2)-\phi(t)(\kappa_1\kappa_2-1)(\kappa_1+\kappa_2)\\
&= 2\alpha f^{\alpha-1}\sum_k\dot f^k(\kappa_k-\kappa_2)(\kappa_k-\kappa_1)+G\left(f^{\alpha-1}\sum_k\dot f^k\kappa_k(2\alpha\kappa_k-(\alpha-1)(\kappa_1+\kappa_2))-\phi(t)(\kappa_1+\kappa_2)
\right)\\
&\geq -CG,
\end{align*}
where $C$ is a bound for the smooth function in the last bracket.
To estimate the remaining terms, we consider the second derivatives of $G$ along a curve on $O(M)$ defined as follows: We let $\gamma$ be any geodesic of $g(t_0)$ in $M$ with $\gamma(0)=x_0$, and define a frame ${\mathbb O}(s)=(e_1(s),\cdots,e_n(s))$ at $\gamma(s)$ by taking $e_i(0)=\bar e_i$ for each $i$, and $\nabla_se_i(s) = \Gamma_{ij}e_j(s)$ for some constant antisymmetric matrix $\Gamma$. Then we compute
\begin{align}\label{eq:d2Gds2}
\frac{d^2}{ds^2}G(x(s),t_0,{\mathbb O}(s))\Big|_{s=0}
&=\kappa_2\nabla^2_sh_{11}+\kappa_1\nabla^2_sh_{22}+2\left(\nabla_sh_{22}\nabla_sh_{11}-(\nabla_sh_{12})^2\right)\notag\\
&\quad\null +4\sum_{p>2}\Gamma_{1p}\kappa_2\nabla_sh_{1p}+4\sum_{p>2}\Gamma_{2p}\kappa_1\nabla_sh_{2p}\notag\\
&\quad\null +2\sum_{p>2}\Gamma_{1p}^2\kappa_2(\kappa_p-\kappa_1)+2\sum_{p>2}\Gamma_{2p}^2\kappa_1(\kappa_p-\kappa_2).
\end{align}
Since $G$ has a minimum at $(x_0,t_0,{\mathbb O}_0)$, the right-hand side of \eqref{eq:d2Gds2} is non-negative for any choice of $\Gamma$. Minimizing over $\Gamma$ gives
\begin{align}\label{eq:D2G}
0&\leq \kappa_2\nabla^2_sh_{11}+\kappa_1\nabla^2_sh_{22}+2\left(\nabla_sh_{22}\nabla_sh_{11}-(\nabla_sh_{12})^2\right)\notag\\
&\quad\null -2\sum_{p>2}\frac{\kappa_2}{\kappa_p-\kappa_1}(\nabla_sh_{1p})^2
-2\sum_{p>2}\frac{\kappa_1}{\kappa_p-\kappa_2}(\nabla_sh_{2p})^2
\end{align}
where we terms on the last line as vanishing if the denominators vanish (since the corresponding component of $\nabla h$ vanishes in that case). This gives
\begin{align}\label{eq:dtGineq}
\frac{\partial}{\partial t}G|_{(x_0,t_0,{\mathbb O}_0)}
&\geq \kappa_1\ddot\Psi(\nabla_2h,\nabla_2h)+\kappa_2\ddot\Psi(\nabla_1h,\nabla_1h)-2\sum_k\dot\Psi^k\left(\nabla_kh_{22}\nabla_kh_{11}-(\nabla_kh_{12})^2\right)\notag\\
&\quad\null +2\sum_k\dot\Psi^k\left(\sum_{p>2}\frac{\kappa_2}{\kappa_p-\kappa_1}(\nabla_kh_{1p})^2
+\sum_{p>2}\frac{\kappa_1}{\kappa_p-\kappa_2}(\nabla_kh_{2p})^2\right)-CG
\end{align}
The right-hand side can be expanded using $\Psi=f^\alpha$ and the identity \eqref{s2:F-ddt}:
\begin{align*}
\frac{f^{1-\alpha}}{\alpha}\left(\frac{d}{dt}G+CG\right)
&\geq
\kappa_2\left( \sum_{k,l}\ddot{f}^{kl}\nabla_1h_{kk}\nabla_1h_{ll}+(\alpha-1)\frac{(\nabla_1f)^2}{f}+\sum_{k\neq l}\frac{\dot{f}^k-\dot{f}^l}{\kappa_k-\kappa_l}(\nabla_1h_{kl})^2
\right)\\
&\quad\null +\kappa_1\left(\sum_{k,l} \ddot{f}^{kl}\nabla_2h_{kk}\nabla_2h_{ll}+(\alpha-1)\frac{(\nabla_2f)^2}{f}+\sum_{k\neq l}\frac{\dot{f}^k-\dot{f}^l}{\kappa_k-\kappa_l}(\nabla_2h_{kl})^2
\right)\\
&\quad\null -2\sum_k\dot f^k\left(\nabla_kh_{22}\nabla_kh_{11}-(\nabla_kh_{12})^2\right)\\
&\quad\null+2\sum_k\dot f^k\left(\sum_{p>2}\frac{\kappa_2}{\kappa_p-\kappa_1}(\nabla_kh_{1p})^2
+\sum_{p>2}\frac{\kappa_1}{\kappa_p-\kappa_2}(\nabla_kh_{2p})^2\right).
\end{align*}
Note that by assumption the function $f$ satisfies the inequalities \eqref{s1:asum-1} and \eqref{s1:asum-2}. By the inequality \eqref{s1:asum-1}, for any $k\neq l$ we have
\begin{align*
\frac{\dot{f}^k-\dot{f}^l}{\kappa_k-\kappa_l}+\frac{\dot{f}^k}{\kappa_l}=&~\frac{\dot{f}^k\kappa_k-\dot{f}^l\kappa_l}{(\kappa_k-\kappa_l)\kappa_l}~\geq 0
\end{align*}
The inequality \eqref{s1:asum-2} is equivalent to
\begin{equation*
\sum_{k,l}\ddot{f}^{kl}y_ky_l\geq~f^{-1}(\sum_{k=1}^n\dot{f}^ky_k)^2-\sum_{k=1}^n\frac{\dot{f}^k}{\kappa_k}y_k^2
\end{equation*}
for all $(y_1,\cdots,y_n)\in \mathbb{R}^n$.
These imply that
\begin{align}\label{s3:Q1}
\frac{f^{1-\alpha}}{\alpha}\left(\frac{dG}{dt}+CG\right) &\geq
\kappa_2\left( \alpha \frac{(\nabla_1f)^2}{f}-\sum_{k=1}^n\frac{\dot{f}^k}{\kappa_k}(\nabla_1h_{kk})^2-\sum_{k\neq l}\frac{\dot{f}^k}{\kappa_l}(\nabla_1h_{kl})^2\right)\nonumber\\
&\quad\null+\kappa_1\left( \alpha \frac{(\nabla_2f)^2}{f}-\sum_{k=1}^n\frac{\dot{f}^k}{\kappa_k}(\nabla_2h_{kk})^2-\sum_{k\neq l}\frac{\dot{f}^k}{\kappa_l}(\nabla_2h_{kl})^2\right)\nonumber\displaybreak[0]\\
&\quad+2\sum_k\dot{f}^k\left(-\nabla_kh_{11}\nabla_kh_{22}+(\nabla_kh_{12})^2\right)\nonumber\\
&\quad+2\sum_{k=1}^n\sum_{p>2}\frac{\dot{f}^k}{\kappa_p}\left(\kappa_2(\nabla_1h_{kp})^2+\kappa_1(\nabla_2h_{kp})^2\right)\nonumber\\
=&~\kappa_2\alpha\frac{(\nabla_1f)^2}{f}+\kappa_1\alpha\frac{(\nabla_2f)^2}{f}+\sum_{k,p=3}^n\frac{\dot{f}^k}{\kappa_p}\left(\kappa_2(\nabla_1h_{kp})^2+\kappa_1(\nabla_2h_{kp})^2\right)\nonumber\\
&-\kappa_2\left(\frac{\dot{f}^1}{\kappa_1}(\nabla_1h_{11})^2+\frac{\dot{f}^2}{\kappa_2}(\nabla_1h_{22})^2+\frac{\dot{f}^1}{\kappa_2}(\nabla_1h_{12})^2+\frac{\dot{f}^2}{\kappa_1}(\nabla_1h_{21})^2\right)\displaybreak[0]\nonumber\\
&-\kappa_1\left(\frac{\dot{f}^1}{\kappa_1}(\nabla_2h_{11})^2+\frac{\dot{f}^2}{\kappa_2}(\nabla_2h_{22})^2+\frac{\dot{f}^1}{\kappa_2}(\nabla_2h_{12})^2+\frac{\dot{f}^2}{\kappa_1}(\nabla_2h_{21})^2\right)\nonumber\\
&+2\dot{f}^1\left(-\nabla_1h_{11}\nabla_1h_{22}+(\nabla_1h_{12})^2\right)+2\dot{f}^2\left(-\nabla_2h_{11}\nabla_2h_{22}+(\nabla_2h_{12})^2\right)\nonumber\\
&+2\dot f^1\sum_{p>2}\left(\frac{\kappa_2}{\kappa_p}(\nabla_1h_{1p})^2+\frac{\kappa_1}{\kappa_p}(\nabla_2h_{1p})^2\right)\nonumber\\
&+2\dot f^2\sum_{p>2}\left(\frac{\kappa_2}{\kappa_p}(\nabla_1h_{2p})^2+\frac{\kappa_1}{\kappa_p}(\nabla_2h_{2p})^2\right)\displaybreak[0]\nonumber\\
&+2\sum_{k>2}\dot{f}^k\left(-\nabla_kh_{11}\nabla_kh_{22}+(\nabla_kh_{12})^2\right).
\end{align}
Since $(x_0,{\mathbb O}_0)$ is a minimum point of $G$ at time $t_0$, we have $\nabla_iG=0$ for $i=1,\cdots,n$, so
\begin{equation}\label{s3:nab-G}
\kappa_2\nabla_ih_{11}+\kappa_1\nabla_ih_{22}=0,\qquad i=1,\cdots,n.
\end{equation}
Substituting \eqref{s3:nab-G} into \eqref{s3:Q1}, the second to the fourth lines on the right of \eqref{s3:Q1} vanish,
the last line becomes $2\sum_{k>2}\dot{f}^k\left(\frac{\kappa_1}{\kappa_2}(\nabla_kh_{22})^2+(\nabla_kh_{12})^2\right)\geq 0$, and the remaining terms are non-negative.
We conclude that $\frac{\partial}{\partial t}G\geq -CG$ at a spatial minimum point, and hence by the maximum principle \cite{Ha1986}*{Lemma 3.5} we have $G\geq {\mathrm e}^{-Ct}\inf_{t=0}G>0$ under the flow \eqref{flow-VMCF}.
\endproof
\section{Proof of Theorem \ref{thm1-1}}\label{sec:thm1-pf}
In this section, we will give the proof of Theorem \ref{thm1-1}.
\subsection{Shape estimate}\label{sec:4-1}
First, we show that the preservation of the volume of $\Omega_t$, together with a reflection argument, implies that the inner radius and outer radius of $\Omega_t$ are uniformly bounded from above and below by positive constants.
\begin{lem}
Denote $\rho_-(t), \rho_+(t)$ be the inner radius and outer radius of $\Omega_t$, the domain enclosed by $M_t$. Then there exist positive constants $c_1,c_2$ depending only on $n,M_0$ such that
\begin{equation}\label{s4:io-radius1}
0<c_1\leq \rho_-(t)\leq \rho_+(t)\leq c_2
\end{equation}
for all time $t\in [0,T)$.
\end{lem}
\proof
We first use the Alexandrov reflection method to estimate the diameter of $\Omega_t$. In \cite{And-Wei2017-2}, the first and the third authors have already used the Alexandrov reflection method in the proof of convergence of the flow. Let $\gamma$ be a geodesic line in $\mathbb{H}^{n+1}$, and let $H_{\gamma(s)}$ be the totally geodesic hyperbolic $n$-plane in $\mathbb{H}^{n+1}$ which is perpendicular to $\gamma$ at $\gamma(s), s\in \mathbb{R}$. We use the notation $H_{s}^+$ and $H_s^-$ for the half-spaces in $\mathbb{H}^{n+1}$ determined by $H_{\gamma(s)}$:
\begin{equation*}
H_s^+:=\bigcup_{s'\geq s}H_{\gamma(s')},\qquad H_s^-:=\bigcup_{s'\leq s}H_{\gamma(s')}
\end{equation*}
For a bounded domain $\Omega$ in $\mathbb{H}^{n+1}$, denote $\Omega^+(s)=\Omega\cap H_s^+$ and $\Omega^-(s)=\Omega\cap H_s^-$. The reflection map across $H_{\gamma(s)}$ is denoted by $R_{\gamma,s}$. We define
\begin{equation*
S_{\gamma}^+(\Omega):=\inf\{s\in \mathbb{R}~|~R_{\gamma,s}(\Omega^+(s))\subset\Omega^-(s)\}.
\end{equation*}
It has been proved in \cite{And-Wei2017-2} that for any geodesic line $\gamma$ in $\mathbb{H}^{n+1}$, $S_{\gamma}^+(\Omega_t)$ is strictly decreasing along the flow \eqref{flow-VMCF} unless $R_{\gamma,\bar s}(\Omega_t)=\Omega_t$ for some $\bar s\in \mathbb{R}$. Note that to prove this property, we only need the convexity of the evolving hypersurface $M_t=\partial\Omega_t$ which is guaranteed by the positivity of the sectional curvature. The readers may refer to \cite{Chow97,Chow-Gul96} for more details on the Alexandrov reflection method.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{fig1}\\
\caption{$\Omega_t$ can not leave out $B_R$
\end{figure}
Choose $R>0$ such that the initial domain $\Omega_0$ is contained in some geodesic ball $B_R(p)$ of radius $R$ and centered at some point $p$ in the hyperbolic space. The above reflection property implies that $\Omega_t\cap B_R(p)\neq \emptyset$ for any $t\in [0,T)$. If not, there exists some time $t$ such that $\Omega_t$ does not intersect the geodesic ball $B_R$. Choose a geodesic line $\gamma(s)$ with the property that there exists a geodesic hyperplace $\Pi=H_{\gamma(s_0)}$ which is perpendicular to $\gamma(s)$ and is tangent to the geodesic sphere $\partial B_R$, and the domain $\Omega_t$ lies in the half-space $H^+_{s_0}$. Then $R_{\gamma,s_0}(\Omega^+_0)=\emptyset\subset \Omega_0^-$. Since $S_{\gamma}^+(\Omega_t)$ is decreasing, we have $R_{\gamma,s_0}(\Omega_t^+)\subset \Omega_t^-$. However, this is not possible because $\Omega_t^-=\Omega_t\cap H_{s_0}^-=\emptyset$ and $R_{\gamma,s_0}(\Omega_t^+)$ is obviously not empty.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{fig2}\\
\caption{Diameter of $\Omega_t$ is bounded
\end{figure}
For any $t\in [0,T)$, let $x_1, x_2$ be points on $M_t=\partial\Omega_t$ such that $d(p,x_1)=\min\{d(p,x): x\in M_t\}$ and $d(p,x_2)=\max\{d(p,x): x\in M_t\}$, where $d(\cdot,\cdot)$ is the distance in the hyperbolic space. Since $\Omega_0$ is contained in the geodesic ball $B_R(p)$ and $\Omega_t\cap B_R(p)\neq \emptyset$, we deduce from $|\Omega_t|=|\Omega_0|$ that $x_1\in B_R(p)$. If $x_2\in B_R(p)$, then the diameter of $\Omega_t$ is bounded from above by $R$. Therefore it suffices to study the case $x_2\notin B_R(p)$. Let $\gamma(s)$ be the geodesic line passing through $x_1$ and $x_2$, i.e., there are numbers $s_1<s_2\in \mathbb{R}$ such that $\gamma(s_1)=x_1$ and $\gamma(s_2)=x_2$. We choose the geodesic plane $\Pi=H_{\gamma(s_0)}$ for some number $s_0\in (s_1,s_2)$ such that $\Pi$ is perpendicular to $\gamma$ and is tangent to the boundary of $B_R(p)$ at $p'\in\partial B_R(p)$. Let $q=\gamma(s_0)$ be intersection point $\gamma\cap\Pi$. By the Alexandrov reflection property, $d(x_2,q)\leq d(q,x_1)$. Then the triangle inequality implies
\begin{align*}
d(p,x_2) \leq &~ d(p,x_1)+d(x_1,x_2) \\
\leq &~ d(p,x_1)+2d(q,x_1)\\
\leq &~ d(p,x_1)+2(d(q,p')+d(p',p)+d(p,x_1))\leq ~7R,
\end{align*}
where we used the fact $x_1\in B_R(p)$. This shows that the diameter of $\Omega_t$ is uniformly bounded along the flow \eqref{flow-VMCF}.
To estimate the lower bound of the inner radius of $\Omega_t$, we project the domain $\Omega_t$ in the hyperbolic space $\mathbb{H}^{n+1}$ to the unit ball in Euclidean space $\mathbb{R}^{n+1}$ as in \cite[\S 5]{And-Wei2017-2}. Denote by $\mathbb{R}^{1,n+1}$ the Minkowski spacetime, that is the vector space $\mathbb{R}^{n+2}$ endowed with the Minkowski spacetime metric $\langle \cdot,\cdot\rangle$ given by $ \langle X,X\rangle=-X_0^2+\sum_{i=1}^nX_i^2$ for any vector $X=(X_0,X_1,\cdots,X_n)\in \mathbb{R}^{n+2}$. Then the hyperbolic space is characterized as
\begin{equation*}
\mathbb{H}^{n+1}=~\{X\in \mathbb{R}^{1,n+1},~~\langle X,X\rangle=-1,~X_0>0\}
\end{equation*}
An embedding $X:M^n\to \mathbb{H}^{n+1}$ induces an embedding $Y:M^n\to B_1(0)\subset \mathbb{R}^{n+1}$ by
\begin{equation*
X~=~\frac{(1,Y)}{\sqrt{1-|Y|^2}}.
\end{equation*}
The induced metrics $g_{ij}^X$ and $g_{ij}^Y$ of $X(M^n)\subset \mathbb{H}^{n+1}$ and $Y(M^n)\subset \mathbb{R}^{n+1}$ are related by
\begin{align*
g_{ij}^X =&\frac 1 {1-|Y|^2}\left(g_{ij}^Y+\frac{\langle Y,\partial_iY\rangle\langle Y,\partial_jY\rangle }{(1-|Y|^2)}\right)
\end{align*}
Let $\tilde{\Omega}_t\subset B_1(0)$ be the corresponding image of $\Omega_t$ in $B_1(0)\subset \mathbb{R}^{n+1}$, and observe that $\tilde\Omega_t$ is a convex Euclidean domain. Then the diameter bound of $\Omega_t$ implies the diameter bound on $\tilde{\Omega}_t$. In particular, $|Y|\leq C<1$ for some constant $C$. This implies that the induced metrics $g_{ij}^X$ and $g_{ij}^Y$ are comparable. So the volume of $\tilde{\Omega}_t$ is also bounded below by a constant depending on the volume of $\Omega_0$ and the diameter of $\Omega_t$. Let $\omega_{\min}(\tilde{\Omega}_t)$ be the minimal width of $\tilde\Omega_t$. Then the volume of $\tilde{\Omega}_t$ is bounded by the a constant times the $\omega_{\min}(\tilde\Omega_t)(\text{\rm diam}(\tilde\Omega_t)^{n}$, since $\tilde\Omega_t$ is contained in a spherical prism of height $\omega_{\min}(\tilde\Omega_t)$ and radius $\text{\rm diam}(\tilde\Omega_t)$. It follows that $\omega_{\min}(\tilde{\Omega}_t)$ is bounded from below by a positive constant $C$. Since $\tilde{\Omega}_t$ is strictly convex, an estimate of Steinhagen \cite{Steinhagen} implies that the inner radius $\tilde{\rho}_-(t)$ of $\tilde{\Omega}_t$ is bounded below by $\tilde{\rho}_-(t)\geq c(n)\omega_{\min}\geq C>0$, from which we obtain the uniform positive lower bound on the inner radius $\rho_-(t)$ of $\Omega_t$. This finishes the proof.
\endproof
By \eqref{s4:io-radius1}, the inner radius of $\Omega_t$ is bounded below by a positive constant $c_1$. This implies that there exists a geodesic ball of radius $c_1$ contained in $\Omega_t$ for each $t\in [0,T)$. The same argument as in \cite[Lemma 4.2]{And-Wei2017-2} yields the existence of a geodesic ball with fixed center enclosed by the flow hypersurface on a suitable fixed time interval.
\begin{lem}\label{s4:lem-inball}
Let $M_t$ be a smooth solution of the flow \eqref{flow-VMCF} on $[0,T)$ with the global term $\phi(t)$ given by \eqref{s1:phit}. For any $t_0\in [0,T)$, let $B_{\rho_0}(p_0)$ be the inball of $\Omega_{t_0}$, where $\rho_0=\rho_-(t_0)$. Then
\begin{equation}\label{s4:inball-eqn1}
B_{\rho_0/2}(p_0)\subset \Omega_t,\quad t\in [t_0, \min\{T,t_0+\tau\})
\end{equation}
for some $\tau$ depending only on $n,\alpha,\Omega_0$.
\end{lem}
\subsection{Upper bound of $F$}\label{sec:4-2}
Now we can use the technique of Tso \cite{Tso85} as in \cite{And-Wei2017-2} to prove the upper bound of $F$ along the flow \eqref{flow-VMCF} provided that $F$ satisfies Assumption \ref{s1:Asum}. The inequality \eqref{s1:asum-1} and the fact that each $M_t$ has positive sectional curvature are crucial in the proof.
\begin{thm}\label{s4:F-bd}
Assume that $F$ satisfies the Assumption \ref{s1:Asum}. Then along the flow \eqref{flow-VMCF} with any $\alpha>0$, we have $F\leq C$ for any $t\in [0,T)$, where $C$ depends on $n,\alpha,M_0$ but not on $T$.
\end{thm}
\proof
For any given $t_0\in [0,T)$, let $B_{\rho_0}(p_0)$ be the inball of $\Omega_{t_0}$, where $\rho_0=\rho_-(t_0)$.
Consider the support function $u(x,t)=\sinh r_{p_0}(x)\langle \partial r_{p_0},\nu\rangle $ of $M_t$ with respect to the point $p_0$, where $r_{p_0}(x)$ is the distance function in $\mathbb{H}^{n+1}$ from the point $p_0$. Since $M_t$ is strictly convex, by \eqref{s4:inball-eqn1},
\begin{equation}\label{s4:sup-1}
u(x,t)~\geq ~\sinh(\frac{\rho_0}2)~=:~2c
\end{equation}
on $M_t$ for any $t\in[t_0,\min\{T,t_0+\tau\})$. On the other hand, the estimate \eqref{s4:io-radius1} implies that $u(x,t)\leq \sinh(2c_2)$ on $M_t$ for all $t\in[t_0,\min\{T,t_0+\tau\})$. Recall that the support function $u(x,t)$ evolves by
\begin{align}\label{s4:evl-u}
\frac{\partial}{\partial t}u =& \dot{\Psi}^{kl}\nabla_k\nabla_lu+\cosh r_{p_0}(x)\left(\phi(t)-\Psi-\dot{\Psi}^{kl}h_{kl}\right)+\dot{\Psi}^{ij}h_i^kh_{kj}u.
\end{align}
as we computed in \cite{And-Wei2017-2}, where $\Psi=F^{\alpha}(\mathcal{W})$. Define the auxiliary function
\begin{equation*
W(x,t)=\frac {\Psi(x,t)}{u(x,t)-c},
\end{equation*}
which is well-defined on $M_t$ for all $t\in [t_0,\min\{T,t_0+\tau\})$. Combining \eqref{s2:evl-Psi} and \eqref{s4:evl-u}, we have
\begin{align*
\frac{\partial}{\partial t}W= &\dot{\Psi}^{ij}\left(\nabla_j\nabla_iW+\frac 2{u-c}\nabla_iu\nabla_jW\right)\nonumber \\
&\quad-\frac{\phi(t)}{u-c}\left( \dot{\Psi}^{ij}(h_i^kh_{k}^j-\delta_i^j)+W\cosh r_{p_0}(x)\right)\nonumber\\
&\quad +\frac{\Psi}{(u-c)^2}(\Psi+\dot{\Psi}^{kl}h_{kl})\cosh r_{p_0}(x)-\frac{c\Psi}{(u-c)^2}\dot{\Psi}^{ij}h_i^kh_{k}^j-W\dot{\Psi}^{ij}\delta_i^j.
\end{align*}
By homogeneity of $\Psi$ and the inverse-concavity of $F$, we have $\Psi+\dot{\Psi}^{kl}h_{kl}=(1+\alpha)\Psi$ and $\dot{\Psi}^{ij}h_i^kh_{k}^j\geq \alpha F^{\alpha+1}$. Moreover, by the inequality \eqref{s1:asum-1} and the fact that $\kappa_1\kappa_2>1$, we have
\begin{align*}
\dot{\Psi}^{ij}(h_i^kh_{k}^j-\delta_i^j)=&~\alpha f^{\alpha-1}\sum_{i=1}^n\dot{f}^i(\kappa_i^2-1)\\
\geq &~ \alpha f^{\alpha-1}\left(\dot{f}^2(\kappa_2^2-1)+\dot{f}^1(\kappa_1^2-1)\right)\displaybreak[0]\\
\geq &~\alpha f^{\alpha-1}\dot{f}^1\left(\frac{\kappa_1}{\kappa_2}(\kappa_2^2-1)+(\kappa_1^2-1)\right)\\
=&~\alpha f^{\alpha-1}\dot{f}^1\kappa_2^{-1}(\kappa_1\kappa_2-1)(\kappa_1+\kappa_2)~\geq~0,
\end{align*}
where we used $\kappa_i\geq 1$ for $i=2,\cdots,n$ in the first inequality. Then we arrive at
\begin{align}\label{s4:evl-W-1-2}
\frac{\partial}{\partial t}W\leq & ~\dot{\Psi}^{ij}\left(\nabla_j\nabla_iW+\frac 2{u-c}\nabla_iu\nabla_jW\right)+(\alpha+1)W^2\cosh r_{p_0}(x)-\alpha cW^2F.
\end{align}
The remaining proof of Theorem \ref{s4:F-bd} is the same with \cite[\S 4]{And-Wei2017-2}. We include it here for convenience of the readers. Using \eqref{s4:sup-1} and the upper bound $r_{p_0}(x)\leq 2c_2$, we obtain from \eqref{s4:evl-W-1-2} that
\begin{align}\label{s4:evl-W-3}
\frac{\partial}{\partial t}W\leq &~ \dot{\Psi}^{ij}\left(\nabla_j\nabla_iW+\frac 2{u-c}\nabla_iu\nabla_jW\right)\nonumber \\
&\quad +W^2\left((\alpha+1)\cosh (2c_2)-\alpha c^{1+\frac 1{\alpha}}W^{1/{\alpha}}\right)
\end{align}
holds on $[t_0,\min\{T,t_0+\tau\})$. Let $\tilde{W}(t)=\sup_{M_t}W(\cdot,t)$. Then \eqref{s4:evl-W-3} implies that
\begin{equation*
\frac d{dt}\tilde{W}(t)\leq \tilde{W}^2\left((\alpha+1)\cosh (2c_2)-\alpha c^{1+\frac 1{\alpha}}\tilde{W}^{1/{\alpha}}\right)
\end{equation*}
from which it follows from the maximum principle that
\begin{equation}\label{s4:evl-W-4}
\tilde{W}(t)\leq \max\left\{ \left(\frac {2(1+\alpha)\cosh(2c_2)}{\alpha}\right)^{\alpha}c^{-(\alpha+1)}, \left(\frac {2}{1+\alpha}\right)^{\frac{\alpha}{1+\alpha}}c^{-1}(t-t_0)^{-\frac{\alpha}{1+\alpha}}\right\}.
\end{equation}
Then the upper bound on $F$ follows from \eqref{s4:evl-W-4} and the facts that
\begin{equation*}
c=\frac 12\sinh(\frac{\rho_0}2)\geq \frac 12\sinh(\frac{c_1}2)
\end{equation*}
and $u-c\leq 2c_2$, where $c_1,c_2$ are constants in \eqref{s4:io-radius1} depending only on $n,M_0$.
\endproof
\subsection{Long time existence and convergence}\label{sec:4-3}
In this subsection, we complete the proof of Theorem \ref{thm1-1}. Firstly, in the case (i) of Theorem \ref{thm1-1}, we can deduce directly a uniform estimate on the principal curvatures of $M_t$. In fact, since $f(\kappa)$ is bounded from above by Theorem \ref{s4:F-bd},
\begin{align}\label{s4:f-condi-1}
C \geq & ~f(\kappa_1,\kappa_2,\cdots,\kappa_n)~ \geq ~f(\kappa_1,\frac 1{\kappa_1},\cdots,\frac 1{\kappa_n}),
\end{align}
where in the second inequality we used the facts that $f$ is increasing in each argument and $\kappa_i\kappa_1>1$ for $i=2,\cdots,n$. Combining \eqref{s4:f-condi-1} and \eqref{thm1-cond1} gives that $\kappa_1\geq C>0$ for some uniform constant $C$. Since the dual function $f_*$ of $f$ vanishes on the boundary of the positive cone $\Gamma_+$ and $f_*(\tau_i)=1/f(\kappa_i)\geq C>0$ by Theorem \ref{s4:F-bd}, the upper bound on $\tau_i=1/{\kappa_i}\leq C$ gives the lower bound on $\tau_i$, which is equivalent to the upper bound of $\kappa_i$. In summary, we obtain uniform two-sided positive bound on the principal curvatures of $M_t$ along the flow \eqref{flow-VMCF} in the case (i) of Theorem \ref{thm1-1}.
The examples of $f$ satisfying Assumption \ref{s1:Asum} and the condition (i) of Theorem \ref{thm1-1} include
\begin{itemize}
\item[a).] $n\geq 2$, $f=n^{-1/k}S_k^{1/k}$ with $k>0$;
\item[b).] $n\geq 3$, $f= E_k^{1/k}$ with $k=1,\cdots,n$;
\item[c).] $n=2, f=(\kappa_1+\kappa_2)/2$.
\end{itemize}
We next consider the case (ii) of Theorem \ref{thm1-1}, i.e., $n=2, f=(\kappa_1\kappa_2)^{1/2}$. In general, the estimate $1\leq \kappa_1\kappa_2=f(\kappa)\leq C$ can not prevent $\kappa_2$ from going to infinity. Instead, we will prove that the pinching ratio $\kappa_2/\kappa_1$ is bounded from above along the flow \eqref{flow-VMCF} with $f=(\kappa_1\kappa_2)^{1/2}$ and $\alpha\in[1/2,2]$. This together with the estimate $1\leq \kappa_1\kappa_2\leq C$ yields the uniform estimate on $\kappa_1$ and $\kappa_2$.
\begin{lem}
In the case $n=2, f=(\kappa_1\kappa_2)^{1/2}$ and $\alpha\in [1/2,2]$, the principal curvatures $\kappa_1,\kappa_2$ of $M_t$ satisfy
\begin{equation}\label{s4:lem-pinch}
0<\frac 1C~\leq \kappa_1\leq \kappa_2\leq C,\quad \forall~t\in [0,T)
\end{equation}
for some positive constant $C$ along the flow \eqref{flow-VMCF}.
\end{lem}
\proof
In this case, $\Psi(\mathcal{W})=\psi(\kappa)=K^{\alpha/2}$, where $K=\kappa_1\kappa_2$ is the Gauss curvature. The derivatives of $\psi$ with respect to $\kappa_i$ are listed in the following:
\begin{align}
\dot{\psi}^1 =&~ \frac{\alpha}2K^{\frac{\alpha}2-1}\kappa_2,\qquad \dot{\psi}^2 = \frac{\alpha}2K^{\frac{\alpha}2-1}\kappa_1\label{s4:psi-d}\\
\ddot{\psi}^{11} =& ~\frac{\alpha}2(\frac{\alpha}2-1)K^{\frac{\alpha}2-2}\kappa_2^2,\qquad
\ddot{\psi}^{22} =~\frac{\alpha}2(\frac{\alpha}2-1)K^{\frac{\alpha}2-2}\kappa_1^2\\
\ddot{\psi}^{12} =&~ \ddot{\psi}^{21}=~\frac{\alpha^2}4K^{\frac{\alpha}2-1}. \label{s4:psi-dd}
\end{align}
Then we have
\begin{align}
\dot{\Psi}^{ij}h_i^kh_k^j= &~ \sum_{i=1}^n\dot{\psi}^i\kappa_i^2=\frac{\alpha}2K^{\alpha/2}H \label{s4:psi-d1}\\
\dot{\Psi}^{ij}\delta_i^j= &~ \sum_{i=1}^n\dot{\psi}^i=\frac{\alpha}2K^{\frac{\alpha}2-1}H,\qquad \dot{\Psi}^{ij}h_i^j=\alpha K^{\alpha/2},\label{s4:psi-d2}
\end{align}
where $H=\kappa_1+\kappa_2$ is the mean curvature.
To prove the estimate \eqref{s4:lem-pinch}, we define a function
\begin{equation*}
G=~K^{\alpha-2}(H^2-4K)
\end{equation*}
and aim to prove that $G(x,t)\leq \max_{M_0}G(x,0)$ by maximum principle. Using \eqref{s4:psi-d1} and \eqref{s4:psi-d2}, the evolution equations \eqref{s2:evl-Psi} and \eqref{s2:evl-h} imply that
\begin{equation}\label{s4:K-evl}
\frac{\partial}{\partial t}K =\dot{\Psi}^{kl}\nabla_k\nabla_lK+(\frac{\alpha}2-1)K^{-1}\dot{\Psi}^{kl}\nabla_kK\nabla_lK+(K^{\alpha/2}-\phi(t))(K-1)H.
\end{equation}
and
\begin{align}\label{s4:H-evl}
\frac{\partial}{\partial t}H =&\dot{\Psi}^{kl}\nabla_k\nabla_lH+\ddot{\Psi}^{kl,pq}\nabla_ih_{kl}\nabla^ih_{pq}\nonumber\\
&\quad +K^{\frac{\alpha}2-1}(K+\frac{\alpha}2(1-k))(H^2-4K)+2K^{\alpha/2}(K-1)\nonumber\\
&\quad -\phi(t)(H^2-2K-2).
\end{align}
By a direct computation using \eqref{s4:K-evl} and \eqref{s4:H-evl}, we obtain the evolution equation of $G$ as follows:
\begin{align}\label{s4:G-evl}
\frac{\partial}{\partial t}G =&\dot{\Psi}^{kl}\nabla_k\nabla_lG-2(\alpha-2)K^{-1}\dot{\Psi}^{kl}\nabla_kK\nabla_lG\nonumber\\
&\quad +(\alpha-2)(\frac{3\alpha}2-2)K^{\alpha-4}\dot{\Psi}^{kl}\nabla_kK\nabla_lK(H^2-4K)\nonumber\\
&\quad -2K^{\alpha-2}\dot{\Psi}^{kl}\nabla_kH\nabla_lH-2(\alpha-2)K^{\alpha-3}\dot{\Psi}^{kl}\nabla_kK\nabla_lK\nonumber\\
&\quad +2HK^{\alpha-2}\ddot{\Psi}^{kl,pq}\nabla_ih_{kl}\nabla^ih_{pq}\nonumber\\
&\quad +2HK^{\frac{3\alpha}2-3}(H^2-4K)-HK^{\alpha-3}(H^2-4K)(\alpha K+2-\alpha)\phi(t)
\end{align}
We will apply the maximum principle to prove that $\max_{M_t}G$ is non-increasing in time along the flow \eqref{flow-VMCF}. We first look at the zero order terms of \eqref{s4:G-evl}, i.e., the terms in the last line of \eqref{s4:G-evl} which we denote by $Q_0$. Since $K=\kappa_1\kappa_2>1$ by Theorem \ref{s3:thm-2}, we have
\begin{equation*}
\phi(t)=\frac 1{|M_t|}{\int_{M_t}K^{\alpha/2}}~\geq 1,\quad \mathrm{and}\quad \alpha K+2-\alpha>2.
\end{equation*}
We also have $H^2-4K=(\kappa_1-\kappa_2)^2\geq 0$. Then
\begin{align*}
Q_0 \leq &~ 2HK^{\frac{3\alpha}2-3}(H^2-4K)-HK^{\alpha-3}(H^2-4K)(\alpha K+2-\alpha) \\
=& ~HK^{\alpha-3}(H^2-4K)\biggl(2K^{\alpha/2}-\alpha K+\alpha-2\biggr).
\end{align*}
For any $K>1$, denote $f(\alpha)=2K^{\alpha/2}-\alpha K+\alpha-2$. Then $f(2)=f(0)=0$ and $f(\alpha)$ is a convex function of $\alpha$. Therefore $f(\alpha)\leq 0$ and $Q_0\leq 0$ provided that $\alpha\in [0,2]$.
At the critical point of $G$, we have $\nabla_iG=0$ for all $i=1,2$, which is equivalent to
\begin{equation}\label{s4:G-grad}
2H\nabla_iH=\left(4(\alpha-1)-(\alpha-2)K^{-1}H^2\right)\nabla_iK.
\end{equation}
Then the gradient terms (denoted by $Q_1$) of \eqref{s4:G-evl} at the critical point of $G$ satisfy
\begin{align}\label{s4:Q1}
Q_1K^{3-\alpha}= & \biggl(-8(\alpha-1)^2\frac{K}{H^2}-2(\alpha-1)(\alpha-2)+(\alpha-1)(\alpha-2)\frac{H^2}K\biggr)\nonumber\\
&\quad \times\dot{\Psi}^{kl}\nabla_kK\nabla_lK+2HK\ddot{\Psi}^{kl,pq}\nabla_ih_{kl}\nabla^ih_{pq}.
\end{align}
Using \eqref{s4:psi-d} -- \eqref{s4:psi-dd}, we have
\begin{align*}
\dot{\Psi}^{kl}\nabla_kK\nabla_lK=&~ \frac{\alpha}2K^{\frac{\alpha}2-1}(\kappa_2(\nabla_1K)^2+\kappa_1(\nabla_2K)^2) \\
\ddot{\Psi}^{kl,pq}\nabla_ih_{kl}\nabla^ih_{pq}=&~\frac{\alpha}2(\frac{\alpha}2-1)K^{\frac{\alpha}2-2}\sum_{i=1}^2(\kappa_2^2(\nabla_ih_{11})^2+\kappa_1^2(\nabla_ih_{22})^2)\\
&\quad +\frac{\alpha^2}2K^{\frac{\alpha}2-1}\sum_{i=1}^2\nabla_ih_{11}\nabla_ih_{22}-\alpha K^{\frac{\alpha}2-1}\sum_{i=1}^2(\nabla_ih_{12})^2\\
=&~\frac{\alpha}2(\frac{\alpha}2-1)K^{\frac{\alpha}2-2}((\nabla_1K)^2+(\nabla_2K)^2)\\
&\quad +\alpha K^{\frac{\alpha}2-1}\biggl(\nabla_1h_{11}\nabla_1h_{22}-(\nabla_1h_{12})^2+\nabla_2h_{11}\nabla_2h_{22}-(\nabla_2h_{12})^2\biggr)
\end{align*}
The equation \eqref{s4:G-grad} implies that $\nabla_ih_{11}$ and $\nabla_ih_{22}$ are linearly dependent, i.e., there exist functions $g_1,g_2$ such that
\begin{equation}\label{s4:G-grad-1}
g_1\nabla_ih_{11}=g_2\nabla_ih_{22}.
\end{equation}
The functions $g_1,g_2$ can be expressed explicitly as follows:
\begin{align*
g_1=&2HK-(4(\alpha-1)K+(2-\alpha)H^2)\kappa_2,\\
g_2=&-2HK+(4(\alpha-1)K+(2-\alpha)H^2)\kappa_1.
\end{align*}
Without loss of generality, we can assume that both $g_1$ and $g_2$ are not equal to zero at the critical point of $G$. In fact, if $g_1=0$, then
\begin{align}\label{s4:g1}
0=g_1 = & ~\biggl((\alpha-2)(H^2-4K)+2H\kappa_1-4K\biggr)\kappa_2\nonumber\\
=&~\biggl((\alpha-2)(\kappa_1-\kappa_2)+2\kappa_1\biggr)(\kappa_1-\kappa_2)\kappa_2.
\end{align}
Since $\alpha\leq 2$, we have $(\alpha-2)(\kappa_1-\kappa_2)+2\kappa_1\geq 2\kappa_1>0$. Thus \eqref{s4:g1} is equivalent to $\kappa_2=\kappa_1$ and we have nothing to prove.
By the equation \eqref{s4:G-grad-1}, we have
\begin{align*}
(\nabla_1K)^2 =& (\kappa_2+g_2^{-1}g_1\kappa_1)^2(\nabla_1h_{11})^2 = g_2^{-2}4H^2K^2(H^2-4K) (\nabla_1h_{11})^2,\\
(\nabla_2K)^2 =& (\kappa_2g_1^{-1}g_2+\kappa_1)^2(\nabla_2h_{22})^2 = g_1^{-2}4H^2K^2(H^2-4K) (\nabla_2h_{22})^2.
\end{align*}
Using the Codazzi identity, the equation \eqref{s4:G-grad-1} also implies that
\begin{align*}
&\nabla_1h_{11}\nabla_1h_{22}-(\nabla_1h_{12})^2+\nabla_2h_{11}\nabla_2h_{22}-(\nabla_2h_{12})^2\\
=&~\nabla_1h_{11}\nabla_1h_{22}-(\nabla_1h_{22})^2+\nabla_2h_{11}\nabla_2h_{22}-(\nabla_2h_{11})^2\\
=&~ g_2^{-2}g_1(g_2-g_1)(\nabla_1h_{11})^2 +g_1^{-2}g_2(g_1-g_2)(\nabla_2h_{22})^2\\
= & (2-\alpha)(H^2-4K)\biggl(g_2^{-2}g_1(\nabla_1h_{11})^2+g_1^{-2}g_2(\nabla_2h_{22})^2\biggr).
\end{align*}
Therefore we can write the right hand side of \eqref{s4:Q1} as linear combination of $(\nabla_{1}h_{11})^2$ and $(\nabla_2h_{22})^2$:
\begin{align}\label{s4:Q1-2}
\frac 2{\alpha}Q_1K^{4-3\alpha/2}= & q_{1}g_2^{-2}(\nabla_{1}h_{11})^2+q_2g_1^{-2}(\nabla_2h_{22})^2,
\end{align}
where the coefficients $q_1,q_2$ satisfy
\begin{align*}
q_1=&\biggl(-8(\alpha-1)^2\frac{K^2}{H^2}+2(\alpha-1)(\alpha-2)K+(\alpha-2)H^2\biggr)4H^2K(H^2-4K)\kappa_2\nonumber\\
&\quad +4(2-\alpha)H^3K^2(H^2-4K)\\
=& -32(\alpha-1)^2K^3(H^2-4K)\kappa_2-4(2-\alpha)(2\alpha-1)H^2K^2(H^2-4K)\kappa_2\\
&\quad -4(2-\alpha)H^2K(H^2-4K)\kappa_2^3
\end{align*}
and
\begin{align*}
q_2=& -32(\alpha-1)^2K^3(H^2-4K)\kappa_1-4(2-\alpha)(2\alpha-1)H^2K^2(H^2-4K)\kappa_1\\
&\quad -4(2-\alpha)H^2K(H^2-4K)\kappa_1^3.
\end{align*}
It can be checked directly that $q_1$ and $q_2$ are both non-positive if $\alpha\in [1/2,2]$. Thus the gradient terms $Q_1$ of \eqref{s4:G-evl} are non-positive at a critical point of $G$ if $\alpha\in [1/2,2]$. The maximum principle implies that $\max_{M_t}G$ is non-increasing in time. It follows that $G(x,t)\leq \max_{M_0}G(x,0)$. Since $1<K\leq C$ for some constant $C>0$ by Theorem \ref{s3:thm-2} and Theorem \ref{s4:F-bd}, we have
\begin{equation}\label{s4:H-bd}
H^2= ~4K+K^{\alpha-2}G\leq ~C.
\end{equation}
Finally, the estimate \eqref{s4:lem-pinch} follows from \eqref{s4:H-bd} and $K>1$ immediately.
\endproof
Now we have proved that the principal curvatures $\kappa_i$ of $M_t$ satisfy the uniform estimate $0<1/C\leq \kappa_i\leq C$ for some constant $C>0$, which is equivalent to the $C^2$ estimate for $M_t$. Since the functions $f$ we considered in Theorem \ref{thm1-1} are inverse-concave, we can apply an argument similar to that in \cite[\S 5]{And-Wei2017-2} to derive higher regularity estimates. The standard continuation argument then implies the long time existence of the flow, and the argument in \cite[\S 6]{And-Wei2017-2} implies the smooth convergence to a geodesic sphere as time goes to infinity.
\section{Horospherically convex regions}\label{sec:h-convex}
In this section we will investigate some of the properties of horospherically convex regions in hyperbolic space (that is, regions which are given by the intersection of a collection of horo-balls). In particular, for such regions we define a horospherical Gauss map, which is a map to the unit sphere, and we show that each horospherically convex region is completely described in terms of a scalar function on the sphere which we call the \emph{horospherical support function}. There are interesting formal similarities between this situation and that of convex Euclidean bodies. For the purposes of this paper the main result we need is that the modified Quermassintegrals are monotone with respect to inclusion for horospherically convex domains. However we expect that the description of horospherically convex regions which we develop here will be useful in further investigations beyond the scope of this paper.
We remark that a similar development is presented in \cite{EGM}, but in a slightly different context: In that paper the `horospherically convex' regions are those which are intersections of complements of horo-balls (corresponding to principal curvatures greater than $-1$ everywhere, while we deal with regions which are intersections of horo-balls, corresponding to principal curvatures greater than $1$. Our condition is more stringent but is more useful for the evolution equations we consider here.
\subsection{The horospherical Gauss map}\label{sec:5-1}
The horospheres in hyperbolic space are the submanifolds with constant principal curvatures equal to $1$ everywhere.
If we identify ${\mathbb H}^{n+1}$ with the future time-like hyperboloid in Minkowski space ${\mathbb R}^{n+1,1}$, then the condition of constant principal curvatures equal to $1$ implies that the null vector $\bar{\mathbf e} := X-\nu$ is constant on the hypersurface, since we have ${\mathcal W}={\mathrm I}$, and hence
$$
D_v\bar{\mathbf e} = D_v(X-\nu) = DX(({\mathrm I}-{\mathcal W})(v)) = 0
$$
for all tangent vectors $v$. Then we observe that
$$
X\cdot\bar{\mathbf e} = X\cdot (X-\nu) = -1,
$$
from which it follows that the horosphere is the intersection of the null hyperplane $\{X:\ X\cdot\bar{\mathbf e}=-1\}$ with the hyperboloid ${\mathbb H}^{n+1}$. The horospheres are therefore in one-to-one correspondence with points $\bar{\mathbf e}$ in the future null cone, which are given by $\{\bar{\mathbf e}=\lambda({\mathbf e},1):\ {\mathbf e}\in S^n,\ \lambda>0\}$, and there is a one-parameter family of these for each ${\mathbf e}\in S^n$. For convenience we parametrise these by their signed geodesic distance $s$ from the `north pole' $N=(0,1)\in{\mathbb H}^{n+1}$, satisfying $-1 = \lambda(\cosh(s)N+\sinh(s)({\mathbf e},0))\cdot ({\mathbf e},1) = -\lambda {\mathrm e}^{s}$. It follows that $\lambda = {\mathrm e}^{-s}$. Thus we denote by $H_{{\mathbf e}}(s)$ the horosphere
$\{X\in{\mathbb H}^{n+1}:\ X\cdot ({\mathbf e},1) = -{\mathrm e}^s\}$. The interior region (called a \emph{horo-ball}) is denoted by
$$
B_{{\mathbf e}}(s) = \{X\in{\mathbb H}^{n+1}:\ 0>X\cdot({\mathbf e},1) >-{\mathrm e}^s\}.
$$
A region $\Omega$ in ${\mathbb H}^{n+1}$ is \emph{horospherically convex} (or h-convex for convenience) if every boundary point $p$ of $\partial\Omega$ has a supporting horo-ball, i.e. a horo-ball $B$ such that $\Omega\subset B$ and $p\in\partial B$. If the boundary of $\Omega$ is a smooth hypersurface, then this implies that every principal curvature of $\partial\Omega$ is greater than or equal to $1$ at $p$. We say that $\Omega$ is \emph{uniformly h-convex} if there is $\delta>0$ such that all principal curvatures exceed $1+\delta$.
Let $M^n=\partial\Omega$ be a hypersurface which is at the boundary of a horospherically convex region $\Omega$. Then the horospherical Gauss map ${\mathbf e}:\ M\to S^n$ assigns to each $p\in M$ the point ${\mathbf e}(p) =\pi(X(p)-\nu(p))\in S^n$, where $\pi(x,y) = \frac{x}{y}$ is the radial projection from the future null cone onto the sphere $S^n\times\{1\}$. We observe that the derivative of ${\mathbf e}$ is non-singular if $M$ is uniformly h-convex: If $v$ is a tangent vector to $M$, then
$$
D{\mathbf e}(v) = D\pi\big|_{X-\nu}\left(({\mathcal W}-{\textrm I})(v)\right).
$$
Here $\tilde v=({\mathcal W}-{\textrm I})(v)$ is a non-zero tangent vector to $M$ since the eigenvalues $\kappa_i$ of ${\mathcal W}$ are greater than $1$. In particular $\tilde v$ is spacelike. On the other hand the kernel of $D\pi|_{X-\nu}$ is the line ${\mathbb R}(X-\nu)$ consisting of null vectors. Therefore $D\pi(\tilde v)\neq 0$. Thus $D{\mathbf e}$ is an injective linear map, hence an isomorphism. It follows that ${\mathbf e}$ is a diffeomorphism from $M$ to $S^n$.
\subsection{The horospherical support function}
Let $M^n=\partial\Omega$ be the boundary of a compact h-convex region. Then for each ${\mathbf e}\in S^n$ we define the \emph{horospherical support function} of $\Omega$ (or $M$) in direction ${\mathbf e}$ by
$$
u({\mathbf e}):= \inf\{s\in{\mathbb R}:\ \Omega\subset B_{{\mathbf e}}(s)\}.
$$
Alternatively, define $f_{\mathbf e}:\ {\mathbb H}^{n+1}\to {\mathbb R}$ by $f_{\mathbf e}(\xi) = \log\left(-\xi\cdot({\mathbf e},1)\right)$. This is a smooth function on ${\mathbb H}^{n+1}$, and we have the alternative characterisation
\begin{equation}\label{eq:defu}
u({\mathbf e}) = \sup\{f_{\mathbf e}(\xi):\ \xi\in\Omega\}.
\end{equation}
The function $u$ is called the \emph{horospherical support function} of the region $\Omega$, and $B_{\mathbf e}(u({\mathbf e}))$ is the \emph{supporting horo-ball} in direction ${\mathbf e}$. The support function completely determines a horospherically convex region $\Omega$, as an intersection of horo-balls:
\begin{equation}\label{eq:utoOmega}
\Omega = \bigcap_{{\mathbf e}\in S^n}B_{\mathbf e}(u({\mathbf e})).
\end{equation}
\subsection{Recovering the region from the support function}
If the region is uniformly h-convex, in the sense that all principal curvatures are greater than $1$, then there is a unique point of $M$ in the boundary of the supporting horo-ball $B_{\mathbf e}(u({\mathbf e}))$. We denote this point by $\bar X({\mathbf e})$. We observe that $\bar X = X\circ {\mathbf e}^{-1}$, so if $M$ is smooth and uniformly h-convex (so that ${\mathbf e}$ is a diffeomorphism) then $\bar X$ is a smooth embedding.
We will show that $\bar X$ can be written in terms of the support function $u$, as follows: Choose local coordinates $\{x^i\}$ for $S^n$ near ${\mathbf e}$. We write $\bar X({\mathbf e})$ as a linear combination of the basis consisting of the two null elements $({\mathbf e},1)$ and $(-{\mathbf e},1)$, together with $({\mathbf e}_j,0)$, where ${\mathbf e}_j = \frac{\partial{\mathbf e}}{\partial x^j}$ for $j=1,\cdots,n$:
$$
\bar X({\mathbf e}) = \alpha (-{\mathbf e},1) + \beta ({\mathbf e},1) + \gamma^j({\mathbf e}_j,0)
$$
for some coefficients, $\alpha$, $\beta$, $\gamma^j$.
Since $\bar X({\mathbf e})\in{\mathbb H}^{n+1}$ we have $|\gamma|^2-4\alpha\beta=-1$, so that $\beta = \frac{1+|\gamma|^2}{4\alpha}$. We also know that $\bar X({\mathbf e})\cdot ({\mathbf e},1) = -{\mathrm e}^{u({\mathbf e})}$ since $\bar X({\mathbf e})\in H_{\mathbf e}(u({\mathbf e}))$, implying that $\alpha = \frac12{\mathrm e}^u$. This gives
$$
\bar X({\mathbf e}) = \frac12{\mathrm e}^{u({\mathbf e})}(-{\mathbf e},1)+\frac12{\mathrm e}^{-u({\mathbf e})}(1+|\gamma|^2)({\mathbf e},1)+\gamma^j({\mathbf e}_j,0).
$$
Furthermore, the normal to $M$ at the point $\bar X({\mathbf e})$ must coincide with the normal to the horosphere $H_{\mathbf e}(u({\mathbf e}))$, which is given by
\begin{equation}\label{eq:nue2}
\nu = \bar X-\bar e = \bar X-{\mathrm e}^{-u({\mathbf e})}({\mathbf e},1).
\end{equation}
Since $|\bar X|^2=-1$ we have $\partial_j\bar X\cdot\bar X=0$, and hence
\begin{align*}
0 &= \partial_j\bar X\cdot \nu\\
&= \partial_j\bar X\cdot\left(\bar X-{\mathrm e}^{-u}({\mathbf e},1)\right)\\
&=-{\mathrm e}^{-u}\partial_jX\cdot ({\mathbf e},1).
\end{align*}
Observing that $({\mathbf e},1)\cdot({\mathbf e},1)=0$ and $({\mathbf e}_i,0)\cdot({\mathbf e},1)=0$, and that $\partial_j{\mathbf e}_i = -\bar g_{ij}{\mathbf e}$ and $\partial_j{\mathbf e}={\mathbf e}_j$, the condition becomes
\begin{align*}
0 &= \partial_jX\cdot({\mathbf e},1)\\
&= \left(\frac12{\mathrm e}^uu_j(-{\mathbf e},1)-\gamma_j({\mathbf e},0)\right)\cdot({\mathbf e},1)\\
&=-{\mathrm e}^uu_j-\gamma_j,
\end{align*}
where $\gamma_j = \gamma^i\bar g_{ij}$ and $\bar g$ is the standard metric on $S^n$.
It follows that we must have $\gamma_j = -{\mathrm e}^uu_j$. This gives the following expression for $\bar X$:
\begin{align}\label{eq:barX2}
\bar X({\mathbf e}) &= \left(-{\mathrm e}^u\bar\nabla u+\left(\frac12{\mathrm e}^u|\bar\nabla u|^2-\sinh u\right){\mathbf e},\frac12{\mathrm e}^u|\bar\nabla u|^2+\cosh u\right)\\
&=-{\mathrm e}^uu_p\bar g^{pg}({\mathbf e}_q,0)+\frac12\left({\mathrm e}^u|\bar\nabla u|^2+{\mathrm e}^{-u}\right)({\mathbf e},1)+\frac12{\mathrm e}^u(-{\mathbf e},1).\label{eq:barX3}
\end{align}
\subsection{A condition for horospherical convexity}\label{sec:5-4}
Given a smooth function $u$, we can use the expression \eqref{eq:barX2} to define a map to hyperbolic space. In this section we determine when the resulting map is an embedding defining a horospherically convex hypersurface.
If order for $\bar X$ to be an immersion, we require the derivatives $\partial_j\bar X$ to be linearly independent. Since we have constructed $\bar X$ in such a way that $\partial_jX$ is orthogonal to the normal vector $\nu$ to the horosphere $B_{{\mathbf e}}(u({\mathbf e}))$, $\partial_j\bar X$ is a linear combination of the basis for the space orthogonal to $\nu$ and $\bar X$ given by the projections $E_k$ of $(e_k,0)$, $k=1,\cdots,n$. Computing explicity, we find
\begin{equation}\label{eq:Ek}
E_k = ({\mathbf e}_k,0)-u_k({\mathbf e},1).
\end{equation}
The immersion condition is therefore equivalent to invertibility of the matrix $A$ define by
$$
A_{jk} = -\partial_j\bar X\cdot E_k.
$$
Given that $A$ is non-singular, we have that $\bar X$ is an immersion with unit normal vector $\nu({\mathbf e})$, and we can differentiate the equation $X-\nu = {\mathrm e}^{-u}({\mathbf e},1)$ to obtain the following:
$$
-(h_j^p-\delta_j^p)\partial_pX = -u_j{\mathrm e}^{-u}({\mathbf e},1)+{\mathrm e}^{-u}({\mathbf e}_j,0).
$$
Taking the inner product with $E_k$ using \eqref{eq:Ek}, we obtain
\begin{equation}\label{eq:Avs2ff}
(h_j^p-\delta_j^p)A_{pk} = {\mathrm e}^{-u}\bar g_{jk}.
\end{equation}
It follows that $A$ is non-singular precisely when ${\mathcal W}-\mathrm{I}$ is non-singular, and is given by
\begin{equation}\label{eq:A-vs-W}
A_{jk} = {\mathrm e}^{-u}\left[\left({\mathcal W}-\mathrm{I}\right)^{-1}\right]_j^p\bar g_{pk}.
\end{equation}
In particular, $A$ is symmetric, and ${\mathcal W}-\mathrm{I}$ is positive definite (corresponding to uniform h-convexity) if and only if the matrix $A$ is positive definite. We conclude that if $u$ is a smooth function on $S^n$, then the map $X$ defines an embedding to the boundary of a uniformly h-convex region if and only if the tensor $A$ computed from $u$ is positive definite.
Computing $A$ explicitly using \eqref{eq:barX3}, we obtain
\begin{align*}
A_{jk} &= \left((\bar\nabla_j({\mathrm e}^u\bar\nabla u),0)-{\mathrm e}^uu_j({\mathbf e},0) -\frac12\partial_j({\mathrm e}^u|\bar\nabla u|^2+{\mathrm e}^{-u})({\mathbf e},1)\right.\\
&\quad\quad\null\left.-\frac12{\mathrm e}^uu_j(-{\mathbf e},1)-\left(\frac12{\mathrm e}^u|\nabla u|^2-\sinh u\right)({\mathbf e}_j,0)\right)\cdot(({\mathbf e}_k,0)-u_k({\mathbf e},1))\\
&=\bar\nabla_j({\mathrm e}^u\bar\nabla_ku)-\frac12{\mathrm e}^u|\bar\nabla u|^2\bar g_{jk}+\sinh u\bar g_{jk}.
\end{align*}
It is convenient to write this in terms of the function $\varphi={\mathrm e}^u$:
\begin{equation}\label{eq:Ainphi}
A_{jk} = \bar\nabla_j\bar\nabla_k\varphi-\frac{|\bar\nabla\varphi|^2}{2\varphi}\bar g_{jk}+\frac{\varphi-\varphi^{-1}}{2}\bar g_{jk}.
\end{equation}
\subsection{Monotonicity of the modified Quermassintegrals}
We will prove that the modified quermassintegrals $\tilde W_k$ are monotone with respect to inclusion by making use of the following result:
\begin{prop}\label{prop:connect-h-convex}
Suppose that $\Omega_1\subset\Omega_2$ are smooth, strictly h-convex domains in ${\mathbb H}^{n+1}$. Then there exists a smooth map $X:\ S^n\times[0,1]\to{\mathbb H}^{n+1}$ such that
\begin{enumerate}
\item $X(.,t)$ is a uniformly h-convex embedding of $S^n$ for each $t$;
\item $X(S^n,0)=\partial\Omega_0$ and $X(S^n,1)=\partial\Omega_1$;
\item The hypersurfaces $M_t=X(S^n,t)$ are expanding, in the sense that $\frac{\partial X}{\partial t}\cdot \nu\geq 0$. Equivalently, the enclosed regions $\Omega_t$ are nested: $\Omega_s\subset\Omega_t$ for each $s\leq t$ in $[0,1]$.
\end{enumerate}
\end{prop}
\begin{proof}
Let $u_0$ and $u_1$ be the horospherical support functions of $\Omega_0$ and $\Omega_1$ respectively, The inclusion $\Omega_0\subset\Omega_1$ implies that $u_0({\mathbf e})\leq u_1({\mathbf e})$ for all ${\mathbf e}\in S^n$, by the characterisation \eqref{eq:defu}.
We define $X({\mathbf e},t) = \bar X[u({\mathbf e},t)]$ according to the formula \eqref{eq:barX2}, where
$$
{\mathrm e}^{u({\mathbf e},t)}=\varphi({\mathbf e},t) := (1-t)\varphi_0({\mathbf e})+t\varphi_1({\mathbf e}),
$$
where $\varphi_i = {\mathrm e}^{u_i}$ for $i=0,1$. Then $u({\mathbf e},t)$ is increasing in $t$, and it follows that the regions $\Omega_t$ are nested, by the expression \eqref{eq:utoOmega}.
We check that each $\Omega_t$ is a strictly h-convex region, by showing that the matrix $A$ constructed from $u(.,t)$ is positive definite for each $t$: We have
\begin{align*}
A_{jk}[u(.,t)] &= \bar\nabla_j\bar\nabla_k\varphi_t -\frac{|\bar\nabla\varphi_t|^2}{2\varphi_t}\bar g_{jk}+\frac{\varphi_t-\varphi_t^{-1}}{2}\bar g_{jk}\\
&= (1-t)A_{jk}[u_0]+tA_{jk}[u_1]\\
&\quad\null +\frac12\left(-\frac{|(1-t)\bar\nabla\varphi_0+t\bar\nabla\varphi_1|^2}{(1-t)\varphi_0+t\varphi_1}+(1-t)\frac{|\bar\nabla\varphi_0|^2}{\varphi_0}+t\frac{|\bar\nabla\varphi_1|^2}{\varphi_1}\right)\bar g_{jk}\\
&\quad\null +\frac12\left(-\frac{1}{(1-t)\varphi_0+t\varphi_1}+\frac{1-t}{\varphi_0}+\frac{t}{\varphi_1}\right)\bar g_{jk}\\
&=(1-t)A_{jk}[u_0]+tA_{jk}[u_1]+t(1-t)\frac{|\varphi_0\bar\nabla\varphi_1-\varphi_1\bar\nabla\varphi_0|^2+|\varphi_1-\varphi_0|^2}{2\varphi_0\varphi_1((1-t)\varphi_0+t\varphi_1)}\bar g_{jk}\\
&\geq (1-t)A_{jk}[u_0]+tA_{jk}[u_1].
\end{align*}
Since $A_{jk}[u_0]$ and $A_{jk}[u_1]$ are positive definite, so is $A_{jk}[u_t]$ for each $t\in[0,1]$, and we conclude that the region $\Omega_t$ is uniformly h-convex.
\end{proof}
\begin{cor}\label{s5:cor}
The modified quermassintegral ${\widetilde W}_k$ is monotone with respect to inclusion for h-convex domains: That is, if $\Omega_0$ and $\Omega_1$ are h-convex domains with $\Omega_0\subset\Omega_1$, then $\widetilde W_k(\Omega_0)\leq\widetilde W_k(\Omega_1)$.
\end{cor}
\begin{proof}
We use the map $X$ constructed in the Proposition \ref{prop:connect-h-convex}. By Lemma \ref{s2:lem3} we have
$$
\frac{d}{dt}\widetilde W_k(\Omega_t) = \int_{M_t}E_k(\lambda)\frac{\partial X}{\partial t}\cdot \nu\,d\mu_t.
$$
Since each $M_t$ is h-convex, we have $\lambda_i>0$ and hence $E_k(\lambda)>0$, and from Proposition \ref{prop:connect-h-convex} we have $\frac{\partial X}{\partial t}\cdot \nu\geq 0$. It follows that $\frac{d}{dt}\widetilde W_k(\Omega_t)\geq 0$ for each $t$, and hence $\widetilde W_k(\Omega_0)\leq \widetilde W_k(\Omega_1)$ as claimed.
\end{proof}
\subsection{Evolution of the horospherical support function}
We end this section with the following observation that the flow \eqref{flow-VMCF-2} of h-convex hypersurfaces is equivalent to an initial value problem for the horospherical support function.
\begin{prop}
The flow \eqref{flow-VMCF-2} of h-convex hypersurfaces in $\mathbb{H}^{n+1}$ is equivalent to the following initial value problem
\begin{equation}\label{s5:flow-u}
\left\{\begin{aligned}
\frac{\partial}{\partial t}\varphi=&~-F((A_{ij})^{-1})+\varphi\phi(t),\\
\varphi(\cdot,0)=&~\varphi_0(\cdot)
\end{aligned}\right.
\end{equation}
on $S^n\times [0,T)$, where $\varphi=e^u$ and $A_{ij}$ is the matrix defined in \eqref{eq:Ainphi}.
\end{prop}
\proof
Suppose that $X(\cdot,t):M\to \mathbb{H}^{n+1}, t\in [0,T)$ is a family of smooth, closed and strictly h-convex hypersurfaces satisfying the flow \eqref{flow-VMCF-2}. Then as explained in \S \ref{sec:5-1}, the horospherical Gauss map $\mathbf{e}$ is a diffeomorphism from $M_t=X(M,t)$ to $S^n$. We can reparametrize $M_t$ such that $\bar{X}=X\circ \mathbf{e}^{-1}$ is a family of smooth embeddings from $S^n$ to $\mathbb{H}^{n+1}$. Then
\begin{equation*}
\frac{\partial}{\partial t}\bar{X}(z,t)= \frac{\partial}{\partial t}{X}(p,t)+ \frac{\partial X}{\partial p_i} \frac{\partial p^i}{\partial t},
\end{equation*}
where $z\in S^n$ and $p=\mathbf{e}^{-1}(z)\in M_t$. Since $\frac{\partial X}{\partial p_i}$ is tangent to $M_t$, we have
\begin{equation}\label{s5:u-1}
\frac{\partial}{\partial t}\bar{X}(z,t)\cdot \nu(z,t)= \frac{\partial}{\partial t}{X}(p,t)\cdot\nu(z,t)=\phi(t)-F(\mathcal{W}-I).
\end{equation}
On the other hand, by \eqref{eq:nue2} we have
\begin{equation}\label{s5:u-1-1}
\bar{X}(z,t)-\nu(z,t)=e^{-u(z,t)}(z,1),
\end{equation}
where $u(\cdot,t)$ is the horospherical support function of $M_t$ and $(z,1)\in \mathbb{R}^{n+1,1}$ is a null vector. Differentiating \eqref{s5:u-1-1} in time gives that
\begin{equation*}
\frac{\partial}{\partial t}\bar{X}(z,t)- \frac{\partial}{\partial t}\nu(z,t)=-e^{-u(z,t)} \frac{\partial u}{\partial t}(z,1).
\end{equation*}
Then
\begin{align}\label{s5:u-2}
\frac{\partial}{\partial t}\bar{X}(z,t)\cdot\nu(z,t) =& -e^{-u(z,t)} \frac{\partial u}{\partial t}(z,1)\cdot \nu(z,t) \nonumber\\
=& -e^{-u(z,t)} \frac{\partial u}{\partial t}(z,1)\cdot (\bar{X}(z,t)-e^{-u(z,t)}(z,1)) \nonumber\\
=&-e^{-u(z,t)} \frac{\partial u}{\partial t}(z,1)\cdot \frac 12e^{u(z,t)}(-z,1) \nonumber\\
=& \frac{\partial u}{\partial t},
\end{align}
where we used \eqref{eq:nue2} and \eqref{eq:barX3}. Combining \eqref{s5:u-1} and \eqref{s5:u-2} implies that
\begin{equation}\label{s5:u-3}
\frac{\partial u}{\partial t}=\phi(t)-F(\mathcal{W}-I).
\end{equation}
Therefore $\varphi=e^u$ satisfies
\begin{align}
\frac{\partial \varphi}{\partial t}=&e^u\phi(t)-F(e^u(\mathcal{W}-I))\nonumber\\
=&\varphi\phi(t)-F((A_{ij})^{-1})
\end{align}
with $A_{ij}$ defined as in \eqref{eq:Ainphi}.
Conversely, suppose that we have a smooth solution $\varphi(\cdot,t)$ of the initial value problem \eqref{s5:flow-u} with $A_{ij}$ positive definite. Then by the discussion in \S \ref{sec:5-4}, the map $\bar{X}$ given in \eqref{eq:barX3} using the function $u=\log \varphi$ defines a family of smooth h-convex hypersurfaces in $\mathbb{H}^{n+1}$. We claim that we can find a family of diffeomorphisms $\xi(\cdot,t): S^n\to S^n$ such that $X(z,t)=\bar{X}(\xi(z,t),t)$ solves the flow equation \eqref{flow-VMCF-2}. Since
\begin{align*}
\frac{\partial}{\partial t}{X}(z,t)=& \frac{\partial}{\partial t}\bar{X}(\xi,t)+\partial_i \bar{X} \frac{\partial \xi^i}{\partial t}\\
=&(\frac{\partial}{\partial t}\bar{X}(\xi,t)\cdot \nu(\xi,t))\nu(\xi,t)+(\frac{\partial}{\partial t}\bar{X}(\xi,t))^{\top}+\partial_i \bar{X} \frac{\partial \xi^i}{\partial t}\\
=& (\phi(t)-F(\mathcal{W}-I))\nu(\xi,t)+(\frac{\partial}{\partial t}\bar{X}(\xi,t))^{\top}+\partial_i \bar{X} \frac{\partial \xi^i}{\partial t},
\end{align*}
where $(\cdot)^{\top}$ denotes the tangential part, it suffices to find a family of diffeomorphisms $\xi:S^n\to S^n$ such that
\begin{equation*}
(\frac{\partial}{\partial t}\bar{X}(\xi,t))^{\top}+\partial_i \bar{X} \frac{\partial \xi^i}{\partial t}=0,
\end{equation*}
which is equivalent to
\begin{equation}\label{s5:u-4}
(\frac{\partial}{\partial t}\bar{X}(\xi,t))^{\top}\cdot E_j-A_{ij}\frac{\partial \xi^i}{\partial t}=0.
\end{equation}
By assumption $A_{ij}$ is positive definite on $S^n\times [0,T)$, the standard theory of the ordinary differential equations implies that the system \eqref{s5:u-4} has a unique smooth solution for the initial condition $\xi(z,0)=z$. This completes the proof.
\endproof
\section{Proof of Theorem \ref{thm1-5}}\label{sec:thm5-pf}
In this section, we will give the proof of Theorem \ref{thm1-5}.
\subsection{Pinching estimate}\label{sec:Pinch}
Firstly, we prove the following pinching estimate for the shifted principal curvatures of the evolving hypersurfaces along the flow \eqref{flow-VMCF-2}.
\begin{prop}
Let $M_t$ be a smooth solution to the flow \eqref{flow-VMCF-2} on $[0,T)$ and assume that $F$ satisfies the assumption in Theorem \ref{thm1-5}. Then there exists a constant $C>0$ depending only on $M_0$ such that
\begin{equation}\label{s5:pinc-1}
\lambda_n~\leq~C\lambda_1
\end{equation}
for all $t\in [0,T)$, where $\lambda_n=\kappa_n-1$ is the largest shifted principal curvature and $\lambda_1=\kappa_1-1$ is the smallest shifted principal curvature.
\end{prop}
\proof
We consider the four cases of $F$ separately.
(i). $F$ is concave and $F$ vanishes on the boundary of the positive cone $\Gamma_+$. Define a function $G=F^{-1}\mathrm{tr}(S)$ on $M\times [0,T)$. Then the equations \eqref{s2:evl-F} and \eqref{s2:evl-S} imply that
\begin{align}\label{s5:G1}
\frac{\partial}{\partial t}G =& ~F^{-1}\frac{\partial}{\partial t}\mathrm{tr}(S)-F^{-2}\mathrm{tr}(S)\frac{\partial}{\partial t}F \displaybreak[0]\nonumber\\
= & ~\dot{F}^{kl}\nabla_k\nabla_lG+2F^{-1}\dot{F}^{kl}\nabla_kF\nabla_lG+F^{-1}\sum_{i=1}^n\ddot{F}^{kl,pq}\nabla_ih_{kl}\nabla^ih_{pq}\displaybreak[0]\nonumber\\
&\quad +\phi(t)f^{-2}\left(\mathrm{tr}(S)\sum_k\dot{f}^k\lambda_k^2-f|S|^2\right)+f^{-1}\left(n\sum_k\dot{f}^k\lambda_k^2-|S|^2\sum_k\dot{f}^k\right).
\end{align}
Since $F$ is concave, by the inequality \eqref{s2:f-conc} we have
\begin{align*}
\mathrm{tr}(S)\sum_k\dot{f}^k\lambda_k^2-f|S|^2=& \sum_{k,l}\left(\dot{f}^k\lambda_k^2\lambda_l-\dot{f}^k\lambda_k\lambda_l^2\right) \\
= &~\frac 12\sum_{k,l}(\dot{f}^k-\dot{f}^l)(\lambda_k-\lambda_l)\lambda_k\lambda_l~\leq~0,
\end{align*}
and
\begin{align*}
n\sum_k\dot{f}^k\lambda_k^2-|S|^2\sum_k\dot{f}^k= & \sum_{k,l}\left(\dot{f}^k\lambda_k^2-\dot{f}^k\lambda_l^2\right) \\
= &~\frac 12\sum_{k,l}(\dot{f}^k-\dot{f}^l)(\lambda_k^2-\lambda_l^2)~\leq~0.
\end{align*}
Thus the zero order terms of \eqref{s5:G1} are always non-positive. The concavity of $F$ also implies that the third term of \eqref{s5:G1} is non-positive. Then we have
\begin{align}\label{s5:G2}
\frac{\partial}{\partial t}G \leq& ~\dot{F}^{kl}\nabla_k\nabla_lG+2F^{-1}\dot{F}^{kl}\nabla_kF\nabla_lG.
\end{align}
The maximum principle implies that the supremum of $G$ over $M_t$ is decreasing in time along the flow \eqref{flow-VMCF-2}. The assumption that $f$ approaches zero on the boundary of the positive cone $\Gamma_+$ then guarantees that the region $\{G(t)\leq \sup_{t=0}G\}\subset \Gamma_+$ does not touch the boundary of $\Gamma_+$. Since $G$ is homogeneous of degree zero with respect to $\lambda_i$, this implies that $\lambda_n\leq C\lambda_1$ for some constant $C>0$ depending only on $M_0$ for all $t\in [0,T)$.
(ii). $F$ is concave and inverse concave. Define a tensor $T_{ij}=S_{ij}-\varepsilon~ \mathrm{tr}(S)\delta_i^j$, where $\varepsilon$ is chosen such that $T_{ij}$ is positive definite initially. Clearly, $0<\varepsilon\leq \frac 1n$. The evolution equation \eqref{s2:evl-S} implies that
\begin{align}\label{s5:evl-T}
\frac{\partial}{\partial t}T_{ij} =&~ \dot{F}^{kl}\nabla_k\nabla_lT_{ij}+\ddot{F}^{kl,pq}\nabla_ih_{kl}\nabla_jh_{pq}-\varepsilon\left(\sum_{i=1}^n\ddot{F}^{kl,pq}\nabla_ih_{kl}\nabla^ih_{pq}\right)\delta_i^j\nonumber\\ &\quad +\left(\sum_{k=1}^n\dot{f}^k\lambda_k^2+2f-2\phi(t)\right)T_{ij} -\left(\phi(t)+\sum_{k=1}^n\dot{f}^k\right)\left(T_i^kT_{kj}+2\varepsilon \mathrm{tr}(S)T_{ij}\right)\nonumber\displaybreak[0]\\
&\quad +\varepsilon \left(\phi(t)+\sum_{k=1}^n\dot{f}^k\right)\left(|S|^2-\varepsilon (\mathrm{tr}(S))^2\right)\delta_i^j+\sum_{k=1}^n\dot{f}^k\lambda_k^2(1-\varepsilon n)\delta_i^j.
\end{align}
We will apply the tensor maximum principle in Theorem \ref{s2:tensor-mp} to show that $T_{ij}$ is positive definite for $t>0$. If not, there exists a first time $t_0>0$ and some point $x_0\in M_{t_0}$ such that $T_{ij}$ has a null vector $v\in T_{x_0}M_{t_0}$, i.e., $T_{ij}v^j=0$ at $(x_0,t_0)$. The second line of \eqref{s5:evl-T} satisfies the null vector condition and can be ignored. The last line of \eqref{s5:evl-T} is also nonnegative, since $0<\varepsilon<\frac 1n$ and $|S|^2\geq (\mathrm{tr}(S))^2/n$. For the gradient terms in \eqref{s5:evl-T}, Theorem 4.1 of \cite{And2007} implies that
\begin{align*}
&\ddot{F}^{kl,pq}\nabla_ih_{kl}\nabla_jh_{pq}v^iv^j-\varepsilon\left(\sum_{i=1}^n\ddot{F}^{kl,pq}\nabla_ih_{kl}\nabla^ih_{pq}\right)|v|^2\displaybreak[0]\\ &+\sup_{\Lambda}2a^{kl}\left(2\Lambda_k^p\nabla_lT_{ip}v^i-\Lambda_k^p\Lambda_l^qT_{pq}\right)\geq 0
\end{align*}
for the null vector $v$ provided that $F$ is concave and inverse concave. Thus by Theorem \ref{s2:tensor-mp}, the tensor $T_{ij}$ is positive definite for $t\in [0,T)$. Equivalently,
\begin{equation*}
\lambda_1~\geq~\varepsilon (\lambda_1+\cdots+\lambda_n)
\end{equation*}
for any $t\in [0,T)$, which implies the pinching estimate \eqref{s5:pinc-1}.
(iii). $F$ is inverse concave and $F_*$ approaches zero on the boundary of $\Gamma_+$. In this case, we define $T_{ij}=S_{ij}-\varepsilon F\delta_i^j$, where $\varepsilon$ is chosen such that $T_{ij}$ is positive definite initially. By \eqref{s2:evl-F} and \eqref{s2:evl-S},
\begin{align}\label{s5:evl-T-i}
\frac{\partial}{\partial t}T_{ij} =&~ \dot{F}^{kl}\nabla_k\nabla_lT_{ij}+\ddot{F}^{kl,pq}\nabla_ih_{kl}\nabla^jh_{pq} +(\dot{f}^{k}\lambda_k^2+2f-2\phi(t))S_{ij}\nonumber\\
&\quad -(\phi(t)+\sum_{k=1}^n\dot{f}^{k}) S_{ik}S_{kj}+\dot{f}^{k}\lambda_k^2\delta_i^j-\varepsilon (F-\phi(t))\dot{f}^{k}\lambda_k(\lambda_k+2)\delta_i^j.
\end{align}
Suppose $v=e_1$ is the null eigenvector of $T_{ij}$ at $(x_0,t_0)$ for some first time $t_0>0$. Denote the zero order terms of \eqref{s5:evl-T-i} by $Q_0$. At the point $(x_0,t_0)$, $\varepsilon F$ is the smallest eigenvalue of $S_{ij}$ with corresponding eigenvector $v$. Then
\begin{align*
Q_0v^iv^j =& (\dot{f}^k\lambda_k^2+2f-2\phi(t))\varepsilon f|v|^2 +(f-\phi(t)-\dot{f}^k\kappa_k)\varepsilon^2f^2|v|^2 \nonumber\\
& +\dot{f}^k\lambda_k^2|v|^2-\varepsilon (f-\phi(t))\dot{f}^k\lambda_k(\lambda_k+2)|v|^2\displaybreak[0]\nonumber\\
=&\dot{f}^k\lambda_k^2(1+\varepsilon \phi(t))|v|^2-\varepsilon^2f^2(\sum_k\dot{f}^k+\phi(t))|v|^2\displaybreak[0]\nonumber\\
=&|v|^2\biggl(\dot{f}^k\lambda_k\varepsilon(\lambda_k-\varepsilon f)\phi(t)+\sum_k\dot{f}^k(\lambda_k^2-\varepsilon^2f^2)\biggr)\geq ~0.
\end{align*}
By Theorem \ref{s2:tensor-mp}, to show that $T_{ij}$ remains positive definite for $t>0$, it suffices to show that
\begin{align*
Q_1: =& \ddot{F}^{kl,pq}\nabla_1h_{kl}\nabla_1h_{pq} +2\sup_{\Lambda}\dot{F}^{kl}\left(2\Lambda_k^p\nabla_lT_{1p}-\Lambda_k^p\Lambda_l^qT_{pq}\right)~\geq~0.
\end{align*}
Note that $T_{11}=0$ and $\nabla_kT_{11}=0$ at $(x_0,t_0)$, the supremum over $\Lambda$ can be computed exactly as follows:
\begin{align*}
2\dot{F}^{kl}\left(2\Lambda_k^p\nabla_lT_{1p}-\Lambda_k^p\Lambda_l^qT_{pq}\right) =&2\sum_{k=1}^n\sum_{p=2}^n\dot{f}^k\left(2\Lambda_k^p\nabla_kT_{1p}-(\Lambda_k^p)^2T_{pp}\right)\\
=&2 \sum_{k=1}^n\sum_{p=2}^n\dot{f}^k\left(\frac{(\nabla_kT_{1p})^2}{T_{pp}}-\left(\Lambda_k^p-\frac{\nabla_kT_{1p}}{T_{pp}}\right)^2T_{pp}\right).
\end{align*}
It follows that the supremum is obtained by choosing $\Lambda_k^p=\frac{\nabla_kT_{1p}}{T_{pp}}$. The required inequality for $Q_1$ becomes:
\begin{align*
Q_1=& \ddot{F}^{kl,pq}\nabla_1h_{kl}\nabla_1h_{pq} +2 \sum_{k=1}^n\sum_{p=2}^n\dot{f}^k\frac{(\nabla_kT_{1p})^2}{T_{pp}}~\geq~0.
\end{align*}
Using \eqref{s2:F-ddt} to express the second derivatives of $F$ and noting that $\nabla_kT_{1p}=\nabla_kh_{1p}-\varepsilon \nabla_kF\delta_1^p=\nabla_kh_{1p}$ at $(x_0,t_0)$ for $p\neq 1$, we have
\begin{align}\label{s5:Q1-0}
Q_1=& \ddot{f}^{kl}\nabla_1h_{kk}\nabla_1h_{ll}+2\sum_{k>l}\frac{\dot{f}^k-\dot{f}^l}{\lambda_k-\lambda_l}(\nabla_1h_{kl})^2+2 \sum_{k=1}^n\sum_{l=2}^n\frac{\dot{f}^k}{\lambda_{l}-\varepsilon F}(\nabla_1h_{kl})^2.
\end{align}
Since $f$ is inverse concave, the inequality \eqref{s2:f-invcon-1} implies that the first term of \eqref{s5:Q1-0} satisfies
\begin{align*}
\ddot{f}^{kl}\nabla_1h_{kk}\nabla_1h_{ll} \geq & ~ 2f^{-1}(\sum_{k=1}^n\dot{f}^k\nabla_1h_{kk})^2-2\sum_k\frac{\dot{f}^k}{\lambda_k}(\nabla_1h_{kk})^2\displaybreak[0]\\
=& ~ 2f^{-1}(\nabla_1F)^2-2\sum_k\frac{\dot{f}^k}{\lambda_k}(\nabla_1h_{kk})^2.
\end{align*}
Then
\begin{align*}
Q_1\geq &~2f^{-1}(\nabla_1F)^2-2\sum_k\frac{\dot{f}^k}{\lambda_k}(\nabla_1h_{kk})^2\\
&\quad +2\sum_{k>l}\frac{\dot{f}^k-\dot{f}^l}{\lambda_k-\lambda_l}(\nabla_1h_{kl})^2+2 \sum_{k=1}^n\sum_{l=2}^n\frac{\dot{f}^k}{\lambda_{l}-\varepsilon F}(\nabla_1h_{kl})^2\displaybreak[0]\\
\geq &~2f^{-1}(\nabla_1F)^2-2\frac{\dot{f}^1}{\lambda_1}(\nabla_1h_{11})^2-2\sum_{k>1}\frac{\dot{f}^k}{\lambda_k}(\nabla_1h_{kk})^2\\
&+2\sum_{k>1}\frac{\dot{f}^k-\dot{f}^1}{\lambda_k-\lambda_1}(\nabla_kh_{11})^2 -2\sum_{k\neq l>1}\frac{\dot{f}^k}{\lambda_l}(\nabla_1h_{kl})^2\displaybreak[0]\\
&+2\sum_{k>1}\frac{\dot{f}^1}{\lambda_{k}-\varepsilon F}(\nabla_kh_{11})^2+2\sum_{k>1,l>1}\frac{\dot{f}^k}{\lambda_{l}-\varepsilon F}(\nabla_1h_{kl})^2\displaybreak[0]\\
= &~2f^{-1}(\nabla_1F)^2-2\frac{\dot{f}^1}{\lambda_1}(\nabla_1h_{11})^2+2\sum_{k>1}\frac{\dot{f}^k}{\lambda_k-\lambda_1}(\nabla_kh_{11})^2\displaybreak[0]\\
&\quad +2\sum_{k>1,l>1}\dot{f}^k\left(\frac{1}{\lambda_l-\varepsilon F}-\frac 1{\lambda_l}\right)(\nabla_1h_{kl})^2\displaybreak[0]\\
\geq &~2\left(\frac 1{\varepsilon^2F}-\frac{\dot{f}^1}{\lambda_1}\right)(\nabla_1h_{11})^2\displaybreak[0]\\
=& ~2\left(\frac {\sum_{k=1}^n\dot{f}^k\lambda_k}{\varepsilon^2F^2}-\frac{\dot{f}^1}{\lambda_1}\right)(\nabla_1h_{11})^2 ~\geq ~0,
\end{align*}
where we used $\lambda_1=\varepsilon F$ and $\nabla_kh_{11}=\varepsilon \nabla_kF$ at $(x_0,t_0)$, and the inequality in \eqref{s2:f-invcon} due to the inverse concavity of $f$. Theorem \ref{s2:tensor-mp} implies that $T_{ij}$ remains positive definite for $t\in [0,T)$. Equivalently, there holds
\begin{equation}\label{s5:iii-1}
\frac 1{\lambda_1}~\leq ~\frac 1{\varepsilon}f(\lambda)^{-1}=\frac 1{\varepsilon} f_*(\frac 1{\lambda_1},\cdots,\frac 1{\lambda_n})
\end{equation}
for all $t\in [0,T)$. Since $f_*$ approaches zero on the boundary of the positive cone $\Gamma_+$, the estimate \eqref{s5:iii-1} and Lemma 12 of \cite{Andrews-McCoy-Zheng} give the pinching estimate \eqref{s5:pinc-1}.
(iv). $n=2$. In this case, we don't need any second derivative condition on $F$. Define
\begin{equation*}
G~=~\left(\frac{\lambda_2-\lambda_1}{\lambda_2+\lambda_1}\right)^2.
\end{equation*}
Then $G$ is homogeneous of degree zero of the shifted principal curvatures $\lambda_1,\lambda_2$. The evolution equation \eqref{s2:evl-S} implies that
\begin{align}\label{s5:G3}
\frac{\partial}{\partial t}G =& \dot{F}^{kl}\nabla_k\nabla_lG+\left(\dot{G}^{ij}\ddot{F}^{kl,pq}-\dot{F}^{ij}\ddot{G}^{kl,pq}\right)\nabla_iS_{kl}\nabla^jS_{pq}\displaybreak[0]\nonumber\\ & -\left(\phi(t)+\sum_k\dot{f}^k\right)\dot{G}^{ij}S_{ik}S_{kj}+(\sum_k\dot{f}^k\lambda_k^2)\dot{G}^{ij}\delta_i^j.
\end{align}
The zero order terms of \eqref{s5:G3} are equal to
\begin{equation*}
Q_0=-4G\frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}\left(\phi(t)+\sum_k\dot{f}^k\right)-\frac{4G}{\lambda_1+\lambda_2}(\sum_k\dot{f}^k\lambda_k^2)~\leq~0.
\end{equation*}
The same argument as in \cite{And2010} gives that the gradient terms of \eqref{s5:G3} are non-positive at the critical point of $G$. Then the maximum principles implies that the supremum of $G$ over $M_t$ are non-increasing in time along the flow \eqref{flow-VMCF-2}. This gives the pinching estimate \eqref{s5:pinc-1} and the strict h-convexity of $M_t$ for all $t\in [0,T)$.
\endproof
\subsection{Shape estimate}
Denote by $\rho_-(t), \rho_+(t)$ the inner radius and outer radius of $\Omega_t$. Then there exists two points $p_1,p_2 \in \mathbb{H}^{n+1}$ such that $ B_{\rho_-(t)}(p_1)\subset \Omega_t\subset B_{\rho_+(t)}(p_2)$. By Corollary \ref{s5:cor}, the modified quermassintegral $\widetilde{W}_l$ is monotone under the inclusion of \emph{h-convex} domains in $\mathbb{H}^{n+1}$. This implies that
\begin{equation*}
\tilde{f}_l(\rho_-(t))=\widetilde{W}_l(B_{\rho_-(t)}(p_1))\leq \widetilde{W}_l(\Omega_t)\leq \widetilde{W}_l(B_{\rho_+(t)}(p_2))=\tilde{f}_l(\rho_+(t)).
\end{equation*}
Along the flow \eqref{flow-VMCF-2}, $\widetilde{W}_l(\Omega_t)=\widetilde{W}_l(\Omega_0)$ is a fixed constant. Therefore, $$\rho_-(t)\leq C\leq \rho_+(t),$$where $C=\tilde{f}_l^{-1}(\widetilde{W}_l(\Omega_0))>0$ depends only on $l,n$ and $\Omega_0$.
On the other hand, since each $\Omega_t$ is \emph{h-convex}, the inner radius and outer radius of $\Omega_t$ satisfy $ \rho_+(t)~\leq ~c(\rho_-(t)+\rho_-(t)^{1/2})$ for some uniform positive constant $c$ (see \cite{Cab-Miq2007,Mak2012}). Thus there exist positive constants $c_1,c_2$ depending only on $n,l, M_0$ such that
\begin{equation}\label{s6:io-radius1}
0<c_1\leq \rho_-(t)\leq \rho_+(t)\leq c_2
\end{equation}
for all time $t\in [0,T)$.
\subsection{$C^2$ estimate}
\begin{prop}\label{s6:F-ub}
Under the assumptions of Theorem \ref{thm1-5} with $\phi(t)$ given in \eqref{s1:phit-2}, we have $F\leq C$ for any $t\in [0,T)$, where $C$ depends on $n,l, M_0$ but not on $T$.
\end{prop}
\proof
For any given $t_0\in [0,T)$, let $B_{\rho_0}(p_0)$ be the inball of $\Omega_{t_0}$, where $\rho_0=\rho_-(t_0)$. Then a similar argument as in \cite[Lemma 4.2]{And-Wei2017-2} yields that
\begin{equation}\label{s6:inball-eqn1}
B_{\rho_0/2}(p_0)\subset \Omega_t,\quad t\in [t_0, \min\{T,t_0+\tau\})
\end{equation}
for some positive $\tau$ depending only on $n,l,\Omega_0$. Consider the support function $u(x,t)=\sinh r_{p_0}(x)\langle \partial r_{p_0},\nu\rangle $ of $M_t$ with respect to the point $p_0$. Then the property \eqref{s6:inball-eqn1} implies that
\begin{equation}\label{s6:sup-1}
u(x,t)~\geq ~\sinh(\frac{\rho_0}2)~=:~2c
\end{equation}
on $M_t$ for any $t\in[t_0,\min\{T,t_0+\tau\})$. On the other hand, the estimate \eqref{s6:io-radius1} implies that $u(x,t)\leq \sinh(2c_2)$ on $M_t$ for all $t\in[t_0,\min\{T,t_0+\tau\})$. Define the auxiliary function
\begin{equation*
W(x,t)=\frac {F(\mathcal{W}-\mathrm{I})}{u(x,t)-c}
\end{equation*}
on $M_t$ for $t\in [t_0,\min\{T,t_0+\tau\})$. Combining \eqref{s2:evl-F} and the evolution equation \eqref{s4:evl-u} for the support function, the function $W$ evolves by
\begin{align}\label{s6:evl-W-1}
\frac{\partial}{\partial t}W= &\dot{F}^{ij}\left(\nabla_j\nabla_iW+\frac 2{u-c}\nabla_iu\nabla_jW\right)\nonumber \\
&\quad-\frac{\phi(t)}{u-c}\left( \dot{F}^{ij}(h_i^kh_{k}^j-\delta_i^j)+W\cosh r_{p_0}(x)\right)\nonumber\\
&\quad +\frac{F}{(u-c)^2}(F+\dot{F}^{kl}h_{kl})\cosh r_{p_0}(x)-\frac{cF}{(u-c)^2}\dot{F}^{ij}h_i^kh_{k}^j-W\dot{F}^{ij}\delta_i^j.
\end{align}
The second line of \eqref{s6:evl-W-1} involves the global term $\phi(t)$ and is clearly non-positive by the h-convexity of the evolving hypersurface. By the homogeneity of $f$ with respect to $\lambda_i=\kappa_i-1$, we have $F+\dot{F}^{kl}h_{kl}=2F+\sum_{k=1}^n\dot{f}^k$ and
\begin{equation*}
\dot{F}^{ij}h_i^kh_{k}^j= \dot{f}^k(\lambda_k+1)^2=\dot{f}^k\lambda_k^2+2f+\sum_k\dot{f}^k~\geq Cf^2,
\end{equation*}
where the last inequality is due to the pinching estimate \eqref{s5:pinc-1}. The last term of \eqref{s6:evl-W-1} is non-positive and can be thrown away. In summary, we arrive at
\begin{align*
\frac{\partial}{\partial t}W\leq & ~\dot{\Psi}^{ij}\left(\nabla_j\nabla_iW+\frac 2{u-c}\nabla_iu\nabla_jW\right)\nonumber \\
&\quad +W^2(2+F^{-1}\sum_{k=1}^n\dot{f}^k)\cosh r_{p_0}(x)- c^2CW^3.
\end{align*}
Note that $\dot{f}^k$ is homogeneous of degree zero, the pinching estimate \eqref{s5:pinc-1} implies that each $\dot{f}^k$ is bounded from above and below by positive constants. Then without loss of generality we can assume that $F^{-1}\sum_{k=1}^n\dot{f}^k\leq 1$ since otherwise $F\leq \sum_{k=1}^n\dot{f}^k\leq C$ for some constant $C>0$. By the upper bound $r_{p_0}(x)\leq 2c_2$, we obtain the following estimate
\begin{align*
\frac{\partial}{\partial t}W\leq &~ \dot{\Psi}^{ij}\left(\nabla_j\nabla_iW+\frac 2{u-c}\nabla_iu\nabla_jW\right)+W^2\left(3\cosh (2c_2)- c^{2}CW\right)
\end{align*}
holds on $[t_0,\min\{T,t_0+\tau\})$. Then the maximum principle implies that $W$ is uniformly bounded from above and the upper bound on $F$ follows by the upper bound on the outer radius in \eqref{s6:io-radius1}.
\endproof
\begin{prop}\label{s6:F-lb}
There exists a positive constant $C$, independent of time $T$, such that $F\geq C>0$.
\end{prop}
\proof
Since the evolving hypersurface $M_{t}$ is strictly h-convex, for each time $t_0\in [0,T)$ there exists a point $p\in \mathbb{H}^{n+1}$ and $x_0\in M_{t_0}$ such that $\Omega_{t_0}\subset B_{\rho_+(t_0)}(p)$ and $\Omega_{t_0}\cap B_{\rho_+(t_0)}(p)=x_0$. By the estimate \eqref{s6:io-radius1} on the outer radius, the value of $F$ at the point $(x_0,t_0)$ satisfies
\begin{equation*}
F(x_0,t_0)\geq \coth \rho_+(t_0)\geq \coth c_2.
\end{equation*}
Recall that the function $F$ satisfies the evolution equation \eqref{s2:evl-F} :
\begin{align}\label{s6:evl-F}
\frac{\partial}{\partial t}F=&~g^{ik}\dot{F}^{ij}\nabla_k\nabla_jF+(F-\phi(t))(\dot{F}^{ij}h_i^kh_k^j-\dot{F}^{ij}\delta_i^j).
\end{align}
By the pinching estimate \eqref{s5:pinc-1} and the upper bound on the curvature proved in Proposition \ref{s6:F-ub}, the equation \eqref{s6:evl-F} is uniformly parabolic and the coefficient of the gradient terms and the lower order terms in \eqref{s6:evl-F} have bounded $C^0$ norm. Then there exists $r>0$ depending only on the bounds on the coefficients of \eqref{s6:evl-F} such that we can apply the Harnack inequality of Krylov and Safonov \cite{KS81} to \eqref{s6:evl-F} in a space-time neighbourhood $B_r(x_0)\times (t_0-r^2,t_0]$ of $x_0$ and deduce the lower bound $F\geq C F(x_0,t_0)\geq C>0$ in a smaller neighbourhood $B_{r/2}(x_0)\times (t_0-\frac 14r^2,t_0]$. Note that the diameter $r$ of the space-time neighbourhood is not dependent on the point $(x_0,t_0)$. Consider the boundary point $x_1\in \partial B_{r/2}(x_0)$. We can look at the equation \eqref{s6:evl-F} in a neighborhood $B_r(x_1)\times (t_0-r^2,t_0]$ of the point $(x_1,t_0)$. The Harnack inequality implies that $F\geq C F(x_1,t_0)\geq C>0$ in $B_{r/2}(x_1)\times (t_0-\frac 14r^2,t_0]$. Since the diameter of each $M_{t_0}$ is uniformly bounded from above, after a finite number of iterations, we conclude that $F\geq C>0$ on $M_{t_0}$ for a uniform constant $C$ independent of $t_0$.
\endproof
The pinching estimate \eqref{s5:pinc-1} together with the bounds on $F$ proven in Proposition \ref{s6:F-ub} and Proposition \ref{s6:F-lb} implies that the shifted principal curvatures $\lambda=(\lambda_1,\cdots,\lambda_n)$ satisfy
\begin{equation*}
0<C^{-1}~\leq ~\lambda_i~\leq ~C
\end{equation*}
for some constant $C>0$ and $t\in [0,T)$. This gives the uniform $C^2$ estimate of the evolving hypersurfaces $M_t$. Moreover, the global term $\phi(t)$ given in \eqref{s1:phit-2} satisfies $0<C^{-1}\leq \phi(t)\leq C$ for some constant $C>0$.
\subsection{Long time existence and convergence}
If $F$ is inverse-concave, by applying the similar argument as \cite{And-Wei2017-2,Mcc2017} (see also \cite{TW2013}) to the equation \eqref{s5:flow-u}, we can first derive the $C^{2,\alpha}$ estimate and then the $C^{k,\alpha}$ estimate for all $k\geq 2$. If $F$ is concave or $n=2$, we write the flow \eqref{flow-VMCF-2} as a scalar parabolic PDE for the radial function as follows: Since each $M_t$ is strictly h-convex, we write $M_{t}$ as a radial graph over a geodesic sphere for a smooth function $\rho$ on $S^n$. Let $\{\theta^i\}, i=1,\cdots,n$ be a local coordinate system on $S^n$. The induced metric on $M_{t_0}$ from $\mathbb{H}^{n+1}$ takes the form
$$ g_{ij}=\bar{\nabla}_i\rho \bar{\nabla}_j\rho+\sinh^2\rho\bar{g}_{ij},$$
where $\bar{g}_{ij}$ denotes the round metric on $S^n$. Up to a tangential diffeomorphism, the flow equation \eqref{flow-VMCF-2} is equivalent to the following scalar parabolic equation
\begin{equation}\label{s6:graph-flow}
\frac{\partial }{\partial t}\rho=~(\phi(t)-F(\mathcal{W}-\mathrm{I}))\sqrt{1+{|\bar{\nabla}\rho|^2}/{\sinh^2\rho}}.
\end{equation}
for the smooth function $\rho(\cdot,t)$ on $S^n$. The Weingarten matrix $\mathcal{W}=(h_i^j)$ can be expressed as
\begin{equation*
h_i^j=~\frac{\coth \rho}{v}\delta_i^j+\frac{\coth \rho}{v^3\sinh^2 \rho}\bar{\nabla}^j\rho\bar{\nabla}_i\rho-\frac {\tilde{\sigma}^{jk}}{v\sinh^2\rho}\bar{\nabla}_{k}\bar{\nabla}_i\rho,
\end{equation*}
where
\begin{equation*
v=\sqrt{1+{|\bar{\nabla}\rho|^2}/{\sinh^2\rho}},\quad \mathrm{and }\quad \tilde{\sigma}^{jk}~=~\sigma^{jk}-\frac{\bar{\nabla}^j\rho\bar{\nabla}^k\rho}{v^2\sinh^2\rho}.
\end{equation*}
Thus we can apply the argument as in \cite{And2004,McC2005} to derive the higher regularity estimate. Therefore, for any $F$ satisfying the assumption of Theorem \ref{thm1-5}, the solution of the flow \eqref{flow-VMCF-2} exists for all time $t\in[0,\infty)$ and remains smooth and strictly h-convex. Moreover, the Alexandrov reflection argument as in \cite[\S 6]{And-Wei2017-2} implies that the flow converges smoothly as time $t\to\infty$ to a geodesic sphere $\partial B_{r_{\infty}}$ which satisfies $\widetilde{W}_l(B_{r_{\infty}})=\widetilde{W}_l(\Omega_{0})$. This finishes the proof of Theorem \ref{thm1-5}.
\section{Conformal deformation in the conformal class of $\bar g$}
In this section we mention an interesting connection (closely related to the results of \cite{EGM}) between flows of h-convex hypersurfaces in hyperbolic space by functions of principal curvatures, and conformal flows of conformally flat metrics on $S^n$. This allows us to translate some of our results to convergence theorems for metric flows, and our isoperimetric inequalities to corresponding results for conformally flat metrics.
The crucial observation is that there is a correspondence between conformally flat metrics on $S^n$ satisfying a certain curvature inequality, and horospherically convex hypersurfaces. To describe this, we recall that the isometry group of $\mathbb H^{n+1}$ coincides with $O_+(n+1,1)$, the group of future-preserving linear isometries of Minkowski space. This also coincides with the M\"obius group of conformal diffeomorphisms of $S^n$, by the following correspodence: If $L\in O_+(n+1,1)$, we define a map $\rho_L$ from $S^n$ to $S^n$ by
$$
\rho_L({\mathbf e}) = \pi(L({\mathbf e},1)),
$$
where $\pi(x,y)=\frac{x}{y}$ is the radial projection from the future null cone to the sphere at height $1$.
This defines a group homomorphism from $O_+(n+1,1)$ to the group of M\"obius transformations. We have the following result:
\begin{prop}
If $L\in O_+(n+1,1)$ and $M\subset{\mathbb H}^{n+1}$ is a horospherically convex hypersurface with horospherical support function $u:\ S^n\to{\mathbb R}$, denote by $u_L$ the horospherical support function of $L(M)$. Then $\rho_L$ is an isometry from ${\mathrm e}^{-2u}\bar g$ to ${\mathrm e}^{-2u_L}\bar g$. That is,
$$
{\mathrm e}^{-2u({\mathbf e})}\bar g_{{\mathbf e}}(v_1,v_2) = {\mathrm e}^{-2u_L(\rho_L({\mathbf e}))}\bar g_{\rho_L({\mathbf e})}(D\rho_L(v_1),D\rho_L(v_2))
$$
for all ${\mathbf e}\in S^n$ and $v_1,v_2\in T_{\mathbf e} S^n$.
\end{prop}
\begin{proof}
We compute:
\begin{align*}
{\mathrm e}^{-u({\mathbf e})}&= -X\cdot ({\mathbf e},1)\\
&= -L(X)\cdot L({\mathbf e},1)\\
&= - L(X)\cdot \mu ({\mathbf e}_L,1)\qquad\text{where\ }\mu = |L({\mathbf e},1)\cdot(0,1)|\\
&=\mu {\mathrm e}^{-u_L({\mathbf e}_L)}.
\end{align*}
On the other hand the M\"obius transformation $\rho_L$ is a conformal transformation with conformal factor
$\mu=|L({\mathbf e},1)\cdot(0,1)|$. The result follows directly.
\end{proof}
\begin{cor}
Isometry invariants of a horospherically convex hypersurface $M$ are M\"obius invariants of the conformally flat metric
$\tilde g={\mathrm e}^{-2u}\bar g$, and vice versa. In particular, Riemannian invariants of $g$ are isometry invariants of $M$.
\end{cor}
Computing explicitly, we find that for $n>2$ the Schouten tensor
$$\tilde S_{ij} = \frac{1}{n-2}\left(\tilde R_{ij}-\frac{\tilde R}{2(n-1)}\tilde g_{ij}\right)$$ of $\tilde g$ (which completely determines the curvature tensor for a conformally flat metric) is given by
\begin{align*}
\tilde S_{ij} &= \frac12\bar g_{ij}+\bar\nabla_i\bar\nabla_ju+u_iu_j-\frac12|\bar\nabla u|^2\bar g_{ij}\\
&= {\mathrm e}^{-u}A_{ij}+\frac12 \tilde{g}_{ij}\\
&= \left[(\mathcal{W}-\mathrm I)^{-1}\right]_i^p \tilde g_{pk}+\frac12 \tilde g_{ij}.
\end{align*}
It follows that the eigenvalues of $\tilde S_{ij}$ (with respect to $\tilde g_{ij}$) are $\frac12 + \frac{1}{\lambda_i}$, where $\lambda_i = \kappa_i-1$. When $n=2$ the tensor $\tilde{S}_{ij}$ defined by the right-hand side of the above equation is by construction M\"obius-invariant, and so gives a M\"obius invariant of $\tilde g$ which is not a Riemannian invariant. This tensor still has the same relation to the principal curvatures of the corresponding h-convex hypersurface.
We observe that this connection between the Schouten tensor of $\tilde g$ and the Weingarten map of the hypersurface leads to a conincidence between the corresponding evolution equations: If a family of h-convex hypersurfaces $M_t=X(M,t)$ evolves according to a curvature-driven evolution equation of the form
$$
\frac{\partial X}{\partial t} = -F({\mathcal W}-{\mathrm I},t)\nu
$$
then the metric $\tilde g$ satisfies $\tilde S>\frac12\tilde g$, and evolves according to the parabolic conformal flow
$$
\frac{\partial\tilde g}{\partial t} = 2F((\tilde S-\frac12\tilde g)^{-1},t)\tilde g.
$$
In particular the convergence theorems for hypersurface flows correspond to convergence theorems for the corresponding conformal flows, and the resulting geometric inequalities for hypersurfaces imply corresponding geometric inequalities for the metric $\tilde g$.
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| 55,264 |
\section{Introduction}
The study of {\it secant varieties} and {\it tangential varieties} is very classical in Algebraic Geometry and goes back to the school of the XIX century. In the last decades, these topics received renewed attention because of their \Alessandro{connections} with more applied sciences \Alessandro{which uses {\it additive decompositions of tensors}}. \Alessandro{For us,} {\it tensors} are multidimensional arrays \Alessandro{of complex numbers and} classical geometric objects as Veronese, Segre, Segre-Veronese varieties, and their tangential varieties, are parametrised by tensors with particular symmetries and structure. \Alessandro{Their $s$-secant variety is the closure of the locus of linear combinations of $s$ many of those} particular tensors. We refer to \cite{L} for an exhaustive description of the fruitful use of classical algebraic geometry in problems regarding tensors decomposition.
A very important invariant \Alessandro{of these varieties} is their {\it dimension}.
\medskip
A rich literature has been devoted to studying dimensions of secant varieties of special projective varieties. In particular, we mention:
\begin{enumerate}
\item[1.] {\it Veronese varieties}, completely solved by J. Alexander and A. Hirschowitz \cite{AH};
\item[2.] {\it Segre varieties}, solved in few specific cases, e.g., see \cite{AOP, CGG02-Segre, CGG11-SegreP1};
\item[3.] {\it Segre-Veronese varieties}, solved in even fewer specific cases, e.g., see \cite{AB13-SegreVeronese, CGG05-SegreVeronese, Abr08};
\item[4.] {\it tangential varieties of Veronese varieties}, completely solved by H. Abo and N. Vannieuwenhoven, see \cite{AV18, BCGI09}.
\end{enumerate}
In this paper, we consider the following question.
\begin{question}\label{question: dimension secants}
What is the dimension of secant varieties of tangential varieties of Segre-Veronese surfaces?
\end{question}
Let $a, b$ be positive integers. We define the {\bf Segre-Veronese embedding} of $\mathbb{P}^1\times \mathbb{P}^1$ in bi-degree $(a,b)$ as the embedding of $\mathbb{P}^1\times\mathbb{P}^1$ with the linear system $\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(a,b)$ of curves of bi-degree $(a,b)$, namely
\begin{eqnarray*}
\nu_{a,b} : &\mathbb{P}^1 \times \mathbb{P}^1 & \rightarrow \quad \quad \quad \mathbb{P}^{(a+1)(b+1)-1},\\
&([s_{0}:s_{1}],[t_{0}:t_{1}]) & \mapsto \quad [s_{0}^at_{0}^b:s_{0}^{a-1}s_{1}t_{0}^b:\ldots:s_{1}^at_{1}^b].
\end{eqnarray*}
The image of $\nu_{a,b}$ is the {\bf Segre-Veronese surface} of bi-degree $(a,b)$, denoted by $SV_{a,b}$, \Alessandro{which is parametrized by {\it decomposable partially symmetric tensors}.}
Let $V_1$ and $V_2$ be two $2$-dimensional $\mathbb{C}$-vector spaces. Let $\mathit{Sym}(V_i) = \bigoplus_{d\geq 0}\mathit{Sym}^d(V_i)$ be the symmetric algebra over $V_i$, for $i = 1,2$. If we fix basis $(x_0,x_1)$ and $(y_0,y_1)$ for $V_1$ and $V_2$, respectively, then we have the identifications of the respective symmetric algebras with the polynomial rings \Alessandro{
\begin{equation}\label{eq: identif}
\begin{array}{r c l c r c l}
\mathit{Sym}(V_1) &\simeq& \mathbb{C}[x_0,x_1] & \text{and} & \mathit{Sym}(V_2) & \simeq & \mathbb{C}[y_0,y_1], \\
v_1 = (v_{10},v_{11}) &\leftrightarrow& \ell_1 = v_{10}x_0+v_{11}x_1 & & v_2 = (v_{20},v_{21}) &\leftrightarrow& \ell_2 = v_{20}y_0+v_{21}y_1 \\
\end{array}
\end{equation}
}
Therefore, $\mathit{Sym}(V_1) \otimes \mathit{Sym}(V_2)$ is identified with the bi-graded ring $\mathcal{S} = \mathbb{C}[x_0,x_1;y_0,y_1] = \bigoplus_{a,b}\mathcal{S}_{a,b}$, where $\mathcal{S}_{a,b}$ is the $\mathbb{C}$-vector space of bi-homogeneous polynomials of bi-degree $(a,b)$, i.e., $\mathcal{S}_{a,b} = \mathit{Sym}^a(V_1) \otimes_{\mathbb{K}} \mathit{Sym}^b(V_2)$. \Alessandro{In particular, we consider the monomial basis of $\mathcal{S}_{a,b}$ given by
$$
\left\{{a \choose i_0}{b \choose j_0}x_0^{i_0}x_1^{i_1}y_0^{j_0}y_1^{j_1} ~|~ \substack{i_0,i_1,j_0,j_1 \geq 0 \\ i_0+i_1 = a, j_0+j_1 = b}\right\}.
$$
In this way, the Segre-Veronese embedding of $\mathbb{P}^1\times\mathbb{P}^1$ in bi-degree $(a,b)$ can be rewritten as }
\begin{eqnarray*}
\nu_{a,b} : & \mathbb{P}(V_1) \times \mathbb{P}(V_2) & \rightarrow \quad \mathbb{P}(\Alessandro{\mathit{Sym}^a(V_1)\otimes\mathit{Sym}^b(V_2)}),\\
&([v_1],[v_2]) & \mapsto [v_1^{\otimes a}\otimes v_2^{\otimes b}].
\end{eqnarray*}
Throughout all the paper\Alessandro{, by \eqref{eq: identif}, we} identify the tensor $v_1^{\otimes a}\otimes v_2^{\otimes b}$ with the polynomial $\ell_1^a\ell_2^b$ \Alessandro{and we view the Segre-Veronese variety of bi-degree $(a,b)$ as the projective variety parametrized by these particular bi-homogeneous polynomials.}
\smallskip
Given any projective variety $X \subset \mathbb{P}^N$, we define the {\bf tangential variety} of $X$ as the Zariski closure of the union of the tangent spaces at smooth points of $X$, i.e., if $U \subset X$ denotes the open subset of smooth points of $X$, then it is
$$
\calT(X) := \overline{\bigcup_{P \in U} T_P(X)} \subset \mathbb{P}^N,
$$
where $T_P(X)$ denotes the tangent space of $X$ at the point $P$.
\smallskip
Given any projective variety $X \subset \mathbb{P}^N$, we define the {\bf $s$-secant variety} of $X$ as the Zariski closure of the union of all linear spans of $s$-tuples of points on $X$, i.e.,
\begin{equation}\label{equation: expected dimension}
\sigma_s(X) := \overline{\bigcup_{P_1,\ldots,P_s \in X} \left\langle P_1,\ldots,P_s \right\rangle} \subset \mathbb{P}^N,
\end{equation}
where $\langle - \rangle$ denotes the linear span of the points.
As we said before, we are interested in the dimension of these varieties. By parameter count, we have an {\bf expected dimension} of $\sigma_s(X)$ which is
$$
{\rm exp}.\dim\sigma_s(X) = \min\{N, s\dim(X) + (s-1)\}.
$$
However, we have varieties whose $s$-secant variety has dimension smaller than the expected one and we call them {\it defective varieties}. In this article we prove that \Alessandro{the} tangential varieties \Alessandro{of all the Segre-Veronese surfaces} are never defective.
\setcounter{section}{4}
\setcounter{theorem}{5}
\begin{theorem}
Let $a,b$ be positive integers with $ab > 1$. Then, the tangential variety of any Segre-Veronese surface $SV_{a,b}$ is non defective, i.e., all secant varieties have the expected dimension; namely,
$$
\dim\sigma_s(\calT(SV_{a,b})) = \min\{(a+1)(b+1), 5s\}-1.
$$
\end{theorem}
\Alessandro{In order to prove our result, we use an approach already used in the literature which involves the study of {\it Hilbert functions} of {\it $0$-dimensional schemes} in the multiprojective space $\mathbb{P}^1\times\mathbb{P}^1$. In particular, first we use a method introduced by the first author, with A. V. Geramita and A. Gimigliano \cite{CGG05-SegreVeronese}, to reduce our computations to the standard projective plane; and second, we study the dimension of particular linear systems of curves with non-reduced base points by using degeneration techniques which go back to G. Castelnuovo, but have been refined by the enlightening work of J. Alexander and A. Hirschowitz \cite{AHb, AHa, AH, AH00}. However, as far as we know, the particular type of degeneration that we are using has not been exploited before in the literature and we believe that it might be useful to approach other similar problems.}
\setcounter{section}{1}
\paragraph*{\bf Structure of the paper.} In Section \ref{sec: secant and 0-dim schemes}, we show how the dimension of secant varieties can be computed by studying the Hilbert function of $0$-dimensional schemes and we introduce the main tools that we use in our proofs, such as the {\it multiprojective-affine-projective method} and {\it la m\'ethode d'Horace diff\'erentielle}. In Section \ref{sec: lemmata}, we consider the cases of small bi-degrees, i.e., when $b \leq 2$. These will be the base steps for our inductive proof of the general case that we present in Section~\ref{sec: main}.
\smallskip
\paragraph*{\bf Acknowledgements.}
The first author was supported by the Universit\`a degli Studi di Genova through the \Alessandro{``FRA (Fondi per la Ricerca di Ateneo) 2015”}. The second author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R\&D (MDM-2014-0445).
\section{Secant varieties and $0$-dimensional schemes}\label{sec: secant and 0-dim schemes}
In this section we recall some basic constructions and we explain how they are used to reduce the problem of computing dimensions of secant varieties to the problem of computing Hilbert functions of special $0$-dimensional schemes.
\subsection{Terracini's Lemma} A standard way to compute the dimension of an algebraic variety is to look at its tangent space at a general point. In the case of secant varieties, the structure of tangent spaces is very nicely described by a classical result of A.~Terracini \cite{T11}.
\begin{lemma}[Terracini's Lemma \cite{T11}]
Let $X$ be a projective variety. Let $P_1,\ldots,P_s$ be general points on $X$ and let $P$ be a general point on their linear span. Then,
$$
T_P\sigma_s(X) = \left\langle T_{P_1}X,\ldots,T_{P_s}X\right\rangle.
$$
\end{lemma}
\noindent \Alessandro{Therefore, in order to understand the dimension of the general tangent space of the secant variety $\sigma_s(X)$, we need to compute the dimension of the linear span of the tangent spaces to $X$ at $s$ general points. We do it in details for the tangential varieties of Segre-Veronese surfaces.}
\smallskip
Let $\ell_1 \in \mathcal{S}_{1,0}$ and $\ell_2 \in \mathcal{S}_{0,1}$. Then, we consider the bi-homogeneous polynomial $\ell_1^a\ell_2^b \in \mathcal{S}_{a,b}$ which represents a point on the Segre-Veronese variety $SV_{a,b}$. Now, if we consider two general linear forms $m_1 \in \mathcal{S}_{1,0}$ and $m_2 \in \mathcal{S}_{0,1}$, then
$$
\left.\frac{d}{dt}\right|_{t=0}(\ell_1+tm_1)^a(\ell_2+tm_2)^b = a\ell_1^{a-1}\ell_2^bm_1 + b\ell_1^a\ell_2^{b-1}m_2;
$$
therefore, we obtain that
$$
T_{[\ell_1^a\ell_2^b]} SV_{a,b} = \mathbb{P}\left( \left\langle \ell_1^{a-1}\ell_2^b \cdot\mathcal{S}_{1,0},~\ell_1^{a}\ell_2^{b-1} \cdot \mathcal{S}_{0,1} \right\rangle \right).
$$
Hence, the tangential variety $\calT(SV_{a,b})$ is the image of the embedding
\begin{eqnarray*}
\tau_{a,b} : & \big(\mathbb{P}(V_1)\times\mathbb{P}(V_2)\big)\times \big(\mathbb{P}(V_1)\times\mathbb{P}(V_2)\big) & \rightarrow ~~~~~~~~\mathbb{P}({\mathcal{S}_{a,b}}),\\
&~~~\big(([\ell_1],[\ell_2])~;~([m_1],[m_2])\big) & \mapsto [\ell_1^{a-1}\ell_2^bm_1+\ell_1^a\ell_2^{b-1}m_2].
\end{eqnarray*}
\begin{remark}\label{rmk: (a,b) = (1,1)}
The variety $SV_{1,1}$ is the Segre surface of $\mathbb{P}^3$ whose tangential variety clearly fills the entire ambient space. For this reason, we will always consider pairs of positive integers $(a,b)$ where at least one is strictly bigger than $1$. Hence, from now on, we assume $ab > 1$.
\end{remark}
Now, fix $\ell_1,m_1 \in \mathcal{S}_{1,0}$ and $\ell_2,m_2 \in \mathcal{S}_{0,1}$. For any $h_1,k_1 \in \mathcal{S}_{1,0}$ and $h_2,k_2\in \mathcal{S}_{0,1}$, we have
\begin{align*}
\left.\frac{d}{dt}\right|_{t=0} (\ell_1 + th_1)^{a-1}&(m_1+tk_1)(\ell_2+th_2)^b + (\ell_1 + th_1)^{a}(\ell_2+th_2)^{b-1}(m_2+tk_2) = \\
& = (a-1)\ell_1^{a-2}h_1m_1\ell_2^b + \ell_1^{a-1}k_1\ell_2^b + b\ell_1^{a-1}m_1\ell_2^{b-1}h_2 + \\ & \quad \quad + a\ell_1^{a-1}h_1\ell_2^{b-1}m_2 + (b-1)\ell_1^a\ell_2^{b-2}h_2m_2 + \ell_1^a\ell_2^{b-1}k_2.
\end{align*}
\Alessandro{Note that, if $b = 1$ (or $a = 1$, resp.), the summand where $\ell_2$ is appearing with exponent $(b-2)$ (or where $\ell_1$ is appearing with exponent $(a-2)$, resp.) vanishes since it appears multiplied by the coefficient $(b-1)$ (or $(a-1)$, resp.).}
Therefore, we have that, if $P = \tau_{a,b}\left(([\ell_1],[\ell_2]),([m_1,m_2])\right)\Alessandro{\in\mathcal{T}(SV_{a,b})}$, then
\begin{align*}
T_{P} (\calT(SV_{a,b})) = \mathbb{P}\left( \left\langle \ell_1^a\ell_2^{b-1}\cdot \mathcal{S}_{0,1},~ \ell_1^{a-1}\ell_2^b \cdot \mathcal{S}_{1,0},~ \right.\right. &\left.\left. \ell_1^{a-2}\ell_2^{b-1}\big((a-1) m_1\ell_2 + a\ell_1 m_2 \big) \cdot \mathcal{S}_{1,0}, \right.\right. \\
& \quad\quad\left.\left. \ell_1^{a-1}\ell_2^{b-2}\big(b m_1\ell_2 + (b-1)\ell_1 m_2 \big)\cdot \mathcal{S}_{0,1} \right\rangle \right)
\end{align*}
From this description of the general tangent space to the tangential variety, we can conclude that the tangential variety $\calT(SV_{a,b})$ has the expected dimension.
\begin{lemma}\label{lemma: dimension tangential}
Let $(a,b)$ be a pair of positive integers with $ab > 1$. Then, $\calT(SV_{a,b})$ is $4$-dimensional.
\end{lemma}
\begin{proof}
Let $P$ be a general point of $\calT(SV_{a,b})$. Now, up to change of coordinates, we may assume
$$
\ell_1 = x_0,~m_1 = x_1 \quad \text{ and } \quad \ell_2 = y_0, m_2 = y_1.
$$
By direct computations, we have that the affine cone over $T_P(\calT(SV_{a,b}))$ is
\begin{align*}
\left\langle x_0^ay_0^b,~ x_0^ay_0^{b-1}y_1,~ x_0^{a-1}x_1y_0^b,~ \right.& (a-1)x_0^{a-2}x_1^2y_0^b + a x_0^{a-1}x_1y_0^{b-1}y_1, \\ & \quad \quad \left.
bx_0^{a-1}x_1y_0^{b-1}y_1 + (b-1)x_0^ay_0^{b-2}y_1^2\right\rangle.
\end{align*}
\Alessandro{As mentioned above, if either $a = 1$ (or $b = 1$, resp.), the summand where $x_0$ is appearing with exponent $(a-2)$ (or $y_0$ is appearing with exponent $(b-2)$, resp.) vanishes because it is multiplied by $(a-1)$ (or $(b-1)$, resp.). Since $ab > 1$, this} linear space clearly has dimension $5$. Thus, the claim follows.
\end{proof}
Therefore, by \eqref{equation: expected dimension}, we have that
\begin{equation}\label{equation: expected dimension tangent}
{\rm exp}.\dim \sigma_s(\calT(SV_{a,b})) = \min\left\{(a+1)(b+1), 5s\right\} - 1.
\end{equation}
\subsection{Apolarity Theory \Alessandro{and Fat points}}\label{section: apolarity}
For higher secant varieties, similar computations as Lemma \ref{lemma: dimension tangential} are not feasible. In order to overcome this difficulty, a classical strategy is to use {\it Apolarity Theory}. Here we recall the basic constructions, but, for exhaustive references on this topic, we refer to \cite{IK, G}.
\smallskip
Let $\mathcal{S} = \mathbb{C}[x_0,x_1;y_0,y_1] = \bigoplus_{(a,b)\in\mathbb{N}^2} \mathcal{S}_{a,b}$ be the bi-graded polynomial ring. Any bi-homogeneous ideal $I$ inherits the grading, namely, we have $I = \bigoplus_{a,b} I_{a,b}$, where $I_{a,b} = I \cap \mathcal{S}_{a,b}$. Fixed a bi-degree $(a,b)$, for any $\alpha = (\alpha_0,\alpha_1)\in \mathbb{N}^2$ and $\beta = (\beta_0,\beta_1) \in \mathbb{N}^2$ such that $|\alpha| = \alpha_0+\alpha_1 = a$ and $|\beta| = \beta_0+\beta_1 = b$, we denote by $\mathrm{x}^\alpha\mathrm{y}^\beta$ the monomial $x_0^{\alpha_0}x_1^{\alpha_1}y_0^{\beta_0}y_1^{\beta_1}$; hence, for any polynomial $f \in \mathcal{S}_{a,b}$, we write $f = \sum_{\substack{\alpha,\beta \in \mathbb{N}^2, \\ |\alpha|=a,~|\beta|=b}} f_{\alpha,\beta}\mathrm{x}^\alpha\mathrm{y}^\beta$. For any bi-degree $(a,b)\in\mathbb{N}^2$, we consider the non-degenerate apolar pairing
$$
\circ : \mathcal{S}_{a,b} \times \mathcal{S}_{a,b} \longrightarrow \mathbb{C}, \quad\quad (f,g) \mapsto \sum_{\substack{\alpha,\beta \in \mathbb{N}^2, \\ |\alpha|=a,~|\beta|=b}} f_{\alpha,\beta}g_{\alpha,\beta}.
$$
Now, given a subspace $W \subset \mathcal{S}_{a,b}$, we denote by $W^\perp$ the perpendicular space with respect to the apolar pairing, i.e, $W^\perp = \{f \in \mathcal{S}_{a,b} ~|~ f \circ g = 0,~\forall g \in W\}$. From this definition, it is easy to prove that, given $W_1,\ldots,W_s \subset \mathcal{S}_{a,b}$, we have
\begin{equation}\label{equation: intersection perp}
\left\langle W_1,\ldots,W_s \right\rangle^\perp = W_1^\perp \cap \ldots \cap W_s^\perp.
\end{equation}
\begin{remark}\label{lemma: tangential 0-dimensional ideal}
Let $P = \tau_{a,b}\big(([\ell_1],[\ell_2]),([m_1],[m_2])\big)\in \calT(SV_{a,b})$ be a general point and let $W$ be the affine cone over the tangent space, i.e., $\mathbb{P}(W) = T_P(\calT(SV_{a,b}))$. Then, we may observe that \Alessandro{$\wp^3_{a,b} \subset W^\perp \subset \wp^2_{a,b}$}, where $\wp$ is the ideal defining the point $([\ell_1],[\ell_2]) \in \mathbb{P}^1\times\mathbb{P}^1$.
Indeed, up to change of coordinates, we may assume
$$
\ell_1 = x_0,~m_1 = x_1 \quad \text{ and } \quad \ell_2 = y_0, ~m_2 = y_1.
$$
Therefore, \Alessandro{as in the proof} of Lemma \ref{lemma: dimension tangential}, we have
\begin{align}
W = \left\langle x_0^ay_0^b,~ x_0^ay_0^{b-1}y_1,~ x_0^{a-1}x_1y_0^b,~ \right.& (a-1)x_0^{a-2}x_1^2y_0^{b} + a x_0^{a-1}x_1y_0^{b-1}y_1, \nonumber \\ & \quad \quad \left. bx_0^{a-1}x_1y_0^{b-1}y_1 + (b-1)x_0^{a}y_0^{b-2}y_1^2\right\rangle \subset \mathcal{S}_{a,b}. \label{eq: explicit tangent}
\end{align}
\Alessandro{It is easy to check that
$$[\wp^2_{a,b}]^\perp = \left\langle x_0^ay_0^b,~ x_0^ay_0^{b-1}y_1,~ x_0^{a-1}x_1y_0^b \right\rangle \subset W \quad \text{and} \quad\wp^3_{a,b} = (x_1^3,x_1^2y_1,x_1y_1^2,y_1^3)_{a,b} \subset W^\perp;$$ therefore, since the apolarity pairing is non-degenerate, we obtain that
$
\wp^{3}_{a,b} \subset W^\perp \subset \wp^{2}_{a,b}.
$}
\end{remark}
\begin{definition}
Let $P \in \mathbb{P}^1\times\mathbb{P}^1$ be a point defined by the ideal $\wp$. We call {\bf fat point} of {\bf multiplicity $m$}, or $m$-fat point, and {\bf support in $P$}, the $0$-dimensional scheme $mP$ defined by the ideal $\wp^m$.
\end{definition}
Therefore, from Remark \ref{lemma: tangential 0-dimensional ideal}, we have that the general tangent space to the tangential variety of the Segre-Veronese surface $SV_{a,b}$ is the projectivisation of a $5$-dimensional vector space whose perpendicular is the bi-homogeneous part in bi-degree $(a,b)$ of an ideal describing a $0$-dimensional scheme of \Alessandro{length} $5$ which is contained in between a $2$-fat point and a $3$-fat point. In the next lemma, we describe better the structure of the latter $0$-dimensional scheme.
\begin{lemma}\label{lemma: explicit tangent}
Let $P = \tau_{a,b}\big(([\ell_1],[\ell_2]),([m_1],[m_2])\big)\in \calT(SV_{a,b})$ be a general point and let $W$ be the affine cone over the tangent space, i.e., $\mathbb{P}(W) = T_P(\calT(SV_{a,b}))$. Then, $W^\perp = [\wp^3 + I^2]_{a,b}$, where $\wp$ is the ideal of the point $P$ and $I$ is the principal ideal $I = (\ell_1m_2+\ell_2m_1)$.
\end{lemma}
\begin{proof}
Up to change of coordinates, we may assume
$$
\ell_1 = x_0,~m_1 = x_1 \quad \text{ and } \quad \ell_2 = y_0,~ m_2 = y_1.
$$
Let $\wp = (x_1,y_1)$ and $I = (x_1y_0+x_0y_1)$. \Alessandro{Now, from \eqref{eq: explicit tangent}, it follows} that
$$
W \subset \Alessandro{[\wp^3_{a,b}]^\perp\cap [I^2_{a,b}]^\perp \subsetneq [\wp^3_{a,b}]^\perp.}
$$
The latter inequality is strict; therefore, since \Alessandro{$\dim [\wp^3_{a,b}]^\perp = 6$, we get that $\dim [\wp^3_{a,b}]^\perp\cap [I^2_{a,b}]^\perp \leq 5$} and, since $\dim W = 5$, equality follows. Hence, since the apolarity pairing is non-degenerate,
$$
W^\perp = [\wp^3+I^2]_{a,b}.
$$
\end{proof}
From these results, we obtain that the general tangent space to the tangential variety to the Segre-Veronese surface $SV_{a,b}$ can be described in terms of a connected $0$-dimensional scheme of \Alessandro{length} $5$ contained in between a \Alessandro{$2$-fat and a $3$-fat} point. Moreover, such a $0$-dimensional scheme is independent from the choice of $a,b$. Therefore, we introduce the following definition.
\begin{definition}\label{def: (3,2)-points P1xP1}
Let $\wp$ be the prime ideal defining a point $P$ in $\mathbb{P}^1\times\mathbb{P}^1$ and let $(\ell)$ the principal ideal generated by a $(1,1)$-form passing through $P$. The $0$-dimensional scheme defined by $\wp^3+(\ell)^2$ is called {\bf $(3,2)$-fat point} with support at $P$. We call $P$ the {\bf support} and $\ell$ the {\bf direction} of the scheme.
\end{definition}
Now, by using Terracini's Lemma, we get that, from Lemma \ref{lemma: explicit tangent} and \eqref{equation: intersection perp},
\begin{equation}\label{equation: dimension of secants and HF}
\dim \sigma_s(\calT(SV_{a,b})) = \dim \mathcal{S}_{a,b}/I(\mathbb{X})_{a,b}-1,
\end{equation}
where $\mathbb{X}$ is the union of $s$ many $(3,2)$-fat points with generic support.
\medskip
Let us recall the definition of {\it Hilbert function}. Since we will use it both in the standard graded and in the multi-graded case, we present this definition in a general setting.
\begin{definition}
Let $\mathcal{S} = \bigoplus_{h\in H}\mathcal{S}_h$ be a polynomial ring graded over a semigroup $H$. Let $I = \bigoplus_{h \in H} \subset \mathcal{S}$ be a $H$-homogeneous ideal, i.e., an ideal generated by homogeneous elements. For any $h\in H$, we call {\bf Hilbert function} of $\mathcal{S}/I$ in degree $h$, the dimension, as vector space, of the homogeneous part of degree $h$ of the quotient ring $\mathcal{S}/I$, i.e.,
$$
\mathrm{HF}_{\mathcal{S}/I}(h) = \dim [\mathcal{S}/I]_{h} = \mathcal{S}_{h}/I_{h}.
$$
Given a $0$-dimensional scheme $\mathbb{X}$ defined by the ideal $I(\mathbb{X}) \subset \mathcal{S}$, we say {\bf Hilbert function} of $\mathbb{X}$ when we refer to the Hilbert function of $\mathcal{S}/I(\mathbb{X})$.
\end{definition}
In the standard graded case we have $H = \mathbb{N}$, while in the bi-graded cases we have $H = \mathbb{N}^2$.
\medskip
Therefore, by \eqref{equation: dimension of secants and HF}, we reduced the problem of computing the dimension of secant varieties to the problem of computing the Hilbert function of a special $0$-dimensional scheme.
\begin{question}\label{question: 0-dim P1xP1}
{\it Let $\mathbb{X}$ be a union of $(3,2)$-fat points with generic support and generic direction.
\centerline{
For any $(a,b) \in \mathbb{N}^2$, what is the Hilbert function of $\mathbb{X}$ in bi-degree $(a,b)$?
}}
\end{question}
\smallskip
Also for this question we have an {\it expected} answer. Since $\mathbb{X}$ is a $0$-dimensional scheme in $\mathbb{P}^1\times\mathbb{P}^1$, if we represent the multi-graded Hilbert function of $\mathbb{X}$ as an infinite matrix $(\mathrm{HF}_\mathbb{X}(a,b))_{a,b\geq 0}$, then it is well-known that, in each row and column, it is strictly increasing until it reaches the degree of $\mathbb{X}$ and then it remains constant. Hence, if we let the support of $\mathbb{X}$ to be generic, we expect the Hilbert function of $\mathbb{X}$ to be the largest possible. Since a $(3,2)$-fat point has degree $5$, if $\mathbb{X}$ is a union of $(3,2)$-fat points with generic support, then
$$
{\rm exp}.\mathrm{HF}_{\mathbb{X}}(a,b) = \min\left\{ (a+1)(b+1), 5s \right\}.
$$
As we already explained in \eqref{equation: dimension of secants and HF}, this corresponds to the expected dimension \eqref{equation: expected dimension tangent} of the $s$-th secant variety of the tangential varieties to the Segre-Veronese surfaces $SV_{a,b}$.
\subsection{\Alessandro{Multiprojective-affine-projective method}}
In \cite{CGG05-SegreVeronese}, the authors defined a very powerful method to study Hilbert functions of $0$-dimensional schemes in multiprojective spaces. The method reduces those computations to the study of the Hilbert function of schemes in standard projective spaces, which might have higher dimensional connected components, depending on the dimensions of the projective spaces defining the multiprojective space. However, in the case of products of $\mathbb{P}^1$'s, we still have $0$-dimensional schemes in standard projective space, as we explain in the following.
\medskip
We consider the birational function
\begin{center}
\begin{tabular}{c c c c c c}
$\phi :$ & $ \mathbb{P}^1 \times \mathbb{P}^1$ & $\dashrightarrow$ & $\mathbb{A}^2$ & $\rightarrow$ & $\mathbb{P}^2$ \\
& $([s_0:s_1],[t_0,t_1])$ & $\mapsto$ & $(\frac{s_1}{s_0},\frac{t_1}{t_0})$ & $\mapsto$ & $[1:\frac{s_1}{s_0}:\frac{t_1}{t_0}] = [s_0t_0:s_1t_0:s_0t_1].$
\end{tabular}
\end{center}
\begin{lemma}{\rm \cite[Theorem 1.5]{CGG05-SegreVeronese}}
Let $\mathbb{X}$ be a $0$-dimensional scheme in $\mathbb{P}^1\times \mathbb{P}^1$ with generic support, i.e., assume that the function $\phi$ is well-defined over $\mathbb{X}$. Let $Q_1 = [0:1:0], Q_2 = [0:0:1] \in \mathbb{P}^2$. Then,
$$
\mathrm{HF}_{\mathbb{X}}(a,b) = \mathrm{HF}_{\mathbb{Y}}(a+b),
$$
where $\mathbb{Y} = \phi(\mathbb{X}) + aQ_1 + bQ_2$.
\end{lemma}
Therefore, in order to rephrase Question \ref{question: 0-dim P1xP1} as a question about the Hilbert function of $0$-dimensional schemes in standard projective spaces, we need to understand what is the image of a $(3,2)$-fat point of $\mathbb{P}^1\times\mathbb{P}^1$ by the map $\phi$.
\smallskip
Let $z_0,z_1,z_2$ be the coordinates of $\mathbb{P}^2$. Then, the map $\phi$ corresponds to the function of rings
\begin{center}
\begin{tabular}{c c c c}
$\Phi :$ & $\mathbb{C}[z_0,z_1,z_2]$ & $\rightarrow$ & $\mathbb{C}[x_0,x_1;y_0,y_1]$; \\
& $z_0$ & $\mapsto$ & $x_0y_0$, \\
& $z_1$ & $\mapsto$ & $x_1y_0$, \\
& $z_2$ & $\mapsto$ & $x_0y_1$. \\
\end{tabular}
\end{center}
By genericity, we may assume that the $(3,2)$-fat point $J$ has support at $P = ([1:0],[1:0])$ and it is defined by $I(J) = (x_1,y_1)^3 + (x_0y_1+x_1y_0)^2$. By construction, we have that $I(\phi(J)) = \Phi(I(J))$ and it is easy to check that
\begin{equation}\label{eq: (3,2)-points construction}
\Phi(I(J)) = (z_1,z_2)^3 + (z_1+z_2)^2.
\end{equation}
Therefore, $\phi(J)$ is a $0$-dimensional scheme obtained by the scheme theoretic intersection of a triple point and a double line passing though it. In the literature also these $0$-dimensional schemes are called {\it $(3,2)$-points}; e.g., see \cite{BCGI09}. We call {\bf direction} the line defining the scheme. This motivates our Definition \ref{def: (3,2)-points P1xP1} which is also a slight abuse of the name, but we believe that it will not rise any confusion in the reader since the ambient space will always be clear in the exposition. We consider a generalization of this definition in the following section. By using these constructions, Question \ref{question: 0-dim P1xP1} is rephrased as follows.
\begin{question}\label{question: 0-dim P2}
{\it Let $\mathbb{Y}$ be a union of $s$ many $(3,2)$-points with generic support and generic direction in $\mathbb{P}^2$.
\centerline{
For any $a,b$, let $Q_1$ and $Q_2$ be generic points and consider $\mathbb{X} = \mathbb{Y} + aQ_1 + bQ_2$.}
\centerline{
What is the Hilbert function of $\mathbb{X}$ in degree $a+b$?
}}
\end{question}
\begin{notation}
Given a $0$-dimensional scheme $\mathbb{X}$ in $\mathbb{P}^2$, we denote by $\mathcal{L}_d(\mathbb{X})$ the linear system of plane curves of degree $d$ having $\mathbb{X}$ in the base locus, i.e., the linear system of plane curves whose defining degree $d$ equation is in the ideal of $\mathbb{X}$. Similarly, if $\mathbb{X}'$ is a $0$-dimensional scheme in $\mathbb{P}^1\times\mathbb{P}^1$, we denote by $\mathcal{L}_{a,b}(\mathbb{X}')$ the linear system of curves of bi-degree $(a,b)$ on $\mathbb{P}^1\times\mathbb{P}^1$ having $\mathbb{X}'$ in the base locus. Therefore, Question \ref{question: 0-dim P1xP1} and Question \ref{question: 0-dim P2} are equivalent of asking the dimension of these types of linear systems of curves. We define the {\bf virtual dimension} as
$$
{\it vir}.\dim\mathcal{L}_d(\mathbb{X}) = {d+2 \choose 2} - \deg(\mathbb{X}) \quad \text{ and } \quad {\it vir}.\dim\mathcal{L}_{a,b}(\mathbb{X}') = (a+1)(b+1) - \deg(\mathbb{X}').
$$
Therefore, the {\bf expected dimension} is the maximum between $0$ and the virtual dimension. We say that a $0$-dimensional scheme $\mathbb{X}$ in $\mathbb{P}^2$ ($\mathbb{X}'$ in $\mathbb{P}^1\times\mathbb{P}^1$) imposes {\bf independent conditions} on $\mathcal{O}_{\mathbb{P}^2}(d)$ (on $\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(a,b)$, respectively) if the dimension of $\mathcal{L}_{d}(\mathbb{X})$ ($\mathcal{L}_{a,b}(\mathbb{X}')$, respectively) is equal to the virtual dimension.
\end{notation}
\subsection{M\'ethode d'Horace diff\'erentielle}
From now on, we focus on Question \ref{question: 0-dim P2}. We use a degeneration method, known as {\it differential Horace method}, which has been introduced by J. Alexander and A. Hirschowitz, by extending a classical idea which was already present in the work of G. Castelnuovo. They introduced this method in order to completely solve the problem of computing the Hilbert function of a union of $2$-fat points with generic support in $\mathbb{P}^n$ \cite{AHb,AHa,AH}.
\begin{definition}
In the algebra of formal functions $\mathbb{C}\llbracket z_1,z_2\rrbracket$, we say that an ideal is {\bf vertically graded} with respect to $z_2$ if it is of the form
$$
I = I_0 \oplus I_1z_2 \oplus I_2z_2^2 \oplus \ldots \oplus I_mz_2^m \oplus (z_2^{m+1}),\quad \text{where $I_i$'s are ideals in $\mathbb{C} \llbracket z_1\rrbracket$.}
$$
If $\mathbb{X}$ is a connected $0$-dimensional scheme in $\mathbb{P}^2$ and $C$ is a curve through the support $P$ of $\mathbb{X}$, we say that $\mathbb{X}$ is {\bf vertically graded} with {\bf base} $C$ if there exist a regular system of parameters $(z_1,z_2)$ at $P$ such that $z_2 = 0$ is the local equation of $C$ and the ideal of $\mathbb{X}$ in $\mathcal{O}_{\mathbb{P}^2,P} \simeq \mathbb{C} \llbracket z_1,z_2 \rrbracket$ is vertically graded.
\end{definition}
Let $\mathbb{X}$ be a vertically graded $0$-dimensional scheme in $\mathbb{P}^2$ with base $C$ and let $j\geq 1$ be a fixed integer; then, we define:
\begin{align*}
j{\bf -th ~Residue}: & \quad {\rm Res}_C^j(\mathbb{X}), \text{ the scheme in $\mathbb{P}^2$ defined by } \mathcal{I}_{\mathbb{X}} + (\mathcal{I}_\mathbb{X} : \mathcal{I}^{j}_C) \mathcal{I}_C^{j-1}. \\
j{\bf -th ~Trace}: & \quad {\rm Tr}_C^j(\mathbb{X}), \text{ the subscheme of $C$ defined by } (\mathcal{I}_\mathbb{X} : \mathcal{I}^{j-1}_C) \otimes \mathcal{O}_{C}.
\end{align*}
Roughly speaking, we have that, in the $j$-th residue, we remove the $j$-th {\it slice} of the scheme $\mathbb{X}$; while, in the $j$-th trace, we consider only the $j$-th {\it slice} as a subscheme of the curve $C$.
In the following example, we can see how we see as vertically graded schemes the $(3,2)$-fat points we have introduced before.
\begin{example}\label{example: (3,2)-point}
Up to a linear change of coordinates, we may assume that the scheme $J$ constructed in \ref{eq: (3,2)-points construction} is defined by the ideal
$
I(J) = (z_1,z_2)^3 + (z_1)^2 = (z_1^2,z_1z_2^2,z_2^3).
$
Therefore, we have that, in the local system of parameters $(z_1,z_2)$, the scheme $J$ is vertically graded with respect to the $z_2$-axis defined by $\{z_1 = 0\}$; indeed, we have the two vertical layers given by
$$
I(X) = I_0 \oplus I_1z_1 \oplus (z_1)^2, \text{ where $I_0 = (z_2^3)$ and $I_1=(z_2^2)$};
$$
at the same time, we have that it is also vertically graded with respect to the $z_1$-axis defined by $\{z_2 = 0\}$; indeed, we have the three horizontal layers given by
$$
I(J) = I_0 \oplus I_1z_2 \oplus I_2z_2^2 \oplus (z_2)^3, \text{ where $I_0 = I_1 = (z_1^2)$ and $I_2=(z_1)$}.
$$
We can visualize $J$ as in Figure \ref{figure: (3,2)-point}, where the black dots correspond to the generators of the $5$-dimensional vector space $\mathbb{C}\llbracket z_1,z_2 \rrbracket / I(J) = \left\langle 1, z_1, z_2, z_1^2, z_1z_2 \right\rangle$.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm,scale = 0.7]
\clip(-2.,-1.) rectangle (3.,3.);
\draw [line width=1.pt,dash pattern=on 1pt off 2pt] (0.,-1.) -- (0.,3.);
\draw [color = black] (-0.5,2.7) node {$z_2$};
\draw [line width=1.pt,dash pattern=on 1pt off 2pt,domain=-2.:3.] plot(\x,{(-0.-0.*\x)/-1.});
\draw [color = black] (2.5,-0.5) node {$z_1$};
\begin{scriptsize}
\draw [fill=black] (1.,0.) circle (3.5pt);
\draw [fill=black] (0.,1.) circle (3.5pt);
\draw [fill=black] (1.,1.) circle (3.5pt);
\draw [fill=black] (0.,2.) circle (3.5pt);
\draw[color=black] (0.22,8.17) node {$f$};
\draw[color=black] (-10.12,0.33) node {$g$};
\draw [fill=black] (0.,0.) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}
\caption{A representation of the $(3,2)$-point $J$, defined by the ideal $I(J) = (z_1,z_2)^3 + (z_1)^2$, as a vertically graded scheme. }
\label{figure: (3,2)-point}
\end{figure}
Therefore, if we consider $J$ as vertically graded scheme with base the $z_2$-axis, we compute the $j$-th residue and trace, for $j = 1,2$, as follows:
\begin{minipage}{0.6\textwidth}
\begin{align*}
{\rm Res}^1_{z_2}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) + (z_1^2,z_1z_2^2,z_2^3) : (z_1) = (z_1,z_2^2); & \\
{\rm Tr}^1_{z_2}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) \otimes \mathbb{C}\llbracket z_1,z_2 \rrbracket / (z_1) = (z_2^3); \\
~ \\
{\rm Res}^2_{z_2}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) + ((z_1^2,z_1z_2^2,z_2^3) : (z_1^2))\cdot (z_1) = (z_1,z_2^3); & \\
{\rm Tr}^2_{z_2}(X) : & \quad \left((z_1^2,z_1z_2^2,z_2^3) : (z_1)\right)\otimes \mathbb{C}\llbracket z_1,z_2 \rrbracket / (z_1) = (z_2^2).
\end{align*}
\end{minipage}
\begin{minipage}{0.35\textwidth}
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm,scale = 0.6]
\clip(-0.5,-0.5) rectangle (7.,5.5);
\draw [line width=1.pt,dash pattern=on 1pt off 2pt] (1.,0.5) -- (1.,5.5);
\draw [line width=1.pt,dash pattern=on 1pt off 2pt] (5.,0.5) -- (5.,5.5);
\begin{scriptsize}
\draw [fill=black] (1.,4.) circle (4.0pt);
\draw [fill=black] (1.,3.) circle (4.0pt);
\draw [fill=black] (1.,2.) circle (4.0pt);
\draw [color=black] (2.,3.) circle (3.5pt);
\draw [color=black] (2.,2.) circle (3.5pt);
\draw [fill=black] (5.,3.) circle (4.0pt);
\draw [fill=black] (5.,2.) circle (4.0pt);
\draw [color=black] (5.98,2.99) circle (3.5pt);
\draw [color=black] (5.98,1.99) circle (3.5pt);
\draw[color=black] (5.166432702358458,6.933351817483886) node {$g_1$};
\draw [color=black] (6.,4.) circle (3.5pt);
\end{scriptsize}
\node[draw] at (1.5,0.05) {$j = 1$};
\node[draw] at (5.5,0.05) {$j = 2$};
\end{tikzpicture}
\end{minipage}
\bigskip
Similarly, if we consider it as vertically graded with respect to the $z_1$-axis, we compute the $j$-th residue and trace, for $j = 1,2,3$, as follows:
\begin{minipage}{0.55\textwidth}
\begin{align*}
{\rm Res}^1_{z_1}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) + (z_1^2,z_1z_2^2,z_2^3) : (z_2) = (z_1^2,z_1z_2,z_2^2); & \\
{\rm Tr}^1_{z_1}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) \otimes \mathbb{C}\llbracket z_1,z_2 \rrbracket / (z_2) = (z_1^2); \\
~ \\
{\rm Res}^2_{z_1}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) + ((z_1^2,z_1z_2^2,z_2^3) : (z_2^2))\cdot (z_2) = (z_1^2,z_1z_2,z_2^2); & \\
{\rm Tr}^2_{z_1}(X) : & \quad \left((z_1^2,z_1z_2^2,z_2^3) : (z_2)\right)\otimes \mathbb{C}\llbracket z_1,z_2 \rrbracket / (z_2) = (z_1^2); \\
~ \\
{\rm Res}^3_{z_1}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) + ((z_1^2,z_1z_2^2,z_2^3): (z_2^3))\cdot (z_2^2) = (z_1^2,z_2^2); & \\
{\rm Tr}^3_{z_1}(X) : & \quad \left((z_1^2,z_1z_2^2,z_2^3): (z_2^2)\right)\otimes \mathbb{C}\llbracket z_1,z_2 \rrbracket / (z_2) = (z_1).
\end{align*}
\end{minipage}
\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm,scale = 0.6]
\clip(-2.,-1.5) rectangle (4.,10.);
\draw [line width=1.pt,dash pattern=on 1pt off 2pt,domain=-2.:4.] plot(\x,{(--1.0207472463132268-0.*\x)/1.});
\draw [line width=1.pt,dash pattern=on 1pt off 2pt,domain=-2.:4.] plot(\x,{(--5.964822902537174-0.*\x)/1.});
\begin{scriptsize}
\draw [color=black] (0.15798866047282512,3.020747246313228) circle (3.5pt);
\draw [color=black] (0.15798866047282512,2.020747246313227) circle (3.5pt);
\draw [fill=black] (0.15798866047282512,1.020747246313226) circle (4.0pt);
\draw [color=black] (1.1579886604728258,2.020747246313227) circle (3.5pt);
\draw[color=black] (-6.874345903049084,1.514396450654374) node {$f_1$};
\draw [color=black] (0.14984696094040872,7.964822902537176) circle (3.5pt);
\draw [color=black] (0.14984696094040872,6.964822902537174) circle (3.5pt);
\draw [fill=black] (0.14984696094040872,5.964822902537174) circle (4.0pt);
\draw [color=black] (1.1498469609404087,6.964822902537174) circle (3.5pt);
\draw [fill=black] (1.1498469609404087,5.964822902537174) circle (4.0pt);
\draw[color=black] (-6.874345903049084,5.859958317926224) node {$f_2$};
\draw [color=black] (1.1549537540078336,3.01286606005846) circle (3.5pt);
\end{scriptsize}
\node[draw] at (0.5,4.8) {$j = 1,2$};
\node[draw] at (0.4,0.1) {$j = 3$};
\end{tikzpicture}
\end{minipage}
\end{example}
\begin{notation}
Let $\mathbb{X} = X_1 + \ldots+X_s$ be a union of vertically graded schemes with respect to the same base $C$. Then, for any vector $\mathbf{j} = (j_1,\ldots,j_s) \in \mathbb{N}_{\geq 1}^s$, we denote
$$
{\rm Res}^\mathbf{j}_C(\mathbb{X}) := {\rm Res}^{\mathbf{j}_1}_C(\mathbb{X}_1) \cap \ldots \cap {\rm Res}^{\mathbf{j}_s}_C(\mathbb{X}_s), \quad \quad \text{ and } \quad \quad {\rm Tr}^\mathbf{j}_C(\mathbb{X}) := {\rm Tr}^{\mathbf{j}_1}_C(\mathbb{X}_1) \cap \ldots \cap {\rm Tr}^{\mathbf{j}_s}_C(\mathbb{X}_s).
$$
\end{notation}
We are now ready to describe the Horace differential method.
\begin{proposition}[Horace differential lemma, \mbox{\cite[Proposition 9.1]{AH}}]\label{proposition: Horace}
Let $\mathbb{X}$ be a $0$-dimensional scheme and let $L$ be a line. Let $Y_1,\ldots,Y_s,\widetilde{Y}_1,\ldots,\widetilde{Y_s}$ be $0$-dimensional connected schemes such that $Y_i \simeq \widetilde{Y_i}$, for any $i = 1,\ldots,s$; $\widetilde{Y_i}$ has support on the line $L$ and is vertically graded with base $L$; the support of $\mathbb{Y} = \bigcup_{i=1}^t Y_i$ and of $\widetilde{\mathbb{Y}} = \bigcup_{i=1}^t \widetilde{Y_i}$ are generic in the corresponding Hilbert schemes.
Let $\mathbf{j} = (j_1,\ldots,j_s)\in \mathbb{N}_{\geq 1}^s$ and $d \in \mathbb{N}$.
\begin{enumerate}
\item If:
\begin{enumerate}
\item ${\rm Tr}^1_L(\mathbb{X}) + {\rm Tr}^\mathbf{j}_L(\widetilde{\mathbb{Y}})$ imposes independent conditions on $\mathcal{O}_{\ell}(d)$;
\item ${\rm Res}^1_L(\mathbb{X}) + {\rm Res}^\mathbf{j}_L(\widetilde{\mathbb{Y}})$ imposes independent conditions on $\mathcal{O}_{\mathbb{P}^2}(d-1)$;
\end{enumerate}
then, $\mathbb{X} + \mathbb{Y}$ imposes independent conditions on $\mathcal{O}_{\mathbb{P}^2}(d)$.
\item If:
\begin{enumerate}
\item $\mathcal{L}_{d,\mathbb{P}^1}\left({\rm Tr}^1_\ell(\mathbb{X}) + {\rm Tr}^\mathbf{j}_\ell(\widetilde{\mathbb{Y}})\right)$ is empty;
\item $\mathcal{L}_{d-1,\mathbb{P}^2}\left({\rm Res}^1_\ell(\mathbb{X}) + {\rm Res}^\mathbf{j}_\ell(\widetilde{\mathbb{Y}})\right)$ is empty;
\end{enumerate}
then $\mathcal{L}_d(\mathbb{X} + \mathbb{Y})$ is empty.
\end{enumerate}
\end{proposition}
The latter result contains all our strategy. Given a $0$-dimensional scheme as in Question \ref{question: 0-dim P2} with generic support, we specialize some of the $(3,2)$-points to have support on a line in such a way the arithmetic allows us to use the conditions of Proposition \ref{proposition: Horace}. Such a specialization will be done in one of the different ways explained in Example \ref{example: (3,2)-point}. Recall that, if a specialized scheme has the expected dimension, then, by semicontinuity of the Hilbert function, also the original general scheme has the expected dimension.
In particular, the residues of $(3,2)$-points have very particular structures. For this reason, we introduce the following definitions.
\begin{definition}
We call {\bf $m$-jet} with {\bf support} at $P$ in the {\bf direction} $L$ the $0$-dimensional scheme defined by the ideal $(\ell,\ell_1^m)$ where $\ell$ is a linear form defining the line $L$ and $(\ell,\ell_1)$ defines the point $P$.
We call {\bf $(m_1,m_2)$-jet} with {\bf support} at $P$ in the {\bf directions} $L_1,L_2$ the $0$-dimensional scheme defined by the ideal $(\ell_1^{m_1},\ell_2^{m_2})$ where $\ell_i$ is a linear form defining the line $L_i$, for $i =1,2$, and $P = L_1 \cap L_2$.
\end{definition}
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(5.,5.) rectangle (7.5,7.5);
\draw [line width=1.pt] (1.,5.) -- (1.,7.5);
\draw [line width=1.pt,domain=5.:7.5] plot(\x,{(--6.-0.*\x)/1.});
\draw [line width=1.pt] (1.,5.) -- (1.,7.5);
\draw [line width=1.pt] (6.,5.) -- (6.,7.5);
\draw [line width=1.pt] (6.5,5.) -- (6.5,7.5);
\draw [line width=1.pt] (6.5,5.) -- (6.5,7.5);
\draw [line width=1.pt,domain=5.:7.5] plot(\x,{(--3.25-0.*\x)/0.5});
\begin{scriptsize}
\draw [fill=black] (1.,7.) circle (4.5pt);
\draw [fill=black] (1.,6.) circle (4.5pt);
\draw [fill=black] (6.,6.5) circle (4.5pt);
\draw [fill=black] (2.,6.) circle (4.5pt);
\draw [fill=black] (6.,6.) circle (4.5pt);
\draw [fill=black] (6.5,6.5) circle (4.5pt);
\draw [fill=black] (6.5,6.) circle (4.5pt);
\draw[color=black] (1.0995805946910735,9.785609888363922) node {$f$};
\draw[color=black] (1.0116847258085508,6.162346848873283) node {$i$};
\draw[color=black] (1.0995805946910735,9.785609888363922) node {$j$};
\draw[color=black] (6.099878913341249,9.785609888363922) node {$k$};
\draw[color=black] (6.597955503675544,9.785609888363922) node {$l$};
\draw[color=black] (6.597955503675544,9.785609888363922) node {$h$};
\draw [fill=black] (6.,6.5) circle (4.5pt);
\draw[color=black] (1.0116847258085508,6.435800663174463) node {$g$};
\end{scriptsize}
\end{tikzpicture}
\quad\quad
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(5.,5.) rectangle (8.,7.5);
\draw [line width=1.pt] (1.,5.) -- (1.,8.);
\draw [line width=1.pt] (1.97546,5.) -- (1.97546,8.);
\draw [line width=1.pt,domain=5.:8.] plot(\x,{(--6.-0.*\x)/1.});
\draw [line width=1.pt] (1.,5.) -- (1.,8.);
\draw [line width=1.pt] (6.,5.) -- (6.,8.);
\draw [line width=1.pt] (6.5,5.) -- (6.5,8.);
\draw [line width=1.pt] (7.,5.) -- (7.,8.);
\draw [line width=1.pt] (6.5,5.) -- (6.5,8.);
\begin{scriptsize}
\draw [fill=black] (1.,7.) circle (4.5pt);
\draw [fill=black] (1.,6.) circle (4.5pt);
\draw [fill=black] (2.00409,7.) circle (4.5pt);
\draw [fill=black] (2.,6.) circle (4.5pt);
\draw [fill=black] (6.,6.) circle (4.5pt);
\draw [fill=black] (7.,6.) circle (4.5pt);
\draw [fill=black] (6.5,6.) circle (4.5pt);
\draw[color=black] (1.0995805946910735,9.785609888363922) node {$f$};
\draw[color=black] (2.1152661906668904,9.785609888363922) node {$g$};
\draw[color=black] (1.0116847258085508,6.162346848873283) node {$i$};
\draw[color=black] (1.0995805946910735,9.785609888363922) node {$j$};
\draw[color=black] (6.099878913341249,9.785609888363922) node {$k$};
\draw[color=black] (6.597955503675544,9.785609888363922) node {$l$};
\draw[color=black] (6.871409317976725,9.785609888363922) node {$m$};
\draw[color=black] (6.597955503675544,9.785609888363922) node {$h$};
\end{scriptsize}
\end{tikzpicture}
\end{center}
\caption{The representation of a $(2,2)$-jet and a $3$-jet as vertically graded schemes.}
\end{figure}
Since $(2,2)$-points will be crucial in our computations, we analyse further their structure with the following two lemmas that are represented also in Figure \ref{fig: (3,2)-point structure}.
\begin{lemma}\label{lemma: degeneration}
For $\lambda > 0$, let $L_1(\lambda)$ and $L_2(\lambda)$ two families of lines, defined by $\ell_1(\lambda)$ and $\ell_2(\lambda)$, respectively, passing through a unique point. Assume also that $m = \lim_{\lambda \rightarrow 0} \ell_i(\lambda)$, for $i = 1,2$, i.e., the families $L_i(\lambda)$ degenerate to the line $M = \{m=0\}$ when $\lambda$ runs to $0$. Fix a generic line $N = \{n = 0\}$ and consider
$$
J_1(\lambda) = (\ell_1(\lambda),n^2) \quad \text{ and } \quad J_2(\lambda) = (\ell_2(\lambda),n^2).
$$
Then, the limit for $\lambda \rightarrow 0$ of the scheme $\mathbb{X}(\lambda) = J_1(\lambda) + J_2(\lambda)$ is the $(2,2)$-jet defined by $(m^2,n^2)$.
\end{lemma}
\begin{proof}
We may assume that
$$
I(J_1(\lambda)) = (z_1+\lambda z_0,z^2_2) \quad \text{and} \quad
I(J_2(\lambda)) = (z_1-\lambda z_0,z^2_2).
$$
Hence, $M$ is the line $\{z_1 = 0\}$. Then, the limit for $\lambda \rightarrow 0$ of the scheme $\mathbb{X}(\lambda)$ is given by
$$
\lim_{\lambda \rightarrow 0} I(\mathbb{X}(\lambda)) = \lim_{\lambda \rightarrow 0} \left[(z^2_2,z_1+\lambda z_0) \cap (z^2_2,z_1-\lambda z_0)\right] = \lim_{\lambda \rightarrow 0} (z_2^2,z_1^2 - \lambda^2z_0^2) = (z_1^2,z_2^2).
$$
\end{proof}
\begin{lemma}\label{lemma: residue}
Let $J$ be a $(2,2)$-jet, defined by the ideal $(\ell_1^2,\ell_2^2)$, with support at the point $P$. Let $L_i = \{\ell_i=0\}$, for $i = 1,2$. Then:
\begin{enumerate}
\item the residue of $J$ with respect to $L_1$ ($L_2$, respectively) is a $2$-jet with support at $P$ and direction $L_1$ ($L_2$, respectively);
\item the residue of $J$ with respect to a line $L = \{\alpha \ell_1 + \beta \ell_2 = 0\}$ passing through $P$ different from $L_1$ and $L_2$ is a $2$-jet with support at $P$ and direction the line $\{\alpha \ell_1 - \beta \ell_2 = 0\}$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item If we consider the residue with respect to $\{\ell_1=0\}$, we get
$$
(\ell_1^2,\ell_2^2) : (\ell_1) = (\ell_1,\ell_2^2).
$$
Analogously, for the line $\{\ell_2 = 0\}$.
\item If we consider the residue with respect to the line $\{\alpha \ell_1 + \beta\ell_2 = 0\}$, we get
$$
(\ell_1^2,\ell_2^2) : (\alpha \ell_1 + \beta \ell_2) = (\ell_1^2, \alpha\ell_1 - \beta \ell_2).
$$
\end{enumerate}
\end{proof}
\begin{figure}[h]
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[height=1.6in]{degeneration.png}
\caption{The degeneration of Lemma \ref{lemma: degeneration}.}
\end{subfigure}%
~
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[height=1.3in]{residue.png}
\caption{The two different residues of Lemma \ref{lemma: residue}.}
\end{subfigure}
\caption{The structure of a $(2,2)$-jet.}
\label{fig: (3,2)-point structure}
\end{figure}
The construction of $(2,2)$-jets as degeneration of $2$-jets, as far as we know, is a type of degeneration that has not been used before in the literature. Similarly as regard the fact that the structure of the residues of $(2,2)$-jets depend on the direction of the lines. These two facts will be crucial for our computation and we believe that these constructions might be used to attack also other similar problems on linear systems.
\section{Lemmata}\label{sec: lemmata}
\subsection{Subabundance and superabundance}\label{sec: super- and sub-abundance}
The following result is well-known for the experts in the area and can be found
in several papers in the literature. We explicitly recall it for convenience of the reader.
\begin{lemma}\label{lemma: super- and sub-abundance}
Let $\mathbb{X}' \subset \mathbb{X} \subset \mathbb{X}''\subset\mathbb{P}^2$ be $0$-dimensional schemes. Then:
\begin{enumerate}
\item if $\mathbb{X}$ imposes independent conditions on $\mathcal{O}_{\mathbb{P}^2}(d)$, then also $\mathbb{X}'$ does;
\item if $\mathcal{L}_{d}(\mathbb{X})$ is empty, then also $\mathcal{L}_{d}(\mathbb{X}'')$ is empty.
\end{enumerate}
\end{lemma}
In Question \ref{question: 0-dim P2}, we consider, for any positive integers $a,b$ and $s$, the scheme
$$
\mathbb{X}_{a,b;s} = aQ_1 + bQ_2 + Y_1 + \ldots Y_s \subset \mathbb{P}^2,
$$
where the $Y_i$'s are general $(3,2)$-points with support at general points $\{P_1,\ldots,P_s\}$ and general directions. The previous lemma suggests that, fixed $a,b$, there are two critical values to be considered firstly, i.e.,
$$
s_1 = \left\lfloor \frac{(a+1)(b+1)}{5} \right\rfloor \quad \quad \text{ and } \quad \quad s_2 = \left\lceil \frac{(a+1)(b+1)}{5} \right\rceil;
$$
namely, $s_1$ is the largest number of $(3,2)$-points for which we expect to have \textit{subabundance}, i.e., where we expect to have positive virtual dimension, and $s_2$ is the smallest number of $(3,2)$-points where we expect to have \textit{superabundance}, i.e., where we expect that the virtual dimension is negative. If we prove that the dimension of linear system $\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s})$ is as expected for $s = s_1$ and $s = s_2$, then, by Lemma \ref{lemma: super- and sub-abundance}, we have that it holds for any $s$.
\subsection{Low bi-degrees}
Now, we answer to Question \ref{question: 0-dim P2} for $b = 1,2$. These will be the base cases of our inductive approach to solve the problem in general. Recall that $a,b$ are positive integers such that $ab > 1$; see Remark \ref{rmk: (a,b) = (1,1)}.
\begin{lemma}\label{lemma: b = 1}
Let $a > b = 1$ be a positive integer. Then,
$$
\dim\mathcal{L}_{a+1}(\mathbb{X}_{a,1;s}) = \max \{0,~ 2(a+1) - 5s\}.
$$
\end{lemma}
\begin{proof}
First of all, note that $a \geq s_2$. Indeed,
$$
a \geq \frac{2(a+1)}{5} \quad \Longleftrightarrow \quad 3a \geq 2.
$$
Now, if $s \leq s_2$, we note that every line $\overline{Q_1P_i}$ is contained in the base locus of $\mathcal{L}_{a+1}(\mathbb{X}_{a,1;s})$. Hence,
$$
\dim\mathcal{L}_{a+1}(\mathbb{X}_{a,1;s}) = \dim\mathcal{L}_{a+1-s}(\mathbb{X}'),
$$
where $\mathbb{X}' = (a-s)Q_1 + Q_2 + 2P_1 + \ldots + 2P_s$. By \cite{CGG05-SegreVeronese},
$$
\dim\mathcal{L}_{a+1}(\mathbb{X}_{a,1;s}) = \dim\mathcal{L}_{a+1-s}(\mathbb{X}') = \max\{0,~2(a+1-s) - 3s\}.
$$
\end{proof}
Now, we prove the case $b = 2$ which is the crucial base step for our inductive procedure. In order to make our construction to work smoothly, we need to consider separately the following easy case.
\begin{lemma}\label{lemma: a = 2}
Let $a = b = 2$. Then,
$$\dim\mathcal{L}_{4}(\mathbb{X}_{2,2;s}) = \max\{0, 9 - 5s\}.$$
\end{lemma}
\begin{proof}
For $s = s_1 = 1$, then it follows easily because the scheme $2Q_1 + 2Q_2 + 3P_1 \supset \mathbb{X}_{2,2;1}$ imposes independent conditions on quartics. For $s = s_2 = 2$, we specialize the directions of the $(3,2)$-points supported at the $P_i$'s to be along the lines $\overline{Q_1P_i}$, respectively. Now, the lines $\overline{Q_1P_i}$ are fixed components and we can remove them. We remain with the linear system $\mathcal{L}_2(2Q_2 + J_1 + J_2)$, where the $J_i$'s are $2$-jets contained in the lines $\overline{Q_1P_i}$, respectively. Since both lines $\overline{Q_2P_1}$ and $\overline{Q_2P_2}$ are fixed components for this linear system, we conclude that the linear system has to be empty.
\end{proof}
\begin{lemma}\label{lemma: b = 2}
Let $a > b = 2$ be a positive integer. Then,
$$
\dim\mathcal{L}_{a+2}(\mathbb{X}_{a,2;s}) = \max \{0,~ 3(a+1) - 5s\}.
$$
\end{lemma}
\begin{proof}
We split the proof in different steps. Moreover, in order to help the reader in following the constructions, we include figures showing the procedure in the case of $a = 15$.
\medskip
\noindent {\sc Step 1.} Note that since $a > 2$, then $a \geq \frac{3(a+1)}{5}$ which implies $a \geq \left\lceil \frac{3(a+1)}{5} \right\rceil = s_2$.
We specialize the $(3,2)$-points with support at the $P_i$'s to have direction along the lines $\overline{Q_1P_i}$, respectively. In this way, for any $s \leq s_2$, every line $\overline{Q_1P_i}$ is a fixed component of the linear system $\mathcal{L}_{a+2}(\mathbb{X}_{a,2;s})$ and can be removed, i.e.,
$$
\dim\mathcal{L}_{a+2}(\mathbb{X}_{a,2;s}) = \dim\mathcal{L}_{(a-s)+2}(\mathbb{X}'),
$$
where $\mathbb{X}' = a'Q_1 + 2Q_2 + J_1+\ldots+J_s$, where $a' = a-s$ and $J_i$ is a $2$-jet contained in $\overline{Q_1P_i}$, for $i = 1,\ldots,s$. Now, as suggested by Lemma \ref{lemma: super- and sub-abundance}, we consider two cases: $s = s_1$ and $s = s_2$. See Figure \ref{fig: Lemma_1}.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.46\textwidth}
\centering
\scalebox{0.55}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
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\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.46\textwidth}
\centering
\scalebox{0.55}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-5.,-3.5) rectangle (8.,4.5);
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\draw [line width=2.pt,domain=-5.:8.] plot(\x,{(--21.9276929872492-3.9572957108762044*\x)/5.41393824616456});
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\end{tikzpicture}
}
\caption{}
\end{subfigure}
\caption{As example, we consider the case $a = 15$ and $s = s_1 = \left\lfloor \frac{16\cdot 3}{5} \right\rfloor = 9$. Here, we represent: ({\sc A}) the first specialization and ({\sc B}) the reduction explained in Step 1.}
\label{fig: Lemma_1}
\end{figure}
\medskip
We fix $a = 5k + c$, with $ 0 \leq c \leq 4$.
\medskip
\noindent {\sc Step 2: case $s = s_1$.} Consider
$$
s_1 = \left\lfloor \frac{3(5k+c+1)}{5} \right\rfloor =
\begin{cases}
3k & \text{ for } c = 0; \\
3k+1 & \text{ for } c = 1,2; \\
3k+2 & \text{ for } c = 3; \\
3k+3 & \text{ for } c = 4.
\end{cases}
$$
Note that the expected dimension is
$$
{\it exp}.\dim\mathcal{L}_{a+2}(\mathbb{X}_{a,2;s_1}) =
\begin{cases}
3 & \text{ for } c = 0; \\
1 & \text{ for } c = 1; \\
4 & \text{ for } c = 2; \\
2 & \text{ for } c = 3; \\
0 & \text{ for } c = 4. \\
\end{cases}
$$
Let $A$ be a reduced set of points of cardinality equal to the expected dimension. Then, it is enough to show that
$$
\dim\mathcal{L}_{a'+2}(\mathbb{X}'+A) = 0.
$$
where
$$
a' = a-s_1 =
\begin{cases}
2k & \text{ for } c = 0,1; \\
2k+1 & \text{ for } c = 2,3,4.
\end{cases}
$$
Let
$$
t_1 =
\begin{cases}
k & \text{ for } c = 0,2;\\
k+1 & \text{ for } c = 1,3,4.
\end{cases}
$$
Note that $s_1 \geq 2t_1$. Now, by using Lemma \ref{lemma: degeneration}, we specialize $2t_1$ lines in such a way that, for $i = 1,\ldots,t_1$, we have:
\begin{itemize}
\item the lines $\overline{Q_1P_{2i-1}}$ and $\overline{Q_1P_{2i}}$ both degenerate to a general line $R_i$ passing through $Q_1$;
\item the point $P_{2i-1}$ and the point $P_{2i}$ both degenerate to a general point $\widetilde{P}_i$ on $R_i$.
\end{itemize}
In this way, from the degeneration of $J_{2i-1}$ and $J_{2i}$, we obtain the $(2,2)$-jet $W_i$ defined by the scheme-theoretic intersection $2R_i \cap 2\overline{P_{2i-1}P_{2i}}$, for any $i = 1,\ldots,t_1$. Note that the directions $\overline{P_{2i-1}P_{2i}}$ are generic, for any $i$; see Figure \ref{fig: Lemma_2}. Note that, these $(2,2)$-jets have general directions.
\begin{figure}[h]
\centering
\scalebox{0.67}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-5.,-4.) rectangle (8.,5.);
\draw [line width=1.pt,dotted,domain=-5.:8.,color=gray] plot(\x,{(-7.514031000906525-3.9569717098083377*\x)/-4.082209051951633});
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\draw [line width=2.pt,domain=-5.:8.] plot(\x,{(--14.585189910082079-7.119222132191011*\x)/1.6461298279383072});
\begin{scriptsize}
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\draw[color=black] (0.,3.) node {\LARGE $6Q_1$};
\draw [fill=black] (5.153549103316712,3.1969296513445316) circle (4.5pt);
\draw[color=black] (6,3.) node {\LARGE $2Q_2$};
\draw[color = black] (-3,1) node {\large in degree $8$};
\draw [fill=black] (-0.22908063715905802,-2.489937091858407) circle (3.5pt);
\draw [fill=black] (2.5443685037831303,-2.5676973481465053) circle (3.5pt);
\draw [fill=black] (5.861219930905668,-1.1391576212981789) circle (3.5pt);
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\draw [fill=black] (4.71199456993304,-1.9943950992312944) circle (3.5pt);
\draw [fill=black] (-2.898985250235947,-0.9693701549990218) circle (3.5pt);
\draw [fill=black] (-2.5973158622124415,-0.9389674963821508) circle (3.5pt);
\draw [fill=black] (-0.16680037975532125,-2.265013060979562) circle (3.5pt);
\draw [fill=black] (0.009619284499902792,-2.280669547358283) circle (3.5pt);
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\draw [fill=black] (2.666250534361207,-2.300643908055881) circle (3.5pt);
\draw [fill=black] (4.580296319385906,-1.7952451528388482) circle (3.5pt);
\draw [fill=black] (5.668255850940832,-0.9576907230338629) circle (3.5pt);
\draw [fill=black] (6.156743519524302,0.4156517211976407) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}
}
\caption{Since $a = 15$, we have $a = 5\cdot 3 + 0$ and $t_1 = 3$. Hence, we specialize three pairs of lines to be coincident and we construct three $(2,2)$-jets.}
\label{fig: Lemma_2}
\end{figure}
Moreover, we specialize further as follows; see Figure \ref{fig: Lemma_3}:
\begin{itemize}
\item every scheme $W_i$, with support on $R_i$, for $i = 1,\ldots,t_1$, in such a way that all $\widetilde{P}_i$'s are collinear and lie on a line $L$;
\item if $c = 0,1,2,3$, we specialize $A$ (the reduced part) to lie on $L$;
\item if $c = 4$, we specialize the double point $2Q_2$ to have support on $L$.
\end{itemize}
Note that this can be done because
$$
s_1 - t_1 =
\begin{cases}
k & \text{for } c =0,3; \\
k-1 & \text{for } c = 1; \\
k+1 & \text{for } c = 2,4;
\end{cases}
$$
and, for $c = 1$, we may assume $k \geq 1$, since $a \geq 2$.
\begin{figure}[h!]
\centering
\scalebox{0.75}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
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\draw [line width=2.pt,domain=-4.:8.5] plot(\x,{(-0.3695083120389595--0.028408168922945176*\x)/0.3571005711237858});
\draw [->,line width=2.pt] (0.3289024955258037,-1.2460623060398182) -- (-0.007560515160232151,-0.4065304409771011);
\draw [line width=2.pt,dotted,domain=-4.:8.5, color = gray] plot(\x,{(--8.607111108960233-5.488725122958664*\x)/0.2637309371017562});
\draw [->,line width=2.pt] (1.4905991442246822,-1.1370380434870988) -- (1.8372807182972242,-0.33224153224727093);
\draw [->,line width=2.pt] (-1.6047720528515872,-1.3227603153116745) -- (-1.8647832334059937,-0.5798712280133718);
\draw [line width=2.pt,dotted,domain=-4.:8.5, color = gray] plot(\x,{(-4.489159349354937-5.241095427192564*\x)/-3.673581225579258});
\begin{scriptsize}
\draw [fill=black] (1.4125904789475021,3.2373534485241113) circle (4.5pt);
\draw[color=black] (0.5,3) node {\LARGE $6Q_1$};
\draw [fill=black] (5.153549103316712,3.1969296513445316) circle (4.5pt);
\draw[color=black] (6,3) node {\LARGE $2Q_2$};
\draw[color=black] (7,0.5) node {\LARGE $A$};
\draw[color = black] (-3,1) node {\large in degree $8$};
\draw [fill=black] (0.10387284793451354,-1.0256046803923533) circle (3.5pt);
\draw [fill=black] (1.4905991442246822,-1.1370380434870988) circle (3.5pt);
\draw [fill=black] (3.1249551362809522,0.3115956767445914) circle (3.5pt);
\draw [fill=black] (-1.716205415946333,-1.0008417108157432) circle (3.5pt);
\draw [fill=black] (3.236388499375698,1.1906810967142496) circle (3.5pt);
\draw [fill=black] (-1.6047720528515872,-1.3227603153116745) circle (3.5pt);
\draw [fill=black] (0.3289024955258037,-1.2460623060398182) circle (3.5pt);
\draw [fill=black] (1.7134658704141736,-0.901789832509303) circle (3.5pt);
\draw [fill=black] (2.295395655464512,0.5839883420873023) circle (3.5pt);
\draw [fill=black] (-1.8498750949904104,-1.181907592294197) circle (3.5pt);
\draw [fill=black] (-1.4927745238666246,-1.1534994233712519) circle (3.5pt);
\draw [fill=black] (0.16974203912965335,-0.8110453305399878) circle (3.5pt);
\draw [fill=black] (0.38458241343333954,-1.0157042832708876) circle (3.5pt);
\draw [fill=black] (1.4867785796410165,-0.9227971620403612) circle (3.5pt);
\draw [fill=black] (1.696281834687383,-0.6653890232810649) circle (3.5pt);
\draw [fill=black] (2.227862500801172,0.7869664406279986) circle (3.5pt);
\draw [fill=black] (3.0173602157974577,0.4954330816223558) circle (3.5pt);
\draw [fill=black] (3.091602635533084,1.3531602863725496) circle (3.5pt);
\draw [fill=black] (6.160542692533723,-0.5446604970705151) circle (3.5pt);
\draw [fill=black] (6.913021105428445,-0.48479911461147396) circle (3.5pt);
\draw [fill=black] (7.712198191933535,-0.42122274563133616) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}
}
\caption{Since $a = 15 = 5\cdot 3 + 0$, we have that $|A| = 3$. Then, we specialize the scheme in such a way the $(2,2)$-jets have support collinear with the support of the set $A$. Recall that they have general directions.}
\label{fig: Lemma_3}
\end{figure}
By abuse of notation, we call again ${\mathbb{X}}$ the scheme obtained after such a specialization.
In this way, we have that
$$
\deg({\mathbb{X}}\cap L) \quad = \quad
\begin{cases}
2k + 3 & \text{for } c = 0,1; \\
2k + 4 & \text{for } c = 2,3,4.
\end{cases} \quad = \quad a'+3.
$$
Therefore, the line $L$ becomes a fixed component of the linear system and can be removed, see Figure \ref{fig: Lemma_4}, i.e.,
$$
\dim\mathcal{L}_{a'+2}({\mathbb{X}}) = \dim\mathcal{L}_{a'+1}(\mathbb{J}),
$$
with $$
\mathbb{J} =
\begin{cases}
a'Q_1 + 2{Q}_2 + {J}_1 + \ldots + {J}_{t_1} + J_{2t_1+1} + \ldots + J_{s_1}, & \text{ for } c = 0,1,2,3; \\
a'Q_1 + {Q}_2 + {J}_1 + \ldots + {J}_{t_1} + J_{2t_1+1} + \ldots + J_{s_1}, & \text{ for } c = 4; \\
\end{cases}
$$
where
\begin{itemize}
\item
by Lemma \ref{lemma: residue}, ${J}_i$ is a $2$-jet with support on $\widetilde{P}_i \in L$ with general direction, for all $i = 1,\ldots,t_1$;
\item
$J_i$ is a $2$-jet contained in the line $\overline{Q_1P_i}$, for $i = 2t_1+1,\ldots,s_1$.
\end{itemize}
\begin{figure}[h]
\centering
\scalebox{0.77}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-3.5,-2.5) rectangle (7.,4.5);
\draw [line width=2.pt,domain=-3.5:7.] plot(\x,{(--6.606070669096441-2.653365106436809*\x)/0.88280517651701});
\draw [line width=2.pt,domain=-3.5:7.] plot(\x,{(--9.676427200671663-2.92575777177952*\x)/1.7123646573334501});
\draw [line width=2.pt,domain=-3.5:7.] plot(\x,{(--8.795388688536372-2.0466723518098617*\x)/1.8237980204281958});
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(--3.6452140416492202-6.69929374398375*\x)/-1.7971916286827576});
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(-0.3695083120389595--0.028408168922945176*\x)/0.3571005711237858});
\draw [->,line width=2.pt] (0.2648321501824796,-1.1618010130637089) -- (-0.007560515160232151,-0.4065304409771011);
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(--8.607111108960233-5.488725122958664*\x)/0.2637309371017562});
\draw [->,line width=2.pt] (1.5772695377428179,-1.0875121043338787) -- (1.8372807182972242,-0.33224153224727093);
\draw [->,line width=2.pt] (-1.6047720528515872,-1.3227603153116745) -- (-1.8647832334059937,-0.5798712280133718);
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(-4.489159349354937-5.241095427192564*\x)/-3.673581225579258});
\begin{scriptsize}
\draw [fill=black] (1.4125904789475021,3.2373534485241113) circle (4.5pt);
\draw[color=black] (0,3) node {\LARGE $6Q_1$};
\draw [fill=black] (5.153549103316712,3.1969296513445316) circle (4.5pt);
\draw[color=black] (6,3) node {\LARGE $2Q_2$};
\draw[color = black] (-2,1) node {\large in degree $7$};
\draw [fill=black] (1.5772695377428179,-1.0875121043338787) circle (3.5pt);
\draw [fill=black] (3.1249551362809522,0.3115956767445914) circle (3.5pt);
\draw [fill=black] (-1.716205415946333,-1.0008417108157432) circle (3.5pt);
\draw [fill=black] (3.236388499375698,1.1906810967142496) circle (3.5pt);
\draw [fill=black] (-1.6047720528515872,-1.3227603153116745) circle (3.5pt);
\draw [fill=black] (0.2648321501824796,-1.1618010130637089) circle (3.5pt);
\draw [fill=black] (2.295395655464512,0.5839883420873023) circle (3.5pt);
\draw [fill=black] (0.15570386993870922,-0.8567726333969978) circle (3.5pt);
\draw [fill=black] (1.7029018983959778,-0.7564612511753777) circle (3.5pt);
\draw [fill=black] (2.227862500801172,0.7869664406279986) circle (3.5pt);
\draw [fill=black] (3.0173602157974577,0.4954330816223558) circle (3.5pt);
\draw [fill=black] (3.0916026355330835,1.3531602863725503) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}
}
\caption{We remove the line $L$.}
\label{fig: Lemma_4}
\end{figure}
Now, since we are looking at curves of degree $a'+1$, the lines $\overline{Q_1P_i}$, for $i = 2t_1+1,\ldots,s_1$, become fixed components and can be removed, see Figure \ref{fig: Lemma_5}, i.e.,
$$
\dim\mathcal{L}_{a'+1}(\mathbb{J}) = \dim\mathcal{L}_{a''+1}(\mathbb{J}'),
$$
with
$$
\mathbb{J}' =
\begin{cases}
a''Q_1 + 2{Q}_2 + {J}_1 + \ldots + {J}_{t_1}, & \text{ for } c = 0,1,2,3; \\
a''Q_1 + {Q}_2 + {J}_1 + \ldots + {J}_{t_1}, & \text{ for } c = 4; \\
\end{cases}
$$
where
$$
a'' = a'-(s_1-2t_1) =
\begin{cases}
2k - (3k-2k) = k& \text{for } c = 0; \\
2k - (3k+1-2(k+1)) = k+1& \text{for } c = 1;\\
(2k+1) - (3k+1- 2k) = k & \text{for } c = 2; \\
(2k+1) - (3k+2 - 2(k+1)) = k+1 & \text{for } c = 3;\\
(2k+1) - (3k+3 - 2(k+1)) = k & \text{for } c = 4. \\
\end{cases}
$$
\begin{figure}[h]
\centering
\scalebox{0.85}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-3.,-2.5) rectangle (6.5,4.);
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(--6.606070669096441-2.653365106436809*\x)/0.88280517651701});
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(--9.676427200671663-2.92575777177952*\x)/1.7123646573334501});
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(--8.795388688536372-2.0466723518098617*\x)/1.8237980204281958});
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(--3.6452140416492202-6.69929374398375*\x)/-1.7971916286827576});
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(-0.3695083120389595--0.028408168922945176*\x)/0.3571005711237858});
\draw [->,line width=2.pt] (0.2648321501824796,-1.1618010130637089) -- (-0.007560515160232151,-0.4065304409771011);
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(--8.607111108960233-5.488725122958664*\x)/0.2637309371017562});
\draw [->,line width=2.pt] (1.5772695377428179,-1.0875121043338787) -- (1.8372807182972242,-0.33224153224727093);
\draw [->,line width=2.pt] (-1.6047720528515872,-1.3227603153116745) -- (-1.8647832334059937,-0.5798712280133718);
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(-4.489159349354937-5.241095427192564*\x)/-3.673581225579258});
\begin{scriptsize}
\draw [fill=black] (1.4125904789475021,3.2373534485241113) circle (4.5pt);
\draw[color=black] (0.,3.) node {\LARGE $3Q_1$};
\draw [fill=black] (5.153549103316712,3.1969296513445316) circle (4.5pt);
\draw[color=black] (6.,3.) node {\LARGE $2Q_2$};
\draw[color=black] (-2.,1.) node {\large in degree $4$};
\draw [fill=black] (1.5772695377428179,-1.0875121043338787) circle (3.5pt);
\draw [fill=black] (-1.716205415946333,-1.0008417108157432) circle (3.5pt);
\draw [fill=black] (-1.6047720528515872,-1.3227603153116745) circle (3.5pt);
\draw [fill=black] (0.2648321501824796,-1.1618010130637089) circle (3.5pt);
\draw [fill=black] (0.15570386993870922,-0.8567726333969978) circle (3.5pt);
\draw [fill=black] (1.7029018983959778,-0.7564612511753777) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}
}
\caption{We remove all the lines passing through $Q_1$ and containing the directions of the three $(2,2)$-jets.}
\label{fig: Lemma_5}
\end{figure}
Now, if $c = 0,1,2,3$, since we are looking at curves of degree $a''+1$, we have that the line $\overline{Q_1Q_2}$ is a fixed component and can be removed. After that, specialize the point $Q_2$ to a general point lying on $L$; see Figure \ref{fig: Lemma_6}.
\begin{figure}[h]
\begin{subfigure}[b]{0.45\textwidth}
\scalebox{0.75}{\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-3.,-2.5) rectangle (7.,4.);
\draw [line width=2.pt,dotted,domain=-3.:7., color = gray] plot(\x,{(--3.6452140416492202-6.69929374398375*\x)/-1.7971916286827576});
\draw [line width=2.pt,dotted,domain=-3.:7., color = gray] plot(\x,{(-0.3695083120389595--0.028408168922945176*\x)/0.3571005711237858});
\draw [->,line width=2.pt] (0.2648321501824796,-1.1618010130637089) -- (-0.007560515160232151,-0.4065304409771011);
\draw [line width=2.pt,dotted,domain=-3.:7., color = gray] plot(\x,{(--8.607111108960233-5.488725122958664*\x)/0.2637309371017562});
\draw [->,line width=2.pt] (1.5772695377428179,-1.0875121043338787) -- (1.8372807182972242,-0.33224153224727093);
\draw [->,line width=2.pt] (-1.6047720528515872,-1.3227603153116745) -- (-1.8647832334059937,-0.5798712280133718);
\draw [line width=2.pt,dotted,domain=-3.:7., color = gray] plot(\x,{(-4.489159349354937-5.241095427192564*\x)/-3.673581225579258});
\draw [line width=2.pt,dotted,domain=-3.:7., color = gray] plot(\x,{(--12.167907574406456-0.04042379717957978*\x)/3.7409586243692097});
\begin{scriptsize}
\draw [fill=black] (1.4125904789475021,3.2373534485241113) circle (4.5pt);
\draw[color=black] (0.25,2.8) node {\LARGE $2Q_1$};
\draw [fill=black] (5.153549103316712,3.1969296513445316) circle (4.5pt);
\draw[color=black] (5.5,2.5) node {\LARGE $Q_2$};
\draw[color=black] (-2.,1.) node {\large in degree $3$};
\draw [fill=black] (1.5772695377428179,-1.0875121043338787) circle (3.5pt);
\draw [fill=black] (-1.716205415946333,-1.0008417108157432) circle (3.5pt);
\draw [fill=black] (-1.6047720528515872,-1.3227603153116745) circle (3.5pt);
\draw [fill=black] (0.2648321501824796,-1.1618010130637089) circle (3.5pt);
\draw [fill=black] (0.15570386993870922,-0.8567726333969978) circle (3.5pt);
\draw [fill=black] (1.7029018983959778,-0.7564612511753777) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}}
\caption{}
\end{subfigure}
~~~
\begin{subfigure}[b]{0.45\textwidth}
\scalebox{0.75}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-3.5,-3.) rectangle (7.,4.5);
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(--3.6452140416492202-6.69929374398375*\x)/-1.7971916286827576});
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(-0.3695083120389595--0.028408168922945176*\x)/0.3571005711237858});
\draw [->,line width=2.pt] (0.2648321501824796,-1.1618010130637089) -- (-0.007560515160232151,-0.4065304409771011);
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(--8.607111108960233-5.488725122958664*\x)/0.2637309371017562});
\draw [->,line width=2.pt] (1.5772695377428179,-1.0875121043338787) -- (1.8372807182972242,-0.33224153224727093);
\draw [->,line width=2.pt] (-1.6047720528515872,-1.3227603153116745) -- (-1.8647832334059937,-0.5798712280133718);
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(-4.489159349354937-5.241095427192564*\x)/-3.673581225579258});
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(--15.681208319673534-3.9398807354301373*\x)/3.12470370179319});
\begin{scriptsize}
\draw [fill=black] (1.4125904789475021,3.2373534485241113) circle (4.5pt);
\draw[color=black] (0.5,3) node {\LARGE $2Q_1$};
\draw [fill=black] (4.537294180740692,-0.7025272869060261) circle (4.5pt);
\draw[color=black] (5.,0.) node {\LARGE $Q_2$};
\draw[color=black] (-2.,1.) node {\large in degree $3$};
\draw [fill=black] (1.5772695377428179,-1.0875121043338787) circle (3.5pt);
\draw [fill=black] (-1.716205415946333,-1.0008417108157432) circle (3.5pt);
\draw [fill=black] (-1.6047720528515872,-1.3227603153116745) circle (3.5pt);
\draw [fill=black] (0.2648321501824796,-1.1618010130637089) circle (3.5pt);
\draw [fill=black] (0.15570386993870922,-0.8567726333969978) circle (3.5pt);
\draw [fill=black] (1.7029018983959778,-0.7564612511753777) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}
}
\caption{}
\end{subfigure}
\caption{In ({\sc A}) we have removed the line $\overline{Q_1Q_2}$ and then, in ({\sc B}), we specialize the point $Q_2$ to lie on $L$.}
\label{fig: Lemma_6}
\end{figure}
Therefore, for any $c$, we reduced to computing the dimension of the linear system
$
\mathcal{L}_{\widetilde{a}+1}({\mathbb{J}}''),
$
where ${\mathbb{J}}'' = a'''Q_1 + Q_2 + {J}_1 + \ldots + {J}_{t_1} $ with
$$
{a}''' =
\begin{cases}
k-1 & \text{for } c = 0,2; \\
k & \text{for } c = 1,3,4;
\end{cases}
$$
and where
\begin{itemize}
\item $Q_1$ is a general point;
\item $Q_2$ is a general point on $L$;
\item $J_i$'s are $2$-jet points with support on $L$ and general direction.
\end{itemize}
Note that this can be done because, for $c \leq 2$, we may assume $k \geq 1$, since $a \geq 3$.
Moreover,
$$
\deg({\mathbb{J}}'' \cap L) = t_1 + 1 \quad = \quad
\begin{cases}
k+1 & \text{for } c = 0,2; \\
k+3 & \text{for } c = 1,3,4.
\end{cases} \quad = \quad {a}'''+2.
$$
Therefore, the line $L$ is a fixed component for $
\mathcal{L}_{{a}'''+1}({\mathbb{J}}'')
$ and can be removed, i.e.,
$$
\dim\mathcal{L}_{{a}'''+1}({\mathbb{J}}'') =
\dim\mathcal{L}_{{a}'''}({a}'''Q_1 + \widetilde{P}_1 + \ldots + \widetilde{P}_{t_1}),
$$
where the $\widetilde{P}_i$'s are collinear. Since $t_1 = {a}'''+1$, this linear system is empty and this concludes the proof of the case $s = s_1$.
\bigskip
{\sc Step 3: case $s = s_2$.} First of all, note that the case $c = 4$ has been already proved since in that case we have $s_1 = s_2$. Moreover, the case $c = 1$ follows easily because we have that the linear system in the case $s = s_1$ has dimension $1$.
Hence, we are left just with the cases $c = 0,2,3$. We have
$$
s_2 = \begin{cases}
3k+1 & \text{for } c = 0;\\
3k+2 & \text{for } c = 2;\\
3k+3 & \text{for } c = 3.\\
\end{cases}, \quad \text{ and } \quad
a' = a-s_2 = \begin{cases}
2k-1 & \text{for } c = 0;\\
2k & \text{for } c = 2,3.
\end{cases},
$$
Now, define
$$
t_2 =
\begin{cases}
k & \text{for } c = 0;\\
k+1 & \text{for } c = 2,3.
\end{cases}
$$
Then, we proceed similarly as before. Note that $s_2 \geq 2t_2$. Now, we specialize $2t_2$ lines in such a way that, for $i = 1,\ldots,t_2$:
\begin{itemize}
\item the lines $\overline{Q_1P_{2i-1}}$ and $\overline{Q_1P_{2i}}$ both degenerate to a general line $R_i$ passing through $Q_1$;
\item the point $P_{2i-1}$ and the point $P_{2i}$ both degenerate to a general point $\widetilde{P}_i$ on $R_i$.
\end{itemize}
In this way, from the degeneration of $J_{2i-1}$ and $J_{2i}$, we obtain the $(2,2)$-jet $W_i$ defined by the scheme-theoretic intersection $2R_i \cap 2\overline{P_{2i-1}P_{2i}}$, for any $i = 1,\ldots,t_2$. Moreover, we specialize the support of the $W_i$'s and the double point $2Q_2$ on a line $L$. Note that this can be done because $s_2 - 2t_2 \geq 0$. By abuse of notation, we call again ${\mathbb{X}}$ the scheme obtained after such a specialization.
In this way, we have that
$$
\deg({\mathbb{X}}\cap L) =
\begin{cases}
2k + 2 & \text{for } c = 0 \\
2(k+1) + 2 & \text{for } c = 2,3;
\end{cases} = a'+3.
$$
Therefore, the line $L$ is a fixed component and can be removed. Hence,
$$
\dim\mathcal{L}_{a'+2}({\mathbb{X}}) = \dim\mathcal{L}_{a'+1}(\mathbb{J}),
$$
where $\mathbb{J} = a'Q_1 + Q_2 + {J}_1 + \ldots + {J}_{t_2} + J_{2t_2+1} + \ldots + J_{s_2}$, where
\begin{itemize}
\item ${J}_i$ is a $2$-jet with support on $\widetilde{P}_i\in L$ with general direction, for all $i = 1,\ldots, t_2$;
\item ${J}_i$ is a $2$-jet contained in $\overline{Q_1P_i}$, for all $i = 2t_2+1,\ldots,s_2$.
\end{itemize}
Now, since we look at curves of degree $a'+1$, the lines $\overline{Q_1P_i}$, for $i = 2t_2+1,\ldots,s_2$, become fixed components and can be removed. Hence, we get
$$
\dim\mathcal{L}_{a'+1}(\mathbb{J}) = \dim\mathcal{L}_{a''+1}(\mathbb{J}'),
$$
where $\mathbb{J}' = a''Q_1 + Q_2 + {J}_1 + \ldots + {J}_{t_2}$ and
$$
a'' = a' - (s_2 - 2t_2) =
\begin{cases}
k-2 & \text{for } c = 0;\\
k & \text{for } c = 2; \\
k -1 & \text{for } c = 3.
\end{cases}
$$
Hence,
$$
\deg(\mathbb{J}' \cap L) = 1+t_2 =
\begin{cases}
k+1 & \text{for } c = 0;\\
k+2 & \text{for } c = 2,3;
\end{cases} > a''+1;
$$
the line $L$ is a fixed component and can be removed. Hence, we are left with the linear system $\mathcal{L}_{a''}(a''Q_1 + \widetilde{P}_1 + \ldots + \widetilde{P}_{t_2})$. Since $t_2 = a''+1$, this linear system is empty and this concludes the proof of the case $s = s_2$.
\end{proof}
\section{Main result}\label{sec: main}
We are now ready to consider our general case. First, we answer to Question \ref{question: 0-dim P2}.
\begin{remark}[\sc Strategy of the proof]\label{rmk: strategy}
Our strategy to compute the dimension of $\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s})$ goes as follows. Fix a general line $L$. Then, we consider a specialization ${\mathbb{X}} = aQ_1 + bQ_2 + {Y}_1 + \ldots + {Y}_s$ of the $0$-dimensional scheme $\mathbb{X}_{a,b;s}$, where the $Y_i$'s are $(3,2)$-points with support at $P_1,\ldots,P_s$, respectively, and such that:
\begin{itemize}
\item ${Y}_1,\ldots,{Y}_x$ have support on $L$ and $\deg({Y}_i \cap L) = 3$, for all $i = 1,\ldots,x$;
\item ${Y}_{x+1},\ldots,{Y}_{x+y}$ have support on $L$, $\deg({Y}_i \cap L) = 2$ and ${\rm Res}_L({Y}_i)$ is a $3$-jet contained in $L$, for all $i = x+1,\ldots,x+y$;
\item ${Y}_{x+y+1},\ldots,{Y}_{x+y+z}$ have support on $L$, $\deg({Y}_i \cap L) = 2$ and ${\rm Res}_L({Y}_i)$ is a $2$-fat point, for all $i = x+y+1,\ldots,x+y+z$;
\item ${Y}_{x+y+z+1},\ldots,{Y}_{s}$ are generic $(3,2)$-points.
\end{itemize}
We will show that it is possible to chose $x,y,z$ in such a way that we start a procedure that allows us to remove twice the line $L$ and the line $\overline{Q_1Q_2}$ because, step-by-step, they are fixed components for the linear systems considered. A similar idea has been used by the authors, together with E. Carlini, to study Hilbert functions of triple points in $\mathbb{P}^1\times\mathbb{P}^1$; see \cite{CCO17}.
Finally, we remain with the linear system of curves of degree $a+b-2$ passing through the $0$-dimensional scheme $(a-2)Q_1+(b-2)Q_2 + {P}_{x+y+1}+\ldots+{P}_{x+y+z} + {Y}_{x+y+z+1} + \ldots +{Y}_{s}$, where the simple points ${P}_{x+y+1},\ldots,{P}_{x+y+z+1}$ are collinear and lie on the line $L$; see Figure \ref{fig: main}, where the degree goes down from $a+b$ to $a+b-2$ and finally to $a+b-4$. Hence, combining a technical lemma to deal with the collinear points (Lemma \ref{lemma: collinear}), we conclude our proof by a two-step induction, using the results of the previous section for $b \leq 2$.
\end{remark}
\begin{figure}[h]
\xymatrix{
\includegraphics[scale=0.36]{main_1.png}
\ar@{=>}[r]<10ex> & \includegraphics[scale=0.36]{main_2.png} \ar@{=>}[d] \\
& \includegraphics[scale=0.36]{main_3.png}
}
\caption{The three main steps of our proof of Theorem \ref{thm: main P2}.}
\label{fig: main}
\end{figure}
This strategy works in general, except for a few number of cases that we need to consider separately.
\begin{lemma}\label{lemma: small cases}
In the same notation as above:
\begin{enumerate}
\item if $(a,b) = (3,3)$, then $\mathrm{HF}_{\mathbb{X}_{3,3;s}}(6) = \max\{0, 16-5s\}$;
\item if $(a,b) = (5,3)$, then $\mathrm{HF}_{\mathbb{X}_{5,3;s}}(8) = \max\{0, 24-5s\}$;
\item if $(a,b) = (4,4)$, then $\mathrm{HF}_{\mathbb{X}_{4,4;s}}(8) = \max\{0, 25-5s\}$.
\end{enumerate}
\end{lemma}
\begin{proof}
{\it (1)} If $(a,b) = (3,3)$, we have to consider $s_1 = 3$ and $s_2 = 4$. Let $\mathbb{X}_{3,3;3} = 3Q_1 + 3Q_2 + Y_1 + Y_2 + Y_3$, where $P_1,P_2,P_3$ is the support of $Y_1,Y_2,Y_3$, respectively. We specialize the scheme such that $\deg(Y_1 \cap \overline{Q_1P_1}) = 3$ and $\deg(Y_2\cap \overline{P_2P_3}) = \deg(Y_3\cap \overline{P_2P_3}) = 3$. Let $A$ be a generic point on the line $\overline{P_2P_3}$. Therefore, we have that the line $\overline{P_2P_3}$ is a fixed component for $\mathcal{L}_{6}(\mathbb{X}_{3,3;3} + A)$ and we remove it. Let $\mathbb{X}' = {\rm Res}_L(\mathbb{X}_{3,3;3}+A)$. Then:
\begin{itemize}
\item the line $\overline{Q_1Q_2}$ is a fixed component for $\mathcal{L}_5({\mathbb{X}}')$;
\item the line $\overline{Q_1P_1}$ is a fixed component for $\mathcal{L}_4({\rm Res}_{\overline{Q_1Q_2}}({\mathbb{X}'}))$;
\item the line $\overline{P_2P_3}$ is a fixed component for $\mathcal{L}_3({\rm Res}_{\overline{Q_1P_1}\cdot \overline{Q_1Q_2}}({\mathbb{X}}'))$.
\end{itemize}
Therefore, we obtain that
$\dim\mathcal{L}_6(\mathbb{X}_{3,3;3}+A) = \dim\mathcal{L}_2(Q_1 + 2Q_2 + J_1)$, where $J_1$ is a $2$-jet lying on $\overline{Q_1P_1}$. The latter linear system is empty and therefore, since the expected dimension of $\mathcal{L}_6(\mathbb{X}_{3,3;3})$ is $1$, we conclude that $\dim\mathcal{L}_6(\mathbb{X}_{3,3;3}) = 1$. As a direct consequence, we also conclude that $\dim\mathcal{L}_6(\mathbb{X}_{3,3;4}) = 0$.
\medskip
{\it (2)} If $(a,b) = (5,3)$, then $s_1 = 4$ and $s_2 = 5$. Let ${\mathbb{X}} = 5Q_1 + 3Q_2 + Y_1 + {Y}_2 + \ldots + {Y}_4$, where we specialize the support of ${Y}_2, Y_3, Y_4$ to be collinear on a line $L$ and $\deg(Y_2 \cap L) = \deg(Y_3 \cap L) = \deg(Y_4 \cap L) = 3$.
Now, $L$ is a fixed component for $\mathcal{L}_8({\mathbb{X}})$ and $\overline{Q_1Q_2}$ is a fixed component for $ \mathcal{L}_7({\rm Res}_L({\mathbb{X}}))$. Let $\mathbb{X}' = {\rm Res}_{\overline{Q_1Q_2}\cdot L}({\mathbb{X}}) = 4Q_1 + 2Q_2 + Y_1 + J_2 + J_3 + J_4$, where $J_i$'s are $2$-jets lying on $L$. Now, consider a generic point $A$ on $L$.
In this way, $L$ is a fixed component for $\mathcal{L}_6(\mathbb{X}' + A)$, $\overline{Q_1Q_2}$ is a fixed component for $\mathcal{L}_5({\rm Res}_{L}({\mathbb{X}}'+A))$ and $\overline{Q_1P_1}$ is a fixed component for $\mathcal{L}_4({\rm Res}_{\overline{Q_1Q_2}\cdot L}(\mathbb{X}'+A))$, where $P_1$ is the support of $Y_1$. Hence, we have
$$
\dim\mathcal{L}_6({\mathbb{X}}'+A) = \dim\mathcal{L}_3(2Q_1 + Q_2 + 2P_1) = 3.
$$
Since the expected dimension of $\mathcal{L}_6(\mathbb{X}')$ is $4$ and $\dim\mathcal{L}_6({\mathbb{X}}'+A) = 3$, we conclude that $\dim\mathcal{L}_{8}(\mathbb{X}_{5,3;4}) = \dim\mathcal{L}_6({\mathbb{X}}') = 4$, as expected.
In the case $s = s_2 = 5$, we consider ${\mathbb{X}} = 5Q_1 + 3Q_2 + Y_1 + Y_2 + {Y}_3 + \ldots + {Y}_5$, where the support of ${Y}_3, Y_4, Y_5$ are three collinear points ${P}_3, {P}_4$ and ${P}_5$, lying on a line $L$, and $\deg(Y_3 \cap L) = \deg(Y_4 \cap L) = \deg(Y_5 \cap L) = 3$. Then, $L$ is a fixed component for $ \mathcal{L}_8({\mathbb{X}})$ and $\overline{Q_1Q_2}$ is a fixed component for $ \mathcal{L}_7({\rm Res}_L({\mathbb{X}}))$. Now, specializing the scheme such that $\deg(Y_1 \cap \overline{Q_1P_1}) = \deg(Y_2 \cap \overline{Q_1P_2}) = 3$, the lines $\overline{Q_1P_1}$ and $\overline{Q_1P_2}$ become also fixed components of the latter linear system and can be removed.
Hence,
$$
\dim \mathcal{L}_8({\mathbb{X}}) = \dim \mathcal{L}_4(\mathbb{Y}),
$$
where $\mathbb{Y} = 2Q_1 + 2Q_2 + J_1 + J_2 + {J}_3 + \ldots + {J}_5$, where $J_1,J_2$ are $2$-jets lying on $\overline{Q_1P_1}$ and $\overline{Q_1P_2}$, respectively, and $J_3,J_4,J_5$ are $2$-jets lying on $L$. Now:
\begin{itemize}
\item $L$ is a fixed component for $\mathcal{L}_4(\mathbb{Y})$;
\item $\overline{Q_1Q_2}$ is a fixed component for $\mathcal{L}_3({\rm Res}_L(\mathbb{Y}))$;
\item $\overline{Q_1P_1}$ and $\overline{Q_1P_2}$ are fixed components for $\mathcal{L}_2({\rm Res}_{\overline{Q_1Q_2}\cdot L}(\mathbb{Y}))$.
\end{itemize}
From this, we conclude that $\dim\mathcal{L}_8(\mathbb{X}) = \dim\mathcal{L}_4(\mathbb{Y}) = \dim\mathcal{L}_0(Q_2) = 0$, as expected.
\medskip
{\it (3)} If $(a,b) = (4,4)$, we have $s_1 = s_2 = 5$. Consider ${\mathbb{X}} = 4Q_1 + 4Q_2 + Y_1+ Y_2+Y_3 + {Y}_4 + {Y}_5$, where the supports of ${Y}_4$ and ${Y}_5$ are collinear with $Q_2$ on a line $L$. In this specialization, we also assume that:
\begin{itemize}
\item $\deg({Y}_4 \cap L) = 3$;
\item $\deg({Y}_5 \cap L) = 2$ and $\deg({\rm Res}_L({Y}_5) \cap L) = 3$.
\end{itemize}
Therefore, we obtain that $L$ is a fixed component for the linear system and can be removed twice. Hence,
$$
\dim\mathcal{L}_8({\mathbb{X}}) = \dim\mathcal{L}_6({\rm Res}_{2L}({\mathbb{X}})),
$$
where ${\rm Res}_{2L}({\mathbb{X}}) = \mathbb{X}_{4,2;3}$. By Lemma \ref{lemma: b = 2}, the latter linear system is empty and we conclude.
\end{proof}
The following lemma is a well-known tool to study the Hilbert function of $0$-dimensional schemes which have some reduced component lying on a line.
\begin{lemma}\label{lemma: collinear}{\rm \cite[Lemma 2.2]{CGG05-SegreVeronese}}
Let $\mathbb{X} \subset \mathbb{P}^2$ be a $0$-dimensional scheme, and let $P_1,\ldots,P_s$ be general points on a line $L$. Then:
\begin{enumerate}
\item if $ \dim\mathcal{L}_{d}(\mathbb{X}+P_1+\ldots+P_{s-1}) > \dim\mathcal{L}_{d-1}({\rm Res}_L(\mathbb{X}))$, then
$$
\dim\mathcal{L}_d(\mathbb{X}+P_1+\ldots+P_s) = \dim\mathcal{L}_d(\mathbb{X}) - s;$$
\item
if $ \dim\mathcal{L}_{d-1}({\rm Res}_L(\mathbb{X}))=0$ and $\dim\mathcal{L}_{d}(\mathbb{X})\leq s$, then
$\dim\mathcal{L}_{d}(\mathbb{X}+P_1+\ldots+P_{s})= 0$.
\end{enumerate}
\end{lemma}
Now, we are ready to prove our first main result.
\begin{theorem}\label{thm: main P2}
Let $a,b$ be positive integers with $ab > 1$. Then, let $\mathbb{X}_{a,b;s} \subset \mathbb{P}^2$ as in the previous section. Then,
$$
\dim \mathcal{L}_{a+b}(\mathbb{X}_{a,b;s}) = \max\{0, (a+1)(b+1) - 5s\}.
$$
\end{theorem}
\begin{proof}
We assume that $a \geq b \geq 3$, since the cases $b = 1$ and $b = 2$ are treated in Lemma \ref{lemma: b = 1}, Lemma \ref{lemma: a = 2} and Lemma \ref{lemma: b = 2}. Moreover, as explained in Section \ref{sec: super- and sub-abundance}, we consider $s \leq s_2$.
First, we show how to choose the numbers $x,y,z$ described in the Remark \ref{rmk: strategy}.
Let $a+b = 5h+c$, with $0\leq c \leq 4$. Then, we fix
\begin{center}
\begin{tabular}{c || c | c | c}
c & x & y & z \\
\hline
0 & h+1 & h-1 & 0 \\
1 & h & h-2 & 3 \\
2 & h+1 & h-1 & 1 \\
3 & h & h-2 & 4 \\
4 & h+1 & h-1 & 2 \\
\end{tabular}
\end{center}
Note that in order to make these choices, we need to assume that $y \geq 0$ for any $c$. In particular, this means that the cases with $a+b$ equal to $6$ and $8$ have to be treated in a different way. Hence, the only cases we have to treat differently from the main strategy are $(a,b) = (3,3), (5,3), (4,4)$, already considered in Lemma \ref{lemma: small cases}.
Note that, for any $h,c$, we have that $x+y+z \leq s_1 = \left\lfloor \frac{(a+1)(b+1)}{5} \right\rfloor$. Indeed, it is enough to see that
$$
\frac{(a+1)(b+1)}{5} - (x+y+z) \geq 0.
$$
By direct computation, we have that
$$
(a+1)(b+1) - 5(x+y+z) = (a-1)(b-1) -
\begin{cases}
0 & \text{ for } c = 0; \\
3 & \text{ for } c = 1; \\
1 & \text{ for } c = 2; \\
4 & \text{ for } c = 3; \\
2 & \text{ for } c = 4.
\end{cases}
$$
Since $a,b \geq 3$, we conclude that $x+y+z \leq s_1$.
With these assumptions, by direct computation, we obtain that, for any $c$,
$$
\deg({\mathbb{X}} \cap L) = 3x+2y+2z = a+b+1 \quad \text{ and } \quad
\deg({\rm Res}_L({\mathbb{X}}) \cap L) = 2x+3y+2z = a+b-1.
$$
Therefore, we can do the following procedure (see Figure \ref{fig: main}):
\begin{itemize}
\item the line $L$ is a fixed component for $\mathcal{L}_{a+b}({\mathbb{X}})$ and it can be removed;
\item the line $\overline{Q_1Q_2}$ is a fixed component for $\mathcal{L}_{a+b-1}({\rm Res}_L({\mathbb{X}}))$ and it can be removed;
\item the line $L$ is a fixed component for $\mathcal{L}_{a+b-2}({\rm Res}_{\overline{Q_1Q_2}\cdot L}({\mathbb{X}}))$ and it can be removed;
\item the line $\overline{Q_1Q_2}$ is a fixed component for $\mathcal{L}_{a+b-3}({\rm Res}_{\overline{Q_1Q_2}\cdot 2L}({\mathbb{X}}))$ and it can be removed.
\end{itemize}
Denote $\mathbb{Y} = {\rm Res}_{2\overline{Q_1Q_2}\cdot 2L}({\mathbb{X}})$ which is the union of $\mathbb{X}_{a-2,b-2;s-(x+y+z)}$ and a set of $z$ general collinear points on $L$. By the previous reductions, we have that
$$
\dim\mathcal{L}_{a+b}({\mathbb{X}}) = \dim\mathcal{L}_{a+b-4}(\mathbb{Y}).
$$
Now, case by case, we prove that the latter linear system has the expected dimension. In these computations, we proceed by induction on $b$ (with base cases $b = 1$ and $b = 2$ proved in Lemma \ref{lemma: b = 1}, Lemma \ref{lemma: a = 2} and Lemma \ref{lemma: b = 2}) to deal with $\mathbb{X}_{a-2,b-2;s-(x+y+z)}$; and by using Lemma \ref{lemma: collinear} to deal with the the $z$ general collinear points that we denote by $A_1,\ldots,A_z$.
\medskip
\noindent {\sc Case $c = 0$.} In this case, since $z = 0$, we conclude just by induction on $b$; indeed
\begin{align*}
\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) & = \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s-2h}) =
\max\{0, (a-1)(b-1) - 5(s-2h) \} = \\
& = \max\{0, (a+1)(b+1) - 5s \} = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s}).
\end{align*}
\medskip
\noindent {\sc Case $c = 1$.} We first consider the case $s = s_1$. We want to use Lemma \ref{lemma: collinear}, so, since $z = 3$, we compute the following:
\begin{align*}
\dim\mathcal{L}_{a+b-4}&(\mathbb{X}_{a-2,b-2;s_1-(2h+1)} + A_1 + A_2) \\
& = \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s_1-(2h+1)}) - 2 & \hfill \text{(since 2 points are always general)}\\
& = \max\{0,(a-1)(b-1)-5(s_1-(2h+1))\}-2 & \hfill \text{(by induction)} \\
& = \max\{0, (a+1)(b+1)-5s_1+1\} = \\
& = (a+1)(b+1)-5s_1+1.
\end{align*}
\begin{align*}
\dim \mathcal{L}_{a+b-5}&({\rm Res}_L(\mathbb{Y})) = \dim\mathcal{L}_{a+b-5}(\mathbb{X}_{a-2,b-2;s_1-(2h+1)}) = \\
& = \dim\mathcal{L}_{a+b-6}(\mathbb{X}_{a-3,b-3;s_1-(2h+1)}) =
& \hfill \text{(since $\overline{Q_1Q_2}$ is a fixed component)} \\
& = \max\{0,(a-2)(b-2)-5(s_1-(2h+1))\} & \hfill \text{(by induction)} \\
& = \max\{0, ab+7-5s_1\}.
\end{align*}
Now, since
\begin{align*}
(a+1)(b+1) - 5s_1+ 1 > 0, \quad \quad \text{ and } \\
\left( (a+1)(b+1) - 5s_1 + 1\right) - (ab+7-5s_1) = a+b-5 > 0,
\end{align*}
by Lemma \ref{lemma: collinear}(1), we have that
\begin{align*}
\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) & = \max\{0, (a-1)(b-1)-5(s_1-(2h+1)) - 3\} = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s_1}).
\end{align*}
Now, consider $s = s_2$. Then, we have
\begin{align*}
\dim \mathcal{L}_{a+b-5}&({\rm Res}_L(\mathbb{Y})) = \dim\mathcal{L}_{a+b-5}(\mathbb{X}_{a-2,b-2;s_2-(2h+1)}) \\
& = \dim\mathcal{L}_{a+b-6}(\mathbb{X}_{a-3,b-3;s_2-(2h+1)}) & \hfill \text{(since $\overline{Q_1Q_2}$ is a fixed component)} \\
& = \max\{0,(a-2)(b-2)-5(s_2-(2h+1))\} & \hfill \text{(by induction)} \\
& = \max\{0, ab+7-5s_2\} = 0,
\end{align*}
where the latter equality is justified by the fact that, by definition of $s_2$ (see Section \ref{sec: super- and sub-abundance}), we have
$$
ab+7-5s_2 \leq ab + 7 - (a+1)(b+1) = 6 - (a+b) \leq 0.
$$
Moreover, by definition of $s_2$,
\begin{align*}
\dim \mathcal{L}_{a+b-4}({\rm Res}_L(\mathbb{Y})) & = \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s_2-(2h+1)}) = \\
& = \max\{0, (a-1)(b-1)-5(s_2-(2h+1))\} = & \hfill \text{(by induction)} \\
& = \max\{0, (a+1)(b+1)-5s_2+3\} \leq 3.
\end{align*}
Hence, we can apply Lemma \ref{lemma: collinear}(2) and conclude that, for $s = s_2$,
$$\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) = 0 = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s_2}).$$
\medskip
\noindent {\sc Case $c = 2$ and $c = 4$.} Since a generic simple point or two generic simple points always impose independent conditions on a linear system of curves, we conclude this case directly by induction on $b$. Indeed, for $c = 2$, we have
\begin{align*}
\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) & = \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s-(2h+1)}) - 1 = \\
& = \max\{0, (a-1)(b-1)-5(s-(2h+1)) \} - 1 = & \hfill \text{(by induction)} \\
& = \max\{0, (a+1)(b+1)-5s\} = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s}).
\end{align*}
Similarly, for $c = 4$, we have
\begin{align*}
\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) & = \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s-(2h+2)}) - 2 = \\
& = \max\{0, (a-1)(b-1)-5(s-(2h+2))\} - 2 = & \hfill \text{(by induction)} \\
& = \max\{0, (a+1)(b+1)-5s\} = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s}).
\end{align*}
\medskip
\noindent {\sc Case $c = 3$.} We first consider the case $s = s_1$. We want to use Lemma \ref{lemma: collinear}, so, since $z = 4$, we compute the following:
\begin{align*}
\dim\mathcal{L}_{a+b-4}& (\mathbb{X}_{a-2,b-2;s_1-(2h+2)} + A_1 + A_2 + A_3) \\
& \geq \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s_1-(2h+2)}) - 3 \\
& = \max\{0, (a-1)(b-1)-5(s_1-(2h+2)) \} - 3 & \hfill \text{(by induction)} \\
& = \max\{0, (a+1)(b+1)-5s_1+1\} = \\
& = (a+1)(b+1)-5s_1+1 \geq 1.
\end{align*}
\begin{align*}
\dim \mathcal{L}_{a+b-5}&({\rm Res}_L(\mathbb{Y})) = \dim\mathcal{L}_{a+b-5}(\mathbb{X}_{a-2,b-2;s_1-(2h+2)}) = \\
& = \dim\mathcal{L}_{a+b-6}(\mathbb{X}_{a-3,b-3;s_1-(2h+2)}) & \hfill \text{(since $\overline{Q_1Q_2}$ is a fixed component)} \\
& = \max\{0,(a-2)(b-2)-5(s_1-(2h+2))\} & \hfill \text{(by induction)} \\
& = \max\{0, ab+8-5s_1\}.
\end{align*}
Since
\begin{align*}
(a+1)(b+1) - 5s_1 + 1 > 0, \quad \quad \text{ and } \\
\left( (a+1)(b+1) - 5s_1 + 1\right) - (ab+8-5s_1) = a+b-6 > 0,
\end{align*}
by Lemma \ref{lemma: collinear}(1), we have that
\begin{align*}
\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) & = \max\{0, (a-1)(b-1)-5(s_1-(2h+2))\} - 4 = \\
& = (a+1)(b+1) - 5s_1 = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s_1}).
\end{align*}
Now, consider $s = s_2$. We have
\begin{align*}
\dim \mathcal{L}_{a+b-5}&({\rm Res}_L(\mathbb{Y})) = \dim\mathcal{L}_{a+b-5}(\mathbb{X}_{a-2,b-2;s_2-(2h+2)}) = \\
& = \dim\mathcal{L}_{a+b-6}(\mathbb{X}_{a-3,b-3;s_2-(2h+2)}) & \hfill \text{(since $\overline{Q_1Q_2}$ is a fixed component)} \\
& = \max\{0,(a-2)(b-2)-5(s_2-(2h+2))\} & \hfill \text{(by induction)} \\
& = \max\{0, ab+8-5s_2\} = 0,
\end{align*}
where the latter equality is justified by the fact that, by definition of $s_2$ (see Section \ref{sec: super- and sub-abundance}), we have
$$
ab+8-5s_2 \leq ab+8-(a+1)(b+1) = 7 - (a+b) \leq 0.
$$
Moreover, by definition of $s_2$,
\begin{align*}
\dim \mathcal{L}_{a+b-4}({\rm Res}_L(\mathbb{Y})) & = \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s_2-(2h+2)}) = \\
& = \max\{0, (a-1)(b-1)-5(s_2-(2h+2))\} = & \hfill \text{(by induction)} \\
& = \max\{0, (a+1)(b+1)-5s_2+4\} \leq 4.
\end{align*}
Hence, we can apply Lemma \ref{lemma: collinear}(2) and conclude that, for $s = s_2$,
$$\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) = 0 = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s_2}).$$
\end{proof}
By multiprojective-affine-projective method, from Theorem \ref{thm: main P2}, we answer to Question \ref{question: 0-dim P1xP1}.
\begin{corollary}\label{corollary: main P1xP1}
Let $a, b$ be integers with $ab > 1$. Let $\mathbb{X}\subset\mathbb{P}^1\times\mathbb{P}^1$ be the union of $s$ many $(3,2)$-points with generic support and generic direction. Then,
$$
\mathrm{HF}_\mathbb{X}(a,b) = \min\{(a+1)(b+1), 5s\}.
$$
\end{corollary}
In conclusion, by the relation between the Hilbert function of schemes of $(3,2)$-points in $\mathbb{P}^1 \times \mathbb{P}^1$ and the dimension of secant varieties of the tangential variety of Segre-Veronese surfaces (see Section \ref{section: apolarity}), we can prove our final result.
\begin{theorem}\label{thm: main}
Let $a,b$ be positive integers with $ab>1$. Then, the tangential variety of any Segre-Veronese surface $SV_{a,b}$ is non defective, i.e., all the secant varieties have the expected dimension.
\end{theorem}
\bibliographystyle{alpha}
\section{Introduction}
The study of {\it secant varieties} and {\it tangential varieties} is very classical in Algebraic Geometry and goes back to the school of the XIX century. In the last decades, these topics received renewed attention because of their \Alessandro{connections} with more applied sciences \Alessandro{which uses {\it additive decompositions of tensors}}. \Alessandro{For us,} {\it tensors} are multidimensional arrays \Alessandro{of complex numbers and} classical geometric objects as Veronese, Segre, Segre-Veronese varieties, and their tangential varieties, are parametrised by tensors with particular symmetries and structure. \Alessandro{Their $s$-secant variety is the closure of the locus of linear combinations of $s$ many of those} particular tensors. We refer to \cite{L} for an exhaustive description of the fruitful use of classical algebraic geometry in problems regarding tensors decomposition.
A very important invariant \Alessandro{of these varieties} is their {\it dimension}.
\medskip
A rich literature has been devoted to studying dimensions of secant varieties of special projective varieties. In particular, we mention:
\begin{enumerate}
\item[1.] {\it Veronese varieties}, completely solved by J. Alexander and A. Hirschowitz \cite{AH};
\item[2.] {\it Segre varieties}, solved in few specific cases, e.g., see \cite{AOP, CGG02-Segre, CGG11-SegreP1};
\item[3.] {\it Segre-Veronese varieties}, solved in even fewer specific cases, e.g., see \cite{AB13-SegreVeronese, CGG05-SegreVeronese, Abr08};
\item[4.] {\it tangential varieties of Veronese varieties}, completely solved by H. Abo and N. Vannieuwenhoven, see \cite{AV18, BCGI09}.
\end{enumerate}
In this paper, we consider the following question.
\begin{question}\label{question: dimension secants}
What is the dimension of secant varieties of tangential varieties of Segre-Veronese surfaces?
\end{question}
Let $a, b$ be positive integers. We define the {\bf Segre-Veronese embedding} of $\mathbb{P}^1\times \mathbb{P}^1$ in bi-degree $(a,b)$ as the embedding of $\mathbb{P}^1\times\mathbb{P}^1$ with the linear system $\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(a,b)$ of curves of bi-degree $(a,b)$, namely
\begin{eqnarray*}
\nu_{a,b} : &\mathbb{P}^1 \times \mathbb{P}^1 & \rightarrow \quad \quad \quad \mathbb{P}^{(a+1)(b+1)-1},\\
&([s_{0}:s_{1}],[t_{0}:t_{1}]) & \mapsto \quad [s_{0}^at_{0}^b:s_{0}^{a-1}s_{1}t_{0}^b:\ldots:s_{1}^at_{1}^b].
\end{eqnarray*}
The image of $\nu_{a,b}$ is the {\bf Segre-Veronese surface} of bi-degree $(a,b)$, denoted by $SV_{a,b}$, \Alessandro{which is parametrized by {\it decomposable partially symmetric tensors}.}
Let $V_1$ and $V_2$ be two $2$-dimensional $\mathbb{C}$-vector spaces. Let $\mathit{Sym}(V_i) = \bigoplus_{d\geq 0}\mathit{Sym}^d(V_i)$ be the symmetric algebra over $V_i$, for $i = 1,2$. If we fix basis $(x_0,x_1)$ and $(y_0,y_1)$ for $V_1$ and $V_2$, respectively, then we have the identifications of the respective symmetric algebras with the polynomial rings \Alessandro{
\begin{equation}\label{eq: identif}
\begin{array}{r c l c r c l}
\mathit{Sym}(V_1) &\simeq& \mathbb{C}[x_0,x_1] & \text{and} & \mathit{Sym}(V_2) & \simeq & \mathbb{C}[y_0,y_1], \\
v_1 = (v_{10},v_{11}) &\leftrightarrow& \ell_1 = v_{10}x_0+v_{11}x_1 & & v_2 = (v_{20},v_{21}) &\leftrightarrow& \ell_2 = v_{20}y_0+v_{21}y_1 \\
\end{array}
\end{equation}
}
Therefore, $\mathit{Sym}(V_1) \otimes \mathit{Sym}(V_2)$ is identified with the bi-graded ring $\mathcal{S} = \mathbb{C}[x_0,x_1;y_0,y_1] = \bigoplus_{a,b}\mathcal{S}_{a,b}$, where $\mathcal{S}_{a,b}$ is the $\mathbb{C}$-vector space of bi-homogeneous polynomials of bi-degree $(a,b)$, i.e., $\mathcal{S}_{a,b} = \mathit{Sym}^a(V_1) \otimes_{\mathbb{K}} \mathit{Sym}^b(V_2)$. \Alessandro{In particular, we consider the monomial basis of $\mathcal{S}_{a,b}$ given by
$$
\left\{{a \choose i_0}{b \choose j_0}x_0^{i_0}x_1^{i_1}y_0^{j_0}y_1^{j_1} ~|~ \substack{i_0,i_1,j_0,j_1 \geq 0 \\ i_0+i_1 = a, j_0+j_1 = b}\right\}.
$$
In this way, the Segre-Veronese embedding of $\mathbb{P}^1\times\mathbb{P}^1$ in bi-degree $(a,b)$ can be rewritten as }
\begin{eqnarray*}
\nu_{a,b} : & \mathbb{P}(V_1) \times \mathbb{P}(V_2) & \rightarrow \quad \mathbb{P}(\Alessandro{\mathit{Sym}^a(V_1)\otimes\mathit{Sym}^b(V_2)}),\\
&([v_1],[v_2]) & \mapsto [v_1^{\otimes a}\otimes v_2^{\otimes b}].
\end{eqnarray*}
Throughout all the paper\Alessandro{, by \eqref{eq: identif}, we} identify the tensor $v_1^{\otimes a}\otimes v_2^{\otimes b}$ with the polynomial $\ell_1^a\ell_2^b$ \Alessandro{and we view the Segre-Veronese variety of bi-degree $(a,b)$ as the projective variety parametrized by these particular bi-homogeneous polynomials.}
\smallskip
Given any projective variety $X \subset \mathbb{P}^N$, we define the {\bf tangential variety} of $X$ as the Zariski closure of the union of the tangent spaces at smooth points of $X$, i.e., if $U \subset X$ denotes the open subset of smooth points of $X$, then it is
$$
\calT(X) := \overline{\bigcup_{P \in U} T_P(X)} \subset \mathbb{P}^N,
$$
where $T_P(X)$ denotes the tangent space of $X$ at the point $P$.
\smallskip
Given any projective variety $X \subset \mathbb{P}^N$, we define the {\bf $s$-secant variety} of $X$ as the Zariski closure of the union of all linear spans of $s$-tuples of points on $X$, i.e.,
\begin{equation}\label{equation: expected dimension}
\sigma_s(X) := \overline{\bigcup_{P_1,\ldots,P_s \in X} \left\langle P_1,\ldots,P_s \right\rangle} \subset \mathbb{P}^N,
\end{equation}
where $\langle - \rangle$ denotes the linear span of the points.
As we said before, we are interested in the dimension of these varieties. By parameter count, we have an {\bf expected dimension} of $\sigma_s(X)$ which is
$$
{\rm exp}.\dim\sigma_s(X) = \min\{N, s\dim(X) + (s-1)\}.
$$
However, we have varieties whose $s$-secant variety has dimension smaller than the expected one and we call them {\it defective varieties}. In this article we prove that \Alessandro{the} tangential varieties \Alessandro{of all the Segre-Veronese surfaces} are never defective.
\setcounter{section}{4}
\setcounter{theorem}{5}
\begin{theorem}
Let $a,b$ be positive integers with $ab > 1$. Then, the tangential variety of any Segre-Veronese surface $SV_{a,b}$ is non defective, i.e., all secant varieties have the expected dimension; namely,
$$
\dim\sigma_s(\calT(SV_{a,b})) = \min\{(a+1)(b+1), 5s\}-1.
$$
\end{theorem}
\Alessandro{In order to prove our result, we use an approach already used in the literature which involves the study of {\it Hilbert functions} of {\it $0$-dimensional schemes} in the multiprojective space $\mathbb{P}^1\times\mathbb{P}^1$. In particular, first we use a method introduced by the first author, with A. V. Geramita and A. Gimigliano \cite{CGG05-SegreVeronese}, to reduce our computations to the standard projective plane; and second, we study the dimension of particular linear systems of curves with non-reduced base points by using degeneration techniques which go back to G. Castelnuovo, but have been refined by the enlightening work of J. Alexander and A. Hirschowitz \cite{AHb, AHa, AH, AH00}. However, as far as we know, the particular type of degeneration that we are using has not been exploited before in the literature and we believe that it might be useful to approach other similar problems.}
\setcounter{section}{1}
\paragraph*{\bf Structure of the paper.} In Section \ref{sec: secant and 0-dim schemes}, we show how the dimension of secant varieties can be computed by studying the Hilbert function of $0$-dimensional schemes and we introduce the main tools that we use in our proofs, such as the {\it multiprojective-affine-projective method} and {\it la m\'ethode d'Horace diff\'erentielle}. In Section \ref{sec: lemmata}, we consider the cases of small bi-degrees, i.e., when $b \leq 2$. These will be the base steps for our inductive proof of the general case that we present in Section~\ref{sec: main}.
\smallskip
\paragraph*{\bf Acknowledgements.}
The first author was supported by the Universit\`a degli Studi di Genova through the \Alessandro{``FRA (Fondi per la Ricerca di Ateneo) 2015”}. The second author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R\&D (MDM-2014-0445).
\section{Secant varieties and $0$-dimensional schemes}\label{sec: secant and 0-dim schemes}
In this section we recall some basic constructions and we explain how they are used to reduce the problem of computing dimensions of secant varieties to the problem of computing Hilbert functions of special $0$-dimensional schemes.
\subsection{Terracini's Lemma} A standard way to compute the dimension of an algebraic variety is to look at its tangent space at a general point. In the case of secant varieties, the structure of tangent spaces is very nicely described by a classical result of A.~Terracini \cite{T11}.
\begin{lemma}[Terracini's Lemma \cite{T11}]
Let $X$ be a projective variety. Let $P_1,\ldots,P_s$ be general points on $X$ and let $P$ be a general point on their linear span. Then,
$$
T_P\sigma_s(X) = \left\langle T_{P_1}X,\ldots,T_{P_s}X\right\rangle.
$$
\end{lemma}
\noindent \Alessandro{Therefore, in order to understand the dimension of the general tangent space of the secant variety $\sigma_s(X)$, we need to compute the dimension of the linear span of the tangent spaces to $X$ at $s$ general points. We do it in details for the tangential varieties of Segre-Veronese surfaces.}
\smallskip
Let $\ell_1 \in \mathcal{S}_{1,0}$ and $\ell_2 \in \mathcal{S}_{0,1}$. Then, we consider the bi-homogeneous polynomial $\ell_1^a\ell_2^b \in \mathcal{S}_{a,b}$ which represents a point on the Segre-Veronese variety $SV_{a,b}$. Now, if we consider two general linear forms $m_1 \in \mathcal{S}_{1,0}$ and $m_2 \in \mathcal{S}_{0,1}$, then
$$
\left.\frac{d}{dt}\right|_{t=0}(\ell_1+tm_1)^a(\ell_2+tm_2)^b = a\ell_1^{a-1}\ell_2^bm_1 + b\ell_1^a\ell_2^{b-1}m_2;
$$
therefore, we obtain that
$$
T_{[\ell_1^a\ell_2^b]} SV_{a,b} = \mathbb{P}\left( \left\langle \ell_1^{a-1}\ell_2^b \cdot\mathcal{S}_{1,0},~\ell_1^{a}\ell_2^{b-1} \cdot \mathcal{S}_{0,1} \right\rangle \right).
$$
Hence, the tangential variety $\calT(SV_{a,b})$ is the image of the embedding
\begin{eqnarray*}
\tau_{a,b} : & \big(\mathbb{P}(V_1)\times\mathbb{P}(V_2)\big)\times \big(\mathbb{P}(V_1)\times\mathbb{P}(V_2)\big) & \rightarrow ~~~~~~~~\mathbb{P}({\mathcal{S}_{a,b}}),\\
&~~~\big(([\ell_1],[\ell_2])~;~([m_1],[m_2])\big) & \mapsto [\ell_1^{a-1}\ell_2^bm_1+\ell_1^a\ell_2^{b-1}m_2].
\end{eqnarray*}
\begin{remark}\label{rmk: (a,b) = (1,1)}
The variety $SV_{1,1}$ is the Segre surface of $\mathbb{P}^3$ whose tangential variety clearly fills the entire ambient space. For this reason, we will always consider pairs of positive integers $(a,b)$ where at least one is strictly bigger than $1$. Hence, from now on, we assume $ab > 1$.
\end{remark}
Now, fix $\ell_1,m_1 \in \mathcal{S}_{1,0}$ and $\ell_2,m_2 \in \mathcal{S}_{0,1}$. For any $h_1,k_1 \in \mathcal{S}_{1,0}$ and $h_2,k_2\in \mathcal{S}_{0,1}$, we have
\begin{align*}
\left.\frac{d}{dt}\right|_{t=0} (\ell_1 + th_1)^{a-1}&(m_1+tk_1)(\ell_2+th_2)^b + (\ell_1 + th_1)^{a}(\ell_2+th_2)^{b-1}(m_2+tk_2) = \\
& = (a-1)\ell_1^{a-2}h_1m_1\ell_2^b + \ell_1^{a-1}k_1\ell_2^b + b\ell_1^{a-1}m_1\ell_2^{b-1}h_2 + \\ & \quad \quad + a\ell_1^{a-1}h_1\ell_2^{b-1}m_2 + (b-1)\ell_1^a\ell_2^{b-2}h_2m_2 + \ell_1^a\ell_2^{b-1}k_2.
\end{align*}
\Alessandro{Note that, if $b = 1$ (or $a = 1$, resp.), the summand where $\ell_2$ is appearing with exponent $(b-2)$ (or where $\ell_1$ is appearing with exponent $(a-2)$, resp.) vanishes since it appears multiplied by the coefficient $(b-1)$ (or $(a-1)$, resp.).}
Therefore, we have that, if $P = \tau_{a,b}\left(([\ell_1],[\ell_2]),([m_1,m_2])\right)\Alessandro{\in\mathcal{T}(SV_{a,b})}$, then
\begin{align*}
T_{P} (\calT(SV_{a,b})) = \mathbb{P}\left( \left\langle \ell_1^a\ell_2^{b-1}\cdot \mathcal{S}_{0,1},~ \ell_1^{a-1}\ell_2^b \cdot \mathcal{S}_{1,0},~ \right.\right. &\left.\left. \ell_1^{a-2}\ell_2^{b-1}\big((a-1) m_1\ell_2 + a\ell_1 m_2 \big) \cdot \mathcal{S}_{1,0}, \right.\right. \\
& \quad\quad\left.\left. \ell_1^{a-1}\ell_2^{b-2}\big(b m_1\ell_2 + (b-1)\ell_1 m_2 \big)\cdot \mathcal{S}_{0,1} \right\rangle \right)
\end{align*}
From this description of the general tangent space to the tangential variety, we can conclude that the tangential variety $\calT(SV_{a,b})$ has the expected dimension.
\begin{lemma}\label{lemma: dimension tangential}
Let $(a,b)$ be a pair of positive integers with $ab > 1$. Then, $\calT(SV_{a,b})$ is $4$-dimensional.
\end{lemma}
\begin{proof}
Let $P$ be a general point of $\calT(SV_{a,b})$. Now, up to change of coordinates, we may assume
$$
\ell_1 = x_0,~m_1 = x_1 \quad \text{ and } \quad \ell_2 = y_0, m_2 = y_1.
$$
By direct computations, we have that the affine cone over $T_P(\calT(SV_{a,b}))$ is
\begin{align*}
\left\langle x_0^ay_0^b,~ x_0^ay_0^{b-1}y_1,~ x_0^{a-1}x_1y_0^b,~ \right.& (a-1)x_0^{a-2}x_1^2y_0^b + a x_0^{a-1}x_1y_0^{b-1}y_1, \\ & \quad \quad \left.
bx_0^{a-1}x_1y_0^{b-1}y_1 + (b-1)x_0^ay_0^{b-2}y_1^2\right\rangle.
\end{align*}
\Alessandro{As mentioned above, if either $a = 1$ (or $b = 1$, resp.), the summand where $x_0$ is appearing with exponent $(a-2)$ (or $y_0$ is appearing with exponent $(b-2)$, resp.) vanishes because it is multiplied by $(a-1)$ (or $(b-1)$, resp.). Since $ab > 1$, this} linear space clearly has dimension $5$. Thus, the claim follows.
\end{proof}
Therefore, by \eqref{equation: expected dimension}, we have that
\begin{equation}\label{equation: expected dimension tangent}
{\rm exp}.\dim \sigma_s(\calT(SV_{a,b})) = \min\left\{(a+1)(b+1), 5s\right\} - 1.
\end{equation}
\subsection{Apolarity Theory \Alessandro{and Fat points}}\label{section: apolarity}
For higher secant varieties, similar computations as Lemma \ref{lemma: dimension tangential} are not feasible. In order to overcome this difficulty, a classical strategy is to use {\it Apolarity Theory}. Here we recall the basic constructions, but, for exhaustive references on this topic, we refer to \cite{IK, G}.
\smallskip
Let $\mathcal{S} = \mathbb{C}[x_0,x_1;y_0,y_1] = \bigoplus_{(a,b)\in\mathbb{N}^2} \mathcal{S}_{a,b}$ be the bi-graded polynomial ring. Any bi-homogeneous ideal $I$ inherits the grading, namely, we have $I = \bigoplus_{a,b} I_{a,b}$, where $I_{a,b} = I \cap \mathcal{S}_{a,b}$. Fixed a bi-degree $(a,b)$, for any $\alpha = (\alpha_0,\alpha_1)\in \mathbb{N}^2$ and $\beta = (\beta_0,\beta_1) \in \mathbb{N}^2$ such that $|\alpha| = \alpha_0+\alpha_1 = a$ and $|\beta| = \beta_0+\beta_1 = b$, we denote by $\mathrm{x}^\alpha\mathrm{y}^\beta$ the monomial $x_0^{\alpha_0}x_1^{\alpha_1}y_0^{\beta_0}y_1^{\beta_1}$; hence, for any polynomial $f \in \mathcal{S}_{a,b}$, we write $f = \sum_{\substack{\alpha,\beta \in \mathbb{N}^2, \\ |\alpha|=a,~|\beta|=b}} f_{\alpha,\beta}\mathrm{x}^\alpha\mathrm{y}^\beta$. For any bi-degree $(a,b)\in\mathbb{N}^2$, we consider the non-degenerate apolar pairing
$$
\circ : \mathcal{S}_{a,b} \times \mathcal{S}_{a,b} \longrightarrow \mathbb{C}, \quad\quad (f,g) \mapsto \sum_{\substack{\alpha,\beta \in \mathbb{N}^2, \\ |\alpha|=a,~|\beta|=b}} f_{\alpha,\beta}g_{\alpha,\beta}.
$$
Now, given a subspace $W \subset \mathcal{S}_{a,b}$, we denote by $W^\perp$ the perpendicular space with respect to the apolar pairing, i.e, $W^\perp = \{f \in \mathcal{S}_{a,b} ~|~ f \circ g = 0,~\forall g \in W\}$. From this definition, it is easy to prove that, given $W_1,\ldots,W_s \subset \mathcal{S}_{a,b}$, we have
\begin{equation}\label{equation: intersection perp}
\left\langle W_1,\ldots,W_s \right\rangle^\perp = W_1^\perp \cap \ldots \cap W_s^\perp.
\end{equation}
\begin{remark}\label{lemma: tangential 0-dimensional ideal}
Let $P = \tau_{a,b}\big(([\ell_1],[\ell_2]),([m_1],[m_2])\big)\in \calT(SV_{a,b})$ be a general point and let $W$ be the affine cone over the tangent space, i.e., $\mathbb{P}(W) = T_P(\calT(SV_{a,b}))$. Then, we may observe that \Alessandro{$\wp^3_{a,b} \subset W^\perp \subset \wp^2_{a,b}$}, where $\wp$ is the ideal defining the point $([\ell_1],[\ell_2]) \in \mathbb{P}^1\times\mathbb{P}^1$.
Indeed, up to change of coordinates, we may assume
$$
\ell_1 = x_0,~m_1 = x_1 \quad \text{ and } \quad \ell_2 = y_0, ~m_2 = y_1.
$$
Therefore, \Alessandro{as in the proof} of Lemma \ref{lemma: dimension tangential}, we have
\begin{align}
W = \left\langle x_0^ay_0^b,~ x_0^ay_0^{b-1}y_1,~ x_0^{a-1}x_1y_0^b,~ \right.& (a-1)x_0^{a-2}x_1^2y_0^{b} + a x_0^{a-1}x_1y_0^{b-1}y_1, \nonumber \\ & \quad \quad \left. bx_0^{a-1}x_1y_0^{b-1}y_1 + (b-1)x_0^{a}y_0^{b-2}y_1^2\right\rangle \subset \mathcal{S}_{a,b}. \label{eq: explicit tangent}
\end{align}
\Alessandro{It is easy to check that
$$[\wp^2_{a,b}]^\perp = \left\langle x_0^ay_0^b,~ x_0^ay_0^{b-1}y_1,~ x_0^{a-1}x_1y_0^b \right\rangle \subset W \quad \text{and} \quad\wp^3_{a,b} = (x_1^3,x_1^2y_1,x_1y_1^2,y_1^3)_{a,b} \subset W^\perp;$$ therefore, since the apolarity pairing is non-degenerate, we obtain that
$
\wp^{3}_{a,b} \subset W^\perp \subset \wp^{2}_{a,b}.
$}
\end{remark}
\begin{definition}
Let $P \in \mathbb{P}^1\times\mathbb{P}^1$ be a point defined by the ideal $\wp$. We call {\bf fat point} of {\bf multiplicity $m$}, or $m$-fat point, and {\bf support in $P$}, the $0$-dimensional scheme $mP$ defined by the ideal $\wp^m$.
\end{definition}
Therefore, from Remark \ref{lemma: tangential 0-dimensional ideal}, we have that the general tangent space to the tangential variety of the Segre-Veronese surface $SV_{a,b}$ is the projectivisation of a $5$-dimensional vector space whose perpendicular is the bi-homogeneous part in bi-degree $(a,b)$ of an ideal describing a $0$-dimensional scheme of \Alessandro{length} $5$ which is contained in between a $2$-fat point and a $3$-fat point. In the next lemma, we describe better the structure of the latter $0$-dimensional scheme.
\begin{lemma}\label{lemma: explicit tangent}
Let $P = \tau_{a,b}\big(([\ell_1],[\ell_2]),([m_1],[m_2])\big)\in \calT(SV_{a,b})$ be a general point and let $W$ be the affine cone over the tangent space, i.e., $\mathbb{P}(W) = T_P(\calT(SV_{a,b}))$. Then, $W^\perp = [\wp^3 + I^2]_{a,b}$, where $\wp$ is the ideal of the point $P$ and $I$ is the principal ideal $I = (\ell_1m_2+\ell_2m_1)$.
\end{lemma}
\begin{proof}
Up to change of coordinates, we may assume
$$
\ell_1 = x_0,~m_1 = x_1 \quad \text{ and } \quad \ell_2 = y_0,~ m_2 = y_1.
$$
Let $\wp = (x_1,y_1)$ and $I = (x_1y_0+x_0y_1)$. \Alessandro{Now, from \eqref{eq: explicit tangent}, it follows} that
$$
W \subset \Alessandro{[\wp^3_{a,b}]^\perp\cap [I^2_{a,b}]^\perp \subsetneq [\wp^3_{a,b}]^\perp.}
$$
The latter inequality is strict; therefore, since \Alessandro{$\dim [\wp^3_{a,b}]^\perp = 6$, we get that $\dim [\wp^3_{a,b}]^\perp\cap [I^2_{a,b}]^\perp \leq 5$} and, since $\dim W = 5$, equality follows. Hence, since the apolarity pairing is non-degenerate,
$$
W^\perp = [\wp^3+I^2]_{a,b}.
$$
\end{proof}
From these results, we obtain that the general tangent space to the tangential variety to the Segre-Veronese surface $SV_{a,b}$ can be described in terms of a connected $0$-dimensional scheme of \Alessandro{length} $5$ contained in between a \Alessandro{$2$-fat and a $3$-fat} point. Moreover, such a $0$-dimensional scheme is independent from the choice of $a,b$. Therefore, we introduce the following definition.
\begin{definition}\label{def: (3,2)-points P1xP1}
Let $\wp$ be the prime ideal defining a point $P$ in $\mathbb{P}^1\times\mathbb{P}^1$ and let $(\ell)$ the principal ideal generated by a $(1,1)$-form passing through $P$. The $0$-dimensional scheme defined by $\wp^3+(\ell)^2$ is called {\bf $(3,2)$-fat point} with support at $P$. We call $P$ the {\bf support} and $\ell$ the {\bf direction} of the scheme.
\end{definition}
Now, by using Terracini's Lemma, we get that, from Lemma \ref{lemma: explicit tangent} and \eqref{equation: intersection perp},
\begin{equation}\label{equation: dimension of secants and HF}
\dim \sigma_s(\calT(SV_{a,b})) = \dim \mathcal{S}_{a,b}/I(\mathbb{X})_{a,b}-1,
\end{equation}
where $\mathbb{X}$ is the union of $s$ many $(3,2)$-fat points with generic support.
\medskip
Let us recall the definition of {\it Hilbert function}. Since we will use it both in the standard graded and in the multi-graded case, we present this definition in a general setting.
\begin{definition}
Let $\mathcal{S} = \bigoplus_{h\in H}\mathcal{S}_h$ be a polynomial ring graded over a semigroup $H$. Let $I = \bigoplus_{h \in H} \subset \mathcal{S}$ be a $H$-homogeneous ideal, i.e., an ideal generated by homogeneous elements. For any $h\in H$, we call {\bf Hilbert function} of $\mathcal{S}/I$ in degree $h$, the dimension, as vector space, of the homogeneous part of degree $h$ of the quotient ring $\mathcal{S}/I$, i.e.,
$$
\mathrm{HF}_{\mathcal{S}/I}(h) = \dim [\mathcal{S}/I]_{h} = \mathcal{S}_{h}/I_{h}.
$$
Given a $0$-dimensional scheme $\mathbb{X}$ defined by the ideal $I(\mathbb{X}) \subset \mathcal{S}$, we say {\bf Hilbert function} of $\mathbb{X}$ when we refer to the Hilbert function of $\mathcal{S}/I(\mathbb{X})$.
\end{definition}
In the standard graded case we have $H = \mathbb{N}$, while in the bi-graded cases we have $H = \mathbb{N}^2$.
\medskip
Therefore, by \eqref{equation: dimension of secants and HF}, we reduced the problem of computing the dimension of secant varieties to the problem of computing the Hilbert function of a special $0$-dimensional scheme.
\begin{question}\label{question: 0-dim P1xP1}
{\it Let $\mathbb{X}$ be a union of $(3,2)$-fat points with generic support and generic direction.
\centerline{
For any $(a,b) \in \mathbb{N}^2$, what is the Hilbert function of $\mathbb{X}$ in bi-degree $(a,b)$?
}}
\end{question}
\smallskip
Also for this question we have an {\it expected} answer. Since $\mathbb{X}$ is a $0$-dimensional scheme in $\mathbb{P}^1\times\mathbb{P}^1$, if we represent the multi-graded Hilbert function of $\mathbb{X}$ as an infinite matrix $(\mathrm{HF}_\mathbb{X}(a,b))_{a,b\geq 0}$, then it is well-known that, in each row and column, it is strictly increasing until it reaches the degree of $\mathbb{X}$ and then it remains constant. Hence, if we let the support of $\mathbb{X}$ to be generic, we expect the Hilbert function of $\mathbb{X}$ to be the largest possible. Since a $(3,2)$-fat point has degree $5$, if $\mathbb{X}$ is a union of $(3,2)$-fat points with generic support, then
$$
{\rm exp}.\mathrm{HF}_{\mathbb{X}}(a,b) = \min\left\{ (a+1)(b+1), 5s \right\}.
$$
As we already explained in \eqref{equation: dimension of secants and HF}, this corresponds to the expected dimension \eqref{equation: expected dimension tangent} of the $s$-th secant variety of the tangential varieties to the Segre-Veronese surfaces $SV_{a,b}$.
\subsection{\Alessandro{Multiprojective-affine-projective method}}
In \cite{CGG05-SegreVeronese}, the authors defined a very powerful method to study Hilbert functions of $0$-dimensional schemes in multiprojective spaces. The method reduces those computations to the study of the Hilbert function of schemes in standard projective spaces, which might have higher dimensional connected components, depending on the dimensions of the projective spaces defining the multiprojective space. However, in the case of products of $\mathbb{P}^1$'s, we still have $0$-dimensional schemes in standard projective space, as we explain in the following.
\medskip
We consider the birational function
\begin{center}
\begin{tabular}{c c c c c c}
$\phi :$ & $ \mathbb{P}^1 \times \mathbb{P}^1$ & $\dashrightarrow$ & $\mathbb{A}^2$ & $\rightarrow$ & $\mathbb{P}^2$ \\
& $([s_0:s_1],[t_0,t_1])$ & $\mapsto$ & $(\frac{s_1}{s_0},\frac{t_1}{t_0})$ & $\mapsto$ & $[1:\frac{s_1}{s_0}:\frac{t_1}{t_0}] = [s_0t_0:s_1t_0:s_0t_1].$
\end{tabular}
\end{center}
\begin{lemma}{\rm \cite[Theorem 1.5]{CGG05-SegreVeronese}}
Let $\mathbb{X}$ be a $0$-dimensional scheme in $\mathbb{P}^1\times \mathbb{P}^1$ with generic support, i.e., assume that the function $\phi$ is well-defined over $\mathbb{X}$. Let $Q_1 = [0:1:0], Q_2 = [0:0:1] \in \mathbb{P}^2$. Then,
$$
\mathrm{HF}_{\mathbb{X}}(a,b) = \mathrm{HF}_{\mathbb{Y}}(a+b),
$$
where $\mathbb{Y} = \phi(\mathbb{X}) + aQ_1 + bQ_2$.
\end{lemma}
Therefore, in order to rephrase Question \ref{question: 0-dim P1xP1} as a question about the Hilbert function of $0$-dimensional schemes in standard projective spaces, we need to understand what is the image of a $(3,2)$-fat point of $\mathbb{P}^1\times\mathbb{P}^1$ by the map $\phi$.
\smallskip
Let $z_0,z_1,z_2$ be the coordinates of $\mathbb{P}^2$. Then, the map $\phi$ corresponds to the function of rings
\begin{center}
\begin{tabular}{c c c c}
$\Phi :$ & $\mathbb{C}[z_0,z_1,z_2]$ & $\rightarrow$ & $\mathbb{C}[x_0,x_1;y_0,y_1]$; \\
& $z_0$ & $\mapsto$ & $x_0y_0$, \\
& $z_1$ & $\mapsto$ & $x_1y_0$, \\
& $z_2$ & $\mapsto$ & $x_0y_1$. \\
\end{tabular}
\end{center}
By genericity, we may assume that the $(3,2)$-fat point $J$ has support at $P = ([1:0],[1:0])$ and it is defined by $I(J) = (x_1,y_1)^3 + (x_0y_1+x_1y_0)^2$. By construction, we have that $I(\phi(J)) = \Phi(I(J))$ and it is easy to check that
\begin{equation}\label{eq: (3,2)-points construction}
\Phi(I(J)) = (z_1,z_2)^3 + (z_1+z_2)^2.
\end{equation}
Therefore, $\phi(J)$ is a $0$-dimensional scheme obtained by the scheme theoretic intersection of a triple point and a double line passing though it. In the literature also these $0$-dimensional schemes are called {\it $(3,2)$-points}; e.g., see \cite{BCGI09}. We call {\bf direction} the line defining the scheme. This motivates our Definition \ref{def: (3,2)-points P1xP1} which is also a slight abuse of the name, but we believe that it will not rise any confusion in the reader since the ambient space will always be clear in the exposition. We consider a generalization of this definition in the following section. By using these constructions, Question \ref{question: 0-dim P1xP1} is rephrased as follows.
\begin{question}\label{question: 0-dim P2}
{\it Let $\mathbb{Y}$ be a union of $s$ many $(3,2)$-points with generic support and generic direction in $\mathbb{P}^2$.
\centerline{
For any $a,b$, let $Q_1$ and $Q_2$ be generic points and consider $\mathbb{X} = \mathbb{Y} + aQ_1 + bQ_2$.}
\centerline{
What is the Hilbert function of $\mathbb{X}$ in degree $a+b$?
}}
\end{question}
\begin{notation}
Given a $0$-dimensional scheme $\mathbb{X}$ in $\mathbb{P}^2$, we denote by $\mathcal{L}_d(\mathbb{X})$ the linear system of plane curves of degree $d$ having $\mathbb{X}$ in the base locus, i.e., the linear system of plane curves whose defining degree $d$ equation is in the ideal of $\mathbb{X}$. Similarly, if $\mathbb{X}'$ is a $0$-dimensional scheme in $\mathbb{P}^1\times\mathbb{P}^1$, we denote by $\mathcal{L}_{a,b}(\mathbb{X}')$ the linear system of curves of bi-degree $(a,b)$ on $\mathbb{P}^1\times\mathbb{P}^1$ having $\mathbb{X}'$ in the base locus. Therefore, Question \ref{question: 0-dim P1xP1} and Question \ref{question: 0-dim P2} are equivalent of asking the dimension of these types of linear systems of curves. We define the {\bf virtual dimension} as
$$
{\it vir}.\dim\mathcal{L}_d(\mathbb{X}) = {d+2 \choose 2} - \deg(\mathbb{X}) \quad \text{ and } \quad {\it vir}.\dim\mathcal{L}_{a,b}(\mathbb{X}') = (a+1)(b+1) - \deg(\mathbb{X}').
$$
Therefore, the {\bf expected dimension} is the maximum between $0$ and the virtual dimension. We say that a $0$-dimensional scheme $\mathbb{X}$ in $\mathbb{P}^2$ ($\mathbb{X}'$ in $\mathbb{P}^1\times\mathbb{P}^1$) imposes {\bf independent conditions} on $\mathcal{O}_{\mathbb{P}^2}(d)$ (on $\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(a,b)$, respectively) if the dimension of $\mathcal{L}_{d}(\mathbb{X})$ ($\mathcal{L}_{a,b}(\mathbb{X}')$, respectively) is equal to the virtual dimension.
\end{notation}
\subsection{M\'ethode d'Horace diff\'erentielle}
From now on, we focus on Question \ref{question: 0-dim P2}. We use a degeneration method, known as {\it differential Horace method}, which has been introduced by J. Alexander and A. Hirschowitz, by extending a classical idea which was already present in the work of G. Castelnuovo. They introduced this method in order to completely solve the problem of computing the Hilbert function of a union of $2$-fat points with generic support in $\mathbb{P}^n$ \cite{AHb,AHa,AH}.
\begin{definition}
In the algebra of formal functions $\mathbb{C}\llbracket z_1,z_2\rrbracket$, we say that an ideal is {\bf vertically graded} with respect to $z_2$ if it is of the form
$$
I = I_0 \oplus I_1z_2 \oplus I_2z_2^2 \oplus \ldots \oplus I_mz_2^m \oplus (z_2^{m+1}),\quad \text{where $I_i$'s are ideals in $\mathbb{C} \llbracket z_1\rrbracket$.}
$$
If $\mathbb{X}$ is a connected $0$-dimensional scheme in $\mathbb{P}^2$ and $C$ is a curve through the support $P$ of $\mathbb{X}$, we say that $\mathbb{X}$ is {\bf vertically graded} with {\bf base} $C$ if there exist a regular system of parameters $(z_1,z_2)$ at $P$ such that $z_2 = 0$ is the local equation of $C$ and the ideal of $\mathbb{X}$ in $\mathcal{O}_{\mathbb{P}^2,P} \simeq \mathbb{C} \llbracket z_1,z_2 \rrbracket$ is vertically graded.
\end{definition}
Let $\mathbb{X}$ be a vertically graded $0$-dimensional scheme in $\mathbb{P}^2$ with base $C$ and let $j\geq 1$ be a fixed integer; then, we define:
\begin{align*}
j{\bf -th ~Residue}: & \quad {\rm Res}_C^j(\mathbb{X}), \text{ the scheme in $\mathbb{P}^2$ defined by } \mathcal{I}_{\mathbb{X}} + (\mathcal{I}_\mathbb{X} : \mathcal{I}^{j}_C) \mathcal{I}_C^{j-1}. \\
j{\bf -th ~Trace}: & \quad {\rm Tr}_C^j(\mathbb{X}), \text{ the subscheme of $C$ defined by } (\mathcal{I}_\mathbb{X} : \mathcal{I}^{j-1}_C) \otimes \mathcal{O}_{C}.
\end{align*}
Roughly speaking, we have that, in the $j$-th residue, we remove the $j$-th {\it slice} of the scheme $\mathbb{X}$; while, in the $j$-th trace, we consider only the $j$-th {\it slice} as a subscheme of the curve $C$.
In the following example, we can see how we see as vertically graded schemes the $(3,2)$-fat points we have introduced before.
\begin{example}\label{example: (3,2)-point}
Up to a linear change of coordinates, we may assume that the scheme $J$ constructed in \ref{eq: (3,2)-points construction} is defined by the ideal
$
I(J) = (z_1,z_2)^3 + (z_1)^2 = (z_1^2,z_1z_2^2,z_2^3).
$
Therefore, we have that, in the local system of parameters $(z_1,z_2)$, the scheme $J$ is vertically graded with respect to the $z_2$-axis defined by $\{z_1 = 0\}$; indeed, we have the two vertical layers given by
$$
I(X) = I_0 \oplus I_1z_1 \oplus (z_1)^2, \text{ where $I_0 = (z_2^3)$ and $I_1=(z_2^2)$};
$$
at the same time, we have that it is also vertically graded with respect to the $z_1$-axis defined by $\{z_2 = 0\}$; indeed, we have the three horizontal layers given by
$$
I(J) = I_0 \oplus I_1z_2 \oplus I_2z_2^2 \oplus (z_2)^3, \text{ where $I_0 = I_1 = (z_1^2)$ and $I_2=(z_1)$}.
$$
We can visualize $J$ as in Figure \ref{figure: (3,2)-point}, where the black dots correspond to the generators of the $5$-dimensional vector space $\mathbb{C}\llbracket z_1,z_2 \rrbracket / I(J) = \left\langle 1, z_1, z_2, z_1^2, z_1z_2 \right\rangle$.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm,scale = 0.7]
\clip(-2.,-1.) rectangle (3.,3.);
\draw [line width=1.pt,dash pattern=on 1pt off 2pt] (0.,-1.) -- (0.,3.);
\draw [color = black] (-0.5,2.7) node {$z_2$};
\draw [line width=1.pt,dash pattern=on 1pt off 2pt,domain=-2.:3.] plot(\x,{(-0.-0.*\x)/-1.});
\draw [color = black] (2.5,-0.5) node {$z_1$};
\begin{scriptsize}
\draw [fill=black] (1.,0.) circle (3.5pt);
\draw [fill=black] (0.,1.) circle (3.5pt);
\draw [fill=black] (1.,1.) circle (3.5pt);
\draw [fill=black] (0.,2.) circle (3.5pt);
\draw[color=black] (0.22,8.17) node {$f$};
\draw[color=black] (-10.12,0.33) node {$g$};
\draw [fill=black] (0.,0.) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}
\caption{A representation of the $(3,2)$-point $J$, defined by the ideal $I(J) = (z_1,z_2)^3 + (z_1)^2$, as a vertically graded scheme. }
\label{figure: (3,2)-point}
\end{figure}
Therefore, if we consider $J$ as vertically graded scheme with base the $z_2$-axis, we compute the $j$-th residue and trace, for $j = 1,2$, as follows:
\begin{minipage}{0.6\textwidth}
\begin{align*}
{\rm Res}^1_{z_2}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) + (z_1^2,z_1z_2^2,z_2^3) : (z_1) = (z_1,z_2^2); & \\
{\rm Tr}^1_{z_2}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) \otimes \mathbb{C}\llbracket z_1,z_2 \rrbracket / (z_1) = (z_2^3); \\
~ \\
{\rm Res}^2_{z_2}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) + ((z_1^2,z_1z_2^2,z_2^3) : (z_1^2))\cdot (z_1) = (z_1,z_2^3); & \\
{\rm Tr}^2_{z_2}(X) : & \quad \left((z_1^2,z_1z_2^2,z_2^3) : (z_1)\right)\otimes \mathbb{C}\llbracket z_1,z_2 \rrbracket / (z_1) = (z_2^2).
\end{align*}
\end{minipage}
\begin{minipage}{0.35\textwidth}
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm,scale = 0.6]
\clip(-0.5,-0.5) rectangle (7.,5.5);
\draw [line width=1.pt,dash pattern=on 1pt off 2pt] (1.,0.5) -- (1.,5.5);
\draw [line width=1.pt,dash pattern=on 1pt off 2pt] (5.,0.5) -- (5.,5.5);
\begin{scriptsize}
\draw [fill=black] (1.,4.) circle (4.0pt);
\draw [fill=black] (1.,3.) circle (4.0pt);
\draw [fill=black] (1.,2.) circle (4.0pt);
\draw [color=black] (2.,3.) circle (3.5pt);
\draw [color=black] (2.,2.) circle (3.5pt);
\draw [fill=black] (5.,3.) circle (4.0pt);
\draw [fill=black] (5.,2.) circle (4.0pt);
\draw [color=black] (5.98,2.99) circle (3.5pt);
\draw [color=black] (5.98,1.99) circle (3.5pt);
\draw[color=black] (5.166432702358458,6.933351817483886) node {$g_1$};
\draw [color=black] (6.,4.) circle (3.5pt);
\end{scriptsize}
\node[draw] at (1.5,0.05) {$j = 1$};
\node[draw] at (5.5,0.05) {$j = 2$};
\end{tikzpicture}
\end{minipage}
\bigskip
Similarly, if we consider it as vertically graded with respect to the $z_1$-axis, we compute the $j$-th residue and trace, for $j = 1,2,3$, as follows:
\begin{minipage}{0.55\textwidth}
\begin{align*}
{\rm Res}^1_{z_1}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) + (z_1^2,z_1z_2^2,z_2^3) : (z_2) = (z_1^2,z_1z_2,z_2^2); & \\
{\rm Tr}^1_{z_1}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) \otimes \mathbb{C}\llbracket z_1,z_2 \rrbracket / (z_2) = (z_1^2); \\
~ \\
{\rm Res}^2_{z_1}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) + ((z_1^2,z_1z_2^2,z_2^3) : (z_2^2))\cdot (z_2) = (z_1^2,z_1z_2,z_2^2); & \\
{\rm Tr}^2_{z_1}(X) : & \quad \left((z_1^2,z_1z_2^2,z_2^3) : (z_2)\right)\otimes \mathbb{C}\llbracket z_1,z_2 \rrbracket / (z_2) = (z_1^2); \\
~ \\
{\rm Res}^3_{z_1}(X) : & \quad (z_1^2,z_1z_2^2,z_2^3) + ((z_1^2,z_1z_2^2,z_2^3): (z_2^3))\cdot (z_2^2) = (z_1^2,z_2^2); & \\
{\rm Tr}^3_{z_1}(X) : & \quad \left((z_1^2,z_1z_2^2,z_2^3): (z_2^2)\right)\otimes \mathbb{C}\llbracket z_1,z_2 \rrbracket / (z_2) = (z_1).
\end{align*}
\end{minipage}
\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm,scale = 0.6]
\clip(-2.,-1.5) rectangle (4.,10.);
\draw [line width=1.pt,dash pattern=on 1pt off 2pt,domain=-2.:4.] plot(\x,{(--1.0207472463132268-0.*\x)/1.});
\draw [line width=1.pt,dash pattern=on 1pt off 2pt,domain=-2.:4.] plot(\x,{(--5.964822902537174-0.*\x)/1.});
\begin{scriptsize}
\draw [color=black] (0.15798866047282512,3.020747246313228) circle (3.5pt);
\draw [color=black] (0.15798866047282512,2.020747246313227) circle (3.5pt);
\draw [fill=black] (0.15798866047282512,1.020747246313226) circle (4.0pt);
\draw [color=black] (1.1579886604728258,2.020747246313227) circle (3.5pt);
\draw[color=black] (-6.874345903049084,1.514396450654374) node {$f_1$};
\draw [color=black] (0.14984696094040872,7.964822902537176) circle (3.5pt);
\draw [color=black] (0.14984696094040872,6.964822902537174) circle (3.5pt);
\draw [fill=black] (0.14984696094040872,5.964822902537174) circle (4.0pt);
\draw [color=black] (1.1498469609404087,6.964822902537174) circle (3.5pt);
\draw [fill=black] (1.1498469609404087,5.964822902537174) circle (4.0pt);
\draw[color=black] (-6.874345903049084,5.859958317926224) node {$f_2$};
\draw [color=black] (1.1549537540078336,3.01286606005846) circle (3.5pt);
\end{scriptsize}
\node[draw] at (0.5,4.8) {$j = 1,2$};
\node[draw] at (0.4,0.1) {$j = 3$};
\end{tikzpicture}
\end{minipage}
\end{example}
\begin{notation}
Let $\mathbb{X} = X_1 + \ldots+X_s$ be a union of vertically graded schemes with respect to the same base $C$. Then, for any vector $\mathbf{j} = (j_1,\ldots,j_s) \in \mathbb{N}_{\geq 1}^s$, we denote
$$
{\rm Res}^\mathbf{j}_C(\mathbb{X}) := {\rm Res}^{\mathbf{j}_1}_C(\mathbb{X}_1) \cap \ldots \cap {\rm Res}^{\mathbf{j}_s}_C(\mathbb{X}_s), \quad \quad \text{ and } \quad \quad {\rm Tr}^\mathbf{j}_C(\mathbb{X}) := {\rm Tr}^{\mathbf{j}_1}_C(\mathbb{X}_1) \cap \ldots \cap {\rm Tr}^{\mathbf{j}_s}_C(\mathbb{X}_s).
$$
\end{notation}
We are now ready to describe the Horace differential method.
\begin{proposition}[Horace differential lemma, \mbox{\cite[Proposition 9.1]{AH}}]\label{proposition: Horace}
Let $\mathbb{X}$ be a $0$-dimensional scheme and let $L$ be a line. Let $Y_1,\ldots,Y_s,\widetilde{Y}_1,\ldots,\widetilde{Y_s}$ be $0$-dimensional connected schemes such that $Y_i \simeq \widetilde{Y_i}$, for any $i = 1,\ldots,s$; $\widetilde{Y_i}$ has support on the line $L$ and is vertically graded with base $L$; the support of $\mathbb{Y} = \bigcup_{i=1}^t Y_i$ and of $\widetilde{\mathbb{Y}} = \bigcup_{i=1}^t \widetilde{Y_i}$ are generic in the corresponding Hilbert schemes.
Let $\mathbf{j} = (j_1,\ldots,j_s)\in \mathbb{N}_{\geq 1}^s$ and $d \in \mathbb{N}$.
\begin{enumerate}
\item If:
\begin{enumerate}
\item ${\rm Tr}^1_L(\mathbb{X}) + {\rm Tr}^\mathbf{j}_L(\widetilde{\mathbb{Y}})$ imposes independent conditions on $\mathcal{O}_{\ell}(d)$;
\item ${\rm Res}^1_L(\mathbb{X}) + {\rm Res}^\mathbf{j}_L(\widetilde{\mathbb{Y}})$ imposes independent conditions on $\mathcal{O}_{\mathbb{P}^2}(d-1)$;
\end{enumerate}
then, $\mathbb{X} + \mathbb{Y}$ imposes independent conditions on $\mathcal{O}_{\mathbb{P}^2}(d)$.
\item If:
\begin{enumerate}
\item $\mathcal{L}_{d,\mathbb{P}^1}\left({\rm Tr}^1_\ell(\mathbb{X}) + {\rm Tr}^\mathbf{j}_\ell(\widetilde{\mathbb{Y}})\right)$ is empty;
\item $\mathcal{L}_{d-1,\mathbb{P}^2}\left({\rm Res}^1_\ell(\mathbb{X}) + {\rm Res}^\mathbf{j}_\ell(\widetilde{\mathbb{Y}})\right)$ is empty;
\end{enumerate}
then $\mathcal{L}_d(\mathbb{X} + \mathbb{Y})$ is empty.
\end{enumerate}
\end{proposition}
The latter result contains all our strategy. Given a $0$-dimensional scheme as in Question \ref{question: 0-dim P2} with generic support, we specialize some of the $(3,2)$-points to have support on a line in such a way the arithmetic allows us to use the conditions of Proposition \ref{proposition: Horace}. Such a specialization will be done in one of the different ways explained in Example \ref{example: (3,2)-point}. Recall that, if a specialized scheme has the expected dimension, then, by semicontinuity of the Hilbert function, also the original general scheme has the expected dimension.
In particular, the residues of $(3,2)$-points have very particular structures. For this reason, we introduce the following definitions.
\begin{definition}
We call {\bf $m$-jet} with {\bf support} at $P$ in the {\bf direction} $L$ the $0$-dimensional scheme defined by the ideal $(\ell,\ell_1^m)$ where $\ell$ is a linear form defining the line $L$ and $(\ell,\ell_1)$ defines the point $P$.
We call {\bf $(m_1,m_2)$-jet} with {\bf support} at $P$ in the {\bf directions} $L_1,L_2$ the $0$-dimensional scheme defined by the ideal $(\ell_1^{m_1},\ell_2^{m_2})$ where $\ell_i$ is a linear form defining the line $L_i$, for $i =1,2$, and $P = L_1 \cap L_2$.
\end{definition}
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(5.,5.) rectangle (7.5,7.5);
\draw [line width=1.pt] (1.,5.) -- (1.,7.5);
\draw [line width=1.pt,domain=5.:7.5] plot(\x,{(--6.-0.*\x)/1.});
\draw [line width=1.pt] (1.,5.) -- (1.,7.5);
\draw [line width=1.pt] (6.,5.) -- (6.,7.5);
\draw [line width=1.pt] (6.5,5.) -- (6.5,7.5);
\draw [line width=1.pt] (6.5,5.) -- (6.5,7.5);
\draw [line width=1.pt,domain=5.:7.5] plot(\x,{(--3.25-0.*\x)/0.5});
\begin{scriptsize}
\draw [fill=black] (1.,7.) circle (4.5pt);
\draw [fill=black] (1.,6.) circle (4.5pt);
\draw [fill=black] (6.,6.5) circle (4.5pt);
\draw [fill=black] (2.,6.) circle (4.5pt);
\draw [fill=black] (6.,6.) circle (4.5pt);
\draw [fill=black] (6.5,6.5) circle (4.5pt);
\draw [fill=black] (6.5,6.) circle (4.5pt);
\draw[color=black] (1.0995805946910735,9.785609888363922) node {$f$};
\draw[color=black] (1.0116847258085508,6.162346848873283) node {$i$};
\draw[color=black] (1.0995805946910735,9.785609888363922) node {$j$};
\draw[color=black] (6.099878913341249,9.785609888363922) node {$k$};
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\draw [fill=black] (6.,6.5) circle (4.5pt);
\draw[color=black] (1.0116847258085508,6.435800663174463) node {$g$};
\end{scriptsize}
\end{tikzpicture}
\quad\quad
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(5.,5.) rectangle (8.,7.5);
\draw [line width=1.pt] (1.,5.) -- (1.,8.);
\draw [line width=1.pt] (1.97546,5.) -- (1.97546,8.);
\draw [line width=1.pt,domain=5.:8.] plot(\x,{(--6.-0.*\x)/1.});
\draw [line width=1.pt] (1.,5.) -- (1.,8.);
\draw [line width=1.pt] (6.,5.) -- (6.,8.);
\draw [line width=1.pt] (6.5,5.) -- (6.5,8.);
\draw [line width=1.pt] (7.,5.) -- (7.,8.);
\draw [line width=1.pt] (6.5,5.) -- (6.5,8.);
\begin{scriptsize}
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\draw[color=black] (6.597955503675544,9.785609888363922) node {$h$};
\end{scriptsize}
\end{tikzpicture}
\end{center}
\caption{The representation of a $(2,2)$-jet and a $3$-jet as vertically graded schemes.}
\end{figure}
Since $(2,2)$-points will be crucial in our computations, we analyse further their structure with the following two lemmas that are represented also in Figure \ref{fig: (3,2)-point structure}.
\begin{lemma}\label{lemma: degeneration}
For $\lambda > 0$, let $L_1(\lambda)$ and $L_2(\lambda)$ two families of lines, defined by $\ell_1(\lambda)$ and $\ell_2(\lambda)$, respectively, passing through a unique point. Assume also that $m = \lim_{\lambda \rightarrow 0} \ell_i(\lambda)$, for $i = 1,2$, i.e., the families $L_i(\lambda)$ degenerate to the line $M = \{m=0\}$ when $\lambda$ runs to $0$. Fix a generic line $N = \{n = 0\}$ and consider
$$
J_1(\lambda) = (\ell_1(\lambda),n^2) \quad \text{ and } \quad J_2(\lambda) = (\ell_2(\lambda),n^2).
$$
Then, the limit for $\lambda \rightarrow 0$ of the scheme $\mathbb{X}(\lambda) = J_1(\lambda) + J_2(\lambda)$ is the $(2,2)$-jet defined by $(m^2,n^2)$.
\end{lemma}
\begin{proof}
We may assume that
$$
I(J_1(\lambda)) = (z_1+\lambda z_0,z^2_2) \quad \text{and} \quad
I(J_2(\lambda)) = (z_1-\lambda z_0,z^2_2).
$$
Hence, $M$ is the line $\{z_1 = 0\}$. Then, the limit for $\lambda \rightarrow 0$ of the scheme $\mathbb{X}(\lambda)$ is given by
$$
\lim_{\lambda \rightarrow 0} I(\mathbb{X}(\lambda)) = \lim_{\lambda \rightarrow 0} \left[(z^2_2,z_1+\lambda z_0) \cap (z^2_2,z_1-\lambda z_0)\right] = \lim_{\lambda \rightarrow 0} (z_2^2,z_1^2 - \lambda^2z_0^2) = (z_1^2,z_2^2).
$$
\end{proof}
\begin{lemma}\label{lemma: residue}
Let $J$ be a $(2,2)$-jet, defined by the ideal $(\ell_1^2,\ell_2^2)$, with support at the point $P$. Let $L_i = \{\ell_i=0\}$, for $i = 1,2$. Then:
\begin{enumerate}
\item the residue of $J$ with respect to $L_1$ ($L_2$, respectively) is a $2$-jet with support at $P$ and direction $L_1$ ($L_2$, respectively);
\item the residue of $J$ with respect to a line $L = \{\alpha \ell_1 + \beta \ell_2 = 0\}$ passing through $P$ different from $L_1$ and $L_2$ is a $2$-jet with support at $P$ and direction the line $\{\alpha \ell_1 - \beta \ell_2 = 0\}$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item If we consider the residue with respect to $\{\ell_1=0\}$, we get
$$
(\ell_1^2,\ell_2^2) : (\ell_1) = (\ell_1,\ell_2^2).
$$
Analogously, for the line $\{\ell_2 = 0\}$.
\item If we consider the residue with respect to the line $\{\alpha \ell_1 + \beta\ell_2 = 0\}$, we get
$$
(\ell_1^2,\ell_2^2) : (\alpha \ell_1 + \beta \ell_2) = (\ell_1^2, \alpha\ell_1 - \beta \ell_2).
$$
\end{enumerate}
\end{proof}
\begin{figure}[h]
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[height=1.6in]{degeneration.png}
\caption{The degeneration of Lemma \ref{lemma: degeneration}.}
\end{subfigure}%
~
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[height=1.3in]{residue.png}
\caption{The two different residues of Lemma \ref{lemma: residue}.}
\end{subfigure}
\caption{The structure of a $(2,2)$-jet.}
\label{fig: (3,2)-point structure}
\end{figure}
The construction of $(2,2)$-jets as degeneration of $2$-jets, as far as we know, is a type of degeneration that has not been used before in the literature. Similarly as regard the fact that the structure of the residues of $(2,2)$-jets depend on the direction of the lines. These two facts will be crucial for our computation and we believe that these constructions might be used to attack also other similar problems on linear systems.
\section{Lemmata}\label{sec: lemmata}
\subsection{Subabundance and superabundance}\label{sec: super- and sub-abundance}
The following result is well-known for the experts in the area and can be found
in several papers in the literature. We explicitly recall it for convenience of the reader.
\begin{lemma}\label{lemma: super- and sub-abundance}
Let $\mathbb{X}' \subset \mathbb{X} \subset \mathbb{X}''\subset\mathbb{P}^2$ be $0$-dimensional schemes. Then:
\begin{enumerate}
\item if $\mathbb{X}$ imposes independent conditions on $\mathcal{O}_{\mathbb{P}^2}(d)$, then also $\mathbb{X}'$ does;
\item if $\mathcal{L}_{d}(\mathbb{X})$ is empty, then also $\mathcal{L}_{d}(\mathbb{X}'')$ is empty.
\end{enumerate}
\end{lemma}
In Question \ref{question: 0-dim P2}, we consider, for any positive integers $a,b$ and $s$, the scheme
$$
\mathbb{X}_{a,b;s} = aQ_1 + bQ_2 + Y_1 + \ldots Y_s \subset \mathbb{P}^2,
$$
where the $Y_i$'s are general $(3,2)$-points with support at general points $\{P_1,\ldots,P_s\}$ and general directions. The previous lemma suggests that, fixed $a,b$, there are two critical values to be considered firstly, i.e.,
$$
s_1 = \left\lfloor \frac{(a+1)(b+1)}{5} \right\rfloor \quad \quad \text{ and } \quad \quad s_2 = \left\lceil \frac{(a+1)(b+1)}{5} \right\rceil;
$$
namely, $s_1$ is the largest number of $(3,2)$-points for which we expect to have \textit{subabundance}, i.e., where we expect to have positive virtual dimension, and $s_2$ is the smallest number of $(3,2)$-points where we expect to have \textit{superabundance}, i.e., where we expect that the virtual dimension is negative. If we prove that the dimension of linear system $\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s})$ is as expected for $s = s_1$ and $s = s_2$, then, by Lemma \ref{lemma: super- and sub-abundance}, we have that it holds for any $s$.
\subsection{Low bi-degrees}
Now, we answer to Question \ref{question: 0-dim P2} for $b = 1,2$. These will be the base cases of our inductive approach to solve the problem in general. Recall that $a,b$ are positive integers such that $ab > 1$; see Remark \ref{rmk: (a,b) = (1,1)}.
\begin{lemma}\label{lemma: b = 1}
Let $a > b = 1$ be a positive integer. Then,
$$
\dim\mathcal{L}_{a+1}(\mathbb{X}_{a,1;s}) = \max \{0,~ 2(a+1) - 5s\}.
$$
\end{lemma}
\begin{proof}
First of all, note that $a \geq s_2$. Indeed,
$$
a \geq \frac{2(a+1)}{5} \quad \Longleftrightarrow \quad 3a \geq 2.
$$
Now, if $s \leq s_2$, we note that every line $\overline{Q_1P_i}$ is contained in the base locus of $\mathcal{L}_{a+1}(\mathbb{X}_{a,1;s})$. Hence,
$$
\dim\mathcal{L}_{a+1}(\mathbb{X}_{a,1;s}) = \dim\mathcal{L}_{a+1-s}(\mathbb{X}'),
$$
where $\mathbb{X}' = (a-s)Q_1 + Q_2 + 2P_1 + \ldots + 2P_s$. By \cite{CGG05-SegreVeronese},
$$
\dim\mathcal{L}_{a+1}(\mathbb{X}_{a,1;s}) = \dim\mathcal{L}_{a+1-s}(\mathbb{X}') = \max\{0,~2(a+1-s) - 3s\}.
$$
\end{proof}
Now, we prove the case $b = 2$ which is the crucial base step for our inductive procedure. In order to make our construction to work smoothly, we need to consider separately the following easy case.
\begin{lemma}\label{lemma: a = 2}
Let $a = b = 2$. Then,
$$\dim\mathcal{L}_{4}(\mathbb{X}_{2,2;s}) = \max\{0, 9 - 5s\}.$$
\end{lemma}
\begin{proof}
For $s = s_1 = 1$, then it follows easily because the scheme $2Q_1 + 2Q_2 + 3P_1 \supset \mathbb{X}_{2,2;1}$ imposes independent conditions on quartics. For $s = s_2 = 2$, we specialize the directions of the $(3,2)$-points supported at the $P_i$'s to be along the lines $\overline{Q_1P_i}$, respectively. Now, the lines $\overline{Q_1P_i}$ are fixed components and we can remove them. We remain with the linear system $\mathcal{L}_2(2Q_2 + J_1 + J_2)$, where the $J_i$'s are $2$-jets contained in the lines $\overline{Q_1P_i}$, respectively. Since both lines $\overline{Q_2P_1}$ and $\overline{Q_2P_2}$ are fixed components for this linear system, we conclude that the linear system has to be empty.
\end{proof}
\begin{lemma}\label{lemma: b = 2}
Let $a > b = 2$ be a positive integer. Then,
$$
\dim\mathcal{L}_{a+2}(\mathbb{X}_{a,2;s}) = \max \{0,~ 3(a+1) - 5s\}.
$$
\end{lemma}
\begin{proof}
We split the proof in different steps. Moreover, in order to help the reader in following the constructions, we include figures showing the procedure in the case of $a = 15$.
\medskip
\noindent {\sc Step 1.} Note that since $a > 2$, then $a \geq \frac{3(a+1)}{5}$ which implies $a \geq \left\lceil \frac{3(a+1)}{5} \right\rceil = s_2$.
We specialize the $(3,2)$-points with support at the $P_i$'s to have direction along the lines $\overline{Q_1P_i}$, respectively. In this way, for any $s \leq s_2$, every line $\overline{Q_1P_i}$ is a fixed component of the linear system $\mathcal{L}_{a+2}(\mathbb{X}_{a,2;s})$ and can be removed, i.e.,
$$
\dim\mathcal{L}_{a+2}(\mathbb{X}_{a,2;s}) = \dim\mathcal{L}_{(a-s)+2}(\mathbb{X}'),
$$
where $\mathbb{X}' = a'Q_1 + 2Q_2 + J_1+\ldots+J_s$, where $a' = a-s$ and $J_i$ is a $2$-jet contained in $\overline{Q_1P_i}$, for $i = 1,\ldots,s$. Now, as suggested by Lemma \ref{lemma: super- and sub-abundance}, we consider two cases: $s = s_1$ and $s = s_2$. See Figure \ref{fig: Lemma_1}.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.46\textwidth}
\centering
\scalebox{0.55}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
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\begin{scriptsize}
\draw [fill=black] (1.3,3.1) circle (4.5pt);
\draw[color=black] (0.,3.0) node {\LARGE $15Q_1$};
\draw [fill=black] (5.153549103316712,3.1969296513445316) circle (4.5pt);
\draw[color=black] (6,3.0) node {\LARGE $2Q_2$};
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}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.46\textwidth}
\centering
\scalebox{0.55}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-5.,-3.5) rectangle (8.,4.5);
\draw [line width=2.pt,domain=-5.:8.] plot(\x,{(-10.638532277887979-3.303223402088166*\x)/-4.817007322775031});
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\begin{scriptsize}
\draw [fill=black] (1.3,3.1) circle (4.5pt);
\draw[color=black] (0.,3.) node {\LARGE $6Q_1$};
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\draw[color=black] (6,3) node {\LARGE $2Q_2$};
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\end{scriptsize}
\end{tikzpicture}
}
\caption{}
\end{subfigure}
\caption{As example, we consider the case $a = 15$ and $s = s_1 = \left\lfloor \frac{16\cdot 3}{5} \right\rfloor = 9$. Here, we represent: ({\sc A}) the first specialization and ({\sc B}) the reduction explained in Step 1.}
\label{fig: Lemma_1}
\end{figure}
\medskip
We fix $a = 5k + c$, with $ 0 \leq c \leq 4$.
\medskip
\noindent {\sc Step 2: case $s = s_1$.} Consider
$$
s_1 = \left\lfloor \frac{3(5k+c+1)}{5} \right\rfloor =
\begin{cases}
3k & \text{ for } c = 0; \\
3k+1 & \text{ for } c = 1,2; \\
3k+2 & \text{ for } c = 3; \\
3k+3 & \text{ for } c = 4.
\end{cases}
$$
Note that the expected dimension is
$$
{\it exp}.\dim\mathcal{L}_{a+2}(\mathbb{X}_{a,2;s_1}) =
\begin{cases}
3 & \text{ for } c = 0; \\
1 & \text{ for } c = 1; \\
4 & \text{ for } c = 2; \\
2 & \text{ for } c = 3; \\
0 & \text{ for } c = 4. \\
\end{cases}
$$
Let $A$ be a reduced set of points of cardinality equal to the expected dimension. Then, it is enough to show that
$$
\dim\mathcal{L}_{a'+2}(\mathbb{X}'+A) = 0.
$$
where
$$
a' = a-s_1 =
\begin{cases}
2k & \text{ for } c = 0,1; \\
2k+1 & \text{ for } c = 2,3,4.
\end{cases}
$$
Let
$$
t_1 =
\begin{cases}
k & \text{ for } c = 0,2;\\
k+1 & \text{ for } c = 1,3,4.
\end{cases}
$$
Note that $s_1 \geq 2t_1$. Now, by using Lemma \ref{lemma: degeneration}, we specialize $2t_1$ lines in such a way that, for $i = 1,\ldots,t_1$, we have:
\begin{itemize}
\item the lines $\overline{Q_1P_{2i-1}}$ and $\overline{Q_1P_{2i}}$ both degenerate to a general line $R_i$ passing through $Q_1$;
\item the point $P_{2i-1}$ and the point $P_{2i}$ both degenerate to a general point $\widetilde{P}_i$ on $R_i$.
\end{itemize}
In this way, from the degeneration of $J_{2i-1}$ and $J_{2i}$, we obtain the $(2,2)$-jet $W_i$ defined by the scheme-theoretic intersection $2R_i \cap 2\overline{P_{2i-1}P_{2i}}$, for any $i = 1,\ldots,t_1$. Note that the directions $\overline{P_{2i-1}P_{2i}}$ are generic, for any $i$; see Figure \ref{fig: Lemma_2}. Note that, these $(2,2)$-jets have general directions.
\begin{figure}[h]
\centering
\scalebox{0.67}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-5.,-4.) rectangle (8.,5.);
\draw [line width=1.pt,dotted,domain=-5.:8.,color=gray] plot(\x,{(-7.514031000906525-3.9569717098083377*\x)/-4.082209051951633});
\draw [line width=2.pt,dotted,domain=-5.:8.,color=gray] plot(\x,{(--7.129839324208257--4.216172564101999*\x)/4.06924900923695});
\draw [line width=1.pt,dotted,domain=-5.:8.,color=gray] plot(\x,{(--2.5853698366721902-5.615857177287773*\x)/-1.555000722588424});
\draw [line width=1.pt,dotted,domain=-5.:8.,color=gray] plot(\x,{(--3.1869066081769386-5.641777262717139*\x)/-1.3735601245828604});
\draw [line width=1.pt,dotted,domain=-5.:8.,color=gray] plot(\x,{(--11.358054097920721-5.693617433575872*\x)/1.2184484183537643});
\draw [line width=1.pt,dotted,domain=-5.:8.,color=gray] plot(\x,{(--11.890854945651789-5.667697348146506*\x)/1.3998890163593283});
\draw [line width=2.pt,domain=-5.:8.] plot(\x,{(--17.373726988940465-5.120315184660661*\x)/3.3860744845036743});
\draw [line width=2.pt,domain=-5.:8.] plot(\x,{(--19.832137077686145-4.265077706727545*\x)/4.5352998454763025});
\draw [line width=2.pt,domain=-5.:8.] plot(\x,{(--19.464113343222074-2.8218644627154124*\x)/5.029734012406387});
\draw [line width=2.pt,domain=-5.:8.] plot(\x,{(-12.14777503007683-6.7563409361798845*\x)/-6.751977851176356});
\draw [line width=2.pt,domain=-5.:8.] plot(\x,{(--3.388023713470546-6.587860380889005*\x)/-1.7105212351646215});
\draw [line width=2.pt,domain=-5.:8.] plot(\x,{(--14.585189910082079-7.119222132191011*\x)/1.6461298279383072});
\begin{scriptsize}
\draw [fill=black] (1.325920085429366,3.125920085429366) circle (4.5pt);
\draw[color=black] (0.,3.) node {\LARGE $6Q_1$};
\draw [fill=black] (5.153549103316712,3.1969296513445316) circle (4.5pt);
\draw[color=black] (6,3.) node {\LARGE $2Q_2$};
\draw[color = black] (-3,1) node {\large in degree $8$};
\draw [fill=black] (-0.22908063715905802,-2.489937091858407) circle (3.5pt);
\draw [fill=black] (2.5443685037831303,-2.5676973481465053) circle (3.5pt);
\draw [fill=black] (5.861219930905668,-1.1391576212981789) circle (3.5pt);
\draw [fill=black] (-2.756288966522267,-0.8310516243789717) circle (3.5pt);
\draw [fill=black] (6.355654097835753,0.30405562271395387) circle (3.5pt);
\draw [fill=black] (-2.7433289238075838,-1.0902524786726335) circle (3.5pt);
\draw [fill=black] (-0.04764003915349429,-2.515857177287773) circle (3.5pt);
\draw [fill=black] (2.7258091017886943,-2.5417772627171393) circle (3.5pt);
\draw [fill=black] (4.71199456993304,-1.9943950992312944) circle (3.5pt);
\draw [fill=black] (-2.898985250235947,-0.9693701549990218) circle (3.5pt);
\draw [fill=black] (-2.5973158622124415,-0.9389674963821508) circle (3.5pt);
\draw [fill=black] (-0.16680037975532125,-2.265013060979562) circle (3.5pt);
\draw [fill=black] (0.009619284499902792,-2.280669547358283) circle (3.5pt);
\draw [fill=black] (2.491906026984575,-2.322548462063668) circle (3.5pt);
\draw [fill=black] (2.666250534361207,-2.300643908055881) circle (3.5pt);
\draw [fill=black] (4.580296319385906,-1.7952451528388482) circle (3.5pt);
\draw [fill=black] (5.668255850940832,-0.9576907230338629) circle (3.5pt);
\draw [fill=black] (6.156743519524302,0.4156517211976407) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}
}
\caption{Since $a = 15$, we have $a = 5\cdot 3 + 0$ and $t_1 = 3$. Hence, we specialize three pairs of lines to be coincident and we construct three $(2,2)$-jets.}
\label{fig: Lemma_2}
\end{figure}
Moreover, we specialize further as follows; see Figure \ref{fig: Lemma_3}:
\begin{itemize}
\item every scheme $W_i$, with support on $R_i$, for $i = 1,\ldots,t_1$, in such a way that all $\widetilde{P}_i$'s are collinear and lie on a line $L$;
\item if $c = 0,1,2,3$, we specialize $A$ (the reduced part) to lie on $L$;
\item if $c = 4$, we specialize the double point $2Q_2$ to have support on $L$.
\end{itemize}
Note that this can be done because
$$
s_1 - t_1 =
\begin{cases}
k & \text{for } c =0,3; \\
k-1 & \text{for } c = 1; \\
k+1 & \text{for } c = 2,4;
\end{cases}
$$
and, for $c = 1$, we may assume $k \geq 1$, since $a \geq 2$.
\begin{figure}[h!]
\centering
\scalebox{0.75}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-4.,-3.5) rectangle (8.5,4.5);
\draw [line width=2.pt,domain=-4.:8.5] plot(\x,{(--6.606070669096441-2.653365106436809*\x)/0.88280517651701});
\draw [line width=2.pt,domain=-4.:8.5] plot(\x,{(--9.676427200671663-2.92575777177952*\x)/1.7123646573334501});
\draw [line width=2.pt,domain=-4.:8.5] plot(\x,{(--8.795388688536372-2.0466723518098617*\x)/1.8237980204281958});
\draw [line width=2.pt,dotted,domain=-4.:8.5, color = gray] plot(\x,{(--3.6452140416492202-6.69929374398375*\x)/-1.7971916286827576});
\draw [line width=2.pt,domain=-4.:8.5] plot(\x,{(-0.3695083120389595--0.028408168922945176*\x)/0.3571005711237858});
\draw [->,line width=2.pt] (0.3289024955258037,-1.2460623060398182) -- (-0.007560515160232151,-0.4065304409771011);
\draw [line width=2.pt,dotted,domain=-4.:8.5, color = gray] plot(\x,{(--8.607111108960233-5.488725122958664*\x)/0.2637309371017562});
\draw [->,line width=2.pt] (1.4905991442246822,-1.1370380434870988) -- (1.8372807182972242,-0.33224153224727093);
\draw [->,line width=2.pt] (-1.6047720528515872,-1.3227603153116745) -- (-1.8647832334059937,-0.5798712280133718);
\draw [line width=2.pt,dotted,domain=-4.:8.5, color = gray] plot(\x,{(-4.489159349354937-5.241095427192564*\x)/-3.673581225579258});
\begin{scriptsize}
\draw [fill=black] (1.4125904789475021,3.2373534485241113) circle (4.5pt);
\draw[color=black] (0.5,3) node {\LARGE $6Q_1$};
\draw [fill=black] (5.153549103316712,3.1969296513445316) circle (4.5pt);
\draw[color=black] (6,3) node {\LARGE $2Q_2$};
\draw[color=black] (7,0.5) node {\LARGE $A$};
\draw[color = black] (-3,1) node {\large in degree $8$};
\draw [fill=black] (0.10387284793451354,-1.0256046803923533) circle (3.5pt);
\draw [fill=black] (1.4905991442246822,-1.1370380434870988) circle (3.5pt);
\draw [fill=black] (3.1249551362809522,0.3115956767445914) circle (3.5pt);
\draw [fill=black] (-1.716205415946333,-1.0008417108157432) circle (3.5pt);
\draw [fill=black] (3.236388499375698,1.1906810967142496) circle (3.5pt);
\draw [fill=black] (-1.6047720528515872,-1.3227603153116745) circle (3.5pt);
\draw [fill=black] (0.3289024955258037,-1.2460623060398182) circle (3.5pt);
\draw [fill=black] (1.7134658704141736,-0.901789832509303) circle (3.5pt);
\draw [fill=black] (2.295395655464512,0.5839883420873023) circle (3.5pt);
\draw [fill=black] (-1.8498750949904104,-1.181907592294197) circle (3.5pt);
\draw [fill=black] (-1.4927745238666246,-1.1534994233712519) circle (3.5pt);
\draw [fill=black] (0.16974203912965335,-0.8110453305399878) circle (3.5pt);
\draw [fill=black] (0.38458241343333954,-1.0157042832708876) circle (3.5pt);
\draw [fill=black] (1.4867785796410165,-0.9227971620403612) circle (3.5pt);
\draw [fill=black] (1.696281834687383,-0.6653890232810649) circle (3.5pt);
\draw [fill=black] (2.227862500801172,0.7869664406279986) circle (3.5pt);
\draw [fill=black] (3.0173602157974577,0.4954330816223558) circle (3.5pt);
\draw [fill=black] (3.091602635533084,1.3531602863725496) circle (3.5pt);
\draw [fill=black] (6.160542692533723,-0.5446604970705151) circle (3.5pt);
\draw [fill=black] (6.913021105428445,-0.48479911461147396) circle (3.5pt);
\draw [fill=black] (7.712198191933535,-0.42122274563133616) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}
}
\caption{Since $a = 15 = 5\cdot 3 + 0$, we have that $|A| = 3$. Then, we specialize the scheme in such a way the $(2,2)$-jets have support collinear with the support of the set $A$. Recall that they have general directions.}
\label{fig: Lemma_3}
\end{figure}
By abuse of notation, we call again ${\mathbb{X}}$ the scheme obtained after such a specialization.
In this way, we have that
$$
\deg({\mathbb{X}}\cap L) \quad = \quad
\begin{cases}
2k + 3 & \text{for } c = 0,1; \\
2k + 4 & \text{for } c = 2,3,4.
\end{cases} \quad = \quad a'+3.
$$
Therefore, the line $L$ becomes a fixed component of the linear system and can be removed, see Figure \ref{fig: Lemma_4}, i.e.,
$$
\dim\mathcal{L}_{a'+2}({\mathbb{X}}) = \dim\mathcal{L}_{a'+1}(\mathbb{J}),
$$
with $$
\mathbb{J} =
\begin{cases}
a'Q_1 + 2{Q}_2 + {J}_1 + \ldots + {J}_{t_1} + J_{2t_1+1} + \ldots + J_{s_1}, & \text{ for } c = 0,1,2,3; \\
a'Q_1 + {Q}_2 + {J}_1 + \ldots + {J}_{t_1} + J_{2t_1+1} + \ldots + J_{s_1}, & \text{ for } c = 4; \\
\end{cases}
$$
where
\begin{itemize}
\item
by Lemma \ref{lemma: residue}, ${J}_i$ is a $2$-jet with support on $\widetilde{P}_i \in L$ with general direction, for all $i = 1,\ldots,t_1$;
\item
$J_i$ is a $2$-jet contained in the line $\overline{Q_1P_i}$, for $i = 2t_1+1,\ldots,s_1$.
\end{itemize}
\begin{figure}[h]
\centering
\scalebox{0.77}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-3.5,-2.5) rectangle (7.,4.5);
\draw [line width=2.pt,domain=-3.5:7.] plot(\x,{(--6.606070669096441-2.653365106436809*\x)/0.88280517651701});
\draw [line width=2.pt,domain=-3.5:7.] plot(\x,{(--9.676427200671663-2.92575777177952*\x)/1.7123646573334501});
\draw [line width=2.pt,domain=-3.5:7.] plot(\x,{(--8.795388688536372-2.0466723518098617*\x)/1.8237980204281958});
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(--3.6452140416492202-6.69929374398375*\x)/-1.7971916286827576});
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(-0.3695083120389595--0.028408168922945176*\x)/0.3571005711237858});
\draw [->,line width=2.pt] (0.2648321501824796,-1.1618010130637089) -- (-0.007560515160232151,-0.4065304409771011);
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(--8.607111108960233-5.488725122958664*\x)/0.2637309371017562});
\draw [->,line width=2.pt] (1.5772695377428179,-1.0875121043338787) -- (1.8372807182972242,-0.33224153224727093);
\draw [->,line width=2.pt] (-1.6047720528515872,-1.3227603153116745) -- (-1.8647832334059937,-0.5798712280133718);
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(-4.489159349354937-5.241095427192564*\x)/-3.673581225579258});
\begin{scriptsize}
\draw [fill=black] (1.4125904789475021,3.2373534485241113) circle (4.5pt);
\draw[color=black] (0,3) node {\LARGE $6Q_1$};
\draw [fill=black] (5.153549103316712,3.1969296513445316) circle (4.5pt);
\draw[color=black] (6,3) node {\LARGE $2Q_2$};
\draw[color = black] (-2,1) node {\large in degree $7$};
\draw [fill=black] (1.5772695377428179,-1.0875121043338787) circle (3.5pt);
\draw [fill=black] (3.1249551362809522,0.3115956767445914) circle (3.5pt);
\draw [fill=black] (-1.716205415946333,-1.0008417108157432) circle (3.5pt);
\draw [fill=black] (3.236388499375698,1.1906810967142496) circle (3.5pt);
\draw [fill=black] (-1.6047720528515872,-1.3227603153116745) circle (3.5pt);
\draw [fill=black] (0.2648321501824796,-1.1618010130637089) circle (3.5pt);
\draw [fill=black] (2.295395655464512,0.5839883420873023) circle (3.5pt);
\draw [fill=black] (0.15570386993870922,-0.8567726333969978) circle (3.5pt);
\draw [fill=black] (1.7029018983959778,-0.7564612511753777) circle (3.5pt);
\draw [fill=black] (2.227862500801172,0.7869664406279986) circle (3.5pt);
\draw [fill=black] (3.0173602157974577,0.4954330816223558) circle (3.5pt);
\draw [fill=black] (3.0916026355330835,1.3531602863725503) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}
}
\caption{We remove the line $L$.}
\label{fig: Lemma_4}
\end{figure}
Now, since we are looking at curves of degree $a'+1$, the lines $\overline{Q_1P_i}$, for $i = 2t_1+1,\ldots,s_1$, become fixed components and can be removed, see Figure \ref{fig: Lemma_5}, i.e.,
$$
\dim\mathcal{L}_{a'+1}(\mathbb{J}) = \dim\mathcal{L}_{a''+1}(\mathbb{J}'),
$$
with
$$
\mathbb{J}' =
\begin{cases}
a''Q_1 + 2{Q}_2 + {J}_1 + \ldots + {J}_{t_1}, & \text{ for } c = 0,1,2,3; \\
a''Q_1 + {Q}_2 + {J}_1 + \ldots + {J}_{t_1}, & \text{ for } c = 4; \\
\end{cases}
$$
where
$$
a'' = a'-(s_1-2t_1) =
\begin{cases}
2k - (3k-2k) = k& \text{for } c = 0; \\
2k - (3k+1-2(k+1)) = k+1& \text{for } c = 1;\\
(2k+1) - (3k+1- 2k) = k & \text{for } c = 2; \\
(2k+1) - (3k+2 - 2(k+1)) = k+1 & \text{for } c = 3;\\
(2k+1) - (3k+3 - 2(k+1)) = k & \text{for } c = 4. \\
\end{cases}
$$
\begin{figure}[h]
\centering
\scalebox{0.85}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-3.,-2.5) rectangle (6.5,4.);
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(--6.606070669096441-2.653365106436809*\x)/0.88280517651701});
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(--9.676427200671663-2.92575777177952*\x)/1.7123646573334501});
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(--8.795388688536372-2.0466723518098617*\x)/1.8237980204281958});
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(--3.6452140416492202-6.69929374398375*\x)/-1.7971916286827576});
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(-0.3695083120389595--0.028408168922945176*\x)/0.3571005711237858});
\draw [->,line width=2.pt] (0.2648321501824796,-1.1618010130637089) -- (-0.007560515160232151,-0.4065304409771011);
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(--8.607111108960233-5.488725122958664*\x)/0.2637309371017562});
\draw [->,line width=2.pt] (1.5772695377428179,-1.0875121043338787) -- (1.8372807182972242,-0.33224153224727093);
\draw [->,line width=2.pt] (-1.6047720528515872,-1.3227603153116745) -- (-1.8647832334059937,-0.5798712280133718);
\draw [line width=2.pt,dotted,domain=-3.:6.5, color = gray] plot(\x,{(-4.489159349354937-5.241095427192564*\x)/-3.673581225579258});
\begin{scriptsize}
\draw [fill=black] (1.4125904789475021,3.2373534485241113) circle (4.5pt);
\draw[color=black] (0.,3.) node {\LARGE $3Q_1$};
\draw [fill=black] (5.153549103316712,3.1969296513445316) circle (4.5pt);
\draw[color=black] (6.,3.) node {\LARGE $2Q_2$};
\draw[color=black] (-2.,1.) node {\large in degree $4$};
\draw [fill=black] (1.5772695377428179,-1.0875121043338787) circle (3.5pt);
\draw [fill=black] (-1.716205415946333,-1.0008417108157432) circle (3.5pt);
\draw [fill=black] (-1.6047720528515872,-1.3227603153116745) circle (3.5pt);
\draw [fill=black] (0.2648321501824796,-1.1618010130637089) circle (3.5pt);
\draw [fill=black] (0.15570386993870922,-0.8567726333969978) circle (3.5pt);
\draw [fill=black] (1.7029018983959778,-0.7564612511753777) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}
}
\caption{We remove all the lines passing through $Q_1$ and containing the directions of the three $(2,2)$-jets.}
\label{fig: Lemma_5}
\end{figure}
Now, if $c = 0,1,2,3$, since we are looking at curves of degree $a''+1$, we have that the line $\overline{Q_1Q_2}$ is a fixed component and can be removed. After that, specialize the point $Q_2$ to a general point lying on $L$; see Figure \ref{fig: Lemma_6}.
\begin{figure}[h]
\begin{subfigure}[b]{0.45\textwidth}
\scalebox{0.75}{\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-3.,-2.5) rectangle (7.,4.);
\draw [line width=2.pt,dotted,domain=-3.:7., color = gray] plot(\x,{(--3.6452140416492202-6.69929374398375*\x)/-1.7971916286827576});
\draw [line width=2.pt,dotted,domain=-3.:7., color = gray] plot(\x,{(-0.3695083120389595--0.028408168922945176*\x)/0.3571005711237858});
\draw [->,line width=2.pt] (0.2648321501824796,-1.1618010130637089) -- (-0.007560515160232151,-0.4065304409771011);
\draw [line width=2.pt,dotted,domain=-3.:7., color = gray] plot(\x,{(--8.607111108960233-5.488725122958664*\x)/0.2637309371017562});
\draw [->,line width=2.pt] (1.5772695377428179,-1.0875121043338787) -- (1.8372807182972242,-0.33224153224727093);
\draw [->,line width=2.pt] (-1.6047720528515872,-1.3227603153116745) -- (-1.8647832334059937,-0.5798712280133718);
\draw [line width=2.pt,dotted,domain=-3.:7., color = gray] plot(\x,{(-4.489159349354937-5.241095427192564*\x)/-3.673581225579258});
\draw [line width=2.pt,dotted,domain=-3.:7., color = gray] plot(\x,{(--12.167907574406456-0.04042379717957978*\x)/3.7409586243692097});
\begin{scriptsize}
\draw [fill=black] (1.4125904789475021,3.2373534485241113) circle (4.5pt);
\draw[color=black] (0.25,2.8) node {\LARGE $2Q_1$};
\draw [fill=black] (5.153549103316712,3.1969296513445316) circle (4.5pt);
\draw[color=black] (5.5,2.5) node {\LARGE $Q_2$};
\draw[color=black] (-2.,1.) node {\large in degree $3$};
\draw [fill=black] (1.5772695377428179,-1.0875121043338787) circle (3.5pt);
\draw [fill=black] (-1.716205415946333,-1.0008417108157432) circle (3.5pt);
\draw [fill=black] (-1.6047720528515872,-1.3227603153116745) circle (3.5pt);
\draw [fill=black] (0.2648321501824796,-1.1618010130637089) circle (3.5pt);
\draw [fill=black] (0.15570386993870922,-0.8567726333969978) circle (3.5pt);
\draw [fill=black] (1.7029018983959778,-0.7564612511753777) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}}
\caption{}
\end{subfigure}
~~~
\begin{subfigure}[b]{0.45\textwidth}
\scalebox{0.75}{
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-3.5,-3.) rectangle (7.,4.5);
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(--3.6452140416492202-6.69929374398375*\x)/-1.7971916286827576});
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(-0.3695083120389595--0.028408168922945176*\x)/0.3571005711237858});
\draw [->,line width=2.pt] (0.2648321501824796,-1.1618010130637089) -- (-0.007560515160232151,-0.4065304409771011);
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(--8.607111108960233-5.488725122958664*\x)/0.2637309371017562});
\draw [->,line width=2.pt] (1.5772695377428179,-1.0875121043338787) -- (1.8372807182972242,-0.33224153224727093);
\draw [->,line width=2.pt] (-1.6047720528515872,-1.3227603153116745) -- (-1.8647832334059937,-0.5798712280133718);
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(-4.489159349354937-5.241095427192564*\x)/-3.673581225579258});
\draw [line width=2.pt,dotted,domain=-3.5:7., color = gray] plot(\x,{(--15.681208319673534-3.9398807354301373*\x)/3.12470370179319});
\begin{scriptsize}
\draw [fill=black] (1.4125904789475021,3.2373534485241113) circle (4.5pt);
\draw[color=black] (0.5,3) node {\LARGE $2Q_1$};
\draw [fill=black] (4.537294180740692,-0.7025272869060261) circle (4.5pt);
\draw[color=black] (5.,0.) node {\LARGE $Q_2$};
\draw[color=black] (-2.,1.) node {\large in degree $3$};
\draw [fill=black] (1.5772695377428179,-1.0875121043338787) circle (3.5pt);
\draw [fill=black] (-1.716205415946333,-1.0008417108157432) circle (3.5pt);
\draw [fill=black] (-1.6047720528515872,-1.3227603153116745) circle (3.5pt);
\draw [fill=black] (0.2648321501824796,-1.1618010130637089) circle (3.5pt);
\draw [fill=black] (0.15570386993870922,-0.8567726333969978) circle (3.5pt);
\draw [fill=black] (1.7029018983959778,-0.7564612511753777) circle (3.5pt);
\end{scriptsize}
\end{tikzpicture}
}
\caption{}
\end{subfigure}
\caption{In ({\sc A}) we have removed the line $\overline{Q_1Q_2}$ and then, in ({\sc B}), we specialize the point $Q_2$ to lie on $L$.}
\label{fig: Lemma_6}
\end{figure}
Therefore, for any $c$, we reduced to computing the dimension of the linear system
$
\mathcal{L}_{\widetilde{a}+1}({\mathbb{J}}''),
$
where ${\mathbb{J}}'' = a'''Q_1 + Q_2 + {J}_1 + \ldots + {J}_{t_1} $ with
$$
{a}''' =
\begin{cases}
k-1 & \text{for } c = 0,2; \\
k & \text{for } c = 1,3,4;
\end{cases}
$$
and where
\begin{itemize}
\item $Q_1$ is a general point;
\item $Q_2$ is a general point on $L$;
\item $J_i$'s are $2$-jet points with support on $L$ and general direction.
\end{itemize}
Note that this can be done because, for $c \leq 2$, we may assume $k \geq 1$, since $a \geq 3$.
Moreover,
$$
\deg({\mathbb{J}}'' \cap L) = t_1 + 1 \quad = \quad
\begin{cases}
k+1 & \text{for } c = 0,2; \\
k+3 & \text{for } c = 1,3,4.
\end{cases} \quad = \quad {a}'''+2.
$$
Therefore, the line $L$ is a fixed component for $
\mathcal{L}_{{a}'''+1}({\mathbb{J}}'')
$ and can be removed, i.e.,
$$
\dim\mathcal{L}_{{a}'''+1}({\mathbb{J}}'') =
\dim\mathcal{L}_{{a}'''}({a}'''Q_1 + \widetilde{P}_1 + \ldots + \widetilde{P}_{t_1}),
$$
where the $\widetilde{P}_i$'s are collinear. Since $t_1 = {a}'''+1$, this linear system is empty and this concludes the proof of the case $s = s_1$.
\bigskip
{\sc Step 3: case $s = s_2$.} First of all, note that the case $c = 4$ has been already proved since in that case we have $s_1 = s_2$. Moreover, the case $c = 1$ follows easily because we have that the linear system in the case $s = s_1$ has dimension $1$.
Hence, we are left just with the cases $c = 0,2,3$. We have
$$
s_2 = \begin{cases}
3k+1 & \text{for } c = 0;\\
3k+2 & \text{for } c = 2;\\
3k+3 & \text{for } c = 3.\\
\end{cases}, \quad \text{ and } \quad
a' = a-s_2 = \begin{cases}
2k-1 & \text{for } c = 0;\\
2k & \text{for } c = 2,3.
\end{cases},
$$
Now, define
$$
t_2 =
\begin{cases}
k & \text{for } c = 0;\\
k+1 & \text{for } c = 2,3.
\end{cases}
$$
Then, we proceed similarly as before. Note that $s_2 \geq 2t_2$. Now, we specialize $2t_2$ lines in such a way that, for $i = 1,\ldots,t_2$:
\begin{itemize}
\item the lines $\overline{Q_1P_{2i-1}}$ and $\overline{Q_1P_{2i}}$ both degenerate to a general line $R_i$ passing through $Q_1$;
\item the point $P_{2i-1}$ and the point $P_{2i}$ both degenerate to a general point $\widetilde{P}_i$ on $R_i$.
\end{itemize}
In this way, from the degeneration of $J_{2i-1}$ and $J_{2i}$, we obtain the $(2,2)$-jet $W_i$ defined by the scheme-theoretic intersection $2R_i \cap 2\overline{P_{2i-1}P_{2i}}$, for any $i = 1,\ldots,t_2$. Moreover, we specialize the support of the $W_i$'s and the double point $2Q_2$ on a line $L$. Note that this can be done because $s_2 - 2t_2 \geq 0$. By abuse of notation, we call again ${\mathbb{X}}$ the scheme obtained after such a specialization.
In this way, we have that
$$
\deg({\mathbb{X}}\cap L) =
\begin{cases}
2k + 2 & \text{for } c = 0 \\
2(k+1) + 2 & \text{for } c = 2,3;
\end{cases} = a'+3.
$$
Therefore, the line $L$ is a fixed component and can be removed. Hence,
$$
\dim\mathcal{L}_{a'+2}({\mathbb{X}}) = \dim\mathcal{L}_{a'+1}(\mathbb{J}),
$$
where $\mathbb{J} = a'Q_1 + Q_2 + {J}_1 + \ldots + {J}_{t_2} + J_{2t_2+1} + \ldots + J_{s_2}$, where
\begin{itemize}
\item ${J}_i$ is a $2$-jet with support on $\widetilde{P}_i\in L$ with general direction, for all $i = 1,\ldots, t_2$;
\item ${J}_i$ is a $2$-jet contained in $\overline{Q_1P_i}$, for all $i = 2t_2+1,\ldots,s_2$.
\end{itemize}
Now, since we look at curves of degree $a'+1$, the lines $\overline{Q_1P_i}$, for $i = 2t_2+1,\ldots,s_2$, become fixed components and can be removed. Hence, we get
$$
\dim\mathcal{L}_{a'+1}(\mathbb{J}) = \dim\mathcal{L}_{a''+1}(\mathbb{J}'),
$$
where $\mathbb{J}' = a''Q_1 + Q_2 + {J}_1 + \ldots + {J}_{t_2}$ and
$$
a'' = a' - (s_2 - 2t_2) =
\begin{cases}
k-2 & \text{for } c = 0;\\
k & \text{for } c = 2; \\
k -1 & \text{for } c = 3.
\end{cases}
$$
Hence,
$$
\deg(\mathbb{J}' \cap L) = 1+t_2 =
\begin{cases}
k+1 & \text{for } c = 0;\\
k+2 & \text{for } c = 2,3;
\end{cases} > a''+1;
$$
the line $L$ is a fixed component and can be removed. Hence, we are left with the linear system $\mathcal{L}_{a''}(a''Q_1 + \widetilde{P}_1 + \ldots + \widetilde{P}_{t_2})$. Since $t_2 = a''+1$, this linear system is empty and this concludes the proof of the case $s = s_2$.
\end{proof}
\section{Main result}\label{sec: main}
We are now ready to consider our general case. First, we answer to Question \ref{question: 0-dim P2}.
\begin{remark}[\sc Strategy of the proof]\label{rmk: strategy}
Our strategy to compute the dimension of $\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s})$ goes as follows. Fix a general line $L$. Then, we consider a specialization ${\mathbb{X}} = aQ_1 + bQ_2 + {Y}_1 + \ldots + {Y}_s$ of the $0$-dimensional scheme $\mathbb{X}_{a,b;s}$, where the $Y_i$'s are $(3,2)$-points with support at $P_1,\ldots,P_s$, respectively, and such that:
\begin{itemize}
\item ${Y}_1,\ldots,{Y}_x$ have support on $L$ and $\deg({Y}_i \cap L) = 3$, for all $i = 1,\ldots,x$;
\item ${Y}_{x+1},\ldots,{Y}_{x+y}$ have support on $L$, $\deg({Y}_i \cap L) = 2$ and ${\rm Res}_L({Y}_i)$ is a $3$-jet contained in $L$, for all $i = x+1,\ldots,x+y$;
\item ${Y}_{x+y+1},\ldots,{Y}_{x+y+z}$ have support on $L$, $\deg({Y}_i \cap L) = 2$ and ${\rm Res}_L({Y}_i)$ is a $2$-fat point, for all $i = x+y+1,\ldots,x+y+z$;
\item ${Y}_{x+y+z+1},\ldots,{Y}_{s}$ are generic $(3,2)$-points.
\end{itemize}
We will show that it is possible to chose $x,y,z$ in such a way that we start a procedure that allows us to remove twice the line $L$ and the line $\overline{Q_1Q_2}$ because, step-by-step, they are fixed components for the linear systems considered. A similar idea has been used by the authors, together with E. Carlini, to study Hilbert functions of triple points in $\mathbb{P}^1\times\mathbb{P}^1$; see \cite{CCO17}.
Finally, we remain with the linear system of curves of degree $a+b-2$ passing through the $0$-dimensional scheme $(a-2)Q_1+(b-2)Q_2 + {P}_{x+y+1}+\ldots+{P}_{x+y+z} + {Y}_{x+y+z+1} + \ldots +{Y}_{s}$, where the simple points ${P}_{x+y+1},\ldots,{P}_{x+y+z+1}$ are collinear and lie on the line $L$; see Figure \ref{fig: main}, where the degree goes down from $a+b$ to $a+b-2$ and finally to $a+b-4$. Hence, combining a technical lemma to deal with the collinear points (Lemma \ref{lemma: collinear}), we conclude our proof by a two-step induction, using the results of the previous section for $b \leq 2$.
\end{remark}
\begin{figure}[h]
\xymatrix{
\includegraphics[scale=0.36]{main_1.png}
\ar@{=>}[r]<10ex> & \includegraphics[scale=0.36]{main_2.png} \ar@{=>}[d] \\
& \includegraphics[scale=0.36]{main_3.png}
}
\caption{The three main steps of our proof of Theorem \ref{thm: main P2}.}
\label{fig: main}
\end{figure}
This strategy works in general, except for a few number of cases that we need to consider separately.
\begin{lemma}\label{lemma: small cases}
In the same notation as above:
\begin{enumerate}
\item if $(a,b) = (3,3)$, then $\mathrm{HF}_{\mathbb{X}_{3,3;s}}(6) = \max\{0, 16-5s\}$;
\item if $(a,b) = (5,3)$, then $\mathrm{HF}_{\mathbb{X}_{5,3;s}}(8) = \max\{0, 24-5s\}$;
\item if $(a,b) = (4,4)$, then $\mathrm{HF}_{\mathbb{X}_{4,4;s}}(8) = \max\{0, 25-5s\}$.
\end{enumerate}
\end{lemma}
\begin{proof}
{\it (1)} If $(a,b) = (3,3)$, we have to consider $s_1 = 3$ and $s_2 = 4$. Let $\mathbb{X}_{3,3;3} = 3Q_1 + 3Q_2 + Y_1 + Y_2 + Y_3$, where $P_1,P_2,P_3$ is the support of $Y_1,Y_2,Y_3$, respectively. We specialize the scheme such that $\deg(Y_1 \cap \overline{Q_1P_1}) = 3$ and $\deg(Y_2\cap \overline{P_2P_3}) = \deg(Y_3\cap \overline{P_2P_3}) = 3$. Let $A$ be a generic point on the line $\overline{P_2P_3}$. Therefore, we have that the line $\overline{P_2P_3}$ is a fixed component for $\mathcal{L}_{6}(\mathbb{X}_{3,3;3} + A)$ and we remove it. Let $\mathbb{X}' = {\rm Res}_L(\mathbb{X}_{3,3;3}+A)$. Then:
\begin{itemize}
\item the line $\overline{Q_1Q_2}$ is a fixed component for $\mathcal{L}_5({\mathbb{X}}')$;
\item the line $\overline{Q_1P_1}$ is a fixed component for $\mathcal{L}_4({\rm Res}_{\overline{Q_1Q_2}}({\mathbb{X}'}))$;
\item the line $\overline{P_2P_3}$ is a fixed component for $\mathcal{L}_3({\rm Res}_{\overline{Q_1P_1}\cdot \overline{Q_1Q_2}}({\mathbb{X}}'))$.
\end{itemize}
Therefore, we obtain that
$\dim\mathcal{L}_6(\mathbb{X}_{3,3;3}+A) = \dim\mathcal{L}_2(Q_1 + 2Q_2 + J_1)$, where $J_1$ is a $2$-jet lying on $\overline{Q_1P_1}$. The latter linear system is empty and therefore, since the expected dimension of $\mathcal{L}_6(\mathbb{X}_{3,3;3})$ is $1$, we conclude that $\dim\mathcal{L}_6(\mathbb{X}_{3,3;3}) = 1$. As a direct consequence, we also conclude that $\dim\mathcal{L}_6(\mathbb{X}_{3,3;4}) = 0$.
\medskip
{\it (2)} If $(a,b) = (5,3)$, then $s_1 = 4$ and $s_2 = 5$. Let ${\mathbb{X}} = 5Q_1 + 3Q_2 + Y_1 + {Y}_2 + \ldots + {Y}_4$, where we specialize the support of ${Y}_2, Y_3, Y_4$ to be collinear on a line $L$ and $\deg(Y_2 \cap L) = \deg(Y_3 \cap L) = \deg(Y_4 \cap L) = 3$.
Now, $L$ is a fixed component for $\mathcal{L}_8({\mathbb{X}})$ and $\overline{Q_1Q_2}$ is a fixed component for $ \mathcal{L}_7({\rm Res}_L({\mathbb{X}}))$. Let $\mathbb{X}' = {\rm Res}_{\overline{Q_1Q_2}\cdot L}({\mathbb{X}}) = 4Q_1 + 2Q_2 + Y_1 + J_2 + J_3 + J_4$, where $J_i$'s are $2$-jets lying on $L$. Now, consider a generic point $A$ on $L$.
In this way, $L$ is a fixed component for $\mathcal{L}_6(\mathbb{X}' + A)$, $\overline{Q_1Q_2}$ is a fixed component for $\mathcal{L}_5({\rm Res}_{L}({\mathbb{X}}'+A))$ and $\overline{Q_1P_1}$ is a fixed component for $\mathcal{L}_4({\rm Res}_{\overline{Q_1Q_2}\cdot L}(\mathbb{X}'+A))$, where $P_1$ is the support of $Y_1$. Hence, we have
$$
\dim\mathcal{L}_6({\mathbb{X}}'+A) = \dim\mathcal{L}_3(2Q_1 + Q_2 + 2P_1) = 3.
$$
Since the expected dimension of $\mathcal{L}_6(\mathbb{X}')$ is $4$ and $\dim\mathcal{L}_6({\mathbb{X}}'+A) = 3$, we conclude that $\dim\mathcal{L}_{8}(\mathbb{X}_{5,3;4}) = \dim\mathcal{L}_6({\mathbb{X}}') = 4$, as expected.
In the case $s = s_2 = 5$, we consider ${\mathbb{X}} = 5Q_1 + 3Q_2 + Y_1 + Y_2 + {Y}_3 + \ldots + {Y}_5$, where the support of ${Y}_3, Y_4, Y_5$ are three collinear points ${P}_3, {P}_4$ and ${P}_5$, lying on a line $L$, and $\deg(Y_3 \cap L) = \deg(Y_4 \cap L) = \deg(Y_5 \cap L) = 3$. Then, $L$ is a fixed component for $ \mathcal{L}_8({\mathbb{X}})$ and $\overline{Q_1Q_2}$ is a fixed component for $ \mathcal{L}_7({\rm Res}_L({\mathbb{X}}))$. Now, specializing the scheme such that $\deg(Y_1 \cap \overline{Q_1P_1}) = \deg(Y_2 \cap \overline{Q_1P_2}) = 3$, the lines $\overline{Q_1P_1}$ and $\overline{Q_1P_2}$ become also fixed components of the latter linear system and can be removed.
Hence,
$$
\dim \mathcal{L}_8({\mathbb{X}}) = \dim \mathcal{L}_4(\mathbb{Y}),
$$
where $\mathbb{Y} = 2Q_1 + 2Q_2 + J_1 + J_2 + {J}_3 + \ldots + {J}_5$, where $J_1,J_2$ are $2$-jets lying on $\overline{Q_1P_1}$ and $\overline{Q_1P_2}$, respectively, and $J_3,J_4,J_5$ are $2$-jets lying on $L$. Now:
\begin{itemize}
\item $L$ is a fixed component for $\mathcal{L}_4(\mathbb{Y})$;
\item $\overline{Q_1Q_2}$ is a fixed component for $\mathcal{L}_3({\rm Res}_L(\mathbb{Y}))$;
\item $\overline{Q_1P_1}$ and $\overline{Q_1P_2}$ are fixed components for $\mathcal{L}_2({\rm Res}_{\overline{Q_1Q_2}\cdot L}(\mathbb{Y}))$.
\end{itemize}
From this, we conclude that $\dim\mathcal{L}_8(\mathbb{X}) = \dim\mathcal{L}_4(\mathbb{Y}) = \dim\mathcal{L}_0(Q_2) = 0$, as expected.
\medskip
{\it (3)} If $(a,b) = (4,4)$, we have $s_1 = s_2 = 5$. Consider ${\mathbb{X}} = 4Q_1 + 4Q_2 + Y_1+ Y_2+Y_3 + {Y}_4 + {Y}_5$, where the supports of ${Y}_4$ and ${Y}_5$ are collinear with $Q_2$ on a line $L$. In this specialization, we also assume that:
\begin{itemize}
\item $\deg({Y}_4 \cap L) = 3$;
\item $\deg({Y}_5 \cap L) = 2$ and $\deg({\rm Res}_L({Y}_5) \cap L) = 3$.
\end{itemize}
Therefore, we obtain that $L$ is a fixed component for the linear system and can be removed twice. Hence,
$$
\dim\mathcal{L}_8({\mathbb{X}}) = \dim\mathcal{L}_6({\rm Res}_{2L}({\mathbb{X}})),
$$
where ${\rm Res}_{2L}({\mathbb{X}}) = \mathbb{X}_{4,2;3}$. By Lemma \ref{lemma: b = 2}, the latter linear system is empty and we conclude.
\end{proof}
The following lemma is a well-known tool to study the Hilbert function of $0$-dimensional schemes which have some reduced component lying on a line.
\begin{lemma}\label{lemma: collinear}{\rm \cite[Lemma 2.2]{CGG05-SegreVeronese}}
Let $\mathbb{X} \subset \mathbb{P}^2$ be a $0$-dimensional scheme, and let $P_1,\ldots,P_s$ be general points on a line $L$. Then:
\begin{enumerate}
\item if $ \dim\mathcal{L}_{d}(\mathbb{X}+P_1+\ldots+P_{s-1}) > \dim\mathcal{L}_{d-1}({\rm Res}_L(\mathbb{X}))$, then
$$
\dim\mathcal{L}_d(\mathbb{X}+P_1+\ldots+P_s) = \dim\mathcal{L}_d(\mathbb{X}) - s;$$
\item
if $ \dim\mathcal{L}_{d-1}({\rm Res}_L(\mathbb{X}))=0$ and $\dim\mathcal{L}_{d}(\mathbb{X})\leq s$, then
$\dim\mathcal{L}_{d}(\mathbb{X}+P_1+\ldots+P_{s})= 0$.
\end{enumerate}
\end{lemma}
Now, we are ready to prove our first main result.
\begin{theorem}\label{thm: main P2}
Let $a,b$ be positive integers with $ab > 1$. Then, let $\mathbb{X}_{a,b;s} \subset \mathbb{P}^2$ as in the previous section. Then,
$$
\dim \mathcal{L}_{a+b}(\mathbb{X}_{a,b;s}) = \max\{0, (a+1)(b+1) - 5s\}.
$$
\end{theorem}
\begin{proof}
We assume that $a \geq b \geq 3$, since the cases $b = 1$ and $b = 2$ are treated in Lemma \ref{lemma: b = 1}, Lemma \ref{lemma: a = 2} and Lemma \ref{lemma: b = 2}. Moreover, as explained in Section \ref{sec: super- and sub-abundance}, we consider $s \leq s_2$.
First, we show how to choose the numbers $x,y,z$ described in the Remark \ref{rmk: strategy}.
Let $a+b = 5h+c$, with $0\leq c \leq 4$. Then, we fix
\begin{center}
\begin{tabular}{c || c | c | c}
c & x & y & z \\
\hline
0 & h+1 & h-1 & 0 \\
1 & h & h-2 & 3 \\
2 & h+1 & h-1 & 1 \\
3 & h & h-2 & 4 \\
4 & h+1 & h-1 & 2 \\
\end{tabular}
\end{center}
Note that in order to make these choices, we need to assume that $y \geq 0$ for any $c$. In particular, this means that the cases with $a+b$ equal to $6$ and $8$ have to be treated in a different way. Hence, the only cases we have to treat differently from the main strategy are $(a,b) = (3,3), (5,3), (4,4)$, already considered in Lemma \ref{lemma: small cases}.
Note that, for any $h,c$, we have that $x+y+z \leq s_1 = \left\lfloor \frac{(a+1)(b+1)}{5} \right\rfloor$. Indeed, it is enough to see that
$$
\frac{(a+1)(b+1)}{5} - (x+y+z) \geq 0.
$$
By direct computation, we have that
$$
(a+1)(b+1) - 5(x+y+z) = (a-1)(b-1) -
\begin{cases}
0 & \text{ for } c = 0; \\
3 & \text{ for } c = 1; \\
1 & \text{ for } c = 2; \\
4 & \text{ for } c = 3; \\
2 & \text{ for } c = 4.
\end{cases}
$$
Since $a,b \geq 3$, we conclude that $x+y+z \leq s_1$.
With these assumptions, by direct computation, we obtain that, for any $c$,
$$
\deg({\mathbb{X}} \cap L) = 3x+2y+2z = a+b+1 \quad \text{ and } \quad
\deg({\rm Res}_L({\mathbb{X}}) \cap L) = 2x+3y+2z = a+b-1.
$$
Therefore, we can do the following procedure (see Figure \ref{fig: main}):
\begin{itemize}
\item the line $L$ is a fixed component for $\mathcal{L}_{a+b}({\mathbb{X}})$ and it can be removed;
\item the line $\overline{Q_1Q_2}$ is a fixed component for $\mathcal{L}_{a+b-1}({\rm Res}_L({\mathbb{X}}))$ and it can be removed;
\item the line $L$ is a fixed component for $\mathcal{L}_{a+b-2}({\rm Res}_{\overline{Q_1Q_2}\cdot L}({\mathbb{X}}))$ and it can be removed;
\item the line $\overline{Q_1Q_2}$ is a fixed component for $\mathcal{L}_{a+b-3}({\rm Res}_{\overline{Q_1Q_2}\cdot 2L}({\mathbb{X}}))$ and it can be removed.
\end{itemize}
Denote $\mathbb{Y} = {\rm Res}_{2\overline{Q_1Q_2}\cdot 2L}({\mathbb{X}})$ which is the union of $\mathbb{X}_{a-2,b-2;s-(x+y+z)}$ and a set of $z$ general collinear points on $L$. By the previous reductions, we have that
$$
\dim\mathcal{L}_{a+b}({\mathbb{X}}) = \dim\mathcal{L}_{a+b-4}(\mathbb{Y}).
$$
Now, case by case, we prove that the latter linear system has the expected dimension. In these computations, we proceed by induction on $b$ (with base cases $b = 1$ and $b = 2$ proved in Lemma \ref{lemma: b = 1}, Lemma \ref{lemma: a = 2} and Lemma \ref{lemma: b = 2}) to deal with $\mathbb{X}_{a-2,b-2;s-(x+y+z)}$; and by using Lemma \ref{lemma: collinear} to deal with the the $z$ general collinear points that we denote by $A_1,\ldots,A_z$.
\medskip
\noindent {\sc Case $c = 0$.} In this case, since $z = 0$, we conclude just by induction on $b$; indeed
\begin{align*}
\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) & = \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s-2h}) =
\max\{0, (a-1)(b-1) - 5(s-2h) \} = \\
& = \max\{0, (a+1)(b+1) - 5s \} = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s}).
\end{align*}
\medskip
\noindent {\sc Case $c = 1$.} We first consider the case $s = s_1$. We want to use Lemma \ref{lemma: collinear}, so, since $z = 3$, we compute the following:
\begin{align*}
\dim\mathcal{L}_{a+b-4}&(\mathbb{X}_{a-2,b-2;s_1-(2h+1)} + A_1 + A_2) \\
& = \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s_1-(2h+1)}) - 2 & \hfill \text{(since 2 points are always general)}\\
& = \max\{0,(a-1)(b-1)-5(s_1-(2h+1))\}-2 & \hfill \text{(by induction)} \\
& = \max\{0, (a+1)(b+1)-5s_1+1\} = \\
& = (a+1)(b+1)-5s_1+1.
\end{align*}
\begin{align*}
\dim \mathcal{L}_{a+b-5}&({\rm Res}_L(\mathbb{Y})) = \dim\mathcal{L}_{a+b-5}(\mathbb{X}_{a-2,b-2;s_1-(2h+1)}) = \\
& = \dim\mathcal{L}_{a+b-6}(\mathbb{X}_{a-3,b-3;s_1-(2h+1)}) =
& \hfill \text{(since $\overline{Q_1Q_2}$ is a fixed component)} \\
& = \max\{0,(a-2)(b-2)-5(s_1-(2h+1))\} & \hfill \text{(by induction)} \\
& = \max\{0, ab+7-5s_1\}.
\end{align*}
Now, since
\begin{align*}
(a+1)(b+1) - 5s_1+ 1 > 0, \quad \quad \text{ and } \\
\left( (a+1)(b+1) - 5s_1 + 1\right) - (ab+7-5s_1) = a+b-5 > 0,
\end{align*}
by Lemma \ref{lemma: collinear}(1), we have that
\begin{align*}
\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) & = \max\{0, (a-1)(b-1)-5(s_1-(2h+1)) - 3\} = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s_1}).
\end{align*}
Now, consider $s = s_2$. Then, we have
\begin{align*}
\dim \mathcal{L}_{a+b-5}&({\rm Res}_L(\mathbb{Y})) = \dim\mathcal{L}_{a+b-5}(\mathbb{X}_{a-2,b-2;s_2-(2h+1)}) \\
& = \dim\mathcal{L}_{a+b-6}(\mathbb{X}_{a-3,b-3;s_2-(2h+1)}) & \hfill \text{(since $\overline{Q_1Q_2}$ is a fixed component)} \\
& = \max\{0,(a-2)(b-2)-5(s_2-(2h+1))\} & \hfill \text{(by induction)} \\
& = \max\{0, ab+7-5s_2\} = 0,
\end{align*}
where the latter equality is justified by the fact that, by definition of $s_2$ (see Section \ref{sec: super- and sub-abundance}), we have
$$
ab+7-5s_2 \leq ab + 7 - (a+1)(b+1) = 6 - (a+b) \leq 0.
$$
Moreover, by definition of $s_2$,
\begin{align*}
\dim \mathcal{L}_{a+b-4}({\rm Res}_L(\mathbb{Y})) & = \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s_2-(2h+1)}) = \\
& = \max\{0, (a-1)(b-1)-5(s_2-(2h+1))\} = & \hfill \text{(by induction)} \\
& = \max\{0, (a+1)(b+1)-5s_2+3\} \leq 3.
\end{align*}
Hence, we can apply Lemma \ref{lemma: collinear}(2) and conclude that, for $s = s_2$,
$$\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) = 0 = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s_2}).$$
\medskip
\noindent {\sc Case $c = 2$ and $c = 4$.} Since a generic simple point or two generic simple points always impose independent conditions on a linear system of curves, we conclude this case directly by induction on $b$. Indeed, for $c = 2$, we have
\begin{align*}
\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) & = \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s-(2h+1)}) - 1 = \\
& = \max\{0, (a-1)(b-1)-5(s-(2h+1)) \} - 1 = & \hfill \text{(by induction)} \\
& = \max\{0, (a+1)(b+1)-5s\} = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s}).
\end{align*}
Similarly, for $c = 4$, we have
\begin{align*}
\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) & = \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s-(2h+2)}) - 2 = \\
& = \max\{0, (a-1)(b-1)-5(s-(2h+2))\} - 2 = & \hfill \text{(by induction)} \\
& = \max\{0, (a+1)(b+1)-5s\} = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s}).
\end{align*}
\medskip
\noindent {\sc Case $c = 3$.} We first consider the case $s = s_1$. We want to use Lemma \ref{lemma: collinear}, so, since $z = 4$, we compute the following:
\begin{align*}
\dim\mathcal{L}_{a+b-4}& (\mathbb{X}_{a-2,b-2;s_1-(2h+2)} + A_1 + A_2 + A_3) \\
& \geq \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s_1-(2h+2)}) - 3 \\
& = \max\{0, (a-1)(b-1)-5(s_1-(2h+2)) \} - 3 & \hfill \text{(by induction)} \\
& = \max\{0, (a+1)(b+1)-5s_1+1\} = \\
& = (a+1)(b+1)-5s_1+1 \geq 1.
\end{align*}
\begin{align*}
\dim \mathcal{L}_{a+b-5}&({\rm Res}_L(\mathbb{Y})) = \dim\mathcal{L}_{a+b-5}(\mathbb{X}_{a-2,b-2;s_1-(2h+2)}) = \\
& = \dim\mathcal{L}_{a+b-6}(\mathbb{X}_{a-3,b-3;s_1-(2h+2)}) & \hfill \text{(since $\overline{Q_1Q_2}$ is a fixed component)} \\
& = \max\{0,(a-2)(b-2)-5(s_1-(2h+2))\} & \hfill \text{(by induction)} \\
& = \max\{0, ab+8-5s_1\}.
\end{align*}
Since
\begin{align*}
(a+1)(b+1) - 5s_1 + 1 > 0, \quad \quad \text{ and } \\
\left( (a+1)(b+1) - 5s_1 + 1\right) - (ab+8-5s_1) = a+b-6 > 0,
\end{align*}
by Lemma \ref{lemma: collinear}(1), we have that
\begin{align*}
\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) & = \max\{0, (a-1)(b-1)-5(s_1-(2h+2))\} - 4 = \\
& = (a+1)(b+1) - 5s_1 = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s_1}).
\end{align*}
Now, consider $s = s_2$. We have
\begin{align*}
\dim \mathcal{L}_{a+b-5}&({\rm Res}_L(\mathbb{Y})) = \dim\mathcal{L}_{a+b-5}(\mathbb{X}_{a-2,b-2;s_2-(2h+2)}) = \\
& = \dim\mathcal{L}_{a+b-6}(\mathbb{X}_{a-3,b-3;s_2-(2h+2)}) & \hfill \text{(since $\overline{Q_1Q_2}$ is a fixed component)} \\
& = \max\{0,(a-2)(b-2)-5(s_2-(2h+2))\} & \hfill \text{(by induction)} \\
& = \max\{0, ab+8-5s_2\} = 0,
\end{align*}
where the latter equality is justified by the fact that, by definition of $s_2$ (see Section \ref{sec: super- and sub-abundance}), we have
$$
ab+8-5s_2 \leq ab+8-(a+1)(b+1) = 7 - (a+b) \leq 0.
$$
Moreover, by definition of $s_2$,
\begin{align*}
\dim \mathcal{L}_{a+b-4}({\rm Res}_L(\mathbb{Y})) & = \dim\mathcal{L}_{a+b-4}(\mathbb{X}_{a-2,b-2;s_2-(2h+2)}) = \\
& = \max\{0, (a-1)(b-1)-5(s_2-(2h+2))\} = & \hfill \text{(by induction)} \\
& = \max\{0, (a+1)(b+1)-5s_2+4\} \leq 4.
\end{align*}
Hence, we can apply Lemma \ref{lemma: collinear}(2) and conclude that, for $s = s_2$,
$$\dim\mathcal{L}_{a+b-4}(\mathbb{Y}) = 0 = {\rm exp}.\dim\mathcal{L}_{a+b}(\mathbb{X}_{a,b;s_2}).$$
\end{proof}
By multiprojective-affine-projective method, from Theorem \ref{thm: main P2}, we answer to Question \ref{question: 0-dim P1xP1}.
\begin{corollary}\label{corollary: main P1xP1}
Let $a, b$ be integers with $ab > 1$. Let $\mathbb{X}\subset\mathbb{P}^1\times\mathbb{P}^1$ be the union of $s$ many $(3,2)$-points with generic support and generic direction. Then,
$$
\mathrm{HF}_\mathbb{X}(a,b) = \min\{(a+1)(b+1), 5s\}.
$$
\end{corollary}
In conclusion, by the relation between the Hilbert function of schemes of $(3,2)$-points in $\mathbb{P}^1 \times \mathbb{P}^1$ and the dimension of secant varieties of the tangential variety of Segre-Veronese surfaces (see Section \ref{section: apolarity}), we can prove our final result.
\begin{theorem}\label{thm: main}
Let $a,b$ be positive integers with $ab>1$. Then, the tangential variety of any Segre-Veronese surface $SV_{a,b}$ is non defective, i.e., all the secant varieties have the expected dimension.
\end{theorem}
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\section{Introduction}
Let $\mathbb{D}$ be the unit disk of the complex plane $\mathbb{C}$. The present paper is concerned with the study of a class of Hilbert spaces of analytic functions on which the backward shift operator acts as a contraction. More precisely, let $\mathcal{H}$ be a Hilbert space of analytic functions such that
\begin{enumerate}[({A.}1)] \item \label{a11} the evaluation $f \mapsto f(\lambda)$ is a bounded linear functional on $\mathcal{H}$ for each $\lambda \in \mathbb{D}$,
\item \label{a12} $\hil$ is invariant under the backward shift operator $L$ given by $$Lf(z) = \frac{f(z)-f(0)}{z},$$ and we have that $$\|Lf\|_{\mathcal{H}} \leq \|f\|_{\mathcal{H}}, \quad f \in \hil,$$
\item \label{a13} the constant function 1 is contained in $\mathcal{H}$ and has the reproducing property $$\ip{f}{1}_\hil = f(0) \quad f \in \hil.$$
\end{enumerate}
Here $\ip{\cdot}{\cdot}_\mathcal{H}$ denotes the inner product in $\mathcal{H}$. By (A.1), the space $\mathcal{H}$ comes equipped with a reproducing kernel $k_\mathcal{H}: \mathbb{D} \times \mathbb{D} \to \mathbb{C}$ which satisfies $$\ip{f}{k_\mathcal{H}(\cdot, \lambda)}_\mathcal{H} = f(\lambda), f \in \mathcal{H}.$$ The condition (A.3) is a normalization condition which ensures that $k_\mathcal{H}(z,0) = 1$ for all $z \in \mathbb{D}$. The first example that comes to mind is a weighted version of the classical Hardy space, where the norm of an element $f(z) = \sum_{k=0}^\infty f_kz^n$ is given by $$\|f\|^2_w = \sum_{k=0}^\infty w_k|f_k|^2$$ with $w = (w_k)_{k=0}^n$ a nondecreasing sequence of positive numbers $w_k$, and $w_0 = 1$. More generally, the conditions are fulfilled in any $L$-invariat Hilbert space of analytic functions in $\mathbb{D}$ with a normalized reproducing kernel on which the forward shift operator $M_z$, given by $M_zf(z) = zf(z)$, is expansive ($\|M_zf\|_\mathcal{H}\ge\|f\|_\mathcal{H}$). Moreover, any $L$-invariant subspace of such a space of analytic functions satisfies the axioms as well if it contains the constants. A more detailed list of examples will be given in the next section. It includes de Branges-Rovnyak spaces, spaces of Dirichlet type and their $L$-invariant subspaces. Despite the conditions being rather general, it turns out that they imply useful common structural properties of these function spaces. The purpose of this paper is to reveal some of those properties and discuss their applications.
The starting point of our investigation is the following formula for the reproducing kernel. The space $\mathcal{H}$ satisfies (A.1)-(A.3) if and only if the reproducing kernel $k_\mathcal{H}$ of $\mathcal{H}$ has the form \begin{equation}k_\mathcal{H}(z, \lambda) = \frac{1-\sum_{i\ge 1} \conj{b_i(\lambda)}b_i(z)}{1-\conj{\lambda}z} = \frac{1-\textbf{B}(z)\textbf{B}(\lambda)^*}{1-\conj{\lambda}z}, \quad \textbf{B}(0)=0, \label{kernelequation} \end{equation}
where $\textbf{B}$ is the analytic row contraction into $l^2$ with entries $(b_i)_{i \ge 1}$. This follows easily from the positivity of the operator $I_\mathcal{H} - LL^*$ and will be proved in \thref{kernel} below.
The representation in \eqref{kernelequation} continues to hold in the case when $\mathcal{H}$ consists of vector-valued functions, with $\textbf{B}$ an analytic operator-valued contraction. Such kernels have been considered in \cite{ballbolotnikov} where they are called de Branges-Rovnyak kernels. This representation of the reproducing kernel in terms of $\textbf{B}$ is obviously not unique. We shall denote throughout by $[\textbf{B}]$ the class of $\textbf{B}$ with respect to the equivalence $\textbf{B}_1\sim\textbf{B}_2\Leftrightarrow \textbf{B}_1(z)\textbf{B}_1^*(\lambda)=\textbf{B}_2(z)\textbf{B}_2^*(\lambda),~z,\lambda\in \mathbb{D}$, and by
$\mathcal{H}[\textbf{B}]$ the Hilbert space of analytic functions satisfying (A.1)-(A.3) for which the reproducing kernel is given by \eqref{kernelequation}.
\begin{definition} \label{defhB} The space $\mathcal{H}[\textbf{B}]$ is of \textit{finite rank} if there exists $\textbf{C} \in [\textbf{B}],~\textbf{C}= (c_1, \ldots, c_n, \ldots)$ and an integer $N\ge 1$ with $c_n=0,~n\ge N$. The \textit{rank} of $\mathcal{H}[\textbf{B}]$ is defined as the minimal number of nonzero terms which can occur in these representations.
\end{definition}
Note that if $\mathcal{H}[\textbf{B}]$ is of finite rank and $\textbf{C} \in [\textbf{B}],~\textbf{C}= (c_1, \ldots, c_n, \ldots)$ has the (nonzero) minimum number of nonzero terms, then these must be linearly independent.
The rank-zero case corresponds to the Hardy space $H^2$, while rank-one spaces are the classical de Branges-Rovnyak spaces. These will be denoted by $\mathcal{H}(b)$.
Our basic tool for the study of these spaces is a model for the contraction $L$ on $\mathcal{H}$. The intuition comes from a simple example, namely a de Branges-Rovnyak space $\mathcal{H}(b)$ where $b$ is a non-extreme point of the unit ball of $H^\infty$. Then there exists an analytic outer function $a$ such that $|b|^2 + |a|^2 = 1$ on $\mathbb{T}$, and we can consider the $M_z$-invariant subspace $U = \{(bh,ah) : h \in H^2\} \subset H^2 \oplus H^2$. It is proved in \cite{sarasonbook}, Section IV-7, that there is an isometric one-to-one correspondence $f \mapsto (f,g)$ between the elements of $\mathcal{H}(b)$ and the tuples in the orthogonal complement of $U$, and the intertwining relation $Lf \mapsto (Lf, Lg)$ holds.
One of the main ideas behind the results of this paper is that, based on the structure of the reproducing kernel, a similar construction can be carried out for any Hilbert space satisfying (A.1)-(A.3). As is to expect, in this generality the objects appearing in our model are more involved, in particular the direct sum $H^2 \oplus H^2$ needs to be replaced by the direct sum of $H^2$ with an $M_z$-invariant subspace of a vector-valued $L^2$-space. Details of this construction, which extend to the vector-valued case as well, will be given in Section 2. This model is essentially a special case of the functional model of Sz.-Nagy-Foias for a general contractive linear operator (see Chapter VI of \cite{nagyfoiasharmop}). Moreover, it has a connection to a known norm formula (see e.g. \cite{afr}, \cite{alemanrichtersimplyinvariant}, \cite{ars}) which played a key role in the investigation of invariant subspaces in various spaces of analytic functions. We explain this connection in Section \ref{formulasubsection}.
The advantage provided by this point of view is a fairly tractable formula for the norm on such spaces. Our main result in this generality is the following surprising approximation theorem. Recall that the disk algebra $\mathcal{A}$ is the algebra of analytic functions in $\mathbb{D}$ which admit continuous extensions to $\text{clos}(\mathbb{D})$.
\begin{thm} \thlabel{thm1}
Let $\mathcal{H}$ be a Hilbert space of analytic functions which satisfies (A.1)-(A.3). Then any backward shift invariant subspace of $\mathcal{H}$ contains a dense set of functions in $\mathcal{A}$.
\end{thm}
The proof of \thref{thm1} is carried out in Section \ref{densitysection} and the argument covers the finite dimensional vector-valued case as well. Our approach is based on ideas of Aleksandrov \cite{aleksandrovinv} and the authors \cite{dbrcont}, however due to the generality considered here the proof is different since it avoids the use of classical theorems of Vinogradov \cite{vinogradovthm} and Khintchin-Ostrowski \cite[Section 3.2]{havinbook}.
Several applications of \thref{thm1} are presented in Section \ref{applications1}. One of them concerns the case when $\mathcal{H}$ satisfies (A.1)-(A.3) and, in addition, it is invariant for the forward shift. We obtain a very general Beurling-type theorem for $M_z$-invariant subspaces which requires the use of \thref{thm1} in the vector-valued case.
\begin{cor} \thlabel{cor-beurling1} Let $\mathcal{H}$ be a Hilbert space of analytic functions which satisfies (A.1)-(A.3) and is invariant for the forward shift $M_z$. For a closed $M_z$-invariant subspace $\mathcal{M}$ of $\mathcal{H}$ with $\dim \mathcal{M}\ominus M_z\mathcal{M}=n<\infty$, let $\varphi_1,\ldots\varphi_n$ be an orthonormal basis in $\mathcal{M}\ominus M_z\mathcal{M}$, and denote by $\phi$ the corresponding row operator-valued function. Then \begin{equation}\mathcal{M}=\phi \mathcal{H}[\textbf{C}], \label{phihceq}\end{equation}
where $ \mathcal{H}[\textbf{C}]$ consists of $\mathbb{C}^n$-valued functions and the mapping $g \mapsto \phi g$ is an isometry from $\mathcal{H}[\textbf{C}]$ onto $\mathcal{M}$. Moreover,
\begin{equation} \{\sum_{i=0}^n\varphi_iu_i:~u_i\in \mathcal{A},~1\le i\le n\}\cap \mathcal{H}\label{cyclic}\end{equation}
is a dense subset of $\mathcal{M}$.
\end{cor}
The dimension of $\mathcal{M} \ominus M_z \ominus$ is the \textit{index} of the $M_z$-invariant subspace $\mathcal{M}$. Note that \eqref{phihceq} continues to hold when the index is infinite. In fact, this dimension can be arbitrary even in the case of weighted shifts (see \cite{esterle}). We do not know whether \eqref{cyclic} holds in the infinite index case as well. Another natural question which arises is whether one can replace in \eqref{cyclic} the disc algebra $\mathcal{A}$ by the set of polynomials. We will show that this can be done in the case that $\mathcal{H}[\textbf{B}]$ has finite rank. For the infinite rank and index case we point to \cite{richterreinedirichlet} where sufficient conditions are given under which \eqref{cyclic} holds with $\mathcal{A}$ replaced by polynomials and $n = \infty$.
\thref{thm1} can be used to investigate reverse Carleson measures on such spaces, a concept which has been studied in recent years in the context of $\mathcal{H}(b)$-spaces and model spaces (see \cite{revcarlesonross}, \cite{revcarlmodelspaces}). If $\mathcal{H} \cap \mathcal{A}$ is dense in $\mathcal{H}$, then a reverse Carleson measure for $\mathcal{H}$ is a measure $\mu$ on $\text{clos}(\mathbb{D})$ such that $\|f\|^2_\mathcal{H} \leq C \int_{\text{clos}(\mathbb{D})} |f|^2 d\mu$ for $f \in \mathcal{H} \cap \mathcal{A}$. In \thref{rctheorem} we prove that if the norm in $\mathcal{H}[\textbf{B}]$ satisfies the identity \begin{equation}
\|Lf\|^2 = \|f\|^2 - |f(0)|^2 \label{normidentity}
\end{equation} then the space cannot admit a reverse Carleson measure unless it is a backward shift invariant subspace of the Hardy space $H^2$. A class of spaces in which this identity holds has been studied in \cite{nikolskiivasyuninnotesfuncmod}, where conditions are given on $\textbf{B}$ which make the identity hold. It holds in all de Branges-Rovnyak spaces corresponding to extreme points of the unit ball of $H^\infty$, so in particular our theorem answers a question in \cite{revcarlesonross}. On the other hand if $\mathcal{H}$ is $M_z$-invariant, then reverse Carleson measures may exist and can be characterized in several ways. For example, in \thref{revcarl2} we show that in this case $\mathcal{H} = \mathcal{H}[\textbf{B}]$ admits a reverse Carleson measure if and only if $g := (1-\sum_{i \in I} |b_i|^2)^{-1} \in L^1(\mathbb{T})$, and the measure $gd\textit{m}$ on $\mathbb{T}$ is essentially the minimal reverse Carleson measure for $\mathcal{H}$. The two conditions considered here yield almost a dichotomy. More precisely, if $\mathcal{H}[\textbf{B}]$ satisfies \eqref{normidentity} then it cannot be $M_z$-invariant unless it equals $H^2$.
Another application of \thref{thm1} gives an approximation result for the orthogonal complements of $M_z$-invariant subspaces of the Bergman space $L^2_a(\mathbb{D})$. These might consist entirely of functions with bad integrability properties. For example, there are such subspaces $\mathcal{M}$ for which $\int_{\mathbb{D}} |f|^{2+\epsilon} dA = \infty$ holds for all $\epsilon > 0$ and $f \in \mathcal{M} \setminus \{0\}$ (see \thref{badorthocomplements}). Note that primitives of such Bergman space functions are not necessarily bounded in the disk. However, \thref{contderivdense} below shows that the set of functions in $\mathcal{M}$ with a primitive in $\mathcal{A}$ is dense in $\mathcal{M}$.
The second part of the paper is devoted to the special case when $\mathcal{H}[\textbf{B}]$ has finite rank, according to the definition above. Intuitively speaking, in this case the structure of the backward shift $L$ resembles more a coisometry. The simplest examples are $H^2$ (rank zero) and the classical de Branges-Rovnyak spaces $\mathcal{H}(b)$ (rank one). Examples of higher rank $\mathcal{H}[\textbf{B}]$-spaces are provided by Dirichlet-type spaces $\mathcal{D}(\mu)$ corresponding to measures $\mu$ with finite support in $\text{clos}(\mathbb{D})$ (see \cite{richtersundberglocaldirichlet}, \cite{alemanhabil}).
It is not difficult to see that the rank of an $\mathcal{H}[\textbf{B}]$-space is unstable under with respect to equivalent Hilbert space norms. In fact, in \cite{ransforddbrdirichlet} it is shown that $\mathcal{D}(\mu)$-spaces corresponding to a measure with finite support on $\mathbb{T}$ admit equivalent norms under which they become a rank one space. This leads to the fundamental question whether the (finite) rank of any $\mathcal{H}[\textbf{B}]$-space can be reduced in this way. This question is addressed in Section \ref{isomorphismsubsec}. In \thref{modulegen} we relate the rank of $\mathcal{H}[\textbf{B}]$ to the number of generators of a certain $H^\infty$-submodule in the Smirnov class. In particular it turns out (\thref{nonsimilaritytheorem}) that there exist $\mathcal{H}[\textbf{B}]$-spaces whose rank cannot be reduced by means of any equivalent norm satisfying (A.1)-(A.3).
In the case of finite rank $\mathcal{H}[\textbf{B}]$-spaces our model becomes a very powerful tool. We use it in order to establish analogues of fundamental results from the theory of $\mathcal{H}(b)$-spaces. Moreover, we improve \thref{cor-beurling1} to obtain a structure theorem for $M_z$-invariant subspaces which is new even in the rank one case. More explicitly, in \thref{shiftinvcriterion} we show that a finite rank $\mathcal{H}[\textbf{B}]$-space is $M_z$-invariant if and only if $\log(1-\|\textbf{B}\|^2_2)$ is integrable on $\mathbb{T}$. If this is the case, then we show in \thref{polydense} that the polynomials are dense in the space. In \thref{Linvsubspaces} we turn to $L$-invariant subspaces and prove the analoque of a result by Sarason in \cite{sarasondoubly}, namely that every $L$-invariant subspace of $\mathcal{H}[\textbf{B}]$ has the form $\mathcal{H}[\textbf{B}] \cap K_\theta$, where $K_\theta = H^2 \ominus \theta H^2$ is a backward shift invariant subspaces of $H^2$. Concerning the structure of $M_z$-invariant subspaces we establish the following result.
\begin{thm} \thlabel{thm4}
Let $\mathcal{H} = \mathcal{H}[\textbf{B}]$ be of finite rank and $M_z$-invariant. If $\mathcal{M}$ is a closed $M_z$-invariant subspace of $\mathcal{H}$, then $\dim \mathcal{M} \ominus M_z\mathcal{M} = 1$ and any non-zero element in $\mathcal{M} \ominus M_z\mathcal{M}$ is a cyclic vector for $M_z|\mathcal{M}$. Moreover, if $\phi \in \mathcal{M} \ominus M_z \mathcal{M}$ is of norm $1$, then there exists a space $\mathcal{H}[\textbf{C}]$ invariant under $M_z$, where $\textbf{C} = (c_1, \ldots, c_k)$ and $k \leq n$, such that \begin{equation*}\mathcal{M} = \phi \mathcal{H}[\textbf{C}] \end{equation*} and the mapping $g \mapsto \phi g$ is an isometry from $\mathcal{H}[\textbf{C}]$ onto $\mathcal{M}$.
\end{thm}
\section{Basic structure} \label{prelimsec}
\subsection{Reproducing kernel.} Throughout the paper, vectors and vector-valued functions will usually be denoted by boldface letters like $\textbf{c, f}$, while operators, matrices and operator-valued functions will usually be denoted by capitalized boldface letters like \textbf{B}, \textbf{A}. The space of bounded linear operators between two Hilbert spaces $X,Y$ will be denoted by $\mathcal{B}(Y,X)$, and we simply write $\mathcal{B}(X)$ if $X=Y$. All appearing Hilbert spaces will be assumed separable. The identity operator on a space $X$ will be denoted by $I_X$. The backward shift operation will always be denoted by $L$, regardless of the space it acts upon, and regardless of if the operand is a scalar-valued function or a vector-valued function. The same conventions will be used for the forward shift operator $M_z$. If $Y$ is a Hilbert space, then we denote by $H^2(Y)$ the Hardy space of analytic functions $\textbf{f}: \mathbb{D} \to Y$ with square-summable Taylor coefficients, and $H^\infty(Y)$ is the space of bounded analytic functions from $\mathbb{D}$ to $Y$. The concepts of \textit{inner} and \textit{outer} functions are defined as usual (see Chapter V of \cite{nagyfoiasharmop}). The space $L^2(Y)$ will denote the space of square-integrable $Y$-valued measurable functions defined on the circle $\mathbb{T}$, and we will identify $H^2(Y)$ as a closed subspace of $L^2(Y)$ in the usual manner by considering the boundary values of the analytic functions $\textbf{f} \in H^2(Y)$. In the case $Y = \mathbb{C}$ we will simply write $H^2$ and $L^2$. The orthogonal complement of $H^2(Y)$ inside $L^2(Y)$ will be denoted by $\conj{H^2_0(Y)}$. The norm in $L^2(Y)$ and its subspaces will be denoted by $\| \cdot \|_2$. The inner product of two elements $f,g$ in a Hilbert space $\mathcal{H}$ will be denoted by $\ip{f}{g}_\mathcal{H}$.
As mentioned in the introduction, we will actually work in the context of vector-valued analytic functions, which will be necessary in order to prove \thref{thm1} in full generality. Thus, let $X, \mathcal{H}$ be Hilbert spaces, where $\mathcal{H}$ consists of analytic functions $\textbf{f}: \mathbb{D} \to X$. The versions of axioms (A.1)-(A.3) in the $X$-valued context are:
\begin{enumerate}[({A}.1')] \item The evaluation $\textbf{f} \mapsto \ip{\textbf{f}(\lambda)}{x}_X$ is a bounded linear functional on $\mathcal{H}$ for each $\lambda \in \mathbb{D}$ and $x \in X$,
\item $\hil$ is invariant under the backward shift operator $L$ given by $$L\textbf{f}(z) = \frac{\textbf{f}(z)-\textbf{f}(0)}{z},$$ and we have that $$\|L\textbf{f}\|_{\mathcal{H}} \leq \|\textbf{f}\|_{\mathcal{H}}, \quad \textbf{f} \in \hil,$$
\item the constant vectors $x \in X$ are contained in $\mathcal{H}$ and have the reproducing property $$\ip{\textbf{f}}{x}_\hil = \ip{\textbf{f}(0)}{x}_X \quad \mathbf{f} \in \hil, x \in X.$$
\end{enumerate}
By (A.1') there exists an operator-valued reproducing kernel $k_\mathcal{H}: \mathbb{D} \times \mathbb{D} \to \mathcal{B}(X)$ such that for each $\lambda \in \mathbb{D}$ and $x \in X$ the identity $$\ip{\textbf{f}}{k_\mathcal{H}(\cdot, \lambda)x}_\mathcal{H} = \ip{\textbf{f}(\lambda)}{x}_X$$ holds for $\textbf{f} \in \mathcal{H}$. Axiom (A.3') implies that $k_\mathcal{H}(z,0) = I_X$ for each $z \in \mathbb{D}$.
\begin{prop} \thlabel{kernel} Let $\hil$ be a Hilbert space of analytic functions which satisfies the axioms (A.1')-(A.3'). Then there exists a Hilbert space $Y$ and an analytic function $\textbf{B}: \mathbb{D} \to \mathcal{B}(Y,X)$ such that for each $z \in \mathbb{D}$ the operator $\textbf{B}(z): Y \to X$ is a contraction, and
\begin{equation}k_\mathcal{H}(z,\lambda) = \frac{I_X-\textbf{B}(z)\textbf{B}(\lambda)^*}{1-\conj{\lambda}z}. \label{kerneleq}
\end{equation} In particular, if $X = \mathbb{C}$, then $\textbf{B}(z)$ is an analytic row contraction into $l^2$, i.e. there exist analytic functions $\{b_i\}_{i\ge 1}$ in $\mathbb{D}$ such that $$k_\mathcal{H}(\lambda,z) = \frac{1 - \sum_{i \ge 1}b_i(z) \conj{b_i(\lambda)}}{1-\conj{\lambda}z},\quad \sum_{i\ge 1}|b_i(z)|^2\le 1,\, z\in \mathbb{D}. $$ If $\dim (I_\mathcal{H} - LL^*)\mathcal{H} = n < \infty$, then there exists a representation as above, with $b_i=0,~i> n$, and such that the functions $b_1, \ldots, b_n$ are linearly independent.
\end{prop}
\begin{proof}
Let $k = k_\mathcal{H}$ be the reproducing kernel of $\mathcal{H}$. Fix a vector $x \in X$ and $\lambda \in \mathbb{D}$. By (A.3') we have \begin{gather*}
\ip{\textbf{f}}{(k(\cdot,\lambda)-I_X)x/\conj{\lambda}}_\mathcal{H} = \ip{(\textbf{f}(\lambda)-\textbf{f}(0))/\lambda}{x}_X = \ip{L\textbf{f}}{k(\cdot,\lambda)x}_\mathcal{H} \\ = \ip{\textbf{f}}{L^*k(\cdot, \lambda)x}_\mathcal{H}.
\end{gather*} It follows that $$L^*k(z, \lambda)x = \frac{(k(z,\lambda) - I_X)x}{\conj{\lambda}}$$ and \begin{equation}(I_\mathcal{H} - LL^*)k(z,\lambda)x = k(z, \lambda)x - \frac{(k(z,\lambda) - I_X)x}{\conj{\lambda}z}. \label{eq32} \end{equation} By (A.2') the operator $P := I_\mathcal{H} - LL^*$ is positive. Therefore $Pk(\cdot, \lambda)$ is a positive-definite kernel and hence it has a factorization $Pk(z,\lambda) = \tilde{\textbf{B}}(z)\tilde{\textbf{B}}(\lambda)^*$ for some operator-valued analytic function $\tilde{\textbf{B}}: \mathbb{D} \to \mathcal{B}(Y,X)$ (see Chapter 2 of \cite{mccarthypick}). We can now solve for $k$ in \eqref{eq32} to obtain $$k(z,\lambda) = \frac{I_X - \textbf{B}(z)\textbf{B}(\lambda)^*}{1-\conj{\lambda}z},$$ where we have set $\textbf{B}(z) := z\tilde{\textbf{B}}(z)$. It is clear from this expression and the positive-definiteness of $k$ that $\textbf{B}(z)$ must be a contraction for every $z \in \mathbb{D}$. If $\dim(I_\mathcal{H} - LL^*) < \infty$, then the last assertion of the proposition follows in a standard manner from \eqref{eq32} and the spectral theorem applied to the finite rank operator $I_\mathcal{H} - LL^*$.
\end{proof}
\subsection{Model.} The space $\mathcal{H}$ is completely determined by the function $\textbf{B}: \mathbb{D} \to \mathcal{B}(Y,X)$ appearing in \thref{kernel}. To emphasize this we will on occasion write $\mathcal{H}[\textbf{B}]$ in place of $\mathcal{H}$. The function $\textbf{B}$ admits a non-tangential boundary value $\textbf{B}(\zeta)$ for almost every $\zeta \in \mathbb{T}$ (convergence in the sense of strong operator topology), and the operator
\begin{equation}\Delta(\zeta) = (I_{\mathcal{H}} - \textbf{B}(\zeta)^*\textbf{B}(\zeta))^{1/2} \label{delta}\end{equation} induces in a natural way a multiplication operator from $H^2(Y)$ to $L^2(Y)$. The space $\text{clos}({\Delta H^2(Y)})$ is a subspace of $L^2(Y)$ which is invariant under the operator $M_\zeta$ given by $M_\zeta\textbf{g}(\zeta) = \zeta\textbf{g}(\zeta), \zeta \in \mathbb{T}.$ This implies that it can be decomposed as \begin{equation}
\text{clos}({\Delta H^2(Y)}) = W \oplus \Theta H^2(Y_1) \label{decomp}
\end{equation} where $M_\zeta$ acts unitarily on $W$, $Y_1$ is an auxilliary Hilbert space, $\Theta: \mathbb{T} \to \mathcal{B}(Y_1, Y)$ is a measurable operator-valued function such that for almost every $\zeta \in \mathbb{T}$ the operator $\Theta(\zeta): Y_1 \to Y$ is isometric, and the functions in $W$ and $\Theta H^2(Y_1)$ are pointwise orthogonal almost everywhere, i.e, $\ip{\textbf{f}(\zeta)}{ \textbf{g}(\zeta)}_Y = 0$ for almost every $\zeta \in \mathbb{T}$ for any pair $\textbf{f} \in W$ and $\textbf{g} \in \Theta H^2(Y_1)$ (see Theorem 9 of Lecture VI in \cite{helsonbook}). It follows that $\Theta(\zeta)^* \textbf{f}(\zeta) = 0$ for all $\textbf{f} \in W$, and consequently $$\Theta^*\text{clos}({\Delta H^2(Y)}) = \Theta^*W \oplus \Theta^* \Theta H^2(Y_1) = \{0\} \oplus H^2(Y_1).$$ The above computation shows that $\Theta^*\Delta$ maps $H^2(Y)$ to a dense subset of $H^2(Y_1)$, and so standard theory of operator-valued functions implies that there exists an analytic outer function $\textbf{A}: \mathbb{D} \to \mathcal{B}(Y, Y_1)$ such that $\textbf{A}(\zeta) = \Theta^*(\zeta)\Delta(\zeta)$ for almost every $\zeta \in \mathbb{T}$.
\begin{thm} \thlabel{modeltheorem}
There exists an isometric embedding $J: \mathcal{H}[\textbf{B}] \to H^2(X) \oplus \text{clos}({\Delta H^2(Y)})$ satisfying the following properties.
\begin{enumerate}[(i)]
\item A function $\textbf{f} \in H^2(X)$ is contained in $\mathcal{H}[\textbf{B}]$ if and only if there exists $\textbf{g} \in \text{clos}({\Delta H^2(Y)})$ such that \[\textbf{B}^*\textbf{f} + \Delta \textbf{g} \in \conj{H^2_0(Y)}.\] If this is the case, then $\textbf{g}$ is unique and \[J\textbf{f} = (\textbf{f}, \textbf{g}).\]
\item If $J\textbf{f} = (\textbf{f}, \textbf{g})$ and $\textbf{g} = \textbf{w} + \Theta \textbf{f}_1$ is the decomposition of $\textbf{g}$ with respect to \eqref{decomp}, then $$JL\textbf{f} = (L\textbf{f}, \, \conj{\zeta}\textbf{w} + \Theta L\textbf{f}_1).$$
\item The orthogonal complement of $J\mathcal{H}[\textbf{B}]$ inside $H^2(X) \oplus \text{clos}({\Delta H^2(Y)})$ is \[ (J\mathcal{H}[\textbf{B}])^\perp = \{(\textbf{B}\mathbf{h}, \Delta \mathbf{h}) : \mathbf{h} \in H^2(Y)\}.\]
\end{enumerate}
\end{thm}
\begin{proof} Let $K = H^2(X) \oplus \text{clos}({\Delta H^2(Y)}), U = \{(\textbf{B}\mathbf{h}, \Delta \mathbf{h}) : \mathbf{h} \in H^2(Y)\}$ and note that $U$ is a closed subspace of $K$. Let $P$ be the projection taking a tuple $(\textbf{f},\textbf{g}) \in K$ to $\textbf{f} \in H^2(X)$. Then $P$ is injective on $K \ominus U$. Indeed, if $(0,\textbf{g})$ is orthogonal to $U$, then $\textbf{g} \in \text{clos}(\Delta H^2(Y))$ is orthogonal to $\Delta H^2(Y)$, and hence $\textbf{g} = 0$. Any analytic function $\textbf{f}$ can thus appear in at most one tuple $(\textbf{f},\textbf{g}) \in K \ominus U$. We define $\mathcal{H}_0 = P(K \ominus U)$ as the space of analytic $X$-valued functions with the norm $$\|\textbf{f}\|_{\mathcal{H}_0}^2 := \|\textbf{f}\|_2^2 + \|\textbf{g}\|^2_2.$$ We will see that $\mathcal{H}[\textbf{B}] = \mathcal{H}_0$ by showing that the reproducing kernels of the two spaces are equal. Note that for each $c \in X$ the tuple
\begin{equation*}
k_{c,\lambda} = \Bigg( \frac{(I_{X} - \textbf{B}(z)\textbf{B}(\lambda)^*)c}{1-\conj{\lambda}z}, \frac{-\Delta(\zeta)\textbf{B}(\lambda)^*c}{1-\conj{\lambda}z}\Bigg) \in K
\end{equation*} is orthogonal to $U$, and therefore its first component $Pk_{c,\lambda}$ defines an element of $\mathcal{H}_0$. Moreover, it follows readily from our definitions that for any $\textbf{f} \in \hil_0$ we have \begin{eqnarray*}
\ip{\textbf{f}}{Pk_{c,\lambda}}_{\mathcal{H}_0} = \ip{\textbf{f}(\lambda)}{c}_{X}.
\end{eqnarray*} Thus the reproducing kernel of $\mathcal{H}_0$ equals the one given by \eqref{kerneleq}, and so $\mathcal{H}[\textbf{B}] = \mathcal{H}_0$.
It is clear from the above paragraph that a function $\textbf{f} \in H^2(X)$ is contained in $\mathcal{H}[\textbf{B}]$ if and only if there exists $\textbf{g} \in \text{clos}({\Delta H^2(Y)})$ such that $(\textbf{f}, \textbf{g})$ is orthogonal to $(\textbf{B}\textbf{h}, \Delta \textbf{h})$ for all $\textbf{h} \in H^2(Y)$, that is to say, if and only if $\textbf{B}(\zeta)^*\textbf{f}(\zeta) + \Delta(\zeta) \textbf{g}(\zeta) \in \conj{H^2_0(Y)}$. If we let $J = P^{-1}$, then $J\textbf{f} = (\textbf{f}, \textbf{g})$, and part (i) follows. Part (iii) holds by construction. In order to prove (ii), it will be sufficient by (i) to show that \begin{equation}
\textbf{B}^*L\textbf{f} + \Delta(\conj{\zeta}\textbf{w} + \Theta L \textbf{f}_1) \in \conj{H^2_0(Y)}. \label{eq39} \end{equation}
Let $\textbf{A} = \Theta^*\Delta$ be the analytic function mentioned above. We have that \begin{equation}\conj{\zeta}(\textbf{B}^*\textbf{f} + \Delta \textbf{g}) = \conj{\zeta} \textbf{B}^*\textbf{f} + \Delta \conj{\zeta} \textbf{w} + \conj{\zeta} \textbf{A}^*\textbf{f}_1 \in \conj{H^2_0(Y)}.\label{eq19} \end{equation} The term $\textbf{A}^* L \textbf{f}_1$ differs from $\conj{\zeta}\textbf{A}^*\textbf{f}_1$ only by a function in $\conj{H^2_0(Y)}$, and the same is true for $\textbf{B}^*L\textbf{f}$ and $\conj{\zeta}\textbf{B}^*\textbf{f}$. Thus \eqref{eq39} follows from \eqref{eq19}.
\end{proof}
\subsection{Analytic model.} The model of \thref{modeltheorem} can be greatly simplifed if $W = \{0\}$ in the decomposition \eqref{decomp}. The condition for when this occurs can be expressed in terms of $L$.
\begin{cor} \thlabel{nowiener} We have $W = \{0\}$ in \eqref{decomp} if and only if $\|L^n\textbf{f}\|_{\mathcal{H}[\textbf{B}]} \to 0$ as $n \rightarrow \infty$, for all $f \in \mathcal{H}[\textbf{B}]$.
\end{cor}
\begin{proof}
Assume first that $W = \{0\}$. If $J\textbf{f} = (\textbf{f}, \Theta \textbf{h})$, then by (ii) of \thref{modeltheorem} we have that $$\|L^n \textbf{f}\|^2_{\mathcal{H}[\textbf{B}]} = \|L^n\textbf{f}\|^2_2 + \|L^n \textbf{h}\|^2_2$$ which clearly tends to 0 as $n \to \infty$.
Conversely, note that the convergence of $L^n\textbf{f}$ to zero implies that if $J\textbf{f} = (\textbf{f}, \textbf{g})$ and $\textbf{g} = \textbf{w} + \Theta \textbf{h}$ is the decomposition of $\textbf{g}$ with respect to \eqref{decomp}, then $\textbf{w} = 0$ (else $\lim_{n \to \infty} \|L^n\textbf{f}\|_{\mathcal{H}[\textbf{B}]} \geq \|\textbf{w}\|_2$ by (ii) of \thref{modeltheorem}). Thus for any $\textbf{w} \in W$ we have $J\textbf{f} \perp \textbf{w}$ for all $\textbf{f} \in \mathcal{H}[\textbf{B}]$, and so $(0,\textbf{w}) = (\textbf{B}\textbf{h}, \Delta \textbf{h})$ for some $\textbf{h} \in H^2(Y)$ by \thref{modeltheorem}. Since $\textbf{B}\textbf{h} = 0$, we deduce that $\textbf{w} = \Delta \textbf{h} = \textbf{h}$, for example by approximating the function $x \mapsto \sqrt{1-x}$ uniformly on $[0,1]$ by a sequence of polynomials $p_n$ with $p_n(0) = 1$, so that $\Delta \textbf{h} = \lim_{n \to \infty} p_n(\textbf{B}^*\textbf{B})\textbf{h} = \textbf{h}$. It follows that $\textbf{w} = \textbf{h} \in H^2(Y)$. Since $\textbf{w}$ was arbitrary, we deduce that $W \subset H^2(Y)$, and since $M_\zeta$ acts unitarily on $W$, we must have $W = \{0\}$.
\end{proof}
Assume we are in the case described by \thref{nowiener}. Then \eqref{decomp} reduces to $$\text{clos}({\Delta H^2(Y)}) = \Theta H^2(Y_1).$$ Thus it holds that $\conj{\Im \Delta (\zeta)} = \Im \Theta(\zeta)$ for almost every $\zeta \in \mathbb{T}$. Since $\Theta(\zeta) \Theta(\zeta)^*$ is equal almost everywhere to the projection of $Y$ onto $\Im \Theta(\zeta)$, we obtain $$\textbf{A}^*(\zeta)\textbf{A}(\zeta) = \Delta(\zeta)\Theta(\zeta)\Theta(\zeta)^*\Delta(\zeta) = \Delta(\zeta)^2$$ and consequently $\textbf{B}(\zeta)^*\textbf{B}(\zeta) + \textbf{A}(\zeta)^*\textbf{A}(\zeta) = 1_Y$ for almost every $\zeta \in \mathbb{T}$. The following theorem is a reformulation of \thref{modeltheorem}. We omit the proof, which can be easily deduced from the proof of \thref{modeltheorem} with $\textbf{A}$ playing the role of $\Delta$.
\begin{thm} \thlabel{modeltheoremanalytic} Let $\mathcal{H}[\textbf{B}]$ be such that $\|L^n\textbf{f}\|_{\mathcal{H}[\textbf{B}]} \to 0$ as $n \rightarrow \infty$, for all $f \in \mathcal{H}[\textbf{B}]$. There exists an auxilliary Hilbert space $Y_1$, an outer function $\textbf{A}: \mathbb{D} \to \mathcal{B}(Y, Y_1)$ such that $$\textbf{B}(\zeta)^*\textbf{B}(\zeta) + \textbf{A}(\zeta)^*\textbf{A}(\zeta) = 1_Y$$ for almost every $\zeta \in \mathbb{T}$, and an isometric embedding $J:\mathcal{H}[\textbf{B}] \to H^2(X) \oplus H^2(Y_1)$ satisfying the following properties.
\begin{enumerate}[(i)]
\item A function $\textbf{f} \in H^2(X)$ is contained in $\mathcal{H}[\textbf{B}]$ if and only if there exists $\textbf{f}_1 \in H^2(Y_1)$ such that $$\textbf{B}^*\textbf{f} + \textbf{A}^* \textbf{f}_1 \in \conj{H^2_0(Y)}.$$ If this is the case, then $\textbf{f}_1$ is unique, and \[J\textbf{f} = (\textbf{f}, \textbf{f}_1).\]
\item If $J\textbf{f} = (\textbf{f}, \textbf{f}_1)$, then\[J\textbf{f} = (L\textbf{f}, L \textbf{f}_1).\]
\item The orthogonal complement of $J\mathcal{H}[\textbf{B}]$ inside $H^2(X) \oplus H^2(Y_1)$ is \[(\mathcal{H}[\textbf{B}])^\perp = \{(\textbf{B}\mathbf{h}, \textbf{A}\mathbf{h}) : \mathbf{h} \in H^2(Y)\}.\]
\end{enumerate}
\end{thm}
\subsection{A formula for the norm.} \label{formulasubsection} As pointed out in the introduction, the model of \thref{modeltheorem} has a connection to a useful formula for the norm in Hilbert spaces of analytic functions.
\begin{prop} \thlabel{formula}
Let $\mathcal{H}$ be a Hilbert space of analytic functions which satisfies (A.1')-(A.3'). For $\textbf{f} \in \mathcal{H}$ we have \begin{equation}
\|\textbf{f}\|^2_{\mathcal{H}} = \|\textbf{f}\|^2_2 + \lim_{r \to 1} \int_\mathbb{T} \Big\|z\frac{\textbf{f}(z)-\textbf{f}(r\lambda)}{z-r\lambda}\Big\|^2_{\mathcal{H}} - r^2\Big\|\frac{\textbf{f}(z)-\textbf{f}(r\lambda)}{z-r\lambda}\Big\|^2_{\mathcal{H}} d\textit{m}(\lambda). \label{eq57}
\end{equation}
\end{prop}
\noindent Note that the formula makes sense even if $\mathcal{H}$ is not invariant for $M_z$, since for $\lambda \in \mathbb{D}$ we have $$z\frac{\textbf{f}(z)-\textbf{f}(\lambda)}{z-\lambda} = \frac{z\textbf{f}(z)- \lambda\textbf{f}(\lambda)}{z-\lambda} - \textbf{f}(\lambda) = (1-\lambda L)^{-1}\textbf{f}(z) - \textbf{f}(\lambda) \in \mathcal{H},$$ and $(1-\lambda L)^{-1}$ exists since $L$ is a contraction on $\mathcal{H}$. Versions of the above formula have been used in a crucial way in several works related to the structure of invariant subspaces, see for example \cite{afr}, \cite{alemanrichtersimplyinvariant} and \cite{ars}. We shall prove the formula by verifying that if $J\textbf{f} = (\textbf{f}, \textbf{g})$ in \thref{modeltheorem}, then the limit in \eqref{eq57} is equal to $\|\textbf{g}\|_2^2$. Actually, we will prove a stronger result than \thref{formula}, one which we will find useful at a later stage. In the next theorem and in the sequel we use the notation $$\L_\lambda := L(1-\lambda L)^{-1}, \lambda \in \mathbb{D}.$$ One readily verifies that $$L_\lambda \textbf{f}(z) = \frac{\textbf{f}(z)-\textbf{f}(\lambda)}{z - \lambda}.$$
\begin{thm} \thlabel{modelformulaconnection} Let $J\textbf{f} = (\textbf{f}, \textbf{w} + \Theta \textbf{f}_1)$ as in \thref{modeltheorem}, or $J\textbf{f} = (\textbf{f}, \textbf{f}_1)$ as in \thref{modeltheoremanalytic}. In both cases, we have that
\begin{enumerate}[(i)]
\item $\|zL_{\lambda}\textbf{f}\|^2_{\mathcal{H}[\textbf{B}]} - \|L_{\lambda}\textbf{f}\|^2_{\mathcal{H}[\textbf{B}]} = \|\textbf{f}_1(\lambda)\|^2_{Y_1} $,
\item $\|\textbf{f}_1\|_2^2 = \lim_{r \to 1} \int_\mathbb{T} \|zL_{r\lambda}\textbf{f}\|^2_{\mathcal{H}[\textbf{B}]} - \|L_{r\lambda}\textbf{f}\|^2_{\mathcal{H}[\textbf{B}]} d\textit{m}(\lambda)$,
\item $\|\textbf{w}\|_2^2 = \lim_{r \to 1} \int_\mathbb{T} (1-r^2)\|L_{r\lambda}\textbf{f}\|^2_{\mathcal{H}[\textbf{B}]} d\textit{m}(\lambda)$.
\end{enumerate}
In particular, \eqref{eq57} holds.
\end{thm}
\begin{proof} We shall only carry out the proof in the context of \thref{modeltheorem}, the other case being similar. First, we claim that for $\lambda \in \mathbb{D}$ we have that \begin{equation} JzL_\lambda \textbf{f} = \big(zL_\lambda \textbf{f}, (1-\lambda \conj{\zeta})^{-1}\textbf{w} + \Theta (zL_\lambda \textbf{f}_1 + \textbf{f}_1(\lambda))\big). \label{eq114} \end{equation} This follows from part (i) of \thref{modeltheorem}. Indeed, we have to check that
\begin{gather*}
\textbf{B}(\zeta)^*\zeta \frac{\textbf{f}(\zeta)- \textbf{f}(\lambda)}{\zeta - \lambda} + \Delta (\zeta) \frac{\textbf{w}(\zeta)}{1-\lambda \conj{\zeta}} + \Delta(\zeta)\Theta(\zeta)\zeta \frac{\textbf{f}_1(\zeta) - \textbf{f}_1(\lambda)}{\zeta - \lambda} + \Delta(\zeta)\Theta (\zeta) \textbf{f}_1(\lambda) \\ = \frac{\textbf{B}(\zeta)^* \textbf{f}(\zeta) + \Delta(\zeta)\textbf{g}(\zeta)}{1-\lambda \conj{\zeta}} - \frac{\textbf{B}(\zeta)^*\textbf{f}(\lambda)}{1-\lambda \conj{\zeta}} - \Big( \frac{\textbf{A}(\zeta)^*\textbf{h}(\lambda)}{1-\lambda\conj{\zeta}} - \textbf{A}(\zeta)^*\textbf{f}_1(\lambda)\Big)\end{gather*}
lies in $\conj{H^2_0(Y)}$, and this is true since each of the three terms in the last line lies in $\conj{H^2_0(Y)}$. Similarly, we have $$JL_\lambda \textbf{f} = \big(L_\lambda \textbf{f}, \conj{\zeta}(1-\lambda \conj{\zeta})^{-1}\textbf{w} + \Theta L_\lambda \textbf{f}_1\big).$$ Actually, this can be seen immediately by applying (ii) of \thref{modeltheorem} to \eqref{eq114}. Since $J$ is an isometry we have that \begin{equation}\|zL_\lambda \textbf{f}\|^2_{\mathcal{H}[\textbf{B}]} = \|zL_\lambda \textbf{f}\|^2_2 + \|(1-\lambda \conj{\zeta})^{-1}\textbf{w}\|^2_2 + \|zL_\lambda \textbf{f}_1 + \textbf{f}_1(\lambda)\|^2_2 \label{eq111} \end{equation} and \begin{equation}\|L_\lambda \textbf{f}\|^2_{\mathcal{H}[\textbf{B}]} = \|L_\lambda \textbf{f}\|^2_2 + \|\conj{\zeta}(1-\lambda \conj{\zeta})^{-1}\textbf{w}\|^2_2 + \|L_\lambda \textbf{f}_1\|^2_2. \label{eq112} \end{equation} The difference of \eqref{eq111} and \eqref{eq112} equals $\|\textbf{f}_1(\lambda)\|^2_{Y_1}$ which gives (i). Part (ii) is immediate from (i). We will deduce part (iii) from \eqref{eq112}. A brief computation involving power series shows that if $T$ is a contraction on a Hilbert space $\mathcal{H}$, then we have $\|T^nx\|_\mathcal{H} \to 0$ if and only if $$\lim_{r\to 1^-} \int_\mathbb{T} (1-r^2)\|T(1-r\lambda T)^{-1}x\|^2_{\mathcal{H}} d\textit{m}(\lambda) = 0.$$ We apply this to the $T = L$ acting on the Hardy spaces, and deduce (iii) from \eqref{eq112}: \begin{gather*}\lim_{r \to 1} \int_\mathbb{T} (1-r^2)\|L_{r\lambda}\textbf{f}\|^2_{\mathcal{H}[\textbf{B}]} d\textit{m}(\lambda) = \lim_{r \to 1} \int_\mathbb{T} (1-r^2) \|(1-r\lambda \conj{\zeta})^{-1} \textbf{w}\|^2_2 d\textit{m}(\lambda) \\ = \lim_{r \to 1} \int_\mathbb{T} \Bigg(\int_\mathbb{T} (1-r^2)|1-r\lambda \conj{\zeta}|^{-2} d\textit{m}(\lambda) \Bigg)\|\textbf{w}(\zeta)\|^2_Y d\textit{m}(\zeta) = \|\textbf{w}\|^2_2. \end{gather*}
\end{proof}
\subsection{Examples.} \label{examplessubsec} We end this section by discussing some examples of spaces which satisfy our assumptions, some of which were already mentioned in the introduction.
\subsubsection{De Branges-Rovnyak spaces.} If $\textbf{B} = b$ is a non-zero scalar-valued function, then $\Delta = (1-|b|^2)^{1/2}$ is an operator on a $1$-dimensional space, and $\text{clos}(\Delta H^2) \subseteq L^2$ is either of the form $\theta H^2$ for some unimodular function $\theta$ on $\mathbb{T}$, or it is of the form $L^2(E) := \{ f \in L^2 : f \equiv 0 \text{ a.e. on } \mathbb{T} \setminus E \}$. The first case corresponds to $b$ which are non-extreme points of the unit ball of $H^\infty$, while the second case corresponds to the extreme points. It is in the first case that the model of \thref{modeltheoremanalytic} applies to $\mathcal{H}(b)$.
\subsubsection{Weighted $H^2$-spaces.} Let $w = (w_n)_{n \geq 0}$ be a sequence of positive numbers and $H^2_w$ be the space of analytic functions in $\mathbb{D}$ which satisfy \[ \|f\|^2_{H^2_w} := \sum_{n=0}^\infty w_n|f_n|^2 < \infty,\] where $f_n$ is the $n$th Taylor coefficient of $f$ at $z = 0$. If $w_0 = 1$ and $w_{n+1} \geq w_n$ for $n \geq 0$, then $H^2_w$ satisfies (A.1)-(A.3). The reproducing kernel of $H^2_w$ is given by \[k(z,\lambda) = \sum_{n = 0}^\infty \frac{\conj{\lambda^n}z^n}{w_n} = \frac{1 - \sum_{n=1}^\infty (1/w_{n-1} - 1/w_n)\conj{\lambda^n}z^n}{1-\conj{\lambda}z}, \] and thus $H^2_w = \mathcal{H}[\textbf{B}]$ with $\textbf{B}(z) = \big( \sqrt{1/w_{n-1} - 1/w_n}z^n \big)_{n=1}^\infty$.
\subsubsection{Dirichlet-type spaces.} Let $\mu$ be a positive finite Borel measure on $\text{clos}(\mathbb{D})$. The space $\mathcal{D}(\mu)$ consists of analytic functions which satisfy \[ \|f\|^2_{\mathcal{D}(\mu)} := \|f\|^2_2 + \int_{\mathbb{T}}\int_{\text{clos}(\mathbb{D})} \frac{|f(z) - f(\zeta)|^2}{|z - \zeta|^2} d\mu(z) d\textit{m}(\zeta).\] The choice of $d\mu = d\textit{m}$ produces the classical Dirichlet space. It is easy to verify directly from this expression that $\mathcal{D}(\mu)$ satisfies axioms (A.1)-(A.3), but it is in general difficult to find an expression for $\textbf{B}$ corresponding to the space as in \thref{kernel}. In the special case that $\mu = \sum_{i=1}^n c_i\delta_{z_i}$ is a positive sum of unit masses $\delta_{z_i}$ at distinct points $z_i \in \text{clos}(\mathbb{D})$ the space $\mathcal{D}(\mu)$ is an $\mathcal{H}[\textbf{B}]$-space of rank $n$. An isometric embedding $J: \mathcal{D}(\mu) \to (H^2)^{n+1}$ satisfying the properties listed in \thref{modeltheoremanalytic} is given by \[f(z) \mapsto \Big(f(z), \frac{f(z)-f(z_1)}{z - z_1}, \ldots, \frac{f(z)-f(z_n)}{z-z_n} \Big). \]
\subsubsection{Cauchy duals.} Let $\mathcal{H}$ be a Hilbert space of analytic functions which contains all functions holomorphic in a neighbourhood of $\text{clos}(\mathbb{D})$, on which the forward shift operator $M_z$ acts as a contraction and such that $\ip{f}{1}_\mathcal{H} = f(0)$ holds for $f \in \mathcal{H}$. Consider the function \[ Uf(\lambda) = \ip{(1-\lambda z)^{-1}}{f(z)}_\mathcal{H} = \sum_{n=0}^\infty \lambda^n \ip{z^n}{f(z)}_{\mathcal{H}}.\] Then $Uf$ is an analytic function of $\lambda$, and $UM_z^*f = LUf$. If $\mathcal{H}^*$ is defined to be the space of functions of the form $Uf$ for $f \in \mathcal{H}$, with the norm $\|Uf\|_{\mathcal{H}^*} = \|f\|_{\mathcal{H}}$, then it is easy to verify that $\mathcal{H}^*$ is a Hilbert space of analytic functions which satisfies (A.1)-(A.3). The space $\mathcal{H}^*$ is the so-called Cauchy dual of $\mathcal{H}$ (see \cite{acppredualsqp}).
\section{Density of functions with continuous extensions to the closed disk} \label{densitysection}
The goal of this section is to prove \thref{conttheorem} and \thref{conttheoremcor}, which are vector-valued generalizations of \thref{thm1} mentioned in the introduction. We will first recall a few facts about the disk algebra $\mathcal{A}$ and the vector-valued Smirnov classes $N^+(Y)$.
\subsection{Disk algebra, Cauchy transforms and the Smirnov class.} \label{dasubsec}
Let $\mathcal{A}$ denote the disk algebra, the space of scalar-valued analytic functions defined in $\mathbb{D}$ which admit continuous extensions to $\text{clos}(\mathbb{D})$. It is a Banach space if given the norm $\|f\|_\infty = \sup_{z \in \mathbb{D}} |f(z)|$, and the dual of $\mathcal{A}$ can be identified with the space $\mathcal{C}$ of Cauchy transforms of finite Borel measures $\mu$ supported on the circle $\mathbb{T}$. A Cauchy transform $f$ is an analytic function in $\mathbb{D}$ which is of the form $f(z) = C\mu(z) := \int_\mathbb{T} \frac{1}{1-z\conj{\zeta}} d\mu(\zeta)$ for some Borel measure $\mu$. The duality between $\mathcal{A}$ and $\mathcal{C}$ is realized by $$\ip{h}{f} = \lim_{r \rightarrow 1^-}\int_\mathbb{T} h(r\zeta)\conj{f(r\zeta)} d\textit{m}(\zeta) = \int_{\mathbb{T}} h \conj{d\mu}, \quad h \in \mathcal{A}, f = C\mu$$ and the norm $\|f\|_\mathcal{C}$ of $f$ as a functional on $\mathcal{A}$ is given by $\|f\|_\mathcal{C} = \inf_{\mu : C\mu = f} \|\mu\|,$ where $\|\mu\|$ is the total variation of the measure $\mu$. The space $\mathcal{C}$ is continuously embedded in the Hardy space $H^p$ for each $p \in (0,1)$. More precisely, we have for each fixed $p \in (0,1)$ the estimate $\|f\|_p = (\int_\mathbb{T} |f|^p d\textit{m})^{1/p} \leq c_p \|f\|_{\mathcal{C}}.$ As a dual space of $\mathcal{A}$, the space $\mathcal{C}$ can be equipped with the weak-star topology, and a sequence $(f_n)$ converges weak-star to $f \in \mathcal{C}$ if and only if $\sup_n \|f_n\|_{\mathcal{C}} < \infty$ and $f_n(z) \to f(z)$ for each $z \in \mathbb{D}$. See \cite{cauchytransform} for more details.
If $Y$ is a Hilbert space, then the Smirnov class $N^+(Y)$ consists of the functions $\textbf{f}: \mathbb{D} \to Y$ which can be written as $\textbf{f} = \textbf{u}/v$, where $\textbf{u} \in H^\infty({Y})$ and $v: \mathbb{D} \to \mathbb{C}$ is a bounded outer function. In the case $Y = \mathbb{C}$ we will simply write $N^+$. The class $N^+(Y)$ satisfies the following \textit{Smirnov maximum principle}: if $\textbf{f} \in N^+(Y)$, then we have that $\int_\mathbb{T} \|\mathbf{f}(\zeta)\|_{Y}^2 d\textit{m}(\zeta) < \infty$ if and only if $\mathbf{f} \in H^2(Y)$ (see Theorem A in Section 4.7 of \cite{rosenblumrovnyakhardyclasses}).
\subsection{Proof of the density theorem.} \label{proofsubsec}
The proof will depend on a series of lemmas. The first two are routine exercises in functional analysis and the proofs of those will be omitted.
\begin{lemma}
\thlabel{weakstarapprox}
Let $B$ be a Banach space, $B'$ be its dual space, and $S \subset B'$ be a linear manifold. If $l \in B'$ annihilates the subspace $\cap_{s \in S} \ker s \subset B$, then $l$ lies in the weak-star closure of $S$.
\end{lemma}
\begin{lemma} \thlabel{convlemma}
Let $\{h_j\}$ be a sequence of scalar-valued analytic functions in $\mathbb{D}$, with $\sup_n \|h_n\|_\infty < \infty$, and which converges uniformly on compacts to the function $h$. If the sequence $\{\textbf{g}_j\}_{j=1}^\infty$ of functions in $L^2(Y)$ converges in norm to $\textbf{g}$, then $h_j\textbf{g}_j$ converges weakly in $L^2(Y)$ to $h\textbf{g}$.
\end{lemma}
The next two lemmas are more involved. Let $\mathcal{A}^n = \mathcal{A} \times \ldots \times \mathcal{A}$ denote the product of $n$ copies of the disk algebra. The dual of $\mathcal{A}^n$ can then be identified with $\mathcal{C}^n$, the space of $n$-tuples of Cauchy transforms $\textbf{f} = (f^1, \ldots, f^n)$, normed by $\|\textbf{f}\|_{\mathcal{C}^n} = \sum_{i=1}^n \|f^i\|_\mathcal{C}.$ The main technical argument needed for the proof of \thref{conttheorem} is contained in the following lemma.
\begin{lemma} \thlabel{multiplyintoh2lemma}
Let $\{\textbf{f}_m\}_{m=1}^\infty$ be a sequence in $\mathcal{C}^n$ which converges weak-star to $\textbf{f}$. There exists a subsequence $\{\textbf{f}_{m_k}\}_{k=1}^\infty$ and a sequence of outer functions $M_k: \mathbb{D} \to \mathbb{C}$ satisfying the following properties:
\begin{enumerate}[(i)]
\item $\|M_k\|_\infty \leq 1$,
\item $M_k$ converges uniformly on compacts to a non-zero outer function $M$,
\item $M_k\textbf{f}_{m_k} = (M_kf^1_{m_k}, M_kf^2_{m_k}, \ldots, M_kf^n_{m_k}) \in (H^2)^n$,
\item the sequence $\{ M_k\mathbf{f_{m_k}}\}_{m=1}^\infty$ converges weakly to $M\mathbf{f}$ in $(H^2)^n$.
\end{enumerate}
\end{lemma}
\begin{proof} Let $\textbf{g} = (g^1, g^2, \ldots, g^n)$ be an $n$-tuple of Cauchy transforms and for some fixed choice of $p \in (1/2, 1)$ let $$s(\zeta) = \max(\sum_{i=1}^n |g^i(\zeta)|^p,1 ), \quad \zeta \in \mathbb{T}.$$ Then $s$ is integrable on the circle and $\int_{\mathbb{T}} s \,d\textit{m} \leq C_1\|\textbf{g}\|_{\mathcal{C}^n}^p,$ where the constant $C_1 > 0$ depends on $p$ and $n$, but is independent of $\textbf{g} \in \mathcal{C}^n$. We let $H$ be the Herglotz transform of $s$, that is $$H(z) = \int_{\mathbb{T}} \frac{\zeta + z}{\zeta - z} s(\zeta) d\textit{m}(\zeta).$$ Note that the real part of $H$ is the Poisson extension of $s(\zeta)$ to $\mathbb{D}$. This shows that $H$ has positive real part (hence is outer), $|H| \geq s \geq 1$ on $\mathbb{T}$ and $|H(z)| \geq 1$ for all $z \in \mathbb{D}$. We also have that $H(0) = \int_\mathbb{T} s \, d\textit{m} \leq C_1\|\textbf{g}\|^p_{\mathcal{C}^n}.$ Let $q = 2 - 2p \in (0,1).$ For each $i \in \{1, \ldots, n\}$ we have the estimate $$\int_\mathbb{T} |g^i/H|^2 d\textit{m} \leq \int_\mathbb{T} |g^i/s|^2 d\textit{m} \leq \int_\mathbb{T} |g^i|^{2-2p} d\textit{m} = \|g^i\|_{q}^{q} \leq C_2 \|g^i\|^q_{\mathcal{C}}.$$ Since the functions $g^i$ are in $\mathcal{C} \subset N^+$, the Smirnov maximum principle implies that $g^i/H \in H^2$, or equivalently $\textbf{g}/H \in (H^2)^n$. Moreover, $\|\textbf{g}/H\|_{(H^2)^n} \leq C\|\textbf{g}\|^q_{\mathcal{C}^n}$, with constant $C > 0$ depending only on the fixed choice of $p$ and the dimension $n$.
Let now $\{\textbf{f}_m\}_{m=1}^\infty$ be a sequence in $\mathcal{C}^n$ which converges weak-star to $\textbf{f}$, meaning that $\textbf{f}_m$ converges pointwise to $\textbf{f}$ in $\mathbb{D}$, and we have $\sup_{m} \|\textbf{f}_m\|_{\mathcal{C}^n} < \infty$. For each integer $m \geq 1$ we construct the function $H = H_m$ as above. By what we have established above, $\{\textbf{f}_m/H_m\}_{m=1}^\infty$ is a bounded sequence in $(H^2)^n$. Since $\|1/H_m\|_{\infty} \leq 1$ and $H_m(0) \leq C_1\|\textbf{f}_m\|^p_{\mathcal{C}^n}$, there exists a subsequence $\{m_k\}_{k=1}^\infty$ such that $M_k = 1/{H_{m_k}}$ converges uniformly on compacts to a non-zero analytic function $M$. Then $M$ has positive real part, since each of the functions $M_k$ has positive real part, and therefore $M$ is outer. The sequence $\{M_k\textbf{f}_{m_k}\}_{k=1}^\infty$ is bounded in $(H^2)^n$ and converges pointwise to the function $M\textbf{f}$, which is equivalent to weak convergence in $(H^2)^n$.
\end{proof}
For the rest of the section, let $\mathcal{H} = \mathcal{H}[\textbf{B}]$ be a fixed space of $X$-valued functions, where $X$ is a finite dimensional Hilbert space. Thus $\textbf{B}$ takes values in $\mathcal{B}(Y,X)$ for some auxilliary Hilbert space $Y$. As before, set $\Delta(\zeta) = (I_Y - \textbf{B}(\zeta)^*\textbf{B}(\zeta))^{1/2}$ for $\zeta \in \mathbb{T}$. Fix an orthonormal basis $\{e_i\}_{i=1}^n$ for the finite dimensional Hilbert space $X$. We can define a map from $H^2(X)$ to $(H^2)^n$ by the formula $\textbf{f} \mapsto (f^i)_{i=1}^n$ where the components $f^i$ are the coordinate functions $f^i(z) = \ip{\textbf{f}(z)}{e_i}_X.$ Then a function $\textbf{f} \in H^2(X)$ has a continuous extension to $\text{clos}(\mathbb{D})$ if and only if all of its coordinate functions $f^i$ are contained in the disk algebra $\mathcal{A}$.
\begin{lemma} \thlabel{Sset}
\thlabel{techlemma} Let $S \subset \mathcal{C}^n \oplus \text{clos}({\Delta H^2(Y)})$ be the linear manifold consisting of tuples of the form $$(\textbf{B} \textbf{f}, \Delta \textbf{f})$$ for some $\textbf{f} \in N^+(Y)$. Then $S$ is weak-star closed in $\mathcal{C}^n \oplus \text{clos}({\Delta H^2(Y)})$.
\end{lemma}
\begin{proof}
Since $\mathcal{A}^n \oplus \text{clos}({\Delta H^2(Y)})$ is separable, Krein-Smulian theorem implies that it is enough to check weak-star sequential closedness of the set $S$. Thus, let the sequence \[\{(\textbf{B} \textbf{f}_m, \Delta \textbf{f}_m)\}_{m=1}^\infty = \{(\textbf{h}_m, \textbf{g}_m)\}_{m=1}^\infty \subset \mathcal{C}^n \oplus \text{clos}({\Delta H^2(Y)})\] converge in the weak-star topology to $(\textbf{h}, \textbf{g}).$ Then $\{ \textbf{g}_m \}_{m=1}^\infty$ converges weakly in the Hilbert space $\text{clos}({\Delta H^2(Y)})$, and by passing to a subsequence and next to the Ces\`aro means of that subsequence, we can assume that the sequence $\{\textbf{g}_m\}_{m=1}^\infty$ converges to $\textbf{g}$ in the norm. By applying \thref{multiplyintoh2lemma} and \thref{convlemma} we obtain a sequence of outer functions $\{M_k\}_{k=1}^\infty$ and an outer function $M$ such that $\{(M_k\textbf{h}_{m_k}, M_k\textbf{g}_{m_k})\}_{k=1}^\infty$ converges weakly in the Hilbert space $H^2(X) \oplus \text{clos}({\Delta H^2(Y)})$ to $(M\textbf{h}, M\textbf{g})$. Note that $$(M_k\textbf{h}_{m_k}, M_k\textbf{g}_{m_k}) = (\textbf{B} M_k \textbf{f}_{m_k}, \Delta M_k \textbf{f}_{m_k}),$$ and $$\int_\mathbb{T} \|M_k\textbf{f}_{m_k}\|^2_Y d\textit{m} = \int_\mathbb{T} \|\textbf{B} M_k\textbf{f}_{m_k}\|^2_X d\textit{m} + \int_\mathbb{T} \|\Delta M_k\textbf{f}_{m_k}\|^2_Y d\textit{m} < \infty.$$ Since $M_k\textbf{f}_{m_k}$ is in $N^+(Y)$, the Smirnov maximum principle implies that we have $M_k\textbf{f}_{m_k} \in H^2(Y)$, and consequently the tuples $(\textbf{B}M_k\textbf{f}_{m_k}, \Delta M_k\textbf{f}_{m_k})$ are contained in the closed subspace $U = \{ (\textbf{B}\textbf{h}, \Delta\textbf{h}) : \textbf{h} \in H^2(Y)\}$. It follows that the weak limit $(M\textbf{h}, M\textbf{g})$ is also contained in $U$, and hence $(M\textbf{h}, M\textbf{g}) = (B \textbf{f}, \Delta \textbf{f})$ for some $\textbf{f} \in H^2(Y)$. Then $$(\textbf{h}, \textbf{g}) = \Big(\textbf{B} \frac{\textbf{f}}{M} , \Delta \frac{\textbf{f}}{M} \Big),$$ where $\textbf{f}/M \in N^+(Y)$.
\end{proof}
\begin{thm}
\thlabel{conttheorem} Assume that $\mathcal{H}[\textbf{B}]$ consists of functions taking values in a finite dimensional Hilbert space. Then the set of functions in $\mathcal{H}[\textbf{B}]$ which extend continuously to $\text{clos}(\mathbb{D})$ is dense in the space.
\end{thm}
\begin{proof} Let $K = H^2(X) \oplus \text{clos}({\Delta H^2(Y)})$. Recall from \thref{modeltheorem} that the space $\mathcal{H}[\textbf{B}]$ is equipped with an isometric embedding $J$ where the tuple $J\textbf{f} = (\textbf{f},\textbf{g}) \in K$ is uniquely determined by the requirement for it be orthogonal to \[ U = \{ (\textbf{B}\textbf{h}, \Delta \textbf{h} : \textbf{h}\in H^2(Y)\} \subset K. \] We identify functions $\textbf{f} \in \mathcal{H}[\textbf{B}]$ with their coordinates $(f^1, \ldots, f^n)$ with respect to the fixed orthonormal basis of $X$. Now assume that $\textbf{f} \in \mathcal{H}[\textbf{B}]$ is orthogonal to any function in $\mathcal{H}[\textbf{B}]$ which extends continuously to $\text{clos}(\mathbb{D})$, i.e., that $\textbf{f}$ is orthogonal to $\mathcal{H}[\textbf{B}] \cap \mathcal{A}^n$. We shall show that $J\textbf{f} = (\textbf{B} \textbf{h}, \Delta \textbf{h})$ for some $\textbf{h} \in H^2(Y)$, which implies that $\textbf{f} = 0$. Consider $J(\mathcal{A}^n \cap \mathcal{H}[\textbf{B}])$ as a subspace of $\mathcal{A}^n \oplus \text{clos}({\Delta H^2(Y)})$. For each $\textbf{h} \in H^2(Y)$, let $$l_\textbf{h} = (\textbf{B} \textbf{h}, \Delta \textbf{h}) \in \mathcal{C}^n \oplus \text{clos}({\Delta H^2(Y)})$$ be a functional on $\mathcal{A}^n \oplus \text{clos}({\Delta H^2(Y)})$, acting as usual by integration on the boundary $\mathbb{T}$. We claim that $$J(\mathcal{A}^n \cap \mathcal{H}[\textbf{B}]) = \cap_{\textbf{h} \in H^2(Y)} \ker l_\textbf{h}.$$ Indeed, if $\textbf{f} \in \mathcal{A}^n \cap \mathcal{H}[\textbf{B}]$, then for any functional $l_\textbf{h}$ we have $$l_\textbf{h}(J\textbf{f}) = \ip{J\textbf{f}}{(\textbf{B} \textbf{h}, \Delta \textbf{h})} = 0,$$ because $J\textbf{f}$ is orthogonal to $U$. Conversely, if the tuple $(\textbf{f},\textbf{g}) \in \mathcal{A}^n \oplus \text{clos}({\Delta H^2(Y)})$ is contained in $\cap_{\textbf{h} \in H^2(Y)} \ker l_\textbf{h}$, then $(\textbf{f},\textbf{g}) \in K$ is orthogonal to $U$, and hence $\textbf{f} \in \mathcal{A} \cap \mathcal{H}[\textbf{B}]$ by \thref{modeltheorem}. Now, viewed as an element of $\mathcal{C}^n \oplus \text{clos}({\Delta H^2(Y)})$, the tuple $J\textbf{f}$ annihilates $J(\mathcal{A}^n \cap \mathcal{H}[\textbf{B}])$, and so by \thref{weakstarapprox} lies in the weak-star closure of linear manifold of functionals of the form $l_\textbf{h}$. Thus \thref{Sset} implies that $J\textbf{f} = (\textbf{B} \textbf{h}, \Delta \textbf{h})$ for some $\textbf{h} \in N^+(Y)$. The Smirnov maximum principle and the computation $$\int_\mathbb{T} \|\textbf{h}\|^2_Y d\textit{m}(\zeta) = \int_\mathbb{T} \|\textbf{B}\textbf{h}\|^2_X d\textit{m} + \int_\mathbb{T} \|\Delta \textbf{h}\|^2_Y d\textit{m} < \infty$$ show that $\textbf{h} \in H^2(Y)$. Hence $J\textbf{f} \in (JH(\textbf{B})^\perp$, so that $\textbf{f} = 0$ and the proof is complete.
\end{proof}
\thref{thm1} of the introduction now follows as a consequence of the next result, which is an easy extension of \thref{conttheorem}.
\begin{cor} \thlabel{conttheoremcor} Assume that $\mathcal{H}[\textbf{B}]$ consists of functions taking values in a finite dimensional Hilbert space. If $M$ is any $L$-invariant subspace of $\mathcal{H}[\textbf{B}]$, then the set of functions in $M$ which extend continuously to $\text{clos}(\mathbb{D})$ is dense in $M$.
\end{cor}
\begin{proof}
If $M$ contains the constant vectors, then \thref{kernel} applies, and hence $M$ is of the type $\mathcal{H}(\textbf{B}_0)$ for some contractive function $\textbf{B}_0$. Then the result follows immediately from \thref{conttheorem}. If constant vectors are not contained in $M$, then let $$M^+ = \{ \textbf{f} + c : \textbf{f} \in M, c \in X\}.$$ The subspace $M^+$ is closed, as it is a sum of a closed subspace and a finite dimensional space. Moreover, closed graph theorem implies that the skewed projection $P: M^+ \to M$ taking $\textbf{f}+c$ to $\textbf{f}$ is bounded. The theorem holds for $M^+$, so if $\textbf{f} \in M$, then there exists constants $c_n$ and functions $\textbf{f}_n \in M$ such that $\textbf{h}_n = \textbf{f}_n + c_n$ is continuous on $\text{clos}(\mathbb{D})$, and $\textbf{h}_n$ tends to $\textbf{f}$ in the norm of $\hil$. Consequently, the functions $\textbf{f}_n = \textbf{h}_n - c_n$ are continuous on $\text{clos}(\mathbb{D})$, and we have that $\textbf{f}_n = P\textbf{h}_n$ tends to $P\textbf{f} = \textbf{f}$ in the norm of $\hil$.
\end{proof}
\section{Applications of the density theorem} \label{applications1}
We temporarily leave the the main subject in order to present applications of \thref{conttheorem} and \thref{conttheoremcor}. All Hilbert spaces of analytic functions will be assumed to satisfy (A.1)-(A.3).
\subsection{$M_z$-invariant subspaces.} \thref{cor-beurling1} stated in the introduction is now an easy consequence of \thref{conttheorem}. We restate the theorem for the reader's convenience.
\begin{cor} Let $\mathcal{H}$ be a Hilbert space of analytic functions which satisfies (A.1)-(A.3) and is invariant for the forward shift $M_z$. For a closed $M_z$-invariant subspace $\mathcal{M}$ of $\mathcal{H}$ with $\dim \mathcal{M}\ominus M_z\mathcal{M}=n<\infty$, let $\varphi_1,\ldots\varphi_n$ be an orthonormal basis in $\mathcal{M}\ominus M_z\mathcal{M}$, and denote by $\phi$ the corresponding row operator-valued function. Then \begin{equation}\mathcal{M}=\phi \mathcal{H}[\textbf{C}], \end{equation}
where $ \mathcal{H}[\textbf{C}]$ consists of $\mathbb{C}^n$-valued functions and the mapping $g \mapsto \phi g$ is an isometry from $\mathcal{H}[\textbf{C}]$ onto $\mathcal{M}$. Moreover,
\begin{equation} \{\sum_{i=0}^n\varphi_iu_i:~u_i\in \mathcal{A},~1\le i\le n\}\cap \mathcal{H}\end{equation}
is a dense subset of $\mathcal{M}$.
\end{cor}
\begin{proof}
For any $f \in \mathcal{M}$ we have that $f(z) - \sum_{i=1}^n \ip{f}{\phi_i}_\mathcal{H} \phi_i(z) \in M_z\mathcal{M}$. Thus the operator $L^\phi: \mathcal{M} \to \mathcal{M}$ given by \[ L^\phi f(z) = \frac{f(z) - \sum_{i=1}^n \ip{f}{\phi_i}_\mathcal{H} \phi_i(z)}{z} \] is well-defined, and it is a contraction since it is a composition of a projection with the contractive operator $L$. A straightforward computation shows that for $\lambda \in \mathbb{D}$ the following equation holds \[ (1-\lambda L^\phi)^{-1} f(z) = \frac{zf(z) - \lambda \sum_{i=1}^n \ip{(1-\lambda L^\phi)^{-1}f}{\phi_i}_\mathcal{H}\phi_i(z)}{z-\lambda}.\] Thus the analytic function in the numerator on the right-hand side above must have a zero at $z = \lambda$. It follows that $f(\lambda) = \sum_{i=1}^n \ip{(1-\lambda L^\phi)^{-1}f}{ \phi_i}_\mathcal{H} \phi_i(\lambda)$. Consider now the mapping $U$ taking $f \in \mathcal{M}$ to the vector $Uf(\lambda) = \Big( \ip{(1-\lambda L^\phi)^{-1}}{\phi_i}_\mathcal{H} \Big)_{i=1}^n$ and let $\mathcal{M}_0 = U\mathcal{M}$ with the norm on $\mathcal{M}_0$ which makes $U:\mathcal{M} \to \mathcal{M}_0$ a unitary mapping. Then $\mathcal{M}_0$ is a space of $\mathbb{C}^n$-valued analytic functions which satisfies (A.1')-(A.3') and to which \thref{conttheorem} applies. The claims in the statement follow immediately from this.
\end{proof}
\subsection{Reverse Carleson measures.} A finite Borel measure on $\text{clos}(\mathbb{D})$ is a \textit{reverse Carleson measure} for $\mathcal{H}$ if there exists a constant $C > 0$ such that the estimate \begin{equation}\|f\|^2_\mathcal{H} \leq C \int_{\text{clos}(\mathbb{D})} |f(z)|^2 d\mu(z) \label{RCineq}\end{equation} holds for $f$ which belong to some dense subset of $\mathcal{H}$ and for which the integral on the right-hand side makes sense, e.g. by the existence of radial boundary values of $f$ on the support of the singular part of $\mu$ on $\mathbb{T}$. For the class of spaces considered in this paper it is natural to require, due to \thref{conttheorem}, that \eqref{RCineq} holds for all functions in $\mathcal{H}$ which admit continuous extensions to $\text{clos}(\mathbb{D})$.
Our main result in this context characterizes the existence of a reverse Carleson measures for spaces which are invariant for $M_z$. If such a measure exists, then we can moreover identify one which is in a sense minimal.
\begin{thm} \thlabel{revcarl2}
Let $\mathcal{H}$ be invariant for $M_z$. Then the following are equivalent.
\begin{enumerate}[(i)]
\item $\mathcal{H}$ admits a reverse Carleson measure.
\item $$\sup_{0 < r < 1} \int_\mathbb{T} \Bigg\| \frac{\sqrt{1-|r\lambda|^2}}{1-r\conj{\lambda}z}\Bigg\|^2_{\mathcal{H}} d\textit{m}(\lambda) < \infty.$$
\item If $k$ is the reproducing kernel of $\mathcal{H}$, then$$\sup_{0 < r < 1} \int_\mathbb{T} \frac{1}{(1-|r\lambda|^2)k(r\lambda,r\lambda)} d\textit{m}(\lambda) < \infty.$$
\end{enumerate}
If the above conditions are satisfied, then $$h_1(\lambda) := \lim_{r \rightarrow 1} \Bigg\| \frac{\sqrt{1-|r\lambda|^2}}{1-r\conj{\lambda}z}\Bigg\|^2_{\mathcal{H}} $$ and $$h_2(\lambda) := \lim_{r \rightarrow 1} \frac{1}{(1-|r\lambda|^2)k(r\lambda, r\lambda)}$$ define reverse Carleson measures for $\hil$. Moreover, if $\nu$ is any reverse Carleson measure for $\hil$ and $v$ is the density of the absolutely continuous part of the restriction of $\nu$ to $\mathbb{T}$, then $h_1d\textit{m}$ and $h_2d\textit{m}$ have the following minimality property: there exist constants $C_i > 0, i =1,2$ such that $$h_i(\lambda) \leq C_i v(\lambda)$$ for almost every $\lambda \in \mathbb{T}.$
\end{thm}
\begin{proof}(i) $\Rightarrow$ (ii): Let $\nu$ be a reverse Carleson measure for $\mathcal{H}$. First, we show that we can assume that $\nu$ is supported on $\mathbb{T}$. For this, we will use the inequality $$\|z^nf\|^2_\mathcal{H} \leq C \int_\mathbb{D} |z^nf(z)|^2 d\nu(z) + C \int_{\mathbb{T}} |f(z)|^2 d\nu(z).$$ Since $Lzf = f$ and $L$ is a contraction, it follows that $\|zf\|_\mathcal{H} \geq \|f\|_\mathcal{H}$, and thus letting $n$ tend to infinity in the above inequality we obtain $$ \|f\|^2_\mathcal{H} \leq C \int_{\mathbb{T}} |f(z)|^2 d(\nu|\mathbb{T})(z).$$ Thus we might replace $\nu$ by $\nu|\mathbb{T}$, as claimed. Next, we note that $H^\infty \subset \mathcal{H}$. Indeed, $\mathcal{H}$ contains $1$ and is $M_z$-invariant, thus contains the polynomials. If $p_n$ is a uniformly bounded sequence of polynomials converging pointwise to $f \in H^\infty$, then the existence of a reverse Carleson measure ensures that the norms $\|p_n\|_\mathcal{H}$ are uniformly bounded, and thus a subsequence of $\{p_n\}$ converges weakly to $f \in \mathcal{H}$. Thus, the function $z \mapsto \frac{1}{1-\conj{\lambda}z}$ is contained in $\mathcal{H}$ for each $\lambda \in \mathbb{D}$. Define $$H(\lambda) =: \int_\mathbb{T} \frac{1-|\lambda|^2}{|1-\conj{\lambda}z|^2} d\nu(z), \quad \lambda \in \mathbb{D},$$ which is positive and harmonic in $\mathbb{D}$, and since $\nu$ is a reverse Carleson measure for $\hil$, there exists a constant $C > 0$ such that \begin{equation} \Bigg\|\frac{\sqrt{1-|\lambda|^2}}{1-\conj{\lambda}z}\Bigg\|_\mathcal{H}^2 \leq CH(\lambda). \label{RCineq2} \end{equation} The implication now follows from the mean value property of harmonic functions.
(ii) $\Rightarrow$ (iii): There exists an orthogonal decomposition $$\frac{1}{1-\conj{\lambda}z} = \frac{1}{1-|\lambda|^2}\frac{k(\lambda, z)}{k(\lambda, \lambda)} + g(z),$$ where $g$ is some function which vanishes at $\lambda$. Thus $$\Big\|\frac{1}{1-\conj{\lambda}z}\Big\|_\mathcal{H}^2 = \frac{1}{(1-|\lambda|^2)^2k(\lambda,\lambda)} + \|g\|^2$$ and consequently \begin{equation}\frac{1}{(1-|\lambda|^2)k(\lambda,\lambda)} \leq \Bigg\|\frac{\sqrt{1-|a|^2}}{1-\conj{\lambda}z}\Bigg\|_\mathcal{H}^2. \label{RCineq4} \end{equation} Thus (iii) follows from (ii).
(iii) $\Rightarrow$ (i): If $\mathcal{H} = \mathcal{H}[\textbf{B}]$ with $\textbf{B} = (b_i)_{i=1}^\infty$, then let $w$ be the outer function with boundary values satisfying $|w(\zeta)|^2 = 1-\sum_{i\in I} |b_i(\zeta)|^2$. The existence of such a function is ensured by (iii). The space $M = wH^2 = \{f = wg : g \in H^2 \}$ normed by $\|f\|_{M} = \|g\|_2$ is a Hilbert space of analytic functions with a reproducing kernel given by $$K_{M}(\lambda, z) = \frac{\conj{w(\lambda)}w(z)}{1-\conj{\lambda}z}.$$ It is not hard to verify that $K = k_\mathcal{H} - K_M$ is a positive-definite kernel. Then $k_\mathcal{H} = K + K_M$, and hence $M$ is contained contractively in $\mathcal{H}$. Thus for any function $f \in M$ we have that $\|f\|^2_{\hil} \leq \|f\|^2_M = \int_\mathbb{T} \frac{|f|^2}{|w|^2} d\textit{m}.$, and $|w|^{-2}d\textit{m}$ will be a reverse Carleson measure if $M$ is dense in $\hil$. But $1/w \in H^2$ by (iii), and thus $\mathcal{H} \cap \mathcal{A} \subset M$, so $M$ is indeed dense in $\mathcal{H}$.
The limits defining $h_1$ and $h_2$ exist as a consequence of general theory of boundary behaviour of subharmonic functions. The inequality $h_1(\lambda) \leq C_1v(\lambda)$ is seen immediately from \eqref{RCineq2} and $h_1(\lambda) \leq C_2v(\lambda)$ is then seen from \eqref{RCineq4}.
\end{proof}
We remark that if $\mathcal{H}$ is not $M_z$-invariant, then the space might admit a reverse Carleson measure even though (iii) is violated. An example is any $L$-invariant proper subspace of $H^2$.
An application of part (iii) of \thref{revcarl2} to $\mathcal{H} = \mathcal{H}(b)$, with $b$ non-extreme, lets us deduce a result essentially contained in \cite{revcarlesonross}, namely that $\mathcal{H}(b)$ admits a reverse Carleson measure if and only if $(1-|b|^2)^{-1} \in L^1(\mathbb{T})$, and the measure $\mu = (1-|b|^2)^{-1}d\textit{m}$ is then a minimal reverse Carleson measure in the sense made precise by the theorem. A second application is to Dirichlet-type spaces. Recall from Section \ref{examplessubsec} that for $\mu$ a positive finite Borel measure supported on $\text{clos}(\mathbb{D})$, the Hilbert space $\mathcal{D}(\mu)$ is defined as the completion of the analytic polynomials under the norm $$\|f\|^2_{\mathcal{D}(\mu)} = \|f\|^2_2 + \int_{\mathbb{T}} \int_{\text{clos}(\mathbb{D})} \frac{|f(z)-f(\lambda)|^2}{|z-\lambda|^2} d\mu(z) d\textit{m}(\lambda).$$
\begin{cor}
The space $D(\mu)$ admits a reverse Carleson measure if and only if $$\int_{\text{clos}(\mathbb{D})} \frac{d\mu(z)}{1-|z|^2} < \infty.$$ The minimal reverse Carleson measure (in the sense of \thref{revcarl2}) is given by $d\nu = hd\textit{m}$ where $$h(\lambda) = 1 + \int_{\text{clos}(\mathbb{D})} \frac{d\mu(z)}{|1 - \conj{\lambda}z|^2}, \lambda \in \mathbb{T}.$$
\end{cor}
\begin{proof} The space $\mathcal{D}(\mu)$ satisfies the assumptions of \thref{revcarl2}. A computation shows that if $k_\lambda(z) = \frac{1}{1-\conj{\lambda}z}$, then $$(1-|\lambda|^2)\|k_\lambda\|^2_{\mathcal{D}(\mu)} = 1 + \int_{\text{clos}(\mathbb{D})} \frac{|\lambda|^2}{|1-\conj{\lambda}z|^2}d\mu(z).$$ The claim now follows easily from (ii) of \thref{revcarl2} and Fubini's theorem.
\end{proof}
It is interesting to note that the condition on $\mu$ above holds even in cases when $\mathcal{D}(\mu)$ is strictly contained in $H^2$ (see \cite{alemanhabil}).
Our last result in the context of reverse Carleson measures is a non-existence result which answers in particular a question posed in \cite{revcarlesonross}.
\begin{thm}\thlabel{rctheorem}
Assume that the identity $$\|Lf\|^2_{\mathcal{H}} = \|f\|^2_{\mathcal{H}} - |f(0)|^2$$ holds in $\mathcal{H}$. If $\mathcal{H}$ admits a reverse Carleson measure, then $\mathcal{H}$ is isometrically contained in the Hardy space $H^2$.
\end{thm}
\begin{proof}
We will use \thref{modeltheorem} and \thref{modelformulaconnection}. The limit in part (ii) of \thref{modelformulaconnection} vanishes, because the norm identity implies that $\|zL_\lambda f\| = \|L_\lambda f\|_{\mathcal{H}}$. It will thus suffice to show that the limit in part (iii) of said theorem also vanishes, namely that \begin{gather} \lim_{r \to 1} \int_{\mathbb{T}} (1-r^2)\|L_{r\lambda}f\|^2_{\mathcal{H}} d\textit{m}(\lambda) = 0. \label{ineq2}\end{gather} If $f \in \mathcal{H}$ is continuous in $\text{clos}(\mathbb{D})$ then so is $L_{r\lambda}f$, and if $\mu$ is a reverse Carleson measure for $\mathcal{H}$, then we have the estimate
\begin{gather}
\lim_{r \to 1}\int_{\mathbb{T}} (1-r^2)\|L_{r\lambda}f\|^2_{\mathcal{H}} d\textit{m}(\lambda) \nonumber \\ \leq C \limsup_{r\to 1}\int_{\text{clos}(\mathbb{D})} \int_\mathbb{T} \frac{1-r^2}{|\zeta - r\lambda|^2}|f(\zeta) - f(r\lambda)|^2 d\textit{m}(\lambda) d\mu(\zeta) \label{ineq3}
\end{gather} Since $f$ is continuous and bounded in $\text{clos}(\mathbb{D})$, we readily deduce from standard properties of the Poisson kernel and bounded analytic functions that the functions $$G_r(\zeta) = \int_\mathbb{T} \frac{1-r^2}{|\zeta - r\lambda|^2}|f(\zeta) - f(r\lambda)|^2 d\textit{m}(\lambda)$$ are uniformly bounded in $\text{clos}(\mathbb{D})$ and that $\lim_{r \to 1} G_r(\zeta) = 0$ for all $\zeta \in \text{clos}(\mathbb{D})$. Dominated convergence theorem now implies that the limit in \eqref{ineq3} is 0, and so \eqref{ineq2} holds.
\end{proof}
From part (ii) of \thref{modeltheorem} we easily deduce that the condition on $L$ of \thref{rctheorem} is equivalent to $\Theta H^2(Y_1) = \{0\}$ in \eqref{decomp}. In the case that $\textbf{B} = b$ is a scalar-valued function, this occurs if and only if $b$ is an extreme point of the unit ball of $H^\infty$. Thus an extreme point $b$ which is not an inner function cannot generate a space $\mathcal{H}(b)$ which admits a reverse Carleson measure. We remark also that \thref{rctheorem} holds, with the same proof, even when $\mathcal{H}$ consists of functions taking values in a finite dimensional Hilbert space $X$, where the norm identity then instead reads $\|L\textbf{f}\|^2 = \|\textbf{f}\|^2 - \|\textbf{f}(0)\|^2_X$, and definition of reverse Carleson measure is extended naturally to the vector-valued setting.
\subsection{Formula for the norm in a nearly invariant subspace.} A space $\mathcal{M}$ is \textit{nearly invariant} if whenever $\lambda \in \mathbb{D}$ is not a common zero of the functions in $\mathcal{M}$ and $f(\lambda) = 0$ for some $f \in \mathcal{M}$, then $\frac{f(z)}{z-\lambda} \in \mathcal{M}$. If $\mathcal{H}$ is $M_z$-invariant, then an example of a nearly invariant subspace is any $M_z$-invariant subspace for which $\dim \mathcal{M} \ominus M_z\mathcal{M} = 1$. The concept of a nearly invariant subspace has appeared in \cite{hittanulus}, and have since been used as a tool in solutions to numerous problems in operator theory.
We will now prove a formula for the norm of functions contained in a nearly invariant subspace $\mathcal{M}$ which is similar to \thref{formula} but which is better suited for exploring the structure of $\mathcal{M}$.
\begin{prop} \thlabel{nearinvnorm} Let $\mathcal{M} \subseteq \mathcal{H}$ be a nearly invariant subspace and $k$ be the common order of the zero at $0$ of the functions in $\mathcal{M}$. Let $\phi \in \mathcal{M}$ be the function satisfying $\ip{f}{\phi}_\mathcal{H} = \frac{f^{(k)}(0)}{\phi^{(k)}(0)}$ for all $f \in \mathcal{M}$, $L^\phi: \mathcal{M} \to \mathcal{M}$ be the contractive operator given by $$L^\phi f(z) = \frac{f(z)-\ip{f}{\phi}_\mathcal{H}\phi(z)}{z},$$ and $L^\phi_\lambda = L^\phi(1-\lambda L^\phi)^{-1}, \lambda \in \mathbb{D}$. If the sequence $\{L^n\}_{n=1}^\infty$ converges to zero in the strong operator topology, then \begin{equation}
\|f\|^2_{\mathcal{H}} = \|f/\phi\|^2_2 + \lim_{r \to 1} \int_{\mathbb{T}} \|zL^\phi_{r\lambda} f\|^2_{\mathcal{H}}-\|L^\phi_{r\lambda} f\|^2_{\mathcal{H}} d\textit{m}(\lambda). \label{eq221}
\end{equation}
\end{prop}
\begin{proof}
The fact that $L^\phi$ maps $\mathcal{M}$ into itself follows easily by nearly invariance of $\mathcal{M}$. The formula
\begin{equation}
\|f\|^2_{\mathcal{H}} = \|f/\phi\|^2_2 + \lim_{r \to 1} \int_{\mathbb{T}} \|zL^\phi_{r\lambda} f\|^2_{\mathcal{H}}-r^2\|L^\phi_{r\lambda} f\|^2_{\mathcal{H}} d\textit{m}(\lambda) \label{eq211}
\end{equation} holds in much more general context and without the assumption on convergence of $\{L^n\}_{n=1}^\infty$ to zero. For its derivation we refer to Lemma 2.2 of \cite{alemanrichtersimplyinvariant}, and the discussion succeeding it. The equation \eqref{eq221} will follow from \eqref{eq211} if we can show that the additional assumption on $L$ implies that $$\lim_{r \to 1} \int_\mathbb{T} (1-r^2) \|L^\phi_{r\lambda} f\|^2_{\mathcal{H}} d\textit{m}(\lambda) = 0.$$ To this end, observe that for $\lambda \in \mathbb{D}$ we have $$L^\phi_{\lambda}f (z) = \frac{f(z) - f(\lambda)}{z-\lambda} - \frac{f(\lambda)}{\phi(\lambda)} \frac{\phi(z)- \phi(\lambda)}{z-\lambda} = L_\lambda f(z) - \frac{f(\lambda)}{\phi(\lambda)}L_\lambda \phi(z),$$ and so if $f/\phi \in H^\infty$, then the argument in the proof of part (iii) of \thref{modelformulaconnection} shows that \begin{gather*}
\lim_{r \to 1^-} \int_\mathbb{T} (1-r^2) \|L^\phi_{r\lambda} f\|^2_{\mathcal{H}} d\textit{m}(\lambda) \\ \lesssim \lim_{r \to 1^-} \int_\mathbb{T} (1-r^2) \|L_{r\lambda} f\|^2_{\mathcal{H}} d\textit{m}(\lambda) + \|f/\phi\|^2_\infty \int_\mathbb{T} (1-r^2) \|L_{r\lambda} \phi \|^2_{\mathcal{H}} d\textit{m}(\lambda) = 0. \end{gather*}
Next, the Hilbert space $\tilde{\mathcal{M}} = \mathcal{M}/\phi = \{f/\phi : f \in \mathcal{M}\}$ with the norm $\|f/\phi\|_{\tilde{\mathcal{M}}} = \|f\|_{\mathcal{H}}.$ It is easy to see that $\tilde{\mathcal{M}}$ satisfies (A.1)-(A.3), and thus by \thref{conttheorem} the functions with continuous extensions to $\text{clos}(\mathbb{D})$ form a dense subset of $\tilde{\mathcal{M}}$. Since multiplication by $\phi$ is a unitary map from $\tilde{\mathcal{M}}$ to $\mathcal{M}$, we see that functions $f \in \mathcal{M}$ such that $f/\phi$ has continuous extension to $\text{clos}(\mathbb{D})$ form a dense subset of $\mathcal{M}$. Finally, consider the mapping \[\textbf{Q}f = \limsup_{r \to 1^{-}} \int_\mathbb{T} (1-r^2) \|L^\phi_{r\lambda} f\|^2_{\mathcal{H}} d\textit{m}(\lambda), \quad f \in \mathcal{M}.\] Then $\textbf{Q}(f+g) \leq \textbf{Q}f + \textbf{Q}g$, and we have shown above that $\textbf{Q} \equiv 0$ on a dense subset of $\mathcal{M}$. A peek at \eqref{eq211} reveals that $\textbf{Q}f \leq \|f\|^2_{\mathcal{H}}$, and so $\textbf{Q}$ is continuous on $\mathcal{M}$. Thus $\textbf{Q} \equiv 0$ since it vanishes on a dense subset.
\end{proof}
We will see an application of \thref{nearinvnorm} in the sequel. For now, we show how it can be used to deduce a theorem of Hitt on the structure of nearly invariant subspaces of $H^2$ (see \cite{hittanulus}).
\begin{cor}
If a closed subspace $\mathcal{M} \subset H^2$ is nearly invariant, then it is of the form $\mathcal{M} = \phi K_\theta$, where $K_\theta = H^2 \ominus \theta H^2$ is an $L$-invariant subspace of $H^2$, and we have the norm equality $\|f/\phi\|_2 = \|f\|_2, f \in \mathcal{M}$.
\end{cor}
\begin{proof}
If $\mathcal{M}$ is nearly invariant then the formula \eqref{eq221} gives $\|f/\phi\|_2 = \|f\|$, since $M_z$ is an isometry on $H^2$. Thus $\mathcal{M}/\phi := \{f/\phi : f \in \mathcal{M}\}$ is closed in $H^2$, and it is easy to see that it is $L$-invariant. Thus $\mathcal{M}/\phi = K_\theta$ by Beurling's famous characterization.
\end{proof}
\subsection{Orthocomplements of shift invariant subspaces of the Bergman space.}
The Bergman space $L^2_a(\mathbb{D})$ consists of functions $g(z) = \sum_{k=0} g_kz^k$ analytic in $\mathbb{D}$ which satisfy $$\|g\|^2_{L^2_a(\mathbb{D})} = \int_\mathbb{D} |g(z)|^2 dA(z) = \sum_{k=0}^\infty (k+1)^{-1}|g_k|^2 < \infty,$$ where $dA$ denotes the normalized area measure on $\mathbb{D}$. The Bergman space is invariant for $M_z$ and the lattice of $M_z$-invariant subspaces of $L^2_a(\mathbb{D})$ is well-known to be very complicated (see for example Chapter 8 and 9 of \cite{durenbergmanspaces} and Chapters 6, 7 and 8 of \cite{hedenmalmbergmanspaces}). Before stating and proving our next result, we will motivate it by showing that the orthogonal complements of $M_z$-invariant subspaces of $L^2_a(\mathbb{D})$ can consist entirely of ill-behaved functions. The following result is essentially due to A. Borichev \cite{borichevpriv}.
\begin{prop} \thlabel{badorthocomplements}
There exists a subspace $\mathcal{M} \subsetneq L^2_a(\mathbb{D})$ which is invariant for $M_z$, with the property that for any non-zero function $g \in \mathcal{M}^\perp$ and any $\delta > 0$ we have $$\int_{\mathbb{D}} |g|^{2+\delta} dA = \infty.$$
\end{prop}
\begin{proof} The argument is based on the existence of a function $f$ such that $f, 1/f \in L^2_a(\mathbb{D})$, yet if $\mathcal{M}$ is the smallest $M_z$-invariant subspace containing $f$, then $\mathcal{M} \neq L^2_a(\mathbb{D})$. Such a function exists by \cite{borichevhedenmalmcyclicity}. Take $g \in \mathcal{M}^\perp$ and assume that $\int_\mathbb{D} |g|^{2+\delta} dA < \infty$ for some $\delta > 0$. Since $\mathcal{M}$ is $M_z$-invariant, for any polynomial $p$ we have that \begin{equation} \int_\mathbb{D} pf\conj{g} dA = 0.\label{eq37} \end{equation} By the assumption on $g$ and H\"older's inequality we have that $f\conj{g} \in L^r(\mathbb{D})$ for $r > 1$ sufficiently close to 1. Let $s = \frac{r}{r-1}$ be the H\"older conjugate index of $r$. Since $1/f \in L^2_a(\mathbb{D})$, the analytic function $f^{-\epsilon}$ is in the Bergman space $L^s_a(\mathbb{D})$ for sufficiently small $\epsilon > 0$, and hence can be approximated in the norm of $L^s_a(\mathbb{D})$ by a sequence $\{p_n\}_{n=1}^\infty$ of polynomials. If $p$ is any polynomial, then \begin{gather*}\lim_{n\to \infty} \int_\mathbb{D} |(p_n-f^{-\epsilon})pfg| dA \\ \lesssim \lim_{n\to \infty} \|p_n-f^{-\epsilon}\|_{L^s(\mathbb{D})}\|pfg\|_{L^r(\mathbb{D})} = 0.\end{gather*} By choosing $\epsilon = 1/\mathcal{M}$ for sufficiently large positive integer $\mathcal{M}$, we see from the above that for any polynomial $p$ we have $$\int_\mathbb{D} pf^{1-1/\mathcal{M}}\conj{g}dA = 0,$$ which is precisely \eqref{eq37} with $\conj{g}$ replaced by $f^{-1/\mathcal{M}}\conj{g}$, and of course we still have $ff^{-1/\mathcal{M}}\conj{g} \in L^r(\mathbb{D})$ for the same choice of $r > 1$. Repeating the argument gives $$\int_\mathbb{D} pf^{1-2/\mathcal{M}}\conj{g} dA = 0,$$ and after $\mathcal{M}$ repetitions of the argument we arrive at $$\int_\mathbb{D} p\conj{g} dA(z) = 0$$ for any polynomial $p$. Then $g = 0$ by density of polynomials in $L^2_a(\mathbb{D})$.
\end{proof}
Despite the rather dramatic situation described by \thref{badorthocomplements}, an application of \thref{conttheoremcor} yields the following result.
\begin{cor} \thlabel{contderivdense}
Let $\mathcal{M}$ be a subspace of $L^2_a(\mathbb{D})$ which is invariant for $M_z$. Then the functions in the orthocomplement $\mathcal{M}^\perp$ which are derivatives of functions in the disk algebra $\mathcal{A}$ are dense in $\mathcal{M}^\perp$.
\end{cor}
\begin{proof} The operator $U$ given by $$Uf(z) = \frac{1}{z}\int_0^z f(w) \,dw$$ is a unitary map between $L^2_a(\mathbb{D})$ and the classical Dirichlet space $\mathcal{D}$, and \thref{conttheoremcor} applies to the latter space. A computation involving the Taylor series coefficients shows that $UM_z^*U^* = L$, and so if $\mathcal{M} \subset L^2_a(\mathbb{D})$ is $M_z$-invariant, then $U\mathcal{M}^\perp \subset \mathcal{D}$ is $L$-invariant. Thus from \thref{conttheoremcor} we infer that the set of functions $f \in \mathcal{M}^\perp$ for which $Uf(z)$ is in the disk algebra $\mathcal{A}$ is dense in $\mathcal{M}^\perp$, and for any function $f$ in this set we have $f(z) = (zUf(z))'$, so $f$ is the derivative of a function in $\mathcal{A}$.
\end{proof}
\section{Finite rank $\mathcal{H}[\textbf{B}]$-spaces}
\subsection{Finite rank spaces.}
The applicability of \thref{modeltheorem} depends highly on the ability to identify the spaces $W$ and $\Theta H^2(Y_1)$ in \eqref{decomp}. In general, $\Delta(\zeta)$ defined by \eqref{delta} is taking values in the algebra of operators on an infinite dimensional Hilbert space. The situation is much more tractable in the case of finite rank $\mathcal{H}[\textbf{B}]$-spaces, for then $\Delta(\zeta)$ acts on a finite dimensional space. In this last section we restrict ourselves to the study of the finite rank case. Thus, we study spaces of the form $\mathcal{H} = \mathcal{H}[\textbf{B}]$ with $\textbf{B} = (b_1, \ldots, b_n)$, where $n$ is the rank of $\mathcal{H}[\textbf{B}]$ was defined in the introduction. It follows that $b_1, \ldots, b_n$ are linearly independent. For convenience, we will also be assuming that $\textbf{B} \neq 0,1$, which correspond to the cases $\mathcal{H}[\textbf{B}] = H^2$ and $\mathcal{H}(\textbf{B} = \{0\}$, respectively.
\subsection{$M_z$-invariance and consequences.} It turns out that the decomposition \eqref{decomp} is particularly simple in the case when $\mathcal{H}[\textbf{B}]$ is invariant for $M_z$.
\begin{prop} \thlabel{noWlemma}
If the finite rank $\mathcal{H}[\textbf{B}]$-space is invariant for $M_z$, then $W = \{0\}$ in the decomposition \eqref{decomp}, and thus \thref{modeltheoremanalytic} applies, with $\textbf{A}$ a square matrix and $\det \textbf{A} \neq 0$.
\end{prop}
\begin{proof}
In the notation of \thref{modeltheorem} we have $Y \simeq \mathbb{C}^n$, thus \eqref{decomp} becomes \begin{equation}
\conj{\Delta H^2(\mathbb{C}^n)} = W \oplus \Theta H^2(\mathbb{C}^m), \label{eq53}
\end{equation} where $\quad m \leq n$. We will see that $W = \{0\}$ by showing something stronger, namely that $m = n$ in \eqref{eq53}. To this end, let $f \in \mathcal{H}[\textbf{B}]$ and consider $Jf = (f,\textbf{g})$ and $JM_zf = (M_zf, \textbf{g}_0)$, $J$ being the embedding given by \thref{modeltheorem}. Let $\textbf{g} = \textbf{w} + \Theta \textbf{h}$ and $\textbf{g}_0 = \textbf{w}_0 + \Theta \textbf{h}_0$ be the decompositions with respect to \eqref{eq53}. Since $LM_zf = f$, we see from part (ii) of \thref{modeltheorem} that $\textbf{w}_0(\zeta) = \zeta\textbf{w}(\zeta)$ and $\textbf{h}_0(\zeta) = \zeta\textbf{h}(\zeta) + \textbf{c}_f$, where $\textbf{c}_f \in \mathbb{C}^m$. By part (i) of \thref{modeltheorem} we have that \begin{gather} \textbf{B}(\zeta)^*\zeta\textbf{f}(\zeta) + \Delta(\zeta) \textbf{g}_0(\zeta) \nonumber \\ = \zeta\Big(\textbf{B}(\zeta)^*\textbf{f}(\zeta) + \Delta(\zeta)\textbf{g}(\zeta)\Big) + \textbf{A}(\zeta)^*\textbf{c}_f \in \conj{H^2_0(\mathbb{C}^n)} \label{eq41}, \end{gather} where, as before, $\textbf{A} = \Theta^*\Delta$. We apply this to the reproducing kernel $k_\lambda(z)$ of $\mathcal{H}[\textbf{B}]$. Recall from Section \ref{prelimsec} that \begin{equation*}
Jk_\lambda = \Bigg( \frac{1 - \textbf{B}(z)\textbf{B}(\lambda)^*}{1-\conj{\lambda}z}, \frac{-\Delta(\zeta)\textbf{B}(\lambda)^*}{1-\conj{\lambda}z}\Bigg) = (k_\lambda, \textbf{g}_\lambda).
\end{equation*} A brief computations shows that $$\textbf{B}^*(\zeta)k_\lambda(\zeta) + \Delta(\zeta)\textbf{g}_\lambda(\zeta) = \conj{\zeta}\frac{\textbf{B}(\zeta)^* - \textbf{B}(\lambda)^*}{\conj{\zeta}-\lambda}.$$ Using \eqref{eq41} we deduce that for each $\lambda \in \mathbb{D}$ there exists $\textbf{c}_\lambda \in \mathbb{C}^m$ such that $$\frac{\textbf{B}(\zeta)^* - \textbf{B}(\lambda)^*}{\conj{\zeta}-\lambda} + \textbf{A}(\zeta)^*\textbf{c}_\lambda \in \conj{H^2_0(\mathbb{C}^n)}.$$ Since $\textbf{B}(0) = 0$ we see that the constant term of the above function equals $$0 = -\textbf{B}(\lambda)^*/\lambda + \textbf{A}(0)^*\textbf{c}_\lambda.$$ By linear independence of the coordinates $\{b_i\}_{i=1}^n$ we conclude that $\textbf{A}(0)^*$ maps an $m$-dimensional vector space onto an $n$-dimensional vector space. Thus it follows that $m = n$, and hence $W = \{0\}$ in \eqref{eq53}. Since $\textbf{A}$ is outer, invertibility of $\textbf{A}(0)$ implies that $\det \textbf{A} \neq 0$, and thus the function $\det \textbf{A}$ is non-zero and outer.
\end{proof}
The following result characterizes $M_z$-invariance in terms of modulus of $\textbf{B}$, and is a generalization of a well-known theorem for $\mathcal{H}(b)$-spaces.
\begin{thm} \thlabel{shiftinvcriterion}
A finite rank $\mathcal{H}[\textbf{B}]$-space is invariant under the forward shift operator $M_z$ if and only if $$\int_\mathbb{T} \log(1-\|\textbf{B}\|^2_2) d\textit{m} = \int_\mathbb{T} \log(1-\sum_{i=1}^n |b_i|^2) d\textit{m} > -\infty.$$
\end{thm}
\begin{proof} Assume that $\mathcal{H}[\textbf{B}]$ is $M_z$-invariant. Then $\det \textbf{A}(z)$ is non-zero by \thref{noWlemma}. In terms of boundary values on $\mathbb{T}$ we have $\textbf{A} = \Theta^* \Delta = \Theta^*(I_n - \textbf{B}^*\textbf{B})^{1/2}$, where $I_n$ is the $n$-by-$n$ identity matrix, and so $\det \textbf{A} = \conj{\det \Theta}(1-\sum_{i=1}^n |b_i|^2)^{1/2}$ on $\mathbb{T}$. Since $\Theta$ is an isometry we have that $|\det \Theta| = 1$, and thus $$\int_\mathbb{T} \log(1-\sum_{i=1}^n |b_i|^2) d\textit{m} = \int_\mathbb{T} \log(|\det \textbf{A}|) d\textit{m}> -\infty,$$ last inequality being a well-known fact for bounded analytic functions.
Conversely, assume that $\int_\mathbb{T} \log(1-\sum_{i=1}^n |b_i|^2) d\textit{m} > -\infty$. Thus $1-\sum_{i=1}^n |b_i|^2 = \det \Delta > 0$ almost everywhere on $\mathbb{T}$. Consider again the decomposition in \eqref{eq53}. We claim that $W = \{0\}$. Assume, seeking a contradiction, that $W \neq \{0\}$. $W$ is invariant under multiplication by scalar-valued bounded measurable functions, and so $W$ contains a function $\textbf{g}$ which is non-zero but vanishes on a set of positive measure. Fix $\textbf{h}_n \in H^2(\mathbb{C}^n)$ such that $\Delta \textbf{h}_n \to \textbf{g}$. Let $\text{adj} (\Delta) := (\det \Delta)\Delta^{-1}$ be the adjugate matrix. Then $\text{adj} (\Delta)$ has bounded entries and thus \begin{equation}\det \Delta \textbf{h}_n = \text{adj} (\Delta) \Delta \textbf{h}_n \to \text{adj} (\Delta) \textbf{g},\label{eq54}\end{equation} in $L^2(\mathbb{C}^n)$. By our assumption, there exists an analytic outer function $d$ with $|d| = |\det \Delta|= (1-\sum_{i=1}^n |b_i|^2)^{1/2}$ almost everywhere on $\mathbb{T}$. Then $\det \Delta = \psi d$ for some measurable function $\psi$ of modulus $1$ almost everywhere on $\mathbb{T}$, and $\det \Delta \textbf{h}_n = \psi d\textbf{h}_n \in \psi H^2(\mathbb{C}^n),$ where $\psi H^2(\mathbb{C}^n)$ is norm-closed and contains no non-zero function which vanishes on a set of positive measure on $\mathbb{T}$. But by \eqref{eq54} we have $\text{adj} (\Delta) \textbf{g} \in \psi H^2(\mathbb{C}^n)$, which is a contradiction. Thus $W = \{0\}$ in \eqref{eq53}, so that $\conj{\Delta H^2(\mathbb{C}^n)} = \Theta H^2(\mathbb{C}^m)$ and \thref{modeltheoremanalytic} applies. We have $1-\sum_{i=1}^n |b_i(\zeta)|^2 > 0$ almost everywhere on $\mathbb{T}$, and therefore $\Delta (\zeta)$ is invertible almost everywhere on $\mathbb{T}$. This implies that $m = n$. Since $\textbf{A}(\zeta) = \Theta(\zeta)^* \Delta(\zeta)$, we see that $\textbf{A}(z)$ is an $n$-by-$n$ matrix-valued outer function. In particular, $\textbf{A}(z)$ is invertible at every $z \in \mathbb{D}$. Let $J$ be the embedding of \thref{modeltheoremanalytic} and $Jf = (f, \textbf{f}_1)$, where $f \in \mathcal{H}[\textbf{B}]$ is arbitrary. We claim that we can find a vector $\textbf{c}_f \in \mathbb{C}^n$ such that \begin{equation}\textbf{B}(\zeta)^*\zeta f(\zeta) + \textbf{A}(\zeta)^* (\zeta\textbf{f}_1(\zeta) + \textbf{c}_f) \in \conj{H^2_0(\mathbb{C}^n)}.\label{eq554} \end{equation} Indeed, by (i) of \thref{modeltheoremanalytic} we have that $\textbf{B}(\zeta)^*f + \textbf{A}(\zeta)^*\textbf{f}_1(\zeta) \in \conj{H^2_0(\mathbb{C}^n)}.$ Let $\textbf{v}$ be the zeroth-order coefficient of the coanalytic function $\zeta \textbf{B}(\zeta)^*f + \zeta \textbf{A}(\zeta)^*\textbf{f}_1(\zeta)$. It then suffices to take $\textbf{c}_f = (\textbf{A}(0)^*)^{-1}\textbf{v}$ for \eqref{eq554} to hold. Thus by (i) of \thref{modeltheoremanalytic} we have that $zf(z) \in \mathcal{H}[\textbf{B}]$, which completes the proof.
\end{proof}
From now on we will be working exclusively with $M_z$-invariant $\mathcal{H}[\textbf{B}]$-spaces of finite rank, and $J$ will always denote the embedding of $\mathcal{H}[\textbf{B}]$-space that appears in \thref{modeltheoremanalytic}.
\begin{lemma} \thlabel{shiftspecrad}
If a finite rank $\mathcal{H}[\textbf{B}]$-space is $M_z$-invariant, then the spectral radius of the operator $M_z$ equals 1.
\end{lemma}
\begin{proof} We have seen above that if $Jf = (f, \textbf{f}_1)$, then $JM_zf = (zf(z), z\textbf{f}_1(z) + \textbf{c}_f),$ where $\textbf{c}_f \in \mathbb{C}^n$ is some vector depending on $f$. This shows that $M_z$ is unitarily equivalent to a finite rank perturbation of the isometric shift operator $(g, \textbf{g}_1) \mapsto (zg, z\textbf{g}_1)$ acting on $H^2 \oplus H^2(\mathbb{C}^n)$. It follows that the essential spectra of the two operators coincide, and so are contained in $\text{clos}(\mathbb{D})$. Thus for $|\lambda| > 1$ the operator $M_z - \lambda$ is Fredholm of index $0$. Since $M_z - \lambda$ is injective, index $0$ implies that that $M_z - \lambda$ is invertible.
\end{proof}
A consequence of \thref{shiftspecrad} is that the operators $(I_{\mathcal{H}[\textbf{B}]} - \conj{\lambda} M_z)^{-1}$ exist for $|\lambda| < 1$. Their action is given by $$(I_{\mathcal{H}[\textbf{B}]} - \conj{\lambda} M_z)^{-1}f(z) = \frac{f(z)}{1-\conj{\lambda}z}.$$ Let $$J\frac{f(z)}{1-\conj{\lambda}z} = \Big(\frac{f(z)}{1-\conj{\lambda}z}, \textbf{g}_{\lambda}(z)\Big)$$ for some $\textbf{g}_\lambda \in H^2(\mathbb{C}^n)$. We compute \begin{gather*}Jf = J(I_{\mathcal{H}[\textbf{B}]}-\conj{\lambda}M_z)(I_{\mathcal{H}[\textbf{B}]}-\conj{\lambda}M_z)^{-1}f = \Big( f, (1-\conj{\lambda}z)\textbf{g}_\lambda(z) + \textbf{c}_f(\lambda) \Big), \end{gather*} where $\textbf{c}_f(\lambda) \in \mathbb{C}^n$ is some vector depending on $f$ and $\lambda$. By re-arranging we obtain $\textbf{g}_\lambda(z) = \frac{\textbf{f}_1(z) - \textbf{c}_f(\lambda)}{1-\conj{\lambda}z},$
and so \begin{equation}
J\frac{f(z)}{1-\conj{\lambda}z} = \Big(\frac{f(z)}{1-\conj{\lambda}z}, \frac{\textbf{f}_1(z) - \textbf{c}_f (\lambda)}{1-\conj{\lambda}z}\Big). \label{eq43}
\end{equation} We will now show that $\textbf{c}_f(\lambda)$ is actually a coanalytic function of $\lambda$ which admits non-tangential boundary values almost everywhere on $\mathbb{T}$. To this end, let $\textbf{u}_f(z) \in \conj{H^2_0(\mathbb{C}^n)}$ be the coanalytic function with boundary values $$\textbf{u}_f(\zeta) = \textbf{B}(\zeta)^*f(\zeta) + \textbf{A}(\zeta)^*\textbf{f}_1(\zeta).$$ By (i) of \thref{modeltheoremanalytic} and \eqref{eq43} we have
\begin{equation}
\frac{\textbf{B}(\zeta)^*f(\zeta) + \textbf{A}(\zeta)^*\textbf{f}_1(\zeta)}{1 - \conj{\lambda}\zeta} - \frac{\textbf{A}(\zeta)^*\textbf{c}_f(\lambda)}{1-\conj{\lambda}\zeta} = \frac{\textbf{u}_f(\zeta)}{1 - \conj{\lambda}\zeta} - \frac{\textbf{A}(\zeta)^*\textbf{c}_f(\lambda)}{1-\conj{\lambda}\zeta}\in \conj{H^2_0(\mathbb{C}^n)}, \label{eq44}
\end{equation} and projecting this equation onto $H^2(\mathbb{C}^n)$ we obtain \begin{equation} \textbf{u}_f(\lambda) = A(\lambda)^*\textbf{c}_f(\lambda).
\label{eq45}
\end{equation} Since $\textbf{A}(\lambda)^*$ is coanalytic and invertible for every $\lambda \in \mathbb{D}$, we get that $$\textbf{c}_f(\lambda) = (\textbf{A}(\lambda)^*)^{-1}\textbf{u}_f(\lambda) = \frac{\text{adj}(\textbf{A}(\lambda)^*)\textbf{u}_f(\lambda)}{\conj{\det \textbf{A}(\lambda)}}$$ is coanalytic. Moreover, the last expression shows that $\textbf{c}_f(\lambda)$ is in the coanalytic Smirnov class $\conj{N^+(\mathbb{C}^n)}$, and thus admits non-tangential limits almost everywhere on $\mathbb{T}$. For the boundary function we have the equality \begin{gather}
\textbf{c}_f(\zeta) = (\textbf{A}(\zeta)^*)^{-1}\textbf{u}_f(\zeta) = (\textbf{A}(\zeta)^*)^{-1}\textbf{B}(\zeta)^*f(\zeta) + \textbf{f}_1(\zeta) \label{eq51}
\end{gather} almost everywhere on $\mathbb{T}$. The following proposition summarizes the discussion above.
\begin{prop} \thlabel{cfprop}
Let $\mathcal{H}[\textbf{B}]$ be of finite rank and $M_z$-invariant. If $f \in \mathcal{H}[\textbf{B}]$, then for all $\lambda \in \mathbb{D}$ we have $\frac{f(z)}{1-\conj{\lambda}z} \in \mathcal{H}[\textbf{B}]$ and $$J\frac{f(z)}{1-\conj{\lambda}z} = \Big(\frac{f(z)}{1-\conj{\lambda}z}, \frac{\textbf{f}_1(z) - \textbf{c}_f (\lambda)}{1-\conj{\lambda}z}\Big),$$ where $\textbf{c}_f$ is a coanalytic function of $\lambda$ which admits non-tangential boundary values almost everywhere on $\mathbb{T}$, and \eqref{eq51} holds.
\end{prop}
\subsection{Density of polynomials.} Since we are assuming that $1 \in \mathcal{H}[\textbf{B}]$, the $M_z$-invariance implies that the polynomials are contained in the space.
\begin{thm}\thlabel{polydense}
If finite rank $\mathcal{H}[\textbf{B}]$ is $M_z$-invariant, then the polynomials are dense in $\mathcal{H}$.
\end{thm}
\begin{proof} Since $J1 = (1, 0)$, an application of \thref{cfprop} and \eqref{eq51} shows that
\begin{equation} J\frac{1}{1-\conj{\lambda}z} = \Big(\frac{1}{1-\conj{\lambda}z}, -\frac{(\textbf{A}(\lambda)^*)^{-1}\textbf{B}(\lambda)^*1}{1-\conj{\lambda}z}\Big), \quad \lambda \in \mathbb{D} \label{eq55}
\end{equation} Assume that $f \in \mathcal{H}[\textbf{B}]$ is orthogonal to the polynomials. Then it follows that $Jf = (f, \textbf{f}_1)$ is orthogonal in $H^2 \oplus H^2(\mathbb{C}^n)$ to tuples of the form given by \eqref{eq55}. Thus
\begin{gather*}
0 = f(\lambda) - \ip{\textbf{f}_1(\lambda)}{(\textbf{A}(\lambda)^*)^{-1}\textbf{B}(\lambda)^*1}_{\mathbb{C}^n} \\ = f(\lambda) - \textbf{B}(\lambda) (\textbf{A}(\lambda))^{-1}\textbf{f}_1(\lambda).
\end{gather*} Set $$\textbf{h}(z) = (\textbf{A}(z))^{-1}\textbf{f}_1(z) = \frac{\text{adj}(\textbf{A}(z))\textbf{f}_1(z)}{\det \textbf{A}(z)}.$$ The last expression shows that $\textbf{h}$ belongs to the Smirnov class $N^+(\mathbb{C}^n)$. By the Smirnov maximum principle we furthermore have $\textbf{h} \in H^2(\mathbb{C}^n)$, since \begin{gather*}\int_{\mathbb{T}} \|\textbf{h}\|^2_{\mathbb{C}^n} d\textit{m}(\zeta) = \int_{\mathbb{T}} \|\textbf{B}\textbf{h}\|^2_{\mathbb{C}^n} d\textit{m} + \int_{\mathbb{T}} \|\textbf{A}\textbf{h}\|^2_{\mathbb{C}^n} d\textit{m} \\ = \int_{\mathbb{T}} |f|^2 d\textit{m} + \int_{\mathbb{T}} \|\textbf{f}_1\|^2_{\mathbb{C}^n} d\textit{m} = \|f\|^2_{\mathcal{H}[\textbf{B}]} < \infty.\end{gather*} Hence $Jf = (f,\textbf{f}_1) = (\textbf{B}\textbf{h}, \textbf{A} \textbf{h})$ with $\textbf{h} \in H^2(\mathbb{C}^n)$, so $Jf \in (J\mathcal{H}[\textbf{B}])^\perp$ and it follows that $f = 0$.
\end{proof}
\subsection{Equivalent norms.} \label{isomorphismsubsec} In the finite rank case we meet a natural but fundamental question whether the rank of an $\mathcal{H}[\textbf{B}]$-space can be reduced with help of an equivalent norm. Here we assume that the space satisfies (A.1)-(A.3) with respect to the new norm. If that is the case, then the renormed space is itself an $\mathcal{H}[\textbf{D}]$-space for some function $\textbf{D}$. We will say that the spaces $\mathcal{H}[\textbf{B}]$ and $\mathcal{H}[\textbf{D}]$ are \textit{equivalent} if they are equal as sets and the two induced norms are equivalent. As mentioned in the introduction, \cite{ransforddbrdirichlet} shows that in the case $\mathcal{H}[\textbf{B}] = \mathcal{D}(\mu)$ for $\mu$ a finite set of point masses the space is equivalent to a rank one $\mathcal{H}(b)$-space.
It is relatively easy to construct a $\textbf{B} = (b_1, b_2)$ such that $\mathcal{H}[\textbf{B}] = \mathcal{H}(b) \cap K_\theta$, where $b$ is non-extreme and $K_\theta = H^2 \ominus \theta H^2$. For appropriate choices of $b$ and $\theta$, the space $\mathcal{H}(b) \cap K_\theta$ cannot be equivalent to a de Branges-Rovnyak space. We shall not go into details of the construction, but we mention that it can be deduced from the proof of \thref{Linvsubspaces}. However, it is less obvious how to obtain an example of an $\mathcal{H}[\textbf{B}]$-space which is $M_z$-invariant and which is not equivalent to a non-extreme de Branges-Rovnyak space. The purpose of this section is to verify existence of such a space, and thus confirm that the class of $M_z$-invariant $\mathcal{H}[\textbf{B}]$-spaces is indeed a non-trivial extension of the $\mathcal{H}(b)$-spaces constructed from non-extreme $b$. The result that we shall prove is the following.
\begin{thm} \thlabel{nonsimilaritytheorem} For each integer $n \geq 1$ there exists $\textbf{B} = (b_1, \ldots, b_n)$ such that $\mathcal{H}[\textbf{B}]$ is $M_z$-invariant and has the following property: if $\mathcal{H}[\textbf{B}]$ is equivalent to a space $\mathcal{H}[\textbf{D}]$ with $\textbf{D} = (d_1, \ldots, d_m)$, then $m \geq n$.
\end{thm}
Our first result in the direction of the proof of \thref{nonsimilaritytheorem} is interesting in its own right and gives a criterion for when two spaces $\mathcal{H}[\textbf{B}]$ and $\mathcal{H}[\textbf{D}]$ are equivalent. Below, $\conj{H^\infty}$ and $\conj{N^+}$ will denote the spaces of complex conjugates of functions in $H^\infty$ and $N^+$, respectively.
\begin{thm} \thlabel{modulegen}
Let $\mathcal{H}[\textbf{B}]$ with $\textbf{B} = (b_1, \ldots, b_n)$ and $\mathcal{H}[\textbf{D}]$ with $\textbf{D} = (d_1, \ldots, d_m)$ be two spaces with embeddings $J_1: \mathcal{H}[\textbf{B}] \to H^2 \oplus H^2(\mathbb{C}^n)$ and $J_2: \mathcal{H}[\textbf{D}] \to H^2 \oplus H^2(\mathbb{C}^m)$ as in \thref{modeltheoremanalytic}, such that \begin{gather*}
J_1\frac{1}{1-\conj{\lambda}z} = \frac{(1,\textbf{c}(\lambda))}{1-\conj{\lambda}z} = \frac{(1, c_1(\lambda), \ldots, c_n(\lambda))}{1-\conj{\lambda}z}, \\ J_2\frac{1}{1-\conj{\lambda}z} = \frac{(1,\textbf{e}(\lambda))}{1-\conj{\lambda}z} = \frac{(1, e_1(\lambda), \ldots, e_m(\lambda))}{1-\conj{\lambda}z}.
\end{gather*} Then the spaces $\mathcal{H}[\textbf{B}]$ and $\mathcal{H}[\textbf{D}]$ are equivalent if and only if the $\conj{H^\infty}$-submodules of $\conj{N^+}$ generated by $\{1, c_1, \ldots, c_n\}$ and by $\{1, e_1, \ldots, e_m\}$ coincide.
\end{thm}
\begin{proof}
Assume that $\mathcal{H}[\textbf{B}]$ and $\mathcal{H}[\textbf{D}]$ are equivalent. Let $i: \mathcal{H}[\textbf{B}] \to \mathcal{H}[\textbf{D}]$ be the identity mapping $if = f$. The subspaces $K_1 = J_1\mathcal{H}[\textbf{B}]$ and $K_2 = J_2\mathcal{H}[\textbf{D}]$ are invariant under the backward shift $L$ by \thref{modeltheoremanalytic}, and if $T = J_2iJ_1^{-1} : K_1 \to K_2$, then $LT = TL$. The commutant lifting theorem (see Theorem 10.29 and Exercise 10.31 of \cite{mccarthypick}) implies that $T$ extends to a Toeplitz operator $T_{\boldsymbol{\Phi}}$, where $\boldsymbol{\Phi}$ is an $(m+1)$-by-$(n+1)$ matrix of bounded coanalytic functions, acting by matrix multiplication followed by the component-wise projection $P_+$ from $L^2$ onto $H^2$: \[ T_{\boldsymbol{\Phi}} (f, f_1, \ldots, f_n)^t = P_+\boldsymbol{\Phi}(f, f_1, \ldots, f_n)^t.\] Thus we have \begin{equation} \frac{(1,\textbf{e}(\lambda))^t}{1-\conj{\lambda}z} = T_{\boldsymbol{\Phi}}\frac{(1,\textbf{c}(\lambda))^t}{1-\conj{\lambda}z} = \frac{\boldsymbol{\Phi}(\lambda)(1, \textbf{c}(\lambda))^t}{1-\conj{\lambda}z} \label{eq600}. \end{equation} By reversing the roles of $\mathcal{H}[\textbf{B}]$ and $\mathcal{H}[\textbf{D}]$, we obtain again a coanalytic Toeplitz operator $T_{\boldsymbol{\Psi}}$ such that \begin{equation}\frac{(1,\textbf{c}(\lambda))^t}{1-\conj{\lambda}z} = T_{\boldsymbol{\Psi}}\frac{(1,\textbf{e}(\lambda))^t}{1-\conj{\lambda}z} = \frac{\boldsymbol{\Psi}(\lambda)(1, \textbf{e}(\lambda))^t}{1-\conj{\lambda}z}. \label{eq601} \end{equation} The equations \eqref{eq600} and \eqref{eq601} show that the $\conj{H^\infty}$-submodule of $\conj{N^+}$ generated by $\{1, c_1, \ldots, c_n\}$ coincides with the $\conj{H^\infty}$-submodule generated by $\{1, e_1, \ldots, e_m\}$.
Conversely, assume that the $\conj{H^\infty}$-submodules of $\conj{N^+}$ generated by $\{1, c_1, \ldots, c_n\}$ and $\{1, e_1, \ldots, e_m\}$ coincide. Then there exists a matrix of coanalytic bounded functions $\boldsymbol{\Phi}$ such that \[\boldsymbol{\Phi}(1,c_1, \ldots, c_n)^t = (1, e_1, \ldots, e_m)^t\] where we can choose the top row of $\boldsymbol{\Phi}$ to equal $(1,0,\ldots, 0)$, and there exists also a matrix of coanalytic bounded functions $\boldsymbol{\Psi}$ such that \[\boldsymbol{\Psi}(1,e_1, \ldots, e_n)^t = (1, c_1, \ldots, c_n)^t\] with top row $(1, 0, \ldots, 0)$. It is now easy to see by density of the elements $\frac{1}{1-\conj{\lambda}z}$ in the spaces $\mathcal{H}[\textbf{B}]$ and $\mathcal{H}[\textbf{D}]$ (which can be deduced from \thref{polydense} and \thref{shiftspecrad}) that, in the notation of the above paragraph, we have $T_{\boldsymbol{\Phi}}K_1 = K_2$, $T_{\boldsymbol{\Psi}}K_2 = K_1$, $T_{\boldsymbol{\Psi}}T_{\boldsymbol{\Phi}} = I_{K_1}$, $T_{\boldsymbol{\Phi}}T_{\boldsymbol{\Psi}} = I_{K_2}$ and thus $\mathcal{H}[\textbf{B}]$ and $\mathcal{H}[\textbf{D}]$ are equivalent.
\end{proof}
\begin{lemma} \thlabel{clemma} Let $\textbf{c} = (c_1, \ldots, c_n)$ be an arbitrary $n$-tuple of functions in $\conj{N^+}$. There exists a $\textbf{B} = (b_1, \ldots, b_n)$ such that $\mathcal{H}[\textbf{B}]$ is $M_z$-invariant and an embedding $J: \mathcal{H}[\textbf{B}] \to H^2 \oplus H^2(\mathbb{C}^n)$ as in \thref{modeltheoremanalytic} such that \[ J\frac{1}{1-\conj{\lambda}z} = \Bigg( \frac{1}{1-\conj{\lambda}z}, \frac{\textbf{c}(\lambda)}{1-\conj{\lambda}z} \Bigg). \]
\end{lemma}
\begin{proof}
Since $\conj{c_i} \in N^+$, there exists a factorization $\conj{c_i} = d_i/u_i$, where $d_i,u_i \in H^\infty$, and $u_i$ is outer. Let $\textbf{D} = (d_1, \ldots, d_n)$ and $\textbf{U} = \text{diag}(u_1, \ldots, u_n)$. The linear manifold \[ V = \{(\textbf{D}\mathbf{h}, \textbf{U}\mathbf{h}) : \mathbf{h} \in H^2(\mathbb{C}^n)\} \subset H^2 \oplus H^2(\mathbb{C}^n)\] is invariant under the forward shift $M_z$. It follows from general theory of shifts and the Beurling-Lax theorem (see Chapter 1 of \cite{rosenblumrovnyakhardyclasses}) that $M_z$ acting on $\text{clos}(V)$ is a shift of multiplicity $n$, and \[\text{clos}(V) = \{(\textbf{B}\mathbf{h}, \textbf{A}\mathbf{h}) : \mathbf{h} \in H^2(\mathbb{C}^n)\} \] for some analytic $\textbf{B}(z) = (b_1(z), \ldots, b_n(z))$ and $n$-by-$n$ matrix-valued analytic $\textbf{A}(z)$ such that the mapping $\textbf{h} \mapsto (\textbf{B}\textbf{h}, \textbf{A}\textbf{h})$ is an isometry from $H^2(\mathbb{C}^n)$ to $H^2 \oplus H^2(\mathbb{C}^n)$. It is easy to see that $\textbf{A}$ must be outer, since $\textbf{U}$ is, and that $\sum_{i=1}^n |b_i(z)|^2 \leq 1$ for all $z \in \mathbb{D}$. If $P$ is the projection from $H^2 \oplus H^2(\mathbb{C}^n)$ onto the first coordinate $H^2$, then as in the proof of \thref{modeltheorem} we set $\mathcal{H}[\textbf{B}]$ to be the image of $\text{clos}(V)^\perp$ under $P$, $\|f\|_{\mathcal{H}(\textbf{B}} = \|P^{-1}f\|_{H^2 \oplus H^2(\mathbb{C}^n)}$ and $J = P^{-1}$. The tuple $( \frac{1}{1-\conj{\lambda}z}, \frac{\textbf{c}(\lambda)}{1-\conj{\lambda}z} \Big)$ is obviously orthogonal to $V$, so $J\frac{1}{1-\conj{\lambda}z} = \Big( \frac{1}{1-\conj{\lambda}z}, \frac{\textbf{c}(\lambda)}{1-\conj{\lambda}z} \Big)$. The forward shift invariance of $\mathcal{H}[\textbf{B}]$ can be seen using the same argument as in the end of the proof of \thref{shiftinvcriterion}.
\end{proof}
The next lemma is a version of a result of R. Mortini, see Lemma 2.8 and Theorem 2.9 of \cite{mortini}. The proofs in \cite{mortini} can be readily adapted to prove our version, we include a proof sketch for the convenience of the reader.
\begin{lemma} \thlabel{mingenideal}
For each $n \geq 1$ there exists an outer function $u \in H^\infty$ and $f_1, \ldots, f_n \in H^\infty$ such that the ideal \[ \Big\{ g_0u + \sum_{i=1}^n g_if_i : g_0, g_1, \ldots, g_n \in H^\infty \Big\} \] cannot be generated by less than $n+1$ functions in $H^\infty$.
\end{lemma}
\begin{proof} Let $\mathcal{M}$ be the maximal ideal space of $H^\infty$. The elements of $H^\infty$ are naturally functions on $\mathcal{M}$, and if $\xi \in \mathcal{M}$, then the evaluation of $f \in H^\infty$ at $\xi$ will be denoted by $f(\xi)$. Let $u$ be a bounded outer function and $I$ be an inner function such that $(u,I)$ is not a corona pair, so that there exists $\xi \in M$ such that $u(\xi) = I(\xi) = 0$. Let $f_k = I^k u^{n-k}$. We claim that the ideal generated by $\{ u^n, f_1, \ldots, f_n \}$ cannot be generated by less than $n+1$ functions.
The proof is split into two parts. In the first part we apply the idea contained in Lemma 2.8 of \cite{mortini} to verify the following claim: if $\phi_0, \phi_1, \ldots, \phi_n \in H^\infty$ are such that $\phi_0 u^n + \phi_1f_1 + \ldots \phi_nf_n = 0$, then $\phi_k(\xi) = 0$ for $0 \leq k \leq n$. To this end, the equality \[ \phi_0 u^n = - (\phi_1 u^{n-1}I + \ldots + \phi_n I^n)\] shows that $\phi_0$ is divisible by $I$, since the right-hand side is, but $u^n$, being outer, is not. It follows that $\phi_0 = Ih_0$ for some $h_0 \in H^\infty$, and therefore $\phi_0(\xi) = I(\xi)h_0(\xi) = 0$. Dividing the above equality by $I$ and re-arranging, we obtain \[(h_0u + \phi_1) u^{n-1} = -(\phi_2u^{n-2}I + \ldots + \phi_n I^{n-1}).\] As above, we must have $h_0u + \phi_1 = h_1I$, with $h_1 \in H^\infty$. Then \[\phi_1(\xi) = h_1(\xi)I(\xi) - h_0(\xi)u(\xi) = 0.\] By repeating the argument we conclude that $\phi_0(\xi) = \phi_1(\xi) = \ldots \phi_n(\xi) = 0$.
The second part of the proof is identical to the proof of Theorem 2.9 in \cite{mortini}. Assuming that the ideal generated by $\{u^n, f_1, \ldots, f_n\}$ is also generated by $\{e_1, \ldots, e_m\}$, we obtain a matrix $\textbf{M}$ of size $m $-by-$ (n+1)$ and a matrix $\textbf{N}$ of size $(n+1) $-by-$ m$, both with entires in $H^\infty$, such that \[ \textbf{M}(u^n, f_1, \ldots, f_n)^t = (e_1, \ldots, e_m)^t, \quad \textbf{N}(e_1, \ldots, e_m)^t = (u^n, f_1, \ldots, f_n)^t.\] Then $(\textbf{NM}- I_{n+1})(u^n, f_1, \ldots, f_n)^t = \textbf{0}$, where $I_{n+1}$ is the identity matrix of dimension $n+1$. By the first part of the proof we obtain that $\textbf{N}(\xi)\textbf{M}(\xi) = I_{n+1}$, where the evaluation of the matrices at $\xi$ is done entrywise. Then the rank of the matrix $\textbf{N}(\xi)$ is at least $n+1$, i.e., $m \geq n+1$.
\end{proof}
We are ready to prove the main result of the section.
\begin{proof}[Proof of \thref{nonsimilaritytheorem}]
By \thref{mingenideal} there exist $u, f_1, \ldots, f_n \in H^\infty$, with $u$ outer, such that the ideal of $H^\infty$ generated by $\{u, f_1, \ldots, f_n\}$ is not generated by any set of size less than $n+1$. Let $c_i = \conj{f_i/u} \in \conj{N^+}$ and apply \thref{clemma} to $\textbf{c} = (c_1, \ldots, c_n)$ to obtain a $\textbf{B} = (b_1, \ldots, b_n)$ and a space $\mathcal{H}[\textbf{B}]$ such that \[ J\frac{1}{1-\conj{\lambda}z} = \Bigg( \frac{1}{1-\conj{\lambda}z}, \frac{\textbf{c}(\lambda)}{1-\conj{\lambda}z} \Bigg). \] If $\mathcal{H}[\textbf{B}]$ is equivalent to $\mathcal{H}[\textbf{D}]$, where $\textbf{D} = (d_1, \ldots, d_m)$ and \[ J_2\frac{1}{1-\conj{\lambda}z} = \frac{(1,\textbf{e}(\lambda))}{1-\conj{\lambda}z} = \frac{(1, e_1(\lambda), \ldots, e_m(\lambda))}{1-\conj{\lambda}z},\] where $J_2$ is the embedding associated to $\mathcal{H}[\textbf{D}]$, then \thref{modulegen} implies that the sets $\{u, f_1, \ldots, f_n \}$ and $\{u, \conj{e_1} u, \ldots, \conj{e_m} u \}$ generate the same ideal in $H^\infty$. Thus $n+1 \geq m+1$.
\end{proof}
\subsection{$M_z$-invariant subspaces.}
Our structure theorem for $M_z$-invariant subspaces will follow easily from \thref{modeltheoremanalytic} after this preliminary lemma.
\begin{lemma} \thlabel{index1prop} Let $\mathcal{H}[\textbf{B}]$ be of finite rank and $M_z$-invariant. If $\mathcal{M}$ is an $M_z$-invariant subspace of $\mathcal{H}[\textbf{B}]$, then for each $\lambda \in \mathbb{D}$ we have that $\dim \mathcal{M} \ominus (M_z -\lambda)\mathcal{M} = 1$, and thus $M$ is nearly invariant.
\end{lemma}
\begin{proof}
Let $f \in \mathcal{M}, h \in \mathcal{M}^\perp$ and $Jf = (f, \textbf{f}_1), Jh = (h, \textbf{h}_1)$. By $M_z$-invariance of $\mathcal{M}$ we have, in the notation of \thref{cfprop}, \begin{equation}
0 = \ip{(I_{\mathcal{H}[\textbf{B}]}-\conj{\lambda} M_z)^{-1}f}{h}_{\mathcal{H}[\textbf{B}]} = \int_\mathbb{T} \frac{f(\zeta)\conj{h(\zeta)} + \ip{\textbf{f}_1(\zeta)}{\textbf{h}_1(\zeta)}_{\mathbb{C}^n}}{1-\conj{\lambda}z}d\textit{m}(\zeta) - \ip{\textbf{c}_f(\lambda)}{\textbf{h}_1(\lambda)}_{\mathbb{C}^n}. \label{eq541}
\end{equation} Let $K_{f,h}$ be the Cauchy transform \begin{equation}K_{f,h}(\lambda) = \lambda \int_\mathbb{T} \frac{f(\zeta)\conj{h(\zeta)} + \ip{\textbf{f}_1(\zeta)}{\textbf{h}_1(\zeta)}_{\mathbb{C}^n}}{z-\lambda} d\textit{m}(\zeta). \label{eq542}\end{equation} Then $K_{f,h}$ is analytic for $\lambda \in \mathbb{D}$ and admits non-tangential boundary values on $\mathbb{T}$. Adding \eqref{eq541} and \eqref{eq542} gives \begin{equation*}K_{f,h}(\lambda) = \int_\mathbb{T} \frac{1-|\lambda|^2}{|\zeta - \lambda|^2} \big( f(\zeta)\conj{h(\zeta)} + \ip{\textbf{f}_1(\zeta)}{\textbf{h}_1(\zeta)}_{\mathbb{C}^n}\big) d\textit{m}(\zeta) - \ip{\textbf{c}_f(\lambda)}{\textbf{h}_1(\lambda)}_{\mathbb{C}^n}. \end{equation*} By taking the limit $|\lambda| \to 1$ and using basic properties of Poisson integrals we see that, for almost every $\lambda \in \mathbb{T}$, we have the equality \begin{gather*}
K_{f,h}(\lambda) = f(\lambda)\conj{h(\lambda)} + \ip{\textbf{f}_1(\lambda)}{\textbf{h}_1(\lambda)}_{\mathbb{C}^n} - \ip{\textbf{c}_f(\lambda)}{\textbf{h}_1(\lambda)}_{\mathbb{C}^n} \\ = f(\lambda)\conj{h(\lambda)} - \ip{(\textbf{A}(\lambda)^*)^{-1}\textbf{B}(\lambda)^*f(\lambda)}{\textbf{h}_1(\lambda)}_{\mathbb{C}^n} \\ =
f(\lambda)\Big( \conj{h(\lambda)} - \ip{(\textbf{A}(\lambda)^*)^{-1}\textbf{B}(\lambda)^*1}{\textbf{h}_1(\lambda)}_{\mathbb{C}^n}\Big),
\end{gather*} where we used \eqref{eq51} in the computation. The meromorphic function $K_{f,h}/f$ thus depends only on $h$, and not on $f$.
Let $f(\lambda) = 0$ for some $\lambda \in \mathbb{D} \setminus \{0\}$ which is not a common zero of $\mathcal{M}$, so that there exists $g \in \mathcal{M}$ with $g(\lambda) \neq 0$. From $K_{f,h}/f = K_{g,h}/g$ we deduce that $K_{f,h}(\lambda)g(\lambda) = K_{g,h}(\lambda)f(\lambda) = 0$, and thus \begin{gather*}
0 = K_{f,h}(\lambda) = \lambda \int_\mathbb{T} \frac{f(\zeta)\conj{h(\zeta)} + \ip{\textbf{f}_1(\zeta)}{\textbf{h}_1(\zeta)}_{\mathbb{C}^n}}{z-\lambda} d\textit{m}(\zeta) \\ = \lambda \int_\mathbb{T} \frac{f(\zeta)\conj{h(\zeta)} + \ip{\textbf{f}_1(\zeta) - \textbf{f}_1(\lambda)}{\textbf{h}_1(\zeta)}_{\mathbb{C}^n}}{z-\lambda} d\textit{m}(\zeta) \\ = \lambda \ip{\tfrac{f(z)}{z-\lambda}}{h}_{\mathcal{H}[\textbf{B}]}.
\end{gather*} Since $h \in \mathcal{M}^\perp$ is arbitrary, we conclude that $\tfrac{f(z)}{z-\lambda} \in \mathcal{M}$, and thus $\dim \mathcal{M} \ominus (M_z - \lambda)\mathcal{M} = 1$. The fact that $\dim \mathcal{M} \ominus M_z\mathcal{M} = 1$ holds also for $\lambda = 0$ or a common zero of the functions in $\mathcal{M}$ follows from basic Fredholm theory. The operators $M_z - \lambda$ are injective semi-Fredholm operators, and thus $\lambda \mapsto \dim \mathcal{M} \ominus (M_z - \lambda)\mathcal{M}$ is a constant function in $\mathbb{D}$.
\end{proof}
The following is our main theorem on $M_z$-invariant subspaces of finite rank $\mathcal{H}[\textbf{B}]$-spaces.
\begin{thm} Let $\mathcal{H} = \mathcal{H}[\textbf{B}]$ be of finite rank and $M_z$-invariant and $\mathcal{M}$ be a closed $M_z$-invariant subspace of $\mathcal{H}$. Then
\begin{enumerate}[(i)]
\item $\dim \mathcal{M} \ominus M_z\mathcal{M} = 1$,
\item any non-zero element in $\mathcal{M} \ominus M_z\mathcal{M}$ is a cyclic vector for $M_z|\mathcal{M}$,
\item if $\phi \in \mathcal{M} \ominus M_z \mathcal{M}$ is of norm $1$, then there exists a space $\mathcal{H}[\textbf{C}]$ invariant under $M_z$, where $\textbf{C} = (c_1, \ldots, c_k)$ and $k \leq n$, such that \begin{equation*}\mathcal{M} = \phi \mathcal{H}[\textbf{C}] \end{equation*} and the mapping $g \mapsto \phi g$ is an isometry from $\mathcal{H}[\textbf{C}]$ onto $\mathcal{M}$,
\item if $J$ is the embedding given by \thref{modeltheoremanalytic}, $\phi \in \mathcal{M} \ominus M_z\mathcal{M}$ with $J\phi = (\phi, \boldsymbol{\phi}_1)$, then \[ \mathcal{M} = \big\{ f \in \mathcal{H}[\textbf{B}] : \tfrac{f}{\phi} \in H^2, \tfrac{f}{\phi}\boldsymbol{\phi_1} \in H^2(\mathbb{C}^n) \big\}.\]
\end{enumerate}
\end{thm}
\begin{proof}
Part (i) has been established \thref{index1prop} and part (ii) follows from (iii) by \thref{polydense}. It thus suffices to prove parts (iii) and (iv).
We verified in \thref{index1prop} that $\mathcal{M}$ is nearly invariant, and thus norm formula \eqref{eq221} of \thref{nearinvnorm} applies. A computation shows that, in the notation of \thref{nearinvnorm}, we have $L^\phi_\lambda f = L_\lambda (f-\tfrac{f(\lambda)}{\phi(\lambda)}\phi)$, at least when $\phi(\lambda) \neq 0$. Thus if $Jf = (f, \textbf{f}_1)$ and $J\phi =(\phi, \boldsymbol{\phi}_1)$, then by \eqref{eq221} and part (i) of \thref{modelformulaconnection} we obtain that \begin{equation}\|f\|^2_{\mathcal{H}[\textbf{B}]} = \|f/\phi\|^2_{H^2} + \|\textbf{g}_1\|^2_{H^2(\mathbb{C}^n)}\label{501} \end{equation} where $\textbf{g}_1(z) = \textbf{f}_1(z) - \tfrac{f(z)}{\phi(z)}\boldsymbol{\phi}_1(z)$. The mapping $If := (f/\phi, \textbf{g}_1)$ is therefore an isometry from $\mathcal{M}$ into $H^2 \oplus H^2(\mathbb{C}^n)$. The identity $IL^\phi f = (L(f/\phi), L\textbf{g}_1)$ shows that $I\mathcal{M}$ is a backward shift invariant subspace of $H^2 \oplus H^2(\mathbb{C}^n) \simeq H^2(\mathbb{C}^{n+1})$. Consequently by the Beurling-Lax theorem we have that $(I\mathcal{M})^\perp = \Psi H^2(\mathbb{C}^k)$, for some $(n+1)$-by-$k$ matrix-valued bounded analytic function such that $\Psi(\zeta): \mathbb{C}^k \to \mathbb{C}^{n+1}$ is an isometry for almost every $\zeta \in \mathbb{T}$. We claim that $k \leq n$. Indeed, in other case $\Psi$ is an $(n+1)$-by-$(n+1)$ square matrix, and hence $\psi(z) = \det \Psi (z)$ is a non-zero inner function. We would then obtain \begin{equation}\psi H^2(\mathbb{C}^{n+1}) = \Psi \text{adj}(\Psi) H^2(\mathbb{C}^{n+1}) \subset \Psi H^2(\mathbb{C}^{n+1}) = (I\mathcal{M})^\perp. \label{eq59} \end{equation} But since $\mathcal{M}$ is shift invariant, the function $p\phi$ is contained in $\mathcal{M}$ for any polynomial $p$, and hence for any polynomial $p$ there exists a tuple of the form $(p, \textbf{g})$ in $I\mathcal{M}$. Together with $
\eqref{eq59}$ this shows that the polynomials are orthogonal to $\psi H^2$, so $\psi = 0$ and we arrive at a contradiction.
Decompose the matrix $\Psi$ as \begin{equation*}
\Psi(z) = \begin{bmatrix}
\textbf{C}(z) \\\textbf{D}(z)
\end{bmatrix}
\end{equation*} where $\textbf{C}(z) = (c_1(z), \ldots, c_k(z))$ and $\textbf{D}(z)$ is an $n$-by-$k$ matrix. Consider the Hilbert space $\tilde{\mathcal{M}} = \mathcal{M}/\phi = \{f/\phi : f \in \mathcal{M}\}$ with the norm $\|f/\phi\|_{\tilde{\mathcal{M}}} = \|f\|_{\mathcal{H}[\textbf{B}]}$. By \eqref{501}, the map \begin{equation} \tilde{I}f/\phi := \Big(f/\phi, \textbf{f}_1 - \tfrac{f}{\phi}\boldsymbol{\phi}_1 \Big) \label{Iemb} \end{equation} is an isometry from $\tilde{\mathcal{M}}$ into $H^2 \oplus H^2(\mathbb{C}^n)$, and \begin{equation}(\tilde{I}\tilde{\mathcal{M}})^\perp = \{ (\textbf{C}\textbf{h}, \textbf{D}\textbf{h}) : \textbf{h} \in H^2(\mathbb{C}^k) \}. \label{eqCD} \end{equation} The argument of \thref{modeltheorem} can be used to see that $$k_{\tilde{\mathcal{M}}}(\lambda,z) = \frac{1- \sum_{i=1}^k \conj{c_i(\lambda)}c_i(z)}{1-\conj{\lambda}z}$$ is the reproducing kernel of $\tilde{\mathcal{M}}$. Thus $\tilde{\mathcal{M}} = \mathcal{H}[\textbf{C}]$, and the proof of part (iii) is complete.
Finally, we prove part (iv). The inclusion of $\mathcal{M}$ in the set given in (iv) has been established in the proof of part (iii) above. On the other hand, assume that $f \in \mathcal{H}[\textbf{B}]$ is contained in that set. We will show that $(f/\phi, \textbf{f}_1 - \tfrac{f}{\phi}\boldsymbol{\phi}_1 )$ is orthogonal to the set given in \eqref{eqCD}, and thus $f/\phi \in \tilde{\mathcal{M}}$, so that $f \in \mathcal{M}$. In order to verify the orthogonality claim, we must show that $\textbf{C}^*\tfrac{f}{\phi} + \textbf{D}^*(\textbf{f}_1 - \tfrac{f}{\phi}\boldsymbol{\phi}_1) \in \overline{H^2_0(\mathbb{C}^n)}.$ According to \thref{cfprop}, we have that \[ J\frac{\phi(z)}{1-\conj{\lambda}z} = \Big( \frac{\phi(z)}{1-\conj{\lambda}z}, \frac{\boldsymbol{\phi}_1(z) - \textbf{c}_\phi(\lambda)}{1-\conj{\lambda}z} \Big) \] for some coanalytic function $\textbf{c}_\phi$ which satisfies $\textbf{c}_\phi = (\textbf{A}^*)^{-1}\textbf{B}^*\phi + \boldsymbol{\phi}_1$ on $\mathbb{T}$. Setting $f(z) = \frac{\phi(z)}{1-\conj{\lambda}z}$ in \eqref{Iemb} we obtain from \eqref{eqCD} that \[ \Big( \frac{1}{1-\conj{\lambda}z} , -\frac{\textbf{c}_\phi(\lambda)}{1-\conj{\lambda}z}\Big) \perp \{ (\textbf{C}\textbf{h}, \textbf{D}\textbf{h}) : \textbf{h} \in H^2(\mathbb{C}^k) \}\] and then it easily follows that $\textbf{c}_\phi(\lambda) = (\textbf{D}^*(\lambda))^{-1}\textbf{C}^*(\lambda)1$. Using \eqref{eq51} we obtain the boundary value equality \[\textbf{C}^* = \textbf{D}^*(\textbf{A}^*)^{-1}\textbf{B}^* \phi + \textbf{D}^*\boldsymbol{\phi}_1,\] and thus on $\mathbb{T}$ we have \begin{equation}\textbf{C}^*\tfrac{f}{\phi} + \textbf{D}^*(\textbf{f}_1 - \tfrac{f}{\phi}\boldsymbol{\phi}_1) = \textbf{D}^* (\textbf{A}^*)^{-1}(\textbf{B}^*f + \textbf{A}^* \textbf{f}_1) \label{eq99}. \end{equation} Since $f \in \mathcal{H}[\textbf{B}]$ we have that $\textbf{B}^*f + \textbf{A}^*g \in \overline{H^2_0(\mathbb{C}^n)}$ and thus \eqref{eq99} represents square-integrable boundary function of a coanalytic function in the Smirnov class. An appeal to the Smirnov maximum principle completes the proof of (iv).
\end{proof}
\subsection{Backward shift invariant subspaces.} The lattice of $L$-invariant subspaces of $\mathcal{H}[\textbf{B}]$-spaces is much less complicated than the lattice of $M_z$-invariant subspaces. The following theorem generalizes a result of \cite{sarasondoubly} for $\mathcal{H}(b)$ with non-extreme $b$. Our method of proof is new, and relies crucially on \thref{modeltheoremanalytic}.
\begin{thm} \thlabel{Linvsubspaces}Any proper $L$-invariant subspace of a $M_z$-invariant finite rank $\mathcal{H}[\textbf{B}]$-space is of the form $$\mathcal{H}[\textbf{B}] \cap K_\theta,$$ where $\theta$ is an inner function and $K_\theta = H^2 \ominus \theta H^2$.
\end{thm}
\begin{proof}
Let $J: \mathcal{H}[\textbf{B}] \to H^2 \oplus H^2({\mathbb{C}^n})$ be the embedding of \thref{modeltheoremanalytic}. If $\mathcal{M}$ is an $L$-invariant subspace of $\mathcal{H}[\textbf{B}]$, then $J\mathcal{M}$ is an $L$-invariant subspace of $H^2 \oplus H^2(\mathbb{C}^n)$ by (iii) of \thref{modeltheoremanalytic}. Thus $(J\mathcal{M})^\perp$ is an $M_z$-invariant subspace containing $(J\mathcal{H}[\textbf{B}])^\perp = \{(\textbf{B}\textbf{h}, \textbf{A}\textbf{h}) : \textbf{h} \in H^2(\mathbb{C}^n)\}$. Because $M_z$ acting on $(J\mathcal{H}[\textbf{B}])^\perp$ is a shift of multiplicity $n$, the multiplicity of $M_z$ acting on $(J\mathcal{M})^\perp$ is at least $n$, and since $(J\mathcal{M})^\perp \subset H^2 \oplus H^2(\mathbb{C}^n)$, it is at most $n+1$. We claim that this multiplicity must equal $n+1$. Indeed, if it was equal to $n$, then it is easy to see that $$(J\mathcal{M})^\perp = \{ (\textbf{Ch}, \textbf{Dh}) : \textbf{h} \in H^2(\mathbb{C}^n)\}$$ for some $\textbf{C}(z) = (c_1(z), \ldots, c_n(z))$ and $\textbf{D}(z)$ an $n$-by-$n$-matrix valued analytic function. The fact that no tuple of the form $(0,\textbf{g})$ is included in $J\mathcal{M}$ implies that $\textbf{D}$ is an outer function, and thus $\textbf{D}(\lambda)$ is an invertible operator for every $\lambda \in \mathbb{D}$. The tuple \begin{equation*} \Big(\frac{1}{1-\conj{\lambda}z}, -\frac{(\textbf{D}(\lambda)^*)^{-1}\textbf{C}(\lambda)^*1}{1-\conj{\lambda}z}\Big)
\end{equation*} is clearly orthogonal to $(J\mathcal{M})^\perp$, and thus $\frac{1}{1-\conj{\lambda}z} \in \mathcal{M}$ for every $\lambda \in \mathbb{D}$. Then $\mathcal{M} = \mathcal{H}[\textbf{B}]$ by the proof of \thref{polydense}. We assumed that $\mathcal{M}$ is a proper subspace, and so multiplicity of $M_z$ on $(J\mathcal{M})^\perp$ cannot be $n$.
Having established that $M_z$ is a shift of multiplicity $n+1$ on $(J\mathcal{M})^\perp$, we conclude that $$(J\mathcal{M})^\perp= \Psi H^2(\mathbb{C}^{n+1}),$$ where $\Psi$ is an $(n+1)$-by-$(n+1)$ matrix-valued inner function, and $\theta = \det \Psi$ is a non-zero scalar-valued inner function. Note that $\theta H^2(\mathbb{C}^{n+1}) = \Psi\,\text{adj}(\Psi) H^2(\mathbb{C}^{n+1}) \subseteq \Psi H^2(\mathbb{C}^{n+1})$. Thus if $f \in \mathcal{M}$, then $Jf = (f,\textbf{f}_1) \perp \Psi H^2(\mathbb{C}^{n+1}) \supseteq \theta H^2(\mathbb{C}^{n+1})$. It follows that $f \in K_\theta$, and thus we have shown that $\mathcal{M} \subseteq \mathcal{H}[\textbf{B}] \cap K_\theta$. Next, consider the $(n+1)$-by-$(n+1)$ matrix \begin{gather*}
\textbf{M}(z) = \begin{bmatrix}
\textbf{B}(z) & \theta(z) \\ \textbf{A}(z) & 0
\end{bmatrix}.
\end{gather*} Then it is easy to see that $f \in \mathcal{H}[\textbf{B}] \cap K_\theta$ if and only if $Jf = (f,\textbf{f}_1) \perp \textbf{M}(z)\textbf{h}(z)$ for all $\textbf{h} \in H^2(\mathbb{C}^{n+1})$. If $\textbf{M}(z) = \textbf{I}(z)\textbf{U}(z)$ is the inner-outer factorization of $\textbf{M}$ into an $(n+1)$-by-$(n+1)$-matrix valued inner function $\textbf{I}$ and an $(n+1)$-by-$(n+1)$-matrix valued outer function $\textbf{U}$, then we also have that \begin{equation}f \in \mathcal{H}[\textbf{B}] \cap K_\theta \text{ if and only if } Jf = (f, \textbf{f}_1) \perp \textbf{I}H^2(\mathbb{C}^{n+1}). \label{eq72} \end{equation} From the containment $\mathcal{M} \subseteq \mathcal{H}[\textbf{B}] \cap K_\theta$ we get by taking orthocomplements that $\textbf{I}H^2(\mathbb{C}^{n+1}) \subseteq \Psi H^2(\mathbb{C}^{n+1})$ and thus there exists a factorization $\textbf{I} = \Psi \textbf{J}$, where $\textbf{J}$ is an $(n + 1)$-by-$(n+1)$-matrix valued inner function. Since $-\theta \det \textbf{A} = \det \textbf{M} = \det \textbf{I} \det \textbf{U}$, we see (by comparing inner and outer factors) that $\det \textbf{I} = \alpha\theta$, with $\alpha \in \mathbb{T}$, and so $\alpha \theta = \det \textbf{I} = \det \Psi \det \textbf{J} = \theta \det \textbf{J}$. We conclude that $\det \textbf{J}$ is a constant, and thus $\textbf{J}$ is a constant unitary matrix. But then $(J\mathcal{M})^\perp = \Psi H^2(\mathbb{C}^{n+1}) = \textbf{I}H^2(\mathbb{C}^{n+1})$, and so the claim follows by \eqref{eq72}.
\end{proof}
\bibliographystyle{siam}
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\section{Introduction}
Remote processing of neural networks in the cloud is not without risk. Traditional encryption techniques can protect the data while sending it to the cloud but the unencrypted data is needed by the computation node to evaluate the neural network. The operator of the computing node can not necessarily be trusted and has access to the raw data of the users. An even greater risk is the compromise of the node by a third party. Recent security breaches such as the leak of personal images stored in Apple iCloud or the abuse of personal data shared on Facebook for political goals have raised public awareness of privacy and security risks.
In this paper we present a technique to obfuscate the data before sending it to the cloud. The obfuscation routine renders the data unintelligible for a human eavesdropper while still retaining enough structure to allow a correct classification by the neural network. We focus on image classification using deep neural networks (DNNs) since this is arguably one of the most common use cases for DNNs but this technique could be applied to other application domains
as well.
Previous approaches to protect the privacy of users in computer vision tasks include extreme downsampling \cite{ryoo2017privacy}, \cite{chen2017semi} and blurring or scrambling \cite{manolakosprivacy} of the inputs. These are hand-crafted heuristics that are able to remove privacy sensitive details but they also have a large penalty on the classification accuracy. The most similar approach to our work is \cite{ren2018learning} where the authors introduce a trainable model that modifies video frames to obfuscate each person's face with minimal effect on action detection performance. The biggest difference is that they train the classification network together with the obfuscation network. The classification network will therefore only work together with the obfuscation network. In contrast, we use pretrained classification networks that were trained on unobfuscated images. We then train an obfuscation network to transform images in order to make them unintelligible for humans while still allowing for a high classification accuracy with the pretrained classification network. We also obfuscate the full image instead of only specific parts of the human face.
\section{Architecture}
Our approach builds upon two recent discoveries in deep learning: Adversarial inputs and Generative Adversarial Networks (GANs). Adversarial inputs \cite{goodfellow2014explaining} are special input samples that have been carefully tweaked to fool neural networks. They are created by making tiny changes to real inputs such that the real and the perturbed versions are indistinguishable to human observers yet the model consistently misclassifies the perturbed input with high confidence. In this paper we are however interested in the exact opposite behaviour, we want to transform images in order to make them unintelligible for human observers yet the neural network should still be able to correctly classify them.
Generative Adversarial Networks (GANs) \cite{goodfellow2014generative} are models that can learn to generate artificial datapoints that follow the same distribution as real datapoints. The model consists of two networks, the generator and the discriminator competing against each other. The task of the generator is to generate new datapoints based on random input. The discriminator tries to distinguish between real data points and generated data. By training both networks together the generator will eventually be able to generate realistically looking datapoints.
Our proposed architecture is shown in figure \ref{fig:architecture}. It consists of three deep neural networks. The pretrained network on the right is a network trained for image classification on normal, unobfuscated images. We do not modify the weights of this network. The obfuscator and the deobfuscator are two autoencoder-like networks. The obfuscator takes the original image as input and generates an obfuscated version that is then fed into the pretrained classification network. The deobfuscator tries to reconstruct the original image based solely on the obfuscated version. The final goal is to train an obfuscator network than can transform the image in order that the classifying network is still able to recognize the object without the deobfuscator being able to reconstruct the original input. We introduce two loss terms to train the architecture. ${L_c}$ is the crossentropy classification loss that is commonly used in classification problems. ${L_r}$ is the reconstruction loss that measures the euclidean distance between the original image and the reconstructed version. The obfuscator is jointly trained to minimize the classification loss and to maximize the reconstruction loss. The deobfuscator is solely trained to minimize the reconstruction loss. Both networks are trained at the same time.
The premise of our approach is to offload the computationally costly classification network to the cloud and to do a local obfuscation step to protect the privacy of the user. It is therefore crucial that the obfuscator network is as small as possible. We use a MobileNet inspired architecture \cite{howard2017mobilenets} with depthwise seperable convolutions to reduce the computational cost. Details of the architecture and training routine can be found in Appendix \ref{app:one}.
\vspace{-0.5cm}
\begin{figure}[H]
\caption{Overview of our architecture.}
\label{fig:architecture}
\centering
\includegraphics[width=0.5\textwidth]{architecture.pdf}
\end{figure}
\vspace{-1cm}
\section{Experiments}
All our experiments were implemented in PyTorch \cite{paszke2017automatic}
We used the CIFAR10 dataset \cite{krizhevsky2009learning} for all these experiments.
Table \ref{tbl:accuracy} shows the accuracy of the classification models on original and on obfuscated images. We find similar results for the different architectures where the accuracy drops by ~5\%. We argue that this is a reasonable price to pay for the added privacy. We also show the overhead of the obfuscation network relative to the classification network both in terms of FLOPS and number of parameters. Table \ref{tbl:images} shows some more qualitative results. We show the original images, the obfuscated and the deobfuscated versions. The classification network was trained on the original images but is also able to classify the obfuscated versions. The deobfuscated versions were included to prove that it is indeed impossible to retrieve the original images from the obfuscated versions. These images show that there is still information on the background color and the location of the object encoded in the obfuscated image but all details are lost.
One disadvantage with our proposed approach is that we need to backpropagate through the classification network to train the obfuscation network. This means that our technique does not treat the classification model as a truly black box since we need the weights of the network which might be unavailable. In our last experiment we examine how transferable the obfuscator networks are. We train the obfuscator network with one classification network and test it with another. The results are shown in Table \ref{tbl:transfer}. There is a large drop in accuracy but surprisingly the accuracy does not drop to the random level for most combinations. This suggests that the obfuscator network can learn a transformation that is not completely overfitted to one classification network but that captures some universal features that are used by different classification networks.
\vspace{-0.5cm}
\begin{table}[H]
\caption{Classification accuracies for plain and obfuscated images. The overhead column shows the cost (in terms of FLOPS and parameters) of the obfuscator network relative to the classification network. Absolute measurements are included in Appendix \ref{app:one}.}
\label{tbl:accuracy}
\begin{center}
\begin{scriptsize}
\begin{sc}
\begin{tabular}{lcc|cc}
\toprul
&\multicolumn{2}{c}{Accuracy}&\multicolumn{2}{c}{Overhead}\\
\midrule
Architecture & Plain & Obfusc. & Flops & Param.\\
\midrule
VGG19 & 93.4\% & 89.3\% & 6.7\% & 1.6\% \\
ResNet18 & 94.8\% & 89.8\% & 4.8\% & 2.9\%\\
ResNet50 & 95.1\% & 90.2\% & 2.1\% & 1.4\%\\
GoogleNet & 95.2\% & 90.5\% & 1.7\% & 5.3\% \\
\bottomrul
\end{tabular}
\end{sc}
\end{scriptsize}
\end{center}
\end{table}
\vspace{-1cm}
\begin{table}[H]
\caption{Original, obfuscated (output of the obfuscator) and reconstructed images (output of the deobfuscator). More examples are shown in Appendix \ref{app:two}}
\vspace{-0.35cm}
\label{tbl:images}
\begin{center}
\begin{scriptsize}
\begin{sc}
\begin{tabular}{ccc|ccc}
\toprule
\rotatebox{45}{Original} & \rotatebox{45}{Obfusc.} & \rotatebox{45}{Reconstr.} & \rotatebox{45}{Original} & \rotatebox{45}{Obfusc.} & \rotatebox{45}{Reconstr.}\\
\midrule
\includegraphics{zzz_original_128_64.png} & \includegraphics{zzz_obfuscated_128_64.png} & \includegraphics{zzz_deobfuscated_128_64.png} & \includegraphics{zzz_original_96_416.png} & \includegraphics{zzz_obfuscated_96_416.png} & \includegraphics{zzz_deobfuscated_96_416.png}\\
\includegraphics{zzz_original_96_64.png} & \includegraphics{zzz_obfuscated_96_64.png} & \includegraphics{zzz_deobfuscated_96_64.png} & \includegraphics{zzz_original_96_96.png} & \includegraphics{zzz_obfuscated_96_96.png} & \includegraphics{zzz_deobfuscated_96_96.png}\\
\bottomrul
\end{tabular}
\end{sc}
\end{scriptsize}
\end{center}
\end{table}
\vspace{-1cm}
\begin{table}[H]
\caption{The classification accuracy when applying the obfuscator network to other networks than it was originally trained with. ResNet18\_1 and ResNet18\_2 are the same architecture, trained in the same way but from a different random initialisation.}
\vspace{-0.35cm}
\label{tbl:transfer}
\begin{center}
\begin{scriptsize}
\begin{sc}
\begin{tabular}{p{1.7cm}|p{0.9cm}p{0.9cm}p{0.9cm}p{0.9cm}p{0.9cm}cc}
\toprul
& \multicolumn{5}{c}{Tested on}\\
\cmidrule{2-6}
\rotatebox{0}{Trained on} & \rotatebox{45}{ResNet18\_1} & \rotatebox{45}{ResNet18\_2} & \rotatebox{45}{ResNet50} & \rotatebox{45}{GoogleNet} & \rotatebox{45}{VGG19}\\
\cmidrule{1-6}
ResNet18\_1 & 89.8\% & 53.8\% & 36.0\% & 30.6\% & 47.4\%\\
ResNet18\_2 & 72.6\% & 90.0\% & 46.7\% & 34.7\% & 46.0\%\\
ResNet50 & 62.1\% & 54.9\% & 90.2\% & 29.1\% & 44.8\%\\
GoogleNet & 71.0\% & 74.0\% & 66.3\% & 90.5\% & 41.5\%\\
VGG19 & 25.8\% & 20.0\% & 20.2\% & 13.0\% & 89.3\%\\
\bottomrul
\end{tabular}
\end{sc}
\end{scriptsize}
\end{center}
\end{table}
\vspace{-0.7cm}
\section{Conclusion and future work}
We introduced a trainable obfuscation step that renders images unintelligible for humans but still allows a high classification accuracy by pretrained networks. Future work will focus on applying this technique to more complex datasets and on improving the transferability of the obfuscator networks between classification networks.
\newpage
\section*{Acknowledgements}
We gratefully acknowledge the support of NVIDIA Corporation with the donation of GPU hardware used in this research.
| 3,019 |
\section{Introduction}
\label{sec:intro}
How will a patient's health be affected by taking a medication \citep{Criado-Perez2020invisible}?
How will a user's question be answered by a search recommendation \citep{noble2018algorithms}?
We can gain insight into these questions by learning about personalized treatment effects.
Estimating personalized treatment effects from observational data is essential when experimental designs are infeasible, unethical, or expensive.
Observational data represent a population of individuals described by a set of pre-treatment covariates (age, blood pressure, socioeconomic status), an assigned treatment (medication, no medication), and a post-treatment outcome (severity of migraines).
An ideal personalized treatment effect is the difference between the post-treatment outcome if the individual receives treatment and the post-treatment outcome if they do not receive treatment.
However, it is impossible to observe both outcomes for an individual; therefore, the difference is estimated between populations instead.
In the setting of binary treatments, data belong to either the \emph{treatment group} (individuals that received the treatment) or the \emph{control group} (individuals who did not).
The personalized treatment effect is the expected difference in outcomes between treated and controlled individuals who share the same (or similar) measured covariates; as an illustration, see the difference between the solid lines in \cref{fig:observational_data} (middle pane).
The use of pre-treatment covariates assembled from high-dimensional, heterogeneous measurements such as medical images and electronic health records is increasing \citep{sudlow2015uk}.
Deep learning methods have been shown capable of learning personalized treatment effects from such data \citep{shalit2017estimating, shi2019adapting,jesson2021quantifying}.
However, a problem in deep learning is data efficiency.
While modern methods are capable of impressive performance, they need a significant amount of labeled data.
Acquiring labeled data can be expensive, requiring specialist knowledge or an invasive procedure to determine the outcome.
Therefore, it is desirable to minimize the amount of labeled data needed to obtain a well-performing model.
Active learning provides a principled framework to address this concern \citep{cohn1996active}.
In active learning for treatment effects \citep{deng2011active, sundin2019active, qin2021budgeted} a model is trained on available labeled data consisting of covariates, assigned treatments, and acquired outcomes.
The model predictions decide the most informative examples from data comprised of only covariates and treatment indicators.
Outcomes are acquired, e.g., by performing a biopsy for the selected patients, and the model is retrained and evaluated.
This process repeats until either a satisfactory performance level is achieved or the labeling budget is exhausted.
At first sight, this might seem simple; however, active learning induces biases that result in divergence between the distribution of the acquired training data and the distribution of the pool set data \citep{farquhar2021on}.
In the context of learning causal effects, such bias can have both positive and negative consequences.
For example, while random acquisition active learning results in an unbiased sample of the training data, we demonstrate how it can lead to over-allocation of resources to the mode of the data at the expense of learning about underrepresented data.
Conversely, while biasing acquisitions toward lower density regions of the pool data can be desirable, it can also lead to outcome acquisition for data with unidentifiable treatment effects, which leads to uninformed, potentially harmful, personalized recommendations.
\begin{wrapfigure}{r}{0.45\textwidth}
\vspace{-1.5em}
\begin{center}
\includegraphics[width=0.45\textwidth]{figures/mainfig.png}
\end{center}
\vspace{-3mm}
\caption{
Observational data.
Top: data density of treatment (right) and control (left) groups.
Middle: observed outcome response for treatment (circles) and control (x's) groups.
Bottom: data density for active learned training set after a number of acquisition steps.
}
\label{fig:observational_data}
\vspace{-2mm}
\end{wrapfigure}
To see how training data bias can benefit treatment effect estimation, consider one difference between experimental and observational data: the treatment assignment mechanism is unavailable for observational data.
In observational data, variables that affect treatment assignment (an untestable condition) may be unobserved.
Moreover, the relative proportion of treated to controlled individuals varies across different sub-populations of the data.
\cref{fig:observational_data} illustrates the latter point, where there are relatively equal proportions of treated and controlled examples for data in region 3.
However, the proportions become less balanced as we move to either the left or the right.
In extreme cases, say if a group described by some covariate values were systematically excluded from treatment, the treatment effect for that group \emph{cannot be known} \citep{petersen2012diagnosing}.
\cref{fig:observational_data} illustrates this in region 1, where only controlled examples reside, and in region 5, where only treated cases occur.
In the language of causal inference, the necessity of seeing both treated and untreated examples for each sub-population corresponds to satisfying the overlap (or positivity) assumption (see \ref{asm:overlap}).
The data available in the pool set limits overlap when treatments cannot be assigned.
In this setting, regions 2 and 4 of \cref{fig:observational_data} are very interesting because while either the treated or control group are underrepresented, there may still be sufficient coverage to estimate treatment effects.
\citet{d2021deconfounding} have described such regions as having weak overlap.
Training data bias towards such regions can benefit treatment effect estimation for underrepresented data by acquiring low-frequency data with sufficient overlap.
We hypothesize that the efficient acquisition of unlabeled data for treatment effect estimation focuses on only exploring regions with sufficient overlap, and uncertainty should be high for areas with non-overlapping support.
The bottom pane of \cref{fig:observational_data} imagines what a resulting training set distribution could look like at an intermediate active learning step.
It is not trivial to design such acquisition functions: naively applying active learning acquisition functions results in suboptimal and sample inefficient acquisitions of training examples, as we show below.
To this end, we develop epistemic uncertainty-aware methods for active learning of personalized treatment effects from high dimensional observational data.
In contrast to previous work that uses only information gain as the acquisition objective, we propose $\rho\textrm{BALD}$~and $\mu\rho\textrm{BALD}$~as ``Causal BALD'' objectives because they consider both the information gain and overlap between treated and control groups.
We demonstrate the performance of the proposed acquisition strategies using synthetic and semi-synthetic datasets.
\section{Background}
\label{sec:background}
\subsection{Estimation of Personalized Treatment-Effects}
Personalized treatment-effect estimation seeks to know the effect of a treatment $\mathrm{T} \in \mathcal{T}$ on the outcome $\mathrm{Y} \in \mathcal{Y}$ for individuals described by covariates $\mathbf{X} \in \mathcal{X}$.
In this work, we consider the random variable (r.v.) $\mathrm{T}$ to be binary ($\mathcal{T} = \{0, 1\}$), the r.v. $\mathrm{Y}$ to be part of a bounded set $\mathcal{Y}$, and $\mathbf{X}$ to be a multi-variate r.v. of dimension $d$ ($\mathcal{X} = \mathbb{R}^d$).
Under the Neyman-Rubin causal model \citep{neyman1923applications, rubin1974estimating}, the individual treatment effect (ITE) for a person $u$ is defined as the difference in potential outcomes $\Y^1(u) - \Y^0(u)$, where the r.v. $\Y^1$ represents the potential outcome were they \emph{treated}, and the r.v. $\Y^0$ represents the potential outcome were they \emph{controlled} (not treated).
Realizations of the random variables $\mathbf{X}$, $\mathrm{T}$, $\mathrm{Y}$, $\Y^0$, and $\Y^1$ are denoted by $\mathbf{x}$, $\mathrm{t}$, $\mathrm{y}$, $\y^0$, and $\y^1$, respectively.
The ITE is a fundamentally unidentifiable quantity, so instead we look at the expected difference in potential outcomes for individuals described by $\mathbf{X}$, or the Conditional Average Treatment Effect (CATE): $\tau(\mathbf{x}) \equiv \mathop{\mathbb{E}}[\Y^1 - \Y^0 \mid \mathbf{X} = \mathbf{x}]$ \cite{abrevaya2015estimating}.
The CATE is identifiable from an observational dataset $\mathcal{D} = \left\{ (\mathbf{x}_i, \mathrm{t}_i, \mathrm{y}_i)\right\}_{i=1}^n$ of samples $(\mathbf{x}_i, \mathrm{t}_i, \mathrm{y}_i)$ from the joint empirical distribution $P_{\mathcal{D}}(\mathbf{X}, \mathrm{T}, \Y^0, \Y^1)$, under the following three assumptions \cite{rubin1974estimating}:
\begin{assumption}
(Consistency) $\mathrm{y} = \mathrm{t} \y^\tf + (1 - \mathrm{t}) \mathrm{y}^{1 - \mathrm{t}}$, i.e. an individual's observed outcome $\mathrm{y}$ given assigned treatment $\mathrm{t}$ is identical to their potential outcome $\y^\tf$.
\label{asm:consistency}
\end{assumption}
\begin{assumption}
(Unconfoundedness) $(\Y^0, \Y^1) \indep \mathrm{T} \mid \mathbf{X}$.
\label{asm:confounded}
\end{assumption}
\begin{assumption}
(Overlap) $0 < \pi_{\mathrm{t}}(\mathbf{x}) < 1: \forall \mathrm{t} \in \mathcal{T}$,
\label{asm:overlap}
\end{assumption}
where $\pi_{\mathrm{t}}(\mathbf{x}) \equiv \mathrm{P}(\mathrm{T} = \mathrm{t} \mid \mathbf{X} = \mathbf{x})$ is the \textbf{propensity for treatment} for individuals described by covariates $\mathbf{X} = \mathbf{x}$.
When these assumptions are satisfied, $\widehat{\tau}(\mathbf{x}) \equiv \mathop{\mathbb{E}}[Y \mid \mathrm{T} = 1, \mathbf{X} = \mathbf{x}] - \mathop{\mathbb{E}}[Y \mid \mathrm{T} = 0, \mathbf{X} = \mathbf{x}]$ is an unbiased estimator of $\tau(\mathbf{x})$ and is identifiable from observational data.
A variety of parametric \citep{robins2000marginal,tian2014simple,shalit2017estimating} and non-parametric estimators \citep{hill2011bayesian,xie2012estimating,alaa2017bayesian,gao2020minimax} have been proposed for CATE.
Here, we focus on parametric estimators for compactness.
Parametric CATE estimators assume that outcomes $\mathrm{y}$ are generated according to a likelihood $p_{\bm{\omega}}(\mathrm{y} \mid \mathbf{x}, \mathrm{t})$, given measured covariates $\mathbf{x}$, observed treatment $\mathrm{t}$, and model parameters $\bm{\omega}$.
For continuous outcomes, a Gaussian likelihood can be used: $\mathcal{N}(\mathrm{y} \mid \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \mathrm{t}), \widehat{\sigma}_{\bm{\omega}}(\mathbf{x}, \mathrm{t}))$.
For discrete outcomes, a Bernoulli likelihood can be used: $\mathrm{Bern}(\mathrm{y} \mid \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \mathrm{t}))$.
In both cases, $\widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \mathrm{t})$ is a parametric estimator of $\mathop{\mathbb{E}}[Y \mid \mathrm{T} = \mathrm{t}, \mathbf{X} = \mathbf{x}]$, which leads to: $\widehat{\tau}_{\bm{\omega}}(\mathbf{x}) \equiv \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, 1) - \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, 0)$, a parametric CATE estimator.
\citet{jesson2020identifying} have shown that Bayesian inference over the model parameters $\bm{\omega}$, treated as stochastic instances of the random variable $\bm{\Omega} \in \mathcal{W}$, yields a model capable of quantifying when assumption \ref{asm:overlap} (overlap) does not hold. Moreover, they show that such models can quantify when there is insufficient knowledge about the treatment effect $\tau(\mathbf{x})$ because the observed value $\mathbf{x}$ lies far from the support of $P_{\mathcal{D}}(\mathbf{X}, \mathrm{T}, \Y^0, \Y^1)$.
Such methods seek to enable sampling from the posterior distribution of the model parameters given the data, $p(\bm{\Omega} \mid \mathcal{D})$.
Each sample, $\bm{\omega} \sim p(\bm{\Omega} \mid \mathcal{D})$ induces a unique CATE function $\widehat{\tau}_{\bm{\omega}}(\mathbf{x})$.
\citet{jesson2020identifying} propose $\mathop{\mathrm{Var}}_{\bm{\omega} \sim p(\bm{\Omega} \mid \mathcal{D})}(\widehat{\mu}_{\bm{\omega}}(\mathbf{x}, 1) - \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, 0))$ as a measure of epistemic uncertainty (i.e., how much the functions ``disagree'' with one another at a given value $\mathbf{x}$) \citep{kendall2017uncertainties} for the CATE estimator.
\subsection{Active Learning}
Formally, an active learning setup consists of an unlabeled dataset $\D_{\mathrm{pool}} = \{\mathbf{x}_i\}_{i = 1}^{n_{\mathrm{pool}}}$, a labeled training set $\D_{\mathrm{train}} = \{\mathbf{x}_i, \mathrm{y}_i\}_{i = 1}^{n_{\mathrm{train}}}$, and a predictive model with likelihood $p_{\bm{\omega}}(\mathrm{y} \mid \mathbf{x})$ parameterized by $\bm{\omega} \sim p(\bm{\Omega} \mid \D_{\mathrm{train}})$.
The setup also assumes that an oracle exists to provide outcomes $\mathrm{y}$ for any data point in $\D_{\mathrm{pool}}$.
After model training, a batch of data $\{ \mathbf{x}^*_i \}_{i=1}^{b}$ is selected from $\D_{\mathrm{pool}}$ using an acquisition function $a$ according to the informativeness of the batch.
By including the treatment, we depart from the standard active learning setting.
For active learning of treatment effects, we define $\D_{\mathrm{pool}} = \{\mathbf{x}_i, \mathrm{t}_i\}_{i = 1}^{n_{\mathrm{pool}}}$, a labeled training set $\D_{\mathrm{train}} = \{\mathbf{x}_i, \mathrm{t}_i, \mathrm{y}_i\}_{i = 1}^{n_{\mathrm{train}}}$, and a predictive model with likelihood $p_{\bm{\omega}}(\mathrm{y} \mid \mathbf{x}, \mathrm{t})$ parameterized by $\bm{\omega} \sim p(\bm{\Omega} \mid \D_{\mathrm{train}})$.
The acquisition function takes as input $\D_{\mathrm{pool}}$ and returns a batch of data $\{\mathbf{x}_i, \mathrm{t}_i\}_{i=1}^{b}$ which are labelled using an oracle and added to $\D_{\mathrm{train}}$. We are specifically examining the case when there is access to only the treatments observed in the pool data $\{\mathrm{t}_i\}_{i = 1}^{n_{\mathrm{pool}}}$: i.e., scenarios where treatment assignment is not possible.
An intuitive way to define informativeness is using the estimated uncertainty of our model.
In general, we can distinguish two sources of uncertainty: epistemic and aleatoric uncertainty \citep{der2009aleatory,kendall2017uncertainties}.
Epistemic (or model) uncertainty, arises from ignorance about the model parameters.
For example, this is caused by the model not seeing similar data points during training, so it is unclear what the correct label would be.
We focus on using epistemic uncertainty to identify the most informative points for label acquisition.
\textbf{Bayesian Active Learning by Disagreement (BALD)} \citep{houlsby2011bayesian} defines an acquisition function based on epistemic uncertainty.
Specifically, it uses the mutual information (MI) between the unknown output and model parameters as a measure of disagreement:
\begin{equation}
\mathrm{I}(\mathrm{Y}; \bm{\Omega} \mid \mathbf{x}, \D_{\mathrm{train}}) = \mathrm{H}(\mathrm{Y} \mid \mathbf{x}, \D_{\mathrm{train}}) - \mathbb{E}_{\bm{\omega} \sim p(\bm{\Omega} \mid \D_{\mathrm{train}})} \left[ \mathrm{H}(\mathrm{Y} \mid \mathbf{x}, \bm{\omega}) \right],
\end{equation}
where $\mathrm{H}$ is the entropy function; a straightforward estimand for discrete outcomes with Bernoulli or Categorical likelihoods.
The general acquisition function based on BALD for acquiring a batch of data points given the pool dataset and the model parameters is given by the joint mutual information between the set $\{\mathrm{Y}_i\}$ and the model parameters \citep{kirsch2019batchbald}:
\begin{equation}
\label{eq:bald_acquisition_f}
a_{\mathrm{BALD}}(\D_{\mathrm{pool}}, p(\bm{\Omega} \mid \D_{\mathrm{train}})) = \argmax_{\{\mathbf{x}_i\}_{i=1}^{b} \subseteq \D_{\mathrm{pool}}} \mathrm{I}(\{\mathrm{Y}_i\}; \bm{\Omega} \mid \{\mathbf{x}_i\}, \D_{\mathrm{train}}).
\end{equation}
This batch acquisition function can be upper-bounded by scoring each point in $\D_{\mathrm{pool}}$ independently and taking the top $b$; however, this bound ignores correlations between the samples.
In fact, for datasets with significant repetition, this approach can perform worse than random acquisition, and computing the joint mutual information (introduced as \emph{BatchBALD}) rectifies the issue \citep{kirsch2019batchbald}.
Estimating the joint mutual information is computationally expensive, as evaluating the joint entropy over all possible outcomes (for classification) or a covariance matrix over all inputs (for regression) is required.
An alternative approach is to use softmax-BALD, which involves importance weighted sampling across $\D_{\mathrm{pool}}$ with the individual importance weights given by BALD \citep{kirsch2022stochastic}.
We use softmax-BALD for batch acquisition because it is computationally more efficient and performs competitively with BatchBALD.
We discuss how BALD maps onto epistemic uncertainty quantification in CATE and the arising complications stemming from the question of overlap in Section \ref{sec:methods}.
\section{Methods}
\label{sec:methods}
In this section: we introduce several acquisition functions, we then analyze how they bias the acquisition of training data, and we show the resulting CATE functions learned from such training data.
We are interested in acquisition functions conditioned on realizations of both $\mathbf{x}$ and $\mathrm{t}$:
\begin{equation}
\label{eq:causal_bald_acquisition_f}
a(\D_{\mathrm{pool}}, p(\bm{\Omega} \mid \D_{\mathrm{train}}))
= \!\!\!\!\! \argmax_{\{\mathbf{x}_i, \mathrm{t}_i\}_{i=1}^{b} \subseteq \D_{\mathrm{pool}}} \mathrm{I}(\bullet \mid \{ \mathbf{x}_i, \mathrm{t}_i \}, \D_{\mathrm{train}}),
\end{equation}
where $\mathrm{I}(\bullet \mid \mathbf{x}, \mathrm{t}, \D_{\mathrm{train}})$ is a measure of disagreement between parametric function predictions given $\mathbf{x}$ and $\mathrm{t}$ over samples $\bm{\omega} \sim p(\bm{\Omega} \mid \mathcal{D})$.
We make \cref{asm:consistency,asm:confounded} (consistency, and unconfoundedness).
We relax \cref{asm:overlap} (overlap) by allowing for its violation over subsets of the support of $\D_{\mathrm{pool}}$.
We present all theorems, proofs, and detailed assumptions in \cref{sec:theoretical_results}.
\subsection{How do naive acquisition functions bias the training data?}
\begin{figure}[t!]
\centering
\begin{subfigure}[]{0.245\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/intuition/bias_random.png}
\caption{Random}
\label{fig:bias_random}
\end{subfigure}
\hfill
\begin{subfigure}[]{0.245\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/intuition/bias_pi.png}
\caption{Propensity}
\label{fig:bias_pi}
\end{subfigure}
\hfill
\begin{subfigure}[]{0.245\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/intuition/bias_tau.png}
\caption{$\tau$BALD}
\label{fig:bias_tau}
\end{subfigure}
\hfill
\begin{subfigure}[]{0.245\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/intuition/bias_mu.png}
\caption{$\mu$BALD}
\label{fig:bias_mu}
\end{subfigure}
\caption{Naive acquisition functions: How the training set is biased and how this effects the CATE function with a fixed budget of 300 acquired points.}
\label{fig:naive_acquisitions}
\vspace{0em}
\end{figure}
To motivate Causal--BALD, we first look at a set of naive acquisition functions.
A random acquisition function selects data points uniformly at random from $\D_{\mathrm{pool}}$ and adds them to $\D_{\mathrm{train}}$.
In \cref{fig:bias_random} we have acquired 300 such examples from a synthetic dataset and trained a deep-kernel Gaussian process \citep{van2021improving} on those labeled examples.
Comparing the top two panels, we see that $\D_{\mathrm{train}}$ (middle) contains an unbiased sample of the data in $\D_{\mathrm{pool}}$ (top).
However, in the bottom panel, we see that while the CATE estimator is accurate and confident near the modes of $\D_{\mathrm{pool}}$, it becomes less accurate as we move to lower-density regions.
In this way, the random acquisition of data reflects the biases inherent in $\D_{\mathrm{pool}}$ and over-allocates resources to the modes of the distribution.
If the mode were to coincide with a region of non-overlap, the function would most frequently acquire uninformative examples.
Next, we look at using the propensity score to bias data acquisition toward regions where the overlap assumption is satisfied.
\begin{definition}
Counterfactual Propensity Acquisition
\begin{equation}
\mathrm{I}(\widehat{\pi}_{\mathrm{t}} \mid \mathbf{x}, \mathrm{t}, \D_{\mathrm{train}}) \equiv 1 - \widehat{\pi}_{\mathrm{t}}(\mathbf{x})
\label{eq:propensity-acquisition}
\end{equation}
\end{definition}
\vspace{-.7em}
Intuitively, this function prefers points where the propensity for observing the counterfactual is high.
We are considering the setup where $\D_{\mathrm{pool}}$ contains observations of both $\mathbf{X}$ and $\mathrm{T}$, so it is straightforward to train an estimator for the propensity, $\widehat{\pi}_{\mathrm{t}}(\mathbf{x})$.
\Cref{fig:bias_pi} shows that while propensity score acquisition matches the treated and control densities in the train set, it still biases data selection towards the modes of $\D_{\mathrm{pool}}$.
The goal of BALD is to acquire data $(\mathbf{x}, \mathrm{t})$ that maximally reduces uncertainty in the model parameters $\bm{\Omega}$ used to predict the treatment effect.
The most direct way to apply BALD is to use our uncertainty over the predicted treatment effect, expressed using the following information theoretic quantity:
\begin{definition}
$\tau\textrm{BALD}$
\begin{equation}
\mathrm{I}(\Y^1 - \Y^0; \bm{\Omega} \mid \mathbf{x}, \mathrm{t}, \D_{\mathrm{train}})
\approx \!\!\!\! \mathop{\mathrm{Var}}_{\bm{\omega} \sim p(\bm{\Omega} \mid \D_{\mathrm{train}})} \left( \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, 1) - \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, 0) \right).
\label{eq:tau_bald_acquisition}
\end{equation}
\end{definition}
\vspace{-.7em}
Building off the result in \citep{jesson2020identifying}, we show how the LHS measure about the \emph{unobservable potential outcomes} is estimated by the variance over $\bm{\Omega}$ of the \emph{identifiable difference in expected outcomes} in Theorem \ref{th:tau-BALD} of the appendix.
\citet{alaa2018bayesian} propose a similar result is for non-parametric models.
Intuitively, this measure represents the information gain for $\bm{\Omega}$ if we observe the difference in potential outcomes $\Y^1 - \Y^0$ for a given measurement $\mathbf{x}$ and $\D_{\mathrm{train}}$.
However, a fundamental flaw with this measure exists: observing labels for the random variable $\Y^1 - \Y^0$ is impossible.
Thus, $\tau\textrm{BALD}$~represents an irreducible measure of uncertainty.
That is, $\tau\textrm{BALD}$~will be high if it is uncertain about the label given the unobserved treatment $\tf^{\prime}$, regardless of its certainty about the outcome given the factual treatment $\mathrm{t}$, which makes $\tau\textrm{BALD}$~highest for low-density regions and regions with no overlap.
\Cref{fig:bias_tau} illustrates these consequences.
We see the acquisition biases the training data away from the modes of the $\D_{\mathrm{pool}}$, where we cannot know the treatment effect (no overlap).
In datasets where we have limited overlap, it leads to uninformative acquisitions.
One remedy to the issues of $\tau$BALD is to only focus on reducible uncertainty:
\begin{definition}
$\mu$BALD
\begin{equation}
\mathrm{I}(\Y^\tf; \bm{\Omega} \mid \mathbf{x}, \mathrm{t}, \D_{\mathrm{train}}) \;\;
\approx \!\!\!\! \mathop{\mathrm{Var}}_{\bm{\omega} \sim p(\bm{\Omega} \mid \D_{\mathrm{train}})} \left( \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \mathrm{t}) \right).
\label{eq:mu-bald-estimate}
\end{equation}
\end{definition}
This measure represents the information gain for the model parameters $\bm{\Omega}$ if we obtain a label for the observed potential outcome $\Y^\tf$ given a data point ($\mathbf{x}, \mathrm{t}$) and $\D_{\mathrm{train}}$.
We give proof for these results in Theorem \ref{th:mu-BALD} of the appendix.
$\mu$BALD only contains observable quantities; however, it does not account for our belief about the counterfactual outcome.
As illustrated in \cref{fig:bias_mu}, this approach can prefer acquiring $(\mathbf{x}, \mathrm{t})$ when we are also very uncertain about $(\mathbf{x}, \tf^{\prime})$, even if $(\mathbf{x}, \tf^{\prime})$ is not in $\D_{\mathrm{pool}}$.
Since we can neither reduce uncertainty over such $(\mathbf{x}, \tf^{\prime})$ nor know the treatment effect, the acquisition function would not be optimally data efficient.
\subsection{Causal--BALD.}
In the previous section, we looked at naive methods that either considered overlap or considered information gain. In this section, we present three measures that account for both factors when choosing a new point to acquire for model training.
\begin{figure}[t!]
\centering
\begin{subfigure}[]{0.32\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/intuition/bias_mu-pi.png}
\caption{$\mu\pi$BALD}
\label{fig:bias_mupi}
\end{subfigure}
\hfill
\begin{subfigure}[]{0.32\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/intuition/bias_rho.png}
\caption{$\rho$BALD}
\label{fig:bias_rho}
\end{subfigure}
\hfill
\begin{subfigure}[]{0.32\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/intuition/bias_mu-rho.png}
\caption{$\mu\rho$BALD}
\label{fig:bias_causal}
\end{subfigure}
\caption{Causal--BALD acquisition functions: How the training set is biased and how this effects the CATE function with a fixed budget of 300 acquired points.}
\vspace{0em}
\end{figure}
A straightforward to combine knowledge about a data point's information gain and overlap is to simply multiply $\mu\textrm{BALD}$ \eqref{eq:mu-bald-estimate} by the propensity acquisition term \eqref{eq:propensity-acquisition}:
\begin{definition}
$\mu\pi$BALD
\begin{equation}
\mathrm{I}(\mu \pi \mid \mathbf{x}, \mathrm{t}, \D_{\mathrm{train}})
\equiv (1 - \widehat{\pi}_{\mathrm{t}}(\mathbf{x})) \mathop{\mathrm{Var}}_{\bm{\omega} \sim p(\bm{\Omega} \mid \D_{\mathrm{train}})} \left( \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \mathrm{t}) \right)
\end{equation}
\end{definition}
We can see in \cref{fig:bias_mupi} that the acquisition of training data results in matched sampling that we saw for propensity acquisition in \cref{fig:bias_pi}. However, the tails of the overlapping distributions extend further into the low-density regions of the pool set support where the overlap assumption is satisfied.
Alternatively, we can take an information-theoretic approach to combine knowledge about a data point's information gain and overlap.
Let $\widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \mathrm{t})$ be an instance of the random variable $\widehat{\mu}_{\bm{\Omega}}^{\mathrm{t}} \in \mathbb{R}$ corresponding to the expected outcome conditioned on $\mathrm{t}$.
Further, let $\widehat{\tau}_{\bm{\omega}}(\mathbf{x})$ be an instance of the random variable $\widehat{\tau}_{\bm{\Omega}} = \widehat{\mu}_{\bm{\Omega}}^{1} - \widehat{\mu}_{\bm{\Omega}}^{0}$ corresponding to the CATE.
Then,
\begin{definition}
$\rho$BALD
\begin{equation}
\mathrm{I}(\Y^\tf; \widehat{\tau}_{\bm{\Omega}} \mid \mathbf{x}, \mathrm{t}, \D_{\mathrm{train}})
\gtrapprox \frac{1}{2}\log{\left(\frac{\mathop{\mathrm{Var}}_{\bm{\omega}} \left( \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \mathrm{t}) \right) - 2\mathop{\mathrm{Cov}}_{\bm{\omega}}(\widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \mathrm{t}), \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \tf^{\prime}))}{\mathop{\mathrm{Var}}_{\bm{\omega}} \left( \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \tf^{\prime}) \right)} + 1\right)}.
\label{eq:rho_def}
\end{equation}
\end{definition}
This measure represents the information gain for the CATE $\tau_{\bm{\Omega}}$ if we observe the outcome $\mathrm{Y}$ for a datapoint ($\mathbf{x}$, $\mathrm{t}$) and the data we have trained on $\D_{\mathrm{train}}$.
We give proof for this result in Theorem \ref{th:rho-BALD}.
In contrast to $\mu$-BALD, this measure accounts for overlap in two ways.
First, $\rho$--BALD will be scaled by the inverse of the variance of the expected counterfactual outcome $\widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \tf^{\prime})$.
This scaling biases acquisition towards examples for which we know about the counterfactual outcome, so we can assume that overlap is satisfied for observed $(\mathbf{x}, \mathrm{t})$.
Second, $\rho$--BALD is discounted by $\mathop{\mathrm{Cov}}_{\bm{\omega}}(\widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \mathrm{t}), \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \tf^{\prime}))$.
This discounting is a concept that we will leave for future discussion.
In \cref{fig:bias_rho} we see that $\rho$--BALD has matched the distributions of the treated and control groups similarly to propensity acquisition in \cref{fig:bias_pi}.
Further, we see that the CATE estimator is more accurate over the support of the data.
There is, however, a shortcoming of $\rho$--BALD that may lead to suboptimal data efficiency.
Consider two examples in $\D_{\mathrm{pool}}$, $(\mathbf{x}_1, \mathrm{t}_1)$ and $(\mathbf{x}_2, \mathrm{t}_2)$ where $\mathop{\mathrm{Var}}_{\bm{\omega}}(\widehat{\mu}_{\bm{\omega}}(\mathbf{x}_1, \mathrm{t}_1)) = \mathop{\mathrm{Var}}_{\bm{\omega}}(\widehat{\mu}_{\bm{\omega}}(\mathbf{x}_1, \tf^{\prime}_1))$ and $\mathop{\mathrm{Var}}_{\bm{\omega}}(\widehat{\mu}_{\bm{\omega}}(\mathbf{x}_2, \mathrm{t}_2)) =\mathop{\mathrm{Var}}_{\bm{\omega}}( \widehat{\mu}_{\bm{\omega}}(\mathbf{x}_2, \tf^{\prime}_2))$: for each point, we are as uncertain about the conditional expectation given the factual treatment as we are uncertain given the counterfactual treatment.
Further, let $\mathop{\mathrm{Cov}}_{\bm{\omega}}(\widehat{\mu}_{\bm{\omega}}(\mathbf{x}_1, \mathrm{t}_1), \widehat{\mu}_{\bm{\omega}}(\mathbf{x}_1, \tf^{\prime}_1)) = \mathop{\mathrm{Cov}}_{\bm{\omega}}(\widehat{\mu}_{\bm{\omega}}(\mathbf{x}_2, \mathrm{t}_2), \widehat{\mu}_{\bm{\omega}}(\mathbf{x}_2, \tf^{\prime}_2))$.
Finally, let $\mathop{\mathrm{Var}}_{\bm{\omega}}(\widehat{\mu}_{\bm{\omega}}(\mathbf{x}_1, \mathrm{t}_1)) > \mathop{\mathrm{Var}}_{\bm{\omega}}(\widehat{\mu}_{\bm{\omega}}(\mathbf{x}_2, \mathrm{t}_2))$: we are more uncertain about the conditional expectation given the factual treatment for data point $(\mathbf{x}_1, \mathrm{t}_1)$ than we are for data point $(\mathbf{x}_2, \mathrm{t}_2)$.
Under these three conditions, $\rho$--BALD would rank these two points equally, and so this method would bias training data to the modes of $\D_{\mathrm{pool}}$ when $(\mathbf{x}_2, \mathrm{t}_2)$ is more frequent than $(\mathbf{x}_1, \mathrm{t}_1)$.
In practice, it may be more data-efficient to choose $(\mathbf{x}_1, \mathrm{t}_1)$ over $(\mathbf{x}_2, \mathrm{t}_2)$ as it would more likely be a point as yet unseen by the model.
To combine the positive attributes of $\mu$--BALD and $\rho$--BALD, while mitigating their shortcomings, we introduce $\mu\rho$BALD.
\begin{definition}
$\mu\rho$BALD
\begin{equation}
\begin{split}
\mathrm{I}(\mu \rho \mid \mathbf{x}, \mathrm{t}, \D_{\mathrm{train}})
&\equiv \mathop{\mathrm{Var}}_{\bm{\omega}}\left( \widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \mathrm{t}) \right) \frac{\mathop{\mathrm{Var}}_{\bm{\omega}}(\widehat{\tau}_{\bm{\omega}}(\mathbf{x}))}{\mathop{\mathrm{Var}}_{\bm{\omega}}(\widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \tf^{\prime}))}.
\end{split}
\end{equation}
\end{definition}
Here, we scale Equation \ref{eq:rho_def}, which has equivalent expression $\frac{\mathop{\mathrm{Var}}_{\bm{\omega}}(\widehat{\tau}_{\bm{\omega}}(\mathbf{x}))}{\mathop{\mathrm{Var}}_{\bm{\omega}}(\widehat{\mu}_{\bm{\omega}}(\mathbf{x}, \tf^{\prime}))}$ by our measure for $\mu$BALD such that in the cases where the ratio may be equal, there is a preference for data points the current model is more uncertain about.
We can see in \cref{fig:bias_causal} that the training data acquisition is distributed more uniformly over the support of the pool data where the overlap assumption is satisfied.
Furthermore, the accuracy of the CATE estimator is highest over that region.
\section{Related Work}
\label{sec:related_work}
\Citet{deng2011active} propose the use of Active Learning for recruiting patients to assign treatments that will reduce the uncertainty of an Individual Treatment Effect model.
However, their setting is different from ours -- we assume that suggesting treatments are too risky or even potentially lethal.
Instead, we acquire patients to reveal their outcome (e.g., by having a biopsy).
Additionally, although their method uses predictive uncertainty to identify which patients to recruit, it does not disentangle the sources of uncertainty; therefore, it will also recruit patients with high outcome variance.
Closer to our proposal is the work from \citet{sundin2019active}.
They propose using a Gaussian process (GP) to model the individual treatment effect and use the expected information gain over the S-type error rate, defined as the error in predicting the sign of the CATE, as their acquisition function.
Although GPs are suitable for quantifying uncertainty, they do not work well on high-dimensional input spaces.
In this work, we use Neural network methods to obtain uncertainty: Deep Ensembles \citep{lakshminarayanan2017simple} and DUE \citep{van2021improving}, a Deep Kernel Learning GP, both of which work well even on high dimensional inputs.
Additionally, the authors assume that noisy observations about the counterfactual treatments are available at training time where we make no such assumptions.
We compare to this in our experiment by limiting the access to counterfactual observations ($\boldsymbol\gamma$ baseline) and adapting it to Deep Ensembles \citep{lakshminarayanan2017simple} and DUE \citep{van2021improving} (we provide more details about the adaptation in Appendix \ref{a:sundin}).
Recent work by \citet{qin2021budgeted} looks at budgeted heterogeneous effect estimation but does not factor weak or limited overlap into their acquisition function.
\section{Experiments}
\label{sec:experiments}
In this section, we evaluate our acquisition objectives on synthetic and semi-synthetic datasets. Code to reproduce these experiments is available at \url{https://github.com/OATML/causal-bald}.
\noindent\textbf{Datasets}
Starting from the hypothesis that different objectives can target different types of imbalances and degrees overlap, we construct a \textbf{synthetic} dataset~\citep{kallus2019interval} demonstrating the various biases.
We depict this dataset graphically in \cref{fig:observational_data}.
We use this dataset primarily for illustrative purposes.
By design, we have constructed a primary data mode and have regions of weak or no overlap.
Additionally, we study the performance of our acquisition functions on the \textbf{IHDP} dataset~\citep{hill2011bayesian, shalit2017estimating}, which is a standard benchmark in causal treatment effect literature.
Finally, we demonstrate that our method is suitable for high-dimensional, large-sample datasets on \textbf{CMNIST}~\cite{jesson2021quantifying}, an MNIST~\citep{lecun1998mnist} based dataset adapted for causal treatment effect studies.
In \cref{fig:cmnist}, we see that CMNIST is an adaptation of the synthetic dataset.
Model inputs are MNIST digits and assigned treatments, and the response surfaces are generated based on a projection of the digits onto a latent 1-dimensional manifold.
The observed digits are high-dimensional proxies for the confounding covariate $\phi$.
Detailed descriptions of each dataset are available in \cref{a:datasets}.
\begin{wrapfigure}{r}{0.45\textwidth}
\vspace{-1.4em}
\begin{center}
\includegraphics[width=0.45\textwidth]{figures/cmnist.png}
\end{center}
\vspace{-0.2em}
\caption{
Visualizing CMNIST dataset.
Model inputs are MNIST digits and assigned treatments.
The MNIST digits are high-dimensional proxies for the latent confounding covariate $\phi$.
Digits are projected onto $\phi$ by ordering them first by image intensity and then by digit class (0 - 9).
Methods must be able to implicitly learn this non-linear mapping in order to predict the conditional expected outcomes.
}
\label{fig:cmnist}
\vspace{-1.2em}
\end{wrapfigure}
\noindent\textbf{Model}
Our objectives rely on methods that are capable of modeling uncertainty and handling high-dimensional data modalities.
DUE \citep{van2021improving} is an instance of Deep Kernel Learning \citep{wilson2016deep} that uses a deep feature extractor to transform the inputs and defines a Gaussian process (GP) kernel over the extracted feature representation.
In particular, DUE uses a variational inducing point approximation \citep{hensman2015scalable} and a constrained feature extractor that contains residual connections and spectral normalization to enable reliable uncertainty.
Due obtains SotA results on IHDP \citep{van2021improving}.
In DUE, we distinguish between the model parameters $\theta$ and the variational parameters $\bm{\omega}$, and we are Bayesian only over the $\bm{\omega}$ parameters.
Since DUE is a GP, we obtain a full Gaussian posterior over outcomes from which we can use the mean and covariance directly.
When necessary, sampling is very efficient and only requires a single forward pass in the deep model.
We describe all hyperparameters in Appendix \ref{a:architecture}.
\noindent\textbf{Baselines}
We compare against the following baselines:
\noindent\textbf{Random.} This acquisition function selects points uniformly at random.
\noindent\textbf{Propensity.} An acquisition function based on the propensity score (Eq.~\ref{eq:propensity-acquisition}).
We train a propensity model on the pool data, which we then use to acquire points based on their propensity score.
Please note that this is a valid assumption as training a propensity model does not require outcomes.
\noindent\textbf{$\boldsymbol\gamma$ (S-type error rate)~\citep{sundin2019active}.} This acquisition function is the S-type error rate based method proposed by~\citet{sundin2019active}.
We have adapted the acquisition function to use with Bayesian Deep Neural Networks.
The objective is defined as $\mathrm{I}(\boldsymbol\gamma;\bm{\Omega}\mid\mathbf{x},\D_{\mathrm{train}})$, where $\gamma(x)=\text{probit}^{-1}\left(-\frac{|\mathbf{E}_{p(\tau\mid \mathbf{x}, \D_{\mathrm{train}})}[\tau]|}{\sqrt{\text{Var}(\tau|\mathbf{x}, \D_{\mathrm{train}})}}\right)$ and $\text{probit}^{-1}(\cdot)$ is the cumulative distribution function of normal distribution.
In contrast to the original formulation, we do not assume access to counterfactual observations at training time.
\subsection{Experimental Results}
\label{sec:results}
For each of the acquisition objectives, dataset, and model we present the mean and standard error of empirical square root of precision in estimation of heterogenous effect (PEHE)~\footnote{$\sqrt{\epsilon_{PEHE}}=\sqrt{ \frac{1}{N} \sum_x{(\hat{\tau}(x)-\tau(x))^2}}$}.
We summarize each active learning setup in ~\cref{table:acquisition_settings}.
The \emph{warm up size} is the number of examples in the initial pool dataset.
\emph{Acquisition size} is the number of examples labeled at each acquisition step.
\emph{Acquisition steps} is the number of times we query a batch of labels.
\emph{Pool size} is the number of examples in the pool dataset.
Finally, \emph{valid size} is the number of examples used for model selection when optimizing the model at each acquisition step.
\begin{table}[t!]
\vspace{0em}
\centering
\label{table:acquisition_settings}
\caption{Summary of active learning parameters for each dataset.}
\begin{tabular}{lccccc}
\toprule
\textbf{Dataset} & \textbf{Warm-up size} & \textbf{Acquisition size} & \textbf{Acquisition steps} & \textbf{Pool Size} & \textbf{Valid Size} \\
\midrule
Synthetic & 10 & 10 & 30 & 10k & 1k \\
IHDP & 100 & 10 & 38 & 471 & 201 \\
CMNIST & 250 & 50 & 55 & 35k & 15k \\
\bottomrule
\end{tabular}
\vspace{0em}
\end{table}
\begin{figure*}[t!]
\centering
\begin{subfigure}[b]{\linewidth}
\includegraphics[width=.9\linewidth]{figures/results/legends.png}
\end{subfigure}
\\~\\
\begin{subfigure}[b]{0.32\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/results/due/synthetic/baselines_expected_pehe.png}
\end{subfigure}
\begin{subfigure}[b]{0.32\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/results/due/ihdp/baselines_expected_pehe.png}
\end{subfigure}
\begin{subfigure}[b]{0.32\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/results/due/cmnist/baselines_expected_pehe.png}
\end{subfigure}
\begin{subfigure}[b]{0.32\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/results/due/synthetic/balds_expected_pehe.png}
\end{subfigure}
\begin{subfigure}[b]{0.32\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/results/due/ihdp/balds_expected_pehe.png}
\end{subfigure}
\begin{subfigure}[b]{0.32\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/results/due/cmnist/balds_expected_pehe.png}
\end{subfigure}
\caption{
$\sqrt{\epsilon_{PEHE}}$ performance (shaded standard error) for DUE models. \textbf{(left to right)} \textbf{synthetic} (40 seeds), and \textbf{IHDP} (200 seeds). We observe that BALD objectives outperform the \textbf{random}, \textbf{$\boldsymbol\gamma$} and \textbf{propensity} acquisition functions significantly, suggesting that epistemic uncertainty aware methods that target reducible uncertainty can be more sample efficient.
}
\label{fig:main_results}
\end{figure*}
In \cref{fig:main_results}, we see that epistemic uncertainty aware ~$\mu\rho\textrm{BALD}$~ outperforms the baselines, random, propensity, and S-Type error rate ($\boldsymbol\gamma$).
As analyzed in section \ref{sec:methods}, we expect this improvement as our acquisition objectives target reducible uncertainty -- that is, epistemic uncertainty when there is overlap between treatment and control.
Additionally, $\mu\rho\textrm{BALD}$ shows superior performance over the other objectives in the high-dimensional dataset CMNIST verifying our qualitative analysis in Figure~\ref{fig:bias_causal}.
Each of ($\mu\textrm{BALD}$, $\rho\textrm{BALD}$, $\mu\pi\textrm{BALD}$, and $\mu\rho\textrm{BALD}$) outperforms the baseline methods on these tasks.
Of note, the performance $\rho\textrm{BALD}$~improves as the dimensionality of the covariates increases.
In contrast, the performance of the propensity score-based $\mu\pi\textrm{BALD}$~worsens as the dimensionality of the covariates increases.
Propensity score estimation is known to be a problem in high-dimensions \citep{d2021overlap}.
We see that both $\mu\textrm{BALD}$~ and $\mu\rho\textrm{BALD}$~ perform consistently as dimensionality increases, with $\mu\rho\textrm{BALD}$~ showing a statistically significant improvement over $\mu\textrm{BALD}$~ on two of the three tasks.
These improvements indicate that $\mu\rho\textrm{BALD}$~ is more robust for data with high-dimensional covariates than $\mu\pi\textrm{BALD}$~; moreover, $\mu\rho\textrm{BALD}$~ does not need an additional propensity score model.
\section{Conclusion}
\label{sec:conclusion}
\vspace{-.5em}
We have introduced a new acquisition function for active learning of individual-level causal-treatment effects from high dimensional observational data, based on Bayesian Active Learning by Disagreement~\cite{houlsby2011bayesian}.
We derive our proposed method from an information-theoretic perspective and compare it with acquisition strategies that either do not consider epistemic uncertainty (i.e., random or propensity-based) or target irreducible uncertainty in the observational setting (i.e., when we do not have access to counterfactual observations).
We show that our methods significantly outperform baselines, while also studying the various properties of each of our proposed objectives in both quantitative and qualitative analyses, potentially impacting areas like healthcare where sample efficiency in the acquisition of new examples implies improved safety and reductions in costs.
\vspace{-.5em}
\section{Broader Impact}
\label{sec:impact}
\vspace{-.5em}
Active Learning for learning treatment effects from observational data is highly actionable research, and there are several sectors where our research can have an impact.
Take, for example, a hospital that needs to decide whom to treat, based on some model.
To these choices, the decision-maker needs to have a confident and accurate treatment effect prediction model.
However, improving the performance of such a model requires data from patients, which might be costly and perhaps even unethical to acquire.
With this work, we assume that we cannot assign new treatments to patients but only perform biopsies or questionnaires post-treatment to reveal the outcome.
We believe that this is an impactful and realistic scenario that will directly benefit from our proposal.
However, our method can also impact fields like computational-advertisement, where the goal is to learn a model to predict the captivate the attention of users, or policymaking where a government wants to decide how to intervene for beneficial or malicious reasons.
Active learning inherently biases the acquisition of training data.
We attempt to show how this can be beneficial or detrimental for learning treatment effects under different acquisition functions under the unconfoundedness assumption.
We are not making guarantees on the overall unbiasedness of our methods.
We guarantee only that our results are conditional on the unconfoundedness assumption.
Unobserved confounding can result in the biased sampling of training data concerning the hidden confounding variable.
This bias can result in performance inequality between groups and a biased estimate of the unconfounded CATE function.
Further, the models’ uncertainty estimates are not informative of when this may occur.
An anonymous reviewer writes, ``one risk is that the method could, e.g., lead to a biased, non-representative sampling in terms of ethnicity and other protected attributes - particularly, if the unconfoundedness assumption this work is based on is blindly trusted''
Recent work by \citet{andrus2021we} does a great job of highlighting the difficulties practitioners face when accounting for algorithmic bias across protected attributes.
In such cases, model uncertainty is not enough to identify non-representative sampling concerning the protected attribute.
They report about a practitioner's use of structural causal models in concert with domain expert feedback as a means to inform clients of potential sources of bias. Perhaps such methodology could be used with causal sensitivity analysis for CATE \citep{kallus2019interval, yadlowsky2018bounds, jesson2021quantifying} as a way to model beliefs about the protected attribute without observing it.
Impactful avenues for future work include relaxing the unconfoundedness assumption, incorporating beliefs about hidden confounding into the acquisition function.
Furthermore, in addition to uncovering the outcome, we think it is interesting to revisit the active treatment assignment problem.
On the active learning side, exploring more rigorous batch acquisition methods could yield improvements over the current stochastic sampling estimation we use.
Finally, in this work, we assume access to a validation set, which may not be available in practice, so exploration of active acquisition of validation data will also have an impact \citep{kossen2021active}.
\begin{ack}
We would like to thank OATML group members and all anonymous reviewers for sharing their valuable feedback and insights.
PT and AK are supported by the UK EPSRC CDT in Autonomous Intelligent Machines and Systems (grant reference EP/L015897/1).
JvA is grateful for funding by the EPSRC (grant reference EP/N509711/1) and by Google-DeepMind.
U.S. was partially supported by the Israel Science Foundation (grant No. 1950/19).
\end{ack}
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\section{Introduction}\label{sec:introduction}}
\IEEEPARstart{I}{t} has been shown that image classification can be performed directly from the discrete cosine transformations used in the standard JPEG compression format, that this procedure is more computationally efficient and maintains performance comparable to state-of-the-art methods for image classification tasks \cite{gueguen2018faster}. In recent years a growing research community has emerged around learned image compression strategies based on deep neural networks, and many results have demonstrated that such approaches can surpass highly engineered approaches to image compression \cite{Agustsson2018, Balle2018, minnen2018joint}.
\begin{figure*}[!ht]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{figures/architecture/JTC.pdf} &
\includegraphics[width=0.50\linewidth]{figures/architecture/downstreamtasks.pdf}
\\
\footnotesize{(a) We jointly train an image compressor.} & \footnotesize{(b) \textbf{New} tasks are trained from \textbf{fixed} compressed format.}
\end{tabular}
\caption{A high level view of our strategy for learning compressible image representations for machine perception. We jointly train a model to compress as well as perform key vision tasks directly from the compressible representation. The approach leads to significant gains compared to both JPEG or naive learned deep compression when training new models for new tasks directly from our learned compressible representations. }
\label{fig:general_view}
\end{figure*}
Neural image compression is normally formulated as an optimization balancing a trade-off between two objectives: the compression ratio and image distortion (or image quality). A common way to achieve this is to produce a representation that contains the information necessary for reconstruction while maintaining low entropy on the latent representation, under some probability model. This latent space can then be compressed to a bit-stream using a lossless compression algorithm, thus achieving a lower BPP compression of the image. In contrast, in our work here we study whether this latent space representation can be further biased or re-structured such that it better represents and retains information that is relevant for subsequent semantically oriented vision tasks such as classification, detection, and segmentation.
Intuitively, our work aims to develop an approach to learning neural image compression in a way that preserves or improves performance on common machine perception tasks and yields representations that likely to retain information required when training models for \emph{new} tasks - since the representation is explicitly learned such that it is capable of reconstructing the original image. In our framework, these different tasks are performed by task specific decoders that operate directly on the (common) learned, compressible representation. Furthermore, we test the hypothesis that our neural compression models learned in this way might be particularly good at adapting to new tasks using a competitive low-show learning benchmark evaluation. Our approach yields state of the art performance on this evaluation. A high level view of our approach is shown in Figure \ref{fig:general_view}.
We explore these ideas by biasing the learned representations of neural encoders specifically constructed to yield quantizable and compressible codes (under arithmetic coding) by jointly training them with alternative decoders for auxiliary tasks. We conjecture that this should lead to compressible representations having comparable rate/distortion characteristics while better preserving and structuring the information necessary for performing computer vision tasks directly from these highly compressible codes. We underscore that this is potentially a challenging setting, since other transfer learning work has shown that computer vision tasks such as auto-encoding and semantic segmentation can be less compatible than others, leading to decreased performance when models are trained on both tasks simultaneously \cite{zamir2018taskonomy}.
The contribution of our work here is to propose and explore a joint training procedure that allows models to learn highly compressible representations, while minimizing the impact on performance on key machine perception tasks. Importantly, in our approach these tasks are performed directly from the compressed representation, without the need for decoding the image. We test our hypothesis that this approach might yield representations that are well suited for new tasks and few-shot learning. We find that they achieve state-of-the-art performance on few-shot image detection when learning directly from our task-informed compressible representations. Furthermore, our representations can be compressed to be \emph{four to ten times smaller than standard JPEG imagery} while improving performance or suffering only a minor degradation on the key vision tasks of classification, detection and segmentation.
\section{Learning Compressible Representations for Machine Perception Tasks}
\subsection{Preliminaries: Rate-Distortion and Perceptual Quality}
The traditional goal of lossy image compression is to find the best trade-off between minimizing the expected length of the bit-stream representing the input image, $x$, and minimizing the expected distortion between the image and its reconstruction $\hat{x}$.
In the information theory literature, this is known as the rate-distortion optimization problem \cite{cover1999elements}.
In the context of learned image compression, this can be formulated as a multi-task loss:
\begin{equation
\mathcal{R}(\mathbf{Q}(f(x))) + \lambda \cdot \mathcal{D}(g(\mathbf{Q}(f(x))), x),
\label{eq:r_and_d}
\end{equation}
where $\mathcal{R}$ represents the rate or
bit-stream length, and $\mathcal{D}$ represents the distortion or reconstruction fidelity, usually implemented by a distance function. To find such an encoding, the non-linear transform coding strategy \cite{goyal2001theoretical} consists of producing a discrete latent representation $z$ from an image $x$ using an \textit{encoder}, i.e. $z = f(x)$. This real-valued representation is quantized, by a quantization operator $\mathbf{Q}$, into $\hat{z}$.
The reconstructed input, $\hat{x}$, is generated by the image decoder $g(\hat{z})$. Because both objectives cannot be completely minimized simultaneously, different values of $\lambda$ lead to a different operating points in the trade-off between rate and distortion. Note that the representation $\hat{z}$ must be an element of a finite set to be losslessly encoded to a bit-stream with a lossless compression algorithm such as arithmetic coding \cite{langdon1984introduction}.
Because perceptual quality is not directly correlated with distortion, a perceptual quality metric is often added to the loss \citep{blau2019rethinking}. This idea has been used to improve the quality of image compression algorithms \citep{mentzer2020high}.
\subsection{Rate-Distortion-Utility Perspective}
The representation $z$ can be encoded in a bit stream
and stored or transmitted, but also used directly to perform visual tasks \textit{without reconstructing} the input image \citep{torfason2018towards}. This common use-case highlights the potential of exploring another dimension beyond the classical rate-distortion optimization trade-off. Rather than thinking of perceptual quality as the difference between reconstructed and the original images, we are interested in the direct utility of the learned compressed representation for machine perception tasks.
This leads to our \emph{key hypothesis}:
For any desired rate, $R$, and distortion level $D$, we conjecture that there exists many different, quantized latent representations, $\hat{z}=\mathbf{Q}(f(x))$, which lead to significant differences in performance on subsequent machine vision tasks. In other words, there exists many representations which achieve a given rate-distortion trade-off, but some representations may be better suited for subsequent semantic tasks than others.
In this context, we propose that the objective function to be optimized should go beyond the classical rate-distortion paradigm, to include a measure of the utility of the representation for concrete machine perception tasks:
\begin{equation}
\mathcal{R}(\mathbf{Q}(f(x)), x) + \lambda_d \cdot \mathcal{D}(g(\mathbf{Q}(f(x))), x) + \lambda_u \cdot \mathcal{U}(\mathbf{Q}(f(x))), \label{eq:RDU}
\end{equation}
where $\mathcal{U}(\cdot)$ is a pragmatically defined utility, or machine perceptual quality metric.
In contrast to the usual notion of human perceptual quality which is often characterized by metrics such as squared error, we can define machine vision perceptual quality much more pragmatically, i.e. based on a multi-task loss:
\begin{equation}
\mathcal{U}(\mathbf{Q}(f(x))) =
\sum_{t\in\mathcal{T}} \lambda_t [L_{t}(h_t(f(x)), y_t)], \label{eq:u_tasks}
\end{equation}
where $L_{t}$ is a loss function for an specific task $t$ with respect to the labels
$y_t$. The functions $h_t$ are the task specific decoders. We conjecture that the optimization of learned compression models for rate $\mathcal{R}$, distortion $\mathcal{D}$, and utility $\mathcal{U}$ as defined in (\ref{eq:RDU}) will result in encoders $f(x)$ that are capable of producing low bit-per-pixel quantized representations $\hat{z}$ that are better structured to facilitate the learning of new machine perception tasks directly from $\hat{z}$.
\subsection{Compression Architecture}
We base our image compression approach on the hierarchical variational-auto-encoder (VAE) formulation proposed by \cite{balle2018variational}. The function $\mathcal{R}$ is computed as the expected marginal probability of the latent \emph{quantized} representation, $\E[-\log_{2}p_{\hat{z}}(\hat{z})]$, where $\hat{z}$ is a the quantized encoding of an input image and the expectation is with respect to the empirical distribution of a training set.
A Gaussian Mean-Scale Model (GMSM) is used for $p_{\hat{z}}(\hat{z})$ as suggested in \citep{balle2018variational,minnen2018joint}. Each element of $\hat{z}$ is modeled by a Gaussian with a mean and a unique scale for a given lossless compression ratio.
Since quantization is not a differentiable operation, a special form of uniform noise is added to the latent variables $z$ during training, which yields a differentiable strategy in which one must ensure that the amplitude of the noise keeps the codes within the same quantization bin. This induces the encoder to learn to output a code that is invariant to integer quantization. Recall that the addition of two independent random variables follows the convolution of their individual distributions. Therefore, the addition of uniform noise to the latent representation leads to a model of the form:
\begin{equation}
p_{\hat{z}}(\hat{z}) = \prod_i \left(\mathcal{N}(\mu_i, \sigma_i ) * u\left(-\frac{1}{2}, \frac{1}{2}\right)\right)(\hat{z}_i),
\end{equation}
where all the elements of $p_{\hat{z}}(\hat{z})$ are independent and `$*$' denotes the convolution operator.
The values of $\mu_i$ and $\sigma_i$ are the mean an variances output by the \textit{hyper-prior} network comprised of a separate VAE which encodes the latent representation $z$ into the quantized hyper-parameter $\hat{w}$ and decodes it into the parameters of the probabilisitic model $p_{\hat{z}}$. See Appendix I for further details.
We use a mean squared error distortion metric, which results in the following loss function for our naive (i.e. in that it is not task informed) compression framework:
\begin{equation}
\label{eq:compression_optimization}
\E[-\log_{2}p_{\hat{z}}(\hat{z})] + \E[-\log_{2}p_{\hat{w }}(\hat{w})] + \lambda \E[\norm{x - g(\hat{z})}^2],
\end{equation}
where $\E[-\log_{2}p_{\hat{z}}(\hat{z})]$ is the expected compression rate of the latent representation $\hat{z}$ and $\E[-\log_{2}p_{\hat{w}}(\hat{w})]$ the expected compression rate of the hyper-latent representation $\hat{w}$.
The loss corresponding to our task informed compression setup is obtained
by combining the multi-task loss from (\ref{eq:u_tasks}) and using our rate-distortion-utility framework to obtain our task informed learned compression optimization framework:
\begin{multline}\label{eq:final_semantic_compression}
\E[-\log_{2}p_{\hat{z}}(\hat{z})] + \E[-\log_{2}p_{\hat{w}}(\hat{w} )] +
\lambda_e \E[\norm{x - g(\hat{x})}^2] \\ +
\sum_{t\in\mathcal{T}} \lambda_t \E[L_{t}(h_t(\hat{z}), y_t)].
\end{multline}
\subsection{A Multi-task Inference and Compression Architecture}
\begin{figure*}[!h]
\centering
\includegraphics[width=1.0\linewidth]{figures/architecture/architecture3.pdf}
\caption {Detailed architecture of the proposed model for jointly training of image compression with different computer vision tasks.}
\label{fig:architecture}
\end{figure*}
Here we illustrate the architecture, $h(z)$, used to produce results for different vision tasks. The same architecture is used both to optimize for the machine perception quality
and perform inference from the compressible representation.
A detailed visualization of our proposed architecture is depicted in Figure \ref{fig:architecture}.
The compression architecture is depicted at the top of the figure (encoder/decoder). This architecture is the same as proposed by \cite{cheng2020learned}. The encoder is composed of four convolutional layers interlaced by so-called Generalized Divisive Normalization (GDN) layers \cite{Balle2016} for activation. The encoder expects an RGB image of size $3 \times W \times H$ and outputs a tensor of shape $256 \times W/16\times H/16$. Note, the rank of the compressible format is one third of the input image rank, but is actually smaller still (in bpp) because of it's redundancy.
In order to perform vision tasks directly from the latent space, it is necessary to adapt the backbone ResNet architecture, commonly used to extract useful features for vision tasks. Compared to the original image, the input is comprised of many channels with lower spatial resolution.
Several solutions have been proposed to this problem. In \citep{gueguen2018faster}, the authors observed that a learned deconvolution yields better results since it helps overcoming the reduction in receptive field. In \cite{torfason2018towards}, the authors proposed to simply remove the the first layers of a Resnet and input the compressed space as is.
This inputs the latent space directly to the first Resnet bottleneck block,
which does not allow a space with bigger receptive field to be included.
Here we observed the necessity of recovering the full
image resolution and propose a pixel shuffle adapter that
outputs directly to the bottleneck blocks of the Resnet.
It was observed in \cite{torfason2018towards}, especially for lower bit rate configurations, a notable loss in performance for tasks that require spatial information such as semantic segmentation.
Pixel shuffle \cite{shi2016real} performs sub-pixel resolution
convolutions and mitigates any potential loss of spatial information.
The architecture, refereed as \textit{sResnet} is depicted Fig. \ref{fig:architecture},
and contains three convolutional blocks (which include ReLUs) interlaced with two pixel shuffle blocks. A complete detailed
description of the architecture is presented in the supplementary material.
\section{Experimental Results and Discussion}
\label{sec:results}
We investigate the potential of performing classification, object detection, few-shot detection and segmentation directly from the different, previously described, informed compressible representations and how well this information generalizes to new tasks and categories. In other words, we investigate whether compression formats can be biased to preserve semantic information. First, we study compression performance and show that our approach is on-par with known state-of-the art CNN-based models and exceeds the performance of popular compression algorithms.
We examine the impact of adding different decoder networks trained to perform computer vision tasks directly from our compressible representation. This includes training directly from compressible codes for semantic segmentation, a held-out task not used to train the representation. We also provide an ablation analysis and examine different model variants. We then explore this approach for learning low shot object detection. Finally, we examine questions of computation efficiency.
\subsection{Visual Results and Implementation Details for Learning Compressed Representations}
In Figure \ref{fig:detail_quality_compare} we show the results of
our proposed rate-distortion-utility approach to learned compression in panels (d) and (e). We compare it with JPEG compression at a much higher bitrate of 0.31 bpp, in (c) and a naive version of our approach in (b) -- which one could consider as an instance of, or comparison with the approach proposed in \cite{balle2018variational}, but implemented within a specific encoder architectures held fixed across these experiments. The architecture is the best one from the broader family of neural compression models discussed and explored in more detail below. Notice that we can see a striking phenomenon: at very high $\lambda_t$, the texture on an object category of interest to the classifier/detector (a bird in this case) is preserved while the background (foliage) is compressed. This is in agreement with the recent finding that convolutional networks tend to focus on texture rather than shape \citep{hermann2019origins}.
\begin{figure*}[!ht]
\centering
\begin{tabular}{ccc}
\hspace{-.25cm}
\multirow{2}{*}[4em]{
\includegraphics[width=0.30\linewidth]{figures/compression/bird_gt.png}}
&
\hspace{-.30cm}
\includegraphics[width=0.30\linewidth]{figures/compression/bird_naive.png} &
\hspace{-.30cm}
\includegraphics[width=0.30\linewidth]{figures/compression/bird_jpeg.png} \\
\multirow{2}{*}[-8.3em]{(c) Uncompressed}
&
\footnotesize{
\hspace{-.35cm}
(b) Naive, $D=26.8$, $R = 0.14$} &
\hspace{-.35cm}
\footnotesize{(c) JPEG, $D=<26$, $R = 0.31$} *
\\
&
\hspace{-.45cm}
\includegraphics[width=0.30\linewidth]{figures/compression/bird_det1000.png} &
\hspace{-.45cm}
\includegraphics[width=0.30\linewidth]{figures/compression/bird_det5000.png} \\
&
\hspace{-.75cm}
\footnotesize{
(d) $\lambda_t=1k$, $D=26.3$, $R = 0.14$} &
\hspace{-.25cm}
\footnotesize{(e) $\lambda_t=5k$, $D=25.1$, $R = 0.11$ }
\end{tabular}
\caption{Visual image quality comparisons, at distortion level $D$ (PSNR), and rate $R$ (bpp). (a) Uncompressed
ground-truth (b) Naive hyper-prior based learned compressor. (c) JPEG compressed
image. (d)-(e) Compressor jointly trained with an object detector using the $\lambda_t$ parameters, respectively, as 1000 and 5000. We observe that increasing the importance of Pascal object detection performance (i.e. the Utility $U$ of the representation) improves the visual quality of the texture features on the birds feathers and reduces the quality of the background (foliage).}
\label{fig:detail_quality_compare}
\end{figure*}
We obtain a compressible space $z$ to perform vision tasks
by applying Equation (\ref{eq:final_semantic_compression}) over the datasets detailed below. For all our experiments we use the Adam optimizer with an initial learning rate of $5 \times 10^{-5}$. The training dataset used to train the image decoder is a subset of 400k images of OpenImages \cite{kuznetsova2018open} that was pre-processed using the strategy proposed by \cite{mentzer2019practical} that downscales the images in order to reduce potential compression artifacts. To test compression quality results, we use the standard losslessly compressed Kodak dataset \cite{kodak1993kodak}.
We use classification and detection losses to approximate a machine perception quality metric. Differently from \citep{torfason2018towards},
we train on the compression and the machine perception tasks from scratch, jointly.
For classification, we use multi-class cross-entropy as the loss function. We train the model using the ImageNet dataset jointly with OpenImages \cite{kuznetsova2018open}. We train both tasks for $70$ epochs dividing the learning rate by a factor of 10 every 30 epochs. In order to sample the same number of images for each task (reconstruction and classification) we
set a batch size of $72$ for image reconstruction, which is substantially
bigger than the batch size used in previous work \cite{balle2018variational}. For the classification task we set a batch size of $256$. Jointly trained with classification we experiment with two \textbf{class informed} models.
We set $\lambda_t=10,\lambda_d=100$ to obtain a bit-rate of $\approx0.4$, $\lambda_t=10,\lambda_d=10$ for a bit-rate of $\approx1.0$.
For detection, the task specific module in this case consist of an architecture based on a Faster-RCNN \citep{ren2015faster} structured module, which contains a region proposal network and bounding box classifier. We use Pascal VOC as the training dataset. Finally, the standard region-proposal regression and detection class loss \citep{ren2015faster} are used as the detection loss function. We built two \textbf{detection informed} models.
We set $\lambda_t=100,\lambda_d=100$ to obtain a bit-rate of $\approx0.4$ $\lambda_t=100,\lambda_d=25$ for a bit-rate of $\approx1.0$.
We also trained \textbf{class and detection informed model} with a bit-rate of $\approx1.0$ using $\lambda_t=5$ for classification $\lambda_t=100$ for detection and $\lambda_d=25$. All the $\lambda$ hyper-parameters were chosen in order to balance the different batch sizes of tasks.
\subsection{Evaluating Our Learned Compression Approach}
We evaluate different strategies for learning informed compressible representations on various downstream tasks.
For all tasks, the size of the input compressible representation is of $256 \times W/16 \times H/16$. We assume a fully-trained input representation capable being decoded back to the input RGB image; and capability to encode a bitstream from the latent space. In practice the second requirement means that the latent representation is invariant to quantization for the purpose of reconstruction. For all cases, we use the sResnet architecture with a ResNet-101 backbone. We added task specific layers, accordingly, for each task.
\textbf{Classification.}
We investigate the classification accuracy on the ImageNet validation set while training in the full ImageNet dataset 2012 \citep{russakovsky2015imagenet}.
We use the default hyper-parameters used for training a Resnet architecture \citep{he2016deep}. This includes, stochastic gradient descent as optimizer and a starting learning rate of 0.1 that is divided by 10 every 30 epochs. We also set the batch size as 256 for training.
\textbf{Detection.}
We also investigate the average precision on a detection task using the Faster-RCNN \citep{ren2016faster} region
proposal and classification heads.
Similarly to segmentation, we initialize
the training with ImageNet weights trained
for the compressed representation.
\textbf{Semantic Segmentation.}
For this task specific module we use the Deep Lab V3 \cite{chen2017rethinking} and an atrous spatial pyramid pooling layer.
We train and evaluate a semantic segmentation network on the Pascal VOC dataset \cite{everingham2010pascal}, measuring the mean intersection over union (mIoU) score. It is important to note that competitive semantic segmentation results can only be obtained when starting from backbone weights that were pre-trained on ImageNet, we found this also to be the case when training from compressible representations. Therefore we initialize training for the semantic segmentation task with the sResNet-101 weights learned by the classification task.
The results when given different representation inputs for training are presented on Table \ref{tbl:representation_v2}.
The main feature we observe is the capability
to improve over the results of an uncompressed
input by using a compressible
representation that is five times smaller. The uncompressed baselines were trained with the JPEG image set provided on the
ImageNet, for classification, and the PascalVOC datasets, for detection and semantic segmentation. We observe the overall best performance on semantic segmentation and image detection, when using
our compression trained with classification
and detection features. This is an interesting boost and proves the generality of our method
since no features from semantic segmentation
were used when training the compressed representation input. Those results corroborate our hypothesis that constraining the representation
learning problem into also being a compressible
representation aids generalization. Also, it is important to notice that, for segmentation and detection,
that the classification and detection informed
results perform better than only detection
or only classification informed compressed formats.
This indicates that increasing the amount of
information for the compressed representation
may improve generalization performance.
\begin{table*}[!ht]
\resizebox{1\linewidth}{!}{
\footnotesize
\centering
\begin{tabular}{@{}p{3mm}p{59mm}cp{12mm}p{1mm}p{5mm}p{1mm}ccccc@{}}
\toprule
&
&& \hspace{-.25cm}Distortion
& \multicolumn{3}{c}{Segmentation$^*$} & \multicolumn{2}{c}{Classification} && \multicolumn{2}{c}{Detection} \\
Rate & Input && PSNR && mIoU && Top 1 & Top 5 && mAP & AP50 \\
\midrule
& RGB from standard 4 to 5 bpp JPEGs && && 70.8 && 77.3 & 93.3 && 53.1 & 80.8 \\
\midrule
\multirow{5}{*}{\rotatebox[origin=c]{90}{bpp $\sim1.0$} }
& RGB from 1 bpp JPEGs && \textcolor{red}{32.0} && \textcolor{red}{67.5} && \textcolor{red}{75.0} & \textcolor{red}{92.2} && \textcolor{red}{52.3} & \textcolor{red}{79.8} \\
& Our Compression - Naively Learned && 36.2 && {67.5} && {74.8} & {92.2} && {52.8} & {80.3} \\
& Our Compression - Class Informed && 34.6 && 69.5 && 75.1 & 92.4 && 54.7 & 81.1 \\
& Our Compression - Detection Informed && 35.6 && 70.5 && 74.9 & 92.1 && 54.4 & 81.4 \\
& Our Compression - Class \& Detection Informed && 34.5 && \textcolor{ForestGreen}{\textbf{71.3}} && 75.1 & 92.2 && \textcolor{ForestGreen}{\textbf{54.6}} & \textcolor{ForestGreen}{\textbf{81.6}} \\
\midrule
\multirow{4}{*}{\rotatebox[origin=c]{90}{bpp $\sim0.4$} }
& RGB from .4 bpp JPEGs && \textcolor{red}{28.0} && \textcolor{red}{57.8} && \textcolor{red}{70.7} & \textcolor{red}{89.8} && \textcolor{red}{51.4} & \textcolor{red}{78.1} \\
& Our Compression - Naive Learned && 30.7 && 68.9 && 74.0 & 91.7 && 52.1 & 79.6 \\
& Our Compression - Class Informed && 30.1 && 69.0 && 75.1 & 92.3 && 52.8 & 80.1 \\
& Our Compression - Detection Informed && 30.5 && 70.2 && 74.8 & 92.1 && 52.9 & 80.3 \\
\bottomrule
\end{tabular}}
\vspace{3mm}
\caption{Comparison of task performance when using different compressible latent formats as input. Classification experiments are on ImageNet, while Detection and Segmentation experiments are on the Pascal VOC. $^*$Note that none of our learned compressed representations were trained for semantic segmentation; however, a model trained from our best 1bpp compressed representation can even outperform training from high quality RGB images (\textcolor{ForestGreen}{\textbf{bold green}}). Importantly, recall that the encoder is frozen and learning is performed directly from our compressed format.}
\label{tbl:representation_v2}
\end{table*}
We can perceive an even clearer benefit when
training from compressed representations when the compression rate (bits/pixels) is larger. The
image classification results obtained
for low compression, 0.4 bpp, fourth row on Table \ref{tbl:representation_v2}, are able to match
the results obtained for mid
compression, 1.0 bpp. That indicates a clearer
benefit of using informed compressed spaces in
lower bit-rate regimes.
As a drawback, we see that the improvement
by using informed compressed representations for
classification is limited, not being able to fully
recover the uncompressed classification results. An insight into
these results can be acquired by
observing the reconstruction of some informed compressed representations on Fig.\ref{fig:detail_quality_compare} (e) where, even though the bit
rate is lower, the features of the object of interest are well maintained.
\subsection{Ablations and Analysis}
Table \ref{tab:subpixel_stem} shows the impact that different
architectural elements have on training
with compressed representations as input.
Those differences are compared based on
the Top 1 classification result on the
ImageNet validation set.
We can see that increasing the number of pixel shuffle blocks has a substantial positive impact on the results. The same is true when adding a residual connection. We also observed some considerable
improvements when using the Mish \citep{misra2019mish} or the
SiLU \citep{elfwing2018sigmoid} activation functions as compared to the standard ReLU activations.
\begin{table*}[h]
\centering
\begin{tabular}{c|c|c|c|c|c}
One pixel shuffle block & Two pixel shuffle blocks & Residual block & Mish & SiLU & Top 1\\
\hline
\checkmark & & & & & 69.2 \\ \hline
\checkmark & & \checkmark & & & 71.2 \\ \hline
& \checkmark & \checkmark & & & 74.8 \\ \hline
& \checkmark & \checkmark & \checkmark& & 75.0\\ \hline
& \checkmark & \checkmark & & \checkmark & 75.1
\end{tabular}
\caption{Ablation analysis and architecture variant comparisons for the classification task.}
\label{tab:subpixel_stem}
\end{table*}
In Table \ref{tab:vs_joint} we show the comparison our super-resolution based architecture with the one proposed by \cite{torfason2018towards}.
A significant impact on the results can be observed, particularly on the classification task. We believe that this is likely due to a better initial receptive field being presented to the ResNet backbone. We can also see the benefit of using a task informed representation both
for the simpler architecture (cResnet) and the one proposed (sResNet).
\begin{table*}[h]
\centering
\begin{tabular}{@{}llcccccccc@{}}
\toprule
& && Segmentation && \multicolumn{2}{c}{Classification} && \multicolumn{2}{c}{Detection} \\
& Input && mIoU && Top 1 & Top 5 && mAP & AP50 \\
\midrule
& RGB from standard 4 to 5 bpp JPGs && 70.8 && 77.3 & 93.3 && 53.1 & 80.8 \\
\midrule
& cResnet (0.4 bpp) \citep{torfason2018towards} && $64.39$ && $70.34$ & $89.75$ && - & - \\
& cResnet Detection Informed (0.4 bpp) && $64.82$ && $71.57$ & $90.47$ && - & - \\
& sResnet (0.4 bpp) && 68.9 && 74.0 & 91.7 && 52.1 & 79.6 \\
& sResnet Detection Informed (0.4 bpp) && \textbf{70.2} && \textbf{75.1} & \textbf{92.3} && \textbf{52.9} &\textbf{ 80.3} \\
\bottomrule
\end{tabular}
\caption{Ablation analysis and comparison of different architectural variants for creating compressed representations on different downstream tasks at low bit-per-pixels.
}
\label{tab:vs_joint}
\end{table*}
\vspace{5mm}
\subsection{New Object Detection Categories in the Low-shot Learning Regime}
The jointly trained compressed representations are particularly interesting for low-shot regimes. We conduct experiments on few-shot detection by reproducing the experiments and evaluation procedure in \citep{wang2020frustratingly} using the their open source code base. The dataset used is Pascal VOC, and Faster-RCNN as the detector architecture \cite{ren2016faster}. The classes are separated into 15 base classes and 5 novel classes. First, the detector is trained on the 15 base classes. Then, in the fine-tuning phase, the detector learns a new bounding-box (bbox) classifier and a new bbox regressor using only $k \in \{1, 2, 3, 5, 10 \}$ samples from the base classes and the novel classes.
The results of our experiments are summarized on Table \ref{tbl:few_shot_detection_pascal}.
For all the compressed representations we
use a bit rate of 1 bpp.
\begin{table*}[!h]
\centering
\resizebox{1\linewidth}{!}{
\begin{tabular}{@{}lcccccccccccccccccc@{}}
\hline
\multirow{2}{*}{Method} && \multicolumn{5}{c}{Novel Set 1} && \multicolumn{5}{c}{Novel Set 2} &&
\multicolumn{5}{c}{Novel Set 3} \\
$ \quad\quad\quad\quad \# $ of Training Examples & $\rightarrow$ & 1 & 2 & 3 & 5 & 10
&& 1 & 2 & 3 & 5 & 10
&& 1 & 2 & 3 & 5 & 10 \\
\midrule
MetaDet \cite{wang2019meta}
&& 18.9 & 20.6 & 30.2 & 36.8 & 49.6 &&
21.8 & 23.1 & 27.8 & 31.7 & 43.0 &&
20.6 & 23.9 & 29.4 & 43.9 & 44.1 \\
Meta R-CNN \cite{yan2019meta}
&& 19.9 & 25.5 & 35.0 & 45.7 & 51.5 &&
10.4 & 19.4 & 29.6 & 34.8 & \textbf{45.4} &&
14.3 & 18.2 & 27.5 & 41.2 & 48.1\\
\midrule
TFA w/fc \citep{wang2020frustratingly}
&& 36.8 & 29.1 & 43.6 & 55.7 & 57.0 &&
18.2 & 29.0 & 33.4 & \textbf{35.5} & 39.0 &&
27.7 & 33.6 & 42.5 & 48.7 & \textbf{50.2}\\
TFA w/cos \citep{wang2020frustratingly}
&& \textbf{39.8} & 36.1 & 44.7 & 55.7 & 56.0 &&
23.5 & 26.9 & 34.1 & 35.1 & 39.1 &&
30.8 & 34.8 & 42.8 & 49.5 & 49.8 \\
\midrule
TFA w/fc, our compression
&& 35.6 & 27.4 & 42.8 & 54.2 & 55.6 &&
16.3 & 28.0 & 31.9 & 33.9 & 37.8 &&
25.0 & 31.5 & 40.6 & 47.0 & 48.4\\
Ours - Naive
&& 33.5 & 26.3 & 40.7 & 53.5 & 55.1 &&
20.2 & 24.1 & 30.5 & 31.1 & 36.8 &&
26.1 & 30.5 & 38.7 & 45.6 & 45.9 \\
Ours - Class. + Det. Informed
&& 38.1 & 37.3 & 44.7 & 56.2 & 57.1 && 23.6 & 27.3 & 34.0 & 35.1 & 38.9 && 31.0 & 33.9 & 42.1 & 48.7 & 48.9 \\
Ours - Classification Informed
&& 38.8 & 37.2 & 44.1 & 55.3 & 56.7 &&
23.7 & 27.2 & 34.5 & 35.3 & 39.2 &&
31.2 & 34.9 & 42.5 & 49.4 & 49.5 \\
Ours - Detection Informed && 39.3 & \textbf{37.5} & \textbf{44.8} & \textbf{56.3} & \textbf{57.1} &&
\textbf{23.7} & \textbf{29.1} & \textbf{34.8} & 35.3 & 39.2 &&
\textbf{31.3} & \textbf{34.9} & \textbf{42.9} & \textbf{49.5} & 49.9 \\
\bottomrule
\end{tabular}
}
\caption{Few-shot detection performance (mAP50) on PASCAL VOC dataset. We compare our results (bottom) when using a informed
compressed representation with other methods from the literature. We obtain a convincingly better performance while using a mid bit-rate representation (bpp 1.0).}
\label{tbl:few_shot_detection_pascal}
\end{table*}
These experiments show that training the few-shot detector from our classification informed compressible representation not only improves overall few-shot performance at 1 bpp, but also achieves state-of-the-art results for many of the experimental configurations. Also, note that the addition of semantic information to the representation has a significant impact, leading to an average increase of 4 points in mAP50. This strongly suggests that, when jointly trained with visual tasks, compressible representations can serve as a regularizer in low-data regimes.
Finally, the performance gap between our method
and the TFA method \citep{wang2020frustratingly}
is even larger when the compression
level is increased on the TFA input to match a bit rate of 1.0.
\subsection{Computational and Memory Cost}
All the performance benchmarks were made on 4 Nvidia V100 GPUs processing a batch in parallel.
One of the motivations for our approach is the fact that there exists an advantage to performing downstream tasks directly from a latent compressed space, besides the obvious saving storage/transmission requirement, the input size is considerably smaller and thus computationally cheaper to forward propagate (fprop).
We typically see performance gains of up to $20\%$ on the classification task. However, it should be noted that our experiments are performed at full 32-bit floating point resolution, yet the compressible representation can be expressed using 4-bits. See Appendix I for more details. The use of 4-bit representations for our compression format could make data processing pipelines even more efficient. Furthermore, if combined with appropriately engineered low-bit decoders we suspect that much more dramatic computational gains are possible for those who desire improved performance in this regard. Given that the field of low bit precision deep learning has been making many advances \cite{courbariaux2015binaryconnect,hubara2016binarized,banner2018scalable,iandola2016squeezenet}, low-bit depth decoders based on this type of compression scheme could be an interesting direction for future work. A recent survey in \cite{gholami2021survey} summarizes some key developments, including extremely fast INT8 and INT4 implementations of canonical Convolutional Neural Network architectures.
\section{Related Work}
Early work using auto-encoder architectures for image compression date back to the late 1980's \cite{cottrell1988imagecb} and many extensions were developed during the following decades \cite{jiang1999image}. They follow the traditional approach of pixel transforms, quantization, and entropy coding. The use of recurrent architectures \cite{Toderici2017,johnston2018improved} was proposed to allow configurable rate-distortion trade-off by progressively encoding image residuals, this can also be improved by tiling \cite{minnen2017spatially}.
Most relevant to our work, is the state-of the art technique of creating a latent representation using a variational auto-encoder and using a hyper-prior network to produce a probabilistic model of the quantized latent representation \cite{balle2018variational, minnen2018joint, cheng2020learned}. We use the quantized latent space of the VAE as an input format for visual tasks, thus the latent space should capture all necessary information for performing these tasks. Those techniques
have also been recently extended to include perceptual quality
metrics, drastically improving the
results from a human perception
perspective \cite{mentzer2020high,agustsson2019generative}. The present work does not focus on improving results from a distortion
or human perception viewpoint, but strives to improve the compression format for machine perception via canonical computer vision tasks such as image classification or object detection as proxies.
Recent work has shown that it is possible to learn compressible representations of features in deep neural networks that do not reconstruct input images \cite{singh2020end}, but which serve to perform classifications. Their work shows that it is possible to create image classifiers using the resulting compressed feature representations that have comparable performance to those trained directly on images.
At the other end of the spectrum, as mentioned in our introduction, other work has used the image representations obtained from the ubiquitous JPEG image compression format as input to convolutional networks. This approach reduces the computational and memory requirements of the overall pipeline of training and classification while achieving a similar accuracy on ImageNet classification \cite{ulicny2017using,fu2016using,gueguen2018faster}.
We also propose the approach of performing tasks directly from compressible representations; however, the aim of our work is to \emph{learn a semantically sensitive compressible image format} which facilitates downstream tasks such as classification and semantic segmentation.
Of course there is considerable prior work that has involved training auto-encoders to learn image features via low-dimensional embeddings or codes \cite{hinton2006reducing} \cite{masci2011stacked}. However, these types of approaches have not involved explicitly compressing the latent representations and evaluating such methods in terms of compression rates and distortion. In contrast, our approach and architectures produce low dimensional feature spaces with reduced entropy through encouraging representations to be robust to the integer quantization necessary for arithmetic coding. This approach thereby leads to measurable gains in yielding compressible image representations.
Because image reconstruction and other semantic tasks can be viewed as competing objectives, our work is highly related to multi-task learning \cite{ruder2017overview}.
To the best of our knowledge the work most closely related to ours is \cite{torfason2018towards}, which was the first to perform semantic inference directly from a learned, compressible representation. However, in their work performance was compromised on semantic tasks. We also explore inference from a compressible representation, however we aim to develop an algorithm that produces \emph{semantically sensitive compressible representations without sacrificing their ability to compute the information necessary to reconstruct the input image}. Our experiments underscore the various advantages of our proposed format, particularly when learning new out-of-domain tasks and for few-shot generalization.
\section{Conclusions}
We have proposed stepping away from the classical rate-distortion paradigm for learned image compression and moving to a rate-distortion-utility framework.
We have shown that this can be achieved by simply jointly training a compressor to also serve as the intermediate representation for multiple tasks.
Our work provides evidence that the combined effects of: (1) compact codes provided by explicit neural compression techniques, and (2) a multi-task learning setup, indeed produces representations that are particularly effective in both: (a) the low shot learning regime, and (b) when training a new task such as semantic segmentation directly from these types of learned compressed representations for machine perception. It should be noted that encouraging compression methods to better preserve certain types of semantic content could introduce biases into which elements of an image are better compressed than others. One must therefore ensure that unwanted biases are not unintentionally induced in the algorithm. If our approach becomes adopted, as time goes on even larger datasets of tasks could address the limitations of training using only the ImageNet, OpenImages and Pascal datasets.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
We thank NSERC, the COHESA Strategic Network, Google, Mila and CIFAR for their support under the AI Chairs program.
| 12,869 |
\section{Introduction and Motivation} \label{sec: intro}
The study of categories of modules and bimodules over unital associative rings or algebras is one of the most developed subjects of modern algebra and its inception might be traced back to the work of R.Dedekind, E.Noether and B.L.van den Waerden, among many others.
\medskip
Multimodules over unital associative rings and algebras are quite a natural generalization of right/left-modules and bimodules that, as far as we know, have been first described in N.Bourbaki~\cite{Bou89}. Since, for most purposes, multimodules are equivalently seen as bimodules over tensor products of rings and algebras, it can be claimed that their investigation essentially reduces to the study of special classes of bimodules and not much attention has been paid to them (we have been able to locate only one specific reference on multimodules~\cite{Ke62} and some sporadic mentioning of them, for example in~\cite[section~0]{Ta87}).
\medskip
The ``substitution'' of multimodules with corresponding bimodules over tensor products turns out to be problematic whenever the category of morphisms is extended with the inclusion of maps that have different covariance properties with respect to the several actions involved. One could still substitute multimodules with bimodules over tensor products of rings, as long as such tensor products of rings are simultaneously equipped with different products (all distributive with respect to the same Abelian group structure), but this essentially amounts to define an ``hyper-algebra structure'' on the tensor product multimodule of the rings (see remark~\ref{rem: hyper}).
\medskip
The basic algebraic material here presented naturally arose as a byproduct in our study of non-commutative generalizations of contravariant calculus.~\footnote{
P.Bertozzini, R.Conti, C.Puttirungroj, Non-commutative Contravariant Differential Calculus (in preparation). \label{foo: ncdc}
}
Since quite surprisingly we have not been able to locate any relevant source dealing with this topic, we thought that the subject deserves an adequate separate treatment.
Specifically (anticipating arguments and motivations pertaining to the aforementioned work) in non-commutative (algebraic) geometry, it is a common thread to look for generalizations of the usual notion of ``differential operator'' to the case of maps between bimodules over a non-commutative algebra ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ and it often happens (for example whenever one is considering ``double derivations'' on ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$) that the spaces of such ``non-commutative differential operators'' are naturally equipped with a multimodule structure over the original non-commutative algebra ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$.
Although of tangential interest for this work, a general definition of first-order differential operators between multimodules, covering in particular all such cases, will be included in appendix~\ref{sec: 1st-ord-multi}.
Further developments in these directions, including investigations of non-commutative vector fields and non-commutative connections on (multi-)modules, will have to wait subsequent works (see the paper in footnote~\ref{foo: ncdc} and references therein for more details).
\bigskip
In short, the specific goals of the present work are to:
\begin{itemize} \it
\item[$\rightsquigarrow$]
define multimodules based over an arbitrary $\Zs$-central bimodule~\footnote{
Where $\Zs$ is a commutative unital associative ring/algebra} (more generally over a \hbox{$\Zs$-central} unital associative ring $\Rs_\Zs$) instead of just an Abelian group:
\end{itemize}
this allows to discuss mutually commuting (right/left) actions that are compatible with a certain fixed $\Zs$-linear structure, but that can still have alternative $\Rs$-linearity properties;
\begin{itemize} \it
\item[$\rightsquigarrow$]
introduce a notion of involution for multimodules that allows for different covariance/contravariance:
\end{itemize}
since involutions for us are just involutive morphisms, this requires an appropriate definition of category of multimodules, where morphisms (necessarily $\Zs$-linear) can have different covariance properties (and even different conjugate-$\Rs_\Zs$-linearity properties) with respect to the different actions involved;
\begin{itemize}\it
\item[$\rightsquigarrow$]
provide a systematic treatment of the several ($\Zs$-linear) duals of multimodules, their associated categorical semi-adjunctions and (under saturation conditions for evaluations) establish transposition dualities:
\end{itemize}
it is already known that in the case of bimodules one needs to separately consider right and left duals in place of the usual notion of dual vector space; in the case of multimodules, the situation is a bit more involved and one can construct different (conjugate)-duals for any choice of subfamilies of left/right actions (and corresponding conjugations of $\Rs_\Zs$); each dual is defined in this work via a universal factorization property and its elements are concretely realized as $\Zs$-multilinear functions that have selective \hbox{$\Rs_\Zs$-(conjugate)}-linearity properties with respect to the specified actions;
\begin{itemize}
\item[$\rightsquigarrow$] \it
introduce universal traces, and more generally contractions, on multimodules;
\end{itemize}
traces of linear operators and contractions of tensors are quite standard operations performed in multilinear algebra; we reframe such notions in the more general context of multimodules, providing again a definition via universal factorization properties;
\begin{itemize}
\item[$\rightsquigarrow$] \it
discuss, for multimodules over involutive rings/algebras, suitable notions of ``inner products'' and (under conditions of non-degeneracy/fullness) establish Riesz isomorphisms:
\end{itemize}
inner products on multimodules also come in several types, each corresponding to a different dual, and are here realized as certain \textit{balanced multi-sesquilinear maps}; involutive algebras are necessary in order to give a meaning to Hermitianity conditions on inner products; every inner-product induces a \textit{canonical Riesz morphisms} of a multimodule into a corresponding dual; non-degeneracy and fullness are required to obtain an isomorphism.
\begin{itemize}\it
\item[$\rightsquigarrow$]
describe first order differential operators between multimodules:
\end{itemize}
the first order condition in non-commutative geometry~\cite[sections $4.\gamma$ and $4.\delta$]{Co94}, usually formulated in the case of operators between bimodules, is here expanded to cover the general setting of multimodules;
\begin{itemize}
\item[$\rightsquigarrow$] \it
make the first steps toward a study of involutive colored properads using multimodules as a template:
\end{itemize}
the material here included is mostly intended to provide a usable language for quite practical situations (some of which have been actually originating from work in categorical non-commutative geometry) where multimodules and their duals might be used and manipulated. As a consequence, we have not been looking for maximal generality in the statements and we kept a rather low sophistication level in the discussion of all the category-theoretical aspects of the subject;
a more detailed study of these topics is under way, but we can already anticipate that it will fall within the scope of certain variants of \textit{involutive colored properads} and \textit{involutive polycategories}.
As stated above, we plan to address more properly these points in subsequent works.
\bigskip
Here below is a more detailed description of the content of the paper.
\medskip
In section~\ref{sec: generalities} we modify the usual setting of bimodules over unital associative rings considering, in place of the initial ring $\ZZ$ a commutative unital associative ring $\Zs$ and instead of rings acting on Abelian groups (\hbox{$\ZZ$-bimodules}), $\Zs$-central algebras ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ acting in a $\Zs$-bilinear way on $\Zs$-central bimodules ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$.
Morphisms are in this case pairs of $\Zs$-linear maps (in place of additive maps) that induce a unital $\Zs$-linear covariant or contravariant grade-preserving homomorphism on the associated ${\mathbb{N}}}\newcommand{\OO}{{\mathbb{O}}}\newcommand{\PP}{{\mathbb{P}}$-graded algebras ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\urcorner:={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\oplus{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\oplus\{0\}\cdots$ of the bimodules.
This kind of environment can immediately describe, as a special case, categories of $\KK$-linear covariant or contravariant morphisms of unital bimodules over $\KK$-algebras, for a certain field $\KK$ in place of $\Zs$.
\medskip
The existence of many situations requiring the usage of non-trivial (involutive auto)morphisms for the base field $\KK$ and the consequent need to deal simultaneously with maps that are not $\KK$-linear, imposes a further refinement of the structure: the common base commutative associative unital ring $\Zs$ is replaced by a \hbox{$\Zs$-central} unital associative ring $\Rs_\Zs$. The family of unital covariant or contravariant $\Zs$-linear homomorphisms $\phi$ of $\Zs\oplus\Rs$ identifies the possible alternative notions of $\phi$-linearity with respect to the base ring $\Rs$.
The paradigmatic situation with $\Rs:=\CC$ and $\Zs:=\RR$ imposes only $\RR$-linearity on morphisms that are further classified as $\CC$-linear and $\CC$-conjugate-linear depending on the choice of the $\CC$-automorphism $\phi$; but the formalism can be used in the case of algebras over arbitrary extensions of fields (or more generally extensions of rings).
\medskip
Section~\ref{sec: multimodules} puts forward our definition of multimodules over families of unital associative algebras over $\Rs_\Zs$. We stress that taking $\Rs=\Zs=\ZZ$, we just reproduce the usual definition of multimodules in~\cite{Bou89} and taking $\Zs\to\Rs$ an extension of fields we obtain multimodules as $\Zs$-vector spaces equipped with $\Zs$-bilinear actions of $\Rs$-algebras, where morphisms can be $\phi$-linear for any $\Zs$-linear automorphism $\phi$ of $\Rs$.
The unavoidability of multimodules (in every context dealing with bimodules) is witnessed by the construction of $\Zs$-central multimodules of $\Zs$-linear maps, and $\Zs$-tensor products, between $\Zs$-central bimodules.
\medskip
In section~\ref{sec: involutions-m} we specialize to the treatment of involutive endomorphisms of $\Zs$-central multimodules over \hbox{$\Zs$-central} $\Rs_\Zs$-algebras and we examine how involutions on bimodules (and multimodules) propagate to involutions for spaces of $\Zs$-linear morphisms and tensor products of multimodules.
\medskip
The main result of the paper is contained in section~\ref{sec: multi-dual-paring} where we introduce definitions of duals of multimodules via universal factorization properties and we prove that transpositions functors in the category of multimodules give rise to contravariant right semi-adjunctions (theorem~\ref{th: sadj}) that, for multimodules satisfying reflexivity, produce dualities.
In general there exist different \textit{conjugate-duals} for a $\Zs$-central multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_B}$ over \hbox{$\Rs_\Zs$-algebras}, each one of them ${}^{(\gamma_i)_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{(\gamma_j)_J}$ specified by certain families $(\gamma_i)_{i\in I}$ and $(\gamma_j)_{j\in J}$ of $\Rs$-conjugations, with arbitrary sets of indexes $I\subset A$ and $J\subset B$.
\medskip
In the first part of section~\ref{sec: traces-ip} we define universal contractions/traces on multimodules via universal factorization properties and we construct them quotienting the original multimodules with respect to certain \textit{commutator sub-multimodules}.
The remaining part of section~\ref{sec: traces-ip} discusses tentative generalizations, to the setting of multimodules over involutive algebras, of the familiar notion of inner product for vector spaces or modules and for each such inner product defines its Riesz ``natural transformation''.\footnote{
Due to the different covariance of the functors involved, a categorical discussion of the ``naturality'' of Riesz morphisms would require the usage of hybrid 2-categories~\cite{BePu14} of multimodules.
}
Under conditions of non-degeneracy and fullness of the inner products, we also provide a multimodule version of Riesz isomorphism theorem.
The inner products here introduced are not necessarily positive: a positivity requirement can be added (at such an abstract level) imposing the existence of positive cones on the algebras.
\medskip
The final outlook section~\ref{sec: outlook} briefly expands on the already mentioned planned utilization of the categories of multimodules, here developed, as a paradigmatic example in the study of the abstract notion of ``involutive colored properad'' and their associated involutive ``convolution hyper-algebroids'' following the lines that some of us have discussed in previous papers~\cite{BCLS20}.
\medskip
In appendix~\ref{sec: functorial-pairing}, we briefly recall the notion of (contravariant) semi-adjunction~\cite{Med74}, a special case of regular full functorial pairings later defined in~\cite{Wis13}, that will be needed to describe the dualities for contravariant trasposition functors in categories of multimodules.
Special attention has been devoted to the explicit characterization of semi-adjunctions for contravariant functors.
\medskip
As already mentioned, the present paper was motivated by an ongoing effort towards the study of non-com\-mu\-ta\-tive vectors fields and contravariant non-commutative differential calculus (see footnote~\ref{foo: ncdc}); in appendix~\ref{sec: 1st-ord-multi} we present the generalization, to the case of multimodules, of a definition of first-order differential operator on bimodules over non-commutative algebras, that has been useful in that context. Further extensions in the direction of differential analysis on multimodules (starting with a theory of connections) are briefly mentioned in the outlook section and will be dealt with elsewhere.
\section{Generalities} \label{sec: generalities}
We start specifying basic settings and definitions; for more details on background material that is not explicitly mentioned, we refer to the texts~\cite{Al09} and~\cite{Bou89}.
\medskip
We assume $\Zs$ to be a commutative unital associative ring.
All the rings $\Rs$ here considered will be unital associative (not necessarily commutative) and \emph{$\Zs$-central rings}: they are equipped with a unital homomorphism of rings $\iota_\Rs:\Zs\to Z(\Rs):=\{r\in\Rs \ | \ \forall x\in\Rs \st r\cdot x=x\cdot r \}$, where $Z(\Rs)$ denotes the center of the ring $\Rs$ (itself a commutative unital associative ring).
\medskip
All the $\Rs$-bimodules ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ considered in this paper are assumed to be unital ($1_\Rs\cdot x=x$, for all $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$) and
\emph{\hbox{$\Zs$-central} $\Rs$-bimodules}, meaning that there is a unital homomorphism of rings $\iota_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}:\Zs\to Z({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^0$, where we define $Z({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^0:=\{r\in Z(\Rs) \ | \ \forall x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}} \st r\cdot x=x\cdot r \}$ as
the center ring of the $\Rs$-bimodule ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ (that is itself a $\Zs$-central unital sub-ring of $\Rs$). Similarly,
$Z({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^1:=\{x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}} \ | \ \forall r\in\Rs \st r\cdot x=x\cdot r \}$ denotes the center module of ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ (itself a $\Zs$-central unital $\Rs$-bimodule).
\medskip
Here \emph{$\Zs$-central $\Rs$-algebras} are defined as $\Zs$-central $\Rs$-bimodules ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}:={}_\Rs{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Rs$ with a distributive multiplication $\circ$ such that:
$(r\cdot x)\circ y=r\cdot(x\circ y)$, $(x\cdot r)\circ y=x\circ(r\cdot y)$ and $x\circ(y\cdot r)=(x\circ y)\cdot r$, for all $x,y\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ and $r\in \Rs$. In this way, multiplication in a $\Zs$-central $\Rs$-algebra is necessarily $\Zs$-bilinear and every $\Zs$-central ring $\Rs$ becomes an (associative unital) \hbox{$\Zs$-central} algebra over itself.
We will usually consider $\Zs$-central $\Rs$-algebras ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ that are unital and associative. We will consider $\Zs$-central ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$-bimodules ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ that are unital and hence become canonically $\Zs$-central $\Rs$-bimodules with action $r\cdot x:=(r\cdot 1_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}})\cdot_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}} x$, for $r\in\Rs$ and $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$.
\medskip
Since $\ZZ$ is initial in the category of unital associative rings, $\ZZ\cdot 1_\Rs\subset \Zs\subset Z(\Rs)$ and the \emph{characteristic of $\Rs$} is the minimum $n\in{\mathbb{N}}}\newcommand{\OO}{{\mathbb{O}}}\newcommand{\PP}{{\mathbb{P}}$ such that $\ke(\iota)=n\cdot\ZZ$, where $\iota:\ZZ\to\Rs$ is the initial unital homomorphism $z\mapsto z\cdot 1_\Rs$. Whenever the characteristic is a prime number, $\Rs$ is actually an ${\mathbb{F}}}\newcommand{\GG}{{\mathbb{G}}}\newcommand{\HH}{{\mathbb{H}}$-algebra over the finite field ${\mathbb{F}}}\newcommand{\GG}{{\mathbb{G}}}\newcommand{\HH}{{\mathbb{H}}:=\ZZ/\ke(\iota)$.
\medskip
Particular attention should be given to the definition of morphisms for bimodules over $\Zs$-central $\Rs$-algebras.
\begin{definition}
Let $\Zs$ be a commutative unital associative ring and $\Rs_\Zs$ a unital associative $\Zs$-central ring.
A map ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{\Phi}\Ns$ between two $\Zs$-central unital bimodules ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}:={\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_\Zs$, $\Ns:=\Ns_\Zs$,
is said to be \emph{$\Zs$-linear} if:
\begin{equation*}
\Phi(x+y)=\Phi(x)+\Phi(y), \quad \quad \Phi(\iota_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}(z)\cdot x)=\iota_\Ns(z)\cdot\Phi(x),\quad \forall x,y\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ z\in\Zs.
\end{equation*}
A map ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\xrightarrow{\phi}\Bs$ between $\Zs$-central unital associative rings ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}:={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Zs$, $\Bs:=\Bs_\Zs$ is
\begin{itemize}
\item
\emph{covariant} if: $\phi(x \circ_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}} y)=\phi(x)\circ_\Bs\phi(y)$, for all $x,y\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$,
\item
\emph{contravariant} if: $\phi(x \circ_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}} y)=\phi(y)\circ_\Bs\phi(x)$, for all $x,y\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$,
\item
\emph{unital} if: $\phi(1_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}})=1_\Bs$,
\item
\emph{homomorphism} if: it is $\Zs$-linear covariant and unital,
\item
\emph{anti-homomorphism} if: it is $\Zs$-linear contravariant and unital.
\end{itemize}
A $\Zs$-linear map ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}:={}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}}\xrightarrow{\Phi}{}_\Bs\Ns_{\Bs}$ between $\Zs$-central unital bimodules over $\Zs$-central unital associative rings ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}:={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Zs,\Bs:=\Bs_\Zs$ is said to be \emph{$\phi$-linear}, for a certain $\Zs$-linear unital homomorphism (anti-homomorphism) ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\xrightarrow{\phi}\Bs$ if:
\begin{align*}
&\Phi(a_1\cdot x\cdot a_2)=\phi(a_1)\cdot \Phi(x)\cdot \phi(a_2), \quad \forall x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ a_1,a_2\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}},
&&\text{in the $\phi$-covariant case},
\\
&\Phi(a_1\cdot x\cdot a_2)=\phi(a_2)\cdot \Phi(x)\cdot \phi(a_1), \quad \forall x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ a_1,a_2\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}},
&&\text{in the $\phi$-contravariant case}.
\end{align*}
A $\phi$-linear covariant (contravariant) \emph{morphism} of $\Zs$-central unital bimodules, over $\Zs$-central unital associative rings, consists of a pair $(\phi,\Phi)$ as above. In the case of $\Zs$-central unital associative algebras over
$\Zs$-central unital associative rings, the morphism ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\xrightarrow{\Phi}\Bs$ must be unital and covariant (contravariant).
\medskip
For $\Zs$-central bimodules ${}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}, {_\Bs}\Ns_\Bs$ over $\Zs$-central unital associative algebras ${}_\Rs{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Rs, {}_\Rs\Bs_\Rs$ over a $\Zs$-central unital associative ring $\Rs_\Zs$, morphisms are still denoted by $(\phi,\Phi)$, where $\Phi:=(\Phi^0,\Phi^1)$ is a pair of $\phi$-linear unital morphisms $\Phi^0:{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\to\Bs$ of algebras and $\Phi^1:{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\to\Ns$ of bimodules, such that $\Phi^1$ is $\Phi^0$-linear:
\begin{equation*}
\xymatrix{
\Zs \ar@{=}[d] \ar[r]^{\iota_\Rs} & \Rs \ar[d]^\phi \ar[r]^{\iota_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}} & {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\ar[d]^{\Phi^0} & {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}} \ar[d]^{\Phi^1}
&\Phi^1(a_1\cdot x\cdot a_2)=\Phi^0(a_1)\cdot \Phi^1(x)\cdot \Phi^0(a_2)
\quad
\text{covariant $\phi$-linear case\phantom{ntra}}
\\
\Zs \ar[r]^{\iota_\Rs} & \Rs \ar[r]^{\iota_\Bs} & \Bs & \Ns
&
\Phi^1(a_1\cdot x\cdot a_2)=\Phi^0(a_2)\cdot \Phi^1(x)\cdot \Phi^0(a_1)
\quad
\text{contravariant $\phi$-linear case}.
}
\end{equation*}
\end{definition}
\begin{remark}
We have a category of $\Zs$-linear maps between $\Zs$-central unital bimodules over $\Zs$-central unital associative $\Rs_\Zs$-algebras. Such category is not $\ZZ_2$-graded with respect to covariance / contravariance, since the same morphism $\Phi$ can be $\phi$-covariant or $\phi$-contravariant depending on the choice of $\phi$.
\medskip
A better alternative consists, as we did, in defining morphisms as triples ${}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\xrightarrow{(\phi,\Phi)}{}_\Bs\Ns_\Bs$ of $\Zs$-linear maps $\phi:\Rs_\Zs\to\Rs_\Zs$, $\Phi^0:{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Rs\to\Bs_\Rs$ and $\Phi^1:{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\to\Ns_\Bs$ with $(\phi,\Phi^0)$ and $(\Phi^0,\Phi^1)$ both $\phi$-linear morphisms.
In this case the category is $\ZZ_2$-graded (by the covariance of the triple) furthermore it is isomorphic to the \hbox{$\ZZ_2$-graded} category of degree zero unital $\Zs$-linear (covariant or contravariant) morphisms $(\phi,\Phi^0,\Phi^1)$ between graded unital associative \hbox{$\Zs$-central} algebras of the form ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\urcorner:=\Rs\oplus{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\oplus {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\oplus\{0\}\cdots$.
\xqed{\lrcorner}
\end{remark}
\begin{definition}
A \emph{covariant} (respectively \emph{contravariant}) \emph{involution} on a $\Zs$-central unital associative ring $\Rs_\Zs$ is a $\Zs$-linear covariant (respectively contravariant) map $\Rs\xrightarrow{\star}\Rs$ that is involutive $(x^\star)^\star=x$, for all $x\in\Rs$.
\medskip
Whenever dealing with $\Zs$-central algebras ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Rs$ over a $\Zs$-central unital associative ring $\Rs_\Zs$, we use the term \emph{$\Rs_\Zs$-conjugation} to denote an involution of the $\Zs$-central unital associative ring $\Rs_\Zs$.
\medskip
A \emph{covariant (contravariant) involution} $\star$ on ${}_\Rs{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Rs$ is said to be \emph{$\gamma$-conjugate-linear} if it is $\gamma$-linear for a certain covariant (contravariant) $\Rs_\Zs$-conjugation $\gamma$, specifically:
$(r_1\cdot x\cdot r_2)^\star=\gamma(r_1)\cdot x^\star\cdot \gamma(r_2)$ in the $\gamma$-covariant case;
$(r_1\cdot x\cdot r_2)^\star=\gamma(r_2)\cdot x^\star\cdot \gamma(r_1)$) in the $\gamma$-contravariant case, for all $r_1,r_2\in\Rs$ and $x\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$.
\end{definition}
\begin{remark}
For $\Zs$-central algebras ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Rs$ over non-commutative rings $\Rs$, covariant (contravariant) involutions can be $\gamma$-conjugate-linear only with respect to a covariant (contravariant) conjugation $\gamma$.
Whenever $\Rs$ is commutative, there is no difference between covariant and contravariant conjugations and hence, for an arbitrary conjugation $\gamma$, we can have covariant or contravariant involutions on ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Rs$ that are $\gamma$-conjugate-linear.
\medskip
Notice that for involutive $\Zs$-central $\Rs$-algebras ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$, we necessarily have $\eta_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}(\Zs)\subset Z({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Rs)^0\cap\{x\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}} \ | \ x^\star=x \}$. It is of course possible, for a certain $\Zs$-central ring $\Rs$ to have involutions $\gamma$ that do not necessarily leave $\eta_\Rs(\Zs)$ invariant or that do not necessarily fix all the elements of $\eta_\Rs(\Zs)$; in this case one can further ``restrict'' the commutative algebra $\Zs$ in order to make $\gamma$ a conjugation: given a certain family $\Gamma$ of additive (covariant or contravariant) involutions of $\Rs$, we see that
$\Zs^\Gamma:=\eta^{-1}_\Rs\left(\bigcap_{\gamma\in\Gamma} \{x\in Z(\Rs) \ | \ \gamma(r)=r \}\right)$ is a unital sub-algebra of $\Zs$ making all the $\gamma\in\Gamma$ conjugations of $\Rs$ as $\Zs^\Gamma$-central ring.
\xqed{\lrcorner}
\end{remark}
\medskip
There are universal ways to reformulate $\gamma$-conjugate-linear unital morphisms of $\Zs$-central $\Rs$-algebras (and also of $\Zs$-central $\Rs$-bimodules) as covariant $\Rs$-linear unital morphisms.
\begin{definition}\label{def: cj-dual}
Given a $\Zs$-central $\Rs$-algebra ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Rs$ and a conjugation $\gamma$ in $\Rs$, a \emph{$\gamma$-conjugate of ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Rs$}
consists of a $\gamma$-conjugate-linear unital morphism of $\Zs$-central $\Rs$-algebras ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\xrightarrow{\eta_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}^\gamma$ that satisfies the universal factorization property:
for any $\gamma$-conjugate-linear unital morphism of $\Zs$-central $\Rs$-algebras ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\xrightarrow{\phi}\Bs$, there exists a unique covariant $\Rs$-linear homomorphism ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}^\gamma\xrightarrow{\hat{\phi}}\Bs$ such that $\phi=\hat{\phi}\circ\eta_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$.
\medskip
In the case of $\Zs$-central unital $\Rs$-bimodules ${}_\Rs{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_\Rs$ the definition of $\gamma$-conjugate ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{\eta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\gamma$ is given via the same universal factorization property diagram of $\Rs$-bimodules, ``forgetting'' the multiplication.
\end{definition}
\begin{remark}\label{rem: cj-dual}
Unicity of $\gamma$-conjugates up to a unique isomorphism compatible with the universal property is standard, their existence can be provided as follows.
\medskip
Given a $\Zs$-central unital associative $\Rs$-algebra ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\Rs$ and a conjugation $\gamma$ in $\Rs$, take as a $\Zs$-central bimodule ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}^\gamma:={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ and define $\eta_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}:{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\to{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}^\gamma$ as the identity map, here denoted as ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\ni x\mapsto \hat{x}\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}^\gamma$. If $\gamma$ is a contravariant conjugation, define $r_1 \ \hat{\cdot}\ \hat{x} \ \hat{\cdot}\ r_2:=\widehat{\gamma(r_2)\cdot x\cdot \gamma(r_1)}$ and $\hat{x}\ \hat{\circ}\ \hat{y}:=\widehat{y\circ x}$, for all $x,y\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ and $r_1,r_2\in\Rs$. If $\gamma$ is a covariant conjugation, define $r_1 \ \hat{\cdot}\ \hat{x} \ \hat{\cdot}\ r_2:=\widehat{\gamma(r_1)\cdot x\cdot \gamma(r_2)}$ and $\hat{x}\ \hat{\circ}\ \hat{y}:=\widehat{x\circ y}$, for all $x,y\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ and $r_1,r_2\in\Rs$.
Notice that in both cases ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}^\gamma$ becomes a $\Zs$-central $\Rs$-bimodule with the new actions $\hat{\cdot}$ and it becomes a $\Zs$-central $\Rs$-algebra with the new product $\hat{\circ}$; furthermore the map $\eta_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}:{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\to{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}^\gamma$ turns out to be a $\Zs$-linear $\gamma$-conjugate-linear contravariant (respectively covariant) unital homomorphism.
For any $\gamma$-conjugate-linear unital contravariant (respectively contravariant) homomorphism $\phi:{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\to\Bs$, we necessarily need to define $\hat{\phi}(\hat{x}):=\phi(x)$, and we verify that $\hat{\phi}:{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}^\gamma\to\Bs$ is an $\Rs$-linear unital covariant homomorphism in both cases.
\xqed{\lrcorner}
\end{remark}
\section{Multimodules Over Unital Associative $\Zs$-central $\Rs$-algebras} \label{sec: multimodules}
We introduce here multimodules \textit{over families of unital associative $\Zs$-central $\Rs$-algebras}.\footnote{This generalizes the special case of multimodules over unital associative $\KK$-algebras over the field $\KK$: in this case one can take $\Rs:=\KK$ and $\Zs$ a subfield of $\KK$ consisting of fixed points for all the relevant conjugations of $\KK$ (in practice it is always possible, in each characteristic $p$, to take $\Zs$ as the initial field of that characteristic: ${\mathbb{Q}}$ in characteristic $0$ and ${\mathbb{F}}}\newcommand{\GG}{{\mathbb{G}}}\newcommand{\HH}{{\mathbb{H}}_p$ for any $p$ prime).}
\medskip
In the following, we adapt the general definition of multimodule from~\cite[section~II.1.14]{Bou89}:
\begin{definition}\label{def: morphism}
Let $\Zs$ be a commutative unital associative ring and $\Rs$ be a unital associative $\Zs$-central ring.
Given two families of unital associative $\Zs$-central $\Rs$-algebras $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$ and $(\Bs_\beta)_{\beta\in B}$, an \emph{$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)$-$(\Bs_\beta)$ multimodule} ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)}$ is a $\Zs$-central bimodule that is a $\Zs$-central unital ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha$-$\Bs_\beta$ bimodule for every $(\alpha,\beta)\in A\times B$ such that every pair of left actions and every pair of right actions commute.\footnote{
We assume the existence of a \textit{common} $\Zs$-central bimodule structure on ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ compatible with all the $\Zs$-bilinear right/left actions.
}
\medskip
A \emph{morphism of multimodules} ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_B}\xrightarrow{(\phi,\eta,\Phi,\zeta,\psi)_f}{}_{(\Cs_\gamma)_C}\Ns_{({\mathscr{D}}_\delta)_D}$, $(A_+,B_+)$-covariant in the sub-families of indexes $A_+\subset A$, $B_+\subset B$ and $(A_-,B_-)$-contravariant in the sub-families of indexes $A_-:=A-A_+$, $B_-:=B-B_+$, consists of:
\begin{itemize}
\item
an injective function $f:A\uplus B\to C\uplus D$, with $A_+=A\cap f^{-1}(C)$, $B_+=B\cap f^{-1}(D)$;
\item
two maps $A\xrightarrow{\phi}\End_\Zs(\Rs)\xleftarrow{\psi}B$
associating to every pair of indexes $\alpha\in A$ and $\beta\in B$ two $\Zs$-linear unital endomorphisms $\phi_\alpha,\psi_\beta$ of $\Rs_\Zs$, covariant for $(\alpha,\beta)\in A_+\times B_+$ and contravariant for $(\alpha,\beta)\in A_-\times B_-$,
\item
for $(\alpha,\beta)\in A_+\times B_+$,
$\Zs$-linear covariant unital homomorphisms ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\xrightarrow{(\phi_\alpha,\eta_\alpha)}\Cs_{f(\alpha)}$, $\Bs_\beta\xrightarrow{(\psi_\beta,\zeta_\beta)}{\mathscr{D}}_{f(\beta)}$;
\item
for $(\alpha,\beta)\in A_-\times B_-$,
$\Zs$-linear contravariant unital homomorphisms ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\xrightarrow{(\phi_\alpha,\eta_\alpha)}{\mathscr{D}}_{f(\alpha)}$, $\Bs_\beta\xrightarrow{(\psi_\beta,\zeta_\beta)}\Cs_{f(\beta)}$;
\item
a $\Zs$-linear map ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{\Phi}\Ns$ such that $\Phi(a\cdot x\cdot b)=\eta_\alpha(a)\cdot \Phi(x)\cdot \zeta_\beta(b)$, for all $(\alpha,\beta)\in A_+\times B_+$ and $\Phi(a\cdot x\cdot b)=\zeta_\beta(b)\cdot \Phi(x)\cdot \eta_\alpha(a)$, for all $(\alpha,\beta)\in A_-\times B_-$, $(a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\times\Bs$ and $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$.
\end{itemize}
\medskip
The \emph{signature} of the morphism is $(\phi,\eta,\zeta,\psi)_f$. The function $f$ is the \emph{covariance of signature} of the morphism and \emph{covariant morphisms} are those for which $f(A)\subset C$ and $f(B)\subset D$. The pair $(\phi,\psi)$ is the \emph{$\Rs$-linearity of the signature} of the morphism and \emph{$\Rs$-linear morphisms} are those for which both $\phi$ and $\psi$ are constant equal to $\id_\Rs$. In some cases we will denote by $\Phi^\sigma$ a morphism $(\phi,\eta,\Phi,\zeta,\psi)_f$with signature $\sigma=(\phi,\eta,\zeta,\psi)_f$.
\medskip
The \emph{composition of morphisms}
${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_B}\xrightarrow{(\phi_2,\eta_2,\Phi_2,\zeta_2,\psi_2)_{f_2}}{}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}'_{\alpha'})_{A'}}\Ns_{(\Bs'_{\beta'})_{B'}}
\xrightarrow{(\phi_1,\eta_1,\Phi_1,\zeta_1,\psi_1)_{f_1}} {}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}''_{\alpha''})_{A''}}\Ps_{(\Bs''_{\beta''})_{B''}}$
of multimodules is given componentwise:
\begin{equation*}
(\phi_1,\eta_1,\Phi_1,\zeta_1,\psi_1)_{f_1}\circ (\phi_2,\eta_2,\Phi_2,\zeta_2,\psi_2)_{f_2}:=
(\phi_1\circ\phi_2,\eta_1\circ\eta_2,\Phi_1\circ\Phi_2,\zeta_1\circ\zeta_2,\psi_1\circ\psi_2)_{f_1\circ f_2}.
\end{equation*}
The \emph{identity} of a multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_B}$ is the morphism
$(\id_\Rs, (\id_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha})_A, \id_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},(\id_{\Bs_\beta})_B,\id_\Rs)_{\id_{A\uplus B}}$.
\end{definition}
\begin{remark}\label{rem: z2}
The map $\Phi:{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\to\Ns$ between multimodules does not have an intrinsic covariance: for every left index $\alpha\in A$ and for every right index $\beta\in B$ the morphisms $(\eta_\alpha,\Phi)$ and $(\zeta_\beta,\Phi)$ are covariant or contravariant depending on the sign $\pm$ indicated in the subsets $A_\pm$ and $B_\pm$.
\medskip
Similarly, the map $\Phi:{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\to\Ns$ between multimodules is always $\Zs$-linear, but it does not have an intrinsic $\phi$-linearity with respect to $\Rs$ for a fixed $\Zs$-linear morphism $\phi$: for every left index $\alpha\in A$ and right index $\beta\in B$, the morphism $(\eta_\alpha,\Phi)$ is $\phi_\alpha$-linear and the morphism $(\zeta_\beta,\Phi)$ is $\psi_\beta$-linear.
\medskip
We have a category $\Mf_{[\Rs_\Zs]}$ of morphisms of $\Zs$-central multimodules over $\Rs_\Zs$-algebras with composition of morphisms defined componentwise.
The subcategories of $\Mf_{[\Rs_\Zs]}$ consisting of $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$-$(\Bs_\beta)_{\beta\in B}$ multimodules, over the same two families of unital associative $\Rs_\Zs$-algebras, and morphisms given by $(\phi,\eta,\Phi,\zeta,\psi)_f$, with $f:=\id_{A\uplus B}$,
$\phi_\alpha:=\id_\Rs=:\psi_\beta$ and $\eta_\alpha=\id_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha}$, $\zeta_\beta=\id_{\Bs_\beta}$ for all $(\alpha,\beta)\in A\times B$, are denoted by ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B}$. In case of ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$-bimodules, we use the notation ${}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\Mf_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$.
\xqed{\lrcorner}
\end{remark}
This essential remark explains why the study of multimodules cannot be ``reduced'' to the theory of bimodules.
\begin{remark}\label{rem: hyper}
If $\Rs=\Zs$, it is common to dismiss the usage of multimodules ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)}$ in favor of their ``equivalent'' description as bimodules ${}_{\bigotimes^\Rs_\alpha{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{\bigotimes^\Rs_\beta\Bs_\beta}$ over tensor product $\Rs$-algebras $\bigotimes^\Rs_{\alpha\in A}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha$ and ${\bigotimes^\Rs_{\beta\in B}\Bs_\beta}$ since:
\begin{quote}
if $\Rs=\Zs$, there is a categorical isomorphism between the sub-category $_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B}$ of \textit{covariant $\Rs$-linear} morphisms of $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A$-$(\Bs_\beta)_B$-multimodules and the category ${}_{\bigotimes^\Rs_\beta{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha}\Mf_{\bigotimes^\Rs_\beta\Bs_\beta}$ of \textit{covariant $\Rs$-linear} morphisms of bimodules over the $\Rs$-balanced tensor product of the $\Rs_\Zs$-algebras.
\end{quote}
As soon as one considers morphisms of multimodules with arbitrary covariance $f$, it is actually impossible to impose a unique unital associative product on the $\Rs$-tensor product algebras in order to obtain a similar equivalent treatment via categories of bimodules.
\medskip
A perfectly possile alternative (that we do not pursue here) would be to work with the category of ``bimodules'' over \textit{hyper-$\Zs$-central $\Rs$-algebras}: $\Zs$-central bimodules $\bigotimes^\Zs_{\alpha\in A}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha$ equipped with many different $\Rs$-actions (on each of the tensor-factors) and different $\Zs$-bilinear associative unital binary product operations suitably compatible with the $\Rs$-actions (see for example~\cite[section~5.3]{BCLS20}); but in this case multimodules need anyway to be used in order to define hyper-algebras.
\xqed{\lrcorner}
\end{remark}
To a certain extent, the usage of general morphisms of multimodules (with arbitrary conjugation and convariance signatures as in definition~\ref{def: morphism}) can be avoided, replacing the target multimodule with a suitable ``twisted version'' (depending on the signatures of the original morphism) and obtaining as a result an $\Rs$-linear covariant morphism into such ``twisted multimodule''. The construction follows similar steps as in definition~\ref{def: cj-dual} and remark~\ref{rem: cj-dual} and it simultaneously extends to multimodules the notions of \textit{conjugate-dual}, \textit{opposite}, \textit{restriction of rings}, \textit{pull-back}.
\begin{definition}\label{def: twisted}
Let ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_B}$ and ${}_{(\Cs_\gamma)_C}\Ns_{({\mathscr{D}}_\delta)_D}$ be two $\Zs$-central multimodules over $\Rs_\Zs$-algebras, and let \hbox{$\sigma:=(\phi,\eta,\zeta,\phi)_f$} be a given signature for multimodule morphisms between ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ and $\Ns$.
\medskip
A \emph{$\sigma$-twisted multimodule}\footnote{
We might also write \textit{$\Phi$-twisted} of $\Ns$, for a morphism ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{\Phi}\Ns$, instead of $\sigma(\Phi)$-twisted, where $\sigma(\Phi)$ denotes the signature of $\Phi$.
} of $\Ns$ consists of a morphism of multimodules ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\Ns^\sigma}_{(\Bs_\beta)_B}\xrightarrow{\Theta^\sigma_\Ns}{}_{(\Cs_\gamma)_C}\Ns_{({\mathscr{D}}_\delta)_D}$, with signature $\sigma$, such that the following universal factorization property is satisfied:
for any other morphism of multimodules ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}_{(\Bs_\beta)_B}\xrightarrow{\Phi}{}_{(\Cs_\gamma)_C}\Ns_{({\mathscr{D}}_\delta)_D}$, with signature $\sigma$, there exists a unique covariant $\Rs$-linear morphism of multimodules
${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}_{(\Bs_\beta)_B}\xrightarrow{\Phi^\sigma}{\Phi}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\Ns^\sigma}_{(\Bs_\beta)_B}$
in the category ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\Mf}_{(\Bs_\beta)_B}$ such that $\Phi=\Theta^\sigma_\Ns\circ \Phi^\sigma$.
\end{definition}
\begin{remark}\label{prop: cj-duals}
As any definition via universal factorizations, $\sigma$-twisted of a given multimodule are unique, up to a unique isomorphism compatible with the factorization property. A construction can be achieved as follows.
Consider $\Ns^\sigma:=\Ns$ as a $\Zs$-central bimodule and $\Theta_\Ns^\sigma:\Ns^\sigma\to\Ns$ as the identity map. For all $x\in \Ns$ we will denote by $x^\sigma\in\Ns^\sigma$ its corresponding element, hence $\Theta^\sigma_\Ns(x^\sigma)=x$, for all $x\in\Ns$. For all $\sigma$-covariant indexes $(\alpha_+,\beta_+)\in A_+\times B_+$, and $\sigma$-contravariant indexes $(\alpha_-,\beta_-)\in A_-\times B_-$, we define new actions on $\Ns^\sigma$:
\begin{gather*}
a\cdot_{\alpha_+} x^\sigma\cdot_{\beta_+} b:= \left(\eta_{\alpha_+}(a)\cdot_{f(a_+)} x\cdot_{f(b_+)} \zeta_{b_+}(b)\right)^\sigma,
\quad \forall (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\alpha_+}\times\Bs_{\beta_+}, \ \forall x^\sigma\in\Ns^\sigma,
\\
a\cdot_{\alpha_-} x^\sigma\cdot_{\beta_-} b:=
\left( \zeta_{b_-}(b)\cdot_{f(b_-)} x \cdot_{f(a_-)} \eta_{\alpha_-}(a)\right)^\sigma,
\quad \forall (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\alpha_-}\times\Bs_{\beta_-}, \ \forall x^\sigma\in\Ns^\sigma,
\end{gather*}
obtaining a multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\Ns^\sigma}_{(\Bs_\beta)_B}$ such that the map $\Theta_\Ns^\sigma:x^\sigma\mapsto x$ is a morphism of multimodules with signature $\sigma$.
Finally, given any other morphism ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}_{(\Bs_\beta)_B}\xrightarrow{\Phi}{}_{(\Cs_\gamma)_C}\Ns_{({\mathscr{D}}_\delta)_D}$ of multimodules with signature $\sigma$, the function $\Phi^\sigma: m\mapsto (\Phi(m))^\sigma\in\Ns^\sigma$, (due to the bijectivity of $\Theta_\Ns^\sigma$) is the unique map that satisfies
$\Theta_\Ns^\sigma(\Phi^\sigma(m))=\Theta_\Ns^\sigma((\Phi(m))^\sigma)=\Phi(m)$, for all $m\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$, and by direct calculation,
we see that it is also a morphism of multimodules with identity signature.
\xqed{\lrcorner}
\end{remark}
As typical of any category of homomorphisms of algebraic structures, sub-structures can be defined via algebraically closed subsets and quotient-structures via congruences.
\begin{definition}
Given a multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_B}$ over $\Zs$-central $\Rs$-algebras,
\begin{itemize}
\item
a \emph{sub-multimodule} of ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ is a subset $\Ns\subset{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ that is \emph{algebraically closed} under all the operations:
\begin{equation*}
0_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\in\Ns, \quad
x,y\in\Ns \imp x+y\in\Ns,
\quad
x\in\Ns \imp a\cdot_\alpha x\cdot_\beta b\in\Ns,
\quad
\forall (\alpha,\beta)\in A\times B, \ (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ x,y\in \Ns;
\end{equation*}
\item
a \emph{multimodule congruence} on ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_B}$ is an equivalence relation $\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}\subset{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ such that:
\begin{equation*}
x\sim_\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G} y \imp (x+z)\sim_\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G} (y+z),
\quad
x\sim_\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G} y \imp (a\cdot_\alpha x\cdot_\beta b)\sim_\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G} (a\cdot_\alpha y\cdot_\beta b),
\quad \forall (\alpha,\beta)\in A\times B, \ (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ x,y,z\in {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}.
\end{equation*}
\end{itemize}
A \emph{quotient multimodule} of ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ by the congruence $\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}$ is the multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}(\frac{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}{\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}})_{(\Bs_\beta)_B}$ consisting of the quotient set ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}/\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}$ equipped with the well-defined addition $[x]_\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}+[y]_\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}:=[x+y]_\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}$, for all $x,y\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$, and the well-defined actions:
$a\cdot_\alpha [x]_\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}\cdot_\beta b:=[a\cdot_\alpha x\cdot_\beta b]_\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}$, for all $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$, $(\alpha,\beta)\in A\times B$ and $(a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta$.
\end{definition}
\begin{remark}
As usual, any multimodule congruence $\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}$ uniquely determines the ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$-sub-multimodule $[0_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}]_\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}$; reciprocally any ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$-sub-multimodule $\Ns$ uniquely determines a multimodule congruence $x\sim y :\iff x-y\in\Ns$ whose equivalence classes, for $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$, are the affine spaces $[x]_\sim=x+\Ns:=\{x+y \ | \ y\in\Ns \}$. The notation $\frac{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}{\Ns}$ is used to identify the quotient of ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ by the congruence uniquely determined by the sub-multimodule $\Ns$.
\medskip
Inclusions of sub-multimodules $\Ns\xrightarrow{\iota}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ and quotients ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{\pi}\frac{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}{\Ns}$ are morphisms in the category ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B}$.
\xqed{\lrcorner}
\end{remark}
\medskip
Despite being rarely mentioned, multimodules naturally appear whenever bimodules are around:
\begin{proposition}\label{prop: homK}
Let ${}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_\Bs$ and ${}_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}'}\Ns_{\Bs'}$ be $\Zs$-central bimodules over $\Zs$-central unital associative $\Rs$-algebras ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}, {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}',\Bs,\Bs'$. The set $\Hom_\Zs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)$ of \hbox{$\Zs$-linear} maps $\phi:{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\to\Ns$ is a left-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}',\Bs)$ right-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}},\Bs')$ multimodule with the following actions, for all $a\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$, $a'\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}'$, $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$, $b\in\Bs$, $b'\in\Bs'$:
\begin{align*}
&\textit{left external action:} &&(a'\cdot \phi)(x):=a'\phi(x),
\\
&\textit{right external action:} &&(\phi\cdot b')(x):=\phi(x)b',
\\
&\textit{left internal action:} &&(b\odot \phi)(x):=\phi(xb),
\\
&\textit{right internal action:} &&(\phi\odot a)(x):=\phi(ax).
\end{align*}
\end{proposition}
\begin{proof}
By direct calculation, $x\mapsto (a'\cdot\phi)(x)$, $x\mapsto (\phi\cdot b')(x)$, $x\mapsto (b\odot \phi)(x)$, $x\mapsto(\phi\odot a)(x)$ are all $\Zs$-linear and the above defined maps are all $\Zs$-bilinear actions.
To prove the multimodule structure on $\Hom_\Zs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)$ we check that the actions pairwise commute, for all $a\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$, $a'\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}'$ and all $b\in\Bs$, $b'\in\Bs'$:
\begin{align*}
&(a'\cdot\phi)\cdot b'=a'\cdot(\phi\cdot b'),
&&
(a'\cdot\phi)\odot a=a'\cdot(\phi\odot a),
\\
&(b\odot\phi)\odot a=b\odot(\phi\odot a),
&&
(b\odot\phi)\cdot b'=b\odot(\phi\cdot b').
\qedhere
\end{align*}
\end{proof}
\begin{remark}\label{rem: Hom-multi}
More generally, if ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)}$ and ${}_{(\Cs_\gamma)}\Ns_{({\mathscr{D}}_\delta)}$ are $\Zs$-central multimodules over $\Zs$-central $\Rs$-algebras, the $\Zs$-central bimodule $\Hom_\Zs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)$ becomes a left-$(\Bs_\beta,\Cs_\gamma)_{\beta\in B,\gamma\in C}$ and a right-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha,{\mathscr{D}}_\delta)_{\alpha\in A,\delta\in D}$ $\Zs$-central multimodule with internal/external actions given by:
\begin{gather*}
(c_\gamma\cdot \phi\cdot d_\delta)(x):=c_\gamma\cdot \phi(x)\cdot d_\delta,
\quad
(b_\beta\odot \phi\odot a_\alpha)(x):=\phi(a_\alpha\cdot x\cdot b_\beta),
\end{gather*}
for all $(\alpha,\beta,\gamma,\delta)\in A\times B\times C\times D$, $(a_\alpha,b_\beta,c_\gamma,d_\delta)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta\times\Cs_\gamma\times{\mathscr{D}}_\delta$, $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ and $\phi\in\Hom_\Zs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)$.
\xqed{\lrcorner}
\end{remark}
Tensor products provide other examples of multimodules~\cite[section~II.3.4]{Bou89}:
\begin{proposition}
Let ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)}$ and ${}_{(\Cs_\gamma)}\Ns_{({\mathscr{D}}_\delta)}$ be $\Zs$-central multimodules over $\Zs$-central $\Rs$-algebras. Their tensor product ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Zs\Ns$ over $\Zs$ is a left-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha,\Cs_\gamma)_{\alpha\in A,\gamma\in C}$ right-$(\Bs_\beta,{\mathscr{D}}_\delta)_{\beta\in B,\delta\in D}$ $\Zs$-central multimodule.
\end{proposition}
\begin{proof}
The definition of tensor product (via universal factorization property for $\Zs$-balanced bi-homomorphism) and its construction are well-known: see for example~\cite[section~II.3, proposition~3]{Bou89}; we only recall here the relevant actions on simple tensors:
\begin{equation*}
a\cdot (x\otimes_\Zs y)=(a\cdot x)\otimes_\Zs y,
\quad
c\cdot (x\otimes_\Zs y)=x\otimes_\Zs (c\cdot y),
\quad
(x\otimes_\Zs y)\cdot b=(x\cdot b)\otimes_\Zs y,
\quad
(x\otimes_\Zs y)\cdot d=x\otimes_\Zs (y\cdot d),
\end{equation*}
for all $(x,y)\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns$, $(a,b,c,d)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta\times\Cs_\gamma\times{\mathscr{D}}_\delta$, $(\alpha,\beta,\gamma,\delta)\in A\times B\times C\times D$.
\end{proof}
\begin{remark}\label{rem: otimes-tr}
One can actually define tensor products of multimodules in much greater generality.
\medskip
Instead of taking only the tensor product over the algebra $\Zs$ of ``scalars'' and use $\Zs$-bilinear maps, we can ``contract'' over arbitrary families of shared $\Zs$-central $\Rs$-algebras acting on the two multimodules and utilize suitable maps that are ``balanced'' over the ``contracted actions'', obtaining multimodules over the remaining ``un-contracted'' actions, as detailed in the following exposition.
\medskip
Let ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_B}$ and ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\gamma)_C}\Ns_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\delta)_D}$ be a pair of multimodules; consider the relation $A\uplus B\xrightarrow{\Sigma}C\uplus D$, defined by $(\xi,\zeta)\in\Sigma \iff {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\zeta$ (where $\xi\in A\uplus B$ and $\zeta\in C\uplus D$), and let $A\uplus B \supset A'\uplus B' \xrightarrow{\Gamma} C'\uplus D'\subset C\uplus D$ be a bijective function between subsets of indexes, such that $\Gamma\subset \Sigma$ (in practice ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\Gamma(\xi)}$, for all $\xi\in A'\uplus B'$).
\medskip
A \emph{tensor product of multimodules} ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ and $\Ns$ over $\Gamma$ consists of:
\begin{itemize}
\item
a left-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi)_{\xi\in (A-A')\uplus(C-C')}$ right-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\zeta)_{\zeta\in (B-B')\uplus(D-D')}$ multimodule
${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi)_{(A-A')\uplus(C-C')}}({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Gamma\Ns)_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\zeta)_{(B-B')\uplus(D-D')}}$,
\item
a \emph{$\Gamma$-balanced bi-morphism} ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns\xrightarrow{\eta} {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Gamma\Ns$, that means a $\Zs$-bilinear map that satisfies:
\begin{align} \notag
&
\eta(a \cdot_\xi x,y)=\eta(x,a\cdot_{\Gamma(\xi)} y), \ \forall a\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi, \ (\xi,\Gamma(\xi))\in A'\times C', \ (x,y)\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns,
\\ \notag
&
\eta(x \cdot_\xi a,y)=\eta(x,y\cdot_{\Gamma(\xi)} a), \ \forall a\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi, \ (\xi,\Gamma(\xi))\in B'\times D', \ (x,y)\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns,
\\ \notag
&\eta(x \cdot_\xi a,y)=\eta(x,a\cdot_{\Gamma(\xi)} y), \ \forall a\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi, \ (\xi,\Gamma(\xi))\in B'\times C', \ (x,y)\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns,
\\ \label{eq: balanced-cong}
&\eta(a \cdot_\xi x,y)=\eta(x,y\cdot_{\Gamma(\xi)} a), \ \forall a\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi, \ (\xi,\Gamma(\xi))\in A'\times D', \ (x,y)\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns,
\\ \notag
&\eta(a \cdot_\xi x,y)=a\cdot_\xi\eta(x,y), \ \forall a\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi, \ \xi\in A-A', \ (x,y)\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns,
\\ \notag
&\eta(x,c \cdot_\xi y)=c\cdot_\xi\eta(x,y), \ \forall c\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi, \ \xi\in C-C', \ (x,y)\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns,
\\ \notag
&\eta(x \cdot_\xi b,y)=\eta(x,y)\cdot_\xi b, \ \forall b\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi, \ \xi\in B-B', \ (x,y)\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns,
\\ \notag
&\eta(x,y \cdot_\xi d)=\eta(x,y)\cdot_\xi d, \ \forall d\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi, \ \xi\in D-D', \ (x,y)\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns,
\end{align}
\end{itemize}
in such a way that the following universal factorization property holds: for any other $\Gamma$-balanced bi-morphism
${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns\xrightarrow{\Phi}\Ps$ into a left-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi)_{(A-A')\uplus(C-C')}$ right-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\zeta)_{\zeta\in (B-B')\uplus(D-D')}$ multimodule $\Ps$, there exists a unique morphism of multimodules ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Gamma\Ns\xrightarrow{\hat{\Phi}}\Ps$ (over the same indexed families of algebras) such that $\Phi=\hat{\Phi}\circ \eta$.
\medskip
Its construction is standard and consists of the quotient of a free multimodule over ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns$ by the congruence generated by the required axioms of $\Gamma$-balanced bi-morphism. More specifically we recall that:
\begin{itemize}
\item
a \emph{free $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$-$(\Bs_\beta)_{\beta\in B}$ multimodule}, over a set $X$, is function $X\xrightarrow{\eta^X}{}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Fg(X)_{(\Bs_\beta)_B}$, with values into a
$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$-$(\Bs_\beta)_{\beta\in B}$ multimodule $\Fg(X)$, such that the following universal factorization property is satisfied:
for any other map $X\xrightarrow{\Phi}{}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_B}$ into an $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$-$(\Bs_\beta)_{\beta\in B}$ multimodule ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$, there exists a unique morphism of multimodules $\Fg(X)\xrightarrow{\hat{\Phi}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ in the category ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B}$ such that $\phi=\hat{\Phi}\circ\eta^X$;
\item
a construction of free multimodule over $X$ can be achieved taking $\bigoplus_{x\in X}[(\otimes^\Zs_{\alpha\in A}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in B}\Bs)]$, the set of finitely supported functions from $X$ into
$(\otimes^\Zs_{\alpha\in A}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in B}\Bs)$ with pointwise addition and pointwise outer target actions as specified in footnote~\ref{foo: free-m}, defining $\eta^X_x(y):=\begin{cases}
(\otimes^\Zs_{\alpha\in A}1_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha})\otimes_\Zs (\otimes^\Zs_{\beta\in B}1_{\Bs_\beta}), \ y=x,
\\
(\otimes^\Zs_{\alpha\in A}0_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha})\otimes_\Zs (\otimes^\Zs_{\beta\in B}0_{\Bs_\beta}), \ y\neq x,
\end{cases}$ for all $x,y\in X$, and checking the universal factorization property;
\item
the congruence $\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}_\Gamma$ generated by the relations in equations~\eqref{eq: balanced-cong} is just the intersection of the set of congruences of $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A$-$(\Bs_\beta)_B$ multimodule in $\Fg({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times \Ns)$, that contain all of the differences between left and right terms in each of the equations~\ref{eq: balanced-cong};
\item
the tensor product consists of the quotient multimodule ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Gamma\Ns:=\frac{\Fg({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns)}{\mathcal{E}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}_\Gamma}$, with the $\Gamma$-balanced bi-morphism $\eta:=\pi\circ\eta^{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns}$, where $\Fg({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns)\xrightarrow{\pi}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Gamma\Ns$ is the quotient morphism.
\xqedhere{3cm}{\lrcorner}
\end{itemize}
\end{remark}
\section{Involutions in Multimodules} \label{sec: involutions-m}
In parallel with the case of morphisms, also the nature of involutions in multimodules is more delicate and
an involution is an involutive endomorphism inducing involutions on the algebras and conjugations on $\Rs$.
\begin{definition}
Let ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_B}$ be a $\Zs$-central multimodule over $\Zs$-central unital associative $\Rs$-algebras.
\medskip
A \emph{multimodule involution} on ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ is a morphism ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_B}\xrightarrow{(\phi,\eta,\star,\zeta,\psi)_f}{}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_B}$ that is involutive:
\begin{itemize}
\item
$A\uplus B\xrightarrow{f}A\uplus B$ is an involutive function $f\circ f=\id_{A\uplus B}$;\footnote{
From the involutivity of $f$, we have $f(A_+)=A_+, \ f(B_+)=B_+, \ f(A_-)=B_-$ and $f(B_-)=A_-$.
}
\item
for all $(\alpha_1,\alpha_2)\in f\cap (A\times A)$, ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\alpha_1}={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\alpha_2}$,
$\dagger_{\alpha_1}:=\phi_{\alpha_1}=\phi_{\alpha_2}$ is a covariant $\Rs_\Zs$-conjugation,
$\ddagger_{\alpha_1}:=\eta_{\alpha_1}=\eta_{\alpha_2}$ is a covariant
$\dagger_{\alpha_1}$-linear involution;
\item
for all $(\beta_1,\beta_2)\in f\cap (B\times B)$, ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\beta_1}={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\beta_2}$,
$\dagger_{\beta_1}:=\psi_{\beta_1}=\psi_{\beta_2}$ is a covariant $\Rs_\Zs$-conjugation,
$\ddagger_{\beta_1}:=\zeta_{\beta_1}=\zeta_{\beta_2}$ is a covariant
$\dagger_{\beta_1}$-linear involution;
\item
for all $(\alpha,\beta)\in f\cap (A\times B)$, ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\alpha}={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\beta}$,
$\dagger_\alpha:=\phi_{\alpha}=\psi_{\beta}=:\dagger_\beta$ is a contravariant $\Rs_\Zs$-conjugation,
$\ddagger_\alpha:=\eta_{\alpha}=\zeta_{\beta}=:\ddagger_\beta$ is a contravariant
$\dagger_\alpha$-linear involution;
\item
${}_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\alpha}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\beta}}\xrightarrow{\star}{}_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\alpha}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\beta}}$ is an involution such that:
\begin{gather*}
\forall (\alpha,\beta)\in A_+\times B_+ \st (a\cdot_\alpha x\cdot_\beta b)^\star= a^{\ddagger_\alpha}\cdot_{f(\alpha)} x^\star\cdot_{f(\beta)} b^{\ddagger_\beta}, \quad \forall (a,b)\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta, \ x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};
\\
\forall (\alpha,\beta)\in A_-\times B_-
\st (a\cdot_\alpha x\cdot_\beta b)^\star=b^{\ddagger_\beta}\cdot_{f(\beta)} x^\star\cdot_{f(\alpha)} a^{\ddagger_\alpha}, \quad \forall (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta, \ x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}.
\end{gather*}
\end{itemize}
If necessary, we will denote an involution of ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}}$ by
$(\dagger^\sigma_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},\ddagger^\sigma_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})_{\sigma\in f_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}$, where:
\begin{equation*}
\vcenter{\xymatrix{
\Zs \ar@{=}[d] \ar[r]^{\iota_\Rs} & \Rs \ar[d]^{\dagger_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\sigma} \ar[r]^{\iota_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}} & {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\sigma_1} \ar[d]^{\ddagger_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\sigma} & {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}} \ar[d]^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}
\\
\Zs \ar[r]^{\iota_\Rs} & \Rs \ar[r]^{\iota_\Bs} & {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\sigma_2} & {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}
}}\quad \sigma:=(\sigma_1,\sigma_2)\in f_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\subset (A_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\uplus B_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})\times(A_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\uplus B_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}).
\end{equation*}
\end{definition}
Here we examine involutions for multimodules of morphisms between involutive bimodules.
\begin{proposition}\label{prop: dagger}
Suppose that ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ and $\Bs$ are both $\Zs$-central $\Rs_\Zs$-algebras with involutions $\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ and $\ddagger_\Bs$ over the respective $\Rs$-conjugations $\dagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ and $\dagger_\Bs$.
If the $\Zs$-central bimodules ${}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ and ${}_{\Bs}\Ns_{\Bs}$ are both involutive with involutions $(\dagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}},\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}},\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})_{f_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}$ and $(\dagger_\Bs,\ddagger_\Bs,\star_\Ns)_{f_\Ns}$,
also the $\Zs$-central multimodule ${}_{\Bs,{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}}\Hom_\Zs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}},\Bs}$, considered in proposition~\ref{prop: homK}, is equipped with an involutive map $\star:T\mapsto T^\star:=\star_\Ns\circ T\circ \star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ and becomes an involutive multimodule with multimodule involution $(\dagger,\ddagger,\star)_f$, defined as follows:
\begin{equation*}
\vcenter{\xymatrix{
\Rs \ar[d]_{\dagger^\rho:=\dagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}} \ar[r]^{\iota_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}} & {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{} \ar[d]^{\ddagger^\rho:=\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}} &
\Hom_\Zs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns) \ar[d]^{\star}
& \Bs_{} \ar[d]_{\ddagger^\sigma:=\ddagger_\Bs} & \Rs \ar[d]^{\dagger^\sigma:=\dagger_\Bs} \ar[l]_{\iota_\Bs}
\\
\Rs \ar[r]^{\iota_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}} & {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{} & \Hom_\Zs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)
& \Bs_{} & \Rs \ar[l]_{\iota_\Bs}
}}
\quad
f:=f_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^*\uplus f_\Ns, \quad \rho\in f_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^*, \ \sigma\in f_\Ns.
\end{equation*}
The involution $\star$ has covariance signature and $\Rs$-linearity signatures that, for inner actions, coincide with those of $\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$; and for outer actions with those of $\star_\Ns$.
\medskip
If $({}_{\Cs}\Ps_{\Cs},\star_\Ps)$ is an involutive $\Zs$-central bimodule over $\Rs_\Zs$-algebras, we have
$(T\circ S)^\star=T^\star\circ S^\star$, for all $(T,S)\in \Hom_\KK(\Ns;\Ps)\times\Hom_\KK({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)$. In particular $(\Hom_\Zs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}),\circ,\star)$ is a unital associative $\Zs$-central algebra with a covariant involution.\footnote{
Notice that the involution $\star$ is multiplicative independently from the convariace/contravariace of the original involutions on ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$.}
\end{proposition}
\begin{proof}
If $T\in\Hom_\Zs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)$ with $\Rs$-linearity signature $\phi_T$, the composition $T^\star:=\star_\Ns\circ T\circ \star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ is $\Zs$-linear and with $\Rs$-linearity signature $\dagger_\Ns\circ\phi_T\circ\dagger_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ and hence
$T\mapsto T^\star$ is well-defined as an endo-map of $\Hom_\Zs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)$.
\medskip
The involutivity of $\star$ follows from: $(T^{\star})^{\star}=\star_\Ns\circ\star_\Ns\circ T\circ \star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\circ \star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}=T$, for all $T\in\Hom_\Zs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)$.
\medskip
For the actions, if $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}},\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}})$ and $(\Bs,\ddagger_\Bs)$ are contravariantly involutive, we necessarily have:
\begin{align*}
(c\cdot b\odot T\odot a\cdot d)^{\star}(x)&=(c\cdot T(a\cdot x^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}\cdot b)\cdot d)^{\star_\Ns}
=(d^{\ddagger_\Bs}\cdot T((b^{\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}}\cdot x\cdot a^{\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}})^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}})^{\star_\Ns}\cdot c^{\ddagger_\Bs})
\\
&=(d^{\ddagger_\Bs}\cdot a^{\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}}\odot T^\star\odot b^{\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}}\cdot c^{\ddagger_\Bs})(x),
\quad \forall a,b\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}, \ c,d\in \Bs, \ x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}.
\end{align*}
Whenever $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}},\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}})$ and $(\Bs,\ddagger_\Bs)$ are covariantly involutive, we obtain:
\begin{align*}
(c\cdot b\odot T\odot a\cdot d)^\star(x)&=(c\cdot T(a\cdot x^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}\cdot b)\cdot d)^{\star_\Ns}
=(c^{\ddagger_\Bs}\cdot T((a^{\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}}\cdot x\cdot b^{\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}})^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}})^{\star_\Ns}\cdot d^{\ddagger_\Bs})
\\
&=(c^{\ddagger_\Bs}\cdot b^{\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}}\odot T^\star\odot a^{\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}}\cdot d^{\ddagger_\Bs})(x),
\quad \forall a,b\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}, \ c,d\in \Bs, \ x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}.
\end{align*}
The remaining two cases with opposite contravariance between $(A,\ddagger_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}})$ and $(\Bs,\ddagger_\Bs)$ are treated similarly.
\medskip
Finally $(T\circ S)^\star=\star_\Ps\circ T\circ S\circ \star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}} =\star_\Ps\circ T\circ \star_\Ns\circ\star_\Ns \circ S\circ \star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}=T^\star\circ S^\star, \ \forall (T,S)\in\Hom_\KK(\Ns;\Ps)\times\Hom_\KK({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)$.
\medskip
Notice that the involution $\star:T\mapsto T^\star$ has $\Rs$-linearity signature $\dagger_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ for the inner actions and the $\Rs$-linearity signature of $\dagger_\Ns$ for the outer actions.
\end{proof}
\begin{remark}
The previous proposition can easily be further generalized: whenever ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)}$ and ${}_{(\Bs_\gamma)}\Ns_{(\Bs_\delta)}$ are $\Zs$-central multimodules over $\Zs$-central $\Rs_\Zs$-algebras, any pair $(\dagger_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},\ddagger_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})_{f_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}$ and $(\dagger_\Ns,\ddagger_\Ns,\star_\Ns)_{f_\Ns}$ of involutions,
induces an involution $\star:T\mapsto T^\star:=\star_\Ns\circ T\circ\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ of $\Hom_\Zs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)$,
that is compatible with all the external and internal actions of the multimodule ${}_{(\Bs_\gamma,{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)}\Hom_\KK({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha,\Bs_\delta)}$ defined in remark~\ref{rem: Hom-multi} and hence, defining $f:=f_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^*\uplus f_\Ns$, $\dagger:=\dagger_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\uplus \dagger_\Ns$, $\ddagger:=\ddagger_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\uplus\ddagger_\Ns$, we see that $(\dagger,\ddagger,\star)_f$ is an involution of the $\Zs$-central multimodule ${}_{(\Bs_\gamma,{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)}\Hom_\KK({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Ns)_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha,\Bs_\delta)}$ over $\Rs_\Zs$-algebras.\footnote{
Here, given two functions $F:A\to B$ and $G:C\to D$ with define $F\uplus G:A\uplus B\to C\uplus D$ the ``disjoint union'' of the two maps.
}
\xqed{\lrcorner}
\end{remark}
The following proposition describes involutions in the case of tensor products of involutive multimodules.
\begin{proposition}\label{prop: inv-ot}
Let $\left({}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_B},((\star_\alpha)_A,\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},(\star_\beta)_B)_f\right)$ and $\left({}_{(\Bs_\gamma)_C}\Ns_{(\Bs_\delta)_D},((\star_\gamma)_C,\star_\Ns,(\star_\delta)_D)_g\right)$ be involutive $\Zs$-central multimodules over $\Rs_\Zs$-algebras;
the tensor product multimodule
${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha,\Bs_\gamma)_{A\uplus C}}({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Zs\Ns)_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta,\Bs_\delta)_{B\uplus D}}$ has an involution $((\star_\alpha,\star_\beta)_{A\uplus C}, \star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Zs\star_\Ns,(\star_\gamma,\star_\delta)_{B\uplus D})_{(f,g)}$.
\end{proposition}
\begin{proof}
Define $A\uplus B\uplus C\uplus D \xrightarrow{(f,g)}A\uplus B\uplus C\uplus D$ as the ``disjoint union'' of the involutions $A\uplus B\xrightarrow{f} A\uplus B$ and $C\uplus D\xrightarrow{g} C\uplus D$. It follows that $(f,g)$ is an involution.
Furthermore, for all $\Ts\in\{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}},\Bs\}$, for all $\tau\in\{\alpha,\beta,\gamma,\delta\}$ we have
$(\Ts_{\tau},\star_{\tau})=(\Ts_{(f,g)(\tau)}\star_{(f,g)(\tau)})$.
\medskip
The $\Zs$-linear map ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Zs\Ns\xrightarrow{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Zs\star_\Ns}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Zs\Ns$, defined by universal factorization property from the $\Zs$-bilinear map
${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns \ni (x,y)\mapsto x^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}\otimes_\Zs y^{\star_\Ns}\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Zs\Ns$, for all $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$, $y\in\Ns$, is involutive.
\medskip
The covariance/contravariance behavior of the involution $\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Zs\star_\Ns$ with respect to the several actions is described as follows, denoting by $\tau_\pm$, for $\tau\in\{\alpha,\beta,\gamma,\delta\}$, the indexes corresponding respectively to covariantly/contravariantly involutive algebras:
\begin{align*}
\forall (\alpha_+,\beta_+,\gamma_+,\delta_+)\in A_+\times B_+\times C_+\times D_+, \
&\forall (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\alpha_+}\times{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\beta_+},\ (c,d)\in\Bs_{\gamma_+}\times\Bs_{\delta_+},\ (x,y)\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns:
\\
(a\cdot_{\alpha_+} c\cdot_{\gamma_+}(x\otimes_\Zs y) \cdot_{\beta_+} b \cdot_{\delta_+} d)^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Zs\star_\Ns}
&=(a\cdot_{\alpha_+}x\cdot_{\beta_+} b)^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}\otimes_\Zs (c\cdot_{\gamma_+} y \cdot_{\delta_+} d)^{\star_\Ns}
\\
&=(a^{\star_{\alpha_+}}\cdot_{f(\alpha_+)}x^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}\cdot_{f(\beta_+)} b^{\star_{\beta_+}})\otimes_\Zs (c^{\star_{\gamma_+}}\cdot_{g(\gamma_+)} y^{\star_\Ns} \cdot_{g(\delta_+)} d^{\star_{\delta_+}})
\\
&=a^{\star_{\alpha_+}}\cdot_{f(\alpha_+)} c^{\star_{\gamma_+}}\cdot_{g(\gamma_+)}
(x\otimes_\Zs y)^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Zs\star_\Ns}
\cdot_{f(\beta_+)} b^{\star_{\beta_+}}\cdot_{g(\delta_+)} d^{\star_{\delta_+}},
\\
\forall (\alpha_-,\beta_-,\gamma_-,\delta_-)\in A_-\times B_-\times C_-\times D_-, \
&\forall (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\alpha_-}\times{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\beta_-},\ (c,d)\in\Bs_{\gamma_-}\times\Bs_{\delta_-},\ (x,y)\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times\Ns:
\\
(a\cdot_{\alpha_-} c\cdot_{\gamma_-}(x\otimes_\Zs y) \cdot_{\beta_-} b \cdot_{\delta_-} d)^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Zs\star_\Ns}
&=(a\cdot_{\alpha_-}x\cdot_{\beta_-} b)^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}\otimes_\Zs (c\cdot_{\gamma_-} y \cdot_{\delta_-} d)^{\star_\Ns}
\\
&=(b^{\star_{\beta_-}}\cdot_{f(\beta_-)}x^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}\cdot_{f(\alpha_-)} a^{\star_{\alpha_-}})\otimes_\Zs (d^{\star_{\delta_-}}\cdot_{g(\delta_-)} y^{\star_\Ns} \cdot_{g(\gamma_-)} c^{\star_{\gamma_-}})
\\
&=b^{\star_{\beta_-}}\cdot_{f(\beta_-)}d^{\star_{\delta_-}}\cdot_{g(\delta_-)}
(x\otimes_\Zs y)^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Zs\star_\Ns}
\cdot_{f(\alpha_-)} a^{\star_{\alpha_-}}
\cdot_{g(\gamma_-)} c^{\star_{\gamma_-}},
\end{align*}
where we used the fact that $(\Ts_{\tau_\pm},\star_{\tau_\pm})=(\Ts_{(f,g)(\tau_\pm)},\star_{(f,g)(\tau_\pm)})$, for $\tau\in\{\alpha,\beta,\gamma,\delta \}$.
\end{proof}
More generally, we can use the tensor product over subfamilies defined in remark~\ref{rem: otimes-tr}.
\begin{remark}
Let $\left({}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_B},((\star_\alpha)_A,\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},(\star_\beta)_B)_{f_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}\right)$ and $\left({}_{(\Bs_\gamma)_C}\Ns_{(\Bs_\delta)_D},((\star_\gamma)_C,\star_\Ns,(\star_\delta)_D)_{f_\Ns}\right)$ be two involutive multimodules. The ``internal tensor product'' ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Gamma\Ns$ over an indexed family $\Gamma\subset \Sigma:=\{(\alpha,\beta) \ | \ {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha=\Bs_\beta \}$ of common subalgebras ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta=\Bs_{\gamma}$, with $(\beta,\gamma)\in \Gamma$, that is stable under $f:=(f_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},f_\Ns)$, the disjoint union of the support involutions $f_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},f_\Ns$:
\begin{equation*}
(\xi,\zeta)\in\Gamma \imp (f(\xi),f(\zeta))\in\Gamma, \quad \forall \xi,\zeta\in A\uplus B\uplus C\uplus D,
\end{equation*}
becomes an involutive multimodule with involution $\star:=\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\otimes_\Gamma\star_\Ns$. The involution $\star$ is well-defined by universal factorization property of tensor products.\footnote{
Apart from checking directly that, under the stability condition, the involution is well-defined, it is also possible to obtain the same result, considering first the involution already defined in proposition~\ref{prop: inv-ot} and making use of proposition~\ref{prop: inv-tr} together with remark~\ref{rem: inv-tr}.}
\xqed{\lrcorner}
\end{remark}
\section{Pairing Dualities in $\Zs$-central Multimodules} \label{sec: multi-dual-paring}
Here we provide an extension, to the case of $\Zs$-central multimodules, of the notion of duality of vector spaces.
\medskip
Although tensor products are always introduced via their universal factorization property, and later used to provide examples of monoidal categories, in the literature duals are almost never defined via universal factorization properties and are rather described either with non-categorical definitions or as dual objects inside suitable monoidal categories.
\medskip
Our main purpose here will be to directly discuss the several pairing dualities for multi-modules.
\medskip
Let us more generally consider the case of $\Zs$-central left-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$ right-$(\Bs_\beta)_{\beta\in B}$ multimodules ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_{\beta\in B}}$.
We can define several notions of duals, one for every subset of indexes $I\times J\subset A\times B$:
\begin{definition}\label{def: dual-multi}
Given a $\Zs$-central multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_{\beta\in B}}$ over $\Rs_\Zs$-algebras $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$-$(\Bs_\beta)_{\beta\in B}$ and a family of indexes $I\times J\subset A\times B$, an \emph{($I,J$)-dual of the multimodule} ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ is a pair $(\Ns,\tau)$, where ${}_{(\Bs_\beta)_{\beta\in B}}\Ns_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}$ is a \hbox{$\Zs$-central} $(\Bs_\beta)_{\beta\in B}$-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$ multimodule over the $\Rs_\Zs$-algebras $(\Bs_\beta)_{\beta\in B}$-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$ and
$\tau:\Ns\times{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\to (\otimes^\Zs_{\alpha\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J}\Bs_\beta)$ is a $\Zs$-multilinear $(A-I,B-J)$-balanced $(I,J)$-multilinear map:\footnote{\label{foo: free-m}
With some abuse of notation, will denote by $\cdot$ the ``outer actions'' on the tensor product multimodule $(\otimes^\Zs_{\alpha\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J}\Bs_\beta)$ given by: $a\cdot_{\beta_o}[(\otimes^\Zs_{\alpha\in I} x_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J} y_\beta)]\cdot_{\alpha_o} b
:=
(\otimes^\Zs_{\alpha\in I} x'_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J} y'_\beta)$,
where $x'_\alpha:=\begin{cases}x_\alpha,\phantom{\ \ \cdot a} \quad \alpha\neq \alpha_o
\\ a\cdot x_{\alpha_o}, \quad \alpha={\alpha_o}
\end{cases}
$ and $y'_\beta:=\begin{cases}
y_\beta,\phantom{\ \ \cdot b} \quad \beta\neq \beta_o
\\ y_{\beta_o}\cdot b, \quad \beta={\beta_o}
\end{cases}
$
for all $(\alpha_o,\beta_o), (\alpha,\beta)\in I\times J$, $(a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\alpha_o}\times\Bs_{\beta_o}$, $(x_\alpha,y_\beta)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta$.
}
$\forall t,x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$,
\begin{gather*}
\tau(t,a\cdot_\alpha x\cdot_\beta b)=a\cdot_\alpha\tau(t,x)\cdot_\beta b,
\quad
\tau(b\cdot_\beta t\cdot_\alpha a,x)=b\cdot_\beta\tau(t,x)\cdot_\alpha a,
\ \forall (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times \Bs_\beta, \ (\alpha,\beta)\in I\times J,
\\
\tau(b\cdot_\beta t\cdot_\alpha a,x)=\tau(t,a\cdot_\alpha x\cdot_\beta b), \quad \forall (a,b)\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \quad (\alpha,\beta)\in(A-I)\times(B-J),
\end{gather*}
satisfying the following universal factorization property:
for any $(A-I,B-J)$-balanced $(I,J)$-multilinear map $\Phi:\widehat{\Ns}\times{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\to (\otimes^\Zs_{\alpha\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J}\Bs_\beta)$
where${}_{(\Bs_\beta)_{\beta\in B}}\widehat{\Ns}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}$ is another $\Zs$-central
$(\Bs_\beta)_{\beta\in B}$-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$ multimodule over $\Rs_\Zs$-algebras,
there exists a unique morphism of multimodules $\hat{\Phi}:\widehat{\Ns}\to\Ns$ such that $\Phi=\tau\circ(\hat{\Phi},\id_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})$. \footnote{
We are assuming here that, in the category of $\Zs$-central $\Rs_\Zs$-multimodules, $\otimes^\Zs_{\alpha\in\varnothing}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha:=\Rs=:\otimes^\Zs_{\beta\in\varnothing}\Bs_\beta$.
}
\end{definition}
Again, if an $(I,J)$-dual exists, it is unique up to a unique isomorphism of $(\Bs_\beta)_{\beta\in B}$-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$ multimodules satisfying the previous universal factorization property.
The existence is provided in the following result.
\begin{theorem}
For every $(I,J)$ with $I\times J\subset A\times B$, there exists an $(I,J)$-dual $({}^{*_{I}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_{J}},\tau)$ of the $\Zs$-central multimodule ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ over $\Rs_\Zs$-algebras $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$-$(\Bs_\beta)_{\beta\in B}$.
\end{theorem}
\begin{proof}
For every $(I,J)$ with $I\times J\subset A\times B$, consider the following set:
\begin{equation*}
{}^{*_{I}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_{J}}:=\left\{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{\phi} (\otimes^\Zs_{\alpha\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J}\Bs_\beta) \ | \ \forall (\alpha,\beta)\in I\times J, \ \forall (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta \st \phi(a\cdot_\alpha x\cdot_\beta b)=a\cdot_\alpha\phi(x)\cdot_\beta b \right\}.
\end{equation*}
We see that ${}^{*_{I}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_{J}}$ is a $\Zs$-central $(\Bs_\beta)_{\beta\in A}$-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$ multimodule defining,
for all $\phi,\psi\in {}^{*_{I}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_{J}}$ and $x\in {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$:\footnote{
\label{foo: free-m2}
To avoid confusion, we will denote by $\bullet$ the ``inner actions'' on the tensor product multimodule $(\otimes^\Zs_{\alpha\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J}\Bs_\beta)$ given by: $b\bullet_{\beta_o}[(\otimes^\Zs_{\alpha\in I} x_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J} y_\beta)]\bullet_{\alpha_o} a
:=
(\otimes^\Zs_{\alpha\in I} x'_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J} y'_\beta)$,
where $x'_\alpha:=\begin{cases}x_\alpha,\phantom{\ \ \cdot a} \quad \alpha\neq \alpha_o
\\ x_{\alpha_o}\cdot a, \quad \alpha={\alpha_o}
\end{cases}
$ and $y'_\beta:=\begin{cases}
y_\beta,\phantom{\ \ \cdot b} \quad \beta\neq \beta_o
\\ b\cdot y_{\beta_o}, \quad \beta={\beta_o}
\end{cases}
$
for all $(\alpha_o,\beta_o), (\alpha,\beta)\in I\times J$, $(a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\alpha_o}\times\Bs_{\beta_o}$, $(x_\alpha,y_\beta)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta$.
}
\begin{gather*}
\phi+\psi: x\mapsto \phi(x)+\psi(x),
\\
b\bullet_\beta \phi\bullet_\alpha a: x\mapsto b\bullet_\beta \phi(x)\bullet_\alpha a,
\quad \forall (\alpha,\beta)\in I\times J, \quad (a,b)\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta,
\\
b\odot_\beta \phi\odot_\alpha a: x\mapsto \phi(a\cdot_\alpha x\cdot_\beta b),
\quad \forall (\alpha,\beta)\in (A-I)\times (B-J), \quad (a,b)\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta.
\end{gather*}
The evaluation map $\tau(\phi,x):=\phi(x)$, for all $\phi\in {}^{*_{I}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_{J}}$ and $x\in {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ turns out to be an $(A-I,B-J)$-balanced $(I,J)$-multilinear map $\tau:{}^{*_{I}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_{J}}\times{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\to
(\otimes^\Zs_{\alpha\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J}\Bs_\beta)$.
\medskip
To every $\Zs$-multilinear $(A-I)$-$(B-J)$-balanced and $(I,J)$-multilinear map
$\Phi:\widehat{\Ns}\times{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}} \to (\otimes^\Zs_{\alpha\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J}\Bs_\beta)$, the usual Curry isomorphism associates the map $\widehat{\Phi}:\widehat{\Ns} \to
[(\otimes^\Zs_{\alpha\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J}\Bs_\beta)]^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ that to every element $t\in\widehat{\Ns}$ associates the map
$\widehat{\Phi}_t:{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\to (\otimes^\Zs_{\alpha\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J}\Bs_\beta)$ given by $\widehat{\Phi}_t(x):=\Phi(t,x)$, for all $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$. The defining properties of $\Phi$ entail that $\widehat{\Phi}_t\in{}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}$, for all $t\in\widehat{\Ns}$ and that the map $\widehat{\Phi}:\widehat{\Ns}\to{}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}$ given by $t\mapsto \widehat{\Phi}_t$ is a morphism of $(\Bs_\beta)_{\beta\in B}$-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$ multimodules. Finally $\Phi(t,x)=\widehat{\Phi}_t(x)=\tau(\widehat{\Phi}_t,x)$, for $t\in\widehat{\Ns}$ and $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$.
\end{proof}
For every pair of families of unital associative $\Zs$-central $\Rs_\Zs$-algebras $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}$ and $(\Bs_\beta)_{\beta\in B}$ and every family of indexes $I\times J\subset A\times B$, $(I,J)$-transposition functors (and evaluation natural transformations) give us a \textit{contravariant right semi-adjunction} according to the definitions fully recalled in appendix~\ref{sec: functorial-pairing}, remark~\ref{rem: s-aj}.
\begin{theorem}\label{th: sadj}
Let ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B}$ be the category with objects $\Zs$-central $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A$-$(\Bs_\beta)_B$ multimodules over unital associative \hbox{$\Rs$-algebras} and with morphism \hbox{$\Zs$-lin}\-ear maps ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_1\xrightarrow{\Phi}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_2$ such that $\Phi(a\cdot_\alpha x\cdot_\beta b)=a\cdot_\alpha \Phi(x)\cdot_\beta b$, for all $(\alpha,\beta)\in A\times B$, $(a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta$, $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$.
\medskip
For every subset $I\times J\subset A\times B$ of indexes, we have a different contravariant right semi-adjoint functorial pairing $\quad \underline{{}_I\flat_J} \ | \stackrel[\theta]{\vartheta}{\leftrightarrows}|\ {}_I\sharp_J \quad$ between the \emph{transposition functors}\footnote{
The apparent distinction between $\flat$ and $\sharp$ is purely formal since they interchange by permuting the sets of indexes: ${}_{I}\flat_{J}={}_{J}\sharp_{I}$.
}
$\xymatrix{{}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B} \rtwocell^{{}_{{}_I\flat_J}}_{{}^{{}_I\sharp_J}}{'} & {}_{(\Bs_\beta)_B}\Mf_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}}$ that
\begin{itemize}
\item
on objects of the respective categories, are given by duals:
\begin{equation*}
{}_{I}\flat_{J}:{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\mapsto {}^{*_{I}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_{J}}, \quad \forall {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\in \Ob_{{}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B}},
\quad \quad
{}_{I}\sharp_{J}:\Ns\mapsto {}^{*_{J}}\Ns^{*_{I}}, \quad \forall \Ns\in \Ob_{{}_{(\Bs_\beta)_B}\Mf_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}};
\end{equation*}
\item
on morphisms $({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_2\xleftarrow{\mu}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_1)\in\Hom_{{}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B}}({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_1;{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_2)$ and $(\Ns_2\xleftarrow{\nu}\Ns_1)\in\Hom_{{}_{(\Bs_\beta)_B}\Mf_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}}(\Ns_1;\Ns_2)$, are respectively given by $\mu$-pull-backs and $\nu$-pull-backs:
\begin{gather} \label{eq: transp}
({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_2)^{{}_{I}\flat_{J}}\xrightarrow{\mu^{{}_{I}\flat_{J}}}({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_1)^{{}_{I}\flat_{J}},
\quad \mu^{{}_{I}\flat_{J}}(\phi):=\phi\circ \mu, \quad \forall \phi\in {}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_2^{*_J},
\\ \notag
(\Ns_2)^{{}_{I}\sharp_{J}}\xrightarrow{\nu^{{}_{I}\sharp_{J}}}(\Ns_1)^{{}_{I}\sharp_{J}},
\quad \nu^{{}_{J}\flat_{I}}(\psi):=\psi\circ \nu,
\quad \forall \psi\in {}^{*_J}\Ns_2^{*_I};
\end{gather}
\end{itemize}
where unit and co-unit of the semi-adjunction are given by the following \emph{natural evaluation transformations}:\footnote{
These evaluations maps are just obtained applying Curry isomorphism to the pairing duality $\tau$ in definition~\ref{def: dual-multi}.
}
\begin{gather*}
\id_{{}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}\Mf_{(\Bs_\beta)_{\beta\in B}}}\xrightarrow{\theta}\sharp\circ\flat,
\quad {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\mapsto \theta^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \quad
{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}} \xrightarrow{\theta^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}} ({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{{}_{I}\flat_{J}})^{{}_{J}\sharp_{I}},
\quad \theta^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_x:\phi\mapsto \phi(x), \quad \forall \phi\in {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{{}_{I}\flat_{J}},\ x\in {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},
\\
\id_{{}_{(\Bs_\beta)_{\beta\in B}}\Mf_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}}\xrightarrow{\vartheta}\flat\circ\sharp,
\quad
\Ns\mapsto \vartheta^\Ns,
\quad
\Ns\xrightarrow{\vartheta^\Ns} (\Ns^{{}_{J}\sharp_{I}})^{{}_{I}\flat_{J}},
\quad \vartheta^\Ns_y:\psi\mapsto \psi(y), \quad \forall \psi\in \Ns^{{}_{J}\sharp_{I}},\ y\in \Ns.
\end{gather*}
Restricting the previous contravariant right semi-adjunction $\ \underline{{}_I\flat_J} \ | \stackrel[\theta]{\vartheta}{\leftrightarrows}|\ {}_I\sharp_J \ $ to the full reflective subcategories (whose objects are those multimodules for which the evaluation natural transformations are isomorphisms), we obtain a categorical duality.
\end{theorem}
\begin{proof}
The contravariant functorial nature of ${}_I\flat_J$ and ${}_I\sharp_J$ is standard from their definitions.
\medskip
By direct computation $\theta^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ is a morphism in ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}\Mf_{(\Bs_\beta)_{\beta\in B}}$ and $\vartheta^\Ns$ is a morphism in ${}_{(\Bs_\beta)_{\beta\in B}}\Mf_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}$ furthermore for every pair of morphisms $({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_1)\xrightarrow{\mu}({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_2)$ in ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}\Mf_{(\Bs_\beta)_{\beta\in B}}$ and $(\Ns_1)\xrightarrow{\nu}(\Ns_2)$ in ${}_{(\Bs_\beta)_{\beta\in B}}\Mf_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}$:
\begin{equation*}
\theta^{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_2}\circ \mu=(\mu^{{}_{I}\flat_{J}})^{{}_{J}\sharp_{I}} \circ \theta^{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_1},
\quad \quad
\vartheta^{\Ns_2}\circ \nu=(\nu^{{}_{J}\sharp_{I}})^{{}_{I}\flat_{J}} \circ \vartheta^{\Ns_1}.
\end{equation*}
Finally we check the right semi-adjunction condition $\underline{{}_I\flat_J} \ | \stackrel
]
}{\leftrightarrows}|\ {}_I\sharp_J$ using formula~\eqref{eq: r-s-aj}:
\begin{align*}
({{}_I\flat_J}(\theta^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}))\circ \vartheta^{({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{{}_{I}\flat_{J}})}&=\iota_{({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{{}_{I}\flat_{J}})},
\quad \forall {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\in\Ob_{{}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B}},
\\
[({{}_I\flat_J}(\theta^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})\circ \vartheta^{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{{}_{I}\flat_{J}}})(\phi)](x)
&
=[(\theta^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_I\flat_J}(\vartheta^{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{{}_{I}\flat_{J}}}_\phi)](x)
=[(\vartheta^{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{{}_{I}\flat_{J}}}_\phi)\circ \theta^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}](x)
= \vartheta^{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{{}_{I}\flat_{J}}}_\phi(\theta^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_x)
\\
&
=\theta_x^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}(\phi)
=\phi(x)
=[\iota_{({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{{}_{I}\flat_{J}})}(\phi)](x),
\quad \quad \forall \phi\in {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{{}_{I}\flat_{J}}, \quad x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}.
\end{align*}
For the full reflective subcategories of the semi-adjunction we have a categorical duality (see remark~\ref{rem: s-aj}).
\end{proof}
\begin{remark}
There is of course the possibility to define also \emph{${}_I\gamma_J$-conjugate duals} of ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_B}$ for any family of $\Rs_\Zs$-conjugations $(\gamma_k)_{k\in I\uplus J}$, for
$(i,j)\in I\times J\subset A\times B$. For this purpose is just enough to repeat the previous construction of duals utilizing maps that are $\gamma_k$-conjugate-$\Rs_\Zs$-linear. Whenever $\gamma_k=\id_{\Rs}$, for all $k\in I\uplus J$, we re-obtain the previous definition as a special case.
\xqed{\lrcorner}
\end{remark}
Here we discuss how our definition of duals relates to already available notions in the case of bimodules.
\begin{remark}
The notion of $I$-$J$ dual of an $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A$-$(\Bs_\beta)_B$ multimodule over $\Rs_\Zs$-algebras that we have just introduced in definition~\ref{def: dual-multi} is a direct generalization of some much more familiar constructs for bimodules.
\medskip
Here below, we consider an ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$-bimodule ${}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ as an ${({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}$-$(\Bs_\beta)_B$-multimodule, with
$A:=\{\alpha_o\}$, $B:=\{\beta_o \}$, $A\times B=\{(\alpha_o,\beta_o)\}$ singleton sets and with ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_{\alpha_o}:={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}=:\Bs_{\beta_o}$.
\medskip
The ``double dual''\footnote{
To be precise, the double dual is obtained choosing here $\Zs:=\KK$; this is a slight generalization that we found particularly useful in our treatment of contravariant non-commutative differential calculus (see footnote~\ref{foo: ncdc}).
}
${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\vee$ of an \hbox{${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$-bimodule} ${}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ (see for example~\cite[section~2.1]{Fer17} for more details) is the central $\Zs$-bimodule ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\vee:=\Hom_{{}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\Mf_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}}({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};{}_{\cdot \ }({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\otimes_\Zs{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}})_{\ \cdot})$ of covariant homomorphisms of bimodules, from ${}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$, with values into ${}_{\cdot \ }({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}\otimes_\Zs{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}})_{\ \cdot}$ seen as an ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$-bimodule with the ``exterior actions'' given by: $a\cdot (x\otimes_\Zs y)\cdot b:=(ax)\otimes_\Zs(yb)$, for all $x,y,a,b\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$; where ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\vee$ is an ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$-bimodule with the actions $(b\odot \phi\odot a)(x):=\phi(a\cdot x\cdot b)$, for all $a,b\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$, $x\in {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$, $\phi\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\vee$.
Taking $I\times J = A\times B$ as a singleton (only one right and only one left action) in definition~\ref{def: dual-multi}, we see that ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\vee={}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}$.
\medskip
The well-known notions (see for example~\cite{Bo97}) of ``right dual'' ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^*:=\Hom_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}};{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}})$ and ``left dual'' ${}^*{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}:=\Hom_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}({}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}})$ of a bimodule ${}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ are just the dual of the right ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$-module ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$ (respectively the dual of the left ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$-module ${}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$), as in~\cite[section~II.3]{Bou89}, equipped with the following actions $(a\cdot \phi\odot b)(x):=a\phi(bx)$, for all $a,b\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$, $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ and $\phi\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^*$ (respectively $(a\odot \phi\cdot b)(x):=\psi(xa)b$, for all $a,b\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$, $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ and $\psi\in{}^*{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$).
When $A\times B$ is a singleton, taking $I:=\varnothing$, $J:=B$, we recover ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^*={}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}$ and, when $I:=B$, $J:=\varnothing$, we get ${}^*{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}={}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}$.
\medskip
Finally the ``scalar dual'' of a bimodule\footnote{
For algebras over $\Rs:=\Zs:=\KK$ this is just the usual dual as a $\KK$-vector space.
} ${}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$, defined as ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}':=\Hom_\Rs({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}};\Rs\otimes_\Zs\Rs)$, equipped with the actions $(b\odot \phi\odot a)(x):=\phi(a\cdot x\cdot b)$, for all $a,b\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}$, $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ and $\phi\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}'$, can be obtained from our definition as ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}'={}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}$, taking $A\times B$ to be, as usual, a singleton and $I:=\varnothing=:J$.
\xqed{\lrcorner}
\end{remark}
In the following we study the ``inclusion relations'' between the different duals of a given multimodule.
\begin{remark}
Consider the auxiliary $(I_2,J_2)$-global $(I_1,J_1)$-dual multimodules, for $I_1\times J_1\subset I_2\times J_2\subset A\times B$:
\begin{equation*}
{}^{I_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J_2}:=
\left\{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{\phi} (\otimes^\Zs_{\alpha\in I_2}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)\otimes_\Zs(\otimes^\Zs_{\beta\in J_2}\Bs_\beta) \ | \ \forall (\alpha,\beta)\in I_1\times J_1, \ \forall (a_\alpha,b_\beta)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta \st \phi(a_\alpha xb_\beta)=a_\alpha\phi(x)b_\beta \right\},
\end{equation*}
equipped with the multimodule actions specified as follows (see footnotes~\ref{foo: free-m} \ref{foo: free-m2}), for all $\phi\in{}^{I_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J_2}$ and $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$:
\begin{gather}\notag
b\bullet_\beta \phi\bullet_\alpha a: x\mapsto b\bullet_\beta \phi(x)\bullet_\alpha a,
\quad \forall (\alpha,\beta)\in I_2\times J_2, \quad (a,b)\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta,
\\ \notag
b\odot_\beta \phi\odot_\alpha a: x\mapsto \phi(a\cdot_\alpha x\cdot_\beta b),
\quad \forall (\alpha,\beta)\in (A-I_1)\times (B-J_1), \quad (a,b)\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta,
\\ \label{eq: extra}
(a\cdot_\alpha \phi\cdot_\beta b):=x \mapsto a\cdot_\alpha\phi(x)\cdot_\beta b, \quad \forall (\alpha,\beta)\in (I_2-I_1)\times(J_2-J_1).
\end{gather}
Notice that whenever $(I_1,J_1)=(I_2,J_2)$, we have ${}^{I_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J_2}={}^{*_{I_1}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_{J_1}}$ as a multimodule and that the extra multimodule actions in line~\eqref{eq: extra} appear only when $I_1\times J_1\neq I_2\times J_2$.
\medskip
If $I_1\times J_1\subset I_1'\times J_1'$ we have natural set theoretic inclusions: $\xymatrix{{}^{I'_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J'_1}_{J_2} \ar@{^(->}[r]^{\eta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}} & {}^{I_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J_2}}$, that are also covariant morphisms of multimodules for all the common actions involved (inner target actions $\bullet$ for indexes in $I_2\times J_2$; internal source actions $\odot$ for indexes in $(A-I_1')\times(B-J_1')$ and external target action
$\cdot \ $ for indexes in the set $(I_2-I_1')\times(J_2-J_1')$).
Keeping $(I_1,J_1)$ fixed, if $I_2\times J_2\subset I_2'\times J_2'$, we define the following embedding map:
\begin{equation*}
{}^{I_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J_2}\xrightarrow{\quad \zeta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}:=(\otimes^\Zs_{\alpha\in I_2'-I_2}1_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha})\otimes_\Zs - \otimes_\Zs(\otimes^\Zs_{\beta\in J_2'-J_2}1_{\Bs_\beta})\quad}{}^{I_1}_{I'_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J'_2},
\end{equation*}
that to every $\phi\in{}^{I_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J_2}$ associates the map ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\ni x\mapsto (\otimes^\Zs_{\alpha\in I_2'-I_2}1_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha})\otimes_\Zs \phi(x) \otimes_\Zs(\otimes^\Zs_{\beta\in J_2'-J_2}1_{\Bs_\beta})$ in ${}^{I_1}_{I'_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J'_2}$, that is actually a covariant morphism of multimodules for all the common relevant actions involved (inner target actions $\bullet$ for indexes in $I_2\times J_2$; internal source actions $\odot$ for indexes in $(A-I_1)\times(B-J_1)$ and external target action $\cdot \ $ for indexes in $(I_2-I_1)\times(J_2-J_1)$).
\xqed{\lrcorner}
\end{remark}
\begin{proposition}
Given a multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_{\beta\in B}}$, for all the inclusions $I_1\times J_1\subset I_2\times J_2\subset A\times B$ of indexes, we have the following natural transformations between contravariant functors from the category ${}_{(\Bs_\beta)_{\beta\in B}}\Mf_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}$ into the category $\Mf_{[\Rs_\Zs]}$ of $\Zs$-central multimodules over $\Rs_\Zs$-algebras:
\begin{equation*}
{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}} \quad \mapsto \quad \left[
{}^{*_{I_2}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_{J_2}}
\xrightarrow{\tiny
\xymatrix{{}^{I'_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J'_1}_{J_2} \ar@{^(->}[r]^{\eta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}& {}^{I_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J_2}}
}
{}^{I_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J_2}
\xleftarrow{\zeta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}:=(\otimes^\Zs_{\beta\in J-J'}1_{\Bs_\beta})\otimes_\Zs -\otimes_\Zs(\otimes^\Zs_{\alpha\in I-I'}1_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha})} {}^{*_{I_1}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_{J_1}}
\right].
\end{equation*}
\end{proposition}
\begin{proof}
The passage associating to a multimodule ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\in\Ob_{{}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B}}$ its $(I_2,J_2)$-global $(I_1,J_1))$-dual multimodule ${}^{I_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J_2}$,
is a contravariant functor acting on morphisms by transposition as in equation~\eqref{eq: transp}:
\begin{equation*}
\left({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{\mu}\Ns \right) \quad \mapsto \left( {}^{I_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J_2} \xleftarrow{\mu^\bullet} {}^{I_1}_{I_2}\Ns^{J_1}_{J_2} \right), \quad \text{where} \quad \mu^\bullet(\phi):=\phi\circ \mu,
\end{equation*}
and from $\eta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\circ\mu^{{}_{I_2}\flat_{J_2}}=\mu^\bullet\circ \eta_\Ns$ and $\mu^\bullet\circ\zeta_\Ns=\zeta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\circ\mu^{{}_{I_1}\flat_{J_1}}$ we see that $\eta$ and $\zeta$ are natural transformations.
\end{proof}
\begin{theorem}
For every inclusion of indexes $I_1\times J_1\subset I_2\times J_2\subset A\times B$, considering the two contravariant right semi-adjunctions
$\left[\underline{{}_{I_k}\flat_{J_k}} \ | \stackrel[{}^{I_k}\theta^{J_k}]{{}^{I_k}\vartheta^{J_k}}{\leftrightarrows}|\ {}_{I_k}\sharp_{J_k} \right]$, for $k=1,2$, as in theorem~\ref{th: sadj}, we have a \emph{morphism of contravariant right semi-adjunctions} defined in the following way:
\begin{itemize}
\item
for all morphisms ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{\mu}\Ns$ in ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B}$
and $\Ps\xrightarrow{\nu}{\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}}$ in ${}_{(\Bs_\beta)_B}\Mf_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}$
we have commutative diagrams:
\begin{equation*}
\xymatrix{
& (\Ps)^{{}_{I_1}\sharp_{J_1}} \ar[dl]_{\zeta_\Ps} & \ar[l]_{\quad \nu^{{}_{I_1}\sharp_{J_1}} } ({\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}})^{{}_{I_1}\sharp_{J_1}}\ar[dr]^{\zeta_{\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}}} &
\\
{}^{I_1}_{I_2}\Ps^{J_1}_{J_2} & & & {}^{I_1}_{I_2}{\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}}^{J_1}_{J_2} \ar[lll]_{\nu^\bullet}
\\
& (\Ps)^{{}_{I_2}\sharp_{J_2}} \ar[ul]^{\eta_\Ps} & \ar[l]_{\quad \nu^{{}_{I_2}\sharp_{J_2}} } ({\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}})^{{}_{I_2}\sharp_{J_2}} \ar[ur]_{\eta_{\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}}} &
}
\quad \quad
\xymatrix{
& ({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_{I_1}\flat_{J_1}} \ar[ld]_{\zeta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}} & \ar[l]_{\quad \mu^{{}_{I_1}\flat_{J_1}} } (\Ns)^{{}_{I_1}\flat_{J_1}} \ar[dr]^{\zeta_\Ns} &
\\
{}^{I_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J_2} & & & {}^{I_1}_{I_2}\Ns^{J_1}_{J_2} \ar[lll]_{\mu^\bullet}
\\
& ({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_{I_2}\flat_{J_2}} \ar[ul]^{\eta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}} & \ar[l]_{\quad \mu^{{}_{I_2}\flat_{J_2}} } (\Ns)^{{}_{I_2}\flat_{J_2}} \ar[ur]_{\eta_\Ns} &
}
\end{equation*}
\item
for every pair of objects ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ in ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B}$ and ${\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}}$ in ${}_{(\Bs_\beta)_B}\Mf_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}$ we have the commuting diagrams:
\begin{equation*}
\xymatrix{
& (({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_{I_1}\flat_{J_1}})^{{}_{I_1}\sharp_{J_1}} \ar[rr]^{\zeta_{({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_{I_1}\flat_{J_1}}}}
& & {}^{J_1}_{J_2}(({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_{I_1}\flat_{J_1}})^{I_1}_{I_2}
\\
{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}} \ar[ur]^{{{}^{I_1}\theta^{J_1}}_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}} \ar[dr]_{{{}^{I_2}\theta^{J_2}}_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}} \ar[rrr]_{\ev^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}} \quad \quad} & & & {}^{J_1}_{J_2}({}^{I_1}_{I_2}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{J_1}_{J_2})^{I_1}_{I_2} \ar[u]_{\zeta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\bullet} \ar[d]^{\eta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\bullet} \ ,
\\
& (({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_{I_2}\flat_{J_2}})^{{}_{I_2}\sharp_{J_2}} \ar[rr]_{\eta_{({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_{I_1}\flat_{J_2}}}}
& & {}^{J_1}_{J_2}(({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_{I_2}\flat_{J_2}})^{I_1}_{I_2}
}
\quad \quad
\xymatrix{
& (({\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}})^{{}_{I_1}\sharp_{J_1}})^{{}_{I_1}\flat_{J_1}} \ar[rr]^{\zeta_{({\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}})^{{}_{I_1}\sharp_{J_1}}}}
& & {}^{J_1}_{J_2}(({\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}})^{{}_{I_1}\sharp_{J_1}})^{I_1}_{I_2}
\\
{\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}} \ar[ur]^{{{}^{I_1}\vartheta^{J_1}}_{\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}}} \ar[dr]_{{{}^{I_2}\vartheta^{J_2}}_{\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}}}\ar[rrr]_{\ev^{\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}}\quad \quad } & & & {}^{J_1}_{J_2}({}^{I_1}_{I_2}{\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}}^{J_1}_{J_2} )^{I_1}_{I_2} \ar[u]_{\zeta_{\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}}^\bullet} \ar[d]^{\eta_{\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}}^\bullet}.
\\
& (({\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}})^{{}_{I_2}\sharp_{J_2}})^{{}_{I_2}\flat_{J_2}} \ar[rr]_{\eta_{({\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}})^{{}_{I_2}\sharp_{J_2}}}}
& & {}^{J_1}_{J_2}(({\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}})^{{}_{I_2}\sharp_{J_2}})^{I_1}_{I_2}
}
\end{equation*}
\end{itemize}
\medskip
Considering the category $\If$ (actually the poset) of pairs $(I,J)$, with $I\times J\subset A\times B$, where for every inclusion $I_1\times J_1\subset I_2\times J_2$ there is a unique morphism of pairs $(I_1,J_1)\xrightarrow{} (I_2,J_2)$,
we have that every multimodule ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ has an associated \emph{dual functor} $\If\xrightarrow{\bigstar_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}\Sf$ into contravariant right semi-adjunctions: $\bigstar_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}:(I,J)\mapsto \left[\underline{{}_I\flat_J} \ | \stackrel[{}^I\theta^J]{{}^I\vartheta^J}{\leftrightarrows}|\ {}_I\sharp_J \right]$.
\end{theorem}
\begin{proof}
In the first pair of diagram, due to the exchange symmetry $\mu\leftrightarrow \nu$ it is sufficient to prove the second.
Taking $\phi\in(\Ns)^{{}_{I_1}\flat_{J_1}}$ and $\psi\in(\Ns)^{{}_{I_2}\flat_{J_2}}$ we immediately get, for $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$:
\begin{align*}
[\zeta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\circ \mu^{{}_{I_1}\flat_{J_1}} (\phi)](x)
&= [\zeta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}(\phi\circ\mu)](x)
=(\otimes^\Zs_{\alpha\in I_2-I_1}1_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha})
\otimes_\Zs\phi(\mu(x))\otimes_\Zs(\otimes^\Zs_{\beta\in J_2-J_1}1_{\Bs_\beta})
\\
&=[\zeta_\Ns (\phi)\circ \mu](x)
=[\mu^\bullet\circ\zeta_\Ns (\phi)](x),
\\
[\eta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\circ \mu^{{}_{I_2}\flat_{J_2}} (\psi)](x)
&= [\eta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}(\psi\circ\mu)](x)
=[\eta_\Ns (\psi)\circ \mu](x)
=\phi(\mu(x))
=[\mu^\bullet\circ\eta_\Ns (\psi)](x).
\end{align*}
\medskip
In the second pair of commuting diagrams, by the exchange symmetry ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\leftrightarrow {\mathscr{Q}}}\newcommand{\Rs}{{\mathscr{R}}}\newcommand{\Ss}{{\mathscr{S}}$, it is enough to prove the first.
Consider $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$, $\phi\in ({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_{I_1}\flat_{J_1}}$ and $\psi\in({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_{I_2}\flat_{J_2}}$:
\begin{align*}
[\zeta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\bullet\circ\ev^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}(x)](\phi)
&=[\zeta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\bullet(\ev^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_x)](\phi)
=\ev^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_x(\zeta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}(\phi))
=(\otimes_{\alpha\in I_2-I_1}^\Zs 1_{{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha})\otimes_\Zs\phi(x)\otimes_\Zs(\otimes_{\beta\in J_2-J_1}^\Zs 1_{\Bs_\beta})
\\
&=[\zeta_{({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_{I_1}\flat_{J_1}}} (({}^{I_2}\theta^{J_2}_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})_x)](\phi),
\\
[\eta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\bullet\circ\ev^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}(x)](\psi)
&=[\eta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^\bullet(\ev^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_x)](\psi)
=\ev^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_x(\eta_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}(\psi))
=\psi(x)
=[\eta_{({\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})^{{}_{I_1}\flat_{J_2}}}({}^{I_2}\theta^{J_2}_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})_x](\psi).
\end{align*}
We define a poset category $\If$ of index pairs via the order relation $(I_1,J_1)\leq(I_2,J_2) \iff (I_1\subset I_2)\wedge (J_1\subset J_2)$.
\medskip
We consider $\Sf_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ the category whose objects are contravariant right semi-adjunctions and whose morphisms are specified by the previous commuting diagrams of natural transformations $\zeta,\eta$.
\medskip
To every index pair $(I,J)\in\Ob_\If$ we associate the contravariant right semi-adjunction $\bigstar_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{(I,J)}:=\left[\underline{{}_I\flat_J} \ | \stackrel[{}^I\theta^J]{{}^I\vartheta^J}{\leftrightarrows}|\ {}_I\sharp_J \right]$
and to every morphism $(I_1,J_1)\leq(I_2,J_2)$ in $\If$ we associate the morphism $\bigstar_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{(I_2,J_2)}\xrightarrow{(\zeta,\eta)}\bigstar_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{(I_1,J_1)}$ of contravariant right semi-adjunctions.
We notice that $\If\xrightarrow{\bigstar_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}\Sf_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ is a contravariant functor.
\end{proof}
\section{Traces and Inner Products on Multimodules} \label{sec: traces-ip}
In the first part of this section we generalize to the setting of multimodules the well-known multilinear algebraic operations producing contractions of tensors (over pairs of contravariant/covariant indexes) over a vector space and hence the equally familiar notion of trace of linear operators.
\medskip
We proceed again introducing the relevant universal factorization properties.
\begin{definition}
Given a $\Zs$-central multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_B}$ over $\Rs_\Zs$-algebras and $\Gamma\subset (A\uplus B)\times (A\uplus B)$ an injective symmetric relation\footnote{We can also assume that $\Gamma$ is irreflexive: $(\xi,\zeta)\in\Gamma \imp \xi\neq \zeta$; since ``tracing an action over itself'' does not have any effect.} such that ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\zeta$, for all $(\xi,\zeta)\in\Gamma$, let $A^\Gamma:=A-\dom(\Gamma)$ and $B^\Gamma:=B-\im(\Gamma)$.
\medskip
A $\Zs$-linear map ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_B}\xrightarrow{T}\Vs$, of $\Zs$-central bimodules, is \emph{$\Gamma$-tracial} if it satisfies the following properties:
\begin{align*}
T(a\cdot_\xi x)&=T(a\cdot_\zeta x), \quad \forall x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ \forall a\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\zeta, \quad
(\xi,\zeta)\in\Gamma\cap (A\times A),
\\
T(x\cdot_\xi a)&=T(x\cdot_\zeta a), \quad \forall x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ \forall a\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\zeta, \quad
(\xi,\zeta)\in\Gamma\cap (B\times B),
\\
T(a\cdot_\xi x)&=T(x\cdot_\zeta a), \quad \forall x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ \forall a\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\zeta, \quad
(\xi,\zeta)\in\Gamma\cap (A\times B).
\end{align*}
\medskip
A \emph{$\Gamma$-contraction of the multimodule} ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_B}$, consists of a $\Gamma$-tracial
morphism of \hbox{$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A^\Gamma}$-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{\beta\in B^\Gamma}$} multimodules
over $\Rs_\Zs$-algebras
${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A^\Gamma}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B^\Gamma}}\xrightarrow{T^\Gamma_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}} {}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A^\Gamma}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\ |\ \Gamma_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B^\Gamma}}$
such that the following universal factorization property is satisfied:
for any $\Gamma$-tracial morphism
${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A^\Gamma}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B^\Gamma}}\xrightarrow{T} {}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A^\Gamma}}\Ns_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B^\Gamma}}$
of multimodules, there exists a unique morphism
${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A^\Gamma}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\ |\ \Gamma_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B^\Gamma}}\xrightarrow{\tilde{T}} {}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A^\Gamma}}\Ns_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B^\Gamma}}$ of multimodules such that $T=\tilde{T}\circ T^\Gamma_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$.
\end{definition}
\begin{remark} \label{rem: contractions}
As usual, $\Gamma$-contractions of $\Zs$-central multimodules are unique up to a unique isomorphisms compatible with the defining factorization property; their existence is provided by the following construction.
\medskip
Given the $\Zs$-central multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B}}$ and the injective symmetric relation $\Gamma\subset (A\uplus B)\times(A\uplus B)$ with ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\zeta$ whenever $(\xi,\zeta)\in\Gamma$, defining $A^\Gamma:=A-\dom(\Gamma)$ and $B^\Gamma:=B-\im(\Gamma)$, consider the $\Zs$-central
$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A^\Gamma}$-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{\beta\in B^\Gamma}$ sub-multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A^\Gamma}}[{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},\Gamma]_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B^\Gamma}}$ of ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A^\Gamma}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B^\Gamma}}$ generated by the elements of the form:
\begin{align*}
a\cdot_\xi x&-a\cdot_\zeta x, \quad \forall x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ \forall a\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\zeta, \quad
(\xi,\zeta)\in\Gamma\cap (A\times A),
\\
x\cdot_\xi a&-x\cdot_\zeta a, \quad \forall x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ \forall a\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\zeta, \quad
(\xi,\zeta)\in\Gamma\cap (B\times B),
\\
a\cdot_\xi x&-x\cdot_\zeta a, \quad \forall x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ \forall a\in {\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\xi={\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\zeta, \quad
(\xi,\zeta)\in\Gamma\cap (A\times B).
\end{align*}
The quotient map $T^\Gamma_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}:{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\to \frac{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}{[{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},\Gamma]}=:{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\ | \ \Gamma$ onto the quotient $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A^\Gamma}$-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B^\Gamma}$ multimodule, satisfies the universal factorization property, since every $\Gamma$-tracial homomorphism ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{T}\Ns$ of $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A^\Gamma}$-$({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B^\Gamma}$ multimodules entails $[{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},\Gamma]\subset \ke(T)$ and hence canonically factorizes via $T^\Gamma_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$.
\medskip
Notice that it is possible to have multimodules that only possess trivial $\Gamma$-traces (for a certain family $\Gamma$) and hence they have trivial universal $\Gamma$-contractions.
\xqed{\lrcorner}
\end{remark}
We briefly examine how involutions in multimodules descend to their contractions.
\begin{definition}
An involution $(\dagger^\sigma_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},\ddagger^\sigma_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})_{\sigma\in f_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}$ on a $\Zs$-central multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B}}$ over $\Rs_\Zs$-algebras is a \emph{$\Gamma$-compatible involution}, where
$\Gamma\subset (A\uplus B)\times(A\uplus B)$ is an injective symmetric relation
on $A\uplus B$, if:
\begin{equation} \label{eq: Gamma}
(\xi,\zeta)\in\Gamma \imp (f(\xi),f(\zeta))\in\Gamma, \quad \forall \xi,\zeta\in A\uplus B.
\end{equation}
\end{definition}
\begin{proposition}\label{prop: inv-tr}
Suppose that $({}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{A}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\beta)_{B}},\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}})$ in an involutive $\Zs$-central multimodule over $\Rs_\Zs$-algebras. If the involution is $\Gamma$-compatible with a $\Gamma$-contraction, there exists a unique contracted involution onto ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\ |\ \Gamma$ and the contraction map ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{T_\Gamma} {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\ | \ \Gamma$ is involutive.
\end{proposition}
\begin{proof}
Condition~\eqref{eq: Gamma} implies that the involution $\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ leaves invariant the submultimodule ${}_{{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)}_{A^\Gamma}}[{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},\Gamma]_{{(\Bs_\beta)}_{B^\Gamma}}$ and hence, defining $(x+[{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},\Gamma])^{\star}:=x^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}+[{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},\Gamma]$, for all $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$, the involution will pass to the quotient multimodule (with the same covariance properties in $(\alpha,\beta)\in A^\Gamma\times B^\Gamma$) and $T_\Gamma(x^{\star_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}})=T_\Gamma(x)^\star$, for all $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$.
\end{proof}
\begin{remark}\label{rem: inv-tr}
Looking at the universal contructions of tensor products of multimodules in remark~\ref{rem: otimes-tr} and of contrations in remark~\ref{rem: contractions}, we obtain the following familiar result:
\begin{equation*}
\left({}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_{\beta\in B}} \right)
\otimes_\Gamma
\left({}_{(\Cs_\gamma)_{\gamma\in C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{D}}_\delta)_{\delta\in D}} \right)
\simeq
T_\Gamma \left({}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_{\alpha\in A}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_{\beta\in B}} \otimes_\Zs
{}_{(\Cs_\gamma)_{\gamma\in C}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{({\mathscr{D}}_\delta)_{\delta\in D}} \right),
\end{equation*}
tensor products over $\Gamma\subset (A\uplus C)\uplus(B\uplus D)$ are naturally isomorphic to $\Gamma$-contracted tensor products over $\Zs$.
\xqed{\lrcorner}
\end{remark}
\bigskip
We pass now to consider the generalization of inner product couplings for multimodules.
\medskip
\begin{definition} \label{def: ip-multi}
Suppose that the unital associative $\Rs_\Zs$-algebras ${\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha$ and $\Bs_\beta$, for $(\alpha,\beta)\in A\times B$, are all contravariantly involutive.
Given a multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_B}$ and sub-indexes $I\times J\subset A\times B$, an \emph{$(I,J)$-right-inner product on ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$} consists of a bi-additive map $\ip{\cdot}{\cdot}_{I-J}: {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\to \left(\bigotimes^\Zs_{i\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_i\right)\otimes_\Zs\left(\bigotimes^\Zs_{j\in J}\Bs_j\right)$ such that:
\begin{align*}
\ip{x}{a\cdot_\alpha y\cdot_\beta b}_{I-J} &= a\cdot_\alpha\ip{x}{y}_{I-J}\cdot_\beta b,
&&
\forall x,y\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ (\alpha,\beta)\in I\times J,
\\
\ip{a\cdot_\alpha x\cdot_\beta b}{y}_{I-J} &=b^*\bullet_\beta\ip{x}{y}_{I-J}\bullet_\alpha a^*,
&&
\forall x,y\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ (\alpha,\beta)\in I\times J,
\\
\ip{a\cdot_\alpha x\cdot_\beta b}{y}_{I-J} &=\ip{x}{a^*\cdot_\alpha y\cdot_\beta b^*}_{I-J},
&&
\forall x,y\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ (\alpha,\beta)\in (A-I)\times (B-J).
\end{align*}
A \emph{$(I,J)$-left-inner product on ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$} is a bi-additive map ${}_{I-J}\ip{\cdot}{\cdot}: {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\times{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\to \left(\bigotimes^\Zs_{i\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_i\right)\otimes_\Zs\left(\bigotimes^\Zs_{j\in J}\Bs_j\right)$ such that:
\begin{align*}
{}_{I-J}\ip{a\cdot_\alpha x\cdot_\beta b}{y} &= a\cdot_\alpha{}_{I-J}\ip{x}{y}\cdot_\beta b,
&&
\forall x,y\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ (\alpha,\beta)\in I\times J,
\\
{}_{I-J}\ip{x}{a\cdot_\alpha y\cdot_\beta b} &=b^*\bullet_\beta{}_{I-J}\ip{x}{y}\bullet_\alpha a^*,
&&
\forall x,y\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ (\alpha,\beta)\in I\times J,
\\
{}_{I-J}\ip{a\cdot_\alpha x\cdot_\beta b}{y} &={}_{I-J}\ip{x}{a^*\cdot_\alpha y\cdot_\beta b^*},
&&
\forall x,y\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ (\alpha,\beta)\in (A-I)\times (B-J).
\end{align*}
\end{definition}
\begin{proposition}
For every right-$(I,J)$-inner product $(x,y)\mapsto\ip{x}{y}_{I-J}$ we have its:
\begin{equation*}
\text{\emph{transpose}} \quad (x,y)\mapsto\ip{y}{x}_{I-J},
\quad
\text{\emph{$*$-conjugate}} \quad (x,y)\mapsto\ip{x}{y}^*_{I-J},
\quad
\text{\emph{$*$-adjoint}} \quad (x,y)\mapsto\ip{y}{x}^*_{I-J}.
\end{equation*}
The transpose and conjugate are left-$(I,J)$-inner products; the adjoint is a right-$(I,J)$-inner product on ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$.
\end{proposition}
\begin{remark}
Without entering into a detailed discussion of ``positivity'' for inner products, we simply mention that stating this condition requires an additional compatible ``order structure'' on the involved rings and algebras. Whenever the commutative unital associative involutive ring $\Zs$ is equipped with a positive cone $\Zs_+$ (that by definition is a pointed subset $0_\Zs\in \Zs_+\subset \Zs$, stable under addition $\Zs_+ + \Zs_+\subset\Zs_+$, stable under multiplication $\Zs_+\cdot\Zs_+\subset\Zs_+$, sharp $\Zs_+\cap(-\Zs_+)=\{0_\Zs\}$ and involutive $\Zs_+^{\star_\Zs}\subset\Zs_+$), any $\Zs$-central unital associative algebra $\Rs$ (and hence any $\Zs$-central $\Rs_\Zs$-algebra) canonically inherits a positive cone $\Rs_+:=\Zs_+\cdot 1_\Rs\subset \Rs$.
In this case, positivity of a right-$(I,J)$-inner product can be imposed requiring $\ip{x}{x}_{I-J}\in\left[ \left(\bigotimes^\Zs_{i\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_i\right)\otimes_\Zs\left(\bigotimes^\Zs_{j\in J}\Bs_j\right)\right]_+$, for all $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$. Similar condition can be imposed for left-$(I,J)$-inner products.
\xqed{\lrcorner}
\end{remark}
\begin{theorem}\label{th: Riesz}
An $(I,J)$-inner product (right or left) induces canonical \emph{Riesz maps}
\begin{gather*}
{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{{}^I\overrightarrow{\Lambda}^J} {}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J},
\quad \quad {}^I\overrightarrow{\Lambda}_x^J:x\mapsto {}^I\overrightarrow{\Lambda}_x^J\quad \quad {}^I\overrightarrow{\Lambda}_x^J:y\mapsto \ip{x}{y}_{I-J},
\\
{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\xrightarrow{{}^I\overleftarrow{\Lambda}^J} {}^{*_I}\tilde{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}^{*_J},
\quad \quad {}^I\overleftarrow{\Lambda}_y^J:y\mapsto {}^I\overleftarrow{\Lambda}_y^J\quad \quad {}^I\overleftarrow{\Lambda}_y^J:x\mapsto \ip{x}{y}_{I-J},
\end{gather*}
where ${}^I\overrightarrow{\Lambda}^J$ is a contravariant morphism of multimodules into ${}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}$, the $(I,J)$-dual of ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$, and respectively ${}^I\overleftarrow{\Lambda}^J$ is a covariant morphism of multimodules into the ${}^I\overrightarrow{\Lambda}^J$-twisted of ${}^{*_I}{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}^{*_J}$, here denoted by ${}^{*_I}\tilde{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}^{*_J}$.
\end{theorem}
\begin{proof}
From definition~\ref{def: ip-multi} we see that ${}^I\overrightarrow{\Lambda}^J_x\in {}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}$, for all $x\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$. The map ${}^I\overrightarrow{\Lambda}^J:{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}\to {}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}$ is a contravariant morphism of multimodules:
\begin{align*}
{}^I\overrightarrow{\Lambda}^J_{a\cdot_\alpha x\cdot_\beta b}(y)
&=\ip{a\cdot_\alpha x\cdot_\beta b}{y}_{I-J}
=b^*\bullet_\beta \ip{x}{y}_{I-J}\bullet_\alpha a^*
=b^*\bullet_\beta {}^I\overrightarrow{\Lambda}^J_x (y)\bullet_\alpha a^*
\\
&=(b^*\bullet_\beta {}^I\overrightarrow{\Lambda}^J_x \bullet_\alpha a^*)(y), \quad
\forall (\alpha,\beta)\in I\times J, \ x,y\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ (a_\alpha,b_\beta)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times \Bs_\beta;
\\
{}^I\overrightarrow{\Lambda}^J_{a\cdot_\alpha x\cdot_\beta b}(y)
&=\ip{a\cdot_\alpha x\cdot_\beta b}{y}_{I-J}
=\ip{x}{a^*\cdot_\alpha y\cdot_\beta b^*}_{I-J}
={}^I\overrightarrow{\Lambda}^J_x(a^*\cdot_\alpha y\cdot_\beta b^*)
\\
&=(b^*\odot_\beta {}^I\overrightarrow{\Lambda}^J_x \odot_\alpha a^*)(y), \quad
\forall (\alpha,\beta)\in (A-I)\times (B-J), \ x,y\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}, \ (a_\alpha,b_\beta)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times \Bs_\beta.
\end{align*}
The proof for the case of ${}^I\overleftarrow{\Lambda}^J$ can be obtained passing to the transpose inner product.
\end{proof}
\begin{definition}
An inner product $\ip{\cdot}{\cdot}_{I-J}$ is \emph{$*$-Hermitian} if it coincides with its $*$-adjoint; \emph{non-degenerate} if both the Riesz maps ${}^I\overrightarrow{\Lambda}^J$ and ${}^I\overleftarrow{\Lambda}^J$ are injective;
\emph{algebraically full} if $\ip{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}_{I-J}=\left(\bigotimes^\Zs_{i\in I}{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_i\right)\otimes_\Zs\left(\bigotimes^\Zs_{j\in J}\Bs_j\right)$; \emph{saturated} if both
${}^I\overrightarrow{\Lambda}^J$ and ${}^I\overleftarrow{\Lambda}^J$ are surjective.
\end{definition}
\begin{remark}
Notice that the Riesz map ${}^I\overrightarrow{\Lambda}^J$ is contravariant and hence, under non-degeneracy and saturation, an $(I,J)$-inner product always induces an anti-isomorphism between ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ and its $(I,J)$ dual ${}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}$.
Under fullness and saturation, ${}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}\xrightarrow{({}^I\overrightarrow{\Lambda}^J)^{-1}}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ is a $({}^I\overrightarrow{\Lambda}^J)^{-1}$-twisted of ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ as defined in~\ref{def: twisted} and~\ref{prop: cj-duals}.
\medskip
The contravariant nature of Riesz maps requires contravariant involutions in the definition of inner products;
alternative possibilities can be explored with ``inner couplings'' on ${\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ with more general signatures.
\xqed{\lrcorner}
\end{remark}
\begin{remark}\label{rem: hybrid}
Thinking of multimodules in the 1-category ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}\Mf_{(\Bs_\beta)_B}$ as ``1-arrows'' $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A\xleftarrow{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}(\Bs_\beta)_B$, in a 2-category of morphisms $\xymatrix{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A & \ltwocell^{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}_{\Ns}{\Phi} (\Bs_\beta)_B}$, we see that, for all $I\times J\in A\times B$, $(I,J)$-duals provide an $\{0,1\}$-contravariant involution $\xymatrix{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A \rrtwocell^{{}^{*_I}\Ns^{*_J}}_{{}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}}{\quad {}^{*_I}\Phi^{*_J}} & & (\Bs_\beta)_B}$, over objects and 1-arrows, in the sense described in~\cite[section~4]{BCLS20}.
Riesz maps can be considered as ``natural transformations'' examples of \textit{hybrid 2-arrows}
$\xymatrix{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A \rrtwocell^{{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}}_{{}^{*_I}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}^{*_J}}{'\ \quad {\Downarrow}\ {}^{*_I}\overrightarrow{\Lambda}^{*_J}} & & (\Bs_\beta)_B}$
following the definition of \textit{hybrid 2-category} described in~\cite{BePu14}. We will pursue such developments elsewhere.
\xqed{\lrcorner}
\end{remark}
\begin{remark}
In definition~\ref{def: ip-multi} in order to keep the closest possible resemblance to the usual axioms for inner products in Hilbert spaces and Hilbert-C*-modules, we have imposed covariance, for certain actions, only one of the two variables and contravariance on the other. It is perfectly possible to consider more general cases, where covariance and contravariance are simultaneously present in both variables (on disjoint sets of indexes):
let $I\times J\subset A\times B$ with $I:=I_l\cup I_r$, $J=J_l\cup J_r$ and
$I_l\cap I_r=\varnothing=J_l\cap J_r$,
a \emph{$(I_l,J_l)$-left $(I_r,J_r)$-right inner product} on ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_B}$ is a bi-additive map $(x,y)\mapsto {}_{I_l-J_l}\ip{x}{y}_{I_r-J_r}$, for $x,y\in {\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$, such that: $\forall x,y\in{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}},$
\begin{align*}
{}_{I_l-J_l}\ip{x}{a\cdot_\alpha y\cdot_\beta b}_{I_r-J_r} &= a\cdot_\alpha\ip{x}{y}_{I_r-J_r}\cdot_\beta b,
&&
\forall (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ (\alpha,\beta)\in I_r\times J_r,
\\
{}_{I_l-J_l} \ip{a\cdot_\alpha x\cdot_\beta b}{y}_{I_r-J_r}&= a\cdot_\alpha\ip{x}{y}_{I_r-J_r}\cdot_\beta b,
&&
\forall (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ (\alpha,\beta)\in I_l\times J_l,
\\
{}_{I_l-J_l}\ip{a\cdot_\alpha x\cdot_\beta b}{y}_{I_r-J_r} &=b^*\bullet_\beta{}_{I_l-J_l}\ip{x}{y}_{I_r-J_r}\bullet_\alpha a^*,
&&
\forall (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ (\alpha,\beta)\in I_r\times J_r,
\\
{}_{I_l-J_l}\ip{x}{a\cdot_\alpha y\cdot_\beta b}_{I_r-J_r} &=b^*\bullet_\beta{}_{I_l-J_l}\ip{x}{y}_{I_r-J_r}\bullet_\alpha a^*,
&&
\forall (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ (\alpha,\beta)\in I_l\times J_l,
\\
{}_{I_l-J_l}\ip{a\cdot_\alpha x\cdot_\beta b}{y}_{I_r-J_r} &={}_{I_l-J_l}\ip{x}{a^*\cdot_\alpha y\cdot_\beta b^*}_{I_r-J_r},
&&
\forall (a,b)\in{\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha\times\Bs_\beta, \ (\alpha,\beta)\in (A-I)\times (B-J).
\end{align*}
Riesz maps ${}^{I_l-J_l}\overrightarrow{\Lambda}^{I_r-J_r}$ and ${}^{I_l-J_l}\overleftarrow{\Lambda}^{I_r-J_r}$ can be similarly defined and a perfect parallel of theorem~\ref{th: Riesz} holds.
\xqed{\lrcorner}
\end{remark}
\section{Outlook} \label{sec: outlook}
Although we are not going here into specific details, that will be subject of a forthcoming work, we preview some of the categorical features making multimodules a quite intriguing playground.
\medskip
The family of multimodules, with their several tensor products, constitutes a paradigmatic example of ``algebraic structure'' consisting of ``many inputs / many outputs nodes'' that can be ``linked'' in many different ways: each multimodule ${}_{({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A}{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}_{(\Bs_\beta)_B}$ should be interpreted as an arrow with sources $(\Bs_\beta)_B$ and targets $({\mathscr{A}}}\newcommand{\Bs}{{\mathscr{B}}}\newcommand{\Cs}{{\mathscr{C}}_\alpha)_A$;\footnote{
We are using here, for the tensor products, the same ``reversed order'' notation of the functional compositions.
}
every tensor product over a subfamily provides a possible ``concatenation'' of arrows and such compositions will be subject to associativity and unitality axioms typical of category theory.
\medskip
At the 1-categorical level (when only multimodules as 1-arrows and tensor products as compositions are considered) the structure seems to be describable as a \textit{colored properad}~\cite{HRY15}, a horizontal categorification (i.e.~a many objects version) of the notion of \textit{properad} introduced by~\cite{Va07}.
\medskip
Dualities of multimodules seem to provide the easiest examples of involutions for arrows in a colored properad and can be taken as a paradigmantic template in order to axiomatize a notion of ``involutive colored properad''.
Contractions can be used to introduce ``sinks'' and ``sources'', hence more general types of ``partial involutions''.
\medskip
As already mentioned in remark~\ref{rem: hybrid}, we plan to further study Riesz dualities as examples of \textit{hybrid natural transformations}, between functors with different covariance, in the context of hybrid \hbox{2-categories} introduced in~\cite{BePu14}.
\medskip
Covariant morphisms of multimodules should be interpreted (exactly as in the usual case of categories of bimodules) as cubical 2-arrows. In this way, one obtains for multimodules a colored properad analog of the usual double category of covariant morphisms of bimodules.
\medskip
It is also possible to iterate the construction of multimodules over multimodules creating a vertical categorification ladder that can be used to define ``involutive higher colored properads'' (possibly requiring the non-commutative exchange property introduced in~\cite{BCLS20}).
\medskip
The purely algebraic theory of $\Zs$-central multimodules over $\Rs_\Zs$-algebras here presented can be subject to a functional analytic treatment as soon as topologies/uniformities are introduced and the actions are required to be continuous in the suitable sense. We will explore in the future the more restrictive axioms for a (higher) C*-algebraic version of this material and obtain (infinite-dimensional) functional analytic generalizations of the (essentially finite-dimensional) \textit{reflexivity} ($\theta^{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ covariant isomorphism) and \textit{self-duality} (${}^J\overrightarrow{\Lambda}^I_{\mathscr{M}}}\newcommand{\Ns}{{\mathscr{N}}}\newcommand{\Os}{{\mathscr{O}}$ contravariant isomorphism) conditions on multimodules.
\medskip
Although the basic definition of first-order differential operator between $\Zs$-central multimodules over non-commutative $\Rs_\Zs$-algebras is included in appendix~\ref{sec: 1st-ord-multi}, much more needs to be done regarding the full differential theoretic theory of multimodules (and also bimodules!), starting with a theory of connections on multimodules and possibly proceeding in the direction of \textit{properadic non-commutative geometry} as a natural extension of our current efforts in categorical non-commutative geometry. An exploration of the interplay between duality (for bimodules) and first-order differential operators associated to covariant differential calculi on a non-commutative $\Zs$-central algebra is carried on in our forthcoming work (mentioned in footnote~\ref{foo: ncdc}).
\bigskip
\emph{Notes and Acknowledgments:}
P.Bertozzini thanks Starbucks Coffee ($1^{\text{st}}$ floor of Emporium Tower, Emquartier Sky Garden, Jasmine City) where he spent most of the time dedicated to this research project; he thanks Fiorentino Conte of ``The Melting Clock'' for the great hospitality during many crucial on-line dinner-time meetings Bangkok-Rome.
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\section{Introduction}
The complex Langevin method (CL), first proposed by Parisi~\cite{Parisi:1980ys} has been successfully adapted and used in various models and approximations of QCD, for instance, the heavy-dense limit of QCD~\cite{Aarts:2016qrv}. Early results have been obtained for QCD with very heavy quarks~\cite{Sexty:2013ica,Scherzer:2020kiu}. Here, we aim to study the QCD phase diagram with significantly lighter quarks ($m_\pi \sim 480\,$MeV). Our simulations span a large range of chemical potentials, ranging up to approximately $6500\,$MeV, and various temperatures, as low as $25\,$MeV.
\section{Complex Langevin}
The (complex) Langevin method evolves the gauge links of a lattice simulation along a new fictitious time $\theta$ for a small step size $\epsilon$. A first order update scheme can be written as
\begin{align*}
U_{x\mu} (\theta + \epsilon) = \exp \Big( \mathrm{i}\, \lambda^a\, ( - \epsilon \, D^a_{x\mu} S[U] + \sqrt{\epsilon} \, \eta_{x,\mu}^a)
\Big) \, U_{x\mu} (\theta),
\end{align*}
where $\lambda^a$ are the Gell-Mann matrices, $D^a_{x\mu} S$ are the derivatives of the action with respect to the gauge link $U_{x\mu}$ and $\eta_{x,\mu}^a$ are the random white noise. The complexification is introduced by enlarging the group manifold from SU$(3)$ to SL$(3,\mathbb{C})$, i.e. by allowing the coefficients of the generators to become complex numbers
\begin{align*}
U_{x\mu} = \exp \Big( \mathrm{i}\, \lambda^a (A_{x\mu}^a + \mathrm{i}\, B_{x\mu}^a) \Big).
\end{align*}
The new time direction created by this procedure is analogue to the Molecular Dynamics time in standard lattice calculations. For sufficient long simulations, an observable $O$ can be obtained by taking an average over the Langevin time $\theta$
\begin{align*}
\langle O \rangle = \frac{1}{\theta_\mathrm{max} - \theta_\mathrm{therm}} \sum_{\theta = \theta_\mathrm{therm}}^{\theta_\mathrm{max}} O[U(\theta)],
\end{align*}
after discarding a sufficient amount $\theta_\mathrm{therm}$ to remove thermalisation effects. As we aim to take the limit of $\epsilon \to 0$, proper treatment of autocorrelation effects is important. We use the automatic autocorrelation method presented in~\cite{Wolff:2003sm}.
\section{Setup}
For our study we use two dynamical flavours of Wilson quarks without a clover term $c_{\mathrm{SW}} = 0$. Our simulation parameters are based on one of the setups used in~\cite{DelDebbio:2005qa}. The gauge coupling is fixed at $\beta = 5.8$ throughout all simulations. This implies a fixed lattice spacing of approximately $a\sim 0.06\,$fm~\cite{DelDebbio:2005qa}. The hopping parameter of $\kappa = 0.1544$ leads to a pion mass of $a\, m_\pi = 0.1458(7)$, which was measured using a HMC simulation at vanishing chemical potential using $N_t =128$. Converting the pion mass into physical units, we find $m_\pi \sim 480\,$MeV. The spatial simulation volume is fixed to $24^3$ and so the product $m_\pi L = 3.5$ is sufficiently large, so that volume effects are expected to be small. The authors of~\cite{DelDebbio:2005qa} found a pion mass of $a\, m_\pi = 0.1481(11)$ for a larger spatial volume of $32^3$. The difference is $1.5\%$, which indicates small volume effects. Different temperatures are realised by varying the temporal lattice extent. In this study, we use up to $128$ points in time, resulting in temperature down to $25\,$MeV. The chemical potential was scaled up to $a\, \mu = 2.0$.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|}
\hline
$\beta = 5.8$ & $\kappa= 0.1544$ \\
\hline
$V = 24^3$ & $m_\pi L =3.5$\\
\hline
$N_t = 4 - 128$ & $T \sim 800 - 25\,$MeV \\
\hline
$a\mu = 0.0 - 2.0 $ & $\mu \sim 0 - 6500\,$MeV \\
\hline
\end{tabular}
\caption{The lattice parameters used in this study.}
\label{tab:params}
\end{table}
Table~\ref{tab:params} shows a summary of the key parameters. To improve the convergence of the complex Langevin simulation, we employ adaptive step size scaling~\cite{Aarts:2009dg}, gauge cooling~\cite{Seiler:2012wz} and dynamic stabilisation~\cite{Attanasio:2018rtq,Attanasio:2020spv}.
\section{Extrapolation at zero chemical potential}
As a first consistency check, we compare HMC simulations with complex Langevin simulations at vanishing chemical potential. Even though this is a purely real setup and complex Langevin simulations are not necessary, we allow and start with configurations that are in the SL$(3,\mathbb{C})$ manifold. As complex Langevin suffers from finite step size corrections, we simulate multiple values of the step size $\epsilon$ and extrapolate to zero. As we employ step size scaling~\cite{Aarts:2009dg}, we measure the average step size $\langle\epsilon\rangle$ and adapt the relevant control parameter such that we achieve different values. \begin{figure}[!htb]
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{gfx/Extra/plaq8.pdf}
\end{minipage}
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{gfx/Extra/plaq32.pdf}
\end{minipage}
\caption{The plaquette as function of the average Langevin step size $\langle\epsilon\rangle$ at zero chemical potential. The lines are different extrapolations towards the HMC result. (Left): High temperature regime, i.e. $N_t = 8$ corresponding to $T \sim 400\,$MeV. (Right): Low temperature regime, i.e. $N_t = 32$ corresponding to $T \sim 100\,$MeV}
\label{fig:extra}
\end{figure}
Figure~\ref{fig:extra} shows the results for the plaquette as a function of the average step sizes for two different setups. The first setup (left panel) corresponds to large temperatures of approximately $400\,$MeV. The second setup (right panel) is at lower temperatures of approximately $100\,$MeV. The data points are augmented by three different fits to the data: a linear (lin), quadratic (quad) and cubic (cub) extrapolation. In the linear extrapolation, the four right most points have been removed for the fitting procedure. As we use a first-order integration scheme, we expect our step size correction to be of first order. This behaviour is clearly visible. Overall, we find very good agreement between the HMC results and the extrapolated result from complex Langevin simulations. The agreement is up to the 4th significant figure or less than a permille.
\section{Simulation at non-zero chemical potential}
To explore the QCD phase diagram, we simulate at different chemical potential and temperatures. Our parameters are chosen such that we span a large region of the QCD phase diagram. To study the phase behaviour, we investigate in particular the fermion density and Polyakov loop.
\begin{figure}[!htb]
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{gfx/Phase/dens.pdf}
\end{minipage}
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{gfx/Phase/dens3.pdf}
\end{minipage}
\caption{The fermion density normalised over $(6 V N_t)$ for each flavour for large set of lattice extends $N_t$ and chemical potentials $\mu$. The two grey vertical lines correspond to $m_\pi/2$ and $m_N/3$. (Left): The full simulation range. (Right:) Zoom into the view on the small chemical potentials and tiny densities. At low temperatures the fermion density remains zero until $m_N/3$, which is a first sign of the Silver Blaze phenomenon.}
\label{fig:dens}
\end{figure}
Figure~\ref{fig:dens} shows the fermion density as a function of the chemical potential. Each data point corresponds to an independent complex Langevin simulation. The left panel shows the entire simulation range, whereas the right panel focuses on smaller chemical potentials. For both, the fermion density is divided by $6\, V\, N_t$ for each flavour, i.e. the saturation density. At large $\mu$ we see aforementioned saturation, which is an intrinsic lattice artefact. This occurs when all lattice sites are filled with fermions, so that Pauli blocking does not allow additional fermions to be added to the system. The right-hand side of figure~\ref{fig:dens} also shows an enlarged view of the fermion density for small chemical potentials and tiny values of the density. Interestingly, the fermion density remains $0$ up to the $m_N/3$, which is a manifestation of the so-called Silver Blaze phenomenon~\cite{Cohen:2003kd}. The Polyakov loop is shown in the left panel of figure~\ref{fig:poly}. Here the same lattice artefact is visible for large chemical potentials.
\begin{figure}[!htb]
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{gfx/Phase/poly.pdf}
\end{minipage}
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{gfx/Phase/poly3.pdf}
\end{minipage}
\caption{(Left): The Polyakov loop as function of chemical potential $\mu$ for the full range of simulations. As before the gray lines indicate $m_\pi/2$ and $m_N/3$ (Right): The Polyakov loop as function of the temperature $T$ for two choices of $\mu$. }
\label{fig:poly}
\end{figure}
In the right panel of figure~\ref{fig:poly} we show the Polyakov loop as function of the temperature for two chemical potentials. We find that the transition to a nonzero Polyakov loop is occurring at lower temperatures for larger chemical potentials. This allow us to quantify the transition(s). Additionally, we also look at the Binder cumulant~\cite{Binder:1981sa}, defined in the following way
\begin{align*}
B = 1 - \frac{\langle O^4\rangle}{3 \langle O^2\rangle^2}.
\end{align*}
The Binder cumulant for the fermion density is shown in the left panel of figure~\ref{fig:binder} and for the Polyakov loop in the right panel. The transition to non-zero values for the Binder cumulant occurs at smaller temperatures as the chemical potential increases, as already seen in the right panel of figure~\ref{fig:poly}.
\begin{figure}[!htb]
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{gfx/Binder/dens2.pdf}
\end{minipage}
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{gfx/Binder/poly32.pdf}
\end{minipage}
\caption{The Binder cumulants for the fermion density (left) and the Polyakov loop (right) for different temperatures and chemical potentials. }
\label{fig:binder}
\end{figure}
We have covered a large range of temperatures and chemical potentials, however, the simulations at low temperatures become increasingly challenging, from a numerical point of view. In particular, the number of conjugate gradient steps increases sharply as the chemical potential increases, in particular for low temperature simulations.
\begin{figure}[!htb]
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{gfx/Phase/iter.pdf}
\end{minipage}
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{gfx/Phase/unit.pdf}
\end{minipage}
\caption{(Left): The number of conjugate gradient iterations necessary for the computation of the drift force in the complex Langevin process.
(Right): The average unitarity norm of the individual simulation, i.e. the distance to the SU(3) manifold. }
\label{fig:numerics}
\end{figure}
This behaviour can be seen in the left panel of figure~\ref{fig:numerics}. On the right panel, the average unitarity norm is shown, which stays sufficiently small for all simulations. For lower temperatures we can see an increase for larger chemical potentials.
\section{Outlook \& Conclusion}
Here we have reported on our ongoing
first-principle study of the QCD phase diagram using complex Langevin simulations. We have performed simulations at various chemical potentials and temperatures using two flavour Wilson quarks with a pion mass of approximately $480\,$MeV. In particular, we have found that the fermion density stays zero in the region of $m_\pi/2$ to $m_N / 3$, i.e. the Silver Blaze phenomenon.
\medskip
In the future, we plan to focus on improved
computations and algorithms, in particular at lower temperatures. The necessary code developments are already underway. Furthermore, we plan to repeat our study with different volumes, in order to quantify the order of the transition(s) by studying the volume scaling.
\acknowledgments
The work of F.A. was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster) and under the Collaborative Research Centre SFB 1225 (ISOQUANT). This work was performed using PRACE resources at Hawk (Stuttgart) with project ID 2018194714. This work used the DiRAC Extreme Scaling service at the University of Edinburgh, operated by the Edinburgh Parallel Computing Centre on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BEIS capital funding via STFC capital grant ST/R00238X/1 and STFC DiRAC Operations grant ST/R001006/1. DiRAC is part of the National e-Infrastructure.
| 4,178 |
\section{Introduction}
The problem of resolving two point sources has been intensively studied as model problem for optical system resolution characterisation for more then a century. For a long time only visual observation of the light coming from the sources was available, thus resolving criteria were based on visual characteristics of intensity distribution \cite{Rayleigh1879,sparrow1916spectroscopic}. Visual criteria, such as Rayleigh criterion, were aimed to distinguish between images of two sources and one, but not to describe the problem of the separation estimation.
Later it became possible to carry out an analysis of the whole intensity distribution in the image plane, that allows to resolve sources beyond visual limits \cite{helstrom1967image}. The efficiency of this approach can be estimated on the basis of the Fisher information (FI)
and the Cram\'er-Rao bound. Measuring intensity distribution is called direct imaging (DI). This technique proved to often lead to vanishing FI in the limit of small separations, this feature is known as "Rayleigh's curse". At the same time analysis of the Quantum Fisher information (QFI), that provides the ultimate limit for resolution regardless of the measurement, shows that in most cases one can increase resolution and avoid "Rayleigh's curse" by choosing more optimal measurement than DI \cite{TsangPRX}.
A fine example of such measurement is spatial mode demultiplexing (SPADE). SPADE proposes to decompose the field in the image plane into spatial modes and measure their intensities. The resolution of SPADE was widely studied for uncorrelated thermal sources. The FI of this measurement, calculated in the limit of small photon numbers, was proved to saturate the ultimate bound set by QFI \cite{TsangPRL,pirandola}. Recently another measure of resolution was proposed --- a sensitivity measure based on method of moments \cite{PRLGessner2019}. This measure allowed to analyse problem of resolving bright uncorrelated thermal sources and proved SPADE to saturate the ultimate limits without additional assumption about sources intensities~\cite{sorelli2021PRL,sorelli2021PRA}.
At the same time an important discussion about the role of sources' mutual coherence took place \cite{Larson:18,tsang2019comment,Larson19reply,Hradil19QFI}.
Two different approaches to describe partial coherence of low brightness sources, departing in their account of losses, have been introduced, leading to different results. Recently it became clear that, for the case of unknown sources' brightness, a loss-less model \cite{Larson:18} can be applied.
On the other hand, if the sources' brightness is assumed to be known, one needs to rigorously account for losses on the finite aperture of the optical system to benefit from dependence of the total registered intensity on estimated separation \cite{tsang2019comment,kurdzialek2021sources}.
Despite these intensive studies of the role of the mutual coherence of the sources there are still gaps in knowledge in this area. This is especially true for bright sources, since most of the cited research is based on the assumption of low intensity of the detected signal. Wherein the study of the QFI of partially correlated thermal light shows a significant influence of the sources' brightness on resolution limit \cite{pirandola}. Thus for practical application of SPADE it is important to develop its description for the case of bright sources and to be able to estimate the sensitivity of specific measurements.
In this study we focus on the fundamentally and practically important case of a pair of perfectly mutually coherent sources. We consider the most general quantum state of emitted light, and make no assumption about sources' absolute and relative brightness and relative phase.
Using the method of moments, we analyse the sensitivity of DI and SPADE for sources' separation estimation and analytically show an advantage of the latter approach. Considering a multiparameter estimation approach we analytically find the separation estimation sensitivity with and without prior knowledge about the brightness of the sources. We show that anti-bunching of the sources' radiation leads to an increase of the sensitivity, but at the same time ignorance of sources brightness wipes out any possible profit from non-classical statistics of the sources.
Finally, we show that the calculated sensitivity saturates the QFI for the cases where it is known. These include non-Gaussian entangled states of the sources, for which the QFI of separation estimation is maximal.
Our approach can be easily generalised to other parameter estimation problems with mutually coherent probes or single-mode probe, like coherent imaging \cite{ferraro2011coherent} or even distributed quantum sensing \cite{guo2020distributed}. Moreover, moment-based estimation gives a practical way to estimate a bound on the sensitivity of specific measurements which can be saturated by a parameter retrieval method that does not require complicated data processing.
\section{Optical scheme}
We consider a traditional optical scheme for parameter estimation problem, where two point-like sources emit light, that passes through diffraction-limited imaging system (see the right part of fig.\ref{fig:scheme}). Then sources' parameters, such as separation $d$, are estimated from measuring the diffracted light.
\begin{figure}[ht]
\includegraphics[width=0.95 \linewidth]{pics/fig1.scheme.pdf}
\caption{\label{fig:scheme} On the left: principal scheme for generating general two-mode coherent state based on beamsplitter with transmissivity $T=\cos^2 \theta$ and phase shifting element $\phi$. On the right: optical scheme for sources' separation estimation, were $\kappa$ is the transmissivity and photon counting is performed in the measurement modes $f_m (\vec r)$ with corresponding field operators~$\hat a_m$.}
\end{figure}
\subsection{Emitted field}
Point-sources emit light in the orthogonal modes with field operators $\hat s_{1,2}$. We only consider mutually coherent sources, meaning that the absolute value of the first order coherence degree of emitted light is $| g^{(1)}|=1.$ One can always find single mode (called principal mode) that fully describe this field configuration. Thus the most general mutually coherent state of the modes $\hat s_{1,2}$ can be considered as the result of splitting some mode $\hat s_0$ on an asymmetric beamsplitter with transmissivity $T=\cos^2 \theta$ and adding phase $\phi$ to one of the output modes $\hat s_{1,2}$ (see the left part of fig.\ref{fig:scheme}). The first order coherency matrix of the modes $\hat s_{1,2}$ reads
\begin{equation}
\label{eqn:cohS}
\langle \hat s^\dagger_j \hat s_k \rangle=
\begin{pmatrix}
N_S \cos^2 \theta & \frac{N_S}{2} \sin 2\theta~e^{i \phi} \\
\frac{N_S}{2} \sin 2\theta~e^{-i \phi} & N_S \sin^2 \theta
\end{pmatrix},
\end{equation}
where $N_S=\operatorname{Tr} (\hat \rho_0 \hat s_0^\dagger \hat s_0 ) $ -- average number of emitted photons and parameter $\theta$ is responsible for sources' intensity asymmetry. The ratio between the sources' brightness $\langle \hat s_2 ^\dagger \hat s_2 \rangle/\langle \hat s_1 ^\dagger \hat s_1 \rangle=\tan^2\theta$. For concreteness we assume parameter $\theta$ to be within a range $0 \le \theta \le \pi/4$, where $\theta=\pi/4$ corresponds to equally bright sources, and $\theta=0$ to all light in mode $\hat s_1$.
Parameter $\gamma=\langle \hat s_1^\dagger \hat s_2 \rangle/\sqrt{\langle \hat s_1 ^\dagger \hat s_1 \rangle \langle \hat s_2 ^\dagger \hat s_2 \rangle}=e^{i\phi}$ is often referred to as the degree of mutual coherence between modes, and in our case it has modulus equal to one.
Note, that no assumptions about state $\hat \rho_0$ of the mode $\hat s_0$ was made so far. If it has Poisson photon-number statistics, the state of the modes $\hat s_{1,2}$ is separable; for thermal statistics of $\hat \rho_0$ the modes $\hat s_{1,2}$ are classically correlated; for Fock state of $\hat s_0$ the output modes are entangled.
For further description of the parameter estimation in terms of the method of moments we will only need the two first moments for the state $\hat \rho_0$: average photon number $N_S$ and its variance $\Delta N_S^2$.
\subsection{Field detection}
The emitted light goes through an imaging system with finite aperture that has transmissivity $\kappa$ and point spread function (PSF) $u_0(\vec r)$. We use paraxial approximation where loss factor does not depend on position of the source. In the image plane the light is detected either directly (DI) or via SPADE. Both cases can be described as measurement over some field modes $f_m( \vec r)$ with corresponding operators $\hat a_m$, where for DI $f_m( \vec r)$ are localized pixel modes and for SPADE $f_m( \vec r)$ are more general non-localized modes, for example Hermite-Gauss. Then the parameters of interest are estimated from measured numbers of photons in detection modes $N_m=\langle \hat a_m^\dagger \hat a_m \rangle$. Particularly, we analyse the estimation of the sources' separation $d$ with and without prior knowledge of the number of emitted photons $N_S$.
Passing through the lossy optical system can be described as mixing of field modes
with
vacuum modes \cite{pirandola}. Taking into account also vacuum mode from beamsplitter $\theta$ the field operators of the measurement modes can be represented as $\hat a_m = A_m \hat s_0 + \hat v_m$, where $A_m$ is a complex coefficients and $\hat v_m$ - non-orthogonal combinations of vacuum modes' field operators
. Thus in measurement modes any normally ordered average has the following property:
\begin{equation}
\label{eqn:property}
\langle : F (\hat a^\dagger_m, \hat a_n): \rangle = \langle : F (A_m^* \hat s_0^\dagger , A_n \hat s_0): \rangle ,
\end{equation}
where $:\hat X:$ denotes normal ordering, $F$ is an arbitrary analytical function of the field operators. Then average number of detected photons in the $m$-th measurement mode reads
\begin{equation}
\label{eqn:Nm}
N_m=\langle \hat a_m^\dagger \hat a_m \rangle=|A_m|^2 N_S.
\end{equation}
Specifically for the scheme on fig.\ref{fig:scheme} the coefficients
\begin{equation}
\label{eqn:Am}
A_m=\sqrt \kappa \int d \vec r~f_m(\vec r) \big[u_0(\vec r - \vec r_1) \cos \theta + u_0 (\vec r- \vec r_2) e^{i \phi} \sin \theta \big]
\end{equation}
depends on the sources positions $\vec r_{1,2}$, their intensity asymmetry (that depends on $\theta$) and the mutual phase $\phi$, optical system transmissivity $\kappa$ and PSF $u_0(\vec r)$ and measurement modes shape $f_m(\vec r)$.
\section{Moment based sensitivity}
To estimate resolution of the considered optical scheme we calculate the moment-based sensitivity (further we call it just sensitivity). Within the framework of the method of moments it was shown that estimation a set of variables $\{q_\alpha\}$ from mean values $\{X_m\}$ of observables $\{\hat X_m\}$ corresponds to sensitivity matrix
\cite{PRLGessner2019}
\begin{equation}
\label{eqn:sensitivity}
M_{\alpha \beta} = \sum_{m,n} (\Gamma^{-1})_{m n} \frac{\partial X_m} {\partial q_\alpha} \frac{\partial X_n} {\partial q_\beta},
\end{equation}
where $\Gamma_{m n}=\langle \hat X_m \hat X_n \rangle-X_m X_n$ is the covariance matrix of the observables. Inverted sensitivity matrix sets a lower bound for estimators' covariances
\begin{equation}
\operatorname{cov}[q_\alpha,q_\beta] \ge \frac{1}{\mu} ( M^{-1})_{\alpha \beta},
\end{equation}
where $\mu$ is the number of measurement repetitions.
Generally estimators that includes all the moments of observables can have lower variance, i.e. the following chain of inequalities holds
\begin{equation}
\label{eqn:inequality}
\mathbf{M} (\{q_\alpha\}, \{\hat X_m\}) \le \mathcal{F} ( \{q_\alpha \}, \{\hat X_m \}) \le \mathcal{F}_Q (\{q_\alpha\}),
\end{equation}
where sensitivity matrix $ \mathbf{M} (\{q_\alpha\}, \{\hat X_m\})$ is given by eqn. \eqref{eqn:sensitivity}, $\mathcal{F} ( \{q_\alpha\}, \{\hat X_m \})$ is Fisher information (FI) matrix, $\mathcal{F}_Q(\{q_\alpha \})$ is quantum Fisher information (QFI) matrix and matrix inequality $\mathbf{A} \le \mathbf{B}$ means that $\vec a^T \mathbf{A} \vec a \le \vec a^T \mathbf{B} \vec a$ for any given column vector $\vec a$. Thus the sensitivity can be considered as a lower bound for the FI. It was shown, that the sensitivity of photon counting in Hermite-Gauss (HG) modes for estimation of separation between equally bright uncorrelated thermal sources saturates the QFI~\cite{sorelli2021PRA}.
Sensitivity matrix also gives a set of linear combinations of observables $\hat X_m$, that contains sufficient information for estimation of the parameters $\{q_\alpha\}$ with precision set by \eqref{eqn:inequality}. These combinations allow to avoid complicated numerical method of parameters inferring such as maximum likelihood estimation.
Using expression \eqref{eqn:sensitivity} one can calculate the sensitivity of photon counting in spatial modes $ f_m (\vec r)$, i.e. use $\hat X_m=\hat a^\dagger_m \hat a_m$ as an observable
. Thus
using property \eqref{eqn:property} and equation \eqref{eqn:Nm} one can find the elements of the photon number covariance matrix
\begin{equation}
\label{eqn:gamma}
\Gamma_{m n}=\delta_{mn} N_m + h N_m N_n,
\end{equation}
where $h=g^{(2)}-1=(\Delta N_S^2 - N_S)/N_S^2$. Matrix \eqref{eqn:gamma} can be analytically inverted with Sherman-Morrison formula \cite{sherman1950adjustment} obtaining
\begin{equation}
\label{eqn:gamma_Inv}
(\Gamma^{-1})_{m n}=\delta_{mn} N_m^{-1} - \frac{ h}{1+h N_D},
\end{equation}
where $N_D=\sum_m N_m$ is the
total number of detected photons. Then the sensitivity matrix \eqref{eqn:sensitivity}
can be expressed as
\begin{equation}
\label{eqn:Mmatrix}
M_{\alpha \beta}=N_D \sum_m \frac{1}{\varepsilon _m} \frac{\partial \varepsilon_m}{\partial q_\alpha} \frac{\partial \varepsilon_m}{\partial q_\beta} + \frac{1}{\Delta N_D^2} \frac{\partial N_D}{\partial q_\alpha} \frac{\partial N_D}{\partial q_\beta} ,
\end{equation}
where $\varepsilon_m=N_m/N_D$ is the relative photon number and
\begin{equation}
\label{eqn:N_D_variance}
\Delta N_D^2=\sum_{mn} \Gamma_{mn}=N_D(1+h N_D)
\end{equation}
is the variance of the total number of detected photons.
\subsection{Single parameter estimation}
If all parameters except separation $d$ are known, the sensitivity matrix become a number
\begin{equation}
\label{eqn:Md}
M_d=N_D \sum_m \frac{1}{\varepsilon _m} \left (\frac{\partial \varepsilon_m}{\partial d} \right )^2 + \frac{1}{\Delta N_D^2} \left( \frac{\partial N_D}{\partial d} \right )^2.
\end{equation}
This expression has two terms, the first term equals $N_D M_\varepsilon$, where
\begin{equation}
\label{eqn:Mepsilon}
M_\varepsilon = \sum_m \frac{1}{\varepsilon _m} \left(\frac{\partial \varepsilon_m}{\partial d} \right)^2
\end{equation}
does not depend on the state $\hat \rho_0$ but strongly depends on the measurement basis $\{f_m(\vec r) \}$.
$M_\varepsilon$ can be called the sensitivity per detected photon of relative intensity measurements (see two-parameters case for the details). In the small photon number limit it coincides with the FI of relative intensity measurement.
The second term
\begin{equation}
\label{eqn:MND}
M_D=\frac{1}{\Delta N_D^2} \left(\frac{\partial N_D}{\partial d} \right)^2,
\end{equation}
does not depend on the individual signals $N_m$, but only on the total one $N_D$, thus it stays the same for any measurement basis $\{f_m(\vec r) \}$ as long as all photons in the image plane are detected. This additional sensitivity $M_D$ occurs due to the interference of mutually coherent sources and the subsequent dependence of the total number of registered photons $N_D$ on the separation $d$.
The expression \eqref{eqn:MND} has a self-consistent structure representing the simple error-propagation formula. Variance of the total number of detected photons \eqref{eqn:N_D_variance} depends on the quantum statistics of the source and growth with source bunching.
In the next subsection we show, that this sensitivity of total photon number measurement $M_D$ vanishes, if sources' brightness $N_S$ is unknown.
Total sensitivity $M_d=N_D M_\varepsilon + M_D$ sets lower bound for separation estimator variance
\begin{equation}
\label{eqn:bound1}
\Delta d^2 \ge \frac{1}{\mu} (M_d)^{-1}=\frac{1}{\mu} \frac{1}{N_D M_\varepsilon + M_D}.
\end{equation}
\subsection{Two parameters estimation}
For better understanding of the physical meaning of values $M_\varepsilon$ and $M_D$ let's consider two-parameter problem, where both separation $d$ and emitted number of photons $N_S$ are unknown. Sensitivity matrix \eqref{eqn:Mmatrix} of estimating a set of variables $q_\alpha=\{d,N_S\}$ reads
\begin{equation}
\label{eqn:Mmatrix2}
M_{\alpha \beta}=
\begin{pmatrix}
N_D M_\varepsilon+M_D & ~~\dfrac{1}{\Delta N^2_D} \dfrac{\partial N_D}{\partial N_S} \dfrac{\partial N_D}{\partial d} \\[12pt]
\dfrac{1}{\Delta N^2_D} \dfrac{\partial N_D}{\partial N_S}\dfrac{\partial N_D}{\partial d} & ~~\dfrac{1}{\Delta N^2_D} \left(\dfrac{\partial N_D}{\partial N_S} \right)^2
\end{pmatrix},
\end{equation}
where $\partial \varepsilon_m/\partial N_S=0$ is taken into account. In this case variance of $d$ estimation is bounded by
\begin{equation}
\label{eqn:bound2}
\Delta d^2 \ge \frac{1}{\mu} (M^{-1})_{11}=\frac{1}{\mu} \frac{1}{N_D M_\varepsilon}.
\end{equation}
This bound shows, that without knowing the number of emitted photons $N_S$ one can benefit only from measurement of relative intensities $\varepsilon_m$.
Comparing it to the bound \eqref{eqn:bound1} one can see that knowing the number of emitted photons $N_S$ increases sensitivity of separation estimation by total photon number sensitivity $M_D$. Thus, ignorance of sources brightness $N_S$ wipes out any possible profit from non-classical statistics of the sources, that are only present in term $M_D$.
If one uses bucket detection, that corresponds to the detection of all the photons in the image plane, and can be described in this particular case as a single mode measurement in the principle mode, then the only relative intensity $\epsilon_0=1$ does not depend on parameter, thus $M_\epsilon=0$. One can not estimate separation from this measurement without knowing the number of emitted photons $N_S$. If $N_S$ is known then, as expected, full sensitivity of bucket detection is provided by total photon number detection $M_d=M_D$.
Note, even though the structure of the sensitivity expression \eqref{eqn:Mmatrix} looks quite intuitive, it was derived specifically for the single mode (fully coherent) case and does not necessarily holds for other cases. For example, the sensitivity of incoherent thermal sources' separation estimation non-linearly depends on the sources brightness \cite{sorelli2021PRA}, although the total number of detected photons does not depend on the separation in this case.
It is worth mentioning, that the used property \eqref{eqn:property} is valid for a quite general class of parameter-estimation schemes, where the parameters are encoded in an arbitrary number of mutually coherent modes, that are subjected to correlated parameter-dependent linear losses. Therefore formula \eqref{eqn:Mmatrix} for the sensitivity matrix of photon counting is valid for this wider class of systems, since the explicit form of coefficients $A_m$ \eqref{eqn:Am} was never used yet. Thus the developed approach can be used for other problems like coherent imaging or quantum sensing in a continuous variable entangled network in the single-mode regime \cite{guo2020distributed}.
\section{Sensitivity of relative intensity measurement $M_\varepsilon$}
Now let us consider separately the two parts of the separation estimation sensitivity \eqref{eqn:Md}. The sensitivity of the relative intensity measurement $M_\varepsilon$ does not depend on the quantum state $\hat \rho_0$, but strongly depends on the measurement basis $\{f_m(\vec r) \}$. As we already mentioned, it equals zero for bucket detection in the principal mode. Here, we consider two more measurement bases.
\subsection{Direct imaging}
For DI the intensity distribution in the image plane reads
\begin{multline}
I(\vec r)=\kappa N_S \big(u_0^2(\vec r-\vec r_1) \cos^2 \theta+ u_0^2(\vec r-\vec r_2) \sin^2\theta+ \\ + u_0(\vec r-\vec r_1)u_0(\vec r-\vec r_2) \sin 2\theta \cos \phi \big).
\end{multline}
Then the sensitivity of a relative intensity measurement in the continuous limit can be calculated as
\begin{equation}
\label{eqn:MDI}
M_\varepsilon^{DI} = \int \frac{1}{i(\vec r)} \left( \frac {\partial i(\vec r)}{\partial d}\right)^2 d \vec r,
\end{equation}
where $i(\vec r)=I(\vec r)/N_D$, in which we introduce the total detected photon number
\begin{equation}
\label{eqn:ND}
N_D=\int I(\vec r)d \vec r=\kappa N_S (1 + \chi \delta).
\end{equation}
In turn, we introduce the parameters
\begin{equation}
\label{eqn:chi}
\chi=\sin 2\theta \cos \phi,
\end{equation}
and $\delta$ the overlap between the images of the sources
\begin{equation}
\delta=\int u_0(\vec r-\vec r_1)u_0(\vec r-\vec r_2) d \vec r.
\end{equation}
Expression \eqref{eqn:ND} is valid for any measurement basis in which one can decompose the image mode. Hereinafter we stick to a soft aperture model with a Gaussian PSF
\begin{equation}
u_0(\vec r)=\sqrt{\frac{1}{2 \pi \sigma^2}} \exp \left [{-\frac{|\vec r|^2}{4 \sigma^2}} \right],
\end{equation}
where $\sigma$ is the width of the PSF. This results in an overlap
\begin{equation}
\label{eqn:delta}
\delta= \exp \left[ - \frac{d^2}{8\sigma^2}\right].
\end{equation}
The sensitivity \eqref{eqn:MDI} can be calculated analytically in the cases of in-phase ($\phi=0$) and anti-phase ($\phi=\pi$) sources
\begin{equation}
\label{eqn:MDI_0pi}
M_\varepsilon^{DI}\Bigg|_{\phi=0,\pi}= \frac{1}{4\sigma^2} \left(1- \chi \delta + \frac{d^2}{4\sigma^2} \frac{\chi \delta}{1+\chi \delta} \right),
\end{equation}
and in the case of fully asymmetric sources ($\theta=0$, which is equivalent to a single source's centroid estimation) giving the well known result
\begin{equation}
\label{eqn:MDI_0theta}
M_\varepsilon^{DI}\Bigg|_{\theta=0}=\frac{1}{4\sigma^2}.
\end{equation}
For this case $\chi=0$, meaning that eqn. \eqref{eqn:MDI_0pi} is also valid for $\theta=0$. For other values of $\phi$ and $\theta$ the sensitivity $M_\varepsilon^{DI}$ is calculated numerically.
The variable $M_\epsilon$ corresponds to the sensitivity per detected photon. The sensitivity per emitted photon is instead given by $N_D M_\epsilon / N_S$. We additionally normalise this value over transmissivity $\kappa$ and multiply by $4\sigma^2$ to remove dependence on these parameters. A plot of the resulting normalised sensitivity
\begin{equation}
\label{eqn:Mepsilon_N}
\mathcal{M}_\varepsilon= \frac{4\sigma^2}{\kappa}\frac{N_D M_\varepsilon}{N_S}
\end{equation}
for the case of DI is presented on fig. \ref{fig:plotDI}.
\begin{figure}[ht]
\begin{tabular}{c}
\includegraphics[width=0.65 \linewidth]{pics/plotDI1.pdf} \\
\includegraphics[width=0.65 \linewidth]{pics/plotDI2.pdf}
\end{tabular}
\begin{tabular}{c}
\includegraphics[width=0.292 \linewidth]{pics/legDI.pdf}
\end{tabular}
\caption{\label{fig:plotDI} Normalised sensitivity per emitted photon of relative intensity distribution direct measurement $\mathcal{M}_\varepsilon^{DI}$. Top panel corresponds to the sources with equal intensity and parameter $\chi=\{1, 1/2, 0,-1/2, -1, -0.99\}$, bottom panel -- to asymmetric case and $\chi=\{0.87, 0.43, 0, -0.43, -0.87, -0.86\}$.}
\end{figure}
On the top inset of fig.~\ref{fig:plotDI} one can see that direct measurement of the relative intensity distribution in the case of equally bright sources leads to low sensitivity for small separation and, as expected, to ``Raleigh's curse'', i.e., vanishingly small sensitivity for infinitely small $d$. In case of asymmetric sources (bottom of fig.~\ref{fig:plotDI}) Raleigh's curse does not occur for direct imaging, although the sensitivity is quite small for small separation, unless the sources are nearly in anti-phase.
\subsection{Spatial demultiplexing}
Another imaging technique under study is SPADE. It was shown that the sensitivity of photon counting in HG modes saturates QFI for the estimation of two equally-bright incoherent thermal sources' separation in case of Gaussian PSF \cite{sorelli2021PRA}. This measurement also beats DI in the asymmetric case (unequally bright sources), though the optimality of the HG basis was never proved for this case.
Here, we analyse the sensitivity of photon counting in HG modes in the case of fully coherent bright sources in an arbitrary quantum state. We assume to know all the sources' parameters except the separation (in \eqref{eqn:Md} and \eqref{eqn:Mmatrix2}), and the total brightness $N_S$ (only in \eqref{eqn:Mmatrix2}). This implies that the position of the centroid $(\vec r_1 + \vec r_2)/2$
and the orientation of the pair of sources are assumed to be known prior to the measurement. Often these parameters, if unknown, can be estimated with an additional preparatory DI measurement. However, one should keep in mind that such an estimation has a finite precision, and, for an asymmetric source, it is correlated with the estimation of the separation itself. Under these assumptions, the sources' positions equal to $\vec r_{1,2}=\{\pm d/2,0 \}$. And the measurement basis can be chosen aligned with the image centroid, i.e. the measurement HG mode basis reads
\begin{equation}
f_m(x,y)= \frac{1}{\sqrt{2^m m!}} H_m \left( \frac{x}{\sqrt{2} \sigma} \right) u_0 \left(\sqrt{x^2+y^2}\right),
\end{equation}
where $H_m$ are Hermite polynomials. Calculating the overlaps with the image modes we find that the coefficients $A_m$ \eqref{eqn:Am} are given by
\begin{equation}
\label{eqn:Am_SPADE}
A^{HG}_m= \sqrt \kappa \Big((-1)^m \cos \theta + e^{i \phi} \sin \theta \Big) ~\beta_m \left( \frac{d}{4 \sigma} \right),
\end{equation}
where
\begin{equation}
\label{eqn:beta_HG}
\beta_m (x_0)
e^{-\frac{x_0^2}{2}} \frac{x_0^m}{\sqrt{m!}}.
\end{equation}
This allows to find the mean photon numbers in the measurement modes \eqref{eqn:Nm}
\begin{equation}
N^{HG}_m=N_S\Big(1+(-1)^m \chi\Big) ~\beta_m^2 \left( \frac{d}{4\sigma} \right).
\end{equation}
These can be normalised with respect to total number of detected photons $N_D$ \eqref{eqn:ND} to calculate the sensitivity \eqref{eqn:Mepsilon} of separation estimation from measured relative photon numbers $\varepsilon_m$. In case of infinitely many HG modes being measured this sensitivity reads
\begin{equation}
\label{eqn:MHG}
M_\varepsilon^{HG}=\frac{1}{4\sigma^2} \left(1- \chi \delta + \frac{d^2}{4\sigma^2} \frac{\chi \delta}{1+\chi \delta} \right).
\end{equation}
Note, that $M_\varepsilon^{HG}$ depends only on the combination $\chi$ of the parameters $\theta$ and $\phi$.
One can notice that the expression for SPADE's sensitivity $M_\varepsilon^{HG}$ \eqref{eqn:MHG} coincides with the sensitivity of DI $M_\varepsilon^{DI}$ in cases of in-phase, anti-phase \eqref{eqn:MDI_0pi}, and fully asymmetric sources \eqref{eqn:MDI_0theta}. It is possible to show, that in the general case
\begin{equation}
\label{eqn:theorem}
M_\varepsilon^{HG} \ge M_\varepsilon^{DI},
\end{equation}
and the inequality is saturated only if $\phi=0$ or $\pi$ or $\theta=0$ (see Appendix \ref{app:HGvsDI} for the proof). Meaning that measurements in the HG basis are always more (or equally) sensitive then DI for separation estimation. However,in the asymmetric case ($\theta \ne \pi/4$), we cannot prove the optimality of this measurement, since the QFI is unknown.
In fig. \ref{fig:plotHG}, we plot the normalised sensitivity \eqref{eqn:Mepsilon_N} of the relative intensity measurement in the HG modes. One can see that Raleigh curse is still present for the symmetric in-phase ($\chi=1$) and anti-phase ($\chi=-1$) cases, which is not surprising, since SPADE for this cases is as sensitive as DI. Note, however, that even a small deviation from the symmetric anti-phase case ($\chi=-1$) leads to a significant sensitivity increase for small separations (see. dashed line on fig. \eqref{fig:plotHG}).
\begin{figure}[ht]
\includegraphics[width=0.65 \linewidth]{pics/plotHG.pdf} \includegraphics[width=0.292 \linewidth]{pics/legHG.pdf}
\caption{\label{fig:plotHG} Normalised sensitivity per emitted photon of relative intensity measurement in HG modes $\mathcal{M}_\varepsilon^{HG}$.}
\end{figure}
\begin{figure*}[ht]
\includegraphics[width=0.28 \linewidth]{pics/plotMD1.pdf}
\includegraphics[width=0.28 \linewidth]{pics/plotMD2.pdf}
\includegraphics[width=0.28 \linewidth]{pics/plotMD3.pdf}
~~\includegraphics[width=0.13 \linewidth]{pics/legHG.pdf}
\caption{\label{fig:plotMD} Normalised sensitivity of total photon number detection $\mathcal{M}_D$. From left to right: Fock ($\kappa=0.2$), coherent and thermal state ($N_S=1.5/\kappa$) of the mode $\hat s_0$.}
\end{figure*}
The sensitivity in the case $\chi=0$ (that can correspond to the mutual phase $\phi=\pi/2$) does not depend on the separation $d$ and coincide with sensitivity and QFI in case of weak uncorrelated thermal sources \cite{pirandola,sorelli2021PRL}. However for incoherent thermal sources the QFI per emitted photon drops with growing number of photons, when $\mathcal{M}_\varepsilon^{HG}$ for correlated sources does not depend on $N_S$ for any photon statistics of the source.
\section{Sensitivity of total intensity measurement $M_D$}
Having an expression for the total number of detected photons \eqref{eqn:ND} one can analytically calculate the total photon-number sensitivity \eqref{eqn:MND}
\begin{equation}
\label{eqn:MD_explicit}
M_D=\frac{\kappa N_S}{4 \sigma^2} \frac{\delta^2 \chi^2}{(1+\delta \chi)+h \kappa N_S(1+\delta \chi)^2} \left(\frac{d}{2\sigma} \right)^2.
\end{equation}
This expression is valid for any detection basis in which the image mode can be decomposed, i.e. for both DI and SPADE. It also includes all quantum states of the sources (as long as the sources are mutually coherent) via the coefficient $h=g^{(2)}-1=(\Delta N_S^2 - N_S)/N_S^2$. From \eqref{eqn:MD_explicit} it is obvious, that anti-bunched states of $\hat s_0$ ($h<0$), leading to entanglement in modes $\hat s_{1,2}$, provide a better sensitivity then bunched states ($h>0$) of $\hat s_0$, which corresponds to classical correlations in $\hat s_{1,2}$. This is a natural result, since a lower photon number variance in $\hat s_0$ leads to a smaller variance of $N_D$ and hence a higher sensitivity of the $N_D$ measurement.
Here we consider the sensitivity of separation estimation from a measured total intensity $N_D$ for different quantum statistics of the sources. We are interested in the normalised sensitivity per emitted photon
\begin{equation}
\label{eqn:MD_N}
\mathcal{M}_D=\frac{4\sigma^2}{\kappa}\frac{M_D}{N_S}.
\end{equation}
The characteristics of the source statistics only appear in the combination $h \kappa N_S$, furthermore $\mathcal{M}_D$ also depend on $\chi$ and the separation $d$.
We explore the impact of the source statistics by studying various common initial states.
\paragraph{Fock state.}
We start by considering the most sensitive case, when the mode $\hat s_0$ is maximally anti-bunched, i.e. the Fock state. In this case $h=-1/N_S$, and the combination $h \kappa N_S=-\kappa$. On the left panel of fig. \ref{fig:plotMD}, we plot the sensitivity $\mathcal{M}_D$ \eqref{eqn:MD_N} with $\kappa=0.2$ (used model of linear losses requires $\kappa \ll 1$). Note that, for the Fock state, the sensitivity per emitted photon does not depend on the number of photons $N_S$.
\paragraph{Poisson statistics.}
If the mode $\hat s_0$ is in the coherent state, i.e. has Poisson statistics, then the state of the modes $\hat s_{1,2}$ is a direct product of coherent states. For this case the parameter $h=0$, and the sensitivity per emitted photon also does not depend on the source intensity. The normalised sensitivity for this case is plotted on the middle inset of fig. \ref{fig:plotMD}.
\paragraph{Thermal statistics.}
Finally we consider a thermal state of the mode $\hat s_0$, that leads to correlated thermal states in modes $\hat s_{1,2}$. For thermal statistics $h=1$. In the small photon number limit ($N_S \to 0$) the sensitivity per emitted photon $\mathcal{M}_D$ coincides with the coherent case, and for high photon number ($N_S \to \infty$) it vanishes ($\mathcal{M}_D \to 0$). We plot the normalised total photon-number sensitivity for correlated thermal sources for $\kappa N_S=1.5$ on the right inset of the fig. \ref{fig:plotMD}.
For all the considered cases the total photon number sensitivity is high in the case of small separation between symmetric anti-phase sources ($\chi=-1$). This occurs due to destructive interference of mutually coherent anti-phase sources, which leads to zero intensity in the image plane if equally bright sources coincide, and non-zero total intensity in presence of finite separation between the sources. For any other case sensitivity $M_D$ vanishes for zero separation. In the case of $\chi=0$ the total photon number $N_D$ \eqref{eqn:ND} does not depend on the parameter and $M_D=0$.
\begin{figure*}[ht]
\includegraphics[width=0.28 \linewidth]{pics/plotM1.pdf}
\includegraphics[width=0.28 \linewidth]{pics/plotM2.pdf}
\includegraphics[width=0.28 \linewidth]{pics/plotM3.pdf}
~~\includegraphics[width=0.13 \linewidth]{pics/legHGQFI.pdf}
\caption{\label{fig:plotMd_Final} Normalised total separation estimation sensitivity $\mathcal{M}^{HG}_d$ via SPADE in HG basis. From left to right: Fock ($\kappa=0.2$), coherent and thermal state ($N_S=1.5/\kappa$) of the mode $\hat s_0$. Cases with known QFI are highlighted with gray, QFI and $M_d^{HG}$ coincide for all of them.}
\end{figure*}
\section{Comparison with Quantum Fisher information}
Here we analyze the total separation estimation sensitivity $M_d=N_D M_\varepsilon + M_D$ and it's normalized version
\begin{equation}
\label{eqn:Md_N}
\mathcal{M}_d= \frac{4\sigma^2}{\kappa}\frac{ M_d}{N_S}=\mathcal{M}_\varepsilon+\mathcal{M}_D.
\end{equation}
We compare the sensitivity of SPADE with the ultimate limit set by the QFI in those cases where the QFI is explicitly known.
\paragraph{Fock state.} The first example we consider is a Fock state of the mode $\hat s_0$. Splitting it on the beamsplitter result in entanglement of modes $\hat s_{1,2}$. Particularly for the symmetric beamsplitter and mutual phase $\phi=0,\pi$ state of the sources takes form
\begin{equation}
\label{eqn:entangled_states}
| \psi \rangle _{s_1 s_2} = \frac{1}{\sqrt{2^{N_S}}} \sum_{j=0}^{N_S} {(\pm 1)^{N_S-j}} | j \rangle_{s_1} | N_S-j \rangle_{s_2}.
\end{equation}
One of these states corresponds to the maximal QFI of separation estimation (which one depends on the transmissivity $\kappa$ and separation $d$) \cite{pirandola}. The analytical expression obtained for the SPADE sensitivity $M_d^{HG}$ coincides with the QFI for the states \eqref{eqn:entangled_states}. For other values of mutual phase $\phi$, or asymmetrically split Fock states, the QFI is not calculated explicitly to the best of our knowledge.
Plots of the sensitivity of SPADE and the QFI for split Fock states are presented on the left inset of fig. \ref{fig:plotMd_Final}.
\paragraph{Poisson statistics.} Since for the Poisson photon number statistics different detection events are independent from each other, this case can be considered in the small photon number limit without loosing any generality and then generalised to higher photon numbers. The QFI for an arbitrary mutual coherence $\gamma$ was explicitly calculated in the ref. \cite{kurdzialek2021sources} in the single-photon subspace, and its analytical expression for any $\gamma=e^{i \phi}$ fully coincides with the sensitivity $M_d$ calculated for Poisson statistics ($h=0$). The dependence of normalised QFI and $\mathcal{M}_d^{HG}$ on the separation are presented in the middle inset of fig. \ref{fig:plotMd_Final}, coinciding with each other. We also plot sensitivity of DI for symmetric sources with Poisson statistics on the fig. \ref{fig:plotMd_DI_Final}
\begin{figure}[h]
\includegraphics[width=0.65 \linewidth]{pics/plotM4DI.pdf}
~~\includegraphics[width=0.27 \linewidth]{pics/legHG.pdf}
\caption{\label{fig:plotMd_DI_Final} Total separation estimation sensitivity of DI $M_d^{DI}$ for equally bright sources with Poisson statistics.}
\end{figure}
\paragraph{Thermal statistics.} Another example we consider is that of correlated thermal sources. The QFI for arbitrarily bright correlated thermal sources that are in-phase or anti-phase is calculated in ref. \cite{pirandola}. It coincides with the sensitivity $M_d^{HG}$ introduced here. With increasing intensity of the correlated thermal source, the sensitivity per photon $\mathcal{M}_d$ drops tending to $\mathcal{M}_\varepsilon$ (fig. \ref{fig:plotHG}).
One can notice, that distinctive behaviour of $\mathcal{M}^{HG}_\epsilon$ and $\mathcal{M}^{HG}_D$ for $\chi=0.99$ cancel each other after summation for any source statistics, and the resulting sensitivity of SPADE $\mathcal{M}^{HG}_d$ is continuous over change of $\chi$, which is not the case for DI.
Fig. \ref{fig:plotMd_Final} also shows, that source statistics does not influence the sensitivity dramatically if the sources are mutually coherent. At the same time comparing fig. \ref{fig:plotMd_Final} and fig. \ref{fig:plotMd_DI_Final} one can see that choice of the measurement basis, i.e. HG modes (SPADE) or pixel modes (DI), result in sufficient difference in sensitivity, unless sources are in-phase or anti-phase.
\section{Conclusion}
We presented a general approach to analyse parameter estimation problems based on photon-counting in mutually coherent modes. The considered sensitivity measure based on the method of moments showed to be a very efficient and practical tool for analyzing this class of problems.
Specifically we have considered in details the problem of separation estimation of two mutually coherent sources. By calculating the moment-based sensitivity, we analytically showed an advantage of the spatial mode demultiplexing measurement over the direct imaging for the considered class of states. We showed that sensitivity of spatial mode demultiplexing saturates the QFI for the cases where it is known, including examples of non-Gaussian entangled states.
Obtained sensitivity expression consists of two terms, that are corresponding to the measurement of the relative photon numbers and the total number of photons. The first one strongly depends on the measurement basis, the second one on quantum statistics of the sources. Moreover, the second term vanishes in case of unknown brightness of the sources, wiping out any profit from the sources anti-bunching. The sensitivity from the total photon number measurement is also negligible for intense bunched states, due to the high noise in the total number of photons.
The moment-based sensitivity approach and results obtained in this research can be applied to other parameter estimation problems with mutually coherent or single-mode sources, like coherent imaging or distributed quantum sensing with mutually coherent probes.
\begin{acknowledgments}
I.K. acknowledges the support from the PAUSE National programme. M.G. acknowledges funding by the LabEx ENS-ICFP:ANR-10-LABX-0010/ANR-10- IDEX-0001-02 PSL*. This work was partially funded by French ANR under COSMIC project (ANR-19-ASTR- 0020-01). This work received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 899587. This work was supported by the European Union’s Horizon 2020 research and innovation programme under the QuantERA programme through the project ApresSF.
\end{acknowledgments}
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\section{Introduction}
Let $F_N=F(A)$ be a free group of finite rank $N\ge 2$, with a fixed free basis $A=\{a_1,\dots, a_N\}$. The \emph{automorphism problem} for $F_N$ asks, given two freely reduced words $w,w'\in F_N=F(A)$, whether there there exists $\varphi\in\mbox{Aut}(F_N)$ such that $w'=\varphi(w)$, that is, whether $\mbox{Aut}(F_N)w=\mbox{Aut}(F_N)w'$. A complete algorithmic solution to this problem was provided in 1936 classic paper of Whitehead~\cite{W36}, via the procedure that came to be called \emph{Whitehead's algorithm}. We briefly recall how this algorithm works, and refer the reader to Section~\ref{s:wh} below.
For an element $g\in F_N$, we denote by $|g|_A$ and by $||g||_A$ the freely reduced length and the cyclically reduced length of $g$ with respect to $A$ accordingly. We also denote by $[g]$ the conjugacy class of $g$ in $F_N$. For $w,w'\in F_N$ we have $\mbox{Aut}(F_N)w=\mbox{Aut}(F_N)w'$ if and only if $\mbox{Out}(F_N)[w]=\mbox{Out}(F_N)[w']$. For that reason we usually think of the automorphism problem in $F_N$ in this latter form, as the question about $[w],[w']$ being in the same $\mbox{Out}(F_N)$-orbit. We denote $\mathcal C_N=\{[g]|g\in F_N\}$.
The group $\mbox{Aut}(F_N)$ has a particularly nice finite generating set $\mathcal W_N$ of so-called \emph{Whitehead automorphisms} or \emph{Whitehead moves} (we use the same terminology for the images of Whitehead automorphisms in $\mbox{Out}(F_N)$); see Definition~\ref{defn:moves} below. Whitehead moves are divided into two types: Whitehead moves $\tau$ of the \emph{first kind} have the form $a_i\mapsto a_{\sigma(i)}^{\pm 1}$ for some permutation $\sigma\in S_n$. They have the property that for each $w\in F_N$ $||\tau(w)||_A=||w||_A$. Whitehead moves \emph{of the second kind} can change the cyclically reduced length of an element of $F_N$.
An element $[g]\in\mathcal C_N$ is called \emph{$\mbox{Out}(F_N)$-minimal} if for every $\varphi\in\mbox{Out}(F_N)$ we have $||g||_A\le ||\varphi(g)||_A$. For $[g]\in \mathcal C_N$ we denote by $\mathcal M([g])$ the set of all $\mbox{Out}(F_N)$-minimal elements in the orbit $\mbox{Out}(F_N)[g]$.
An element $[g]\in\mathcal C_N$ is called \emph{Whitehead-minimal} if for every Whitehead move $\tau\in\mathcal W_N$ we have $||g||_A\le ||\tau(g)||_A$. Whitehead's ``peak reduction lemma" implies the following two key facts: If $[w]$ is not $\mbox{Out}(F_N)$-minimal, then there exists $\tau\in\mathcal W_N$ such that $||\tau(w)||_A<||w||_A$. This fact already has the following important implication: an element $[w]\in \mathcal C_N$ is $\mbox{Out}(F_N)$-minimal if and only if $[w]$ is Whitehead minimal.
The second fact says that for two $\mbox{Out}(F_N)$-minimal $[u],[u']\in \mathcal C_N$ we have $\mbox{Out}(F_N)[u]=\mbox{Out}(F_N)[u']$ if and only if $||w||_A=||w'||_A$ and there exists a finite \emph{length-stable} chain of Whitehead moves $\tau_1\dots, \tau_k\in \mathcal W_N$ (with $k\ge 0$) such that $\tau_k\dots \tau_1[u]=[u']$ and that $||\tau_i\dots \tau_1[u]||_A=||u||_A$ for all $i\le k$. Whitehead's algorithm on the input $([w],[w'])$ consists of two parts. The first one, the \emph{Whitehead minimization algorithm}, starting from $[w]\in \mathcal C_N$ consists of iteratively looking for a Whitehead move that decreases the cyclically reduced length of an element. Once we have arrived at $[u]$ where no such moves are available, we know that $[u]$ is a Whitehead-minimal and hence $\mbox{Out}(F_N)$-minimal element of the orbit $\mbox{Out}(F_N)[w]$. Since the set $\mathcal W_N$ is finite and fixed, this process runs in at most quadratic time in terms of $||w||_A$. Also, do the same thing to $[w']$ to produce an $\mbox{Out}(F_N)$-minimal element $[u']$ of the orbit $\mbox{Out}(F_N)[w]$. If $||u||_A\ne ||u'||_A$, then $\mbox{Out}(F_N)[w]\ne \mbox{Out}(F_N)[w']$ and we are done. The second, hard, part of Whitehead's algorithm, that we call \emph{Whitehead's stabilization algorithm}, deals with the case where $||u||_A=||u'||_A=n\ge 1$. In this case one looks for a length-stable chain $\tau_1\dots, \tau_k\in \mathcal W_N$ of Whitehead's move which satisfies $\tau_k\dots \tau_1[u]=[u']$. Since the ball of radius $n$ in $F_N(A)$ has exponential size in $n$, this second process has a priori exponential in $n$ time complexity. Although a few incremental improvements have been obtained over the years (e.g. see \cite{Ci07,Kh04,Lee1,Lee,MH06,MS03,RVW}), the questions about the computational complexity of the automorphism problem in $F_N$ and about the actual worst-case complexity of Whitehead's algorithm remain wide open and the exponential time bound is the best one known in general. The only exception is the case of rank $N=2$ where it is known that Whitehead's algorithm works in polynomial (in fact, quadratic) time~\cite{MS03,Kh04}. For the general case $N\ge 2$, the best known partial results are due to Donghi Lee~\cite{Lee1,Lee}, who proved that Whitehead's algorithm terminates on $w\in F_N$ in polynomial time (with degree of the polynomial depending on $N$), if some $\mbox{Out}(F_N)$-minimal element $[u]\in \mathcal M([w])$ satisfies a certain technical condition.
In \cite{KSS06} Kapovich, Schupp and Shpilrain initiated a probabilistic study of Whitehead's algorithm, that is, its behavior on ``random" or ``generic" inputs in $F_N$. In that paper ``generic" meant for a large $n\ge 1$, either choosing a uniformly at random freely reduced word of length $n$ in $F(A)$, or taking a a uniformly at random cyclically reduced word of length $n$ in $F(A)$. It turned out that on such ``generic" input both parts of Whitehead's algorithm work very fast. As defined in \cite{KSS06}, an element $[w]\in \mathcal C_N$ is called \emph{strictly minimal} if for every non-inner $\tau\in \mathcal W_N$ of the second kind we have $||w||_A<||\tau(w)||_A$. Thus, in particular, a strictly minimal element is Whitehead-minimal and therefore $\mbox{Out}(F_N)$-minimal. Thus the Whitehead minimization algorithm on $[w]$ terminates in a single step with $[u]=[w]$, and takes linear time in $||w||_A$. Also, in this case (for $[w]$ strictly minimal), if $[u']$ is another $\mbox{Out}(F_N)$-minimal element with $||u'||_A=||u||_A=n$ and with $\mbox{Out}(F_N)[u]=\mbox{Out}(F_N)[u']$ any length-stable chain $\tau_1,\dots,\tau_k$ connecting $[w]=[u]$ to $[u']$ consists only of inner automorphisms and Whitehead moves of the first kind. By composing them we see that $\mbox{Out}(F_N)[u]=\mbox{Out}(F_N)[u']$ if and only if there exists $\tau\in \mathcal W_N$ of the second kind such that $\tau[u]=[u']$. Thus in this case the Whitehead stabilization algorithm also terminates in linear time in $n=||w||_A$. The overall complexity
of Whitehead's algorithm on the input $[w],[w']$, where $[w]$ is strictly minimal, is $O(\max\{||w||_A,||w'||_A^2\})$. A key probabilistic result of \cite{KSS06} says that a ``generic" (in the above basic sense of taking a
uniformly random freely reduced or cyclically reduced word of length $n$) element $[w]\in \mathcal C_N$ is strictly minimal. Therefore, if both $[w],[w']$ are ``generic" in this sense, Whitehead's algorithm on input $[w],[w']$ runs in $O(\max\{||w||_A,||w'||_A\})$ time; and if $[w]$ is generic and $[w']$ is arbitrary, it runs in $O(\max\{||w||_A,||w'||_A^2\})$ time. The results of \cite{KSS06} were generalized in \cite{RVW} for the version of Whitehead's algorithm for $\mbox{Out}(F_N)$-orbits of conjugacy classes of finitely generated subgroups of $F_N$.
The proof in \cite{KSS06} that ``generic" $[w]$ in $F_N$ is strictly minimal crucially relied on the fact that for such $[w]$ the weights (normalized by $||w||_A$) on edges in the Whitehead graph of $w$ are close to being uniform. Roughly, that means that frequencies of 1-letter and 2-letter subwords in $[w]$ are close to being uniform (e.g. that for $i=1,\dots, N$ the frequency of each $a_i^{\pm 1}$ in $[w]$ is close to $\frac{1}{N}$). This close-to-uniform property of frequencies no longer holds if $[w]$ generated by other random processes.
\begin{ex}\label{ex:1}
For example, consider the case $N=2$ and $F_2=F(a,b)$. Let $w_n$ be a positive word in $\{a,b\}$ of length $n$, where every letter is chosen independently, with probability $p(a)=1/10$ and $p(b)=9/10$. Then the frequency of $a$ in a "random" $w_n$ will tend to $1/10$ as $n\to\infty$. Moreover, it is not hard to see that $w_n$ will not be strictly minimal. Here is an informal argument. In this case $w_n$ will contain $\frac{81}{1000} n+o(n)$ occurrences of $ab^2$, as well as $\frac{1}{1000} n+o(n)$ occurrences of $a^3$ and $\frac{9}{1000} n+o(n)$ occurrences of $aba$. Consider the Whitehead move $\tau(a)=ab^{-1}, \tau(b)=b$. Note that $\tau(ab^2)=ab$. The portion of $w_n$ covered by the $\frac{81}{1000} n+o(n)$ occurrences of $ab^2$ has total length $\frac{243}{1000} n+o(n)$ but its image under $\tau$ has total length $\frac{162}{1000} n+o(n)$. Since $\tau(aba)=aab^{-1}$, the image of the portion of $w_n$ covered by occurrences of $aba$ in $w_n$ does not change in length in $\tau(w_n)$. The portion of $w_n$ covered by the $\frac{1}{1000} n+o(n)$ occurrences of $a^3$ has total length $\frac{3}{1000} n+o(n)$, and its image in $\tau(w_n)$ has total length $\frac{6}{1000} n+o(n)$ there. One can conclude from here that $||w_n||_A-||\tau(w_n)||_A\ge \frac{78}{1000} n+o(n)$. Thus $||\tau(w_n)||_A<||w_n||_A$ and, moreover (since $||w_n||_A=n$), $\frac{||\tau(w_n)||_A}{||w_n||_A}\le \frac{922}{1000}+o(1)$. Hence $w_n$ badly fails to be strictly minimal.
\end{ex}
In the present paper we consider the generic-case behavior of Whitehead's algorithm on random inputs for much more general types of random processes than in the \cite{KSS06} setting (in particular, including Example~\ref{ex:1}). Our results have some similarities to the results from \cite{KSS06} but, of course, with important differences that are inherently necessary, as demonstrated by Example~\ref{ex:1}.
The main notions replacing strict minimality are the notions of an \emph{$(M,\lambda,\varepsilon)$-minimal element} $[u]\in \mathcal C_N$ and of \emph{$(M,\lambda,\varepsilon, \mathcal W_N)$-minimal element} $[u]\in \mathcal C_N$ (where $M\ge 1$, $\lambda>1$ and $0<\varepsilon<1$); see Definition~\ref{d:MLE} and Definition~\ref{d:MLEW} below. Roughly, $[u]$ being $(M,\lambda,\varepsilon)$-minimal means that $[u]$ belongs to a subset $S\subseteq \mbox{Out}(F_N)[u]$ of cardinality at most $M$ such that for each element $[u']$ of $S$ an arbitrary $\varphi\in\mbox{Out}(F_N)$ either increases the length $||u'||_A$ by a factor of at least $\lambda$, or distorts the length multiplicatively by a factor $\varepsilon$-close to $1$. For $[u]$ being $(M,\lambda,\varepsilon, \mathcal W_N)$-minimal the definition is similar, but instead of arbitrary $\varphi\in\mbox{Out}(F_N)$ we only require these conditions to hold for arbitrary $\tau\in\mathcal W_N$. From the definitions we see that being $(M,\lambda,\varepsilon)$-minimal directly implies being $(M,\lambda,\varepsilon, \mathcal W_N)$-minimal. The converse is almost true: given $\lambda,\varepsilon$, for all "sufficiently stringent" $\lambda'>\lambda$ and $0<\varepsilon'<\varepsilon$ being $(M,\lambda',\varepsilon', \mathcal W_N)$-minimal implies being $(M,\lambda,\varepsilon)$-minimal. See Proposition~\ref{prop:cont} below for a precise statement. For fixed $M,\lambda,\varepsilon$, deciding if an element $[u]\in\mathcal C_N$ is $(M,\lambda',\varepsilon', \mathcal W_N)$-minimal can be done in linear time in $||u||_A$, while the algorithm for deciding if $[u]$ is $(M,\lambda,\varepsilon)$-minimal has a priori exponential time complexity. See Section~\ref{s:alg} below for details.
We summarize the main results of the present paper:
$\bullet$ We show in Theorem~\ref{t:WHM} that Whitehead's algorithm on an input $[w],[w']$ works fast if at least one of the inputs is $(M,\lambda,\varepsilon)$-minimal: Whitehead minimization algorithm works in linear time on any $(M,\lambda,\varepsilon)$-minimal element $[w]$; if both $[w],[w']$ are $(M,\lambda,\varepsilon)$-minimal, the full Whitehead algorithm the input $([w],[w'])$ works in linear time; if $[w]$ is $(M,\lambda,\varepsilon)$-minimal and $[w']$ is arbitrary, the full Whitehead algorithm the input $([w],[w'])$ works in time $O( ||w||_A, ||w'||_A^2)$. Also, for an $(M,\lambda,\varepsilon)$-minimal $[w]$ the stabilizer $Stab_{\mbox{Out}(F_N)}([w])$ has uniformly bounded rank rank (the smallest cardinality of a generating set).
$\bullet$ We exhibit a rich source of $(M,\lambda,\varepsilon)$-minimal elements. We prove that if $\nu\in\Curr(\FN)$ is a ``filling'' geodesic current then there exist $M\ge 1,\lambda>1, \varepsilon>0$, a neighborhood $U$ of $[\nu]$ in $\mathbb P\Curr(\FN)$, and a finite set $\mathfrak W\subseteq \mbox{Out}(F_N)$ of $\le M$ ``shortening" automorphisms with the following property: For every $[w]$ which belongs in $U$ (when $[w]$ is viewed as a projective current), and every $\varpsi\in\mathfrak W$, the element $[\varpsi(w)]$ is $(M,\lambda,\varepsilon)$-minimal, and, moreover $\mathcal M([w])\subseteq \mathfrak W([w])$. We also produce several sources of ``filling" currents.
$\bullet$ We define the notion of an $F_N$-valued random process $\mathcal W=W_1, W_2,\dots $ being \emph{adapted} to a current $0\ne \nu\in \Curr(\FN)$. We prove that if $\mathcal W$ is adapted to a filling current $\nu$ then Whitehead's algorithm has low complexity when one or both of $[w],[w']$ are ``randomly'' generated by $\mathcal W$. These conclusions, obtained in Theorem~\ref{t:A} and Theorem~\ref{t:A'}, are the main genericity results of this paper.
$\bullet$ We show, in Theorem~\ref{t:rwa}, that for a large class of ``group random walks'' $\mathcal W=W_1, W_2,\dots $ on $F_N$, the walk is adapted to some filling current $[\nu]$ and hence Theorem~\ref{t:A} and Theorem~\ref{t:A'} apply.
$\bullet$ We also show that for a large class of ``graph non-backtracking random walks" $\mathcal W=W_1, W_2,\dots $ on $F_N$, the walk is adapted to some filling current $[\nu]$ (see Theorem~\ref{t:cl}, Theorem~\ref{t:cla} and Proposition~\ref{p:XF}) and hence again Theorem~\ref{t:A} and Theorem~\ref{t:A'} apply.
In \cite{KSS06} the probabilistic results about Whitehead's algorithm are stated in terms of \emph{generic-case complexity}. This notion, introduced in \cite{KMSS03}, is designed to capture practically observable (as distinct from worst-case and average-case) behavior of various algorithms. In \cite{KMSS03} generic-case complexity in the $F_N=F(A)$ context is defined via asymptotic density, that is, essentially, the uniform probability measure on large spheres or balls in $F(A)$; the same definition is still used in \cite{KSS06}. Since then the notion of generic-case complexity has been significantly expanded and generalized, to allow for more general and more natural models of random generation of inputs; see~\cite{MSU} for some background and further details. All the probabilistic complexity results about Whitehead's algorithm obtained in this paper are, in fact, generic-case complexity results. However, to be precise, we state all these results exactly, precisely and explicitly (including quantification of various constants) in terms of the random processes involved, rather than using the language of generic-case complexity. Note that the case of a simple non-backtracking random walk on $F(A)$, which was the context of the results in \cite{KSS06}, is a very special case of the random process $\breve {\mathcal W}$ considered Theorem~\ref{t:cla}.
We expect the results of Section~\ref{s:MLE} about $(M,\lambda,\varepsilon)$-minimal elements to be of independent interest, apart from any probabilistic applications.
Geodesic currents provide a measure-theoretic generalization of the notion of a conjugacy class. Geodesic currents, originally introduced by Bonahon~\cite{Bo88} in the context of hyperbolic surfaces, proved particularly useful in recent years in the study of $\mbox{Out}(F_N)$ and of the Culler-Vogtmann Outer space, see e.g. \cite{CP12,Ha14,BR15,Hor16}. A key tool in the theory is the notion of a \emph{geometric intersection form} between currents and points of the Thurston-like closure of the Outer space. The intersection form was developed by Kapovich and Lustig~\cite{KL09,KL10}. The connection between currents and generic-case complexity was first pointed out in our article \cite{Ka07}, but this connection is explored in detail for the first time in the present paper. In particular, the intersection form defines, for every $\nu\in\Curr(\FN)$, the ``length'' $||\nu||_A\ge 0$ of $\nu$ with respect to $A$. For $1\ne w\in F_N$, we have $||\eta_w||_A=||w||_A$, where $\eta_w\in\Curr(\FN)$ is the ``counting'' current associated with $[w]$.
\begin{ex}[Simple non-backtracking random walk on $F_N$]
Consider the simple nonbacktracking random walk $\mathcal W=W_1, W_2,\dots, W_n,\dots $ of $F_N$ with respect to $A$, as is done in \cite{KSS06}. This means that the $W_n=X_1\dots X_n$ is a freely reduced word of length $n$ in $F(A)$, where the first letter $X_1$ is chosen uniformly at random from $A^{\pm 1}$ with probability $\frac{1}{2N}$ each; and if the $i$-th letter $X_i=a\in A^{\pm 1}$ is already chosen, the letter $X_{i+1}$ is chosen uniformly at random from $A^{\pm 1}-\{a\}$, with probability $\frac{1}{2N-1}$ for each element there. Thus $W_n$ induces the uniform probability distribution on the $n$-sphere in $F(A)$, where every element of the sphere has probability $\frac{1}{2N(2N-1)^{n-1}}$. We will explain here the properties of the walk $\mathcal W$ in the terminology of this paper, omitting the detailed justification of these properties.
For a.e. trajectory $w_1,w_2,\dots $ of the walk $\mathcal W$ we have $\lim_{n\to\infty}\frac{1}{n}\eta_{w_n}=\nu_A$ in $\Curr(\FN)$, where $\nu_A$ is the \emph{uniform current} on $F_N$ corresponding to $A$ (see Definition~\ref{d:uc}). Thus, in the language of the present paper, the walk $\mathcal W$ is adapted to $\nu_A$. Moreover, $\nu_A$ has full support in $\partial^2 F_N$ and therefore $\nu_A$ is filling, by Proposition~\ref{p:fs}. Also, we have $||\nu_A||_A=1$. Let $\Im\subseteq \mbox{Out}(F_N)$ be the set of all Whitehead automorphisms of the first kind. Put $\mathfrak W=\Im\subseteq \mbox{Out}(F_N)$. The results of \cite{Ka07,KKS07} imply that for any $\varphi\in\mbox{Out}(F_N)$, we have $\varphi(\nu_A)=\nu_A$ if $\varphi\in \Im$, and
\[
||\varphi\nu_A||_A=\frac{||\varphi(\nu_A)||_A}{||\nu_A||_A}\ge \lambda_0=1+\frac{2N-3}{2N^2-N}>1
\]
if $\varphi\not\in\Im$. This fact implies that if we choose and fix any $1<\lambda<\lambda_0$, then any $[w]$ that is sufficiently close to $[\nu_A]$ in $\mathbb P\Curr(\FN)$ is strictly minimal (in particular $||\varphi(w)||_A=||w||_A$ for every $\varphi\in \Im$), and, moreover, for any $\varphi\not\in\Im$ we have $||\varphi([w])||_A/||w||_A\ge \lambda$. Then for any sufficiently small $\varepsilon>0$, for $n\to\infty$ our ``random" $w_n$ is $(M,\lambda,\varepsilon)$-minimal with $M=\#\Im$, and the set $S_n=\Im([w_n])=\mathfrak W([w_n])$ is $(M,\lambda,\varepsilon)$-minimazing (in the sense of Definition~\ref{d:MLE}). This example is the simplest case illustrating how our definitions and results work.
\end{ex}
\begin{rem}[A note on the speed of convergence]
In \cite{KMSS03,KSS06} the main results are stated in terms of ``strong genericity", meaning that various probabilities converging to $1$ do so exponentially fast as $n\to\infty$. Parts (b) of Theorem~\ref{t:A} and Theorem~\ref{t:A'} are also stated in terms of probabilities of various events at step $n$ converging to $1$ as $n\to\infty$. We do not include the speed of convergence estimates there because for the moment our main new "group random walks" application, Theorem~\ref{t:rwa}, does not come with a speed of convergence estimate. The reason is that the proof of this theorem relies on the use of a recent result of Gekhtman~\cite[Theorem~1.5]{Ge17} about approximating harmonic measure by counting currents along a random walk on a word-hyperbolic group acting on a $CAT(-1)$ space does not have any speed of convergence estimates. We expect that Gekhtman's result actually holds in much greater generality (e.g. for an arbitrary geometric action of a nonelementary word-hyperbolic group $G$, and with much milder assumptions on the measure $\mu$ defining the walk), with exponentially fast convergence. Once that is proved, the applications of Theorem~\ref{t:A} and Theorem~\ref{t:A'} to the group random walk context can be supplied with the speed of convergence estimates. (Definition~\ref{d:adapt} of a random process adapted to a current would have to be refined to include quantificantion by the speed of convergence.) On the other hand, in the context of our results about graph-based non-backtracking random walks, namely Theorem~\ref{t:cl}, Theorem~\ref{t:cla}, one can already show that the convergence is either exponentially or slightly subexponentially fast.
\end{rem}
We are extremely grateful to Vadim Kaimanovich and Joseph Maher for many helpful discussions about random walks, for help with the references and for clarifying some random walks arguments. In particular the proof of Proposition~\ref{p:supp} was explained to us by Kaimanovich. We are also grateful to the organizers of the March 2019 Dagstuhl conference "Algorithmic Problems in Group Theory" for providing impetus and motivation for completing this paper.
\section{Whitehead's algorithm}\label{s:wh}
Our main background reference for Whitehead's algorithm Lyndon and Schupp, Chapter~I.4~\cite{LS}, and we refer the reader there for additional details.
Some other useful details and complexity results are available in \cite{KSS06,RVW}.
We recall the basic definitions and results here.
In this section we fix a free group $F_N=F(A)$ of rank $N\ge 2$, with a fixed free basis $A=\{a_1,\dots, a_N\}$. Put $\Sigma_A=A\sqcup A^{-1}$.
We will also denote by $\mathcal C_N$ the set of all $F_N$-conjugacy classes $[g]$ where $g\in F_N$.
\begin{defn}[Whitehead automorphisms]\label{defn:moves}
A \emph{Whitehead automorphism} of $F_N$ with respect to $A$ is an automorphism $\tau\in \mbox{Aut}(F_N)$ of
$F_N$ of one of the following two types:
(1) There is a permutation $t$ of $\Sigma_A$ such that
$\tau|_{\Sigma_A}=t$. In this case $\tau$ is called a \emph{relabeling
automorphism} or a \emph{Whitehead automorphism of the first
kind}.
(2) There is an element $a\in \Sigma_A$, the \emph{multiplier}, such
that for any $x\in \Sigma_A$
\[
\tau(x)\in \{x, xa, a^{-1}x, a^{-1}xa\}.
\]
In this case we say that $\tau$ is a \emph{Whitehead automorphism of
the second kind}. (Note that since $\tau$ is an automorphism of $F_N$,
we always have $\tau(a)=a$ in this case).
We also refer to the images of Whitehead automorphisms in $\mbox{Out}(F_N)$ as \emph{Whitehead moves} and sometimes again as \emph{Whitehead automorphisms}. We denote by $\mathcal W_N$ the set of all Whitehead moves $\tau\in\mbox{Out}(F_N)$ such that $\tau\ne 1$ in $\mbox{Out}(F_N)$.
\end{defn}
Note that for any $a\in \Sigma_A$ the inner automorphism $ad(a)\in\mbox{Aut}(F_N)$ is a
Whitehead automorphism of the second kind. Note also that if $\tau\in \mathcal W_N$ then $\tau^{-1}\in \mathcal W_N$.
To simplify the exposition, we formulate all the definitions and results related to Whitehead's algorithm in terms of conjugacy classes of elements of $F_N$.
In this context we usually think of an input $[w]\in \mathcal C_N$ as given by a cyclically reduced word $w\in F(A)$ and the complexity of various algorithms is estimated in terms of $||w||_A$. Since for $w\in F_N$ we have $||w||_A\le |w|_A$, and since it takes linear time in $|w|_A$ to find a cyclically reduced form of $w\in F(A)$ (see \cite{KSS06} for additional discussion on this topic), the same complexity estimates hold in terms of $|w|_A$.
\begin{defn}[Minimal and Whitehead-minimal elements]
A conjugacy class $[w]\in\mathcal C_N$ is \emph{$Out(F_N)$-minimal} with respect to $A$ if for every $\varphi\in\mbox{Out}(F_N)$ we have $||w||_A\le ||\varphi(w)||_A$.
A conjugacy class $[w]\in\mathcal C_N$ is \emph{Whitehead-minimal} with respect to $A$ if for every Whitehead move $\tau\in\mathcal W_N$ we have $||w||_A\le ||\tau(w)||_A$.
For $[w]\in \mathcal C_N$, denote $\mathcal M([w])=\{[u]\in \mbox{Out}(F_N)[w]| [u] \text{ is $Out(F_N)$-minimal}\}$.
\end{defn}
Note that, by definition, an $Out(F_N)$-minimal $[w]$ is necessarily Whitehead-minimal.
\begin{defn}[Automorphism graph]
The \emph{automorphism graph} of $F_N$ is an oriented labelled graph $\mathcal T$ defined as follows.
The vertex set $V\mathcal T$ is $\mathcal C_N$, the set of all conjugacy classes $[w]$ where $w\in F_N$.
The edges of $\mathcal T$ are defined as follows. Suppose that $[w]\ne [w']\in V\mathcal T$ are such that $||w||_A=||w'||_A=n\ge 0$. If there exists a Whitehead move $\tau\in \mathcal W_N$ such that $\tau([w])=[w']$ (and hence $\tau^{-1}[w']=[w]$, with $\tau^{-1}\in \mathcal W_N$) there is a topological edge $e$ connecting $[w]$ and $[w']$. There are two possible orientations on $e$ resulting in mutually inverse oriented edges: the edge with the orientation from $[w]$ to $[w']$ is labelled by $\tau$, and the edge $e$ with the orientation from $[w']$ to $[w]$ is labelled by $\tau^{-1}$.
Also, for $n\ge 0$ denote by $\mathcal T_n$ the subgraph of $\mathcal T$ spanned by all vertices $[w]\in V\mathcal T$ with $||w||_A=n$. For a vertex $[w]$ of $\mathcal T_n$ denote by $\mathcal T_n[w]$ the connected component of $\mathcal T_n$ containing $[w]$.
\end{defn}
We first state the following simplified version of Whitehead's ``peak reduction" lemma (see \cite[Proposition~1.2]{KSS06}):
\begin{prop}\label{p:wpr}
The following hold:
\begin{enumerate}
\item An element $[w]\in \mathcal C_N$ is $\mbox{Out}(F_N)$-minimal if and only if $[w]$ is Whitehead-minimal. (Thus if $[w]$ is not $\mbox{Out}(F_N)$-minimal then there exists $\tau\in\mathcal W_N$ such that $||\tau(w)||_A<||w||_A$).
\item Suppose that $[w]\ne [w']$ are both $\mbox{Out}(F_N)$-minimal. Then $\mbox{Out}(F_N)[w]=\mbox{Out}(F_N)[w']$ if and only if $||w||_A=||w'||_A=n\ge 0$, and there exists a finite sequence $\tau_1,\dots\tau_k\in\mathcal W_N$ such that $\tau_k\dots \tau_1[w]=[w']$ and that for $i=1,\dots, k$ we have
\[
||\tau_i\dots \tau_1(w)||_A=n.
\]
\end{enumerate}
\end{prop}
Proposition~\ref{p:wpr} implies that if $[w]\in \mathcal C_N$ is $\mbox{Out}(F_N)$-minimal with $||w||_A=n$ then $\mathcal M([w])=\mathcal T_n[w]$.
We also record the following more general version of "peak reduction":
\begin{prop}\label{p:wprs}\cite[Proposition~4.17]{LS}
Let $[w],[w']\in \mathcal C_N$ and $\varphi\in\mbox{Out}_N$ be such that $[w']=\varphi([w])$ and that $||w'||_A\le ||w||_A$.
Then there exists a factorization $\varphi=\tau_k\dots \tau_1$ in $\mbox{Out}(F_N)$, where $\tau_i\in \mathcal W$ and where $||\tau_i\dots\tau_1w||_A\le ||w||_A$ for $i=1,\dots, k$.
\end{prop}
\begin{defn}[Whitehead algorithm]
Let $F_N=F(A)$ be free of rank $N\ge 2$, with a fixed free basis $A$.
$\bullet$ The \emph{Whitehead minimization algorithm} is the following process. Given $[w]\in \mathcal C_N$ put $[w_1]=[w]$. If $[w_i]$ is already constructed, check if there exists $\tau\in \mathcal W_N$ such that $||\tau(w_i)||_A<||w_i||_A$. If not, declare that $[w_i]\in\mathcal M([w])$ (that is $[w_i]$ is an $\mbox{Out}(F_N)$-minimal element in $\mbox{Out}(F_N)[w]$ and terminate the algorithm. Put $[w_{i+1}]=[\tau(w_i)]$.
$\bullet$ The \emph{Whitehead stabillization algorithm} is the following process. Suppose that $[w]\in \mathcal C_N$ is Whitehead-minimal (and therefore $\mbox{Out}(F_N)$-minimal) with $||w||_A=n\ge 0$. Construct the component $\mathcal T_n([w])$ of $\mathcal T_n$ using the ``breadth-first" stabilization process. Start with $S_1=\{[w]\}$. Now if a finite collection $S_i$ of conjugacy classes with $||.||_A=n$ is already constructed, for each element $[u]\in S_i$ and each $\tau\in \mathcal W_N$, put
\[
S_{i+1}=S_i\cup\{\tau([u])| [u]\in S_i, \tau\in \mathcal W_N \text{ and } ||\tau(u)||_A=n\}.
\]
Terminate the process with the output $S_i$ for the smallest $i\ge 1$ such that $S_{i+1}=S_i$. Declare that $S_{i}=V\mathcal T_n([w])=\mathcal M([w])$.
$\bullet$ The \emph{Whitehead algorithm} is the following process. Given $[w], [w']\in \mathcal C_N$, first apply the Whitehead minimization process to each of $[w], [w']$ to output elements $[u], [u']$ accordingly. Declare that $[u]\in \mathcal M([w])$ and $[u']\in \mathcal M([w'])$. If $||u||_A\ne ||u'||_A$, declare that $\mbox{Out}(F_N)[w]\ne \mbox{Out}(F_N)[w']$ and terminate the process. Suppose that $||u||_A= ||u'||_A=n\ge 1$. Apply the Whitehead stabilization algorithm to $[u]$ to produce the set $S$. Declare that $S=\mathcal T_n([u])=\mathcal M([w])$. Then check wither $[u']\in S$. If $[u']\in S$, declare that $\mbox{Out}(F_N)[w]=\mbox{Out}(F_N)[w']$, and if $[u']\not\in S$, declare that $\mbox{Out}(F_N)[w]\ne\mbox{Out}(F_N)[w']$, and terminate the process.
\end{defn}
\begin{rem}\label{rem:com}
Part (1) of Proposition~\ref{p:wpr} implies that the Whitehead minimization algorithm on an input $[w]\in \mathcal C_N$ always terminates in $O(||w||_A^2)$ time (where we assume that $[w]$ is given to us as a cyclically reduced word in $F(A)$) and indeed outputs an element of $\mathcal M([w])$. The quadratic time bound arises since going from $[w_i]$ to $[w_{i+1}]$ takes a priori linear time in $||w_i||_A$, and since $||w_1||_A>||w_2||_A>\dots$, the process terminates with some $[w_i]$ with $i\le ||w||_A$.
Part (2) of Proposition~\ref{p:wpr} implies that the Whitehead minimization algorithm on an $\mbox{Out}(F_N)$-minimal input $[w]$ in $F_N=F(A)$ with $||w||_A=n$, always terminates in $O(\#V\mathcal T_n([w]))$ time, and indeed outputs the set $\mathcal M([w])=\mathcal T_n([w])$. Taken together, Proposition~\ref{p:wpr} implies that the Whitehead algorithm on the input $[w], [w']\in \mathcal C_N$ does correctly decide whether or not $\mbox{Out}(F_N)[w]= \mbox{Out}(F_N)[w']$.
Overall, the a priori worst-case complexity of Whitehead's algorithm on the input $[w],[w']$ is exponential in $\max\{||w||_A, ||w'||_A\}$ because for $[u]\in V\mathcal T_n$ the cardinality $\#V\mathcal T_n([u])$ is at most exponential in $n$.
\end{rem}
\begin{defn}
Suppose that $\mathfrak W\subseteq \mbox{Out}(F_N)$ is a fixed finite set of ``auxiliary'' automorphisms.
\begin{itemize}
\item The \emph{$\mathfrak W$-speed-up} of the Whitehead minimization algorithm consists in taking the input $[w]\ne 1$, computing $\mathfrak W[w]=\{\varpsi[w]|\varpsi\in\mathfrak W\}$ first and then applying the Whitehead minimization algorithm, in parallel to $[w]$ and each of the elements of $\mathfrak W([w])$. The result is again an element of $\mathcal M([w])$.
\item The \emph{$\mathfrak W$-speed-up} of the Whitehead's algorithm consists in doing the following. Given $[w], [w']\ne 1$, first apply the $\mathfrak W$-speed-up of the Whitehead minimization algorithm to both $[w]$ and $[w']$ to find $[u]\in \mathcal M([w])$ and $[u']\in \mathcal M([w'])$. Then proceed exactly as in Whitehead's algorithm to decide whether or not $\mbox{Out}(F_N)[w]=\mbox{Out}(F_N)[w']$.
\end{itemize}
\end{defn}
Since $\mathfrak W$ is finite an fixed, the a priori complexity estimates for these speed-up versions are the same as in Remark~\ref{rem:com}, although with worse multiplicative constants.
\section{$(M,\lambda,\varepsilon)$-minimality and Whitehead's algorithm}\label{s:MLE}
Let $F_N=F(A)$ be free of rank $N\ge 2$ where $A=\{a_1,\dots, a_N\}$ is a fixed free basis of $F_N$.
\subsection{Main definitions}
\begin{defn}\label{d:MLE}
Let $M\ge 1$ be an integer, let $\lambda>1$ and let $0\le \varepsilon<\lambda-1$.
A finite set $S$ of conjugacy classes of nontrivial elements of $F_N$ is called \emph{$(M,\lambda,\varepsilon)$-minimizing} if it satisfies the following properties:
\begin{enumerate}
\item We have $\#(S)\le M$.
\item For any $[u],[u']\in S$ we have $\mbox{Out}(F_N)[u]=\mbox{Out}(F_N)[u']$.
\item For any $[u],[u']\in S$ we have $1-\varepsilon\le \frac{||u'||_A}{||u||_A}\le 1+\varepsilon$.
\item For every $[u]\in S$ and every $\varphi\in \mbox{Out}(F_N)$ such that $\varphi([u])\not\in S$ we have $\frac{||\varphi(u)||_A}{||u||_A}\ge \lambda>1+\varepsilon$.
\end{enumerate}
In this case for any $[u]\in S$ we also say that $S$ is a \emph{$(M,\lambda,\varepsilon)$-minimizing set for $[u]$}.
We say that a nontrivial conjugacy class $[u]$ in $F_N$ is \emph{$(M,\lambda,\varepsilon)$-minimal} if there exists an $(M,\lambda,\varepsilon)$-minimizing set $S$ for $[u]$ (and thus $[u]\in S$).
\end{defn}
Note that if $S$ is a $(M,\lambda,\varepsilon)$-minimizing set and if $[u]\in S$ then for $\varphi\in \mbox{Out}(F_N)$ either $\varphi(u)\in S$ or $\frac{||\varphi(u)||_A}{||u||_A}\ge \lambda$, and these outcomes are mutually exclusive.
We record the following useful immediate corollary of the above definition:
\begin{lem}\label{lem:m}
Let $M\ge 1$ be an integer, let $\lambda>1$, let $0\le \varepsilon<\lambda-1$ and let $S$ be an $(M,\lambda,\varepsilon)$-minimizing set of conjugacy classes in $F_N$.
Then for any $[u]\in S$ and $\varphi\in \mbox{Out}(F_N)$ such that $||\varphi(u)||_A\le (1+\varepsilon) ||u||_A$ we have $\varphi([u])\in S$.
\end{lem}
\hfill $\qed$
\begin{defn}\label{d:MLEW}
Let $M\ge 1$ be an integer, let $\lambda>1$ and let $0\le \varepsilon<\lambda-1$.
A finite set $S\subseteq \mathcal C_N$ of conjugacy classes of nontrivial elements of $F_N$ is called \emph{$(M,\lambda,\varepsilon, \mathcal W_N)$-minimizing} if it satisfies the following properties:
\begin{enumerate}
\item We have $\#(S)\le M$.
\item For any $[u],[u']\in S$ we have $\mbox{Out}(F_N)[u]=\mbox{Out}(F_N)[u']$.
\item For any $[u],[u']\in S$ we have $1-\varepsilon\le \frac{||u'||_A}{||u||_A}\le 1+\varepsilon$.
\item For any $[u]\in S$ and $\tau\in \mathcal W_N$ exactly one of the following occurs:
\begin{itemize}
\item[(i)] We have $\tau([u])\in S$.
\item[(ii)] We have $\tau([u])\not\in S$ and $\frac{||\tau(u)||_A}{||u||_A}\ge \lambda>1+\varepsilon$.
\end{itemize}
\end{enumerate}
In this case for any $[u]\in S$ we also say that $S$ is a \emph{$(M,\lambda,\varepsilon, \mathcal W)$-minimizing set for $[u]$}.
We say that a nontrivial conjugacy class $[u]$ in $F_N$ is \emph{$(M,\lambda,\varepsilon, \mathcal W_N)$-minimal} if there exists an $(M,\lambda,\varepsilon, \mathcal W_N)$-minimizing set $S$ for $[u]$ (and thus $[u]\in S$).
\end{defn}
\begin{lem}\label{lem:aux}
Let $M\ge 1$ be an integer, let $\lambda>1$, $0<\varepsilon<1$ be such that $\varepsilon<\lambda-1$ and $\lambda (1-\varepsilon)>1$. Let $S\subseteq \mathcal C_N$ be a finite set of conjugacy classes of nontrivial elements of $F_N$ such that $S$ is $(M,\lambda,\varepsilon, \mathcal W_N)$-minimizing.
\begin{enumerate}
\item For any $[u]\in S$ and $\tau\in\mathcal W_N$ such that $||\tau(u)||_A\le (1+\varepsilon) ||u||_A$ we have $\tau([u])\in S$.
\item For any $[u]\in S$ and $\varphi\in\mbox{Out}(F_N)$ such that $||\varphi(u)||_A\le ||u||_A$ we have $\varphi([u])\in S$.
\end{enumerate}
\end{lem}
\begin{proof}
Part (1) follows from conditions (3), (4) of Definition~\ref{d:MLEW}.
For (2), suppose that $[u]\in S$ and $\varphi\in\mbox{Out}(F_N)$ are such that $||\varphi(u)||_A\le ||u||_A$. By Proposition~\ref{p:wprs}, there exist $\tau_1,\dots,\tau_k\in \mathcal W_N$ such that $\varphi=\tau_k\dots\tau_1$ and that for $[u_0]=[u]$, $[u_i]=\tau_i\dots \tau_1([u])$ for $i=1,\dots, k$ we have $||u_i||_A\le ||u||_A$. Note that $[u_k]=\varphi([u])$.
We argue by induction on $i$ that $[u_i]\in S$ for $i=1,\dots, k$.
We have $[u]=[u_0]\in S$. Suppose now $0\le i<k$ and $[u_i]\in S$. We need to show that $[u_{i+1}]=\tau_{i+1}[u_i]\in S$. Suppose, on the contrary, that $[u_{i+1}]\not\in S$. Then $||u_{i+1}||_A/||u_i||_A\ge \lambda$. Since $[u], [u_i]\in S$, also have $||u_i||_A/||u||_A\ge 1-\varepsilon$. Therefore $||u_{i+1}||_A/||u||_A\ge \lambda(1-\varepsilon)$, so that $||u_{i+1}||_A\ge \lambda(1-\varepsilon)||u||_A>||u||_A$ since $\lambda(1-\varepsilon)>1$. This contradicts the choice of $\tau_1,\dots, \tau_k$. Thus $[u_{i+1}]\in S$, as required.
Hence $[u_k]=\varphi([u])\in S$, and part (2) of the lemma is verified.
\end{proof}
The definitions directly imply that an $(M,\lambda,\varepsilon, \mathcal W_N)$-minimizing set $S$ is $(M,\lambda,\varepsilon)$-minimizing. It turns out that the converse also holds, but with slightly smaller $\varepsilon$ and slightly bigger $\lambda$.
\begin{prop}\label{prop:cont}
Let $M\ge 1$ be an integer, let $\lambda>1$ and let $0\le \varepsilon<\lambda-1$. Let $0<\varepsilon'<\varepsilon$ and $\lambda'>\lambda>1$ be such that be such that $\lambda'(1-\varepsilon')>\lambda$.
Let $S\subseteq \mathcal C_N$ be a finite set of conjugacy classes of nontrivial elements of $F_N$ be such that $S$ is $(M,\lambda',\varepsilon', \mathcal W_N)$-minimizing.
Then $S$ is $(M,\lambda,\varepsilon)$-minimizing.
\end{prop}
\begin{proof}
We need to verify that conditions (1)-(4) of Definition~\ref{d:MLE} of an $(M,\lambda,\varepsilon)$-minimizing set hold for $S$.
Since $S$ is $(M,\lambda',\varepsilon', \mathcal W_N)$-minimizing, it follows that $\#(S)\le M$, any two elements of $S$ are in the same $\mbox{Out}(F_N)$-orbit, and for any $[u],[u']\in S$ we have $\frac{||u'||_A}{||u||_A}\in [1-\varepsilon',1+\varepsilon']\subseteq [1-\varepsilon,1+\varepsilon]$. Thus we only need to verify condition (4) of Definition~\ref{d:MLE} for $S$.
Let $[u]\in S$ and let $\varpsi\in \mbox{Out}(F_N)$ be such that $\varphi([u])\not\in S$. Part (2) of Lemma~\ref{lem:aux} implies that $||u||_A<||\varphi(u)||_A$.
Therefore by
Proposition~\ref{p:wprs}, there exist $\tau_1,\dots,\tau_k\in \mathcal W_N$ such that $\varphi=\tau_k\dots\tau_1$ and that for $[v_0]=[u]$, $[v_i]=\tau_i\dots \tau_1([u])$ for $i=1,\dots, k$ we have $||v_i||_A\le ||\varphi(u)||_A$. Note that $[v_k]=\varphi([u])$. Since $[v_k]\not\in S$, the set $\{i\ge 0| [v_i]\not\in S\}$ is nonempty. Put $j=\min\{i\ge 0| [v_i]\not\in S\}$. Since $[v_0]=[u]\in S$, we have $j\ge 1$, and $[v_i]\in S$ for all $0\le i<j$.
Since $[v_{j-1}], [u]\in S$, we have $||v_{j-1}||_A\ge (1-\varepsilon') ||u||_A$. Since $[v_{j-1}]\in S$ and $[v_{j}]=\tau_j([v_{j-1}])\not\in S$, it follows that $||v_{j}||_A\ge \lambda ||v_{j-1}||_A$ and therefore $||v_j||_A\ge \lambda(1-\varepsilon') ||u||_A$. We also have $||\varphi(u)||_A\ge ||v_j||_A$ and hence
\[
||\varphi(u)||_A\ge ||v_j||_A\ge \lambda'(1-\varepsilon') ||u||_A\ge \lambda ||u||_A.
\]
Thus the set $S$ is $(M,\lambda,\varepsilon)$-minimizing, as required.
\end{proof}
\begin{defn}
Let $M\ge 1$ be an integer, let $\lambda>1$ and let $0\le \varepsilon<\lambda-1$.
Let $1\ne w\in F_N$. We say that $[w]$ is \emph{$(M,\lambda,\varepsilon)$-minimizable} in $F_N=F(A)$ if there exists a subset $\mathfrak W\subseteq \mbox{Out}(F_N)$ such that $\#(\mathfrak W)\le M$ and that the set $S=\mathfrak W [w]$ is $(M,\lambda,\varepsilon)$-minimizing (or, equivalently, if the orbit $\mbox{Out}(F_N)[w]$ contains a $(M,\lambda,\varepsilon)$-minimal element). In this case we say that $\mathfrak W$ is \emph{$(M,\lambda,\varepsilon)$-reducing} for $[w]$.
\end{defn}
Note that for $1\ne w\in F_N$ the conjugacy class $[w]$ is $(M,\lambda,\varepsilon)$-minimizable if and only if the orbit $\mbox{Out}(F_N)[w]$ contains a $(M,\lambda,\varepsilon)$-minimal element.
\subsection{Behavior of Whitehead's algorithm}
We now have:
\begin{prop}\label{p:MLE}
Let $\lambda>1$, let $0\le \varepsilon<\lambda-1$, let $[u]\in\mathcal C_N$ be a $(M,\lambda,\varepsilon)$-minimal element and let $S\subseteq \mathcal C_N$ be an $(M,\lambda,\varepsilon)$-minimizing set for $[u]$ (so that $[u]\in S$). Then the following hold:
\begin{enumerate}
\item We have $\mathcal M([u])\subseteq S$, and, in particular, $\# \mathcal M([u])\le M$.
\item For every $[u']\in \mathcal M([u])$ we have $1\le \frac{||u||_A}{||u'||_A}\le 1+\varepsilon$.
\item For every $[u']\in \mathcal M([u])$ we have $\mathcal M([u])=V\mathcal T_n([u'])$ where $n=||u'||_A$.
\item If $\tau\in \mathcal W_N$ is a Whitehead automorphism such that $||\tau(u)||_A<||u||_A$ then $\tau([u] )\in S$.
\item If $\tau_1,\tau_2,\dots \tau_k\in \mathcal W_N$ are such that
\[
||u||_A>||\tau_1(u)||_A>||\tau_2\tau_1(u)||_A>\dots > ||\tau_k\dots \tau_2\tau_1(u)||_A
\]
then $k\le M-1$ and we have $[u_i]:=\tau_i\dots \tau_1([u])\in S$ for $i=1,\dots, k$.
\item If such a sequence $\tau_1,\tau_2,\dots \tau_k\in \mathcal W_N$ as in (4) is such that is $[u_k]$ is $\mathcal W_N$-minimal then $[u_k]\in \mathcal M([u])$.
\end{enumerate}
\end{prop}
\begin{proof}
Parts (1) and (2) follow directly from Definition~\ref{d:MLE}. Part~(3) holds by the general peak reduction properties of Whitehead's algorithm. Part (4) follows from property (4) in Definition~~\ref{d:MLE}.
Suppose now that $\tau_1,\tau_2,\dots \tau_k\in \mathcal W_N$ are as in part (5) of the proposition. Since the cyclically reduced lengths $||u||_A>||u_1||_A>\dots >||u_k||_A$ are strictly decreasing, the conjugacy classes $[u], [u_1], \dots [u_k]$ are distinct. Since by assumption $[u]\in S$, part (4) of the proposition implies that $[u],[u_1],\dots, [u_k]\in S$. Since $\#S\le M$, it follows that $k\le M-1$.
Part (6) follows from part (5) since every $\mathcal W_N$-minimal conjugacy class is $\mbox{Out}(F_N)$-minimal.
\end{proof}
Proposition~\ref{p:MLE} then directly implies:
\begin{cor}
Let $\lambda>1$, let $0\le \varepsilon<\lambda-1$ and let $[w]\in \mathcal C_N$ be $(M,\lambda,\varepsilon)$-minimizable, with a $(M,\lambda,\varepsilon)$-reducing for $[w]$ set $\mathfrak W\subseteq \mbox{Out}(F_N)$. Let $S=\mathfrak W[w]$ (so that $S$ is $(M,\lambda,\varepsilon)$-minimizing). Then:
\begin{enumerate}
\item We have $\mathcal M([w])\subseteq S$, and, in particular, $\# \mathcal M([w])\le M$.
\item For every $[u']\in \mathcal M([w])$ we have $\mathcal M([w])=V\mathcal T_n([u'])$ where $n=||u'||_A$.
\end{enumerate}
\end{cor}
\hfill $\qed$
\begin{defn}
Let $M\ge 1$, $\lambda>1$, and $0\le \varepsilon<\lambda-1$.
\begin{enumerate}
\item We denote by $U_N(M,\lambda,\varepsilon)$ the set of all $1\ne u\in F_N$ such that $[u]$ is $(M,\lambda,\varepsilon)$-minimal.
\item We denote by $Y_N(M,\lambda,\varepsilon)$ the set of all $1\ne w\in F_N$ such that there exists $[u]\in \mbox{Out}(F_N)[w]$ such that $[u]$ is $(M,\lambda,\varepsilon)$-minimal. [That is, $Y_N(M,\lambda,\varepsilon)$ is the set of all $1\ne w\in F_N$ such that $[w]$ is $(M,\lambda,\varepsilon)$-minimizable.]
\item Let $\varpsi\in \mbox{Out}(F_N)$. Denote by $U_N(M,\lambda,\varepsilon; \varpsi)$ the set of all $1\ne w\in F_N$ such that $\varpsi([w])$ is $(M,\lambda,\varepsilon)$-minimal.
\end{enumerate}
\end{defn}
\begin{lem}\label{lem:YN}
Let $u\in Y_N(M,\lambda,\varepsilon)$ and let $[u']\in \mathcal M([u])$ (that is, $[u']$ is an $\mbox{Out}(F_N)$-minimal element in the orbit $\mbox{Out}(F_N)[u]$). Then $u'\in U_N(M,\lambda,\varepsilon)$ (that is, $[u']$ is $(M,\lambda,\varepsilon)$-minimal).
\end{lem}
\begin{proof}
Since $u\in Y_N(M,\lambda,\varepsilon)$, there exists $\varphi\in\mbox{Out}(F_N)$ such that $\varphi([u])$ is $(M,\lambda,\varepsilon)$-minimal, so that $\varphi([u])$ belongs to some $(M,\lambda,\varepsilon)$-minimizing set $S$.
Part (1) of Proposition~\ref{p:MLE} implies that $\mathcal M(\varphi([u]))\subseteq S$, so that every element of $\mathcal M(\varphi([u]))$ is $(M,\lambda,\varepsilon)$-minimal. Since $\mathcal M(\varphi([u]))=\mathcal M([u])$, the statement of the lemma follows.
\end{proof}
We now summarize algorithmic properties of $(M,\lambda,\varepsilon)$-minimal in relation to Whitehead's algorithm.
\begin{thm}\label{t:WHM} Let $M\ge 1$, $\lambda>1$, and $0\le \varepsilon<\lambda-1$. Then there exists a constant $K\ge 1$ such that the following hold:
\begin{enumerate}
\item[(a)] For any $u\in U_N(M,\lambda,\varepsilon)$ the Whitehead minimization algorithm on the input $u$ terminates time $\le K|u|_A$ and produces an element of $\mathcal M([u])$.
\item[(b)] For any $u_1,u_2\in U_N(M,\lambda,\varepsilon)$, the Whitehead algorithm for the automorphic equivalence problem in $F_N$ terminates in time at most $K\max \{|u_1|_A,|u_2|_A\}$, on the input $(u_1,u_2)$.
\item[(c)] For any $u_1\in Y_N(M,\lambda,\varepsilon)$ and any $1\ne u_2\in F_N$, the Whitehead algorithm for the automorphic equivalence problem in $F_N$ terminates in time at most $K\max \{|u_1|_A^2,|u_2|_A^2\}$, on the input $(u_1,u_2)$.
\item[(d)] For any $u_1\in U_N(M,\lambda,\varepsilon)$ and any $1\ne u_2\in F_N$, the Whitehead algorithm for the automorphic equivalence problem in $F_N$ terminates in time $K\max \{|u_1|_A,|u_2|_A^2\}$, on the input $(u_1,u_2)$.
\item[(e)] Let $\varpsi\in\mbox{Out}(F_N)$ be a fixed element. Then there is $K'=K'\ge 1$ such that for any $u_1,u_2\in U_N(M,\lambda,\varepsilon; \varpsi)$, the $\varpsi$-speed-up of Whitehead's algorithm decides in time at most $K'\max \{|u_1|_A,|u_2|_A\}$, whether or not $\mbox{Out}(F_N)[u_1]=\mbox{Out}(F_N)[u_2]$.
\item[(f)] Let $\varpsi\in\mbox{Out}(F_N)$ be a fixed element. Then there is $K'=K'\ge 1$ such that for any $u_1\in U_N(M,\lambda,\varepsilon;\varpsi)$ and any $1\ne u_2\in F_N$, the $\varpsi$-speed-up of Whitehead's algorithm decides
in time at most $K'\max \{|u_1|_A,|u_2|_A^2\}$, whether or not $\mbox{Out}(F_N)[u_1]=\mbox{Out}(F_N)[u_2]$.
\end{enumerate}
\end{thm}
\begin{proof}
(a) Let $u\in U_N(M,\lambda,\varepsilon)$ be arbitrary. Let $S$ be an $(M,\lambda,\varepsilon)$-minimizing set containing $[u]$. Thus $\#S\le M$. By Lemma~\ref{lem:m}, if $[v]\in S$ and $\tau\in\mathcal W_N$ is a Whitehead move such that $||\tau(v)||_A<||v||_A$ then $\tau([v])\in S$. Therefore starting with $u$ and iteratively looking for Whitehead moves that decrease the $||.||_A$-length terminates after a chain of $\le M$ such moves with a conjugacy class that is Whitehead-minimal and therefore is $\mbox{Out}(F_N)$-minimal, that is, an element of $\mathcal M([u])$. This process takes at most time $C_1|u|_A$ for some constant $C_1>0$ depending only on $N,M,\lambda,\varepsilon$.
(b) Let $u_1,u_2\in U_N(M,\lambda,\varepsilon)$ so that $[u_1],[u_2]$ are $(M,\lambda,\varepsilon)$-minimal. By part (a) above, applying the Whitehead minimization algorithm to $[u_i]$ terminates in at most $M$ steps with an $\mbox{Out}(F_N)$-minimal element $[u_i']$ such that $n_i=||u_i'||_A\le ||u_i||_A\le |u_i|_A$. Each of these $\le M$ takes at most linear times in $|u_i|_A$ since the number of whitehead automorphisms in $\mathcal W_N$ is finite and fixed. Thus it takes linear time in $|u_i|_A$ to produce $[u_i']\in \mathcal M([u_i])$. If $n_1\ne n_2$ then $\mbox{Out}(F_N)[u_1]\ne \mbox{Out}(F_N)[u_2]$ and we are done. Suppose that $n=n_1=n_2$. By part (3) of Proposition~\ref{p:MLE} we have $\mathcal M([u_i])=V\mathcal T_n([u_i'])$ for $i=1,2$. Moreover, by part (1) of Proposition~\ref{p:MLE} we have $\#V\mathcal T_n([u_i'])\le M$ here. Since $M$ is fixed, it takes linear time in $|u_i|_A$ to construct the graph $\mathcal T_n([u_i'])\le M$ from $u_i'$. Then $\mbox{Out}(F_N)[u_1]=\mbox{Out}(F_N)[u_2]$ if and only if $\#V\mathcal T_n([u_1'])\cap \#V\mathcal T_n([u_2'])\ne \varnothing$, and this condition can be checked in linear time in $\max\{|u_1|_A,|u_2|_A\}$. Summing up we get that the total running time of the Whitehead algorithm for the automorphic equivalence problem in $F_N$ is time at most $C_2\max \{|u_1|_A,|u_2|_A\}$, for some constant $C_2>0$ depending only on $N,M,\lambda,\varepsilon$.
(c) Now suppose that $u_1\in Y_N(M,\lambda,\varepsilon)$ and $1\ne u_2\in F_N$. We first apply the Whitehead minimization algorithm to each of $u_1,u_2$ to find $Out(F_N)$-minimal elements $[u_i']\in Out(F_N)[u_i]$ for $i=1,2$. Producing $u_i'$ from $u_i$ takes quadratic time in terms of $|u_i|_A$. Note that by Lemma~\ref{lem:YN} the element $[u_1']$ is $(M,\lambda,\varepsilon)$-minimal, that is $u_1'\in U_N(M,\lambda,\varepsilon)$.
Again put $n_i=||u_i|_A$. If $n_1\ne n_2$ then $\mbox{Out}(F_N)[u_1]\ne \mbox{Out}(F_N)[u_2]$ and we are done. Suppose that $n=n_1=n_2$. Since $u_1'$ is $\mbox{Out}(F_N)$-minimal and $(M,\lambda,\varepsilon)$-minimal, by parts (3) and (1) of Proposition~\ref{p:MLE} we have $\mathcal M([u_1])=\mathcal M([u_1'])=V\mathcal T_n([u_1'])$ and $\#V\mathcal T_n([u_1'])\le M$. Then, since $M$ is fixed, it takes at most linear time in $n=||u_1'||_A\le |u_1|_A$ to construct the graph $\mathcal T_n([u_1'])$. Recall also that $[u_2']\in \mathcal M([u_2])$ and $||u_2||_A=n$. Then we have $\mbox{Out}(F_N)[u_1]=\mbox{Out}(F_N)[u_2]$ if and only if $[u_2']\in \mathcal T_n([u_1'])$. This last condition can be checked in linear time in $n$. Again, summing up we see that the total running time of the Whitehead algorithm on $(u_1,u_2)$ is at most $C_3\max \{|u_1|_A^2,|u_2|_A^2\}$, for some constant $C_3>0$ depending only on $N,M,\lambda,\varepsilon$.
(d) Now let $u_1\in U_N(M,\lambda,\varepsilon)$ and $1\ne u_2\in F_N$. We first apply the Whitehead minimization algorithm to each of $u_1,u_2$ to find $Out(F_N)$-minimal elements $[u_i']\in Out(F_N)[u_i]$ for $i=1,2$. As in (b), producing $u_1'$ from $u_1$ takes linear time in $|u_1|_A$, because $u_1$ is $(M,\lambda,\varepsilon)$-minimal. Producing $u_2'$ from $u_2$ takes at most quadratic time in $|u_2|_A$, by the general Whitehead's minimization algorithm properties. After that we proceed exactly in (2) above to decide if $[u_1']$ and $[u_2']$ are $\mbox{Out}(F_N)$-equivalent. Summing up we see that the total running time of the Whitehead algorithm on $(u_1,u_2)$ is at most $C_4\max \{|u_1|_A,|u_2|_A^2\}$ in this case, for some constant $C_4>0$ depending only on $N,M,\lambda,\varepsilon$.
(e) Choose an automorphism $\Psi\in \mbox{Aut}(F_N)$ in the outer automorphism class $\varpsi$ and put $C=\max_{i=1}^N |\Psi(a_i)|_A$. Since $\varphi$ and $\Psi$ are fixed, given $u_1,u_2\in U_N(M,\lambda,\varepsilon; \varpsi)$, for $i=1,2$ it takes linear time in $|u_i|_A$ to compute the element $\Psi(u_i)$, and $|\Psi(u_i)|_A\le C|u_i|_A$. Moreover, the assumption that $u_1,u_2\in U_N(M,\lambda,\varepsilon; \varpsi)$ implies that $\Psi(u_1), \Psi(u_2)$ are $(M,\lambda,\varepsilon)$-minimal. Then by part (b) above, the Whitehead algorithm on the input $(\Psi(u_1), \Psi(u_2))$ terminates in linear time in $C\max\{|u_1|_A,|u_2|_A\}$ and decides whether or not $\mbox{Aut}(F_N)\Psi(u_1)=\mbox{Aut}(F_N)\Psi(u_2)$, that is, whether or not $\mbox{Out}(F_N)[u_1]=\mbox{Out}(F_N)[u_2]$. The total running time required for this process is at most $C_5\max\{|u_1|_A,|u_2|_A\}$, for some constant $C_5>0$ depending only on $N,M,\lambda,\varepsilon$ and $\Psi$.
(f) We chose $\Psi\in \mbox{Aut}(F_N)$ and $C>0$ as in the proof of part (e) above. Given any $u_1\in U_N(M,\lambda,\varepsilon;\varpsi)$ and any $1\ne u_2\in F_N$ we first compute, in linear time in $|u_1|_A$, the element $\Psi(u_1)$ and again observe that $\Psi(u_1)$ is $(M,\lambda,\varepsilon)$-minimal and that $|\Psi(u_1)|_A\le C|u_1|_A$. We then apply to the pair $(\Psi(u_1),u_2)$ the algorithm from part (d) of this proposition to decide whether or not $\mbox{Out}(F_N)[u_1]=\mbox{Out}(F_N)[u_2]$. The overall running time is of this process is at most $C_6\max \{|u_1|_A,|u_2|_A^2\}$, for some constant $C_6>0$ depending only on $N,M,\lambda,\varepsilon$ and $\Psi$.
\end{proof}
We recall another basic fact related to Whitehead's algorithm which describes $Out(F_N)$-stabilizers of conjugacy classes in $F_N$:
\begin{prop}\label{prop:LSST}
Let $1\ne u\in F_N$ be such that $[u]$ is $Out(F_N)$-minimal, and let $n=||u||_A$. Then for $\varphi\in\mbox{Out}(F_N)$ we have $\varphi([u])=[u]$ if and only if there exists a sequence of Whitehead automorphisms $\tau_1,\dots, \tau_k\in \mathcal W_N$ such that for $u_i=\tau_{i}\dots \tau_1([u])$, we have $||u_i||_A=n$ for $i=1,\dots, k$ and $[u_k]=[u]$ and such that $\varphi=\tau_k\dots \tau_1$ in $\mbox{Out}(F_N)$.
\end{prop}
Recall that for $n\ge 1$ the oriented edges of the graph $\mathcal T_n$ are labelled by Whitehead moves $\tau\in \mathcal W_N$. Thus oriented edge-paths in $\mathcal T_n$ are labelled by products of Whitehead moves. Recall also that for a vertex $[u]$ of $\mathcal T_n$, the graph $\mathcal T_n[u]$ is the connected component of $\mathcal T_n$ containing $[u]$. Therefore we get a natural labelling homomorphism $\rho_{[u]}: \pi_1(\mathcal T_n[u], [u])\to \mbox{Out}(F_N)$ where a closed loop at $[u]$ in $\mathcal T_n$ is mapped to the element of $\mbox{Out}(F_N)$ given by the label of this loop in $\mathcal T_n$.
Note also that, since the set of possible edge labels $\mathcal W_N$ is finite, the rank of the free group $\pi_1(\mathcal T_n, [u])$ is bounded above by some constant $K=K(N, \#V\mathcal T_n[u])$.
Proposition~\ref{prop:LSST} now directly implies:
\begin{cor}\label{cor:st}
Let $1\ne u\in F_N$ be such that $[u]$ is $Out(F_N)$-minimal, and let $n=||u||_A$. Then:
\begin{enumerate}
\item We have $Stab_{\mbox{Out}(F_N)}([u])=\rho_{[u]}\left(\pi_1(\mathcal T_n[u], [u])\right)$.
\item We have ${\rm rank}\ Stab_{\mbox{Out}(F_N)}([u])\le K(N, \#V\mathcal T_n[u])$.
\end{enumerate}
\end{cor}
\begin{prop}\label{prop:st}
Let $M\ge 1$, $\lambda>1$, and $0\le \varepsilon<\lambda-1$. Let $u\in Y_N(M,\lambda,\varepsilon)$ (that is $1\ne u\in F_N$ and the orbit $\mbox{Out}(F_N)[u]$ contains an $(M,\lambda,\varepsilon)$-minimal element).
Then
\[
{\rm rank}\ Stab_{\mbox{Out}(F_N)}([u])\le K(N, M).
\]
\end{prop}
\begin{proof}
Let $[u']\in \mbox{Out}(F_N)[u]$ be an $(M,\lambda,\varepsilon)$-minimal element. Let $[u'']$ be an $\mbox{Out}(F_N)$-minimal element in $\mbox{Out}(F_N)[u]=\mbox{Out}(F_N)[u']$ and put $n=||u''||_A$. Part (1) of Proposition~\ref{p:MLE} implies that $[u'']$ is also $(M,\lambda,\varepsilon)$-minimal and that $V\mathcal T_n[u'']=\mathcal M[u]=\mathcal M[u']=\mathcal M[u'']$ has cardinality $\le M$. Then by Corollary~\ref{cor:st} we have ${\rm rank}\ Stab_{\mbox{Out}(F_N)}([u])\le K(N, M)$, as claimed.
\end{proof}
\subsection{Algorithmic detectability}\label{s:alg}
For a finite nonempty subset $S\subseteq \mathcal C_N$ denote $||S||_A=\max\{||u||_A|[u]\in S\}$.
\begin{rem}
Let an integer $M\ge 1$ and rational numbers $0<\varepsilon<1$ and $\lambda>1+\varepsilon$ be fixed.
(1) Since the set $\mathcal W_N\subseteq \mbox{Out}(F_N)$ of Whitehead moves is finite and fixed, given a subset $S\subseteq \mathcal C_N-\{1\}$ of cardinality $\le M$ we can check in linear time in $||S||_A$ whether or not $S$ is $(M,\lambda,\varepsilon, \mathcal W_N)$-minimizing.
(2) In this setting, given such $S$ it is also possible to algorithmically check whether or not $(M,\lambda,\varepsilon)$-minimizing, but only with the exponential time, in terms of $||S||_A$, complexity estimate. Indeed, we first check (in linear time), if conditions (1) and (3) of Definition~\ref{d:MLE} of an $(M,\lambda,\varepsilon)$-minimizing set hold for $S$. We then use Whitehead's algorithm to check if condition (2) of Definition~\ref{d:MLE} also holds. Suppose they do (otherwise $S$ is not $(M,\lambda,\varepsilon)$-minimizing). Then for every $[u]\in S$ compute, using Whitehead's algorithm, the (finite) set $\mathcal F[u]=\{[u']\in \mbox{Out}(F_N)[u]| ||u'||_A\le ||u||_A\}$. For each $[u]\in S$ check whether $\mathcal F[u]\subseteq S$. If not then $S$ is not $(M,\lambda,\varepsilon)$-minimizing. So suppose that for all $[u]\in S$ $\mathcal F[u]\subseteq S$.
Since balls in the Cayley graphs of $F_N=F(A)$ are finite and since $||w||_A\in \mathbb Z_{\ge 0}$ for all $w\in F_N$, we can then use Whitehead's algorithm to compute, for each $[u]\in S$ the number $\rho([u]):=\min \{\frac{||u'||_A}{||u||_A}\big| [u']\in \mbox{Out}(F_N)[u] \text{ and } ||u'||_A>||u||_A\}$. Then compute $\rho=\min_{[u]\in S} \rho([u])$. Then $S$ is $(M,\lambda,\varepsilon)$-minimizing if and only if $\rho\ge \lambda$.
The complexity of this procedure for deciding if a subset $S\subseteq \mathcal C_N$ with $\#S\le M$ is $(M,\lambda,\varepsilon)$-minimizing is exponential time in $||S||_A$ (when $M,\lambda,\varepsilon$ are fixed).
(3) We can then also decide, given $[u]\in \mathcal C_N$, whether or not $[u]$ is $(M,\lambda,\varepsilon)$-minimal, that is, whether or not $[u]$ belongs to some $(M,\lambda,\varepsilon)$-minimizing subset. Namely, list all subsets $S\subseteq \mathcal C_N$ of cardinality $\le M$ containing $[u]$ and with $||S||_A\le (1+\varepsilon)||u||_A$ and for each of them run the algorithm from (2) to decide if $S$ is $(M,\lambda,\varepsilon)$-minimizing. Again, since $M,\lambda,\varepsilon$ are fixed, this check can be done in exponential time in $||S||_A$.
\end{rem}
It turns out that deciding whether an element $[u]$ is $(M,\lambda,\varepsilon, \mathcal W_N)$-minimal can be done in linear time in $||u||_A$ (under slightly more stringent assumptions in $\lambda,\varepsilon$).
\begin{lem}\label{lem:aux1}
Let $M\ge 1$ be an integer, let $\lambda>1$, $0<\varepsilon<1$ be such that $\varepsilon<\lambda-1$ and $\lambda \frac{1-\varepsilon}{1+\varepsilon}>1$. Let $S\subseteq \mathcal C_N$ be a finite set of conjugacy classes of nontrivial elements of $F_N$ such that $S$ is $(M,\lambda,\varepsilon, \mathcal W_N)$-minimizing. Let $[u]\in S$.
Then for $[u']\in \mathcal C_N, [u']\ne [u]$ the following conditions are equivalent:
\begin{enumerate}
\item We have $[u']\in S$.
\item There exists a chain $\tau_1,\dots, \tau_k\in \mathcal W_N$ such that $k\le M$, that $\tau_k\dots\tau_1[u]=[u']$ and that with $[u_0]=[u]$ $[u_i]=\tau_i\dots\tau_1[u]$ we have $||u_{i+1}||_A\le (1+\varepsilon) ||u_i||_A$ for all $i\le k$.
\end{enumerate}
\end{lem}
\begin{proof}
Part (2) implies part (1) by Lemma~\ref{lem:aux}(1).
We now need to show that (1) implies (2).
Suppose that $[u']\in S$. Then $||u'||_A\le (1+\varepsilon)||u||_A$.
Since $\mbox{Out}(F_N)[u]=\mbox{Out}(F_N)[u']$, there is $\varphi\in \mbox{Out}(F_N)$ such that $\varphi[u]=[u']$. Proposition~\ref{p:wprs} implies that there exist $\tau_1,\dots,\tau_k\in \mathcal W_N$ such that for $[u_0]=[u]$, $[u_i]=\tau_i\dots \tau_1([u])$ for $i=1,\dots, k$ we have \[||u_i||_A\le (1+\varepsilon)||u||_A\] and that $[u_k]=[u']$. We can assume that we have eliminated repetitions among $[u_i]$, so that $[u]=[u_0], [u_1],\dots, [u_k]=[u']$ are distinct.
{\bf Case 1.}
Suppose first that for every $i\ge 1$ we have $||u_{i}||_A\le (1+\varepsilon)||u_{i-1}||_A$. Since $[u_0]=[u]\in S$, Lemma~\ref{lem:aux}(1) then implies that $[u_i]\in S$ for all $i=1,\dots, k$. Since $\#S\le M$ and all $[u_i]$ are distinct, it follows that $k\le M$. Thus the conclusion of part (2) of the lemma holds in this case.
{\bf Case 2.} Suppose there is some $i\ge 1$ we have $||u_{i}||_A>(1+\varepsilon)||u_{i-1}||_A$. Let $i_0$ be the smallest among such $i$. Then for all $j<i_0$ we have $||u_{j}||_A\le (1+\varepsilon)||u_{j-1}||_A$, and we also have $||u_{i_0}||_A>(1+\varepsilon)||u_{i_0-1}||_A$. Again by Lemma~\ref{lem:aux}(1) we conclude that $[u_j]\in S$ for all $j<i_0$. In particular $[u_{i_0-1}]\in S$. Since for $[u_{i_0}]=\tau_{i_0}[u_{i_0-1}]$ we have $||u_{i_0}||_A>(1+\varepsilon)||u_{i_0-1}||_A$, condition (3) of Definition~\ref{d:MLEW} implies that $[u_{i_0}]\not\in S$. Therefore by part (4)(ii) of Definition~\ref{d:MLEW} we have $||u_{i_0}||_A\ge \lambda||u_{i_0-1}||_A$. Since $[u],[u_{i_0-1}]\in S$, we have $||u_{i_0}||_A\ge (1-\varepsilon)||u||_A$. Therefore $||u_{i_0}||_A\ge \lambda(1-\varepsilon) ||u||_A> (1+\varepsilon)||u||_A$, yielding a contradiction. Thus Case~2 is impossible.
Therefore the conclusion of part (2) of the lemma holds, as required.
\end{proof}
\begin{cor}\label{c:alg}
Let $M\ge 1$ be an integer, let $\lambda>1$, $0<\varepsilon<1$ be rational numbers such that $\varepsilon<\lambda-1$ and $\lambda \frac{1-\varepsilon}{1+\varepsilon}>1$.
Then there is an algorithm that, given $1\ne [u]\in \mathcal C_N$ decides in linear time in $||u||_A$ whether or not $[u]$ is $(M,\lambda,\varepsilon, \mathcal W_N)$-minimal (that is, whether $[u]$ belongs to some $(M,\lambda,\varepsilon, \mathcal W_N)$-minimizing set $S$).
\end{cor}
\begin{proof}
Suppose we are given an input $1\ne [u]\in \mathcal C_N$. We need to decide if there exists an $(M,\lambda,\varepsilon, \mathcal W_N)$-minimizing set $S$ containing $[u]$.
We first enumerate all chains of $k\le M$ Whitehead moves as in part (2) of Lemma~\ref{lem:aux1} and collect all $[u']$ reachable from $[u]$ by applying such chains. Denote the resulting subset of $\mathcal C_N$ by $S'$. Computing $S'$ from $[u]$ takes at most linear time in $||u||_A$ since $M$ is fixed and the set $\mathcal W_N$ is also finite and fixed.
Lemma~\ref{lem:aux1} implies that if $[u]$ belongs to some $(M,\lambda,\varepsilon, \mathcal W_N)$-minimizing set $S$ then $S=S'$.
We then check if conditions (1)-(4) of Definition~\ref{d:MLEW} hold for $S'$. Again this can be done in linear time in $||u||_A$ since $M$ is fixed.
We conclude that $[u]$ is $(M,\lambda,\varepsilon, \mathcal W_N)$-minimal if and only if conditions (1)-(4) of Definition~\ref{d:MLEW} do hold for $S'$.
\end{proof}
\section{Geodesic currents on free groups}
We provide some basic background on geodesic currents on $F_N$ here and refer the reader to \cite{Ka06,KL09,KL10} for further details. For the remainder of this section let $F_N$ be a free group of finite rank $N\ge 2$. We denote by $\partial F_N$ the hyperbolic boundary of $F_N$ and denote $\partial^2 F_N:=\{(x,y)|x,y\in\partial F_N, x\ne y\}$.
We give $\partial^2 F_N$ the subspace topology from $\partial F_N\times \partial F_N$ and endow $\partial^2 F_N$ with the natural diagonal translation action of $F_N$ by homeomorphisms. The space $\partial^2 F_N$ also comes with a natural ``flip" involution $\varpi:\partial^2F_N\to\partial^2 F_N$, $\varpi:(x,y)\mapsto (y,x)$. The boundary $\partial F_N$ is homeomorphic to the Cantor set, and $\partial^2 F_N$ is a locally compact totally disconnected but non-compact metrizable topological space.
\subsection{Basic notions}
\begin{defn}
A \emph{geodesic current} on $F_N$ is a locally finite (i.e. finite on compact subsets) positive Borel measure $\nu$ on $\partial^2 F_N$ such that $\nu$ is $F_N$-invariant and flip-invariant. The set of all geodesic currents on $F_N$ is denoted $\Curr(\FN)$.
\end{defn}
The set $\Curr(\FN)$ is equipped with the weak-* topology, which makes $\Curr(\FN)$ locally compact. Any automorphism $\Phi\in\mbox{Out}(F_N)$ is a quasi-isometry of $F_N$ and hence extends to a homeomorphism, which we still denote by $\Phi:\partial F_N\to\partial F_N$. Diagonally extending this homeomorphism we also get a homeomorphism $\Phi:\partial^2 F_N\to\partial^2 F_N$. There is a natural left action of $\mbox{Aut}(F_N)$ by homeomorphisms on $\Curr(\FN)$, where for $\Phi\in\mbox{Aut}(F_N)$ and $\nu\in\Curr(\FN)$ we have $(\Phi\nu)(S)=\nu(\Phi^{-1}(S))$ for $S\subseteq \partial^2 F_N$. The subgroup ${\rm Inn}(F_N)\le \mbox{Aut}(F_N)$ is contained in the kernel of this action, and therefore the action descends to the action of $\mbox{Out}(F_N)$ on $\Curr(\FN)$. There is also a multiplication by a scalar action of $\mathbb R_{>0}$ in $\Curr(\FN)-\{0\}$, with the quotient space $\mathbb P\Curr(\FN)=(\Curr(\FN)-\{0\})/\mathbb R_{>0}$, equipped with the quotient topology. The space $\mathbb P\Curr(\FN)$ is compact, although infinite dimensional. For $0\ne \nu\in\Curr(\FN)$ we denote the $\mathbb R_{>0}$-equivalence class of $\nu$ by $[\nu]$. Thus $[\nu]=\{c\nu| c\in \mathbb R_{>0}\}$ and $[\nu]\in\mathbb P\Curr(\FN)$. We call elements of $\mathbb P\Curr(\FN)$ \emph{projectivized geodesic currents} on $F_N$.
Let $1\ne g\in F_N$. Then $g$ determines a pair of distinct ``poles'' $g^{-\infty}, g^{\infty}\in \partial F_N$, where $g^\infty=\lim_{n\to\infty} g^n$ and $g^{-\infty}=\lim_{n\to\infty} g^{-n}$ in $F_N\cup\partial F_N$. Thus $(g^{-\infty},g^\infty)\in\partial^2 F_N$. For $h\in F_N$ we have $hg^\infty=(hgh^{-1})^\infty$, and we also have $g^{-\infty}=(g^{-1})^\infty$.
\begin{defn}[Counting and rational currents]
Let $1\ne g\in F_N$.
Then
\[
\eta_g:=\sum_{h\in F_N/\langle g\rangle} \delta_{h(g^{-\infty},g^\infty)}+\delta_{h(g^{\infty},g^{-\infty})}
\]
is a geodesic current on $F_N$ called the \emph{counting current} for $g$. We call currents of the form $c\eta_g\in\Curr(\FN)$, where $c>0$ and $1\ne g\in F_N$, \emph{rational currents}.
\end{defn}
It is known that the set of all rational currents is a dense subset of $\Curr(\FN)$, and that for any $1\ne g\in F_N$ and any $u\in F_N$ we have $\eta_g=\eta_{ugu^{-1}}=\eta_{g^{-1}}$. Therefore we also denote $\eta_{[g]}:=\eta_g$ where $[g]$ is the conjugacy class of $g$ in $F_N$. Moreover, for $\varphi\in\mbox{Aut}(F_N)$ and $1\ne g\in F_N$, one has $\varphi\eta_g=\eta_{\varphi(g)}$.
\subsection{Simplicial charts and weights}
We adopt the conventions of~\cite{DKT15} regarding graphs. All graphs are 1-cell complexes, where 0-cells are called vertices and 1-cells are called topological edges. Every topological edge is homeomorphic to an interval $(0,1)$ and thus admits exactly two orientations. An \emph{oriented edge} of a $e$ graph is a topological edge with a choice of an orientation. The same topological edge with the opposite orientation is denoted $e^{-1}$. The set of all oriented edges of a graph $\Delta$ is denoted $E\Delta$. We also denote by $V\Delta$ the set of all vertices of $\Delta$. Unless specified otherwise, by an edge of a graph we always mean an oriented edge. Every oriented edge $e\in E\Delta$ has an \emph{initial vertex} denoted $o(e)\in V\Delta$ and a \emph{terminal vertex} $t(e)\in E\Delta$. We also have $o(e^{-1})=t(e)$ and $t(e^{-1})=o(e)$. An \emph{edge-path} $\gamma$ of length $n\ge 1$ in $\Delta$ is a sequence of edges $e_1,\dots, e_n$ such that $t(e_i)=o(e_{i+1})$. We also consider a vertex $v$ of $\Delta$ to be a path of length $0$. An edge-path $\gamma$ in $\Delta$ is \emph{reduced} if it does not contain subpaths of the form $e,e^{-1}$ where $e\in E\Delta$. We denote by $|\gamma|$ the length of an edge-path $\gamma$.
\begin{defn}[Simplicial chart]
Let $F_N$ be free of rank $N\ge 2$. A \emph{simplicial chart} on $F_N$ is a pair $(\Gamma, \kappa)$ where $\Gamma$ is a finite connected oriented graph with all vertices of degree $\ge 3$ and with the first betti number $b(\Gamma)=N$, and that where $\kappa: F_N\to \pi_1(\Gamma,x_0)$ is a group isomorphism (with $x_0\in V\Gamma$ some base-vertex), called a \emph{marking}.
\end{defn}
When talking about simplicial charts, we usually suppress explicit mention of $\kappa$.
We equip $\Gamma$ and $T_0=\widetilde {(\Gamma,x_0)}$ with simplicial metrics, where every edge has length $1$.
In this setting we denote by $\Omega(\Gamma)$ the set of all semi-infinite reduced edge-paths $e_1,e_2,\dots, $ in $\Gamma$. For $n\ge 1$ denote by $\Omega_n(\Gamma)$ the set of all reduced edge-paths $e_1,e_2,\dots, e_n$ of length $n$ in $\Gamma$. Also denote $\Omega_\ast=\cup_{n=1}^\infty \Omega_n(\Gamma)$.
If $A=\{a_1,\dots, a_N\}$ is a free basis of $F_N$, then the graph $R_A$, with a single vertex $x_0$ and with $N$ petal-edges marked $a_1,\dots, a_N$, is a simplicial chart on $F_N$. In this case the corresponding covering tree $T_A:=\widetilde R_A$ is exactly the Cayley tree of $F_N$ with respect to $A$. We refer to such simplicial chart $R_A$ as an \emph{$N$-rose}.
For a simplicial chart $\Gamma$, the marking $\kappa$ induces an $F_N$-equivariant quasi-isometry $F_N\to T_0$, which we use to identify $\partial F_N$ with $\partial T_0$. For $(x,y)\in \partial^2F_N$ denote by $\gamma_{x,y}$ the bi-infinite geodesic in $T_0$ from $x$ to $y$. The group $F_N=\pi_1(\Gamma,x_0)$ acts on $T_0=\widetilde \Gamma$ by covering transformations, which is a free and isometric discrete action with $T_0/F_N=\Gamma$.
\begin{defn}[Cylinders and weights]
Let $\Gamma$ be a simplicial chart on $F_N$, with $T_0=\widetilde \Gamma$.
(1) For two distinct vertices $p,q\in T_0$ denote by $Cyl_\Gamma([p,q])$ the set of all $(x,y)\in\partial^2 F_N$ such that the bi-infinite geodesic $\gamma_{x,y}$ contains $[p,q]$ as a subsegment. The set $Cyl_\Gamma([p,q])\subseteq \partial^2 F_N$ is called the \emph{cylinder set} corresponding to $[p,q]$.
For any $g\in F_N$ and any $p,q\in VT_0, p\ne q$ we have $gCyl_\Gamma([p,q])=Cyl_\Gamma([gp,gq])$. The cylinder sert $Cyl_\Gamma([p,q])\subseteq \partial^2 F_N$ are compact and open, and the collection of all such cylinder sets forms a basis for the subspace topology on $\partial^2 F_N$ defined above.
(2) For a geodesic current $\eta\in\Curr(\FN)$ denote by $\langle v,\eta\rangle_\Gamma:=\eta\left( Cyl_\Gamma([p,q])\right)$ where $[p,q]$ is any lift of $v$ to $T_0$. The number $0\le \langle v,\eta\rangle_\Gamma<\infty$ is called the \emph{weight} of $v$ in $\eta$ with respect to $\Gamma$.
\end{defn}
If $\Gamma=R_A$ is an $N$-rose, we use the subscript $A$ rather than $R_A$ for chart-related notations. E.g. $\langle v,\eta\rangle_A:=\langle v,\eta\rangle_{R_A}$, etc.
\begin{prop}\cite{Ka06}
Let $F_N$ be free of rank $N\ge 2$ and let $\Gamma$ be a simplicial chart on $F_N$. Then:
\begin{enumerate}
\item For $\eta, \eta_n\in \Curr(\FN)$, where $n=1,2,\dots$, we have $\lim_{n\to\infty} \eta_n=\eta$ in $\Curr(\FN)$ if and only if for every $v\in\Omega_\ast(\Gamma)$ we have
\[
\lim_{n\to\infty} \langle v,\eta_n\rangle_\Gamma=\langle v, \eta\rangle_\Gamma.
\]
\item Let $\eta\in \Curr(\FN)$. Then for every $k\ge 1$ and every $v\in\Omega_k(\Gamma)$ we have
\[
\langle v, \eta\rangle_\Gamma=\sum_{e\in E\Gamma \text{ with } ve\in \Omega_{k+1}(\Gamma)}\langle ve, \eta\rangle_\Gamma=\sum_{e'\in E\Gamma \text{ with } e'v\in \Omega_{k+1}(\Gamma)}\langle e'v, \eta\rangle_\Gamma. \tag{$\ddag$}
\]
Moreover, any system of finite nonnegative weights on $\Omega_\ast(\Gamma)$ satisfying uniquely determines a current $\eta\in\Curr(\FN)$ realizing these weights.
\end{enumerate}
\end{prop}
Condition $(\ddag)$ is often called the \emph{switch condition} for $\Gamma$.
For $v\in \Omega_\ast(\Gamma)$ and a nondegenerate closed reduced and cyclically reduced edge-path $w$ in $\Gamma$, denote by $\langle v,w\rangle_\Gamma$ the number of ways in which $v$ can be read, reading forwards or backwards, in a circle of length $|w|$ labelled by $w$. The number $\langle v,w\rangle_\Gamma\ge 0$ is called the \emph{number of occurrences} of $v$ in $w$.
A key useful fact that follows from the definitions is:
\begin{lem}
Let $F_N$ be free of rank $N\ge 2$ and let $\Gamma$ be a simplicial chart on $F_N$. Let $v\in \Omega_\ast(\Gamma)$ and let $w$ be a nondegenerate closed reduced and cyclically reduced edge-path in $\Gamma$. Then $\langle v,w\rangle_\Gamma=\langle v,\eta_w\rangle_\Gamma$.
\end{lem}
\hfill $\qed$
\begin{defn}[Uniform current]\label{d:uc}
Let $F_N=F(A)$ be free of rank $N\ge 2$ with a free basis $A$. The \emph{uniform current} $\nu_A\in\Curr(\FN)$ corresponding to $A$ is the current given by the weights $\langle v,\nu_A\rangle_A=\frac{1}{N(2N-1)^{k-1}}$ for every $1\ne v\in F_N$ with $|v|_A=k\ge 1$.
\end{defn}
For a current $\eta\in\Curr(\FN)$ the \emph{support} $\mbox{Supp}(\eta)\subseteq \partial^2 F_N$ is
\[
\mbox{Supp}(\eta):=\partial^2 F_N- \cup \{U\subseteq \partial^2 F_N| U \text{ is open and } \eta(U)=0\}.
\]
Thus $\mbox{Supp}(\eta)$ is a closed $F_N$-invariant subset of $\partial^2 F_N$.
\begin{rem}\label{r:sup}
Let $\Gamma$ be a simplicial chart on $F_N$. If $\eta\in\Curr(\FN)$ and $(x,y)\in\partial^2 F_N$ then $(x,y)\in\mbox{Supp}(\mu)$ if and only if every finite nondegenerate edge subpath of $\gamma_{x,y}$ projects to a reduced edge-path $v$ in $\Gamma$ with $\langle v,\eta\rangle_\Gamma>0$.
\end{rem}
\subsection{Geometric intersection form}
We refer the reader to \cite{B,FM,KL10,V15} for the background and basic info regarding the Outer space, and only recall a few facts and definitions here.
Denote by $cv_N$ the (unprojectivized) Culler-Vogtmann Outer space for $F_N$. Elements of $cv_N$ are equivariant $F_N$-isometry classes of free and discrete minimal isometric actions of $F_N$ on $\mathbb R$-trees. In particular, if $\Gamma$ is a simplicial chart on $F_N$ then $T_0=\widetilde \Gamma$ defines a point of $cv_N$.
There is a natural ``axes'' topology on $cv_N$ and a (right) action of $\mbox{Out}(F_N)$ on $cv_N$ by homeomorphisms. Moreover, the closure $\mbox{\overline{cv}}_N$ of $cv_N$ in the axes topology is known to consist of all minimal nontrivial "very small" isometric actions on $F_N$ on $\mathbb R$-trees (again considered up to $F_N$-equivariant isometry), and the action of $\mbox{Out}(F_N)$ extends to $\mbox{\overline{cv}}_N$. For $T\in\mbox{\overline{cv}}_N$ and $g\in F_N$ denote by $||g||_T$ the \emph{translation length} of $g$ in $T$, that is $||g||_T=\inf_{x\in T} d_T(x,gx)$.
A key result of Kapovich and Listing~\cite{KL09} is:
\begin{prop}\label{p:int}
Let $F_N$ be free of finite rank $N\ge 2$. Then there exists a continuous \emph{geometric intersection form}
\[
\langle -\, , \, - \rangle:\mbox{\overline{cv}}_N\times \Curr(\FN)\to \mathbb R_{\ge 0}
\]
satisfying the following properties:
\begin{enumerate}
\item The map $\langle -\, , \, - \rangle$ is $\mathbb R_{\ge 0}$-homogeneous with respect to the first argument and $\mathbb R_{\ge 0}$-linear with respect to the second argument.
\item For every $\varphi\in\mbox{Out}(F_N)$, every $T\in\mbox{\overline{cv}}_N$ and every $\eta\in\Curr(\FN)$ we have
\[
\langle T,\varphi\eta\rangle=\langle T\varphi,\eta\rangle.
\]
\item For every $1\ne g\in F_N$ and every $T\in\mbox{\overline{cv}}_N$ we have $\langle T,\eta_g\rangle=||g||_T$.
\end{enumerate}
\end{prop}
In view of the above proposition, for $T\in\mbox{\overline{cv}}_N$ and $\eta\in\Curr(\FN)$ we denote $||\eta||_T=\langle T,\eta\rangle$.
For every $T\in\mbox{\overline{cv}}_N$ there is an associated \emph{dual lamination} $L(T)\subseteq \partial^2 F_N$, which is a certain closed $F_N$-invariant and flip-invariant subset of $\partial^2 F_N$ recording the information about sequences of elements of $F_N$ with translation length in $T$ converging to $0$. We refer the reader to \cite{KL10} for the precise definition of $L(T)$ and additional details.
We need the following key result of \cite{KL10}:
\begin{prop}\label{p:KL100}
Let $T\in\mbox{\overline{cv}}_N$ and $\eta\in\Curr(\FN)$. Then $||\eta||_T=0$ if and only if $\mbox{Supp}(\eta)\subseteq L(T)$.
\end{prop}
\section{Filling geodesic currents}
\begin{defn} Let $F_N$ be free of rank $N\ge 2$.
\begin{enumerate}
\item An element $g\in F_N$ is \emph{filling} in $F_N$ if for every $T\in\mbox{\overline{cv}}_N$ we have $||g||_T>0$.
\item A current $\eta\in\Curr(\FN)$ is \emph{filling} in $F_N$ if for every $T\in\mbox{\overline{cv}}_N$ we have $||\eta||_T>0$.
\end{enumerate}
\end{defn}
Thus an element $1\ne g\in F_N$ is filling if and only if $\eta_g$ is a filling.
One of the main results of \cite{KL10} is:
\begin{prop}\cite[Corollary~1.6]{KL10}\label{p:fs}
Let $\eta\in\Curr(\FN)$ be such that $\mbox{Supp}(\eta)=\partial^2 F_N$. Then $\eta$ is filling in $F_N$.
\end{prop}
We will sometimes say that a current $\eta\in \Curr(\FN)$ \emph{has full support} if $\mbox{Supp}(\eta)=\partial^2 F_N$.
Remark~\ref{r:sup} directly implies:
\begin{prop}
Let $\Gamma$ be a simplicial chart on $F_N$.
Then $\eta\in\Curr(\FN)$ has full support if and only if for every nondegenerate edge-path $v$ in $\Gamma$ we have $\langle v,\eta\rangle_\Gamma>0$.
\end{prop}
\begin{lem}\label{lem:w}
Let $0\ne \nu\in\Curr(\FN)$.
Let $w$ be a nondegenerated closed reduced and cyclically reduced edge-path in $\Gamma$ such that for every $n\ge 1$ we have $\langle w^n,\nu\rangle_\Gamma>0$. Then $\mbox{Supp}(\eta_w)\subseteq \mbox{Supp}(\nu)$.
\end{lem}
\begin{proof}
Since geodesic currents are flip-invariant, the assumptions of the lemma imply that the points $p_+=(w^{-\infty},w^{\infty}), p_-= (w^{\infty},w^{-\infty})\in \partial^2 F_N$ belong to $\mbox{Supp}(\nu)$. Since $\mbox{Supp}(\eta_w)=\cup_{h\in F_N}h\{p_+,p_-\}$, we then have $\mbox{Supp}(\eta_w)\subseteq \mbox{Supp}(\nu)$.
\end{proof}
\begin{lem}\label{lem:f}
Let $1\ne g\in F_N$ be a filling element and let $0\ne \nu\in \Curr(\FN)$ be a current such that $\mbox{Supp}(\eta_g)\subseteq \mbox{Supp}(\nu)$. Then $\nu$ is a filling current.
\end{lem}
\begin{proof}
Suppose, on the contrary, that $\nu$ is not filling. Then there exists $T\in cv_N$ such that $\langle T,\nu\rangle=0$.
By \cite[Theorem~1.1]{KL10} this implies that $\mbox{Supp}(\nu)\subseteq L(T)$. Hence $\mbox{Supp}(\eta_g)\subseteq L(T)$ as well. Therefore, again by \cite[Theorem~1.1]{KL10}, we have $0=\langle T,\eta_g\rangle=||g||_T$, which contradicts that $g$ is filling.
\end{proof}
\begin{cor}\label{c:z}
Let $z$ be a nondegenerated closed reduced and cyclically reduced edge-path in $\Gamma$ representing the conjugacy class of a filling element $g\in F_N$.
Let $0\ne \nu\in\Curr(\FN)$ be such that for every $n\ge 1$ we have $\langle z^n,\nu\rangle_\Gamma>0$. Then $\nu$ is a filling current.
\end{cor}
\begin{proof}
Lemma~\ref{lem:w} implies that $\mbox{Supp}(\eta_g)\subseteq \mbox{Supp}(\nu)$.
Therefore, by Lemma~\ref{lem:f}, the current $\nu$ is filling.
\end{proof}
\begin{prop}\label{p:ai}
Let $0\ne\nu\in\Curr(\FN)$ be such that for some free basis $A=\{a_1,\dots, a_N\}$ the following holds. For $i=1,\dots, N$ let $w_i$ be a closed reduced and cyclically reduced edge-path in $\Gamma$ representing the conjugacy class of $a_i$ in $F_N$. For $1\le i<j\le N$ let $w_{i,j}$ be a closed reduced and cyclically reduced edge-path in $\Gamma$ representing the conjugacy class of $a_i$ in $F_N$. Suppose that we have $\langle w_i^n,\nu\rangle_\Gamma>0$ for $i=1,\dots, N$ and that we have $\langle w_{ij}^n,\nu\rangle_\Gamma>0$ for all $1\le i<j\le N$ and all $n\ge 1$. Then the current $\nu_X\in\Curr(\FN)$ is filling.
\end{prop}
\begin{proof}
Indeed, suppose $\nu$ is not filling. Then there exists $T\in\mbox{\overline{cv}}_N$ such that $\langle T,\nu\rangle=0$. By \cite[Theorem~1.1]{KL10} this implies that $\mbox{Supp}(\nu)\subseteq L(T)$.
Lemma~\ref{lem:w} implies that for all $i=1,\dots, N$ we have $\mbox{Supp}(\eta_{a_i})\subseteq \mbox{Supp}(\nu)$, and for all $1\le i<j\le N$ we have $\mbox{Supp}(\eta_{a_ia_j})\subseteq \mbox{Supp}(\nu)$.
Since $\mbox{Supp}(\nu)\subseteq L(T)$, \cite[Theorem~1.1]{KL10} implies that for $i=1,\dots, N$
\[
0=\langle T, \eta_{a_i}\rangle=||a_i||_T
\]
and for all $1\le i<j\le N$ we have
\[
0=\langle T, \eta_{a_ia_j}\rangle=||a_ia_j||_T
\]
Thus all $a_i$ and $a_ia_j$ act elliptically on $T$ and so have nonempty fixed sets in $T$.
For $1\le i<j\le N$, the elements $a_i,a_j,a_{ij}$ act elliptically on $T$, and therefore, by \cite[Proposition~1.8]{Pa}, $Fix_T(a_i)\cap Fix_T(a_j)\ne\varnothing$. Thus $Fix_T(a_1),\dots Fix_T(a_N)$ are nonempty subtrees of $T$ with pairwise nonempty intersections. Therefore $\cap_{i=1}^N Fix_T(a_i)\ne\varnothing$. Hence $F_N$ has a global fixed point in $T$, which contradicts the fact that $T\in\mbox{\overline{cv}}_N$ is a nontrivial $F_N$-tree.
\end{proof}
\begin{prop}
Let $F_N=F(A)$ (where $N\ge 2$) and let $w\in F(A)$ be a freely and cyclically reduced word such that for every $v\in F(A)$ with $|v|_A=3$, the word $v$ occurs as a subword of some cyclic permutation of $w$ or of $w^{-1}$. Then:
\begin{enumerate}
\item The element $w\in F_N$ is filling.
\item If $0\ne \nu\in \Curr(\FN)$ is such that for all $n\ge 1$ $\langle w^n,\nu\rangle_A>0$, then the current $\nu$ is filling in $F_N$.
\end{enumerate}
\end{prop}
\begin{proof}
Part (1) is exactly \cite[Corollary~5.6]{CM15}.
Now [art (1) implies part (2) by Corollary~\ref{c:z}.
\end{proof}
\section{Filling currents and $(M,\lambda,\varepsilon)$-minimality}
Also, as before, we denote by $T_A$ the Cayley graph of $F_N$ with respect to the free basis $A$. Thus $T_A$ is a simplicial tree with all edges of length $1$.
\begin{defn}
Let $0\ne \nu\in Curr(F_N)$. The \emph{automorphic distortion spectrum} of $\nu$ with respect to the free basis $A$ of $F_N$ is the set
\[
D_A(\nu):=\{\langle T_A, \varphi \nu \rangle| \varphi\in \mbox{Out}(F_N)\}
\]
Also denote $J_A(\nu):=\inf D_A(\nu)$.
\end{defn}
\begin{rem}
Thus $D_A(\nu)\subseteq \mathbb R_{>0}$. Since $\langle T_A, \varphi \nu,\rangle=\langle T_A\varphi,\nu\rangle=\langle T_{\varphi(A)}, \nu\rangle$, it is easy to see that the set $D_A(\nu)$ is independent of the choice of a free basis $A$ of $F_N$ and depends only on $\nu$. Nevertheless, we will keep the subscript $A$ in the notation $D_A(\nu)$ since for our purposes the fixed choice of $A$ is important.
Note also that for $1\ne w\in F_N$ and $\varphi\in \mbox{Out}(F_N)$ we have $\langle T_A, \varphi \eta_w \rangle=\langle T_A, \eta_\varphi(w)\rangle=||\varphi(w)||_A$. Therefore in this case
$D_A(\eta_w)=\{||\varphi(w)||_A| \varphi\in \mbox{Out}(F_N)\}\subseteq \mathbb Z_{>0}$, and $J_A(\eta_w)$ is the smallest $||.||_A$-length of elements in the orbit $\mbox{Out}(F_N)[w]$.
\end{rem}
We need the following useful result essentially proved in \cite[Theorem~1.2]{KL10}:
\begin{prop}\label{p:KL10}
Let $\nu\in \Curr(\FN)$ (where $N\ge 2$) be a filling current and let $A$ be a free basis of $F_N$. Then:
\begin{enumerate}
\item The set $D_A(\nu)$ is a discrete unbounded subset of $[0,\infty)$.
\item For every $C>0$ the set $\{\varphi\in \mbox{Out}(F_N)| \langle T_A, \varphi \nu \rangle\le C\}$ is finite.
\end{enumerate}
\end{prop}
\begin{proof}
The proof is a verbatim copy of the proof of \cite[Theorem~11.2]{KL10} where the same result was established under the assumption that $\nu\in\Curr(\FN)$ is filling. The only place in the proof of Theorem~11.2 in \cite{KL10} where the filling assumption on $\nu$ was used is at the bottom of page 1461 in \cite{KL10}, to show that $\langle T_\infty,\nu\rangle\ne 0$ for a certain tree $T_\infty\in\mbox{\overline{cv}}_N$ constructed earlier in the proof. However, in our case $\langle T_\infty,\nu\rangle\ne 0$ since $\nu$ is assumed to be filling in the present proposition.
\end{proof}
Proposition~\ref{p:KL10} immediately implies:
\begin{cor}
For $F_N$ and $A$ as in Proposition~\ref{p:KL10} let $\nu\in Curr(F_N)$ be a filling current. Then:
\begin{enumerate}
\item We have $J_A(\nu)\in D_A(\nu)$, so that $J_A(\nu)=\min D_A(\nu)$.
\item The set $\Delta_A(\nu)=\{\varphi\in\mbox{Out}(F_N)| \langle T_A, \varphi\nu\rangle=J_A(\nu)\}$ is finite and nonempty.
\end{enumerate}
\end{cor}
\begin{defn}
Let $F_N$ be free of finite rank $N\ge 2$, let $A$ be a free basis of $A$ and let $\nu\in Curr(F_N)$ be a filling current.
We call the set $\Delta_A(\nu):=\{\varphi\in\mbox{Out}(F_N)| \langle T_A, \varphi\nu\rangle=J_A(\nu)\}$ the \emph{$A$-minimizing set for $\nu$} and we call the integer $M_A(\nu):=\#\Delta_A(\nu)\ge 1$ the \emph{minimizing multiplicity} for $\nu$ with respect to $A$. Also put
Also let $J'_A(\nu)=\min (D_A(\nu)\setminus \{J_A(\nu)\})$ and let $\lambda_A(\nu)=\frac{J'_A(\nu)}{J_A(\nu)}$, so that $\lambda_A(\nu)>1$. We call $\lambda_A(\nu)$ the \emph{distortion threshold} for $\nu$ with respect to $A$. Finally denote $\Im_A(\nu)=\Delta_A(\nu)\nu=\{\varphi\nu|\varphi\in \Delta_A(\nu)\}\subseteq Curr(F_N)$ and call $\Im_A$ the \emph{orbit floor} for $\nu$.
\end{defn}
The following statement is a key technical result of this paper:
\begin{thm}\label{t:key1}
Let $F_N$ be free of finite rank $N\ge 2$, let $A$ be a free basis of $A$ and let $\nu\in \Curr(\FN)$ be a filling current. Let $\lambda$ be such that $1<\lambda<\lambda_A(\nu)$ and let $0<\varepsilon<1$ be such that $\lambda_A(\nu) >\lambda>1+\varepsilon$.
Let $\mathfrak W=\Delta_A(\nu)$ and let $M=M_A(\nu)=\#\mathfrak W$.
Then there exists a neighborhood $U=U([\nu], \lambda,\varepsilon)$ of $[\nu]$ in $\mathbb P\Curr(\FN)$ such that for every $1\ne w\in F_N$ with $[\eta_w]\in U$ the set $S=\mathfrak W[w]\subseteq \mathcal C_N$ is $(M,\lambda,\varepsilon, \mathcal W_N)$-minimizing.
\end{thm}
\begin{proof}
Denote $\Im=\Im_A(\nu)=\mathfrak W \nu \subseteq \Curr(\FN)$. Since $\Im=\mathfrak W \nu$ and $S=\mathfrak W [w]$, it follows that $\#\Im\le M$ and $\#S\le M$.
Also, by construction, $S\subseteq \mbox{Out}(F_N)[w]$. Therefore for every $[u],[u']\in S$ we have $\mbox{Out}(F_N)[u]=\mbox{Out}(F_N)[u']$. Thus conditions (1) and (2) of Definition~\ref{d:MLEW} hold for $S$.
We also have $||\nu'||_A=J_A(\nu)$ for all $\nu'\in \Im$. Moreover, if $\nu'\in \Im$ and $\varpsi\in \mbox{Out}(F_N)$ is such that $\varpsi\nu'\not\in \Im$ then $||\varpsi\nu'||_A/J_A(\nu)\ge \lambda_A(\nu)>1$. In particular, the latter statement holds whenever $\tau\in \mathcal W_N$ is a Whitehead move such that $\tau\nu'\not\in \Im$.
For each $\nu'\in \Im$ denote \[R_\Im(\nu)=\{\varpsi\in\mbox{Out}(F_N)|\varpsi\nu'\in \Im\}.\] Proposition~\ref{p:KL10} also implies that for each $\nu'\in \Im$ the set $R_\Im(\nu')$ is finite.
Moreover, for every $\nu', \nu''\in \Im$ there are $\varphi', \varphi''\in \mathfrak W$ such that $\varphi'\nu=\nu'$ and $\varphi''\nu=\nu''$ so that $\varphi''(\varphi')^{-1}\in R_\Im(\nu')$ and $\varphi'(\varphi'')^{-1}\in R_\Im(\nu'')$. Therefore for every $\nu'\in \Im$ we have $R_\Im(\nu')\nu'=\Im$.
For exactly the same reason, if $[u]=\varphi'[w]$, where $\varphi'\in \mathfrak W$ and $\nu'=\varphi'\nu\in \Im$ then
\[
R_\Im[u]=S.\tag{!}
\]
Since $\lambda_A(\nu) (1-2\varepsilon)>\lambda>1+\varepsilon$, we can choose $\lambda<\lambda_1<\lambda_A(\nu)$ so that
\[
\lambda_A(\nu)>\lambda_1>\lambda_1(1-\varepsilon)>\lambda>1+\varepsilon.
\]
By continuity of the intersection form $\langle - , - \rangle$ and of the action of $\mbox{Out}(F_N)$ on $\mathbb PCurr(F_N)$, there exist neighborhoods $U([\nu'])$ of $[\nu']$ in $\mathbb PCurr(F_N)$, where $\nu'\in S$, and there exists a neighborhood $U([\nu])$ of $[\nu]$, such that the following hold:
\begin{itemize}
\item[(a)] If $\nu'\in \Im$ and $[\eta]\in U([\nu'])$ then for every $\varpsi\in R_\Im(\nu')$ with $\nu''=\varpsi\nu\in \Im$ we have $\varpsi[\eta]\in U([\nu''])$ and
\[
1-\varepsilon \le \frac{||\varpsi\eta||_A}{||\eta||_A}\le 1+\varepsilon.
\]
\item[(b)] If $\nu'\in \Im$, $[\eta]\in U([\nu'])$ and $\tau\in \mathcal W_N$ is a Whitehead move such that $\tau\nu'\not\in \Im$ then
\[
\frac{||\tau\nu'||_A}{||\nu'||_A}\ge \lambda_1
\]
\item[(c)] For every $\varphi\in \mathfrak W$ (so that $\varpsi\nu\in \Im$) we have $\varphi U([\nu]) \subseteq U([\varphi\nu])$.
\end{itemize}
Recall that $S=\mathfrak W [w]=\{\varphi([w])| \varphi\in \mathfrak W\}$.
We will show that for every $1\ne w\in F_N$ with $[\eta_w]\in U([\nu])$ the set $S$ is $(M,\lambda,\varepsilon, \mathcal W_N)$-minimizing, that is that $U([\nu])$ satisfies the conclusion of this proposition.
Thus suppose that $1\ne w\in F_N$ is such that $[\eta_w]\in U([\nu])$.
We have already seen above that conditions (1) and (2) of Definition~\ref{d:MLEW} hold for $S$.
Let $[u]\in S$ be arbitrary. Thus $[u]=\varphi'[w]$ for some $\varphi\in \mathfrak W$, with $\nu'=\varphi'\nu\in \Im$.
By property (c) $\varphi'[\eta_{w}]\in U([\nu'])$. We also have $\eta_{u}=\varphi'\eta_{w}$. Thus $\varphi'[\eta_{w}]=[\eta_{u}]\in U([\nu'])$.
{\bf Claim 0.} For any $[x]\in S$ we have $||x||_A/||u||_A\in [1-\varepsilon, 1+\varepsilon]$.
Indeed, let $[x]\in S$, so that $[x]=\varphi'[w]$ for some $\varpsi'\in \mathfrak W$, so that $\nu''=\varphi''\nu\in \Im$. Then, as for $[u]$, we have $\varphi''[\eta_{w}]=[\eta_{x}]\in U([\nu''])$. Then $\varpsi=\varphi''\varphi^{-1}\nu'=\nu''$, so that $\varpsi\in R_\Im(\nu')$. We also have $[x]=\varpsi[u]$. Then property (a) implies that $||x||_A/||u||_A\in [1-\varepsilon, 1+\varepsilon]$, as required.
Claim~0 shows that conditions (3) of Definition~\ref{d:MLEW} hold for $S$.
{\bf Claim 1.} For any Whitehead move $\tau\in \mathcal W_N$ exactly one of the following occurs:
(i) We have $||\tau(u)||_A/||u||_A\in [1-\varepsilon, 1+\varepsilon]$, $\tau\in R_\Im(\nu')$, $\tau\nu'\in \Im$ and $\tau[u]\in S$.
(ii) We have $||\tau(u)||_A/||u||_A\ge \lambda_1>\lambda>1+\varepsilon$ and $\tau\not\in R_\Im(\nu')$ and $\tau[u]\not\in S$.
Indeed, suppose first that $\tau\in\mathbb R_\Im(\nu')$. Thus $\nu''=\tau\nu'\in \Im$. Since $[\eta_{u}]\in U([\nu'])$, property (a) implies that $\tau[\eta_{u}]\in U([\nu''])$ and that $||\tau(u)||_A/||u||_A\in [1-\varepsilon, 1+\varepsilon]$.
Suppose now that $\tau\not\in\mathbb R_\Im(\nu')$, so that $\tau\nu'\not\in \Im$. Since $[\eta_{u}]\in U([\nu'])$ property (b) implies that $||\tau(u)||_A/||u||_A\ge \lambda_1>\lambda>1+\varepsilon$. Since $||\tau(u)||_A/||u||_A>1+\varepsilon$, Claim~0 now implies that $\tau[u]\not\in S$. Thus Claim~1 is verified.
Claim~1 now implies that condition (4) of Definition~\ref{d:MLE} hold for $S$.
Thus $S$ is $(M,\lambda,\varepsilon, \mathcal W_N)$-minimizing, as required.
\end{proof}
\begin{cor}\label{cor:key1}
Let $F_N$ be free of finite rank $N\ge 2$, let $A$ be a free basis of $A$ and let $\nu\in \Curr(\FN)$ be a filling current. Let $\lambda$ be such that $1<\lambda<\lambda_A(\nu)$ and let $0<\varepsilon<1$ be such that $\lambda_A(\nu) >\lambda>1+\varepsilon$.
Let $\mathfrak W=\Delta_A(\nu)$ and let $M=M_A(\nu)=\#\mathfrak W$.
Then there exists a neighborhood $U_1=U_1([\nu], \lambda,\varepsilon)$ of $[\nu]$ in $\mathbb P\Curr(\FN)$ such that for every $1\ne w\in F_N$ with $[\eta_w]\in U$ the set $S=\mathfrak W[w]\subseteq \mathcal C_N$ is $(M,\lambda,\varepsilon)$-minimizing.
\end{cor}
\begin{proof}
First choose $\lambda'$ such that $\lambda_A(\nu)>\lambda'>\lambda>1$. Then choose $\varepsilon'$ such that $0<\varepsilon'<\varepsilon$ and
that $\lambda'(1-\varepsilon')>\lambda$. By Theorem~\ref{t:key1}, there exists a neighborhood $U=U([\nu], \lambda',\varepsilon')$ of $[\nu]$ in $\mathbb P\Curr(\FN)$ such that for every $1\ne w\in F_N$ with $[\eta_w]\in U$ the set $S=\mathfrak W[w]$ is $(M,\lambda',\varepsilon',\mathcal W_N)$-minimizing. Therefore, by Proposition~\ref{prop:cont}, the set $S$ is $(M,\lambda,\varepsilon)$-minimizing. Therefore $U_1:=U$ satisfies the requirements of the corollary.
\end{proof}
\begin{rem}\label{rem:v}
Suppose we are in the context of Theorem~\ref{t:key1} and that $U\subseteq\mathbb P\Curr(\FN)$ is a neighborhood of $[\nu]$ in $\mathbb P\Curr(\FN)$ provided by the conclusion of Theorem~\ref{t:key1}.
Then $U$ contains a ``basic'' neighborhood $U_0\subseteq U$ of $[\nu]$ defined as follows. There exist a finite collection $\mathbb V\subseteq F(A)-\{1\}$ and $\varepsilon_0>0$ such that for $[\eta]\in \mathbb P\Curr(\FN)$ we have $[\eta]\in U_0$ if and only if for every $v\in \mathbb V$
\[
\left| \frac{\langle v,\eta\rangle}{||\eta||_A} -\frac{\langle v,\nu\rangle}{||\nu||_A}\right| \le \varepsilon_0.
\]
Therefore, if $1\ne w\in F_N$ is such that for all $v\in\mathbb V$
\[
\left| \frac{\langle v,w\rangle}{||w||_A} -\frac{\langle v,\nu\rangle}{||\nu||_A}\right| \le \varepsilon_0 \tag{$\spadesuit$}
\]
then $[\eta_w]\in U_0\subseteq U$ and the conclusion of Theorem~\ref{t:key1} applies to $w$.
\end{rem}
\begin{defn}\label{d:adapt}
Let $\mathcal W=W_1, W_2, \dots, W_n, \dots$ be a sequence of $F_N$-valued random variables.
\begin{enumerate}
\item We say that $\mathcal W$ is \emph{tame} if for some (equivalently, any) free basis $A$ of $F_N$ there exists $C>0$ such that we always have $|W_n|_A\le Cn$ where $n\ge 1$.
\item Let $0\ne \nu\in \Curr(\FN)$. We say that the sequence $\mathcal W$ is \emph{$\nu$-adapted} if a.e. trajectory $w_1,w_2,\dots, w_n, \dots $ of $\mathcal W$ we have:
\[
\lim_{n\to\infty} [\eta_{w_n}]=[\nu]
\]
in $\mathbb PCurr(F_N)$.
\end{enumerate}
\end{defn}
In Definition~\ref{d:adapt} above, a random trajectory of $\mathcal W$ is implicitly required to satisfy $w_n\ne 1$ for all sufficiently large $n$ (which is needed in order for $\eta_{w_n}$ to be defined), but we do not require $||w_n||_A\to\infty$ as $n\to\infty$.
In particular, if $\nu=\eta_w$ for some $1\ne w\in F_N$, and the random process $\mathcal W$ always outputs $W_n=\eta_w$ for all $n\ge 1$, then $\mathcal W$ is $\nu$-adapted.
The following statement is key for our paper:
\begin{prop}\label{prop:key}
Let $F_N$ be free of finite rank $N\ge 2$, let $A$ be a free basis of $A$ and let $\nu\in \Curr(\FN)$ be a filling current. Let $\lambda$ be such that $1<\lambda<\lambda_A(\nu)$ and let $\varepsilon>0$ be such that $\lambda_A(\nu) >\lambda>1+\varepsilon$.
Let $\mathfrak W=\Delta_A(\nu)$ and let $M=M_A(\nu)=\#\mathfrak W$.
Let $\mathcal W=W_1, W_2, \dots, W_n, \dots$ be a $\nu$-adapted sequence of $F_N$-valued random variables. Then the following hold:
\begin{enumerate}
\item For a.e. trajectory $\xi=(w_1,w_2,\dots, w_n, \dots )$ of $\mathcal W$ there exists $n_0=n_0(\xi)\ge 1$ such that for all $n\ge n_0$ the set $S_n=\mathfrak W [w_n]$ is $(M,\lambda,\varepsilon)$-minimizing.
\item We have
\[
\lim_{n\to\infty} Pr\big( \mathfrak W[W_n] \text{ is $(M,\lambda,\varepsilon)$-minimizing}\big)=1.
\]
\end{enumerate}
\end{prop}
\begin{proof}
Let $U_1=U_1([\nu],\lambda,\varepsilon)\subseteq \mathbb P\Curr(\FN)$ be a neighborhood of $[\nu]$ in $\mathbb P\Curr(\FN)$ whose existence is provided by Corollary~\ref{cor:key1}. Choose a basic neighborhood $U_0\subseteq U_1$ of $[\nu]$ in $\mathbb P\Curr(\FN)$ defined by some $\varepsilon>0$ and some finite collection $\mathbb V\subseteq F(A)-\{1\}$, as in Remark~\ref{rem:v}
Since $\mathcal W$ is adapted to $\nu$, for a.e. trajectory $\xi=(w_1,w_2,\dots, w_n, \dots )$ of $\mathcal W$ we have $[\eta_{w_n}]\in U_0$ and, since a.e. convergence implies convergence in probability, we also have
\[
\lim_{n\to\infty} Pr\big( \mathfrak W[\eta_{W_n}]\in U_0\big)=1.
\]
Now Corollary~\ref{cor:key1} implies that statements (1) and (2) of Proposition~\ref{prop:key} hold.
\end{proof}
Note in the context of Proposition~\ref{prop:key}, if $1\ne w\in F_N$ is such that $S=\mathfrak W [w]$ is $(M,\lambda,\varepsilon)$-minimizing then $\#S\le M$ and every element of $S$ is $(M,\lambda,\varepsilon)$-minimal.
\begin{thm}\label{t:A}
Let $F_N=F(A)$ be a free group of finite rank $N\ge 2$ with a free basis $A$.
Let $\mathcal W=W_1, W_2,\dots$ be a sequence of $F(A)$-valued random variables. Let $0\ne\nu\in \Curr(\FN)$ be a filling geodesic current such that $\mathcal W$ is adapted to $\nu$.
Then there exist $M\ge 1$, $\lambda>1$ and a subset $\mathfrak W\subseteq \mbox{Out}(F_N)$ with $\#\mathfrak W\le M$ such that for every $0<\varepsilon<1$ with $\lambda>1+\varepsilon$ the following hold:
\begin{itemize}
\item[(a)] For a.e. trajectory $\xi=(w_1,w_2,\dots, w_n, \dots )$ of $\mathcal W$ there exists $n_0=n_0(\xi)\ge 1$ the following holds for all $n\ge n_0$:
\begin{enumerate}
\item The set $S_n=\mathfrak W [w_n]$ is $(M,\lambda,\varepsilon)$-minimizing.
\item For for every $\varphi\in \mathfrak W$ the conjugacy class $\varphi[w_n]\in S_n$ is $(M,\lambda,\varepsilon)$-minimal.
\item We have $\mathcal M([w_n])\subseteq \mathfrak W[w_n]$, and in particular, $\#\mathcal M([w_n])\le M$.
\item We have ${\rm rank}\ Stab_{\mbox{Out}(F_N)}([w_n])\le K(N, M)$, where $K(N,M)\ge 1$ is some constant depending only on $N$ and $M$.
\end{enumerate}
\item[(b)] The probability of each of the following events tends to $1$ as $n\to\infty$:
\begin{enumerate}
\item The set $\mathfrak W [W_n]$ is $(M,\lambda,\varepsilon)$-minimizing.
\item For for every $\varphi\in \mathfrak W$ the conjugacy class $\varphi[W_n]$ is $(M,\lambda,\varepsilon)$-minimal.
\item We have $\mathcal M([W_n])\subseteq \mathfrak W[W_n]$, and $\#\mathcal M([W_n])\le M$.
\item We have ${\rm rank}\ Stab_{\mbox{Out}(F_N)}([W_n])\le K(N, M)$, where $K(N,M)\ge 1$ is some constant depending only on $N$ and $M$.
\end{enumerate}
\end{itemize}
\end{thm}
\begin{proof}
Put $\mathfrak W=\Delta_A(\nu)$ and let $M=M_A(\nu)=\#\mathfrak W$.
Choose $\lambda$ be such that $1<\lambda<\lambda_A(\nu)$. L $0<\varepsilon<1$ be such that $\lambda>1+\varepsilon$.
Now Proposition~\ref{prop:key} implies that statemenst (a)(1) and (b)(1) of Theorem~\ref{t:A} holds, which, by definition of $(M,\lambda,\varepsilon)$-minimality, implies that (a)(2) and (b)(2) hold as well.
Now Proposition~\ref{p:MLE}(1) implies that statements (a)(3) and (b)(3) of Theorem~\ref{t:A} holds. Finally, Proposition~\ref{prop:st} implies that statements (a)(4) and (b)(4) of Theorem~\ref{t:A} hold.
\end{proof}
\begin{thm}\label{t:A'}
Let $F_N=F(A)$, $\nu$, $\mathfrak W\subseteq \mbox{Out}(F_N)$ and $\mathcal W=W_1, W_2,\dots$ and be as in Theorem~\ref{t:A}. Assume also that $\mathcal W$ is tame.
Then there exists $K_0\ge 1$ such that the following hold:
\begin{itemize}
\item[(a)] For a.e. independently chosen trajectories $\xi=w_1,w_2,\dots $ and $\xi'=w_1',w_2',\dots$ of $\mathcal W$ there exist $n_0,m_0\ge 1$ such that the following hold:
\begin{enumerate}
\item For all $n\ge n_0$, the $\mathfrak W$-speed-up of Whitehead's minimization algorithm on the input $w_n$ terminates in time at most $K_0 n$ and produces an element of $\mathcal M([w_n])$.
\item If $n\ge n_0$, then for any $u\in F_N$ the $\mathfrak W$-speed-up of Whitehead's algorithm decides in time at most $K_0\max\{n,|u|_A^2\}$, whether or not $\mbox{Aut}(F_N)w_n=\mbox{Aut}(F_N)u$.
\item For all $n\ge n_0, m\ge m_0$, the $\mathfrak W$-speed-up of Whitehead's algorithm decides in time at most $K_0\max \{n,m\}$, whether or not $\mbox{Aut}(F_N)w_n=\mbox{Aut}(F_N)w_m'$.
\end{enumerate}
\item[(b)] The probability of each of the following events tends to $1$ as $n\to\infty$:
\begin{enumerate}
\item The $\mathfrak W$-speed-up of Whitehead's minimization algorithm on the input $W_n$ terminates in time at most $K_0n$ and produces an element of $\mathcal M([W_n])$.
\item The element $W_n$ has the property that for any $u\in F_N$ the $\mathfrak W$-speed-up of Whitehead's algorithm decides in time at most $K_0\max\{n,|u|_A^2\}$, whether or not $\mbox{Aut}(F_N)W_n=\mbox{Aut}(F_N)u$.
\item Let $\mathcal W'=W_1', W_2',\dots$ be an independent copy of $\mathcal W$. Let $n_i,m_i\ge 1$ be such that $\lim_{i\to\infty} \min\{n_i,m_i\}=\infty$. Then the probability of the following event tends to $1$ as $i\to\infty$:
The $\mathfrak W$-speed-up of Whitehead's algorithm decides in time at most $K_0\max \{n_i,m_i\}$, whether or not $\mbox{Aut}(F_N)W_{n_i}=\mbox{Aut}(F_N)W_{m_i}'$.
\end{enumerate}
\end{itemize}
\end{thm}
\begin{proof}
Since $\mathcal W$ is tame, there exists $C>0$ such that for all $n\ge 1$ we always have $|W_n|_A\le Cn$.
Let $K\ge 1$ be the constant provided by Theorem~\ref{t:WHM}.
We will show that part (a) of Theorem~\ref{t:A'} holds as the proof of part (b) is essentially the same.
By part (1) of Theorem~\ref{t:A} we know that for all big enough $n\ge n_0$ (where $n_0$ depends on the random trajectory $\xi$) the set $S_n=\mathfrak W[w_n]$ is $(M,\lambda,\varepsilon)$-minimizing and every element of $S_n$ is $(M,\lambda,\varepsilon)$-minimal. The same is true for all $S_m'=\mathfrak W[w_m']$ for all $m\ge m_0=m_0(\xi')$.
(a)(1) Pick an element $\varpsi\in\mathfrak W$. Thus there is $C'\ge 1$ such that for every $g\in F_N$ we have $||\varpsi(g)||_A\le C'||g||_A$.
We first compute $[u_n]=\varpsi([w_n])\in S_n$. Thus $||u_n||_A\le C'||w_n||_A\le CC'n$, and $[u_n]$ is $(M,\lambda,\varepsilon)$-minimal. Therefore, by part (a) of Theorem~\ref{t:WHM}, the Whitehead minimization algorithm on $[u_n]$ terminates in time $\le K||u_n||_A\le KCC'n$ and and produces an element of $\mathcal M([u_n])=\mathcal M([w_n])$.
(a)(2) Let $n\ge n_0$. Again choose any $\varpsi\in \mathfrak W$. Then we have $w_n\in U_N(M,\lambda,\varepsilon;\varpsi)$. Therefore, by part (f) of Theorem~\ref{t:WHM}, the $\varpsi$-speed-up of Whitehead's algorithm decides in time at most $K\max \{|w_n|_A,|u|_A^2\}\le KC\max\{n,|u|_A^2\}$, whether or not $\mbox{Aut}(F_N)w_n=\mbox{Aut}(F_N)u$.
(a)(3) Suppose $n\ge n_0, m\ge m_0$. Thus $S_n, S_m'$ are $(M,\lambda,\varepsilon)$-minimizing and their elements are $(M,\lambda,\varepsilon)$-minimal. Choose any $\varpsi\in \mathfrak W$.
Since $\varpsi[w_n]\in S_n$ and $\varpsi[w_m']\in S_m'$ and since $S_n,S_m'$ are $(M,\lambda,\varepsilon)$-minimizing, it follows that $w_n,w_m'\in U_N(M,\lambda,\varepsilon;\varpsi)$. Thus, by part (e) of Theorem~\ref{t:WHM}, the $\varpsi$-speed-up of Whitehead's algorithm decides in time at most $K\max \{|w_n|_A,|w_m'|_A\}\le KC\max\{m,n\}$, whether or not $\mbox{Aut}(F_N)w_n=\mbox{Aut}(F_N)w_m'$.
\end{proof}
\section{Group random walks as a source of $(M,\lambda,\varepsilon)$minimality}
\begin{conv}[Terminology regarding random processes]
Let $B$ be a set with the discrete topology (such as a discrete group, the set of vertices of a graph, words in a finite alphabet, etc).
For any infinite sequence of $B$-valued random variables $\mathcal W=W_1, W_2, \dots, W_n, \dots$ we assume that the sample space $\Omega=B^\omega$ (as usual given the product topology for the discrete topologies on the factors $B$) is a probability space equipped with a Borel probability measure $Pr$. We will usually suppress the explicit mention of this probability measure $Pr$.
Thus a trajectory of $\mathcal W$ is a sequence $\zeta=(w_1, w_2,\dots, w_n, \dots)\in\Omega$, where all $w_i\in B$. We say that some property $\mathcal E$ \emph{holds for a.e. trajectory} of $\mathcal W$ if \[Pr(\zeta\in \Omega| \zeta \text{ satisfies } \mathcal E)=1.\]
\end{conv}
\begin{conv}
For a discrete probability measure $\mu:G\to [0,1]$ on a group $G$, we denote by $\langle \supp(\mu)\rangle_+$ the subsemigroup of $G$ generated by the support $\mbox{Supp}(\mu)$ of $\mu$. Note that we have $\langle \supp(\mu)\rangle_+=\cup_{n=1}^\infty \mbox{Supp}(\mu^{(n)})$ where $\mu^{(n)}$ is the $n$-fold convolution of $\mu$.
Thus for $g\in G$ we have $g\in \langle \supp(\mu)\rangle_+$ if and only if there exist $n\ge 1$ and $g_1,\dots, g_n\in G$ such that $g=g_1\dots g_n$ and $\mu(g_i)>0$ for $i=1,\dots, n$.
\end{conv}
\begin{defn}[Group random walk]
Let $G$ be a group and let $\mu:G\to [0,1]$ be a discrete probability measure on $G$. Let $X_1, X_2, \dots, X_n,\dots$ be a sequence of $G$-valued i.i.d. random variables, where each $X_i$ has distribution $\mu$. Put $W_n=X_1\dots X_n\in G$, where $n=1,2\dots$. The random process
\[
\mathcal W=W_1,W_2,\dots, W_n,\dots
\]
is called the \emph{random walk on $G$} defined by $\mu$.
\end{defn}
Recall that if $G$ is a group acting on a set $X$, and $\mu$ is a discrete probability measure on $G$, then a measure $\lambda$ on $X$ is called \emph{$\mu$-stationary} if $\lambda=\sum_{g\in G} \mu(g) g\lambda$.
If $G$ is a non-elementary word-hyperbolic group, a discrete probability measure $\mu$ on $G$ is called \emph{non-elementary} if $\langle \supp(\mu)\rangle_+$ contains some two independent loxodromic elements of $G$ (which, for a word-hyperbolic $G$ means some two elements $g_1,g_2\in G$ of infinite order such that $\langle g_1\rangle\cap \langle g_2\rangle=\{1\}$).
We need the following well-known fact (see, e.g. \cite[Theorem~1.1]{MT18} for the most general version of this statement for random walks on groups acting on Gromov-hyperbolic spaces; see \cite[Theorem 7.6]{Kaim00} specifically for the case of a word-hyperbolic $G$):
\begin{prop}\label{p:ne}
Let $G$ be a non-elementary word-hyperbolic group and let $\mu$ be a non-elementary discrete probability measure on $G$. Let $\mathcal W=W_1,W_2,\dots, W_n,\dots$ be the random walk on $G$ defined by $\mu$. Then:
\begin{enumerate}
\item For a.e. trajectory $w_1,w_2,\dots $ of $\mathcal W$ there exists $x\in\partial G$ such that $\lim_{n\to\infty} w_n=p$ in $G\cup \partial G$.
\item Putting, for $S\subseteq \partial G$, $\lambda(S)$ to be the probability that a trajectory of $\mathcal W$ converges to a point of $S$, defines a $\mu$-stationary Borel probability measure $\lambda$ on $\partial G$.
\end{enumerate}
\end{prop}
This measure $\lambda$ is called the \emph{exit measure} or the \emph{hitting measure} for $\mathcal W$.
Recall also that if $G$ is a word-hyperbolic group and $H\le G$ is a non-elementary subgroup, then $\partial G$ contains a unique nonempty minimal closed $H$-invariant subset $\Lambda(H)\subseteq \partial G$ called the \emph{limit set of $H$} (see \cite{KS96,KB02} for details).
We need the following fact which seems be general folklore knowledge, although it does not seem to appear in the literature. We include a proof, explained to us by Vadim Kaimanovich, for completeness.
\begin{prop}\label{p:supp}
Let $G$ be a non-elementary word-hyperbolic group, let $\mu$ be a non-elementary discrete probability measure on $G$, and let $\lambda$ be the exit measure on $\partial G$ for the random walk on $G$ defined by $\mu$.
Suppose $H\le G$ is a non-elementary subgroup such that $H\subseteq \langle \supp(\mu)\rangle_+$. Then $\Lambda(H)\subseteq \mbox{Supp}(\lambda)$.
In particular if $\Lambda(H)=\partial G$ then $\mbox{Supp}(\lambda)=\partial G$.
\end{prop}
\begin{proof}
Let $\lambda$ be the exit measure on $\partial G$ for the random walk determined by $\mu$.
For any $k\ge 1$, the measure $\lambda$ is also an exit measure for the random walk based on $\mu^{(k)}$, and therefore $\lambda$ is $\mu^{(k)}$-stationary.
Thus for every $n\ge 1$ we have $\lambda=\sum_{g\in G} \mu^{(n)}(g)\cdot g\lambda$. Hence $\lambda$ dominates $g\lambda$ whenever $n\ge 1$ and $\mu^{(n)}(g)>0$, that is, whenever $g\in\langle \supp(\mu)\rangle_+$. Since $H\subseteq \langle \supp(\mu)\rangle_+$, it follows that $\lambda$ dominates $h\lambda$ for every $h\in H$. Since $H$ is a subgroup of $G$, this implies that for all $h\in H$ the measures $\lambda$ and $h\lambda$ are in the same measure class. Hence for every $h\in H$ $\mbox{Supp}(\lambda)=h\mbox{Supp}(\lambda)$. Thus $\mbox{Supp}(\lambda)$ is a nonempty closed $H$-invariant subset of $\partial G$, and therefore $\Lambda(H)\subseteq \mbox{Supp}(\lambda)$, as claimed.
\end{proof}
Note that if $\langle \supp(\mu)\rangle_+$ contains a subgroup $H$ of $G$ such that $H$ has finite index in $G$, or such that $H$ is an infinite normal subgroup of $G$, then $\Lambda(H)=\partial G$ (see \cite{KS96}) and therefore we get $\mbox{Supp}(\lambda)=\partial G$ in the conclusion of Proposition~\ref{p:supp}.
\begin{thm}\label{t:rwa}
Let $F_N=F(A)$ be a free group of finite rank $N\ge 2$ with a free basis $A$. Let $\mu:F_N\to [0,1]$ be a finitely supported probability measure such that $\langle \supp(\mu)\rangle_+=F_N$.
Let $\mathcal W=W_1, W_2,\dots$ be the random walk on $F_N$ defined by $\mu$. Then $\mathcal W$ is tame, and there exists a filling current $0\ne \nu\in\Curr(\FN)$ such that $\mathcal W$ is adapted to $\nu$.
\end{thm}
\begin{proof}
Let $T_A$ be the Cayley graph of $F_N$ with respect to $A$. Thus $T_A$ is a $2N$-regular simplicial tree.
Since $\mu$ is finitely, supported, we have $C:=\max\{|g|_A\big| g\in F_N, \mu(g)>0\}<\infty$. Then for all $n$ we have $|W_n|_A\le Cn$. Thus $\mathcal W$ is tame.
As usual, define by $\check\mu:F_N\to [0,1]$ the probability measure on $F_N$ given by the formula $\check\mu(g)=\mu(g^{-1})$ for $g\in F_N$.
Note that $\mbox{Supp}(\check\mu)=(\mbox{Supp}(\mu))^{-1}=\{g^{-1}|g\in F_N, \mu(g)>0\}$. Enumerate $\mbox{Supp}(g)$ as $\mbox{Supp}(g)=\{g_1,\dots, g_r\}$ for some $r\ge 1$. Since $\langle \supp(\mu)\rangle_+=F_N$, for each basis element $a_i\in A$ (where $i=1,\dots, N$) and for each $\varepsilon\in\{\pm 1\}$ there exists a positive word $U_{i,\varepsilon}(x_1,\dots,x_r)$ such that $a_i^{\varepsilon}=U_{i,\varepsilon}(g_1,\dots,g_r)$ in $F_N$. Then $a_i^{-\varepsilon}=U_{i,\varepsilon}^R(g_1^{-1},\dots, g_r^{-1})$, where $U_{i,\varepsilon}^R(x_1,\dots, x_r)$ is the word $U_{i,\varepsilon}$ read in the reverse (but without inverting the letters). Since $g_1^{-1},\dots, g_r^{-1}\in \mbox{Supp}(\check\mu)$, and $1\le i\le N$, $\varepsilon=\pm 1$ were arbitrary, it follows that $a_1^{\pm 1},\dots, a_r^{\pm 1}$ belong to $\langle \mbox{Supp}(\check\mu)\rangle_+$. Hence this $\langle \mbox{Supp}(\check\mu)\rangle_+=F_N$.
Let $\lambda$ and $\check\lambda$ be the exit measures on $\partial F_N$ for the random walks defined by $\mu$ and $\check\mu$ accordingly. Then, by Proposition~\ref{p:supp}, we have $\mbox{Supp}(\lambda)=\mbox{Supp}(\check\lambda)=\partial F_N$.
The Cayley tree $T_A$ of $F_N$ is a proper $CAT(-1)$ geodesic metric space equipped with a properly discontinuous cocompact isometric action of $F_N$. Therefore by a result of Gekhtman \cite[Theorem~1.5]{Ge17} there exists a geodesic current $0\ne \nu\in\Curr(\FN)$ such that $\mathcal W$ is adapted to $\nu$, and, moreover, $\nu$ belongs to the measure class $\check\lambda \times \lambda$ on $\partial^2F_N$. Since both $\lambda$ and $\check\lambda$ have full support on $\partial F_N$, it follows that $\nu$ has full support on $\partial^2 F_N$. Therefore by \cite[Corollary~1.6]{KL10} the current $\nu$ is filling in $F_N$.
This completes the proof of Theorem~\ref{t:rwa}.
\end{proof}
We can now conclude that algebraic and algorithmic conclusions of Theorem~\ref{t:A} and Theorem~\ref{t:A'} apply in the case of $\mu$-random walks on $F_N$, where $\mu$ has finite support with $\langle \supp(\mu)\rangle_+=F_N$:
\begin{cor}
Let $F_N=F(A)$ be free of rank $N\ge 2$, with a free basis $A$. Let $\mu:F_N\to [0,1]$ be a finitely supported discrete probability measure such that $\langle \supp(\mu)\rangle_+=F_N$.
Let $\mathcal W=W_1,\dots, W_n, \dots $ be the random walk on $F_N$ defined by $\mu$.
Then there exist $M\ge 1$, $0<\varepsilon<1$ and $\lambda>1+\varepsilon$ and a subset $\mathfrak W\subseteq \mbox{Out}(F_N)$ with $\#\mathfrak W\le M$ such that the conclusions of Theorem~\ref{t:A} and Theorem~\ref{t:A'} hold for $\mathcal W$ with these choices of $M,\lambda,\varepsilon,\mathfrak W$.
\end{cor}
\section{Finite-state Markov chains and the frequency measures}
We recall some basic notions and facts regarding finite state Markov chains here and refer the reader to \cite{DZ98,Ga13,Ki15,Ku18} for proofs and additional details.
\subsection{Finite-state Markov chains.} Recall that a \emph{finite-state Markov chain, or FSMC} $\mathcal X$ is defined by a finite nonempty set $S$ of \emph{states} and by a family of \emph{transition probabilities} $p_\mathcal X(s,s')\ge 0$, where $s,s'\in S$ such that for every $s\in S$ $\sum_{s'\in S} p_\mathcal X(s,s')=1$. Then for every integer $n\ge 1$ we also get the \emph{$n$-step transition probabilities} $p_\mathcal X^{(n)}(s,s')$ where $p_\mathcal X^{(1)}(s,s')=p_\mathcal X(s,s')$ and where for $n\ge 2$ and $s,s'\in S$ we have
\[
p_\mathcal X^{(n)}(s,s')=\sum_{s''\in S} p_\mathcal X^{(n-1)}(s,s'')p_\mathcal X(s'',s').
\]
The \emph{sample space} associated with $\mathcal X$ is the product space $S^\mathbb N=\{\xi=(s_1,s_2,s_3,\dots, s_n, \dots)| s_i\in S \text{ for } i\ge 1\}$. The set $S$ is given the discrete topology and $S^\mathbb N$ is given the corresponding product topology, which makes $S^\mathbb N$ a compact metrizable totally disconnected topological space. For $i\ge 1$ we denote by $X_i:S^\mathbb N\to S$ the function picking out the $i$-th coordinate of an element of $S^\mathbb N$.
The \emph{transition matrix} $M=M(\mathcal X)$ is an $S\times S$ matrix where for $s,s'\in S$ the entry $M(s,s')$ of $M$ is defined as $M(s,s')=p_\mathcal X^{(n)}(s,s')$. Thus $M(\mathcal X)$ is a nonnegative matrix, where the sum of the entries in each row is equal to $1$. Also, for all $n\ge 1$ and $s,s'\in S$ we have $p_\mathcal X^{(n)}(s,s')=(M^n)(s,s')$. A FSMC $\mathcal X$ as above is called \emph{irreducible} if for all $s,s'\in S$ there exists $n\ge 1$ such that $p_\mathcal X^{(n)}(s,s')>0$. Thus $\mathcal X$ is irreducible if and only if the nonnegative matrix $M(\mathcal X)$ is irreducible in the sense of Perron-Frobenius theory.
For an FSMC $\mathcal X$, given a \emph{initial probability distribution} $\mu$ on $S$, we obtain the corresponding \emph{Markov Process} $\mathcal X_{\mu}=X_1,\dots, X_n,\dots $ where each $X_i$ is an $S$-valued random variable with probability distribution $\mu_i$ on $S$, where $\mu_1=\mu$ and where for $i\ge 2$ and $s'\in S$ we have $\mu_i(s')=\sum_{s\in S} \mu_{i-1}(s)p_\mathcal X(s,s')$. An initial distribution $\mu$ on $S$ is called \emph{stationary} for $\mathcal X$ if $\mu_i=\mu$ for all $i\ge 1$ (equivalently, if $\mu_2=\mu$).
It is well-known, by the basic result of Perron-Frobenius theory, that if $\mathcal X$ is an irreducible finite-state Markov chain with state set $S$, then there is a unique stationary probability distribution $\mu$ on $S$ for $\mathcal X$, and that it satisfies $\mu(s)>0$ for all $s\in S$. In this case the vector $(\mu(s))_{s\in S}$ is the Perron-Frobenius eigenvector of $||.||_1$-norm $1$ for the matrix $M(\mathcal X)$ with eigenvalue $\lambda=1$, and, moreover, $\lambda=1$ is the Perron-Frobenius eigenvalue for $M(\mathcal X)$. In particular, the eigenvalue $\lambda=1$ is simple and is equal to the spectral radius of $M(\mathcal X)$.
For an FSMC $\mathcal X$ with state set $S$, a word $w=s_1\dots s_n\in S^n$ of length $n\ge 2$ is called \emph{feasible} if $p_\mathcal X(s_1,s_2)\dots p_\mathcal X(s_{n-1},s_n)>0$. Also, we consider all words $w=s\in S$ of length $n=1$ to be \emph{feasible}. (Hence every nonempty subword of a feasible word is also feasible). An element $\xi=(s_1,s_2,\dots )\in S^\mathbb N$ is \emph{feasible} for $\mathcal X$ if for every $n\ge 1$ the word $s_1\dots s_n$ is feasible. Denote by $(S^\mathbb N)_+$ the set of all feasible $\xi\in S^\mathbb N$. Also, for every $n\ge 1$ denote by $(S^n)_+$ the set of all feasible $s_1\dots s_n\in S^n$.
For a word $w=s_1\dots s_n\in S^n$ (where $n\ge 2$) put
\[
p_\mathcal X(w):=p_\mathcal X(s_1,s_2)\dots p_\mathcal X(s_{n-1},s_n)
\]
Any initial probability distribution $\mu$ on $S$ defines a Borel probability $\mu_\infty$ via the standard convolution formulas. Namely, if $n\ge 1, s_1,\dots s_n\in S$ then
\[
\mu_\infty \left(Cyl(s_1\dots s_n)\right):=\mu(s_1)p_\mathcal X(s_1,s_2)\dots p_\mathcal X(s_{n-1},s_n)=\mu(s_1)p_\mathcal X(s_1\dots s_n)
\]
where $Cyl(s_1\dots s_n)=\{\xi\in S^\mathbb N| X_i(\xi)=s_i \text{ for } i=1,\dots, n\}$.
If $\mu$ is strictly positive on $S$, then the support $supp(\mu_\infty)$ of $\mu_\infty$ is equal to $(S^\mathbb N)_+$. In particular, that is the case if $\mathcal X$ is an irreducible FSMC and $\mu$ is the unique stationary probability distribution on $S$.
\begin{defn}[Occurrences and frequencies]\label{d:occ}
Let $\mathcal X$ be an irreducible finite-state Markov chain with state set $S$.
(1) For a word $w=s_1\dots s_n\in S^n$ (where $n\ge 1$) and an element $s\in S$ we denote by $\langle s,w\rangle$ the number of those $i\in\{1,\dots, n\}$ such that $s_i=s$. We call $\langle s,w\rangle$ the \emph{number of occurrences of $s$ in $w$}. We also put $\theta_s(w)=\frac{\langle s,w\rangle}{|w|}$, where $|w|=n$ is the length of $w$. We call $\theta_s(w)$ the \emph{frequency} of $s$ in $w$.
(2) The above notions can be extended from $s$ to arbitrary nonempty words $v\in S^\ast$ as follows. Let $v=y_1\dots y_m\in S^m$ where $y_j\in S$ for $j=1,\dots, m$. Also denote by $w^\infty$ the semi-infinite word $w^\infty=wwww\dots $. For an arbitrary integer $i\ge 1$ we still denote by $s_i\in S$ the $i$-th letter of $w^\infty$. Now define $\langle v,w\rangle$ to be the number of $i\in\{1,\dots, n\}$ such that in $w^\infty$ we have $s_i=y_1, s_{i+1}=y_2,\dots, s_{i+m-1}=y_m$. We call $\langle v,w\rangle$ the \emph{number of occurrences of $v$ in $w$}, and we call $\theta_v(w)=\frac{\langle v,w\rangle}{|w|}$ the \emph{frequency of $v$ in $w$}.
\end{defn}
We record the following immediate corollary of the above definition (which holds since we defined the numbers of occurrences in $w$ cyclically).
\begin{lem}\label{lem:occ}
Let $w\in S^n$ where $n\ge 1$. Then the following hold:
\begin{enumerate}
\item We have $n=|w|=\sum_{s\in S} \langle s,w\rangle$ and $1=\sum_{s\in S} \theta_s(w)$.
\item For every $m\ge 1$ we have $n=|w|=\sum_{v\in S^m} \langle v,w\rangle$ and $1=\sum_{v\in S^m} \theta_v(w)$.
\item For every $m\ge 1$ and every $v\in S^m$ we have
\[
\langle v,w\rangle=\sum_{s\in S} \langle vs,w\rangle=\sum_{s'\in S} \langle s'v,w\rangle.
\]
and
\[
\theta_v(w)=\sum_{s\in S} \theta_{vs}(w)=\sum_{s'\in S} \theta_{s'v}(w).
\]
\end{enumerate}
\end{lem}
For a finite-state Markov chain $\mathcal X$ with state set $S$ and an element $\xi=(s_1,s_2,s_3,\dots, s_n, \dots)$ of $S^\mathbb N$, we denote $w_n=s_1\dots s_n\in S^n$, where $n\ge 1$.
The Strong Law of Large numbers applies to a finite-state Markov chain implies:
\begin{prop}\label{prop:fr}
Let $\mathcal X$ be an irreducible finite-state Markov chain with state set $S$ and let $\mu_0$ be the unique stationary probability distribution on $S$. Let $\mu$ be an arbitrary initial distribution on $S$ defining the corresponding Markov process $\mathcal X_{\mu}=X_1,\dots, X_n,\dots $. Then the following hold:
\begin{enumerate}
\item For every $s\in S$ and for $\mu_\infty$-a. e. trajectory $\xi=(s_1,s_2,s_3,\dots, s_n, \dots)\in S^\mathbb N$ of $\mathcal X_{\mu}$, we have
\[
\lim_{n\to\infty} \theta_s(w_n)=\mu_0(s).
\]
\item For every $0<\varepsilon\le 1$ and every $s\in S$
\[
\lim_{n\to\infty} Pr_{\mu_\infty}( |\theta_s(w_n)-\mu_0(s)|<\varepsilon)=1
\]
and the convergence in this limit is exponentially fast as $n\to\infty$.
\end{enumerate}
\end{prop}
\subsection{Iterated Markov Chains}
Let $\mathcal X$ be a finite-state Markov chain with state set $S$. Let $k\ge 1$ be an integer. Consider a finite-state Markov chain $\mathcal X[k]$ with the state set $(S^k)_+$ and with transition probabilities defined as follows. Suppose $s_1\dots s_k\in (S^k)_+$ and $s\in S$ are such that $p_\mathcal X(s_k,s)>0$ (so that $s_1\dots s_k s\in S^{k+1}$ is feasible for $\mathcal X$, and $s_2\dots s_ks\in (S^k)_+$). Then put $p_{\mathcal X[k]}(s_1\dots s_k, s_2\dots s_ks)=p_\mathcal X(s_k,s)$. Set all other transition probabilities in $\mathcal X[k]$ to be $0$. Note that we have $\mathcal X[1]=\mathcal X$.
It is not hard to see that if $\mathcal X$ as above is irreducible then for every $k\ge 1$ the FSMC $\mathcal X[k]$ is also irreducible. Moreover, in this case there is a natural canonical homeomorphism between the set of infinite feasible trajectories $(S^N)_+$ of $\mathcal X$ and the set $\left(((S^k)_+)^\mathbb N\right)_+$ of infinite feasible trajectories for $\mathcal X[k]$. Under this homeomorphism a sequence $\xi=(s_1,\dots, s_n\dots)\in (S^\mathbb N)_+$ goes to $(v_1,v_2\dots, v_n,\dots)\in \left(((S^k)_+)^\mathbb N\right)_+$ where $v_i=s_is_{i+1}\dots s_{i+k-1}$.
Moreover, if $\mu_0$ is the unique stationary distribution for $\mathcal X$ on $S$ then
\[
\mu_0[k](s_1\dots s_k)=\mu_0(s_1)p_\mathcal X(s_1,s_2)\dots p_\mathcal X(s_{k-1},s_k), \tag{*}
\] where $s_1\dots s_k\in (S^k)_+$, is the unique stationary probability distribution for $\mathcal X[k]$. Using these facts and the application of Proposition~\ref{prop:fr}, standard results about Markov chains imply the following statement; see \cite[Proposition~3.13]{CMah} for a more detailed version of this statement, with explicit speed of convergence estimates:
\begin{prop}\label{prop:fr1}
Let $\mathcal X$ be a finite-state Markov chain with state set $S$ and let $\mu_0$ be the unique stationary probability distribution on $S$. Let $\mu$ be an arbitrary strictly positive initial distribution on $S$ defining the corresponding Markov process $\mathcal X_{\mu}=X_1,\dots, X_n,\dots $. Let $k\ge 1$ be an integer and let $\mu_0[k]$ be the distribution on $(S^k)_+$ defined by (*) above. We extend $\mu_0[k]$ to $S^k$ by setting $\mu_0[k](v)=0$ for all $v\in S^k\setminus (S^k)_+$.
Then the following hold:
\begin{enumerate}
\item For every $v\in S^k$ and for $\mu_\infty$-a. e. trajectory $\xi=(s_1,s_2,s_3,\dots, s_n, \dots)\in S^\mathbb N$ of $\mathcal X_{\mu}$, we have
\[
\lim_{n\to\infty} \theta_v(w_n)=\mu_0[k](v).
\]
\item For every $0<\varepsilon\le 1$ and every $v\in S^k$
\[
\lim_{n\to\infty} Pr_{\mu_\infty}( |\theta_v(w_n)-\mu_0[k](v)|<\varepsilon)=1
\]
and the convergence in this limit is exponentially fast as $n\to\infty$.
\end{enumerate}
\end{prop}
\hfill $\qed$
\begin{cor}\label{cor:freq}
Let $X$, $S$, $\mu_0$ and $\mu$ be as in Proposition~\ref{prop:fr1} above. Then:
\begin{enumerate}
\item For every $m\ge 1$ we have $1=\sum_{v\in S^m} \mu_0[k](v)$.
\item For every $m\ge 1$ and every $v\in S^m$ we have
\[
\mu_0[k](v)=\sum_{s\in S} \mu_0[k](vs)=\sum_{s'\in S}\mu_0[k](s'v)
\]
\end{enumerate}
\end{cor}
\begin{proof}
Take a random trajectory $\xi=(s_1,s_2,s_3,\dots, s_n, \dots)\in S^\mathbb N$ of $\mathcal X_{\mu}$ to which the conclusion of Proposition~\ref{prop:fr1} applies and put $w_n=s_1\dots s_n$ for all $n\ge 1$.
The conclusion of part (1) of the corollary now follows directly from part (1) of Proposition~\ref{prop:fr1} and from part (1) of Lemma~\ref{lem:occ}.
Now let $m\ge 1$ and let $v\in S^m$. Then by part (3) of Lemma~\ref{lem:occ} we have
\[
\theta_v(w_n)=\sum_{s\in S} \theta_{vs}(w_n)=\sum_{s'\in S} \theta_{s'v}(w_n).
\]
By passing to the limit as $n\to\infty$ and applying part (1) of Proposition~\ref{prop:fr1} , we obtain part (2) of the corollary.
\end{proof}
\subsection{Quasi-inversions}
We also need the following, somewhat technical to state but mathematically fairly straightforward, statement to later rule out the situation where a random reduced path in a finite graph is closed but far from being cyclically reduced.
We say that a FSMC $\mathcal X$ with state set $S$ is \emph{tight} if $p_\mathcal X(s,s')<1$ for all $s,s'\in S$.
Let $\mathcal X$ be a FSMC with state set $S$ where $\#S\ge 2$. Let $\iota: S'\to S$ be an injective function where $S'\subseteq S$. For a word $w\in (S')^\ast$, $w=s_1\dots s_n$ with $s_i\in S'$ put $\iota(w)=\iota(s_1)\dots\iota(s_n)$. For $w\in S^\ast\setminus (S')^\ast$ put $\iota(w)=\varepsilon$, the empty word.
Also, for a word $w\in S^\ast$ denote by $w^R$ the reverse word. That is, if $w=s_1\dots s_n$ with $s_i\in S$ then $w^R=s_n\dots s_1$.
\begin{prop}\label{p:tight}
Let $\mathcal X$ be an irreducible tight finite-state Markov chain with state set $S$ where $\#S\ge 2$, and let $\mu_0$ be the unique stationary probability distribution on $S$.
Put $\sigma=\max_{s,s'}p_\mathcal X(s,s')$ (so that $0<\sigma<1$).
Let $\mu$ be an arbitrary initial distribution on $S$ defining the corresponding Markov process $\mathcal X_{\mu}=X_1,\dots, X_n,\dots $. Let $\iota: S'\to S$ be an injective function where $S'\subseteq S$ (with $\iota:S^\ast\to S^\ast$ extended as above as well). Then the following hold for a trajectory $\xi=(s_1,s_2,s_3,\dots, s_n, \dots)\in S^\mathbb N$ of $\mathcal X_{\mu}$:
\begin{enumerate}
\item We have \[
Pr \left(s_1s_2\dots s_{\lfloor\sqrt{n}\rfloor} = \left(\iota(s_{n-\lfloor\sqrt{n}\rfloor+1}...s_n)\right)^R\right)\le \sigma^{\lfloor\sqrt{n}\rfloor}
\]
and, in particular,
\[
\lim_{n\to\infty} Pr \left(s_1s_2\dots s_{\lfloor\sqrt{n}\rfloor} = \left(\iota(s_{n-\lfloor\sqrt{n}\rfloor+1}...s_n)\right)^R\right)=0.
\]
\item Let $a,b\in S$ be two states. Then for the conditional probability, conditioned on $s_1=a, s_n=b$, we have:
\[
Pr \left(s_1s_2\dots s_{\lfloor\sqrt{n}\rfloor} = \left(\iota(s_{n-\lfloor\sqrt{n}\rfloor+1}...s_n)\right)^R\big| s_1=a, s_n=b\right)\le \sigma^{\lfloor\sqrt{n}\rfloor-2}
\]
and, in particular,
\[
\lim_{n\to\infty} Pr \left(s_1s_2\dots s_{\lfloor\sqrt{n}\rfloor} = \left(\iota(s_{n-\lfloor\sqrt{n}\rfloor+1}...s_n)\right)^R\big| s_1=a, s_n=b\right)=0.
\]
\end{enumerate}
\end{prop}
\begin{proof}
(1) For a trajectory $\xi=(s_1,s_2,s_3,\dots, s_n, \dots)\in S^\mathbb N$ of $\mathcal X_{\mu}$ denote $w_n=w_n(\xi)=s_1\dots s_n$. For $m\le n$ denote by $\alpha_m(w_n)$ the initial segment of $w_n$ of length $m$.
Let $n\ge 1$ and let $E_n$ be the event that $s_1s_2\dots s_{\lfloor\sqrt{n}\rfloor} = \left(\iota(s_{n-\lfloor\sqrt{n}\rfloor+1}...s_n)\right)^R$. For $u=y_1\dots y_{n-\lfloor\sqrt{n}\rfloor}\in S^{n-\lfloor\sqrt{n}\rfloor}$ let $t(u)=y_{n-\lfloor\sqrt{n}\rfloor}\in S$ be the last letter of $u$.
For any fixed $u=y_1\dots y_{n-\lfloor\sqrt{n}\rfloor}\in S^{n-\lfloor\sqrt{n}\rfloor}$ the conditional probability $Pr(w_n\in E_n| w_{n-\lfloor\sqrt{n}\rfloor}=u)$ is equal to
\begin{gather*}
Pr(w_n\in E_n| w_{n-\lfloor\sqrt{n}\rfloor}=u)=p_\mathcal X(t(u),\iota y_{\lfloor\sqrt{n}\rfloor})p_\mathcal X(\iota y_{\lfloor\sqrt{n}\rfloor}, \iota y_{\lfloor\sqrt{n}\rfloor-1})\dots p_\mathcal X(\iota y_{2}, \iota y_{1})\le \sigma^{\lfloor\sqrt{n}\rfloor}
\end{gather*}
Then
\begin{gather*}
Pr(E_n)=\sum_{u\in S^{n-\lfloor\sqrt{n}\rfloor}} Pr\left(\alpha_{n-\lfloor\sqrt{n}\rfloor}(w_n)=u\right)Pr(w_n\in E_n| w_{n-\lfloor\sqrt{n}\rfloor}=u) \le\\
\sigma^{\lfloor\sqrt{n}\rfloor}\sum_{u\in S^{n-\lfloor\sqrt{n}\rfloor}} Pr\left(\alpha_{n-\lfloor\sqrt{n}\rfloor}(w_n)=u\right)=\sigma^{\lfloor\sqrt{n}\rfloor}.
\end{gather*}
Thus part (1) is verified. The proof of part (2) is similar and we leave the details to the reader.
\end{proof}
\begin{rem}
In fact, the assumption that $\mathcal X$ be tight is not crucial in Proposition~\ref{p:tight} and a similar result holds if we assume that $X$ is an irreducible FSMC with $\#S\ge 2$. We make the tightness assumption to simplify the argument.
\end{rem}
\section{Graph-based non-backtracking random walks}
\begin{conv}\label{conv:g}
In this section we will assume that $F_N=F(A)$ is a free group of finite rank $N\ge 2$, that $\Gamma$ is a finite connected oriented graph with all vertices of degree $\ge 3$ and with the first betti number $b(\Gamma)=N$, and that $\alpha: F_N\to \pi_1(\Gamma,x_0)$ is a fixed isomorphism, where $x_0\in V\Gamma$ is some base-vertex. We equip $\Gamma$ and $T_0=\widetilde \Gamma$ with simplicial metrics, where every edge has length $1$.
\end{conv}
Note that for $\Gamma$ as above we always have $\#E\Gamma\le 6N$.
\begin{defn}
Under the above convention, a FSMC $\mathcal X$ with state set $S$ is \emph{$\Gamma$-based} if the following hold:
\begin{enumerate}
\item We have $S\subseteq E\Gamma$, with $\#S\ge 2$.
\item Whenever $e,e'\in S$ are such that $p_\mathcal X(e,e')>0$ then $t(e)=o(e')$ in $\Gamma$ and $e'\ne e^{-1}$.
\end{enumerate}
\end{defn}
Thus for a $\Gamma$-based FSMC $\mathcal X$ as above, the space of feasible trajectories $(S^\mathbb N)_+$ can be thought of as a subset of the set $\Omega(\Gamma)$ of all reduced semi-infinite edge-paths $\gamma=e_1,e_2,\dots $ in $\Gamma$. Similarly, $(S^n)_+$ can be thought of as a subset of the set $\Omega_n(\Gamma)$ of all reduced length $n$ edge-paths $e_1,e_2,\dots, e_n$ in $\Gamma$.
\begin{prop}\label{p:n}
Let $\mathcal X$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mu_0$ be the unique stationary probability distribution on $S$. For every $k\ge 1$ we extend $\mu_0[k]$ to $\Omega_k(\Gamma)$ by setting $\mu_0[k](v)=0$ for every $v\in \Omega_k(\Gamma)-(S^k)_+$.
There exists a unique geodesic current $\nu$ on $F_N$ with the following properties:
\begin{enumerate}
\item For every $k\ge 1$ and every $v\in \Omega_k(\Gamma)$ we have $\langle v,\nu\rangle_\Gamma=\mu_0[k](v)+\mu_0[k](v^{-1})$.
\item We have $\langle T_0, \nu\rangle=1$.
\end{enumerate}
\end{prop}
\begin{proof}
We use the formulas in part (1) of the proposition to define a system of weights $\nu$ on $\cup_{n\ge 1} \Omega_n(\Gamma)$.
Note that these weights are already symmetrized since the defining equations for the weights in (1) give the same answers for $v$ and $v^{-1}$.
Now Corollary~\ref{cor:freq} implies that these $\nu$ weights satisfy the switch conditions. Therefore they do define a geodesic current $\nu\in \Curr(\FN)$.
Also, part (1) of Corollary~\ref{cor:freq} implies that $\sum_{e\in E\Gamma} \mu_0[1](e)=1=\sum_{e\in E\Gamma} \mu_0[1](e^{-1})$ and therefore
\[
\langle T_0, \nu\rangle=\frac{1}{2} \sum_{e\in E\Gamma} (\mu_0[1](e)+\mu_0[1](e^{-1}))=\frac{1}{2} \sum_{e\in E\Gamma}\langle e, \nu_0\rangle_\Gamma=1.
\]
Thus part (1) of the proposition holds and, in particular $\nu\ne 0$ in $\Curr(\FN)$.
\end{proof}
\begin{defn}[Characteristic current]
Let $\mathcal X$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $0\ne\nu\in\Curr(\FN)$ be the geodesic current constructed in Proposition~\ref{p:n} above.
We call $\nu$ the \emph{characteristic current} of $\mathcal X$ and denote it $\nu=\nu_\mathcal X$
\end{defn}
\begin{defn}[$\mathcal X$-directed random walk on $\Gamma$]
Let $\mathcal X$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mu$ be any initial probability distribution on $S$
defining the corresponding Markov process $\mathcal X_{\mu}=X_1,\dots, X_n,\dots $. For every $n=1,2,\dots $ put $W_n=X_1\dots X_n$ so that $W_n$ takes values in $S^n$.
The random process $\mathcal W_{\mu}=W_1,\dots, W_n,\dots $ is called the \emph{$\mathcal X$-directed non-backtracking random walk} on $\Gamma$ corresponding to $\mu$.
\end{defn}
Note that for $\mathcal W_{\mu}$ and any $n\ge 1$ the only feasible values of $X_n$ are contained in $S^n\cap \Omega_n(\Gamma)$.
Since in general $\Gamma$ may have more than one vertex, a reduced edge-path in $\Gamma$ (such as, for example, the length-$n$ path given by $W_n$ in the above setting) is not necessarily closed and thus may not define a conjugacy class in $\pi_1(\Gamma,x_0)$. To get around this issue, we modify $\mathcal W_{\mu}$ slightly, in two different ways to output closed paths in $\Gamma$.
\begin{defn}[Closing path system]
Let $\Gamma$ be as in Convention~\ref{conv:g}. A \emph{closing path system} for $\Gamma$ is a family $\mathcal B=(\beta_{e,e'})_{e,e'\in E\Gamma}$ of reduced edge-paths in $\Gamma$ such that for every $e,e'\in E\Gamma$ $e\beta_{e,e'}e'$ is a reduced edge-path in $\Gamma$.
For a non-degenerate reduced edge-path $\gamma$ in $\Gamma$ define the \emph{$\mathcal B$-closing} $\widehat \gamma$ of $\gamma$ as $\widehat\gamma=\gamma\beta{e,e'}$ where $e$ is the last edge of $\gamma$ and $e'$ is the first edge of $\gamma$. Note also that for any nondegenerate reduced edge-path $\gamma$ in $\Gamma$ the $\mathcal B$-closing $\widehat\gamma$ is a reduced and cyclically reduced closed edge-path in $\Gamma$.
\end{defn}
Note that $\mathcal B$ is above, if $e,e'\in E\Gamma$ then $t(e)=o(\beta_{e,e'})$ and $o(e')=t(\beta_{e,e'})$. It is also easy to see that for every $\Gamma$ some closing path system $\mathcal B=(\beta_{e,e'})_{e,e'\in E\Gamma}$ exists, and we can always choose $\mathcal B$ so that $|\beta_{e,e'}|\le |E\Gamma|\le 6N$ for all $e,e'\in E\Gamma$.
\begin{defn}[$\mathcal B$-closing of a non-backtracking walk on $\Gamma$]
Let $\mathcal X$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mathcal B=(\beta_{e,e'})_{e,e'\in E\Gamma}$ be a closing path system for $\Gamma$.
Let $\mu$ be any initial probability distribution on $S$ and let $\mathcal W_{\mu}=W_1,\dots, W_n,\dots $ be the $\mathcal X$-directed non-backtracking random walk on $\Gamma$ corresponding to $\mu$.
Define the random process $\widehat {\mathcal W_{\mu}}=\widehat W_1,\dots, \widehat W_n,\dots $, where $\widehat W_n$ the $\mathcal B$-closing of $W_n$.
We call $\widehat {\mathcal W_{\mu}}$ the \emph{$\mathcal B$-closing} of $\mathcal W_{\mu}$.
\end{defn}
An advantage of using $\widehat {\mathcal W_{\mu}}$ is that it always outputs reduced and cyclically reduced closed paths $\widehat W_n$ of length $n\le |\widehat W_n|\le n+C$, where $C=\max_{e,e'}|\beta_{e,e'}|$. However, a weakness of this approach is that the path $W_n$ is already a closed edge-path in $\Gamma$, with asymptotically positive probability as $n\to\infty$ (if $\mathcal X$ is irreducible and $\Gamma$-based). Therefore we offer a variation of the $\widehat {\mathcal W_{\mu}}$ approach which takes this fact into account.
For a reduced nondegenerate closed edge-path $\gamma$ in $\Gamma$ denote by $cyc(\gamma)$ the subpath of $\gamma$ obtained from $\gamma$ by a maximal cyclic reduction. Thus $cyc(\gamma)$ is a nondegenerate closed reduced and cyclically reduced edge-path in $\Gamma$.
\begin{nota}
Let $\mathcal B$ be a closing path system for $\Gamma$. For a nondegenerate reduced edge-path $\gamma$ in $\Gamma$ let $\breve\gamma:=cyc(\gamma)$ is $\gamma$ is a closed path, and let $\breve\gamma:=\widehat\gamma$ otherwise. Thus in both cases $\breve\gamma$ is a closed reduced and cyclically reduced edge-path in $\Gamma$ (but it may now have length $<n$). We call $\breve\gamma$ the \emph{modified $\mathcal B$-closing} of $\gamma$.
\end{nota}
\begin{defn}
Let $\mathcal X$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mathcal B=(\beta_{e,e'})_{e,e'\in E\Gamma}$ be a closing path system for $\Gamma$.
Let $\mu$ be any initial probability distribution on $S$. Let $\mathcal W_{\mu}=W_1,\dots, W_n,\dots $ be the $\mathcal X$-directed non-backtracking random walk on $\Gamma$ corresponding to $\mu$.
Define the random process $\breve {\mathcal W_{\mu}}=\breve W_1,\dots, \breve W_n,\dots $, where $\breve W_n$ the modified $\mathcal B$-closing of $W_n$.
We call $\breve {\mathcal W_{\mu}}$ the \emph{modified $\mathcal B$-closing} of $\mathcal W_{\mu}$.
\end{defn}
\begin{rem}
It is easy to see that the random processes $\widehat{\mathcal W_{\mu}}$, $\breve {\mathcal W_{\mu}}$ considered above always satisfy condition~(1) of Definition~\ref{d:adapt}.
\end{rem}
\begin{thm}\label{t:cl}
Let $\mathcal X$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mu$ be any initial probability distribution on $S$. Let $\mathcal B=(\beta_{e,e'})_{e,e'\in E\Gamma}$ be a closing path system for $\Gamma$. Let $\nu_\mathcal X\in\Curr(\FN)$ be the characteristic current for $\mathcal X$.
Let $\widehat {\mathcal W_{\mu}}=\widehat W_1,\dots, \widehat W_n,\dots $ be the $\mathcal B$-closing of $\mathcal W_{\mu}$.
Then $\widehat {\mathcal W_{\mu}}$ is tame and adapted to $\nu_\mathcal X$.
\end{thm}
\begin{proof}
Put $C=\max_{e,e'}|\beta_{e,e'}|$. We have $|\widehat W_n|\le n+C$ for all $n\ge 1$, which implies that $\widehat {\mathcal W_{\mu}}$ is tame.
Let $\xi=e_1,\dots, e_n,\dots $ be a random trajectory of $\mathcal X_{\mu}$, where $e_i\in S\subseteq E\Gamma$ for all $i\ge 1$. Denote $w_n=e_1\dots e_n$, so that the $\mathcal B$-closing of $w_n$ is $\widehat w_n=e_1\dots e_n\beta_{e_n,e_1}$. Thus $\widehat w_n$ is a closed reduced and cyclically reduced edge-path in $\Gamma$ with $n\le |\widehat w_n|\le n+C$. We can then also think, via the marking isomorphism, of $\widehat w_n$ as defining a nontrivial conjugacy class in $F_N$. Recall that for a nontrivial reduced edge-path $v$ in $\Gamma$ of length $|v|=k\ge 1$ we have $\langle v,\eta_{\widehat w_n}\rangle_\Gamma=\langle v, \widehat w_n\rangle+\langle v^{-1}, \widehat w_n\rangle$, where the latter two terms are the numbers of occurrences of $v^{\pm 1}$ in $ \widehat w_n$ in the sense of Definition~\ref{d:occ}.
Now Proposition~\ref{prop:fr1} implies that
\[
\lim_{n\to\infty}\frac{\langle v,\eta_{\widehat w_n}\rangle_\Gamma}{n}=\lim_{n\to\infty}\frac{\langle v, \widehat w_n\rangle+\langle v^{-1}, \widehat w_n\rangle}{n}=\mu_0[k](v)+\mu_0[k](v^{-1})=\langle v,\nu_\mathcal X\rangle_\Gamma
\]
Therefore $\lim_{n\to\infty} \frac{1}{n}\eta_{\widehat w_n}=\nu_\mathcal X$ in $\Curr(\FN)$, and hence $\lim_{n\to\infty} [\eta_{\widehat w_n}]=[\nu_\mathcal X]$ in $\mathbb P\Curr(\FN)$. This means that $\widehat {\mathcal W_{\mu}}$ is adapted to $\nu_\mathcal X$, as required.
\end{proof}
\begin{thm}\label{t:cla}
Let $\mathcal X$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mu$ be any initial probability distribution on $S$. Let $\mathcal B=(\beta_{e,e'})_{e,e'\in E\Gamma}$ be a closing path system for $\Gamma$. Let $\nu_\mathcal X\in\Curr(\FN)$ be the characteristic current for $\mathcal X$.
Let $\breve {\mathcal W_{\mu}}=\breve W_1,\dots, \breve W_n,\dots $ be the modified $\mathcal B$-closing of $\mathcal W_{\mu}$.
Then $\breve{\mathcal W_{\mu}}$ is tame and adapted to $\nu_\mathcal X$.
\end{thm}
\begin{proof}
Again put $C=\max_{e,e'}|\beta_{e,e'}|$. We have $|\breve W_n|\le n+C$ for all $n\ge 1$, which implies that $\breve {\mathcal W_{\mu}}$ is tame.
Let $\xi=e_1,\dots, e_n,\dots $ be a random trajectory of $\mathcal X_{\mu}$, where $e_i\in S\subseteq E\Gamma$ for all $i\ge 1$. Denote $w_n=e_1\dots e_n$. Thus the corresponding trajectory of $\breve {\mathcal W_{\mu}}$ is $\breve w_1, \breve w_2,\dots, \breve w_n,\dots$. We need to show that $\lim_{n\to\infty} [\eta_{\breve w_n}]=[\nu_\mathcal X]$ in $\mathbb P\Curr(\FN)$.
For every $n\ge 1$ such that $w_n$ is a non-closed path, we have $\breve w_n=\widehat w_n$, and the conclusion of Proposition~\ref{p:cl} applies. Thus it remains to show that for any infinite increasing sequence $n_i$ of indices such that $w_{n_i}$ is a closed path we have $\lim_{i\to\infty} [\eta_{\breve w_{n_i}}]=[\nu_\mathcal X]$ in $\mathbb P\Curr(\FN)$.
Let $n_i$ be such a sequence. Then for all $i\ge 1$ we have $\breve w_{n_i}=cyc(w_{n_i})$.
Let $S_1=\{e\in S| \overline{e}\in S\}$ and define $\iota:S_1\to S$ as $\iota(e)=e^{-1}$ for $e\in S_1$.
Since $\mathcal X$ is tight, Proposition~\ref{p:tight} implies that we have
\[
|\breve w_{n_i}|=|cyc(w_{n_i})|\ge n_i-2\sqrt{n_i}
\]
for $i\to\infty$. Recall also that $cyc(w_{n_i})$ is a subpath of $w_{n_i}$.
Let $v$ be an arbitrary nondegenerate reduced edge-path in $\Gamma$ of length $k\ge 1$. Then we have
\[
|\langle v, w_{n_i}\rangle-\langle v, cyc(w_{n_i})\rangle|\le 2\sqrt{n_i}+2k
\]
and
\[
|\frac{\langle v, w_{n_i}\rangle}{n_i}-\frac{\langle v, cyc(w_{n_i})\rangle}{n_i}|\le 2\frac{1}{\sqrt{n_i}}+\frac{2k}{n_i}\to_{i\to\infty} 0.
\]
Then
Now Proposition~\ref{prop:fr1} implies that
\begin{gather*}
\lim_{i\to\infty}\frac{\langle v,\eta_{\breve w_{n_i}}\rangle_\Gamma}{n_i}=\lim_{i\to\infty}\frac{\langle v, cyc(w_{n_i})\rangle+\langle v^{-1}, cyc(w_{n_i})\rangle}{n_i}=\\
\lim_{i\to\infty}\frac{\langle v, w_{n_i}\rangle+\langle v^{-1}, w_{n_i}\rangle}{n_i}=
\mu_0[k](v)+\mu_0[k](v^{-1})=\langle v,\nu_\mathcal X\rangle_\Gamma.
\end{gather*}
Therefore $\lim_{i\to\infty} \frac{1}{n_i}\eta_{\widehat w_{n_i}}=\nu_\mathcal X$ in $\Curr(\FN)$, and hence $\lim_{i\to\infty} [\eta_{\widehat w_{n_i}}]=[\nu_\mathcal X]$ in $\mathbb P\Curr(\FN)$. As noted above, this implies that $\breve {\mathcal W_{\mu}}$ is adapted to $\nu_\mathcal X$, as required.
\end{proof}
In summary, we get:
\begin{cor}\label{c:sum}
Let $\mathcal X$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\mu$ be any initial probability distribution on $S$. Let $\mathcal B=(\beta_{e,e'})_{e,e'\in E\Gamma}$ be a closing path system for $\Gamma$. Let $\nu_\mathcal X\in\Curr(\FN)$ be the characteristic current for $\mathcal X$. Suppose that $\nu_X$ is filling in $F_N$.
Then $\widehat W_\mu$ and $\breve W_\mu$ are adapted to the characteristic current $\nu_X$. Therefore Theorem~\ref{t:A} and Theorem~\ref{t:A'} apply to $\widehat W_\mu$ and $\breve W_\mu$.
\end{cor}
We next explain several situations where one can guarantee that the current $\nu_\mathcal X\in\Curr(\FN)$ is filling.
\begin{prop}\label{p:XF}
Let $\mathcal X$ be an irreducible $\Gamma$-based FSMC with state set $S\subseteq E\Gamma$. Let $\nu_\mathcal X\in \Curr(\FN)$ be the characteristic current for $\mathcal X$.
\begin{enumerate}
\item Suppose that $\mathcal X$ has the property that $S=E\Gamma$ and that for every $e,e'\in E\Gamma$ such that $ee'$ is a reduced edge-path in $\Gamma$ we have $p_\mathcal X(e,e')>0$. Then the current $\nu_X\in\Curr(\FN)$ is filling.
\item Suppose that $\Gamma=R_A$, the $N$-rose corresponding to a free basis $A=\{a_1,\dots, a_N\}$ of $F_N$ (so that we can identify $E(R_A)=A^{\pm 1}$). Suppose that $\mathcal X$ is such $A\subseteq S$ and that for all $1\le i,j\le N$ we have $p_\mathcal X(a_i,a_j)>0$. Then the current $\nu_X\in\Curr(\FN)$ is filling.
\item Suppose there exists a nondegenerate reduced cyclically reduced closed edge-path $w$ in $\Gamma$ such that $w$ represents a filling element of $F_N$ and that for every $n\ge 2$ we have $p_\mathcal X(w^n)>0$. Then the current $\nu_X\in\Curr(\FN)$ is filling.
\item Suppose there exists a free basis $A=\{a_1,\dots, a_k\}$ such that the following hold. For $i=1,\dots, N$ let $w_i$ be a closed reduced and cyclically reduced edge-path in $\Gamma$ representing the conjugacy class of $a_i$ in $F_N$. For $1\le i<j\le N$ let $w_{i,j}$ be a closed reduced and cyclically reduced edge-path in $\Gamma$ representing the conjugacy class of $a_i$ in $F_N$. Suppose that we have $p_\mathcal X(w_i^2)>0$ for $i=1,\dots, N$ and that we have $p_\mathcal X(w_{ij}^2)>0$ for all $1\le i<j\le N$. Then the current $\nu_X\in\Curr(\FN)$ is filling.
\end{enumerate}
\end{prop}
\begin{proof}
Let $\mu_0$ be the unique stationary probability distribution on $S$ for $\mathcal X$.
(1) The assumption on $\mathcal X$ implies that for every reduced edge-path $v$ in $\Gamma$ of length $k\ge 1$ we have $\mu_0[k](v)>0$, and therefore, by definition of $\nu_\mathcal X$, we also have $\langle v,\nu_\mathcal X\rangle_\Gamma >0$. Thus $\nu_X\in\Curr(\FN)$ has full support and therefore $\nu_X$ is filling in $F_N$.
(2) The assumptions on $\mathcal X$ (with $\Gamma=R_A$) imply that for all $n\ge 1$ and all $1\le i,j\le N$ we have $\mu_0[n](a_i^n), \mu_0[n](a_j^n), \mu_0[2n]((a_ia_j)^n)>0$. Therefore, by definition of $\nu_X$, we also have $\langle a^i,\nu_\mathcal X\rangle_A>0$ and $\langle (a_ia_j)^n,\nu_\mathcal X\rangle_A>0$. Therefore, by Proposition~\ref{p:ai}, the current $\nu_X\in\Curr(\FN)$ is filling.
(3) Again, similarly to (1) and (2) we see that for every $n\ge 1$ $\langle w^n,\nu_\mathcal X\rangle_\Gamma>0$. Therefore by Corollary~\ref{c:z} the current $\nu_X\in\Curr(\FN)$ is filling.
(4) Recall that for a reduced edge-path $v$ in $\Gamma$ of length $k\ge 2$ and starting with $e_1\in E\Gamma$ we have $\mu_0[k](v)=\mu_0(e_1)p_\mathcal X(v)$. Thus $\mu_0[k](v)>0$ if and only if $e_1\in S$ and the transition probabilities $p_\mathcal X(e',e'')$ are $>0$ for all length-2 subpaths $e'e''$ of $v$. Note also that if for the 2-nd edge $e_2$ of $v$ we have $p_\mathcal X(e_1,e_2)>0$ then $e_1,e_2\in S$.
Hence the assumptions in part (4) imply that for $i=1,\dots, N$ and all $n\ge 1$ we have $\mu_0[n](w_i^n)>0$, and, similarly, for all $1\le i<j\le N$ and all $n\ge 1$ we have $\mu_0[n](w_{i,j}^n)>0$. Therefore, by definition of $\nu_X$, we have $\langle w_i^n, \nu_\mathcal X\rangle_\Gamma>0$, and, also, for all $1\le i<j\le N$ and all $n\ge 1$ we have $\langle w_{ij}^n, \nu_\mathcal X\rangle_\Gamma>0$. Therefore, by Proposition~\ref{p:ai}, the current $\nu_X\in\Curr(\FN)$ is filling.
\end{proof}
\begin{ex}
Let $A=\{a_1,\dots, a_N\}$ be a free basis of $F_N=F(A)$ and let $\Gamma=R_A$ be the corresponding $N$-rose.
(1) Consider an $R_A$-based FSMC $\mathcal X$ with state set $S=A^{\pm 1}$ and transition probabilities $p_\mathcal X(a_i^{\varepsilon}, a_j^{\delta})=\frac{1}{2N-1}$ if $a_i^{\varepsilon}\ne a_j^{-\delta}$ and $p_\mathcal X(a_i^{\varepsilon}, a_i^{-\varepsilon})=0$, where $\varepsilon,\delta=\pm 1$. Then $\mathcal X$ is irreducible and tight. The stationary distribution $\mu_0$ is the uniform probability distribution on $A^{\pm 1}$. Then $\mathcal X$, with an initial distrubution $\mu$ on $A^{\pm 1}$, defines the standard non-backtracking simple random walk $\mathcal W_{\mu}=W_1,W_2\dots$ on $F_N=F(A)$. In this case the characteristic current $\nu_\mathcal X$ is the uniform current $\nu_A$ corresponding to $A$. The current $\nu_\mathcal X=\nu_A$ has full support and therefore is filling. Since $R_A$ has only one vertex, the edge-path $W_n$ is always closed, and we have $\breve W_n=W_n$. Using a closing path system $\mathcal B$ produces cyclically reduced words $\widehat W_n=W_n\beta$, where $\beta\in\mathcal B$ is an appropriate closing path. The fact that $W_n$ is adapted to $\nu_A$ is explained in more detail in \cite{Ka07} and exploited in the context of Whitehead's algorithm there. In this case $\nu_A$ already has the "strict minimality" properties similar to those of strictly minimal elements of $F_N$. Again see \cite{Ka07} for details.
(2) Let $\Gamma$ be a simplicial chart on $F_N$. Consider a $\Gamma$-based FSMC $\mathcal X$ with state set $S=E\Gamma$ and transition probabilities satisfying $p_\mathcal X(e,e')>0$ if and only if $ee'$ is a reduced length-2 edge-path in $\Gamma$. Then $\mathcal X$ is irreducible, tight. The characteristic current $\nu_X$ has full support, and therefore is filling.
(3) Let $\mathcal X$ be an $R_A$-based FSMC with state set $S=A$ and transition probabilities satisfying $p_\mathcal X(a_i,a_j)>0$ for all $1\le i,j\le N$. Then $\mathcal X$ is irreducible and tight.The characteristic current $\nu_\mathcal X$ has the property that for $1\ne v\in F(A)$ we have $\langle v,\nu_\mathcal X\rangle>0$ if and only if $v$ or $v^{-1}$ is a positive word over $A$. The current $\nu_\mathcal X$ is filling in $F_N$ by Proposition~\ref{p:ai}. We again have $W_n=\breve W_n$ in this case, and moreover, $W_n$ is already cyclically reduced because it is a positive word.
(4) Let $\Gamma$ be a "fan of lollipops". Namely, $\Gamma$ is a graph with a central vertex $x_0$ with $N$ non-closed oriented edges $e_1,\dots e_N$ emanating from $x_0$ with $N$ distinct end-vertices $y_i=t(e_i)$, $i=1,\dots, N$. For each of these $e_i$ at the vertex $y_i$ there is a closed oriented loop edge $f_i$ attached, with label $a_i\in A$ indicating the marking (so that $f_i^{-1}$ is marked $a_i^{-1}$). Consider a $\Gamma$-based FSMC $\mathcal X$ with state set $S=E\Gamma-\{f_1^{-1}, \dots, f_N^{-1}\}$. The transition probabilities satisfy $p_\mathcal X(e,e')>0$ whenever $e,e'\in S$ and $ee'$ is a reduced length-2 edge-path in $\Gamma$. Then again $\mathcal X$ is irreducible and tight. Moreover, the characteristic current $\nu_X\in\Curr(\FN)$ is filling by Proposition~\ref{p:ai}.
In (1), (2), (3) and (4) above, the processes $\widehat W_\mu$ and $\breve W_\mu$ (where $\mu$ is any initial distribution on the state set $S$ of $\mathcal X$) are adapted to the characteristic current $\nu_X$ of the defining tight irreducible FSMC, and $\nu_X$ is filling in $F_N$. Therefore Theorem~\ref{t:A} and Theorem~\ref{t:A'} apply to $\widehat W_n$ and $\breve W_n$ in these cases.
\end{ex}
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\section{Introduction}
Learning hyperbolic representation of entities have gained recent interest in the machine learning community \cite{bronstein17a,muscoloni17a,nickel17a,sala18a,gu19a}. In particular, hyperbolic embeddings have been shown to be well-suited for various natural language processing problems \cite{nickel17a,dhingra18a,ganea19a} that require modeling hierarchical structures such as knowledge graphs, hypernymy hierarchies, organization hierarchy, and taxonomies, among others. The reason being learning representations in the hyperbolic space provides a principled approach for integrating structural information encoded in such (discrete) entities into continuous space.
The hyperbolic space is a non-Euclidean space and has constant negative curvature. The latter property enables it to \textit{grow} exponentially even in dimension as low as two. Hence, the hyperbolic space has been considered to model trees and complex networks, among others \cite{krioukov10a,hamann18a}. Figure~\ref{fig:hyperbolicDiagrams}(a) is an example of representing a part of mammal taxonomy tree in a hyperbolic space (two-dimensional Poincar{\'e} ball). Hyperbolic embeddings (numerical representations of tasks) have been considered in several applications such as question answering system \cite{tay18a}, recommender systems \cite{vinh18a,chamberlain19a}, link prediction \cite{nickel18a,ganea18a}, natural language inference \cite{ganea18b}, vertex classification \cite{chamberlain17a}, and machine translation \cite{gulcehre19a}.
In this paper, we consider the setting in which an additional low-rank structure may also exist among the learned hyperbolic embeddings. Such a setting may arise when the hierarchical entities are closely related. We propose to learn a low-rank approximation of the given (high dimensional) hyperbolic embeddings. Conceptually, we model high dimensional hyperbolic embeddings as a product of a low-dimensional subspace and low-dimensional hyperbolic embeddings. The optimization problem is cast on the product of the Stiefel and hyperbolic manifolds. We develop an efficient Riemannian trust-region algorithm for solving it. We evaluate the proposed approach on real-world datasets: on the problem of reconstructing taxonomy hierarchies from the embeddings. We observe that the performance of proposed approach match the original embeddings even in low-rank settings.
The outline of the paper is as follows. Section~\ref{sec:background} discusses two popular models of representing hyperbolic space in the Euclidean setting. In Section~\ref{sec:low-rank}, we present our formulation to approximate given hyperbolic embeddings in a low-rank setting. The optimization algorithm is discussed in Section~\ref{sec:optimization}. The experimental results are presented in Section~\ref{sec:experiment} and Section~\ref{sec:conclusion} concludes the paper.
\section{Background}\label{sec:background}
In this section, we briefly discuss the basic concepts of hyperbolic geometry. Interested readers may refer \cite{anderson05,ratcliffe06} for more details.
The hyperbolic space (of dimension $\geq2$) is a Riemannian manifold with a constant negative sectional curvature. Similar to the Euclidean or spherical spaces, it is isotropic. However, the Euclidean space is flat (zero curvature) and the spherical space is positively curved. As a result of negative curvature, the circumference and area of a circle in hyperbolic space grow exponentially with the radius. In contrast, the circumference and area of a circle in the hyperbolic space grow linearly and quadratically, respectively, in Euclidean setting. Hence, hyperbolic spaces \textit{expand} faster than the Euclidean spaces. Informally, hyperbolic spaces may be viewed as a continuous counterpart to discrete trees as the metric properties of a two-dimensional hyperbolic space and a $b$-ary tree (a tree with branching factor $b$) are similar. Hence, trees can be embedded into a two-dimensional hyperbolic space while keeping the overall distortion arbitrarily small. In contrast, Euclidean spaces cannot attain this result even with unbounded number of dimensions.
Since hyperbolic models cannot be represented within Euclidean space without distortion, several (equivalent) models exist for representing hyperbolic spaces for computation purpose. The models are conformal to the Euclidean space and points in one model can be transformed to be represented in another model, while preserving geometric properties such as isometry. However, no model captures all the properties of the hyperbolic geometry. Two hyperbolic models, in particular, have received much interest recently in the machine learning community: the Poincar{\'e} ball model and the hyperboloid model.
\begin{figure*}[t]\centering
{
\includegraphics[width=0.33\columnwidth]{poincare-hierarchy-mammals.jpg}%
\hspace*{\fill}
\includegraphics[width=0.33\columnwidth]{poincare-disk-2.pdf}%
\hspace*{\fill}
\includegraphics[width=0.34\columnwidth]{hyperboloid_new.pdf}%
}
{\ThreeLabels {(a)} {(b)} {(c)}}
\caption{(a) An example of the hyperbolic space ($2$-dimensional Poincar{\'e} ball model $\mathbb{B}^2$) being used to represent a mammal taxonomy. This taxonomy is a part of WordNet \cite{miller95}; (b) A tree is embedded in $\mathbb{B}^2$. The two subtrees from the root are regular trees. All the edges have the same hyperbolic length, computed using (\ref{eqn:poincare-distance}); (c) The Poincar{\'e} disk ($\mathbb{B}^2$) may be viewed as a stereoscopic projection of the hyperboloid model ($\mathbb{H}^2$). Points $p$ and $q$ lie on $\mathbb{H}^2$ and points $u$ and $v$ are their projections, respectively, onto the $\mathbb{B}^2$. The maroon curve $\mathbb{H}^2$ is the geodesic between $p$ and $q$, which projects to the blue geodesic path between $u$ and $v$ on $\mathbb{B}^2$. Figure best viewed in color.}\label{fig:hyperbolicDiagrams}
\end{figure*}
\subsection{Poincar{\'e} ball model}
The Poincar{\'e} ball is a $n$-dimensional hyperbolic space defined as the interior of the $n$-dimensional unit (Euclidean) ball:
\begin{equation*}
\mathbb{B}^n=\{\mathbf u\in\mathbb{R}^n|\norm{\mathbf u}<1\},
\end{equation*}
where $\norm{\cdot}$ denotes the Euclidean norm. The distance between two points $\mathbf u,\mathbf v\in\mathbb{B}^n$ in the Poincar{\'e} ball model is given by
\begin{equation}\label{eqn:poincare-distance}
d_{\mathbb{B}}(\mathbf u,\mathbf v)=\mathrm{arccosh}\left(1+2\frac{\norm{\mathbf u-\mathbf v}^2}{(1-\norm{\mathbf u}^2)(1-\norm{\mathbf v}^2)}\right)
\end{equation}
and the Poincar{\'e} norm is given by
\begin{equation*}
\norm{\mathbf u}_\mathbb{B}\coloneqq d_{\mathbb{B}}({\mathbf 0},\mathbf u)=2\,\mathrm{arctanh}(\norm{\mathbf u}).
\end{equation*}
We observe that the distance between a pair of points near the boundary of the Poincar{\'e} ball (Euclidean norm close to unity) grows much faster than distance between the points close to the center (Euclidean norm close to zero).
In addition, the distance within the Poincar{\'e} ball varies smoothly with respect to points $\mathbf u$ and $\mathbf v$. These properties are helpful embedding discrete hierarchical structures such as trees in hyperbolic spaces and obtain continuous embeddings which respect the tree metric structure. For instance, the origin of the Poincar{\'e} ball may be mapped to the root node of the tree as the root node is relatively closer to all other nodes (points). The leaf nodes can be places near the boundary to ensure they are relatively distant from other leaf nodes. Additionally, the shortest path between a pair of points is usually via a point closer to the origin, just as the shortest path between two nodes in a tree is via their parent nodes. Figure~\ref{fig:hyperbolicDiagrams}(b) shows a Poincar{\'e} disk ($\mathbb{B}^2$) embedding a tree with two regular subtrees.
\subsection{Hyperboloid model}
Let $\bar{\mathbf u},\bar{\mathbf v}\in\mathbb{R}^{n+1}$ such that
$\bar{\mathbf u}=\begin{bmatrix}
u_0\\\mathbf u
\end{bmatrix}$ and
$\bar{\mathbf v}=\begin{bmatrix}
v_0\\\mathbf v
\end{bmatrix}$ and $\mathbf u,\mathbf v\in\mathbb{R}^n$.
The Lorentz scalar product $\inner{\cdot,\cdot}_\mathcal{L}$ of two vectors $\bar{\mathbf u}$ and $\bar{\mathbf v}$ is defined as
\begin{equation}\label{eqn:lorentz_metric}
\inner{\bar{\mathbf u},\bar{\mathbf v}}_\mathcal{L}=\bar{\mathbf u}^\top \mathbf L\bar{\mathbf v}=-u_0v_0 + \mathbf u^\top\mathbf v,
\end{equation}
where $\mathbf L$ is a $(n+1)$-dimensional diagonal matrix
\begin{equation*}
\mathbf L= \begin{bmatrix} -1 & {\mathbf 0}_n^\top \\ {\mathbf 0}_n & {\mathbf I}_n \end{bmatrix},
\end{equation*}
${\mathbf 0}_n$ is the $n$-dimensional zero column vector, and ${\mathbf I}_n$ is the $n$-dimensional identity matrix.
The hyperboloid model, also known as the Lorentz model of hyperbolic geometry, is given by
\begin{equation*}
\mathbb{H}^n=\{\bar{\mathbf u}\in\mathbb{R}^{n+1}|\inner{\bar{\mathbf u},\bar{\mathbf u}}_{\mathcal{L}}=-1,u_0>0\}.
\end{equation*}
The model represents the upper sheet of an $n$-dimensional hyperboloid. From the constraint set, it can be observed that if $\bar{\mathbf u}\in\mathbb{H}^n$, then $u_0=\sqrt{1+\mathbf u^\top\mathbf u}$.
The distance between two points $\bar{\mathbf u},\bar{\mathbf v}\in\mathbb{H}^n$ in the hyperboloid model is given by
\begin{equation}\label{eqn:distance_hyperbolic}
d_{\mathbb{H}}(\bar{\mathbf u},\bar{\mathbf v})=\mathrm{arccosh}(-\inner{\bar{\mathbf u},\bar{\mathbf v}}_\mathcal{L}).
\end{equation}
As stated earlier, both the Poincar{\'e} ball and the hyperboloid models are equivalent and a mapping exists from one model to another \cite{wilson18a}.
Points on the hyperboloid can be mapped to the Poincar{\'e} ball by
\begin{equation*}
h:\mathbb{H}^n\rightarrow\mathbb{B}^n, h(\bar{\mathbf u})=\frac{\mathbf u}{u_0+1}, \bar{\mathbf u}\in\mathbb{H}^n.
\end{equation*}
The reverse mapping, $h^{-1}:\mathbb{B}^n\rightarrow\mathbb{H}^n$ is defined as follows:
\begin{equation*}
h^{-1}({\mathbf w})=\frac{1}{(1-\norm{{\mathbf w}}^2)}\begin{bmatrix}
(1+\norm{{\mathbf w}}^2)\\ 2{\mathbf w}
\end{bmatrix}, {\mathbf w}\in\mathbb{B}^n.
\end{equation*}
Figure~\ref{fig:hyperbolicDiagrams}(c) shows a two-dimensional hyperboloid model $\mathbb{H}^2$. It can be observed that the Poincar{\'e} disk $\mathbb{B}^2$ is obtained as a stereoscopic projection of $\mathbb{H}^2$.
\section{Low-rank parameterization in\\ hyperbolic space}\label{sec:low-rank}
As discussed earlier, hyperbolic embeddings are typically suitable for representing elements of hierarchical structures such as nodes of trees \cite{nickel17a} and complex networks \cite{krioukov10a} to name a few.
When the task involves closely related hierarchical concepts, additional low-rank structure may also exist among such hyperbolic embeddings.
In this section, we propose a novel low-rank parameterization for hyperbolic embeddings.
It should be noted that, unlike the Euclidean embeddings, incorporating a low-rank structure in the hyperbolic framework is non-trivial because of the hyperboloid constraints.
Let $\bar{{\mathbf X}}$ be a $(n+1)\times m$ matrix whose columns represent $n$-dimensional hyperbolic embeddings corresponding to $m$ elements from a given hierarchical structure.
For notational convenience, we represent $\bar{{\mathbf X}}$ and its $i$-th column $\bar{{\mathbf x}}_i$ as follows:
\begin{equation*}
\bar{{\mathbf X}}=\begin{bmatrix}
{\mathbf x}_0 \\ {\mathbf X}
\end{bmatrix} \textup{ and }
\bar{{\mathbf x}}_i=\begin{bmatrix}
x_{0i}\\ {\mathbf x}_i
\end{bmatrix}.
\end{equation*}
We propose to approximate $\bar{{\mathbf x}}_i$ as a low-dimensional hyperbolic embedding such that ${\mathbf x}_i$ shares a latent low-dimensional subspace with ${\mathbf x}_j$ (corresponding to $\bar{{\mathbf x}}_j$), for all $i,j=1,\ldots,m$.
Mathematically, we propose the following $(r+1)$-rank approximation for $\bar{{\mathbf x}}_i$:
\begin{equation*}
\bar{{\mathbf x}}_i
=\begin{bmatrix}
x_{0i}\\ {\mathbf x}_i
\end{bmatrix}
\approx\begin{bmatrix}
z_{0i}\\ {\mathbf U}{\mathbf z}_i
\end{bmatrix}=\hat{{\mathbf z}}_i\ \ \forall i=1,\ldots,m,
\end{equation*}
where $\hat{{\mathbf z}}_i\in\mathbb{H}^n$, ${\mathbf z}_i\in\mathbb{R}^r$, ${\mathbf U}\in\mathbb{R}^{n\times r}$, and ${\mathbf U}^\top{\mathbf U}={\mathbf I}_r$. We discuss below the consequences of the proposed model.
Firstly, we obtain $\bar{{\mathbf z}}_i=\begin{bmatrix}
z_{0i}\\ {\mathbf z}_i
\end{bmatrix}\in\mathbb{H}^r$. This is because $\hat{{\mathbf z}}_i\in\mathbb{H}^n$ implies
\begin{equation*}
-z_{0i}^2 + ({\mathbf U}{\mathbf z}_i)^\top({\mathbf U}{\mathbf z}_i) = -1.
\end{equation*}
$\bar{{\mathbf z}}_i\in\mathbb{H}^r$ follows from the above equality as ${\mathbf U}^\top{\mathbf U}={\mathbf I}_r$.
Secondly, the matrix ${\mathbf X}$ (corresponding to $\bar{{\mathbf X}}$) is modeled as a low-rank matrix as we approximate ${\mathbf X}$ as ${\mathbf U}{\mathbf Z}$, where ${\mathbf Z}=[{\mathbf z}_1,\ldots,{\mathbf z}_m]$.
Thirdly, the space complexity of embeddings reduces from $O(nm)$ (for $\bar{{\mathbf X}}$) to $O(nr+mr)$ (for ${\mathbf U},{\mathbf Z}$ and ${\mathbf z}_0$).
We propose to learn the proposed low-rank paramterization of $\bar{{\mathbf X}}$ by solving the optimization problem:
\begin{equation}\label{eqn:generalOptimization}
\begin{aligned}
& \mathop{\rm min}\limits_{\substack{{\mathbf U}\in\mathbb{R}^{n\times r},\\ \bar{{\mathbf z}}_i\in\mathbb{R}^{n+1} \forall i}} & & \underbrace{\sum_{i=1}^m \ell(\bar{{\mathbf x}_i},\hat{{\mathbf z}}_i)}_{f({\mathbf U},\bar{{\mathbf Z}};\bar{{\mathbf X}})} \\
& \text{subject to} & & {\mathbf U}^\top{\mathbf U}={\mathbf I}_r, \bar{{\mathbf z}}_i\in\mathbb{H}^r, \\
& & &\hat{{\mathbf z}}_i=\begin{bmatrix} 1 & {\mathbf 0}_r^\top \\ {\mathbf 0}_n & {\mathbf U} \end{bmatrix}\bar{{\mathbf z}}_i\ \forall i=1,\ldots,m,
\end{aligned}
\end{equation}
where $\ell: \mathbb{R}^{n+1}\times \mathbb{R}^{n+1}\rightarrow [0,\infty)$ is a \textit{loss function} that measures the quality of the proposed approximation.
Let function $f$ denote the objective function in (\ref{eqn:generalOptimization}), \textit{i.e.}, $f({\mathbf U},\bar{{\mathbf Z}})=\sum_{i=1}^m \ell(\bar{{\mathbf x}_i},\hat{{\mathbf z}}_i)$.
We discuss the following three choices of $f$:
\begin{enumerate}
\item $f({\mathbf U},\bar{{\mathbf Z}};\bar{{\mathbf X}})=\|{\mathbf X}-{\mathbf U}{\mathbf Z}\|_F^2$:\newline we penalize the Euclidean distance between ${\mathbf X}$ and ${\mathbf U}{\mathbf Z}$. This is because ${\mathbf x}_0$ and ${\mathbf z}_0$ are determined from the hyperboloid constraint given ${\mathbf X}$ and ${\mathbf U}{\mathbf Z}$, respectively. We obtain a closed-form solution of (\ref{eqn:generalOptimization}) with this loss function and the solution involves computing a rank-$r$ singular value decomposition of ${\mathbf X}$.
In Section~\ref{sec:experiment}, we denote this approach by the term Method-1.
\item $f({\mathbf U},\bar{{\mathbf Z}};\bar{{\mathbf X}})=\|\bar{{\mathbf X}}-\hat{{\mathbf Z}}\|_F^2$:\newline we penalize the Euclidean distance between the (full) hyperbolic embeddings (matrices) $\bar{{\mathbf X}}$ and $\hat{{\mathbf Z}}=\begin{bmatrix} 1 & {\mathbf 0}_r^\top \\ {\mathbf 0}_n & {\mathbf U} \end{bmatrix}\bar{{\mathbf Z}}$. This approach is denoted by the term Method-2 in Section~\ref{sec:experiment}.
\item $f({\mathbf U},\bar{{\mathbf Z}};\bar{{\mathbf X}})=\sum_i \mathrm{arccosh}(-\inner{\bar{{\mathbf x}}_i,\hat{{\mathbf z}}_i}_\mathcal{L})^2$:\newline since the columns of $\bar{{\mathbf X}}$ and $\hat{{\mathbf Z}}$ are hyperbolic embeddings, we penalize the hyperbolic distance (\ref{eqn:distance_hyperbolic}) between the corresponding embeddings. We denote it by the term Method-3.
\end{enumerate}
It should be noted that the problem (\ref{eqn:generalOptimizationf}) is a nonlinear and non-convex optimization problem, but has well-studied structured constraints. In particular, the structured constraints are cast has Riemannian manifolds. In the next section, we propose a Riemannian trust-region algorithm for solving (\ref{eqn:generalOptimization}) with the loss function discussed in options 2) and 3) above.
\section{Optimization}\label{sec:optimization}
It should be noted that the variable ${\mathbf U}$ in (\ref{eqn:generalOptimization}) belongs to the Stiefel manifold $\Stiefel{r}{n} \coloneqq \{ {\mathbf U} \in \mathbb{R}^{n\times r} : {\mathbf U}^\top{\mathbf U}={\mathbf I}_r \}$ \cite{edelman98a} and the variable $\bar{{\mathbf z}}_i $ belongs to the $r$-dimensional hyperbolic manifold $ \mathbb{H}^r \coloneqq \{ \bar{{\mathbf z}}_i \in \mathbb{R}^{r+1} : -z_{0i}^2 + {\mathbf z}_i^\top {\mathbf z} _i= -1,z_{0i} > 0 \}$ for all $i = \{1,\ldots,m\}$. Consequently, the constraint set of the proposed optimization problem (\ref{eqn:generalOptimization}) is a smooth manifold $\mathcal{M} \coloneqq \Stiefel{r}{n} \times \mathbb{H}^r \times\ldots \times \mathbb{H}^r $, which is the Cartesian product of the Stiefel and $m$ hyperbolic manifolds of dimension $r$. The problem (\ref{eqn:generalOptimization}), therefore, now boils down to the manifold optimization problem:
\begin{equation}\label{eqn:manifold_optimization}
\begin{array}{lll}
\mathop{\rm min}\nolimits\limits_{y \in \mathcal{M}} \quad f(y),
\end{array}
\end{equation}
where $y$ has the representation $y \coloneqq ({\mathbf U},\bar{{\mathbf z}}_1,\ldots, \bar{{\mathbf z}}_m )$ and $f: \mathcal{M} \rightarrow \mathbb{R} : y \mapsto f(y) = \sum_{i=1}^n \ell(\bar{{\mathbf x}_i},\hat{{\mathbf z}}_i)$ is a smooth function.
We tackle the problem (\ref{eqn:manifold_optimization}) in the Riemannian optimization framework that translates it into an unconstrained optimization problem over the nonlinear manifold $\mathcal{M}$, now endowed with a Riemannian geometry \cite{absil08a}. In particular, the Riemannian geometry on manifolds imposes a metric (inner product) structure on $\mathcal{M}$, which in turn allows to generalize notions like the shortest distance between points (on the manifold) or the translation of vectors on manifolds. Following this framework many of the standard nonlinear optimization algorithms in the Euclidean space, e.g., steepest descent and trust-regions, generalize well to Riemannian manifolds in a systematic manner. The Riemannian framework allows to develop computationally efficient algorithms on manifolds~\cite{absil08a}.
Both the Stiefel and hyperbolic manifolds are Riemannian manifolds, and their geometries have been individually well-studied in the literature \cite{nickel18a,absil08a}. Subsequently, the manifold of interest $\mathcal{M}$ also has a Riemannian structure.
Below we list some of the basic optimization-related notions that are required to solve (\ref{eqn:manifold_optimization}) with the Riemannian \emph{trust-region} algorithm that exploits second-order information. The development of those notions follow the general treatment of manifold optimization discussed in \cite[Chapter~7]{absil08a}. The Stiefel manifold related expressions follow from \cite{absil08a}. The hyperobolic related expressions follow from \cite{nickel18a}.
\subsection{Metric and tangent space notions}\label{sec:metric}
Optimization on $\mathcal{M}$ is worked out on the tangent space, which is the linearization of $\mathcal{M}$ at a specific point. It is a vector space associated with each element of the manifold.
As $\mathcal{M}$ is a product space, its tangent space is also the product space of the tangent spaces of the Stiefel $\Stiefel{r}{n}$ and hyperbolic $\mathbb{H}^r$ manifolds. The characterization of the tangent space has the form:
\begin{equation}\label{eqn:tangent_space}
\begin{array}{lll}
T_y \mathcal{M}:= T_{\mathbf U} \Stiefel{r}{n} \times T_{\bar{{\mathbf z}}_1}\mathbb{H}^r \times\ldots \times T_{\bar{{\mathbf z}}_m} \mathbb{H}^r \\
= \{(\xi_{{\mathbf U}}, \xi_{\bar{{\mathbf z}}_i}, \ldots,\xi_{\bar{{\mathbf z}}_m} ) : {\mathrm {symm}}({\mathbf U}^\top \xi_{{\mathbf U}})= {\mathbf 0}_r \ \text{ and } \\
\quad \quad \inner{{\bar{{\mathbf z}}}_i,\xi_{\bar{{\mathbf z}}_i}}_\mathcal{L} = 0 \text{ for all }i
\},
\end{array}
\end{equation}
where $\mathrm{symm}$ extracts the symmetric part of a matrix.
As discussed above, to impose a Riemannian structure on $\mathcal{M}$, a smooth metric (inner product) definition is required at each element of the manifold. A natural choice of the metric $g_y : T_y\mathcal{M} \times T_y\mathcal{M} \rightarrow \mathbb{R} : (\xi_y, \eta_y) \mapsto g_y(\xi_y, \eta_y)$ on $\mathcal{M}$ is the summation of the individual Riemannian metrics on the Stiefel and hyperbolic manifolds. More precisely, we have
\begin{equation}\label{eqn:metric}
\begin{array}{llll}
g_y(\xi_y, \eta_y) \coloneqq \langle \xi_{{\mathbf U}}, \eta_{{\mathbf U}} \rangle + \ \sum_{i=1}^m \inner{\xi_{\bar{{\mathbf z}}_i},\eta_{\bar{{\mathbf z}}_i}}_\mathcal{L},
\end{array}
\end{equation}
where $\xi_y = (\xi_{{\mathbf U}},\xi_{\bar{{\mathbf z}}_i}, \ldots, \xi_{\bar{{\mathbf z}}_m} )$, $\eta_y = (\eta_{{\mathbf U}},\eta_{\bar{{\mathbf z}}_i}, \ldots, \eta_{\bar{{\mathbf z}}_m} )$, $\langle \cdot, \cdot \rangle$ is the standard inner product, and $\langle \cdot, \cdot \rangle_\mathcal{L}$ is the Lorentz inner product (\ref{eqn:lorentz_metric}).
It should be emphasized that the metric in (\ref{eqn:metric}) endows the manifold $\mathcal{M}$ with a Riemannian structure and allows to develop various other notions of optimization in a straightforward manner.
One important ingredient required in optimization is the notion of an orthogonal projection operator $\Pi_y$ from the space $\mathbb{R}^{n\times r}\times \mathbb{R}^{n+1}\ldots \times \mathbb{R}^{n+1}$ to the tangent space $T_y \mathcal{M}$. Exploiting the product and Riemannian structure of $\mathcal{M}$, the projection operator characterization is obtained as the Cartesian product of the individual tangent space projection operator on the Stiefel and hyperbolic manifolds, both of which are well known. Specifically, if $(\zeta_{{\mathbf U}}, \zeta_{\bar{{\mathbf z}}_i}, \ldots,\zeta_{\bar{{\mathbf z}}_m} ) \in \mathbb{R}^{n\times r}\times \mathbb{R}^{n+1}\ldots \times \mathbb{R}^{n+1}$, then its projection onto the tangent space $T_y \mathcal{M}$ is given by
\begin{align}\label{eqn:projection}
\Pi_y(\zeta_{{\mathbf U}}, \zeta_{\bar{{\mathbf z}}_i}, \ldots,\zeta_{\bar{{\mathbf z}}_m} ) \coloneqq &
( \zeta_{{\mathbf U}} - {\mathbf U} \mathrm{symm}({\mathbf U} ^\top \zeta_{{\mathbf U}}), \\
&\zeta_{\bar{{\mathbf z}}_i} + \bar{{\mathbf z}}_i \inner{{{{\mathbf z}}}_i,\zeta_{\bar{{\mathbf z}}_i}}_\mathcal{L}, \nonumber\\
&\ldots, \nonumber\\
&\zeta_{\bar{{\mathbf z}}_m} + \bar{{\mathbf z}}_m \inner{{{{\mathbf z}}}_m,\zeta_{\bar{{\mathbf z}}_m}}_\mathcal{L}).\nonumber
\end{align}
\subsection{Retraction}
An optimization algorithm on manifold requires computation of search direction and then following along it. While the computation of the search direction follows from the notions in Section \ref{sec:metric}, in this section we develop the notion of ``moving'' along a search direction on the manifold. This is characterized by the \emph{retraction} operation, which is the generalization of the the exponential map (that follows the geodesic) on the manifold. The retraction operator $R_y$ takes in a tangent vector at $T_y\mathcal{M}$ and outputs an element on the manifold by approximating the geodesic \cite[Definition~4.1.1]{absil08a}.
Exploiting the product space of $\mathcal{M}$, a natural expression of the retraction operator is obtained by the Cartesian product of the individual retraction operations on the Stiefel and hyperbolic manifolds. If $\xi_y \in T_y \mathcal{M}$, then the retraction operation is given by
\begin{align}\label{eqn:retraction}
R_y(\xi_y) \coloneqq &
(\mathrm{uf}({\mathbf U} + \xi_{{\mathbf U}}), \\
& {\bar{{\mathbf z}}}_i\mathrm{cosh}(\| \xi_{{\bar{{\mathbf z}}}_i}\|_\mathcal{L})+ {\xi_{\bar{{\mathbf z}}}}_i\mathrm{sinh}(\| \xi_{{\bar{{\mathbf z}}}_i}\|_\mathcal{L})/\| \xi_{{\bar{{\mathbf z}}}_i}\|_\mathcal{L}, \nonumber \\
& \ldots, \nonumber \\
& {\bar{{\mathbf z}}}_m\mathrm{cosh}(\| \xi_{{\bar{{\mathbf z}}}_m}\|_\mathcal{L})+ {\xi_{\bar{{\mathbf z}}}}_m\mathrm{sinh}(\| \xi_{{\bar{{\mathbf z}}}_m}\|_\mathcal{L})/\| \xi_{{\bar{{\mathbf z}}}_m}\|_\mathcal{L} ), \nonumber
\end{align}
where $\xi_y = (\xi_{{\mathbf U}},\xi_{\bar{{\mathbf z}}_i}, \ldots, \xi_{\bar{{\mathbf z}}_m} )$, $\| \xi_{{\bar{{\mathbf z}}}_i}\|_\mathcal{L} = \sqrt{\inner{\xi_{{\bar{{\mathbf z}}}_i},\xi_{\bar{{\mathbf z}}_i}}_\mathcal{L}}$ for all $i$, $\mathrm{uf}(\cdot)$ extracts the orthogonal factor of a matrix, \textit{i.e.}, $\mathrm{uf}(\mathbf A) = \mathbf A (\mathbf A^\top \mathbf A)^{-1/2}$ .
\subsection{Riemannian gradient and Hessian computations}\label{sec:gradient_Hessian_computations}
Finally, we require the expressions of the Riemannian gradient and Hessian of $f$ on $\mathcal{M}$. To this end, we first compute the derivatives of $f$ in the Euclidean space. Let $\nabla_y f$ be the first derivative of $f$ and its Euclidean directional derivative along $\xi_y$ is $\mathrm{D} \nabla_y f [\xi_y]$. The expressions of the partial derivatives for the squared Euclidean distance based loss functions mentioned in Section \ref{sec:low-rank} are straightforward to compute. When the loss function is based on the squared hyperbolic distance (\ref{eqn:distance_hyperbolic}), the expressions for $\nabla_y f$ and $\mathrm{D} \nabla_y [\xi_y]$ are discussed in \cite{pennec17a}.
Once the partial derivatives of $f$ are known, converting them to their Riemannian counterparts on $\mathcal{M}$ follows from the theory of Riemannian optimization \cite[Chapter~3]{absil08a}. The expressions are
\begin{align}\label{eqn:Riemannian_gradient_Hessian}
\text{Riemannian gradient } &\mathop{\rm grad}\nolimits_y f = \Pi_y(\nabla_y f),\ \textup{ and } \\
\text{Riemannian Hessian } &\mathrm{Hess}_y f[\xi_y] = \Pi_y ( \mathrm{D}\mathop{\rm grad}\nolimits_y f [\xi_y]), \nonumber
\end{align}
where $\Pi_y$ is the orthogonal projection operator (\ref{eqn:projection}).
\subsection{Riemannian trust-region algorithm}
The Riemannian trust-region (TR) algorithm approximates the function $f$ with a \emph{second-order} model at every iteration. The second-order model (which is called the trust-region sub-problem) makes use of the Riemannian gradient and Hessian computations as shown in Section \ref{sec:gradient_Hessian_computations}. The trust-region sub-problem is then solved efficiently (using an iterative quadratic optimization solver, e.g., with the truncated conjugate gradient algorithm) to obtain a candidate search direction. If the candidate search leads to an appreciable decrease in the function $f$, then it is accepted else it is rejected \cite[Chapter~7]{absil08a}. Algorithm \ref{alg:trust_region} summarizes the key steps of the proposed trust-region algorithm for solving (\ref{eqn:manifold_optimization}).
\begin{algorithm}[tb]
\caption{Riemannian trust-region algorithm for (\ref{eqn:manifold_optimization})}
\label{alg:trust_region}
{\center
\begin{tabular}{ l l }
\multicolumn{2}{l}{{\bfseries Input:} $n$-dimensional hyperbolic embeddings ${\mathbf X}$ and rank $r$. } \\
\multicolumn{2}{l}{ Initialize $y\in\mathcal{M} \coloneqq \Stiefel{r}{n} \times \mathbb{H}^r \times\ldots \times \mathbb{H}^r $.} \\
\multicolumn{2}{l}{{\bfseries repeat}} \\
\multicolumn{2}{l}{\ \ \ \ \ \textbf{1:} Compute $\nabla_y f$.} \vspace{4pt}\\
\multicolumn{2}{l}{\ \ \ \ \ \multirow{2}{500pt}{\textbf{2:} \textbf{Riemannian TR step:} compute a search direction $\xi_y$ which minimizes the trust region sub-problem. It makes use of $\nabla_y f $ and its directional derivative, and their Riemannian counterparts (\ref{eqn:Riemannian_gradient_Hessian}).} } \vspace{4pt}\\
\ \ \ \ \ & \\
\ \ \ \ \ & \vspace{1pt}\\
\multicolumn{2}{l}{\ \ \ \ \ \textbf{3:} Update $y_+ = R_y(\xi_{y})$ (retraction step) from (\ref{eqn:retraction}).} \\
\multicolumn{2}{l}{{\bfseries until} convergence} \\% {\color{red} /*$\|\nabla_{{\mathbf U}} g({\mathbf U}\bU^\top)\|_F\leq \epsilon$*/}
\multicolumn{2}{l}{{{\bfseries Output:} $y = ({\mathbf U}, {\bar{{\mathbf z}}}_1, \ldots, {\bar{{\mathbf z}}}_m)$ and $\hat{{\mathbf Z}}=\begin{bmatrix} 1 & {\mathbf 0}_r^\top \\ {\mathbf 0}_n & {\mathbf U} \end{bmatrix}\bar{{\mathbf Z}}$.}}
\end{tabular}
}
\end{algorithm}
\subsection{Computational complexity}
The manifold-related ingredients cost $O(nr^2 + mr)$. For example, the computation of the Riemannian gradient in (\ref{eqn:Riemannian_gradient_Hessian}) involves only the tangent space projection operation that costs $O(nr^2 + mr)$. Similarly, the retraction operation costs $O(nr^2 + mr + r^3)$.
The computations of $f$ and its derivatives cost $O(nmr)$ (for all the three choices of the loss function in Section \ref{sec:low-rank}). The overall computational cost per iteration of our implementation is, therefore, $O(nmr)$.
\subsection{Numerical implementation}
We use the Matlab toolbox Manopt \cite{boumal14a} to implement Algorithm \ref{alg:trust_region} for (\ref{eqn:manifold_optimization}). Manopt comes with a well-implemented generic Riemannian trust-region solver, which can be used appropriately to solve (\ref{eqn:manifold_optimization}) by providing the necessary optimization-related ingredients mentioned earlier. The Matlab codes are available at \url{https://pratikjawanpuria.com}.
\section{Experiments}\label{sec:experiment}
In this section, we evaluate the performance of the proposed low-rank parameterization of hyperbolic embeddings. In particular, we compare the quality of the low-rank hyperbolic embeddings obtained by minimizing the three different loss functions discussed in Section~\ref{sec:low-rank}.
\subsection*{Experimental setup and evaluation metric} We are provided with the hyperbolic embeddings corresponding to a hierarchical entity such as nodes of a tree or a graph. We also have the ground truth information of the given tree (or graph). Let $(\mathcal{G},\mathcal{E})$ represents the ground truth, where $\mathcal{G}$ is the set of nodes and $\mathcal{E}=\{(u,v)\}$ be the set of edges between the nodes ($u,v\in\mathcal{G}$).
Hyperbolic embeddings can be employed to reconstruct the ground truth since a low hyperbolic distance (\ref{eqn:distance_hyperbolic}) between a pair of nodes implies a high probability of an edge between them.
However, such a reconstruction may also incorporate errors such as missing out on an edge or adding a non-existent edge.
We measure the quality of the hyperbolic embeddings as follows: let $u$ and $v$ be a pair of nodes in $\mathcal{G}$ such that $(u,v)\in\mathcal{E}$. Let $\bar{\mathbf u}$ and $\bar{\mathbf v}$ be the hyperbolic embeddings corresponding to $u$ and $v$, respectively. We compute the hyperbolic distance $d_{\mathbb{H}}(\bar{\mathbf u},\bar{\mathbf v)}$ (\ref{eqn:distance_hyperbolic}) and rank it among the distance corresponding to all untrue edges from $u$, \textit{i.e.}, $\{d_{\mathbb{H}}(u,w)|(u,w)\notin\mathcal{E}\}$. We then compute the mean average precision (MAP) of the ranking. The MAP score is a commonly employed metric for evaluating graph embeddings \cite{nickel17a,nickel18a,bordes13a,nickel16a}. Overall, we compare the quality of the proposed low-rank approximation by comparing the MAP score of the original high dimensional embeddings and the low-rank embeddings.
We obtain the original hyperbolic embeddings from the implementation provided by \cite{nickel17a}. It should be noted that \cite{nickel17a} learns the hyperbolic embeddings from the Poincar{\'e} model and we employ the transformation discussed in Section~\ref{sec:background} to obtain embeddings corresponding to the hyperboloid model. It should be mentioned that though \cite{nickel18a} directly learns hyperbolic embeddings from the hyperboloid model, its implementation is not available.
\begin{table}\centering
\caption{Mean average precision (MAP) score obtained by the proposed approaches on the mammal dataset.}\label{table:mammal-results}
\begin{tabular}[t]{rccc}
\toprule
\multicolumn{1}{c}{Rank} & \multicolumn{1}{c}{Method-1} & \multicolumn{1}{c}{Method-2} & \multicolumn{1}{c}{Method-3}\\
\midrule
$5$ & $0.8274$ & $0.8416$ & $0.1516$ \\
$10$ & $0.9106$ & $0.9143$ & $0.9004$ \\
$20$ & $0.9388$ & $0.9390$ & $0.9388$ \\
$50$ & $0.9488$ & $0.9486$ & $0.9488$ \\
$100$ & $0.9504$ & $0.9504$ & $0.9504$ \\
$200$ & $0.9502$ & $0.9502$ & $0.9502$\\
$300$ & $0.9501$ & $0.9501$ & $0.9501$\\
\bottomrule
\end{tabular}
\end{table}
\subsection*{Datasets}
We perform experiments on the mammal and noun subtrees of the WordNet database\cite{miller95}. WordNet is a lexical database and among other things, it also provides relations between pairs of concepts.
The `mammal' dataset has \textit{mammal} as the root node, with `is-a' (hypernymy) relationship defining the edges. As an example, it has relationships such as `rodent' \textit{is-a} `mammal', `squirrel' \textit{is-a} `rodent', \textit{etc}. Hence, there exists an edge from the `mammal' node to `rodent' node and from `rodent' node to `squirrel' node. The WordNet mammal subtree consists of $|\mathcal{G}|=1\,180$ nodes and $|\mathcal{E}|=6\,540$ edges. A part of this subtree is displayed in Figure~\ref{fig:hyperbolicDiagrams}(a).
Similarly, the `noun' dataset is also a subtree of WordNet database. Examples in this subtree include `photograph' \textit{is-a} `object', 'bronchitis' \textit{is-a} `disease', `disease' \textit{is-a} `entity', \textit{etc}.
It consists of $|\mathcal{G}|=82\,115$ nodes and $|\mathcal{E}|=743\,086$ edges.
\subsection*{Results}
We compare the performance of the proposed low-rank approximation of hyperbolic embeddings with the three loss functions discussed in Section~\ref{sec:low-rank}.
Table~\ref{table:mammal-results} reports the results on the mammal dataset with different values of rank $r=\{5,10,20,50,100,200,300\}$. The original $300$-dimensional hyperbolic embeddings for the mammal subtree achieve a MAP score of $0.9501$. We observe that all the three methods are able to obtain MAP scores very close to the original embeddings with rank $r\geq50$. In addition, Method-1 and Method-2 perform well even in very low-rank setting ($r=5$). This hints that penalizing with the Euclidean distance may be more suitable than compared to the hyperbolic distance (\ref{eqn:distance_hyperbolic}) for approximating hyperbolic embeddings when the given rank is very small.
The results on the noun dataset are reported in Table~\ref{table:noun-results}. This dataset is challenging because of its scale and relatively low reconstruction performance of the original hyperbolic embeddings. The original $100$-dimensional hyperbolic embeddings for the noun subtree achieve a MAP score of $0.8070$. We observe that at rank $r=20$ our methods are able to get within $90\%$ of the performance obtained by the original embeddings.
\begin{table}\centering
\caption{Mean average precision (MAP) score obtained by the proposed approaches on the noun dataset.}\label{table:noun-results}
\begin{tabular}[t]{rccc}
\toprule
\multicolumn{1}{c}{Rank} & \multicolumn{1}{c}{Method-1} & \multicolumn{1}{c}{Method-2} & \multicolumn{1}{c}{Method-3}\\
\midrule
$5$ & $0.5343$ & $0.5422$ & $0.5343$ \\
$10$ & $0.6742$ & $0.6796$ & $0.6742$ \\
$20$ & $0.7425$ & $0.7449$ & $0.7425$ \\
$50$ & $0.7887$ & $0.7891$ & $0.7887$ \\
$100$ & $0.8070$ & $0.8070$ & $0.8070$ \\
\bottomrule
\end{tabular}
}
\end{table}
\section{Conclusion and Future work}\label{sec:conclusion}
Recently, hyperbolic embeddings have gained popularity in many machine learning applications because of their ability to model complex networks. In this paper, we have looked at scenarios where hyperbolic embeddings are potentially high dimensional and how to compress them using a low-rank factorization model. While low-rank decomposition of Euclidean embeddings are well-known, that of hyperbolic embeddings has not been well-studied. To this end, we have proposed a systematic approach to compute low-rank approximations of hyperbolic embeddings. Our approach allows to decompose a high dimensional hyperbolic embedding ($\bar{{\mathbf x}}$) into a product of low-dimensional subspace (${\mathbf U}$) and a smaller dimensional hyperbolic embedding ($\bar{{\mathbf z}}$).
We modeled the learning problem as an optimization problem on manifolds. Various optimization-related notions were presented to implement a Riemannian trust-region algorithm. Our experiments showed the benefit of the proposed low-rank approximations on real-world datasets.
As a future research direction, we would like to explore how low-rank hyperbolic embeddings are useful in downstream applications. Another research direction could be on developing methods to compute a ``good'' rank of hyperbolic embeddings.
\bibliographystyle{unsrt}
| 12,544 |
\section{Introduction and main results}
\label{sect-introduction} \setcounter{equation}{0}
If you think about a linear diffusion equation, probably the first one that will come to your mind is the classical heat equation
\begin{equation} \label{heat.equation}
u_t = \Delta u.
\end{equation}
This equation is naturally associated with the energy
\begin{equation}
\label{eq:energy.heat}
E(u) = \int \frac{|\nabla u|^2}{2},
\end{equation}
in the sense that \eqref{heat.equation} is the gradient flow associated to $E(u)$, see \cite{Evans}.
If you go one step further and consider nonlocal diffusion problems,
one popular choice is
\begin{equation} \label{eq:nonlocal.equation}
u_t (x,t) = \int_{\mathbb{R}^N} J(x-y)(u(y,t)-u(x,t))
\, {\rm d}y,
\end{equation}
where $J: \mathbb{R}^N \to \mathbb{R}$ is a nonnegative, radial function with
$\int_{\mathbb{R}^N} J =1$. Notice that the diffusion of the density $u$ at a point $x$ and time $t$ depends on the values of $u$ at all points in the set $x+\mathop{\rm supp\, }J$, which is what makes the diffusion operator nonlocal. Evolution equations of this form
and variations of it have been recently widely used to model
diffusion processes, see \cite{BCh, BFRW, CF, C, CERW, delia1, F, FW, S, W, Z}. As stated in \cite{F}, if $u(x,t)$ is thought of
as the density of a single population at the point $x$ at time $t$, and
$J(x-y)$ is regarded as the probability distribution of jumping from location
$y$ to location~$x$, then the
rate at which individuals are arriving to position $x$ from all other places is given by $\int_{\mathbb{R}^N} J(y-x)u(y,t)\,{\rm d}y$, while the rate at which they are
leaving location $x$ to travel to all other sites is given by $-\int_{\mathbb{R}^N} J(y-x)u(x,t)\, {\rm d}y=-u(x,t)$. Therefore, in the
absence of external or internal sources, the
density $u$ satisfies equation (\ref{eq:nonlocal.equation}).
In this case there is also an energy that governs the
evolution problem, namely
\begin{equation}
\label{eq:energy.nonlocal}
E(u) = \frac14 \iint J(x-y) (u(y) - u(x))^2 \, {\rm d}x{\rm d}y.
\end{equation}
In the present paper we consider an energy which is local in certain subdomain and nonlocal in the complement, and study the associated gradient flow.
We will show that the Cauchy, Neumann and Dirichlet problems, in the last two cases with zero boundary data, for this equation are well posed. Moreover, we will prove that the solutions to these problems share several properties with their local and nonlocal counterparts~\eqref{heat.equation} and~\eqref{eq:nonlocal.equation}: conservation of mass, for the Cauchy and Neumann problems, comparison principles, and asymptotic behaviour as $t\to \infty$.
\subsection{The Cauchy problem}
Let $\Gamma$ be a smooth hypersurface that divides the space $\mathbb{R}^N$ in two smooth domains $\Omega_\ell$ and $\Omega_{n\ell}$.
We introduce the energy \begin{equation}\label{def:funcional_cauchy.intro}
E(u):=\underbrace{\int_{\Omega_{\ell}} \frac{|\nabla u|^2}{2}}_{\mathcal{L}(u)} + \underbrace{\frac{\alpha}{2}\int_{\Omega_{n\ell}}\int_{\mathbb{R}^N} J(x-y)(u(y)-u(x))^2\, {\rm d}y{\rm d}x}_{\mathcal{N}(u)},
\end{equation}
where $\alpha$ is a fixed positive constant. Thus, the energy functional has two parts, a local one $\mathcal{L}(u)$, that resembles the energy functional~\eqref{eq:energy.heat} for the equation~\eqref{heat.equation}, and a nonlocal part, $\mathcal{N}(u)$, similar to the energy~\eqref{eq:energy.nonlocal} associated with the nonlocal heat equation~\eqref{eq:nonlocal.equation}.
We would like our equation to be the the gradient flow of the energy functional~\eqref{def:funcional_cauchy.intro}. To be more precise, $u$ will be the solution of the ODE (in an infinite dimensional space)
$u'(t)= -\partial E[u(t)]$, $t\ge0$, $u(0)=u_0$, where $\partial E[u]$ denotes the subdifferential of $E$ at the point $u$. To compute the subdifferential, we obtain the derivative of $E$ at $u$ in the direction of $\varphi\in C^\infty_0(\mathbb{R}^N)$,
$$
\begin{aligned}
\displaystyle \partial_\varphi E(u)=& \displaystyle\lim\limits_{h\downarrow 0}\frac{E(u+h\varphi) - E(u)}{h}
\\[6pt]
=& \displaystyle \int_{\Omega_\ell} \nabla u\cdot\nabla \varphi+\alpha\int_{\Omega_{n\ell}}\int_{\mathbb{R}^N} J(x-y)(u(y)-u(x))(\varphi(y)-\varphi(x))\, {\rm d}y{\rm d}x.
\end{aligned}
$$
Thus, if $u$ were smooth, we would have
$$
\begin{aligned}
\displaystyle \partial_\varphi& E(u)= \int_{\Gamma}\varphi\partial_\eta u -\int_{\Omega_\ell} \left\lbrace \Delta u (x) + \alpha \int_{\Omega_{n\ell}} J(x-y)(u(y)-u(x))\,{\rm d}y \right\rbrace\varphi(x)\, {\rm d}x
\\[6pt]
\displaystyle&- \alpha\int_{\Omega_{n\ell}} \varphi(x) \int_{\Omega_\ell} J(x-y)(u(y)-u(x))\, {\rm d}y {\rm d}x
- 2 \alpha\int_{\Omega_{n\ell}}\varphi(x) \int_{\Omega_{n\ell}} J(x-y)(u(y)-u(x))\,{\rm d}y {\rm d}x,
\end{aligned}
$$
where $\eta(x)$ denotes the unit normal at $x\in\Gamma$ pointing towards $\Omega_{n\ell}$ and $\partial_\eta u$ stands for derivativve in the direction of $\eta$ coming from the local part.
Since $\langle \partial E[u],\varphi\rangle=\partial_\varphi E(u)$, we arrive to a problem consisting of a local heat equation with a nonlocal source term in the \lq\lq local'' part of the domain,
\begin{equation} \label{eq:main.Cauchy.local}
\begin{cases}
\displaystyle u_t(x,t)=\Delta u +\alpha\int_{\Omega_{n\ell}} J(x-y)(u(y,t)-u(x,t))\,{\rm d}y,\qquad&x\in \Omega_\ell,\ t>0,
\\[6pt]
\displaystyle \partial_\eta u(x,t)=0,\qquad & x\in \Gamma, \ t>0,
\end{cases}
\end{equation}
together with a nonlocal heat equation in the \lq\lq nonlocal'' part of the domain,
\begin{equation} \label{eq:main.Cauchy.nonlocal}
\begin{aligned}
u_t(x,t)= \alpha\int_{\Omega_\ell} J(x-y)(u(y,t)-u(x,t))\, {\rm d}y + 2\alpha\int_{\Omega_{n\ell}} J(x-y) (u(y,t)-& u(x,t))\, {\rm d}y,
\\[6pt]
&
\quad x \in \Omega_{n\ell},\ t>0,
\end{aligned}
\end{equation}
plus an initial condition
$u(\cdot,0) = u_0 (\cdot)$ in $\mathbb{R}^N$.
From a probabilistic viewpoint (particle systems) in this model particles may jump (according with the probability density $J(x-y)$) when the initial point or the target point, $x$ or $y$,
belongs to the nonlocal region $\Omega_{n\ell}$, and also move according to Brownian motion (with a reflection at $\Gamma$) in the local region
$\Omega_\ell$. Notice that there is some interchange of mass between $\Omega_{n\ell}$ and $\Omega_\ell$ since particles may jump across $\Gamma$.
Notice that we do not impose any continuity to solutions of~\eqref{eq:main.Cauchy.local}--\eqref{eq:main.Cauchy.nonlocal} across the interface $\Gamma$ that separates the local and nonlocal domains.
In fact, solutions can be discontinuous across $\Gamma$ even if the initial condition is smooth.
As precedents for our study we quote \cite{delia2,delia3,Du}. In \cite{delia2} local and nonlocal problems are coupled trough a prescribed region in which
both kind of equations overlap (the data from the nonlocal domain is used as a Dirichlet boundary condition for the local part and viceversa). This kind of coupling gives some continuity in the overlapping region but does not preserve the total mass.
In \cite{delia2} and \cite{Du} numerical schemes using local and nonlocal equations are developed and used in order to improve the computational
accuracy when approximating a purely nonlocal problem.
For this problem we have the following result:
\begin{Theorem}\label{thm:existence_uniqueness_modelo_1.intro}
Given $u_0\in L^1(\mathbb{R}^N)$, there exists a unique $u\in C([0,\infty): L^1(\mathbb{R}^N))$ solving~\eqref{eq:main.Cauchy.local}--\eqref{eq:main.Cauchy.nonlocal} such that $u(\cdot,0)=u_0 (\cdot)$.
The mass of the solution is conserved, $\int_{\mathbb{R}^N} u(\cdot,t)= \int_{\mathbb{R}^N} u_0 $. Moreover, a comparison
principle holds: if $u_0 \geq v_0$ then the corresponding solutions verify $u \geq v$ in $\mathbb{R}^N \times \mathbb{R}_+$.
\end{Theorem}
\subsection{The Neumann problem}
Let us now present the version of this problem with boundary condition in dimension $N\geq 1$. Let us take a bounded smooth domain $\Omega\subset \mathbb{R}^N$ that is itself divided into two other subsets $\Omega_\ell$ and $\Omega_{n\ell}$ by a smooth hypersurface $\Gamma$. Again we can define an energy functional
\begin{equation}\label{def:funcional_neumann.intro}
E(u):=\int_{\Omega_\ell} \frac{|\nabla u|^2}{2} + \frac{\alpha}{2}\int_{\Omega_{n\ell}}\int_{\Omega} J(x-y)(u(y)-u(x))^2\,{\rm d}y{\rm d}x.
\end{equation}
Associated with this energy we obtain an evolution problem with a \lq\lq local'' part
\begin{equation}
\label{eq:main.Neumann.local}
\begin{cases}
\displaystyle u_t(x,t)=\Delta u (x,t) + \alpha\int_{\Omega_{n\ell}} J(x-y)(u(y,t)-u(x,t))\,{\rm d}y,\ &x\in \Omega_\ell,\ t>0,
\\[6pt]
\displaystyle \partial_\eta u(x,t)=0,\qquad & x\in \partial \Omega_\ell, \ t>0,
\end{cases}
\end{equation}
and a \lq\lq nonlocal'' one,
\begin{equation}
\label{eq:main.Neumann.nonlocal}
\begin{aligned}
\displaystyle u_t(x,t)=\alpha\int_{\Omega_\ell} J(x-y)(u(y,t)-u(x,t))\, {\rm d}y
+ 2\alpha\int_{\Omega_{n\ell}}\!\! J(x-y)(u(y,t)-&u(x,t))\, {\rm d}y, \\[6pt]
&
\quad x \in \Omega_{n\ell},\ t>0,
\end{aligned}
\end{equation}
plus an initial condition $u(\cdot,0) = u_0(\cdot)$ in $\Omega$.
Notice that in this model there are no individuals that may jump into $\Omega$ coming from the outside side $\mathbb{R}^N \setminus \Omega$ nor individuals that jump
from $\Omega$ into the exterior side $\mathbb{R}^N \setminus \Omega$. It is in this sense that we call~\eqref{eq:main.Neumann.local}--\eqref{eq:main.Neumann.nonlocal} a Neumann
type problem.
For this problem we also have existence and uniqueness of solutions and a comparison principle. Moreover, as it is expected for Neumann
boundary conditions, we also have conservation of mass.
\begin{Theorem}\label{thm:existence_uniqueness_neumann.intro}
Given $u_0\in L^1(\Omega)$, there exists a unique $u\in C([0,\infty): L^1(\Omega))$ solving~\eqref{eq:main.Neumann.local}--\eqref{eq:main.Neumann.nonlocal} such that $u(\cdot,0)=u_0 (\cdot)$. This solution conserves mass.
Moreover, a comparison principle holds.
\end{Theorem}
\subsection{The Dirichlet problem}
As for the Dirichlet case let us take a bounded smooth domain $\Omega\subset \mathbb{R}^N$ that is itself divided into two subsets $\Omega_\ell$ and $\Omega_{n\ell}$ by a smooth hypersurface $\Gamma$.
The Dirichlet version of the functional reads as
\begin{equation}\label{def:funcional_dirichlet.intro}
E(u):=\int_{\Omega_\ell} \frac{|\nabla u|^2}{2} + \frac{\alpha}{2}\int_{\Omega_{n\ell}}\int_{\mathbb{R}^N} J(x-y)(u(y)-u(x))^2\, {\rm d}y{\rm d}x,
\end{equation}
extending $u$ by zero outside $\Omega$ (and hence also on $\partial \Omega$). Notice that in the nonlocal part we have integrated $y$ in the whole $\mathbb{R}^N$.
The associated evolution problem again has a local part,
\begin{equation} \label{eq:main.Dirichlet.local}
\begin{cases}
\displaystyle u_t(x,t)=\Delta u (x,t) + \alpha\int_{\mathbb{R}^N\setminus \Omega_\ell} J(x-y)(u(y,t)-u(x,t))\,{\rm d} y, &x\in \Omega_\ell,\, t>0,
\\[6pt]
\displaystyle \partial_\eta u(x,t)=0,\qquad & x\in \Gamma, \ t>0,
\end{cases}
\end{equation}
plus a nonlocal one
\begin{equation} \label{eq:main.Dirichlet.nonlocal}
\begin{aligned}
\displaystyle u_t(x,t)=\alpha\int_{\mathbb{R}^N\setminus \Omega_{n\ell}} J(x-y)(u(y,t)-u(x,t))\, {\rm d}y
+ 2\alpha\int_{\Omega_{n\ell}} J(x-y) & (u(y,t)-u(x,t))\, {\rm d}y,
\\[6pt]
& \quad x \in \Omega_{n\ell},\ t>0,
\end{aligned}
\end{equation}
plus the Dirichlet \lq\lq boundary'' condition
\begin{equation} \label{eq:main.Dirichlet.boundary}
\displaystyle u(x,t)= 0,\quad x\in \mathbb{R}^N\setminus \Omega,\ t>0,
\end{equation}
and the initial condition $u(\cdot,0)=u_0 (\cdot)$ in $\Omega$.
In this model we have that individuals may jump outside $\Omega$ but they instantaneously die there since we have that the density
$u$ vanishes identically in $(\mathbb{R}^N\setminus \Omega)\times \mathbb{R}_+$.
For this problem we also have existence and uniqueness of solutions and a comparison principle, but, of course, there is no
conservation of mass.
\begin{Theorem}\label{thm:existence_uniqueness_Dirichlet.intro}
Given $u_0\in L^1(\Omega)$, there exists a unique $u\in C([0,\infty): L^1(\Omega))$ solving~\eqref{eq:main.Dirichlet.local}--\eqref{eq:main.Dirichlet.boundary} such that $u(\cdot,0)=u_0(\cdot)$. Moreover, a comparison principle holds.
\end{Theorem}
\subsection{Asymptotic behavior.} It is well known that solutions of the local heat equation~\eqref{heat.equation} have a polynomial time decay or the Cauchy problem and an exponential decay
(to the mean value of the initial condition or to zero) for the Neumann and the Dirichlet problems. The same is true for solutions of the nonlocal heat equation~\eqref{eq:nonlocal.equation}, though in the case of the Cauchy problem we have to ask the second moment of the kernel
\begin{equation*}
\label{eq:def.second.moment}
M_2(J):=\int_{\mathbb{R}^N}J(z)|z|^2\,{\rm d}z
\end{equation*}
to be finite as in~\cite{Chasseigne-Chaves-Rossi-2006}. Our local/nonlocal model reproduces these behaviours.
\begin{Theorem}\label{thm:asymp.intro}
{\rm (a)} Let $u$ be a solution of the Cauchy problem~\eqref{eq:main.Cauchy.local}--\eqref{eq:main.Cauchy.nonlocal} in $\mathbb{R}^N$ with integrable initial data. If $M_2(J)<\infty$, for any $p\in(1,\infty)$ there is a constant $C$ such that
\begin{equation} \label{decay.cauchy}
\|u(\cdot, t ) \|_{L^{p} (\mathbb{R}^N)} \leq C t^{-\frac{N}{2} (1 - \frac{1}{p})}.
\end{equation}
\noindent{\rm (b)} Let $u$ be a solution of the Dirichlet problem problem~\eqref{eq:main.Dirichlet.local}--\eqref{eq:main.Dirichlet.boundary} in an $N$-dimensional domain with integrable initial data. For any $p\in[1,\infty)$ there are positive constants $C$ and $\lambda$ such that
\begin{equation} \label{decay.Dirichlet}
\|u(\cdot, t ) \|_{L^{p} (\Omega)} \leq C e^{-\lambda t}.
\end{equation}
\noindent{\rm (c)} Let $u$ be a solution of the Neumann problem~\eqref{eq:main.Neumann.local}--\eqref{eq:main.Neumann.nonlocal} in an $N$-dimensional domain. For any $p\in[1,\infty)$ there are positive constants $C$ and $\beta>0$ such that
\begin{equation} \label{decay}
\left\|u(\cdot, t ) - |\Omega|^{-1}\int_{\Omega} u_0 \right\|_{L^{p} (\Omega)} \leq C e^{-\beta t}.
\end{equation}
\end{Theorem}
\subsection{Rescaling the kernel.} Our aim is to recover the usual local problems from our nonlocal ones when
we rescale the kernel according to
\begin{equation}
\label{eq:definition.J.epsilon}
J^\varepsilon(z)=\varepsilon^{-(N+2)}J(z/\varepsilon).
\end{equation}
It is at this point where we choose the constant $\alpha$ that appears in front of our nonlocal terms as
$$
\alpha=1/M_2(J).
$$
Now, for a fixed initial condition $u_0$ and for each $\varepsilon>0$ our evolution problems (Cauchy, Neumann or Dirichlet) have a solution.
Our goal is to look for the limit as $\varepsilon \to 0$ of these solutions to recover in this limit procedure the local heat equation.
Notice that as $\varepsilon$ becomes small the support of the kernel $J^\varepsilon$ shrinks, hence the non locality of the operator
becomes weaker as $\varepsilon$ becomes smaller.
As precedents where this kind of limit procedure is performed we quote
\cite{BLGneu,BLGdir,ElLibro,BChQ,Canizo-Molino-2018,Chass,CER,Cortazar-Elgueta-Rossi-Wolanski,Andres}.
One of the main difficulties here is that we do not have any continuity of the solutions to our nonlocal
equations across the interface that separates the local and nonlocal domains, while the expected limit is smooth
across the interface (being a solution to the heat equation in the whole domain).
\begin{Theorem}\label{thm:rescales.intro} Let $u_0\in L^2(\Omega)$ (with $\Omega=\mathbb{R}^N$ in the case of the Cauchy problem). For each $\varepsilon>0$, let $u^\varepsilon$ be the solution of any of the three previously mentioned problems, Cauchy, Neumann, or Dirichlet with initial data $u_0$. Then, as $\varepsilon \to 0$,
\begin{equation} \label{conver.intro}
u^\varepsilon \to u \quad \mbox{strongly in } L^2,
\end{equation}
where $u$ is the solution to the corresponding problem (Cauchy, Neumann, or Dirichlet, in the two last cases with zero boundary condition) for the local heat equation~\eqref{heat.equation} in $\Omega\times\mathbb{R}_+$ with the same initial condition.
\end{Theorem}
\medskip
The rest of the paper is organized as follows: in Section \ref{sect-prelim} we collect some preliminary results
and prove an inequality that will be the key in our arguments;
in the following sections we prove our main results concerning existence, uniqueness and properties of the model in its three versions
(Cauchy, Neumann, Dirichlet).
We gather the results according to the problem we deal with and hence
in Section \ref{sect-Cauchy} we study the Cauchy problem (including its asymptotic behaviour and the limit when we rescale the kernel); in Section \ref{sect-Neumann} the Neumann problem and
finally in Section \ref{sect-Dirichlet} the Dirichlet problem.
\section{Preliminaries} \label{sect-prelim}
\setcounter{equation}{0}
First, we present a very useful result that we state in its more general form. This result says that we can control
the purely nonlocal energy by our local/nonlocal one.
\begin{Lemma}\label{lemma:preliminar}
Let $\Omega\subset\mathbb{R}^N$ be a smooth domain, $\Omega_\ell\subset \Omega$ a smooth convex subdomain and $\Omega_{n\ell}=\Omega\setminus \overline{\Omega_\ell}$. Let $u\in\mathcal{H}:=\{u\in L^2(\Omega): u|_{\Omega_{\ell}}\in H^1(\Omega_{\ell}) \}$. Then, for any $k\in (0,1/M_2(J))$,
\begin{equation}
\label{eq:fundamental.inequality}
\begin{array}{l}
\displaystyle
\int_{\Omega_\ell} |\nabla u|^2 + \frac{1}{2M_2(J)}\int_{\Omega_{n\ell}}\int_{\Omega} J(x-y)(u(y)-u(x))^2\, {\rm d}y{\rm d}x
\\ [4mm]
\qquad \geq \displaystyle k\int_{\Omega}\int_{\Omega} J(x-y)(u(y)-u(x))^2\, {\rm d}y{\rm d}x.
\end{array}
\end{equation}
\end{Lemma}
\begin{proof}
Thanks to the symmetry of the kernel $J$, our inequality is equivalent to showing that
\begin{equation}
\int_{\Omega_\ell} |\nabla u|^2 -k\int_{\Omega_\ell}\int_{\Omega_\ell} J(x-y)(u(y)-u(x))^2\, {\rm d}y {\rm d}x\geq 0,
\end{equation}
if $k$ is small enough.
This is important because we stick to the domain $\Omega_\ell$, where $u$ belongs to $H^1$, so we can express it as the integral of a derivative and make computations with it.
After a change of variables, using Jensen's inequality we get
\begin{equation}
\begin{aligned}
\displaystyle\int_{\Omega_\ell}\int_{\Omega_\ell} J(x-y)&(u(y)-u(x))^2\,{\rm d}y{\rm d}x = \int_{\Omega_\ell}\int_{\Omega_\ell-x} J(z)(u(x+z)-u(x))^2\,{\rm d}z{\rm d}x
\\[6pt]
\displaystyle &=\int_{\Omega_\ell}\int_{\Omega_\ell-x} J(z)\left(\int_0^{1} \nabla u(x+sz)\cdot z\, {\rm d}s\right)^2\, {\rm d}z{\rm d}x\\[6pt]
\displaystyle&\leq \int_{\Omega_\ell}\int_{\Omega_\ell-x}\int_0^{1} J(z)|z|^2 |\nabla u(x+sz)|^2\, {\rm d}s{\rm d}z{\rm d}x=:\mathcal{A}.
\end{aligned}
\end{equation}
If we define the sets $Z:=\{z\in\mathbb{R}^N:\text{there exists }x\in \Omega_\ell \text{ such that } x+z\in\Omega_\ell\}$ and $\Omega_{z,s}:=\{x\in\Omega_\ell:x+sz \in\Omega_\ell\}$ then we can apply Fubini to see that
$$
\mathcal{A}=\int_{Z} \int_{0}^1\int_{\Omega_{z,s}} J(z)|z|^2 |\nabla u(x+sz)|^2\, {\rm d}x{\rm d}s{\rm d}z.
$$
Now we define, for fixed $z,s$ the variable $w:=x+sz$, which means that the set $\Omega_{z,s}$ can be described as $W_{z,s}:=\{w\in\Omega_\ell : \text{ there exists } x\in\Omega_\ell \text{ such that } w=x+sz \}\subset \Omega_\ell$. Hence,
$$
\begin{aligned}
\mathcal{A}=\int_{Z} \int_{0}^1\int_{W_{z,s}} &J(z)|z|^2 |\nabla u(w)|^2\,{\rm d}w{\rm d}s{\rm d}z \leq\int_{Z} \int_{0}^1\int_{\Omega_\ell} J(z)|z|^2 |\nabla u(w)|^2\,{\rm d}w{\rm d}s{\rm d}z
\\[6pt]
\displaystyle &\leq\int_{\mathbb{R}^N} \int_{0}^1\int_{\Omega_\ell} J(z)|z|^2 |\nabla u(w)|^2\,{\rm d}w{\rm d}s{\rm d}z = M_2(J)\int_{\Omega_\ell} |\nabla u|^2.
\end{aligned}
$$
The result follows taking $k$ small enough.
\end{proof}
It is worth noting that estimate~\eqref{eq:fundamental.inequality} scales well with the rescaled version of the kernel $J^\varepsilon$ given by~\eqref{eq:definition.J.epsilon},
since $M_2(J^\varepsilon)=M_2(J)$. Hence we can take the same constant $k$ for all $\varepsilon$. This will be helpful when studying the asymptotic behaviour of the solutions of these problems and also for the convergence of these problems to the corresponding local ones.
The following lemma will be needed later on to study the limit behaviour under rescales of the kernel for the three different problems and is an adaptation of the results that can be found in~\cite{BLGneu,BLGdir, ElLibro}. In the following we will denote by $\bar{f}$ the extension by zero of a function $f$ outside our domain $\Omega$.
\begin{Lemma}\label{lemma:basic_convergence_lemma_for_rescaled}
Let $\Omega$ be a bounded domain and $\{f^\varepsilon\}$ a sequence of functions in $L^2(\Omega)$ such that
$$
\displaystyle \int_{\Omega}\int_{\Omega} J^\varepsilon(x-y)(f^\varepsilon(y)-f^\varepsilon(x))^2\, {\rm d}y{\rm d}x \leq C
$$
for a positive constant $C$ and $\{f^\varepsilon\}$ is weakly convergent in $L^2(\Omega)$ to $f$ as $\varepsilon$ goes to 0.
Then $$|\nabla f|\in L^2(\Omega)$$ and moreover
$$
\lim\limits_{\varepsilon\to 0} \left( J^{1/2}(z) \chi_{\Omega}(x+\varepsilon z) \frac{\bar{f}^\varepsilon(x+\varepsilon z) - f^\varepsilon(x)}{\varepsilon} \right) = J^{1/2}(z)z\nabla f(x)
$$
weakly in $L^2_x(\Omega)\times L^2_z(\mathbb{R}^N)$.
\end{Lemma}
\begin{proof}
Changing variables $y=x+\varepsilon z$ we obtain
$$
\displaystyle \int_{\Omega}\int_{\mathbb{R}^N} J(z) \chi_{\Omega}(x+\varepsilon z) \frac{(\bar{f}^\varepsilon(x+\varepsilon z,t)-f^\varepsilon(x,t))^2}{\varepsilon^2}\,{\rm d}z{\rm d}x \leq C,
$$
which already provides, for a certain function $h=h(x,z)$, the stated weak convergence to a $J^{1/2}(z)h(x,z)$. Having this weak convergence we can multiply the quantity
$$
J^{1/2}(z) \chi_{\Omega}(x+\varepsilon z) \frac{\bar{f}^\varepsilon(x+\varepsilon z) - f^\varepsilon(x)}{\varepsilon}
$$
by two test functions $\varphi(x)\in C_c^\infty(\Omega)\subset L^2(\mathbb{R}^N)$ and $\psi(z)\in C_c^\infty(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)$, integrate and pass to the limit to obtain that
\begin{equation}\label{eq:lema_preliminar_1}
\begin{aligned}
\displaystyle \int_{\mathbb{R}^N} &J^{1/2}(z)\psi(z) \int_{\Omega} \chi_{\Omega}(x+\varepsilon z) \frac{\bar{f}^\varepsilon(x+\varepsilon z) - f^\varepsilon(x)}{\varepsilon} \varphi(x) \,{\rm d}x{\rm d}z
\\[6pt]
\displaystyle &\to \int_{\mathbb{R}^N} J^{1/2}(z)\psi(z) \int_{\Omega} h(x,z)\varphi(x)\,{\rm d}x{\rm d}z,
\end{aligned}
\end{equation}
and it is now when we note that since $\psi(z)$ has compact support the integral over $\mathbb{R}^N$ in $z$ is really an integral over a compact set, so there exists $\varepsilon$ small enough such that
$$
J^{1/2}(z) \chi_{\Omega}(x+\varepsilon z) = J^{1/2}(z)
$$
for all $z\in \mathbb{R}^N$ and all $x\in \mathop{\rm supp}(\varphi)$. Then
$$
\begin{aligned}
\displaystyle \int_{\mathbb{R}^N} &J^{1/2}(z)\psi(z) \int_{\Omega} \chi_{S}(x+\varepsilon z) \frac{\bar{f}^\varepsilon(x+\varepsilon z) - f^\varepsilon(x)}{\varepsilon} \varphi(x) \,{\rm d}x{\rm d}z
\\[6pt]
&= \displaystyle \int_{\mathbb{R}^N} J^{1/2}(z)\psi(z) \int_{\mathop{\rm supp}(\varphi)}\frac{\bar{f}^\varepsilon(x+\varepsilon z) - f^\varepsilon(x)}{\varepsilon} \varphi(x) \,{\rm d}x{\rm d}z
\\[6pt]
&= \displaystyle - \int_{\mathbb{R}^N} J^{1/2}(z)\psi(z) \int_{\Omega}f^\varepsilon(x)\frac{\varphi(x) - \bar{\varphi}(x-\varepsilon z)}{\varepsilon} \,{\rm d}x{\rm d}z.
\end{aligned}
$$
Using~\eqref{eq:lema_preliminar_1} and the fact that $\{f^\varepsilon\}$ converges weakly to $f$ in $L^2(S)$ we have that
$$
\displaystyle - \int_{\mathbb{R}^N} J^{1/2}(z)\psi(z) \int_{\Omega}f\nabla\varphi(x) \,{\rm d}x{\rm d}z = \displaystyle \int_{\mathbb{R}^N} J^{1/2}(z)\psi(z) \int_{\Omega} h(x,z)\varphi(x)\,{\rm d}x{\rm d}z
$$
and from this point it is easy to conclude what remains of the lemma.
\end{proof}
\section{The Cauchy problem} \label{sect-Cauchy}
\setcounter{equation}{0}
In this section, we will prove existence and uniqueness of solutions to the Cauchy problem~\eqref{eq:main.Cauchy.local}--\eqref{eq:main.Cauchy.nonlocal} with initial data $u_0\in L^1(\mathbb{R}^N)$, conservation of mass, the asymptotic polynomial decay of the $L^p$ norms in time, and the convergence of this problem to the local one when we rescale the kernel.
\subsection{Existence and uniqueness}
To prove existence and uniqueness of a solution the idea is to use a fixed point argument as follows: Given a function $v$ defined for $\Omega_{\ell}$ we solve for $\Omega_{n\ell}$, in a function we will call $z$. With the obtained function we solve back for $\Omega_{\ell}$ in a function $w$. This can be regarded as an operator $T$ that satisfies $T(v)=w$. This is the operator for which we will look for a fixed point via contraction in adequate norms (we will use this technique several times), meaning that there must exist a $v=T(v)$ solving the equation for $\Omega_{\ell}$ with its corresponding $z$ solving the equation for $\Omega_{n\ell}$.
We will write this argument for an initial condition $u_0\in L^1(\mathbb{R}^N)$. Let us define, for a fixed finite $t_0>0$ the norms
$$
\|v\|_\ell= \sup\limits_{t\in [0,t_0]} \|v(\cdot, t)\|_{L^1(\Omega_\ell)} \qquad \text{and}
\qquad \|z\|_{n\ell} = \sup\limits_{t\in [0,t_0]} \|z(\cdot, t)\|_{L^1(\Omega_{n\ell})}.
$$
Given $t_0>0$ to be chosen later, we define an operator $T_1:L^1(\Omega_\ell\times (0,t_0)))\to L^1(\Omega_{n\ell}\times(0,t_0))$ as $T_1(v)=z$, where $z$ is the unique solution to
\begin{equation}\label{eq:solucion_z}
\begin{cases}
\displaystyle z_t(x,t)=\alpha\int_{\Omega_\ell} J(x-y)(v(y,t)-z(x,t))\,{\rm d}y
+ 2\alpha \int_{\Omega_{n \ell}} J(x-y)(z(y,t)-z(x,t))\,{\rm d}y,
\\[6pt]
\displaystyle \hskip10cm\qquad x\in\Omega_{n \ell},\ t\in(0,t_0),
\\[6pt]
\displaystyle z(x,0) = u_0(x),\qquad x\in \Omega_{n \ell} .
\end{cases}
\end{equation}
Let us check that this problem has indeed a unique solution. In addition, we will study its dependence on the data $v$.
\begin{Lemma} Let $t_0\in (0,1/(5\alpha))$.
Given $v\in L^1(\Omega_\ell\times (0,t_0)))$ there exists a unique $z\in L^1(\Omega_{n\ell}\times(0,t_0))$ that solves~\eqref{eq:solucion_z} for some $t_0$ small enough. Moreover, if $z_1$ and $z_2$ are the solutions corresponding respectively to $v_1$ and $z_2$, then
\begin{equation}
\label{eq:contraction.z.v}
\|z_1-z_2\|_{n\ell} \leq \frac{\alpha t_0}{1-5\alpha t_0} \|v_1-v_2\|_\ell.
\end{equation}
\end{Lemma}
\begin{proof}
To show existence and uniqueness we use a fixed point argument. We define an operator $S_v:L^1(\Omega_{n\ell}\times (0,t_0)))\to L^1(\Omega_{n\ell}\times(0,t_0))$ through
$$
\begin{array}{ll} \displaystyle
S_v(z)(x,t):=u_0(x) + \alpha\int_0^t \int_{\Omega_\ell} J(x-y)(v(y,s)-z(x,s))\,{\rm d}y{\rm d}s
\\ [4mm]
\displaystyle\qquad \qquad \qquad + 2\alpha\int_0^t \int_{\Omega_{n\ell}} J(x-y)(z(y,s)-z(x,s))\,{\rm d}y{\rm d}s \quad\text{for }x\in \Omega_{n\ell},\, t\in(0,t_0).
\end{array}
$$
An easy computation shows that
$$
\begin{array}{ll}
\displaystyle
\|S_v(z_1)-S_v(z_2)\|_{n\ell} \leq \alpha \sup\limits_{t\in[0,t_0]} \int_{\Omega_{n\ell}}
\int_0^t \int_{\Omega_\ell} J(x-y)|z_2(x,s)-z_1(x,s)|\,{\rm d}y{\rm d}s{\rm d}x
\\ [4mm]
\displaystyle\qquad\qquad\qquad\qquad\qquad \qquad+ 2\alpha \sup\limits_{t\in[0,t_0]} \int_{\Omega_{n\ell}} \int_0^t \int_{\Omega_{n\ell}}
J(x-y)|z_1(y,s)-z_2(y,s)|\,{\rm d}y{\rm d}s{\rm d}x
\\ [4mm]
\displaystyle\qquad \qquad\qquad\qquad\qquad\qquad + 2\alpha\sup\limits_{t\in[0,t_0]} \int_{\Omega_{n\ell}}\int_0^t
\int_{\Omega_{n\ell}} J(x-y)|z_2(x,s)-z_1(x,s)|\,{\rm d}y{\rm d}s{\rm d}x,
\end{array}
$$
but here we recall that the integral of the kernel $J$ is always lesser or equal than 1, and apply Fubini's theorem to obtain
$$
\|S_v(z_1)-S_v(z_2)\|_{n\ell} \leq 5\alpha t_0\|z_1-z_2\|_{n\ell}.
$$
Choosing $t_0\leq 1/(5\alpha)$, $S_v$ is a strict contraction, and hence has a unique fixed point.
As for the dependence on the data, since $z_1=S_{v_1}(z_1)$ and $z_2=S_{v_2}(z_2)$, a computation similar to the one we have just performed gives
$$
\|z_1-z_2\|_{n\ell} \leq 5\alpha t_0\|z_1-z_2\|_{n\ell} + \alpha t_0\|(v_1-v_2)(x,t)\mathbb{R}
$$
which yields~\eqref{eq:contraction.z.v}.
\end{proof}
Now it is time to go back to $\Omega_\ell$. We define
$T_2:L^1(\Omega_{n\ell}\times (0,t_0)))\to L^1(\Omega_{\ell}\times(0,t_0))$ as $T_2(z)=w$, where $w$ is the unique solution to
\begin{equation}\label{eq:solucion_w}
\begin{cases}
\displaystyle w_t(x,t)= \Delta w (x,t) -C_1w(x,t)A(x)+C_2 B_z(x,t),\quad&(x,t)\in \Omega_{\ell}\times(0,t_0),
\\[6pt]
\displaystyle -\partial_\eta w(x,t)=0,\quad & x\in \Gamma, \, t\in(0,t_0),
\\[6pt]
\displaystyle w(x,0) = u_0(x),\qquad &x\in\Omega_\ell,
\end{cases}
\end{equation}
with
$$
A(x):=\int_{\Omega_{n\ell}} J(x-y)\, {\rm d}y\quad\text{and }B_z(x,t) := \int_{\Omega_{n\ell}} J(x-y)z(y,t)\,{\rm d}y.
$$
\begin{Lemma} Let $t_0\in (0,1/(2\alpha))$.
Given $z\in L^1(\Omega_{n\ell}\times (0,t_0)))$ there exists a unique $w\in L^1(\Omega_{\ell}\times(0,t_0))$ that solves~\eqref{eq:solucion_z} for some $t_0$ small enough. Moreover, if $w_1$ and $w_2$ are the solutions corresponding respectively to $z_1$ and $z_2$, then
\begin{equation}
\label{eq:contraction.z.v}
\|w_1-w_2\|_{n\ell} \leq \frac{\alpha t_0}{1-\alpha t_0} \|z_1-z_2\|_\ell.
\end{equation}
\end{Lemma}
\begin{proof}
Existence and uniqueness of solutions is well known, see \cite{Evans}. The contraction property follows in a similar way as before, taking into account the condition at the boundary and the source term. This time we obtain the estimate
$$
\|w_1-w_2\|_\ell\leq \frac{\alpha t_0}{1-\alpha t_0}\|z_1-z_2\|_{n\ell},
$$
which is a contraction given $t_0\leq 1/(2\alpha)$.
\end{proof}
Thus, combining the two previous lemmas, we have obtained the following theorem.
\begin{Theorem}\label{thm:existence_uniqueness_modelo_1}
Given $u_0\in L^1(\mathbb{R}^N)$, there exists a unique solution to problem~\eqref{eq:main.Cauchy.local}--\eqref{eq:main.Cauchy.nonlocal} which has $u_0$ as initial datum.
\end{Theorem}
\begin{proof}
First, we keep $t_0\leq 1/(6\alpha)$).
If we compose now the two operators
$$w=T(v):=T_2(T_1(v))$$
is easy to obtain
$$
\|w_1-w_2\|_\ell \leq \frac{\alpha^2t_0^2}{(1-\alpha t_0)(1-5\alpha t_0)}\|v_1-v_2\|_\ell,
$$
which again is contraction given $t_0\leq 1/(6\alpha)$. Therefore, there is a fixed point
that gives us a unique solution in $(0,t_0)$. Now, using that the fixed point argument can be
iterated we obtain a global solution for our problem.
\end{proof}
There is also an alternative approach to prove existence of solutions for this problem. Applying the linear semi-group theory, see~\cite{ElLibro}, we can define the operator
$$
B_J(u)= \begin{cases}
\displaystyle
-\Delta u (x) - \alpha\int_{\Omega{n\ell}} J(x-y)(u(y)-u(x))\,{\rm d}y & \text{if }x\in\Omega_\ell, \\[8pt]
\displaystyle -2\alpha\int_{\Omega_{n\ell}} J(x-y)(u(y)-u(x))\,{\rm d}y - \alpha\int_{\Omega_\ell} J(x-y)(u(y)-u(x))\,{\rm d}y & \text{if }x\in\Omega_{n\ell}
\end{cases}
$$
with the constraint that $\partial_\eta u=0$ on $\Gamma$ and let $D(B_J)$ denote its domain. Following~\cite{ElLibro} we will see that this operator is completely accretive and satisfies the range condition $L^2(\mathbb{R}^N)\in R(I+B_J)$. This will imply that for any $\phi\in L^2(\mathbb{R}^N)$ there exists a $u\in D(B_J)$ such that $u+B_J(u)=\phi$ and the resolvent $(I+B_J)^{-1}$ is a contraction in $L^p(\mathbb{R}^N)$ for every $1\leq p\leq \infty$. After that, Crandall-Ligget's Theorem and the linear semi-group theory will give existence and uniqueness of a \textit{mild} solution of our evolution problem. In what follows we will use notations form semi-group theory, see~\cite{ElLibro}.
\begin{Theorem}
The operator $B_J(u)$ is completely accretive and satisfies the range condition $$L^2(\mathbb{R}^N)\in R(I+B_J).$$
\end{Theorem}
\begin{proof}
To show that the operator is completely accretive it is enough to see that for every given $u_i\in D(B_J)$, $i=1,2$, and $q\in C^\infty(\mathbb{R})$ such that $0\leq q'\leq 1$, $\mathop{\rm supp}(q)$ is compact and $0\not\in \mathop{\rm supp}(q)$ we have that
$$
\int_{\mathbb{R}^N}\left(B_J (u_1(x)) - B_J (u_2(x))\right)\cdot q(u_1(x)-u_2(x))\,{\rm d}x\geq 0.
$$
To see this is not difficult through a change of variables $x \leftrightarrow y$, Fubini and the Mean Value Theorem that give us
$$
q(u_1(x)-u_2(x)) = q' (\xi)\cdot (u_1(x)-u_2(x))
$$
for some real intermediate real number $\xi$.
To show the range condition let us take first $\phi \in L^\infty(\mathbb{R}^N)$ and define the auxiliary operator
$$
A_{n,m}(u):= T_c(u) + B_J(u) + \frac{1}{n}u^+ - \frac{1}{m}u^-
$$
where $T_c(u):=\min(c, \max(u,-c))$ is the function $u$ truncated between $-c$ and $c$. This operator is continuous monotone, and more importantly it is easy to check that it is coercive in $L^2(\mathbb{R}^N)$. Then, by~\cite{Brezis-1968}, there exists a $u_{n,m}\in L^2(\mathbb{R}^N)$ such that
$$
T_c(u_{n,m}) + B_J(u_{n,m}) + \frac{1}{n}u_{n,m}^+ - \frac{1}{m}u_{n,m}^-=\phi.
$$
Let us also define the following relation. We will write $f\ll g$ if and only if
$$
\int_{\mathbb{R}^N} j(f) \leq \int_{\mathbb{R}^N} j(g)
$$
for every $j:\mathbb{R}\to [0,\infty]$ convex, lower semi-continuous and with $j(0)=0$.
Using the monotonicity of $$ B_J(u_{n,m}) + \frac{1}{n}u_{n,m}^+ - \frac{1}{m}u_{n,m}^-$$ we have that $T_c(u_{n,m})\ll \phi$, so taking $c>\|\phi\|_{L^\infty(\mathbb{R}^N)}$ we see that $u_{n,m}\ll \phi$ and
$$
u_{n,m} + B_J(u_{n,m}) + \frac{1}{n}u_{n,m}^+ - \frac{1}{m}u_{n,m}^-=\phi.
$$
Now we will see that $u_{n,m}$ is non-decreasing in $n$ and non-increasing in $m$ in order to pass to the limits. We will show the ideas for the monotonicity in $n$, since is similar for $m$. We define $w:=u_{n,m} - u_{n+1,m}$ and this $w$ satisfies
$$
w+B_J(w)+ \frac{1}{n}u_{n,m}^+ - \frac{1}{n+1}u_{n+1,m}^+ + \frac{1}{m}u_{n+1,m}^- - \frac{1}{m}u_{n,m}^- =0.
$$
We can now multiply by $w^+$ and integrate to obtain
$$
\int_{\mathbb{R}^N}(w^+)^2+\int_{\mathbb{R}^N}B_J(w)w^+\leq 0.
$$
Through already mentioned techniques is easy to check that the second integral is positive, meaning that necessarily $w^+=0$, meaning that $u_{n,m} \leq u_{n+1,m}$. Since for the parameter $m$ is similar we have the mentioned monotonicity. Thus, using that $u_{n,m}\ll \phi$, this monotonicity and monotone convergence for the term $B_J(u_{n,m})$, we pass to the limit $n\to \infty$ to obtain
$$
u_{m} + B_J(u_{m}) - \frac{1}{m}u_{m}^-=\phi
$$
and $u_m\ll\phi$. Passing again to the limit in $m$ we obtain
$$
u + B_J(u)=\phi.
$$
Now let $\phi\in L^2(\mathbb{R}^N)$ and $\phi_n$ a sequence in $L^\infty(\mathbb{R}^N)$ such that $\phi_n\to\phi$ in $L^2(\mathbb{R}^N)$. Then we have existence for $u_n=(I+B_J)^{-1}\phi_n$ by the previous steps and due to the complete accretiveness of the operator $u_n\to u$ in $L^2(\mathbb{R}^N)$ and $B_J(u_n)\to B_J(u)$ in $L^2(\mathbb{R}^N)$ (since the dual of $L^2$ is itself). We conclude then that $u+B_J(u)=\phi_n$.
\end{proof}
\subsection{Conservation of mass}
As expected, this model preserves the total mass of the solution. Formally, we have
$$
\begin{aligned}
\partial_t \int_{\mathbb{R}^N}u(x,t)\,{\rm d}x =& \int_{\Omega_\ell} \Delta u (x,t)\,{\rm d}x + \alpha\int_{\Omega_\ell} \int_{\Omega_{n\ell}} J(x-y)(u(y,t)-u(x,t))\,{\rm d}y{\rm d}x
\\[6pt]
&+ \alpha\int_{\Omega_{n\ell}} \int_{\Omega_\ell} J(x-y)(u(y,t)-u(x,t))\,{\rm d}y{\rm d}x
\\[6pt]
&+
2 \alpha\int_{\Omega_{n\ell}} \int_{\Omega_{n\ell}} J(x-y)(u(y,t)-u(x,t))\, {\rm d}y{\rm d}x=0.
\end{aligned}
$$
The first integral is 0 thanks to the boundary condition on $u_x(0,t)$, the second and third ones add up to 0, changing variables $x$ and $y$ and using Fubini, and the last integral is equal to 0 due to the domain of integration and the symmetry of the kernel $J$. With all this and multiplying by a suitable test function our solution it is easy to prove the following theorem.
\begin{Theorem} \label{teo.conserva.masa.Cauchy}
The solution $u$ of problem~\eqref{eq:main.Cauchy.local}--\eqref{eq:main.Cauchy.nonlocal} with initial data $u_0\in L^1(\mathbb{R}^N)$ satisfies
$$
\int_{\mathbb{R}^N} u(x,t) \, {\rm d}x =\int_{\mathbb{R}^N} u_0(x) \, {\rm d}x\qquad \text{for all } t\geq 0.
$$
\end{Theorem}
\subsection{Comparison principle}
If we have two different solutions of the Cauchy problem problem~\eqref{eq:main.Cauchy.local}--\eqref{eq:main.Cauchy.nonlocal} then thanks to the linearity of the operator the difference between them is also a solution. It is also easy to see that given a non-negative initial data $u_0$ the solution keeps this non-negativity
(this follows from the fixed point construction of the solution or from the accretivity of the associated operator).
With this in mind, we state the following result.
\begin{Theorem}
If $u_0\geq 0$ then $u\geq 0$ in $\mathbb{R}^N\times\mathbb{R}_+$. Moreover, given two initial data $u_0$ and $v_0$, if $u_0\geq v_0$ then
$u\geq v$ in $\mathbb{R}^N\times\mathbb{R}_+$.
\end{Theorem}
\subsection{Asymptotic decay}
To study the decay of this problem we need a result that can be found in~\cite{Canizo-Molino-2018}.
\begin{Proposition}[\cite{Canizo-Molino-2018}]\label{proposition:molino}
Take the energy functional
$$
D_p^J(u)=\frac{p}{2}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}J(x-y)(u(y,t)-u(x,t))(|u|^{p-2}u(y,t)-|u|^{p-2}u(x,t))\,{\rm d}y{\rm d}x.
$$
Then, for every $u\in L^1\cap L^{\infty}(\mathbb{R}^N)$ and $p\geq 2$ there exists a positive constant $C$ such that
$$
D_p^J(u)\geq C\min\{\|u\|_{L^1(\mathbb{R}^N)}^{-p\gamma} \|u\|_{L^p(\mathbb{R}^N)}^{p(1+\gamma)}, \|u\|_{L^p(\mathbb{R}^N)}^p\}
$$
where $\gamma=2/N(p-1)$, and this bound provides a decay of the solutions of the evolution problem $$u_t=-(D_p^J)'(u)$$ of the form
$$
\|u(\cdot,t)\|_{L^p(\mathbb{R}^N)}^p \leq Ct^{-\frac{N(p-1)}{2}}
$$
for all $t\geq 0$ and another different positive constant $C$.
\end{Proposition}
The following proposition will be needed to study the case $p>2$. Its proof is left to the reader.
\begin{Proposition}\label{proposition:desigualdad_numerica}
For every pair of real numbers $a$ and $b$ there exists a positive finite constant $C$ such that for every $p\geq 2$
$$
(a-b)(|a|^{p-2}a - |b|^{p-2}b)\geq C(|a|^{p/2}-|b|^{p/2})^2
$$
\end{Proposition}
Thanks to these propositions we can prove the following theorem.
\begin{Theorem}
Every solution $u$ of problem~\eqref{eq:main.Cauchy.local}--\eqref{eq:main.Cauchy.nonlocal} satisfies
$$
\|u(\cdot, t)\|_{L^p(\mathbb{R}^N)}^p \leq Ct^{-\frac{N(p-1)}{2}}
$$
for every $p\in [1,\infty)$.
\end{Theorem}
\begin{Remark} {\rm
This bound coincides with the behaviour of the solutions of the local heat equation~\eqref{heat.equation}
and also with the behaviour of the solutions to the nonolocal evolution equation~\eqref{eq:nonlocal.equation}, see
\cite{Chasseigne-Chaves-Rossi-2006}.}
\end{Remark}
\begin{proof}
Inequality~\eqref{eq:fundamental.inequality} and the previous proposition provide the result when $p=2$, since $E(u)\geq \Omega_{n\ell}^J(u)$. In fact, we can just multiply by $u$ the equation and integrate to obtain
$$
\partial_t \|u\|_{L^2(\mathbb{R}^N)}^2 = - \partial_u E(u) \leq - k\int_{\mathbb{R}^N}\int_{\mathbb{R}^N} J(x-y)(u(y)-u(x))^2\, {\rm d}y{\rm d}x
$$
Using the previous proposition with $p=2$ we get
$$
\partial_t \|u\|_{L^2(\mathbb{R}^N)}^2
\leq - C
\min\Big\{\|u\|_{L^1(\mathbb{R}^N)}^{-4/N} \|u\|_{L^2(\mathbb{R}^N)}^{2(1+2/N)}, \|u\|_{L^2(\mathbb{R}^N)}^2\Big\},
$$
from where it follows that
$$
\|u(\cdot, t)\|_{L^2(\mathbb{R}^N)}^2 \leq Ct^{-\frac{N}{2}}
$$
using the conservation of mass.
The decay for $p\in(1,2)$ can be obtained through interpolation between the previous inequality and the conservation of mass property. For every $p\in(1,2)$ there exists a $\theta\in(0,1)$ such that
$$
\frac{1}{p} = \frac{1-\theta}{1} + \frac{\theta}{2}\quad\text{and}\quad \|u(\cdot,t)\|_{L^p(\mathbb{R}^N)}\leq \|u(\cdot,t)\|_{L^1(\mathbb{R}^N)}^{1-\theta}\|u(\cdot,t)\|_{L^2(\mathbb{R}^N)}^\theta.
$$
Therefore, since the mass of our solutions is constant and $\theta = 2-2/p$ we obtain that
$$
\|u(\cdot,t)\|_{L^p(\mathbb{R}^N)}\leq C\|u(\cdot,t)\|_{L^2(\mathbb{R}^N)}^{2-\frac{2}{p}}\leq C'\left( t^{\frac{-N}{4}} \right)^{2-\frac{2}{p}} = C' t^{- \frac{N(p-1)}{2p}}.
$$
Finally the case in the case $p>2$. One can check that
$$
\begin{aligned}
\partial_t \|u(\cdot,t)\|_{L^p (\mathbb{R}^N)}^p =& -\int_{\Omega_\ell}\left| \nabla (u^{p/2}(\cdot,t)) \right|^2
\\[6pt]
&- \alpha\int_{\Omega_{n\ell}}\int_{\mathbb{R}^N}J(x-y)(u(y,t)-u(x,t))(u^{p-1}(y,t)-u^{p-1}(x,t))\,{\rm d}y{\rm d}x.
\end{aligned}$$
Using Proposition~\ref{proposition:desigualdad_numerica} and Lemma~\ref{lemma:preliminar} we arrive to
$$
\partial_t \|u(\cdot,t)\|_{L^p(\mathbb{R}^N)}^p \leq - C\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}J(x-y)(u^{p/2}(y,t)-u^{p/2}(x,t))\, {\rm d}y{\rm d}x
$$
and using~Proposition \ref{proposition:molino} with $v=u^{p/2}$ and $p=2$ (this is the $p$ from the proposition, not the $p$ of the norm we are studying) we arrive to
$$
\partial_t \|u(\cdot,t)\|_{L^p(\mathbb{R}^N)}^p \leq C\|u(\cdot,t)\|_{L^p(\mathbb{R}^N)}^{p(1+\gamma)},
$$
with $\gamma=2/N(p-1)$.
From here it is easy to finish the proof.
\end{proof}
\subsection{Rescaling the kernel}
In this part we will study how, trough a limit procedure rescaling in the kernel $J$, we can obtain the local problem.
In fact we will show that solutions to the Cauchy problem for the local heat equation~\eqref{heat.equation}
can be obtained as the limit as $\varepsilon\to 0$ of solutions $u^\varepsilon$ to the problem~\eqref{eq:main.Cauchy.local}--\eqref{eq:main.Cauchy.nonlocal} with kernel $J^\varepsilon$ given by~\eqref{eq:definition.J.epsilon} and the same initial data.
We will prove convergence of the solutions in $L^2(\mathbb{R}^N)$ for finite times with Brezis-Pazy Theorem through Mosco's convergence result and this is one of the reasons why we presented another existence of solutions result for this problem based on semi-group theory for m-accretive operators. The associated energy functional to the rescaled problem reads
$$
E^\varepsilon(u):=\int_{\Omega_\ell} \frac{|\nabla u|^2}{2} + \frac{\alpha}{2\varepsilon^{N+2}}\int_{\Omega_{n\ell}}\int_{\mathbb{R}^N} J\left(\frac{x-y}{\varepsilon}\right)(u(y)-u(x))^2\,{\rm d}y{\rm d}x
$$
if $u \in D(E^\varepsilon):=L^2(\mathbb{R}^N)\cap H^1(\Omega_\ell)$ and $E^\varepsilon(u):=\infty$ if not. Analogously, we define the limit energy functional as
$$
E'(u):=\int_{\mathbb{R}^N} \frac{|\nabla u|^2}{2}
$$
if $u \in D(E'):= H^1(\mathbb{R}^N)$ and $E'(u):=\infty$ if not.
Now, given $u_0\in L^2(\mathbb{R}^N)$, for each $\varepsilon>0$, let $u^\varepsilon$ be the solution to the evolution problem
associated with the energy $E^\varepsilon$ and initial datum $u_0$ and $u$ be the solution associated with $E'$ with the same initial condition.
\begin{Theorem}
Under the above assumptions, the functions $u^\varepsilon$ converge to solutions to \eqref{heat.equation}. For any finite $T>0$ we have that
$$
\lim\limits_{\varepsilon\to 0}\left(\sup\limits_{t\in[0,T]} \| u^\varepsilon (\cdot, t) - u(\cdot, t) \|_{L^2(\mathbb{R}^N)}\right)=0.
$$
\end{Theorem}
\begin{proof}
We will make use of the Brezis-Pazy Theorem for the sequence of m-accretive operators $B_{J^\varepsilon}$ in $L^2(\mathbb{R}^N)$ defined previously in the existence and uniqueness subsection. To apply this result we will need to show that the resolvent operators satisfy
$$
\lim\limits_{\varepsilon\to 0} (I+B_{J^\varepsilon})^{-1}\phi = (I+A)^{-1}\phi
$$
for every $\phi\in L^2(\mathbb{R}^N)$ where $A(u):=(-\Delta) u$ is the classic operator for the heat equation in this theory. If we have this then the theorem gives convergence of $u^\varepsilon$ to $u$ in $L^2(\mathbb{R}^N)$ uniformly in $[0,T]$. To prove this convergence of the resolvents we will use Mosco's result, where we only have to prove two things:
\begin{itemize}
\item[(i)] For every $u\in D(E')$ there exists a $\{u^\varepsilon\}\in D(E^\varepsilon)$ such that $u^\varepsilon\to u$ in $L^2(\mathbb{R}^N)$ and
$$
E'(u)\geq \limsup\limits_{\varepsilon\to 0} E^\varepsilon(u^\varepsilon).
$$
\item[(ii)] If $u^\varepsilon \to u$ weakly in $L^2(\mathbb{R}^N)$ then
$$
E'(u)\leq \liminf\limits_{\varepsilon\to 0} E^\varepsilon(u^\varepsilon).
$$
\end{itemize}
For more information, see~\cite{ElLibro}, Appendix 7, Theorems A.3 and A.38.
Let us start with (i). Given $u\in H^1(\mathbb{R}^N)$ we know that there exists a sequence $\{v_n\}\in C_c^\infty (\mathbb{R}^N) $ such that $v_n\to u$ in $L^2(\mathbb{R}^N)$. On the other hand, through Taylor's expansion it is not hard to see that
$$
\lim\limits_{\varepsilon\to 0} \left(\lim\limits_{\varepsilon\to 0}E^\varepsilon(v_n)\right) = E'(u).
$$
This means, by a diagonal argument, that there exists a subsequence of $\varepsilon_j$ such that
$$
\lim\limits_{\varepsilon_j\to 0}E^{\varepsilon_j}(v_j) = E'(u),
$$
showing (i).
To see (ii) from the sequence of $u^\varepsilon$ that converges weakly to $u$ we extract a subsequence $\varepsilon_n$ such that
$$
\lim\limits_{\varepsilon\to 0} E^{\varepsilon_n}(u^{\varepsilon_n}) =\liminf\limits_{\varepsilon\to 0} E^{\varepsilon_n}(u^{\varepsilon_n}).
$$
We will suppose that this inferior limit is finite, since if it is not there is nothing to prove.
Let us now take a ball of radius centered at 0, say $B^R$ and define $B^R_\ell:=B^R\cap\Omega_\ell$ and respectively $B^R_{n\ell}$ an define
$$
E_R^\varepsilon(u^\varepsilon):=\int_{B^R_\ell} \frac{|\nabla u^\varepsilon|^2}{2} + \frac{\alpha}{2\varepsilon^{N+2}}\int_{B^R_{n\ell}}\int_{B^R} J\left(\frac{x-y}{\varepsilon}\right)(u^\varepsilon(y)-u^\varepsilon(x))^2\, {\rm d}y{\rm d}x.
$$
Since the inferior limit is finite there must exist a $\varepsilon_0$ such that this quantity is bounded by a constant $M$ that only depends on $R$ for al $\varepsilon<\varepsilon_0$ and we can apply Lemma~\ref{lemma:preliminar} to obtain that there exists a positive constant $k$ not depending on $\varepsilon$ such that
$$
k\frac{\alpha}{2\varepsilon^{N+2}}\int_{B^R}\int_{B^R} J\left(\frac{x-y}{\varepsilon}\right)(u^\varepsilon(y)-u^\varepsilon(x))^2\, {\rm d}y{\rm d}x<M.
$$
Now on this domain we apply Lemma~\ref{lemma:basic_convergence_lemma_for_rescaled} to obtain a subsequence of $u^\varepsilon$, denoted by itself for simplicity, that converges to $u$ in $L^2(B^R)$ and such that
$$
\lim\limits_{\varepsilon\to 0} \left( \frac{k}{\alpha}J^{1/2}(z) \chi_{B^R}(x+\varepsilon z)\chi_{\Omega}(x) \frac{\bar{u}^\varepsilon(x+\varepsilon z) - \bar{u}^\varepsilon(x)}{\varepsilon} \right) = k\alpha J^{1/2}(z)h(x,z)
$$
weakly in $L^2_x(\mathbb{R}^N)\times L^2_z(\mathbb{R}^N)$ with $h(x,z)=z\nabla u(x)$ for all $(x,z)\in B^{R}\times \mathbb{R}^N$. Using now the lower semi-continuity of the norm for sequences that converge weakly we have that
$$
\left(\int_{\mathbb{R}^N} J(z)z^2\ dz\right) \int_{B^{R}}|\nabla u|^2
\leq \displaystyle\liminf\limits_{\varepsilon\to 0} \frac{k}{\alpha}\int_{B^{R}}\int_{B^{R}} J(x-y)\frac
{(u^\varepsilon(y,t)-u^\varepsilon(x,t))^2}{\varepsilon^2}\,{\rm d} y{\rm d}x,
$$
which means, using again Lemma~\ref{lemma:preliminar}, that
$$
\displaystyle\int_{B^{R}}|\nabla u|^2\leq \lim\limits_{\varepsilon\to 0} E_R^{\varepsilon}(u^\varepsilon)\leq \lim\limits_{\varepsilon\to 0} E^{\varepsilon}(u^\varepsilon)
$$
and we finish just by making $R$ go to $\infty$.
\end{proof}
\section{The Neumann problem} \label{sect-Neumann}
\setcounter{equation}{0}
In this section we discuss the Neumann problem~\eqref{eq:main.Neumann.local}--\eqref{eq:main.Neumann.nonlocal}.
\subsection{Existence, uniqueness and conservation of mass}
The ideas presented for the Cauchy problem can be applied \textit{mutatis mutandis} to this problem.
Therefore, using the fixed point argument, or the alternative approach by semigroup theory we obtain the following result whose
proof is left to the reader.
\begin{Theorem}\label{thm:existence_uniqueness_neumann}
Given $u_0\in L^1(\Omega)$ there exists a unique solution to problem~\eqref{eq:main.Neumann.local}--\eqref{eq:main.Neumann.nonlocal} with initial datum $u_0$. This solution conserves its mass along the evolution.
\end{Theorem}
\subsection{Comparison principle}
Also arguing as we did for the Cauchy problem we have a comparison result for the Neumann case.
\begin{Theorem}
If $u_0\geq 0$ then $u\geq 0$ in $\Omega\times\mathbb{R}_+$. Moreover, given two initial data $u_0$ and $v_0$, with $u_0\geq v_0$ then the corresponding solutions satisfy $u\geq v$ in $\Omega\times\mathbb{R}_+$.
\end{Theorem}
\subsection{Asymptotic behaviour}
In this occasion we expect the solution to converge to the average of the initial condition in every $L^p$. In fact what we are going to show is that the function
$$
v=u-|\Omega|^{-1}\int_{\Omega} u_0
$$
converges to 0 exponentially fast in $L^p$ norm.
\begin{Theorem}
The function $v$ satisfies
$$
\|v(\cdot, t)\|_{L^p(\Omega)}\leq C_1e^{-C_2 t}
$$
for every $p\in[1,\infty)$ where $C_1$ and $C_2$ are positive finite constants ($C_2$ can be taken independent of $u_0$).
\end{Theorem}
\begin{Remark}
{\rm This behaviour coincides with the behaviour of the solutions of the Heat Equation and with the
behaviour of the solutions to the nonlocal evolution equation when the integrals are considered in the domain $\Omega$, see
\cite{Chasseigne-Chaves-Rossi-2006} (we have exponential convergence to the mean value
of the initial condition, but notice that the constants and the exponents can be different for the three cases, local, nonlocal and our mixed
local/nonlocal problems).}
\end{Remark}
\begin{proof}
We will prove the result for $p=2$. The result for $p\in[1,\infty)$ comes from the use of H\"{o}lder's inequality and for $p>2$ from
$$
\begin{aligned}
\partial_t \|u(\cdot,t)\|_{L^p(\Omega)}^p = &-\int_{\Omega_\ell}\left| \nabla (u^{p/2})(\cdot,t) \right|^2\\[6pt]
&- \alpha\int_{\Omega_{n\ell}}\int_{\mathbb{R}^N}J(x-y)(u(y,t)-u(x,t))(u^{p-1}(y,t)-u^{p-1}(x,t))\,{\rm d}y{\rm d}x,
\end{aligned}
$$
the Proposition~\ref{proposition:desigualdad_numerica}, Lemma~\ref{lemma:preliminar} and the fact that
$$
\nabla u\cdot \nabla(|u|^{p-1}u) = |\nabla (|u|^{p/2})|^2
$$
so we can rename $u^{p/2}$ as another function $w$ and apply the case $p=2$ to obtain the cases $p>2$. This case is again left for the reader
and is somehow similar to the analogous case for the Cauchy problem.
So for $p=2$ we compute, after some calculations
$$
\partial_t\int_{\Omega}v^2(x,t) \, {\rm d}x= -2\int_{\Omega_\ell}|\nabla v|^2 - 2C\int_{\Omega_{n\ell}}\int_\Omega J(x-y)(v(y,t)-v(x,t))^2\, {\rm d}y{\rm d}x.
$$
Using Lemma~\ref{lemma:preliminar} and a result from \cite{ElLibro} that shows that for every function $v$ with zero average we have that
$$
\int_{\Omega}\int_\Omega J(x-y)(v(y,t)-v(x,t))^2\, {\rm d}y{\rm d}x \geq \beta \int_{\Omega}v^2 (x,t) \, {\rm d}x
$$
for some positive $\beta$ we obtain
$$
\partial_t\int_{\Omega}v^2 (x,t) \, {\rm d}x \leq -k\int_{\Omega}v^2 (x,t) \, {\rm d}x
$$
for another constant $k$. From this point the result follows trivially.
\end{proof}
\begin{Remark} {\rm
One can define what is the analogous to the first non-zero eigenvalue for this problem as
$$
\beta_1 (\Omega) = \inf_{\int_\Omega u = 0} \frac{ \displaystyle \int_{\Omega_\ell} |\nabla u|^2 + \alpha
\int_{\Omega_{n\ell}}\int_{\Omega} J(x-y)(u(y)-u(x))^2\, {\rm d}y{\rm d}x}{ \displaystyle \int_\Omega u^2 }
$$
and show that $\beta_1 (\Omega)$ is positive (otherwise the exponential decay does not hold
with a constant $C_2$ independent of $u_0$). The existence of an eigenfunction (a function that achieves the
infimum) is not straightforward. We leave this fact open.
}
\end{Remark}
\subsection{Rescaling the kernel}
As before, we want to study the convergence of the solutions $u^\varepsilon$ of the Neumann problem~\eqref{eq:main.Neumann.local}--\eqref{eq:main.Neumann.nonlocal} with rescaled kernel $J^\varepsilon$ given by~\eqref{eq:definition.J.epsilon} to the solution of the Neumann problem for the local heat equation with the same initial datum.
\begin{Theorem}\label{thm:rescaled_Neumann}
For any finite $T>0$ we have that
$$
\lim\limits_{\varepsilon\to 0}\left(\sup\limits_{t\in[0,T]} \| u^\varepsilon (\cdot, t) - u(\cdot, t) \|_{L^2(\mathbb{R}^N)}\right)=0
$$
where this $u$ is the solution of the Neumann problem for the Heat Equation in $\Omega$ with the same initial data $u_0$.
\end{Theorem}
The proof of this theorem is analogous to the one we did for the Cauchy problem (see also~\cite{BLGneu}). Again we use the already mentioned Brezis-Pazy Theorem through convergence of the resolvents. Notice that this approach uses the linear semi-group theory in $L^2(\Omega)$ mentioned in the Cauchy section (that also works just fine in this case).
\section{The Dirichlet problem} \label{sect-Dirichlet}
\setcounter{equation}{0}
In this section we devote our attention to the Dirichlet problem~\eqref{eq:main.Dirichlet.local}--\eqref{eq:main.Dirichlet.boundary}.
\subsection{Existence and uniqueness}
Again, we have the following result whose proof can be obtained as in the previous cases (again
we have two proofs, one using a fixed point argument and another one using semigroup theory).
\begin{Theorem}\label{thm:existence_uniqueness_dirichlet}
Given $u_0\in L^1(\Omega)$, there exists a unique solution to problem~\eqref{eq:main.Dirichlet.local}--\eqref{eq:main.Dirichlet.boundary} which has $u_0$ as initial datum.
\end{Theorem}
\begin{Remark} {\rm For this problem there is a loss of mass trough the boundary. In fact, assume that $u_0$ is nonnegative
(and hence $u(x,t)$ is nonnegative for every $t>0$). Then integrating in $\Omega$ we get
$$
\begin{aligned}
\partial_t \int_\Omega u(x,t) \, {\rm d}x =& \int_{\partial \Omega \cap \partial \Omega_\ell}
\partial_\eta u(x,t) \, {\rm d}\sigma + \alpha \int_{\Omega_\ell} \int_{\mathbb{R}^N\setminus \Omega_\ell} J(x-y)(u(y,t)-u(x,t))\, {\rm d}y{\rm d}x
\\[6pt]&+ \alpha\int_{\Omega_{n\ell}} \int_{\mathbb{R}^N\setminus \Omega_{n\ell}} J(x-y)(u(y,t)-u(x,t))\,{\rm d}y{\rm d}x
\\[6pt]
=& \int_{\partial \Omega \cap \partial \Omega_\ell}
\frac{\partial u}{\partial \eta}(x,t) \, d\sigma(x) - \alpha \int_{\Omega_\ell} \int_{\mathbb{R}^N\setminus \Omega} J(x-y)u(x,t) \, {\rm d}y{\rm d}x
\\[6pt]
& - \alpha \int_{\Omega_{n\ell}} \int_{\mathbb{R}^N\setminus \Omega} J(x-y)u(x,t) \, {\rm d}y{\rm d}x <0.
\end{aligned}
$$
}
\end{Remark}
\subsection{Comparison principle}
As for the previous cases we have a comparison result.
\begin{Theorem}
If $u_0\geq 0$ then $u\geq 0$ in $\Omega\times\mathbb{R}_+$. Moreover, given two initial data $u_0$ and $v_0$ with $u_0\geq v_0$, then the corresponding solutions satisfy
$u\ge v$ in $\Omega\times\mathbb{R}_+$.
\end{Theorem}
\subsection{Asymptotic decay}
The result here is analogous to the one in corresponding section for the Neumann problem. The only extra tool needed is a result that was proved in \cite{ElLibro} that shows that there exists a constant $\beta$ such that for every function $u$ that satisfies $u(x,t)\equiv 0$ for every $x\in\mathbb{R}^N\setminus \Omega$ we have that
$$
\int_{\mathbb{R}^N}\int_\Omega J(x-y)(u(y,t)-u(x,t))^2\,{\rm d}y{\rm d}x \geq \beta \int_{\Omega}u^2 (x)\, {\rm d}x ,
$$
similarly to the previous section for functions with zero average. At this point the proof for the following theorem is straightforward.
\begin{Theorem}
The solution $u$ of the Dirichlet problem problem~\eqref{eq:main.Dirichlet.local}--\eqref{eq:main.Dirichlet.boundary} with initial datum $u_0$ satisfies
$$
\|u(\cdot, t)\|_{L^p(\Omega)}\leq C_1e^{-C_2 t}
$$
for every $p\in[1,\infty)$ where $C_1$ and $C_2$ are positive finite constants ($C_2$ can be chosen independent of $u_0$).
\end{Theorem}
\begin{Remark} {\rm
This coincides with the behaviour of the solutions to the Heat Equation and with the
behaviour of the solutions to the corresponding nonlocal evolution equation with zero exterior condition, see
\cite{Chasseigne-Chaves-Rossi-2006} (we have exponential decay, but notice that again here
the constants and the exponents can be different for the three cases).}
\end{Remark}
\begin{Remark} {\rm Again for this case we have an associated eigenvalue problem. Let us consider
$$
\lambda_1 (\Omega) = \inf_{u|_{\mathbb{R}^N \setminus \Omega} \equiv 0} \frac{ \displaystyle \int_{\Omega_\ell} |\nabla u|^2 + \alpha
\int_{\Omega_{n\ell}}\int_{\mathbb{R}^N} J(x-y)(u(y)-u(x))^2\, {\rm d}y{\rm dx}}{ \displaystyle \int_\Omega u^2}.
$$
One can prove that $\lambda_1 (\Omega)$ is positive using our control of the nonlocal energy, \eqref{eq:fundamental.inequality}
and the results in \cite{ElLibro}. Again in this case the existence of an eigenfunction (a function that achieves the
infimum) is left open.
}
\end{Remark}
\subsection{Rescaling the kernel}
In this part we will study how we can obtain the solution to the Dirichlet problem with zero boundary datum for the local heat equation as the limit as $\varepsilon\to 0$ of solutions $u^\varepsilon$ of the Dirichlet problem~\eqref{eq:main.Dirichlet.local}--\eqref{eq:main.Dirichlet.boundary} with rescaled kernel $J^\varepsilon$ as in~\eqref{eq:definition.J.epsilon} and the same initial datum.
\begin{Theorem}
For any finite $T>0$ we have that
$$
\lim\limits_{\varepsilon\to 0}\left(\sup\limits_{t\in[0,T]} \| u^\varepsilon (\cdot, t) - u(\cdot, t) \|_{L^2(\mathbb{R}^N)}\right)=0
$$
where this $u$ is the solution of the Dirichlet problem for the Heat Equation in $\Omega$ with the same initial data $u_0$ and zero boundary data.
\end{Theorem}
The proof of this theorem is analogous to the previous ones (we also refer to~\cite{BLGdir} here), see the comments
about Theorem~\ref{thm:rescaled_Neumann} in
the previous section.
\section{Comments on possible extensions}
\setcounter{equation}{0}
In this final section we briefly comment on possible extensions of our results.
\begin{itemize}
\item Our results could be extended to cover singular kernels including, for example, fractional Laplacians.
In this case the associated energy for the Cauchy problem looks like
$$
E(u):=\int_{\Omega_\ell} \frac{|\nabla u|^2}{2}+
\frac{C}{2}\int_{\mathbb{R}^N_-}\int_{\mathbb{R}^N} \frac{(u(y)-u(x))^2}{|x-y|^{N+2s}}
\, {\rm d}y{\rm d}x.
$$
The abstract semigroup theory seems the right way to obtain existence and uniqueness of a solution.
One interesting problem is to couple two different fractional Laplacians and look for the asymptotic behaviour
of the solutions to the corresponding Cauchy problem. We will tackle this kind of extension
in a future paper.
\item One can look for moving interfaces, making that $\Gamma$ depends on $t$. To show existence and uniqueness
of solutions for a problem like this seems a challenging problem. In this framework one is tempted to consider free boundary problems
in which we have an unknown interface that evolves with time and we impose that solutions have conservation of the total mass plus some continuity across the free boundary.
\item Finally, we mention that an interesting problem is to look at nonlinear diffusion equations (coupling, for example, a
local $p-$Laplacian with a nonlocal $q-$Laplacian, see \cite{ElLibro} for a definition of the last operator). A possible
energy for this problem is
$$
E(u):=\int_{\Omega_\ell} \frac{|\nabla u|^p}{p} +
\frac{C}{q}\int_{\Omega_{n\ell}}\int_{\mathbb{R}^N} \frac{|u(y)-u(x)|^q}{|x-y|^{N+qs}}
\,{\rm d}y{\rm d}x.
$$
This problem involves new difficulties, especially when one looks
for scaled versions of the kernel and tries to see whether there is a limit.
\end{itemize}
\medskip
\noindent{\large \textbf{Acknowledgments}}
\noindent The first two authors were partially supported by the Spanish project {MTM2017-87596-P}, and the third one by CONICET grant PIP GI No 11220150100036CO
(Argentina), by UBACyT grant 20020160100155BA (Argentina) and by the Spanish project MTM2015-70227-P.
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No.\,777822.
Part of this work was done during visits of JDR to Madrid and of AG and FQ to Buenos Aires. The authors want to thank these institutions for the nice and stimulating working atmosphere.
| 24,486 |
\section{Introduction}
This paper continues the project from \cite{pushforward,cut,lgdual} of making the theory of affine B-twisted topological Landau-Ginzburg models \emph{constructive}, in the sense of deriving formulas and algorithms which compute the fundamental categorical structures from coefficients of the potential and the differentials of matrix factorisations. For example, the units and counits of adjunction in the bicategory of Landau-Ginzburg models \cite{lgdual} and the pushforward and convolution operations \cite{pushforward} may be described in terms of various flavours of Atiyah classes.
We study \emph{idempotent finite $A_\infty$-models} of DG-categories of matrix factorisations over arbitrary $\mathbb{Q}$-algebras $k$. An idempotent finite model of a DG-category $\mathcal{A}$ over $k$ is a pair $(\mathcal{B}, E)$ consisting of an $A_\infty$-category $\mathcal{B}$ over $k$ which is Hom-finite, in the sense that for each pair $X,Y$ of objects the complex $\mathcal{B}(X,Y)$ is a finitely generated projective $k$-module, together with a diagram of $A_\infty$-functors
\[
\xymatrix@C+3pc{
\mathcal{A} \ar@<1ex>[r]^I & \mathcal{B} \ar@(ur,dr)^E \ar@<1ex>[l]^P
}
\]
with $A_\infty$-homotopies $P \circ I \simeq 1$ and $I \circ P \simeq E$. Using the finite pushforward construction of \cite{pushforward} and Clifford actions of \cite{cut} we explain how to define an idempotent finite $A_\infty$-model of the DG-category of matrix factorisations $\mathcal{A}$ of any potential $W$ over $k$. When $k$ is a field this is constructive in the sense that we give algorithms for computing the entries in the higher $A_\infty$-operations on $\mathcal{B}$ and the components of $E$, viewed as matrices, from the Atiyah class of $\mathcal{A}$ and a Gr\"obner basis of the defining ideal of the critical locus of $W$.
We note that our goal is \emph{not} to construct a minimal model: $\mathcal{B}$ has nonzero differential. When $k$ is a field the usual approach to finding a Hom-finite $A_\infty$-model of $\mathcal{A}$ is to equip $H^*(\mathcal{A})$ with an $A_\infty$-structure, so that there is a diagram of $A_\infty$-functors
\[
\xymatrix@C+3pc{
\mathcal{A} \ar@<1ex>[r]^I & H^*(\mathcal{A}) \ar@<1ex>[l]^P
}
\]
and $A_\infty$-homotopies $P \circ I \simeq 1$ and $I \circ P \simeq 1$. The problem is that the minimal model is only under good control for those matrix factorisations $X,Y$ where we happen to know a good cohomological splitting on $\mathcal{A}(X,Y)$. In our approach the idempotent finite model $(\mathcal{B},E)$ is constructed directly from $\mathcal{A}$, and if we happen to know a cohomological splitting then this provides the data necessary to split $E$. An important example is the case where $X = Y = k^{\operatorname{stab}}$ is the standard generator, where there is a natural cohomological splitting and the resulting $A_\infty$-structure has been studied by Seidel \cite[\S 11]{seidel_hms}, Dyckerhoff \cite[\S 5.6]{d0904.4713}, Efimov \cite[\S 7]{efimov} and Sheridan \cite{sheridan}. We explain how to split our idempotent $E$ in this special case and recover the minimal $A_\infty$-model of $\mathcal{A}(k^{\operatorname{stab}},k^{\operatorname{stab}})$ in Section \ref{section:generator}.
\\
To explain in more detail, let $\mathcal{A} = \mfdg(R,W)$ be the DG-category of finite-rank matrix factorizations of a potential $W \in R = k[x_1,\ldots,x_n]$ over an arbitrary $\mathbb{Q}$-algebra $k$. This is a DG-category over $R$, which we can view as a $\mathds{Z}_2$-graded $R$-module
\[
\mathcal{H}_{\mathcal{A}} = \bigoplus_{X,Y} \mathcal{A}(X,Y)
\]
equipped with some $R$-linear structure, namely, the differential $\mu_1$ and composition operator $\mu_2$. Let $\varphi: k \longrightarrow R$ denote the inclusion of constants. We can ask if the restriction of scalars $\varphi_*( \mathcal{H}_{\mathcal{A}} )$ is $A_\infty$-homotopy equivalent over $k$ to an $A_\infty$-category which is Hom-finite over $k$, and can such a finite model be described constructively? This is related to the problem of pushforwards considered in \cite{pushforward} and, here as there, it is unclear that in general a \emph{direct} construction of the finite pushforward exists.
Instead, following \cite{pushforward} the conceptual approach we adopt here is to seek an algorithm which constructs both a larger object $\mathcal{B}$ which is a Hom-finite $A_\infty$-category over $k$ together with an idempotent $E: \mathcal{B} \longrightarrow \mathcal{B}$ which splits to $\mathcal{A}$. More precisely, the larger object is obtained by adjoining to $\mathcal{A}$ a number of odd supercommuting variables $\theta_1,\ldots,\theta_n$ and completing along the the critical locus of $W$ to obtain the DG-category
\[
\mathcal{A}_\theta = \bigwedge F_\theta \otimes_k \mathcal{A} \otimes_R \widehat{R}\,,
\]
where $F_\theta$ is the $\mathds{Z}_2$-graded $k$-module $\bigoplus_{i=1}^n k \theta_i$ which is concentrated in odd degree, and $\widehat{R}$ denotes the $I$-adic completion where $I = (\partial_{x_1} W, \ldots, \partial_{x_n} W)$. The differential on $\mathcal{A}_\theta$ is the one inherited from $\mathcal{A}$. Note that the completion $\mathcal{A} \otimes_R \widehat{R}$ can be recovered from $\mathcal{A}_\theta$ by splitting the idempotent DG-functor on $\mathcal{A}_\theta$ which arises from the morphism of algebras
\begin{equation}\label{eq:projector_e}
\bigwedge F_\theta \longrightarrow k \cdot 1 \longrightarrow \bigwedge F_\theta
\end{equation}
which projects onto the identity by sending all $\theta$-forms of nonzero degree to zero.
One should think of $F_\theta$ as the normal bundle to the critical locus of $W$, and $\mathcal{A}_\theta$ can be equipped with a natural strong deformation retract which arises from a connection $\nabla$ ``differentiating'' in the normal directions to the critical locus. This strong deformation retract can be interpreted as an analogue in algebraic geometry of the deformation retract associated to the Euler vector field of a tubular neighborhood; see Appendix \ref{section:formaltub}. Applying the homological perturbation lemma results in a Hom-finite $A_\infty$-category $\mathcal{B}$ with
\[
\mathcal{B}(X,Y) = R/I \otimes_R \operatorname{Hom}_R(X,Y)\,.
\]
By construction there is an idempotent finite $A_\infty$-model
\[
\xymatrix@C+3pc{
\mathcal{A} \otimes_R \widehat{R}\, \ar@<1ex>[r]^-I & \mathcal{B} \ar@(ur,dr)^E \ar@<1ex>[l]^-P
}
\]
of the completion $\mathcal{A} \otimes_R \widehat{R}$, which we show is homotopy equivalent to $\mathcal{A}$ over $k$ (in particular $\mathcal{A}$ and $\mathcal{A} \otimes_R \widehat{R}$ are both DG-enhancements of the $k$-linear triangulated category $\hmf(R,W)$, with the latter being analogous to working with matrix factorisations over the power series ring $k\llbracket \bold{x} \rrbracket$). The $A_\infty$-idempotent $E$ arises in the obvious way as the transfer to $\mathcal{B}$ of the idempotent \eqref{eq:projector_e}.
\vspace{0.3cm}
\textbf{Outline of the paper.} In Section \ref{section:the_model} we give the details of the above sketch of the construction of the idempotent finite model of $\operatorname{mf}(R,W)$, although the proofs are collected in Appendix \ref{section:proofs}. The algorithmic content of the theory is summarised in Section \ref{section:the_algorithms}, but developed over the course of Section \ref{section:towards} and Section \ref{section:feynman_diagram} culminating in the Feynman rules of Section \ref{section:feynman_diagram_4}. Some of the geometric intuition for the central strong deformation retract is developed in Section \ref{section:formaltub}. In Appendix \ref{section:noetherian} we give some technical observations necessary to remove a Noetherian hypothesis from \cite{cut}.
\vspace{0.3cm}
\textbf{Related work.} The approach we develop here seems to be related to ideas developed in the complex analytic setting by Shklyarov \cite{shklyarov_cy} in order to put a Calabi-Yau structure on $\mathcal{A}$, although we do not understand the precise connection. For applications to string field theory it is important to construct \emph{cyclic} $A_\infty$-minimal models; see for example \cite{carqueville}. For a recent approach to this problem for the endomorphism DG-algebra of $k^{\operatorname{stab}}$ see \cite{tu}. We do not understand the interplay between cyclic $A_\infty$-structures and idempotent finite $A_\infty$-models. This project began as an attempt to understand the work on deformations of matrix factorisations and effective superpotentials in the mathematical physics literature \cite{baumgartl1, baumgartl2, baumgartl3, baumgartl4, carqueville2, carqueville3, knapp} which should be better known to mathematicians. Some ideas developed here were inspired by old work of Herbst-Lazaroiu \cite{herbst} that has now culminated in a new approach to non-affine Landau-Ginzburg models \cite{babalic}.
\vspace{0.3cm}
\textbf{Acknowledgements.} Thanks to Nils Carqueville for introducing me to $A_\infty$-categories, Calin Lazaroiu for encouragement and the opportunity to present the results at the workshop ``String Field Theory of Landau-Ginzburg models'' at the IBS Center for Geometry and Physics in Pohang. The author was supported by the ARC grant DP180103891.
\section{Background}
Throughout $k$ is a commutative $\mathbb{Q}$-algebra and unless specified otherwise $\otimes$ means $\otimes_k$. If $\bold{x} = (x_1,\ldots,x_n)$ is a sequence of formal variables then $k[\bold{x}]$ denotes $k[x_1,\ldots,x_n]$ and similarly for power series rings. Given $M \in \mathbb{N}^n$ we write $x^M$ for $x_1^{M_1} \cdots x_n^{M_n}$.
Let $R$ be a commutative ring. Given finite-rank free $\mathds{Z}_2$-graded $R$-modules $M, N$ and $\phi \in \operatorname{Hom}_R(M,N)$ we say that $\phi$ is \emph{even} (resp. \emph{odd}) if $\phi(M_i) \subseteq N_i$ (resp. $\phi(M_i) \subseteq N_{i+1}$) for all $i \in \mathds{Z}_2$. This makes $\operatorname{Hom}_R(M,N)$ into a $\mathds{Z}_2$-graded $R$-module. Given two homogeneous operators $\psi, \phi$ the \emph{graded commutator} is
\begin{equation}
[\phi, \psi] = \phi \psi - (-1)^{|\phi||\psi|} \psi \phi\,.
\end{equation}
In this note all operators are given a $\mathds{Z}_2$-grading and the commutator always denotes the graded commutator. We briefly recall some important operators on exterior algebras
\[
\bigwedge F_\xi = \bigwedge \bigoplus_{i=1}^r k \xi_i
\]
where $F_\xi = \bigoplus_{i=1}^r k \xi_i$ denotes a free $k$-module of rank $r$ with basis $\xi_1,\ldots,\xi_r$. We give $F_\xi$ a $\mathds{Z}_2$-grading by assigning $|\xi_i| = 1$, that is, $F_\xi \cong k^{\oplus r}[1]$. The inherited $\mathds{Z}_2$-grading on $\bigwedge F_\xi$ is the reduction mod $2$ of the usual $\mathds{Z}$-grading on the exterior algebra, e.g. $|\xi_1 \xi_2| = 0$.
We define odd operators $\xi_j \wedge (-), \xi_j^* \,\lrcorner\, (-)$ on $\bigwedge F_\xi$ by wedge product and contraction, respectively, where contraction is defined by the formula
\begin{align*}
\xi_j^* \,\lrcorner\, \Big( \xi_{i_1} \wedge \cdots \wedge \xi_{i_s} \Big) = \sum_{l=1}^s (-1)^{l-1} \delta_{j, i_l} \xi_{i_1} \wedge \cdots \wedge \widehat{ \xi_{i_l} } \wedge \cdots \wedge \xi_{i_s}\,.
\end{align*}
Often we will simply write $\xi_j$ for $\xi_j \wedge (-)$ and $\xi_j^*$ for $\xi_j^* \,\lrcorner\, (-)$. Clearly with this notation, as operators on $\bigwedge F_\xi$, we have the commutator (as always, graded)
\begin{equation}\label{eq:wedge_contract_comm}
\big[ \xi_i, \xi_j^* \big] = \xi_i \xi_j^* + \xi_j^* \xi_i = \delta_{ij} \cdot 1
\end{equation}
and also $[ \xi_i, \xi_j ] = [\xi_i^*, \xi_j^*] = 0$.
\subsection{$A_\infty$-categories}\label{section:ainfcat}
For the theory of $A_\infty$-categories we follow the notational conventions of \cite[\S 2]{lazaroiu}, which we now recall. Another good reference is Seidel's book \cite{seidel}. A small $\mathds{Z}_2$-graded $A_\infty$-category $\cat{A}$ over $k$ is specified by a set of objects $\operatorname{ob}(\cat{A})$ and $\mathds{Z}_2$-graded $k$-modules $\mathcal{A}(a,b)$ for any pair $a,b \in \operatorname{ob}(\cat{A})$ together with $k$-linear maps
\[
\mu_{a_n,\ldots,a_0}: \mathcal{A}(a_{n-1}, a_n) \otimes \cdots \otimes \mathcal{A}(a_0, a_1) \longrightarrow \mathcal{A}(a_0,a_n)
\]
of degree $2 - n \equiv n$ for every sequence of objects $a_0,\ldots,a_n$ with $n \ge 0$. If the objects involved are clear from the context, we will write $\mu_n$ for this map. These maps are required to satisfy the following equation for $n \ge 1$
\begin{equation}\label{eq_ainf_constraints}
\sum_{\substack{i \ge 0, j \ge 1 \\ 1 \le i + j \le n}} (-1)^{ij + i + j + n} \mu_{n-j+1}\Big( x_n \otimes \cdots \otimes x_{i+j+1} \otimes \mu_j( x_{i+j} \otimes \cdots \otimes x_{i+1} ) \otimes x_i \otimes \cdots \otimes x_1 \Big) = 0
\end{equation}
In particular we have a degree zero map
\[
\mu_2 = \mu_{cba}: \mathcal{A}(b,c) \otimes \mathcal{A}(a,b) \longrightarrow \mathcal{A}(a,c)
\]
which satisfies the $n = 3$ equation
\begin{align*}
- &\mu_2( x \otimes \mu_2( y \otimes z ) ) + \mu_2( \mu_2( x \otimes y ) \otimes z ) + \mu_1 \mu_3( x \otimes y \otimes z )\\
& + \mu_3( \mu_1(x) \otimes y \otimes z ) + \mu_3( x \otimes \mu_1(y) \otimes z ) + \mu_3( x \otimes y \otimes \mu_1(z) ) = 0
\end{align*}
expressing that $\mu_2$ is associative up to the homotopy $\mu_3$ relative to the differential $\mu_1$. The operators $\mu_n$ are sometimes referred to as \emph{higher operations}. Any DG-category is an $A_\infty$-category where $\mu_1$ is the differential, $\mu_2$ is the composition and $\mu_n = 0$ for $n \ge 3$. Note that a DG-category has identity maps $u_a \in \mathcal{A}^0(a,a)$ for all objects $a$, and these make $\cat{A}$ a \emph{strictly unital} $A_\infty$-category \cite[\S 2.1]{lazaroiu}, \cite[\S I.2]{seidel}.
To minimise the trauma of working with $A_\infty$-categories, it is convenient to adopt a different point of view on the higher operations, which eliminates most of the signs: from the $\mu$ we can define \emph{suspended forward compositions} \cite[\S 2.1]{lazaroiu}
\begin{equation}
r_{a_0,\ldots,a_n}: \mathcal{A}(a_0, a_1)[1] \otimes \cdots \otimes \mathcal{A}(a_{n-1}, a_n)[1] \longrightarrow \mathcal{A}(a_0,a_n)[1]
\end{equation}
for which the $A_\infty$-constraints \eqref{eq_ainf_constraints} take the more attractive form
\[
\sum_{\substack{i \ge 0, j \ge 1 \\ 1 \le i + j \le n}} r_{a_0, \ldots, a_i, a_{i+j}, \ldots, a_n} \circ \Big( \operatorname{id}_{a_0a_1} \otimes \cdots \otimes \operatorname{id}_{a_{i-1}a_i} \otimes r_{a_i,\ldots,a_{i+j}} \otimes \operatorname{id}_{a_{i+j}a_{i+j+1}} \otimes \cdots \otimes \operatorname{id}_{a_{n-1}a_n} \Big) = 0
\]
As before we write $r_n$ for $r_{a_0,\ldots,a_n}$ if the indices are clear. Note that while $\mu_n$ has $\mathds{Z}_2$-degree $n$, the $r_n$'s are all odd operators. The $\mathds{Z}_2$-degree of a homogeneous element $x \in \mathcal{A}(a,b)$ will be denoted $|x|$ and we write $\widetilde{x} = |x| + 1$ for the degree of $x$ viewed as an element of $\mathcal{A}(a,b)[1]$. Sometimes we refer to this as the \emph{tilde grading}. We refer the reader to \cite{lazaroiu} for the definition of the suspended forward compositions, but note $\mu_1(x) = r_1(x)$ and
\begin{equation}\label{eq:mu2vsr2}
\mu_2(x_1 \otimes x_2) = (-1)^{\widetilde{x}_1\widetilde{x}_2 + \widetilde{x}_1 + 1} r_2(x_2 \otimes x_1)\,.
\end{equation}
The Koszul sign rule always applies when we evaluate the application of a tensor product of homogeneous linear maps on a tensor, for example since $r_2$ is odd
\[
1 \otimes r_2: \mathcal{A}(a,b)[1] \otimes \mathcal{A}(b,c)[1] \otimes \mathcal{A}(c,d)[1] \longrightarrow \mathcal{A}(a,b)[1] \otimes \mathcal{A}(b,d)[1]
\]
applied to a tensor $x_3 \otimes x_2 \otimes x_1$ is
\[
(1 \otimes r_2)(x_3 \otimes x_2 \otimes x_1) = (-1)^{\widetilde{x_3}} x_3 \otimes r_2( x_2 \otimes x_1 )
\]
where we had to know that the domain involved $\mathcal{A}(a,b)[1]$ rather than $\mathcal{A}(a,b)$ to know that we were supposed to use the tilde grading on $x_3$.
We also use the sector decomposition of \cite[\S 2.2]{lazaroiu}. We associate to $\cat{A}$ the $k$-module
\[
\mathcal{H}_{\cat{A}} = \bigoplus_{a,b \in \operatorname{ob}(\mathcal{A})} \mathcal{A}(a,b)
\]
equipped with the induced $\mathds{Z}_2$-grading. Let $Q$ be the commutative associative $k$-algebra (without identity) generated by $\epsilon_a$ for $a \in \operatorname{ob}(\cat{A})$ subject to the relations $\epsilon_a \epsilon_b = \delta_{ab} \epsilon_a$ (this non-unital algebra is denoted $R$ in \cite{lazaroiu}). Then $\mathcal{H}_{\cat{A}}$ has a $Q$-bimodule structure in which $\epsilon_a$ acts on the left by the projector of $\mathcal{H}_{\mathcal{A}}$ onto the subspace $\bigoplus_{b \in \operatorname{ob}(\mathcal{A})} \mathcal{A}(a,b)$ and $\epsilon_b$ acts on the right by the projector of $\mathcal{H}_{\mathcal{A}}$ onto the subspace $\bigoplus_{a \in \operatorname{ob}(\mathcal{A})} \mathcal{A}(a,b)$. The $n$-fold tensor product of the $Q$-bimodule $\mathcal{H}_{\mathcal{A}}$ over $Q$ is
\begin{equation}\label{eq:bimodule_tensor_hh}
\mathcal{H}_{\mathcal{A}}^{\otimes_Q n} = \bigoplus_{a_0,\ldots,a_n \in \operatorname{ob}(\mathcal{A})} \mathcal{A}(a_0,a_1) \otimes \mathcal{A}(a_1,a_2) \otimes \cdots \otimes \mathcal{A}(a_{n-1},a_n)
\end{equation}
with the obvious $Q$-bimodule structure involving the values of $a_0, a_n$, so that the forward suspended product $r_n$ is an odd $Q$-bilinear map from $\mathcal{H}_{\mathcal{A}}[1]^{\otimes_Q n} \longrightarrow \mathcal{H}_{\mathcal{A}}[1]$.
\section{The idempotent finite model}\label{section:the_model}
Throughout $k$ is a commutative $\mathbb{Q}$-algebra and all $A_\infty$-categories are $k$-linear.
\begin{definition} An $A_\infty$-category $\mathcal{C}$ is called \emph{Hom-finite} if for every pair $a,b$ of objects the underlying $k$-module of $\mathcal{C}(a,b)$ is a finitely generated and projective $k$-module.
\end{definition}
\begin{definition} An \emph{idempotent finite $A_\infty$-model} of a DG-category $\mathcal{A}$ is a pair $(\mathcal{B}, E)$ consisting of a Hom-finite $A_\infty$-category $\mathcal{B}$ and $A_\infty$-functor $E: \mathcal{B} \longrightarrow \mathcal{B}$, and a diagram
\[
\xymatrix@C+3pc{
\mathcal{A} \ar@<1ex>[r]^I & \mathcal{B} \ar@(ur,dr)^E \ar@<1ex>[l]^P
}
\]
of $A_\infty$-functors and $A_\infty$-homotopies $P \circ I \simeq 1$ and $I \circ P \simeq E$.
\end{definition}
We recall from \cite{lgdual} the definition of a potential:
\begin{definition}\label{defn:potential} A polynomial $W \in k[x_1,\ldots,x_n]$ is a \textsl{potential} if
\begin{itemize}
\item[(i)] $\partial_{x_1} W,\ldots,\partial_{x_n} W$ is a quasi-regular sequence;
\item[(ii)] $k[x_1,\ldots,x_n]/(\partial_{x_1} W,\ldots,\partial_{x_n} W)$ is a finitely generated free $k$-module;
\item[(iii)] the Koszul complex of $\partial_{x_1} W,\ldots,\partial_{x_n} W$ is exact except in degree zero.
\end{itemize}
\end{definition}
A typical example is a polynomial $W \in \mathbb{C}[x_1,\ldots,x_n]$ with isolated critical points \cite[Example 2.5]{lgdual}. As shown in \cite{lgdual}, these hypotheses on a potential are sufficient to produce all the properties relevant to two-dimensional topological field theory, even if $k$ is not a field. If $k$ is Noetherian then (iii) follows from (i).
\begin{definition} Given a potential $W \in R = k[x_1,\ldots,x_n]$ the DG-category $\mathcal{A} = \mfdg(R,W)$ has as objects \emph{matrix factorisations} of $W$ over $R$ \cite{EisenbudMF} that is, the pairs $(X, d_X)$ consisting of a $\mathbb{Z}_2$-graded free $R$-module $X$ of finite rank and an odd $R$-linear operator $d_X: X \longrightarrow X$ satisfying $d_X^2 = W \cdot 1_X$. We define
\begin{gather*}
\mathcal{A}(X,Y) = \big( \operatorname{Hom}_R(X,Y) \,, d_{\operatorname{Hom}} \big)\,,\\
d_{\operatorname{Hom}}(\alpha) = d_Y \circ \alpha - (-1)^{|\alpha|} \alpha \circ d_X\,.
\end{gather*}
The composition is the usual composition of linear maps.
\end{definition}
Throughout we set $I = ( \partial_{x_i} W, \ldots, \partial_{x_n} W )$ to be the defining ideal of the critical locus, and write $\widehat{R}$ for the $I$-adic completion. Let $W \in R$ be a potential and let $\mathcal{A}$ be a full sub-DG-category of the DG-category of matrix factorisations $\mfdg( R, W )$. The first observation is that we can replace $\mathcal{A}$ by the completion $\mathcal{A} \otimes_R \widehat{R}$.
\begin{lemma}\label{lemma:completion_he} Let $W \in R$ be a potential. The canonical DG-functor
\[
\mathcal{A} \longrightarrow \mathcal{A} \otimes_R \widehat{R}
\]
is a $k$-linear homotopy equivalence, that is, for every pair of matrix factorisations $X,Y$
\[
\operatorname{Hom}_R(X,Y) \longrightarrow \operatorname{Hom}_R(X,Y) \otimes_R \widehat{R}
\]
is a homotopy equivalence over $k$.
\end{lemma}
\begin{proof}
See Appendix \ref{section:noetherian}.
\end{proof}
To construct an idempotent finite model of $\mathcal{A}$ we form the extension
\begin{equation}\label{eq:defn_AAtheta}
\mathcal{A}_{\theta} = \bigwedge( k \theta_1 \oplus \cdots \oplus k \theta_n ) \otimes \mathcal{A} \otimes_R \widehat{R}
\end{equation}
which is a DG-category with the same objects as $\mathcal{A}$ and mapping complexes
\[
\mathcal{A}_{\theta}( X, Y ) = \bigwedge( k \theta_1 \oplus \cdots \oplus k \theta_n ) \otimes \mathcal{A}(X,Y) \otimes_R \widehat{R}\,.
\]
The differentials in $\mathcal{A}_{\theta}$ are induced from $\mathcal{A}$ and the composition rule is obtained from multiplication in the exterior algebra and composition in $\mathcal{A}$, taking into account Koszul signs when moving $\theta$-forms past morphisms in $\mathcal{A}$. Next we consider the $\mathds{Z}_2$-graded modules $\mathcal{B}(X,Y)$ and the $Q$-bimodule $\mathcal{H}_{\mathcal{B}}$ defined in Section \ref{section:ainfcat}, namely
\begin{gather*}
\mathcal{B}(X,Y) = R/I \otimes_R \operatorname{Hom}_R(X,Y)\\
\mathcal{H}_{\mathcal{B}} = R/I \otimes_R \mathcal{H}_{\mathcal{A}} = \bigoplus_{X,Y \in \mathcal{A}} \mathcal{B}(X,Y)\,.
\end{gather*}
At the moment this has no additional structure: it is just a module, not an $A_\infty$-category. But we note that since $\operatorname{Hom}_R(X,Y)$ is a free $R$-module of finite rank, and $R/I$ is free of finite rank over $k$, the spaces $\mathcal{B}(X,Y)$ are free $k$-modules of finite rank. The goal of this section is to construct higher $A_\infty$-operations $\rho_k$ on $\mathcal{H}_{\mathcal{B}}$.
\begin{setup}\label{setup:overall} Throughout we adopt the following notation:
\begin{itemize}
\item $R = k[\bold{x}] = k[x_1,\ldots,x_n]$.
\item $F_\theta = \bigoplus_{i=1}^n k\theta_i$ is a free $\mathbb{Z}_2$-graded $k$-module of rank $n$, with $|\theta_i| = 1$.
\item $t_1,\ldots,t_n$ is a quasi-regular sequence in $R$, such that with $I = (t_1,\ldots,t_n)$
\begin{itemize}
\item $R/I$ is a finitely generated free $k$-module
\item each $t_i$ acts null-homotopically on $\mathcal{A}(X,Y)$ for all $X,Y \in \mathcal{A}$
\item the Koszul complex of $t_1,\ldots,t_n$ over $R$ is exact except in degree zero.
\end{itemize}
\item We choose a $k$-linear section $\sigma: R/I \longrightarrow R$ of the quotient map $R \longrightarrow R/I$ and as in Appendix \ref{section:formaltub} we write $\nabla$ for the associated connection with components $\partial_{t_i}$.
\item $\lambda_i^X$ is a null-homotopy for the action of $t_i$ on $\mathcal{A}(X,X)$ for each $X \in \mathcal{A}, 1 \le i \le n$.
\item We choose for $X \in \mathcal{A}$ an isomorphism of $\mathds{Z}_2$-graded $R$-modules
\[
X \cong \coeff{X} \otimes R
\]
where $\coeff{X}$ is a finitely generated free $\mathds{Z}_2$-graded $k$-module. Hence
\begin{equation}
\mathcal{A}(X,Y) = \operatorname{Hom}_R(X,Y) \cong \operatorname{Hom}_k(\coeff{X},\coeff{Y}) \otimes R\,.\label{eq:chosenCiso}
\end{equation}
\end{itemize}
\end{setup}
\begin{remark} By the hypothesis that $W$ is a potential, the sequence $\bold{t} = (\partial_{x_1} W, \ldots, \partial_{x_n} W)$ satisfies the hypotheses and we may choose $\lambda_i^X$ to be the operator $\partial_{x_i}(d_X)$ defined by choosing a homogeneous basis for $X$ and differentiating entry-wise the matrix $d_X$ in that basis. However some choices of $\bold{t}$ and the $\lambda_i^X$ may be better than others, in the sense that they lead to simpler Feynman rules
\end{remark}
To explain the construction of the higher operations on $\mathcal{B}$, it is convenient to switch to an alternative presentation of the spaces $\mathcal{A}(X,Y), \mathcal{B}(X,Y)$. Consider the following $\mathds{Z}_2$-graded $k$-modules, where the grading comes only from $\bigwedge F_\theta$ and the Hom-space:
\begin{gather*}
\mathcal{A}'(X,Y) = R/I \otimes \operatorname{Hom}_k(\widetilde{X},\widetilde{Y}) \otimes k\llbracket \bold{t} \rrbracket\,,\\
\mathcal{A}'_\theta(X,Y) = \bigwedge F_\theta \otimes R/I \otimes \operatorname{Hom}_k(\widetilde{X},\widetilde{Y}) \otimes k\llbracket \bold{t} \rrbracket\,, \\
\mathcal{B}'(X,Y) = R/I \otimes \operatorname{Hom}_k(\widetilde{X},\widetilde{Y})\,.
\end{gather*}
Using \eqref{eq:chosenCiso} there is an isomorphism of $\mathds{Z}_2$-graded $R$-modules $\mathcal{B}'(X,Y) \cong \mathcal{B}(X,Y)$. By Lemma \ref{prop_algtube} there is a $k\llbracket \bold{t} \rrbracket$-linear isomorphism $\sigmastar: R/I \otimes k \llbracket \bold{t} \rrbracket \longrightarrow \widehat{R}$ and combined with \eqref{eq:chosenCiso} this induces an isomorphism of $\mathds{Z}_2$-graded $k\llbracket \bold{t} \rrbracket$-modules
\begin{equation}\label{eq:transfer_iso_intro}
\xymatrix@C+2pc{
\sigmastar: R/I \otimes \operatorname{Hom}_k(\coeff{Y},\coeff{X}) \otimes k\llbracket \bold{t} \rrbracket \ar[r]^-{\cong} &
\operatorname{Hom}_R(Y,X) \otimes_R \widehat{R}
}
\end{equation}
which induces an isomorphism of $\mathds{Z}_2$-graded $k\llbracket \bold{t} \rrbracket$-modules
\[
\xymatrix@C+2pc{
\mathcal{A}'_\theta(X,Y) \ar[r]_-{\cong}^{\sigmastar} & \mathcal{A}_\theta(X,Y)
}\,.
\]
Hence there are induced isomorphisms $\mathcal{H}_{\mathcal{A}'_\theta} \cong \mathcal{H}_{\mathcal{A}_\theta}$ and $\mathcal{H}_{\mathcal{B}} \cong \mathcal{H}_{\mathcal{B}'}$. Using these identifications we transfer operators on $\mathcal{A}_\theta,\mathcal{B}$ to their primed cousins, usually without a change in notation. For example we write $d_\mathcal{A}$ for
\[
\xymatrix@C+2pc{
\mathcal{H}_{\mathcal{A}'_\theta} \ar[r]^-{\sigmastar}_-{\cong} & \mathcal{H}_{\mathcal{A}_\theta} \ar[r]^-{d_{\mathcal{A}}} & \mathcal{H}_{\mathcal{A}_\theta} \ar[r]^-{(\sigmastar)^{-1}}_-{\cong} & \mathcal{H}_{\mathcal{A}'_\theta}
}\,.
\]
This map is the differential in a $k\llbracket \bold{t} \rrbracket$-linear DG-category structure on $\mathcal{A}'_\theta$, with the forward suspended composition $r_2$ in this DG-structure given by
\[
\xymatrix@C+2pc{
\mathcal{H}_{\mathcal{A}'_\theta}[1] \otimes_Q \mathcal{H}_{\mathcal{A}'_\theta}[1] \ar[r]^-{\cong} & \mathcal{H}_{\mathcal{A}_\theta}[1] \otimes_Q \mathcal{H}_{\mathcal{A}_\theta}[1] \ar[r]^-{r_2} & \mathcal{H}_{\mathcal{A}_\theta}[1] \ar[r]^-{\cong} & \mathcal{H}_{\mathcal{A}'_{\theta}}[1]
}
\]
where the unlabelled isomorphisms are $\sigmastar \otimes \sigmastar$ and $(\sigmastar)^{-1}$. Going forward when we refer to $\mathcal{A}'_\theta$ as a DG-category this structure is understood. Finally the tensor product of the inclusions $k \subset \bigwedge F_\theta$ and $k \subset k\llbracket \bold{t} \rrbracket$, respectively the projections $\bigwedge F_\theta \longrightarrow k$ and $k \llbracket \bold{t} \rrbracket \longrightarrow k$ define $k$-linear maps $\sigma$ and $\pi$ as in the diagram
\[
\xymatrix@C+2pc{
R/I \otimes \operatorname{Hom}_k(\widetilde{X},\widetilde{Y}) \ar@<1ex>[r]^-{\sigma} & \bigwedge F_\theta \otimes R/I \otimes \operatorname{Hom}_k(\widetilde{X},\widetilde{Y}) \otimes k \llbracket \bold{t} \rrbracket\ar@<1ex>[l]^-{\pi}
}
\]
and hence degree zero $k$-linear maps
\[
\xymatrix@C+3pc{
\mathcal{H}_{\mathcal{B}'} \ar@<1ex>[r]^-{\sigma} & \mathcal{H}_{\mathcal{A}'_\theta}\ar@<1ex>[l]^-{\pi}
}\,.
\]
\begin{definition}\label{defn:atiyah_class} The \emph{critical Atiyah class} of $\mathcal{A}$ is the operator on $\mathcal{H}_{\mathcal{A}'_\theta}$ given by
\[
\vAt_{\mathcal{A}} = [ d_{\mathcal{A}}, \nabla ] = d_{\mathcal{A}} \nabla + \nabla d_{\mathcal{A}}
\]
where $\nabla = \sum_{i=1}^n \theta_i \partial_{t_i}$ is the connection of Section \ref{section:formaltub}. This is a closed $k\llbracket \bold{t} \rrbracket$-linear operator, independent up to $k$-linear homotopy of the choice of connection.
\end{definition}
We call $\vAt_{\mathcal{A}}$ the \emph{critical} Atiyah class since it is defined using the connection $\nabla$, which is a kind of derivative in the directions normal to the critical locus, and some name seems useful to distinguish $\vAt_{\mathcal{A}}$ from various other Atiyah classes also playing a role in the theory of matrix factorisations, for example the associative Atiyah classes of \cite{lgdual}.
\begin{definition}\label{definition:zeta} Since $\mathcal{H}_{\mathcal{A}'_\theta}$ is a module over $\bigwedge F_\theta \otimes k\llbracket \bold{t} \rrbracket$ we may define
\begin{equation}
(\boldsymbol{\theta}, \bold{t}) \mathcal{H}_{\mathcal{A}'_\theta} \subseteq \mathcal{H}_{\mathcal{A}'_\theta}
\end{equation}
where $(\boldsymbol{\theta}, \bold{t})$ is the two-sided ideal spanned by the $\theta_i, t_j$. We define the $k$-linear operator
\begin{gather*}
\zeta: (\boldsymbol{\theta}, \bold{t}) \mathcal{H}_{\mathcal{A}'_\theta} \longrightarrow (\boldsymbol{\theta}, \bold{t}) \mathcal{H}_{\mathcal{A}'_\theta}\\
\zeta\big( \omega \otimes z \otimes \alpha \otimes f \big) = \sum_{\delta \in \mathbb{N}^n} \frac{1}{|\omega| + |\delta|} \omega \otimes z \otimes \alpha \otimes f_\delta t^\delta
\end{gather*}
for a homogeneous $\theta$-form $\omega$ and $f = \sum_{\delta \in \mathbb{N}^n} f_\delta t^\delta \in k\llbracket \bold{t} \rrbracket$. Evaluated on polynomial $f$ this is the inverse of the grading operator for \emph{virtual degree} which is a $\mathds{Z}$-grading $\Vert - \Vert$ in which $\Vert\theta_i\Vert = \Vert t_i \Vert = 1$ for $1 \le i \le n$ and $\Vert z \Vert = \Vert \alpha \Vert = 0$ for $z \in R/I, \alpha \in \operatorname{Hom}_k(\widetilde{X},\widetilde{Y})$.
\end{definition}
\begin{definition}\label{defn:important_operators} We introduce the following operators:
\begin{align*}
\sigma_\infty &= \sum_{m \ge 0} (-1)^m (\zeta \vAt_{\mathcal{A}})^m \sigma: \mathcal{H}_{\mathcal{B}'} \longrightarrow \mathcal{H}_{\mathcal{A}'_\theta}\\
\phi_\infty &= \sum_{m \ge 0} (-1)^m (\zeta \vAt_{\mathcal{A}})^m \zeta \nabla: \mathcal{H}_{\mathcal{A}'_\theta} \longrightarrow \mathcal{H}_{\mathcal{A}'_\theta}\\
\delta &= \sum_{i=1}^n \lambda^\bullet_i \theta_i^*: \mathcal{H}_{\mathcal{A}'_\theta} \longrightarrow \mathcal{H}_{\mathcal{A}'_\theta}
\end{align*}
where $\lambda_i^\bullet$ acts on $\operatorname{Hom}_R(X,Y)$ by post-composition
\[
\lambda_i^\bullet(\alpha) = \lambda_i^Y \circ \alpha\,.
\]
Note that the sums involved are all finite, since $\vAt_{\mathcal{A}}$ has positive $\theta$-degree. The $\mathds{Z}_2$-degrees of these operators are $|\delta| = |\sigma_\infty| = 0$ and $|\phi_\infty| = 1$.
\end{definition}
We now have the notation to state the main theorem. Let $\cat{BT}_k$ denote the set of all valid plane binary trees with $k$ inputs (in the sense of Appendix \ref{section:trees}). Given such a tree $T$, we add some additional vertices and then \emph{decorate} the tree by inserting operators at each vertex. The \emph{denotation} of such an operator decorated tree is defined by reading the tree as a ``flowchart'' with inputs inserted at the leaves and the output read off from the root. For example the tree $T$ in Figure \ref{fig:opdectree} has for its denotation the operator
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{dia13}
\end{center}
\centering
\caption{Example of an operator decorated tree.}\label{fig:opdectree}
\end{figure}
\begin{equation}\label{eq:explicit_tree_operator}
\pi e^{-\delta} r_2\Big( e^{\delta}\sigma_\infty \otimes e^{\delta} \phi_\infty e^{-\delta} r_2\Big( e^{\delta} \sigma_\infty \otimes e^{\delta} \sigma_\infty \Big) \Big)\,.
\end{equation}
See Appendix \ref{section:trees} for our conventions on trees, decorations and denotations. We note that these denotations involve Koszul signs when evaluated, arising from the $\mathds{Z}_2$-degree (with respect to the tilde grading) of the involved operators (recall for example that $r_2$ is odd). See Section \ref{section:ainfcat} for the definition of the ring $Q$ and the $Q$-bimodule structure on $\mathcal{H}_{\mathcal{B}}$, and note that we write $e_i(T)$ for the number of internal edges in a tree $T$.
\begin{theorem}\label{theorem:main_ainfty_products} Define the odd $Q$-bilinear map
\begin{equation}\label{eq:rho_kintermsoftrees}
\rho_k = \sum_{T \in \cat{BT}_k} (-1)^{e_i(T)} \rho_T : \mathcal{H}_{\mathcal{B}'}[1]^{\otimes_Q k} \longrightarrow \mathcal{H}_{\mathcal{B}'}[1]
\end{equation}
where $\rho_T$ is the denotation of the decoration with coefficient ring $Q$ which assigns $\mathcal{H}_{\mathcal{A}'_\theta}[1]$ to every leaf and $\mathcal{H}_{\mathcal{B}'}[1]$ to every edge, and to
\begin{center}
\begin{itemize}
\item \textbf{inputs:} $e^\delta \sigma_\infty$
\item \textbf{internal edges:} $e^\delta \phi_\infty e^{-\delta}$
\item \textbf{internal vertices:} $r_2$
\item \textbf{root:} $\pi e^{-\delta}$
\end{itemize}
\end{center}
Then $( \mathcal{B}, \rho = \{ \rho_k \}_{k \ge 1} )$ is a strictly unital $A_\infty$-category and there are $A_\infty$-functors
\[
\xymatrix@C+3pc{
\mathcal{A}_\theta \ar@<1ex>[r]^F & \mathcal{B} \ar@<1ex>[l]^G
}
\]
and an $A_\infty$-homotopy $G \circ F \simeq 1_{\mathcal{A}_{\theta}}$.
\end{theorem}
\begin{proof}
The full details are given in Appendix \ref{section:proofs}, but in short this is the usual transfer of $A_\infty$-structure via homological perturbation applied to a particular choice of strong deformation retract arising from the connection $\nabla$ and the isomorphism $e^{\delta}$.
\end{proof}
The projector $e$ of \eqref{eq:projector_e} can be written in terms of creation and annihilation operators
\[
e = \theta_n^* \cdots \theta_1^* \theta_1 \cdots \theta_n: \bigwedge F_\theta \longrightarrow \bigwedge F_\theta
\]
where $\theta_i$ denotes the operator $\theta_i \wedge (-)$ and $\theta_i^*$ denotes contraction $\theta_i^* \,\lrcorner\, (-)$. This is a morphism of algebras and induces a functor of DG-categories $e: \mathcal{A}_\theta \longrightarrow \mathcal{A}_\theta$.
\begin{definition} Let $E$ be the following composite of $A_\infty$-functors
\[
\xymatrix@C+2pc{
\mathcal{B} \ar[r]^-G & \mathcal{A}_\theta \ar[r]^-{e} & \mathcal{A}_\theta \ar[r]^-F & \mathcal{B}
}\,.
\]
\end{definition}
\begin{corollary}\label{corollary:idempotent_finite_model} The tuple $(\mathcal{B}, \rho, E)$ is an idempotent finite $A_\infty$-model of $\mathcal{A} \otimes_R \widehat{R}$.
\end{corollary}
\begin{proof}
Consider the diagram
\[
\xymatrix@C+3pc{
\mathcal{A} \otimes_R \widehat{R} \ar@<1ex>[r]^-{i} & \mathcal{A}_\theta \ar@<1ex>[l]^-{p} \ar@<1ex>[r]^F & \mathcal{B} \ar@<1ex>[l]^G
}
\]
where $i$ is the natural inclusion and $p$ is the projection, so that $p \circ i = 1$ and $i \circ p = e$. These are both DG-functors. We define $I = F \circ i$ and $P = p \circ G$ as $A_\infty$-functors. Since we have an $A_\infty$-homotopy $G \circ F \simeq 1$ we have an $A_\infty$-homotopy
\[
P I = p G F i \simeq p i = 1
\]
and by definition $E = I \circ P$.
\end{proof}
At a cohomological level the pushforward of matrix factorisations is expressed in terms of residues and null-homotopies $\lambda$, see for example the results in \cite[\S 11.2]{pushforward} on Chern characters. These residues can be understood as traces of products of commutators with the connection $\nabla$ \cite[Proposition B.4]{pushforward}. The results just stated extend this ``closed sector'' or cohomological level analysis of pushforwards via residues to the ``open sector'' or categorical level, where the supertraces are removed and the higher operations of the idempotent finite model $\mathcal{B}$ are described directly in terms of the commutators $\vAt_{\mathcal{A}} = [d_{\mathcal{A}}, \nabla]$ and homotopies $\lambda$. Moreover, these formulas arise from homological perturbation applied to a kind of tubular neighborhood of the critical locus, so it seems natural to interpret $(\mathcal{B}, E)$ as a kind of ``$A_\infty$-categorical residue'' of $\mathcal{A}$ along the subscheme $\Spec(R/I)$.
\vspace{0.2cm}
Recall that the purpose of the idempotent $E$ is that it encodes the information necessary to ``locate'' $\mathcal{A}$ within the larger object $\mathcal{B}$. The information in the lowest piece $E_1$ of this $A_\infty$-idempotent is the simplest, as it locates $\mathcal{A}$ as a subcomplex within $\mathcal{B}$.
\begin{definition}\label{defn:gamma_anddagger} Let $\gamma_i, \gamma_i^\dagger$ be the $k$-linear cochain maps
\begin{gather*}
\xymatrix@C+2pc{
\mathcal{H}_{\mathcal{B}'} \ar[r]^-{G_1} & \mathcal{H}_{\mathcal{A}'_{\theta}} \ar[r]^-{\theta_i^*} & \mathcal{H}_{\mathcal{A}'_{\theta}} \ar[r]^-{F_1} & \mathcal{H}_{\mathcal{B}'}
}\\
\xymatrix@C+2pc{
\mathcal{H}_{\mathcal{B}'} \ar[r]^-{G_1} & \mathcal{H}_{\mathcal{A}'_{\theta}} \ar[r]^-{\theta_i} & \mathcal{H}_{\mathcal{A}'_{\theta}} \ar[r]^-{F_1} & \mathcal{H}_{\mathcal{B}'}\,.
}
\end{gather*}
respectively.
\end{definition}
\begin{theorem}\label{theorem:homotopy_clifford} There is a $k$-linear homotopy
\begin{equation}
E_1 \simeq \gamma_n \cdots \gamma_1 \gamma_1^\dagger \cdots \gamma_n^\dagger
\end{equation}
and $k$-linear homotopies $\gamma_i \simeq \vAt_i$ and
\begin{equation}
\gamma_i^\dagger \simeq -\lambda_i - \sum_{m \ge 1} \sum_{q_1,\ldots,q_m} \frac{1}{(m+1)!} \big[ \lambda_{q_m}\,, \big[ \lambda_{q_{m-1}}, \big[ \cdots [ \lambda_{q_1}, \lambda_i ] \cdots \big] \At_{q_1} \cdots \At_{q_m}
\end{equation}
where $\vAt_i = [ d_{\mathcal{A}}, \partial_{t_i} ]$ denotes the $i$th component of the Atiyah class $\vAt_{\mathcal{A}}$, viewed as an odd closed $k$-linear operator on $\mathcal{H}_{\mathcal{B}'}$.
\end{theorem}
\begin{proof}
This is essentially immediate from \cite{cut}, see Appendix \ref{section:proofs} for details.
\end{proof}
\subsection{Algorithms}\label{section:the_algorithms}
When $k$ is a field there are algorithms which compute the $A_\infty$-functors $I, P, E$ and the higher $A_\infty$-products $\rho$ in the sense that once we choose a $k$-basis for $R/I$ and homogeneous $R$-bases for the matrix factorisations, for each fixed $k \ge 1$ there is an algorithm computing the coefficients in the matrices $I_k, P_k, E_k, \rho_k$. We explain this algorithm in detail only for $\rho_k$ as the algorithms for $I_k, P_k, E_k$ are variations on the same theme using \cite{markl_transfer}.
The algorithm is implicit in the presentation of $\rho_k$ as the sum of denotations of operator decorated trees, provided we have algorithms for computing the section $\sigma$, Atiyah classes $\vAt_{\mathcal{A}}$ and homotopies $\lambda_i$ as operators on $\mathcal{H}_{\mathcal{A}'_\theta}$. If $k$ is a field, then by choosing a Gr\"obner basis of the ideal $I$ we obtain such algorithms; see Remark \ref{remark:compute_rdelta} and Remark \ref{remark:grobner}. Over the course of Section \ref{section:towards} and Section \ref{section:feynman_diagram} we present the details of this algorithm in the case where the matrix factorisations are of Koszul type, using Feynman diagrams.
\begin{remark} For general $k$ the algorithmic content of the theory depends on the availability of a replacement for Gr\"obner basis methods. One important case where such methods are available is the example of potentials $W \in k[x_1,\ldots,x_n]$ with $k = \mathbb{C}[u_1,\ldots,u_v]$, using Gr\"obner systems \cite{weispfenning} and constructible partitions.
\end{remark}
\begin{remark} Finding a Hom-finite $A_\infty$-category $A_\infty$-homotopy-equivalent to $\mathcal{A}$ is equivalent to \emph{splitting} the idempotent $E$ within Hom-finite $A_\infty$-categories. We do not know a general algorithm which performs this splitting. However, this can be done when we have the data of a chosen cohomological splitting, for example in the case of the endomorphism DG-algebra of the standard generator when $k$ is a field; see Section \ref{section:generator}.
\end{remark}
\section{Towards Feynman diagrams}\label{section:towards}
In this section we collect some technical lemmas needed in the presentation of the Feynman rules, in the next section. Throughout the conventions of Setup \ref{setup:overall} remain in force. See Appendix \ref{section:trees} for our conventions on trees, decorations and denotations. Given a binary plane tree $T$ we denote by $T'$ the \emph{mirror} of $T$, which is obtained by exchanging the left and right branch at every vertex. Associated to a decoration $D$ of $T$ is a mirror decoration $D'$ of $T'$. Given a plane tree $T$ decorated by $D$ as explained in Theorem \ref{theorem:main_ainfty_products} let $\operatorname{eval}_{D'}$ be the mirror decoration evaluated without Koszul signs (Definition \ref{defn:evaluation_tree}).
\begin{lemma}\label{prop:replacer2} We have
\begin{equation}\label{eq:proprhoT}
\rho_T( \beta_1, \ldots, \beta_k ) = (-1)^{\sum_{i < j} \widetilde{\beta}_i \widetilde{\beta}_j + \sum_i \widetilde{\beta}_i P_i + k + 1} \operatorname{eval}_{D'}( \beta_k, \ldots, \beta_1 )\,.
\end{equation}
where $P_i$ is the number of times the path from the $i$th leaf in $T$ (counting from the left) enters a trivalent vertex as the right-hand branch on its way to the root, and $k+1$ is the number of internal vertices in $T$.
\end{lemma}
\begin{proof}
Let us begin with the special case given in Figure \ref{fig:opdectree}, using
\begin{equation}\label{eq:mu2vsr2_v2}
r_2( \beta_1, \beta_2 ) = (-1)^{\widetilde{\beta_1} \widetilde{\beta_2} + \widetilde{\beta_2} + 1}
\mu_2(\beta_2 \otimes \beta_1)
\end{equation}
and the operator given in \eqref{eq:explicit_tree_operator} to compute that
\begin{align*}
\rho_T( \beta_1, \beta_2, \beta_3 ) &= \pi e^{-\delta} r_2\Big( e^{\delta}\sigma_\infty \otimes e^{\delta} \phi_\infty e^{-\delta} r_2\Big( e^{\delta} \sigma_\infty \otimes e^{\delta} \sigma_\infty \Big) \Big)( \beta_1 \otimes \beta_2 \otimes \beta_3 )\\
&= (-1)^{a} \pi e^{-\delta} r_2\Big( e^{\delta}\sigma_\infty(\beta_1) \otimes e^{\delta} \phi_\infty e^{-\delta} r_2\Big( e^{\delta} \sigma_\infty(\beta_2) \otimes e^{\delta} \sigma_\infty(\beta_3) \Big) \Big)
\end{align*}
where $a = \widetilde{\beta_1}( |\phi_\infty| + |r_2| ) \equiv 0$ gives the Koszul sign arising from moving the inputs ``into position''. Note that since $|\delta| = |\sigma_\infty| = 0$ and every $r_2$ decorating the tree $T$, except for the one adjacent to the root, is followed immediately by a $\phi_\infty$, this sign is always $+1$.
Hence the signs that arise in computing $\rho_T(\beta_1,\ldots,\beta_k)$ in terms of $\mu_2$ on the mirrored tree arise entirely from \eqref{eq:mu2vsr2_v2}. If we continue to calculate, we find
\begin{align*}
&= (-1)^{b} \pi e^{-\delta} \mu_2\Big( e^{\delta} \phi_\infty e^{-\delta} r_2\Big( e^{\delta} \sigma_\infty(\beta_2) \otimes e^{\delta} \sigma_\infty(\beta_3) \Big) \otimes e^{\delta}\sigma_\infty(\beta_1) \Big)\\
&= (-1)^{b + c} \pi e^{-\delta} \mu_2\Big( e^{\delta} \phi_\infty e^{-\delta} \mu_2\Big( e^{\delta} \sigma_\infty(\beta_3) \otimes e^{\delta} \sigma_\infty(\beta_2) \Big) \otimes e^{\delta}\sigma_\infty(\beta_1) \Big)\\
&= (-1)^{b + c} \operatorname{eval}_{D'}( \beta_3, \beta_2, \beta_1 )
\end{align*}
where
\[
b = \widetilde{\beta_1}( \widetilde{\beta_2} + \widetilde{\beta_3} ) + \widetilde{\beta_2} + \widetilde{\beta_3} + 1\,, \qquad c = \widetilde{\beta_2}\widetilde{\beta_3} + \widetilde{\beta_3} + 1\,.
\]
This verifies the sign when $P_1 = 0, P_2 = 1, P_3 = 2$. By induction on the height of tree, it is easy to check that in general there is a contribution to the sign of a $\widetilde{\beta_i}\widetilde{\beta_j}$ at the vertex where the path from the $i$th and $j$th leaves to the root meet for the first time, and a $\widetilde{\beta_i}$ every time the path from the $i$th leaf enters a trivalent vertex on the right branch (of the original tree $T$), as claimed.
\end{proof}
\subsection{Transfer to $R/I \otimes k\llbracket \bold{t} \rrbracket$}\label{section:transfer_to_ri}
Recall that given a choice of section $\sigma: R/I \longrightarrow R$, which we have fixed above in Setup \ref{setup:overall}, there is by Lemma \ref{prop_algtube} an associated $k\llbracket \bold{t} \rrbracket$-linear isomorphism
\[
\sigmastar: R/I \otimes k \llbracket \bold{t} \rrbracket \longrightarrow \widehat{R}\,.
\]
From this we obtain \eqref{eq:transfer_iso_intro} which is used to transfer operators on $\operatorname{Hom}_R(X,Y) \otimes_R \widehat{R}$ (such as the differential or the homotopies $\lambda$) to operators on $R/I \otimes \operatorname{Hom}_k(\widetilde{X}, \widetilde{Y}) \otimes k\llbracket \bold{t} \rrbracket$. Since this introduces various complexities we should first justify why such transfers are necessary: that is, why do we prefer the left hand side of \eqref{eq:transfer_iso_intro} to the right hand side?
Recall that the higher products $\rho_k$ on $\mathcal{H}_{\mathcal{B}}$ are defined in terms of operators on the larger space $\mathcal{H}_{\mathcal{A}_\theta}$. If we are to reason about these higher products using Feynman diagrams, then to the extent that it is possible, the operators involved should be written as polynomials in creation and annihilation operators for either bosonic or fermionic Fock spaces (that is, in terms of multiplication by or the derivative with respect to ordinary polynomial variables $t$ or odd Grassmann variables $\theta$). It is not obvious \emph{a priori} how to do this: recall that in order to ensure that the connection $\nabla$ existed we had to pass from $R = k[x_1,\ldots,x_n]$ to the $I$-adic completion $\widehat{R}$, which in general is not a power series ring. For example, it is not clear how to express the operation of multiplication by $r \in R$, which we denote by $r^{\#}$, in terms of creation and annihilation operators on $\operatorname{Hom}_R(X,Y) \otimes_R \widehat{R}$.
The purpose of this section is then to explain how the isomorphism $\sigma_{\bold{t}}$ is the canonical means by which to express $r^{\#}$ in terms of creation operators for ``bosonic'' degrees of freedom, here represented by polynomials in the $t_i$.
\\
In what follows we fix a chosen $k$-basis of $R/I$, which we denote
\[
R/I = k z_1 \oplus \cdots \oplus k z_\mu\,.
\]
When $k$ is a field there is a natural monomial basis for $R/I$ associated to any choice of a monomial ordering on $k[x_1,\ldots,x_n]$ and Gr\"obner basis for $I$, see Remark \ref{remark:grobner}. Since $\sigmastar$ is not, in general, an algebra isomorphism (see Lemma \ref{prop_algtube}) there is information in the transfer of the multiplicative structure on $\widehat{R}$ to an operator on $R/I \otimes k\llbracket \bold{t} \rrbracket$, and we record this information in the following tensor:
\begin{definition}\label{defn_gamma} Let $\Gamma$ denote the $k$-linear map
\[
\xymatrix@C+2pc{
R/I \otimes R/I \ar[r]^-{ \sigma \otimes \sigma } & \widehat{R} \otimes \widehat{R} \ar[r]^-{m} & \widehat{R} \ar[r]^-{(\sigmastar)^{-1}} & R/I \otimes k\llbracket \bold{t} \rrbracket
}
\]
where $m$ denotes the usual multiplication on $\widehat{R}$. We define $\Gamma$ as a tensor via the formula
\[
\sigma(z_i)\sigma(z_j) = \sum_{k=1}^\mu \sum_{\delta \in \mathbb{N}^n} \Gamma^{ij}_{k \delta} \sigma(z_k) t^\delta\,.
\]
\end{definition}
\begin{definition}\label{defn:rsharp} Given $r \in R$ we write $r_{(i,\delta)}$ for the unique collection of coefficients in $k$ with the property that in $\widehat{R}$ there is an equality
\[
r = \sum_{i = 1}^\mu \sum_{\delta \in \mathbb{N}^n} r_{(i,\delta)} \,\sigma(z_i) t^{\delta}\,.
\]
Given $r \in R$ we denote by $r^{\#}$ the $k\llbracket \bold{t} \rrbracket$-linear operator
\[
\xymatrix@C+2pc{
R/I \otimes k\llbracket \bold{t} \rrbracket \ar[r]^-{\sigmastar} & \widehat{R} \ar[r]^{r} & \widehat{R} \ar[r]^-{(\sigmastar)^{-1}} & R/I \otimes k\llbracket \bold{t} \rrbracket
}
\]
where $r: \widehat{R} \longrightarrow \widehat{R}$ denotes multiplication by $r$.
\end{definition}
\begin{remark}\label{remark:compute_rdelta} For the overall construction of the idempotent finite model to be \emph{constructive} in the sense elaborated above, it is crucial that we have an algorithm for computing these coefficients $r_{(i, \delta)}$. In the notation of Section \ref{section:formaltub}, $r_{(i, \delta)}$ is the coefficient of $z_i$ in the vector $r_\delta \in R/I$, so it suffices to understand how to compute the $r_\delta$.
As a trivial example, if $\bold{t} = (x_1,\ldots,x_n)$ then $R/I = k$ so $\mu = 1$ and $r_{(1,\delta)}$ is just the coefficient of the monomial $t^{\delta} = x^{\delta}$ in the polynomial $r$. In general, when $k$ is a field there is an algorithm for computing $r_\delta$, see Remark \ref{remark:grobner}.
\end{remark}
\begin{lemma}\label{lemma:rsharp_explicit} The operator $r^{\#}$ is given in terms of the tensor $\Gamma$ by the formula
\begin{equation}
r^{\#}(z_i) = \sum_{l=1}^\mu \sum_{\delta \in \mathbb{N}^n} \Big[ \sum_{\alpha + \beta = \delta } \sum_{k=1}^\mu r_{(k,\alpha)} \Gamma^{ki}_{l\beta} \Big] z_l \otimes t^\delta\,.
\end{equation}
\end{lemma}
\begin{proof}
We have
\begin{align*}
\sigmastar r^{\#}(z_i \otimes 1) &= r \sigma(z_i)\\
&= \sum_{k, \alpha} r_{(k,\alpha)} [ \sigma(z_k) \sigma(z_i) ] t^{\alpha}\\
&= \sum_{k,\alpha,l,\beta} r_{(k,\alpha)} \Gamma^{ki}_{l \beta} \sigma(z_l) t^{\alpha + \beta}\\
&= \sum_\delta \sum_{k,l} \sum_{\alpha + \beta = \delta} r_{(k,\alpha)} \Gamma^{ki}_{l \beta} \sigma(z_l)t^\delta
\end{align*}
as claimed.
\end{proof}
\subsection{The operator $\zeta$}\label{section:propagator}
One of the most complex aspects of calculating the $A_\infty$-products described by Theorem \ref{theorem:main_ainfty_products} are the scalar factors contributed by the operator $\zeta$ which is the inverse of the grading operator for the virtual degree. In this section we provide a closer analysis of these factors.
While the virtual degree of Definition \ref{definition:zeta} is not a genuine $\mathds{Z}$-grading because $\mathcal{H}_{\mathcal{A}'_\theta}$ involves power series, for any given tree our calculations of higher $A_\infty$-product on $\mathcal{B}$ only involve polynomials in the $t_i$, so there is no harm in thinking about the virtual degree as a $\mathds{Z}$-grading and $\zeta$ as its inverse. Observe that that the critical Atiyah class $\vAt_{\mathcal{A}} = [ d_{\mathcal{A}}, \nabla ]$ is not homogeneous with respect to this grading, because while $\nabla$ is homogeneous of degree zero with respect to the virtual degree (since $\theta_i$ has virtual degree $+1$ and $\partial_{t_i}$ has virtual degree $-1$) the operator $d_{\mathcal{A}}$ involves multiplications by polynomials $r$ which need not have a consistent degree (viewed as operators $r^{\#}$ on $R/I \otimes k\llbracket \bold{t} \rrbracket$ as in the previous section). To analyse this operator on $\mathcal{H}_{\mathcal{A}'_\theta}$ we write
\[
d_{\mathcal{A}} = \sum_{\delta \in \mathbb{N}^n} d_{\mathcal{A}}^{\,(\delta)} t^\delta
\]
for some $k$-linear odd operators $d_{\mathcal{A}}^{\,(\delta)}$ on $\bigoplus_{X,Y} R/I \otimes \operatorname{Hom}_k(\widetilde{X},\widetilde{Y})$. Then
\begin{align*}
(\zeta \vAt_{\mathcal{A}})^m &= \sum_{\delta_1,\ldots,\delta_m} \zeta [ d^{\,(\delta_1)}_{\mathcal{A}} t^{\delta_1}, \nabla ] \cdots \zeta [ d^{\,(\delta_m)}_{\mathcal{A}} t^{\delta_m}, \nabla ]\\
&= \sum_{i_1,\ldots,i_m} \sum_{\delta_1,\ldots,\delta_m} \zeta \Big\{ \theta_{i_1} \partial_{t_{i_1}}(t^{\delta_1}) d_{\mathcal{A}}^{\,(\delta_1)} \Big\} \cdots \zeta \Big\{ \theta_{i_m} \partial_{t_{i_m}}(t^{\delta_m}) d_{\mathcal{A}}^{\,(\delta_m)} \Big\}\,.
\end{align*}
Evaluated on a tensor $\alpha$ of virtual degree $a = \Vert \alpha \Vert$ this gives
\begin{align*}
\sum_{i_1, \ldots, i_m} \sum_{\delta_1,\ldots,\delta_m} Z^{\,\rightarrow}(|\alpha|,|\delta_1|,\ldots,|\delta_m|) \Big\{ \theta_{i_1} \partial_{t_{i_1}}(t^{\delta_1}) d_{\mathcal{A}}^{\,(\delta_1)} \Big\} \cdots \Big\{ \theta_{i_m} \partial_{t_{i_m}}(t^{\delta_m}) d_{\mathcal{A}}^{\,(\delta_m)} \Big\}(\alpha)
\end{align*}
where the scalar factor is computed by
\begin{definition}\label{defn:Z_factors} Given integers $a > 0$ and a sequence $d_1, \ldots, d_m > 0$ we define
\begin{equation}\label{eq:defn_Z_factor_oriented}
Z^{\,\rightarrow}(a,d_1,\ldots,d_m) = \frac{1}{a + d_1} \frac{1}{a + d_1 + d_{2}} \cdots \frac{1}{a + d_1 + \cdots + d_m}
\end{equation}
and a symmetrised version
\begin{equation}
Z(a,d_1,\ldots,d_m) = \sum_{\sigma \in S_m} Z^{\,\rightarrow}(a,d_{\sigma 1},\ldots,d_{\sigma m})\,
\end{equation}
\end{definition}
In general there is no more to say, and the generic factors contributed by $\zeta$ to Feynman diagrams have the form given in \eqref{eq:defn_Z_factor_oriented} above. However, there is a useful special case:
\begin{lemma}\label{lemma:technical_antic} Let $\mathcal{K} \subseteq \mathcal{H}_{\mathcal{A}'_\theta}$ be a subspace with the following properties
\begin{itemize}
\item[(a)] $\mathcal{K}$ is closed under $d_{\mathcal{A}}^{\,(\delta)}$ for every $\delta \neq \bold{0}$.
\item[(b)] As operators on $\mathcal{K}$, we have $\big[ d_{\mathcal{A}}^{\,(\delta)}, d_{\mathcal{A}}^{\, (\gamma)} \big] = 0$ for all $\delta, \gamma \neq \bold{0}$.
\end{itemize}
Then for any $\alpha \in \mathcal{K}$, $(\zeta \vAt_{\mathcal{A}})^m(\alpha)$ is equal to
\begin{equation}
\sum_{i_1 < \ldots < i_m} \sum_{\delta_1,\ldots,\delta_m} (-1)^{\binom{m}{2}} \theta_{i_1} \cdots \theta_{i_m} Z(|\alpha|,|\delta_{1}|,\ldots,|\delta_{m}|) \prod_{r=1}^m \Big\{ \partial_{t_{i_r}}(t^{\delta_{r}}) d_{\mathcal{A}}^{\,(\delta_{r})} \Big\} (\alpha)\,.
\end{equation}
\end{lemma}
\begin{proof}
Only sequences of distinct $\theta$'s contribute, so we find that $(\zeta \vAt_{\mathcal{A}})^m(\alpha)$ is equal to
\begin{align*}
&\sum_{\substack{i_1,\ldots,i_m \\ \delta_1,\ldots,\delta_m}} Z^{\,\rightarrow}(|\alpha|,|\delta_1|,\ldots,|\delta_m|) (-1)^{\binom{m}{2}} \theta_{i_1} \cdots \theta_{i_m} \prod_{r=1}^m \Big\{ \partial_{t_{i_r}}(t^{\delta_r}) d_{\mathcal{A}}^{\,(\delta_r)} \Big\} (\alpha)\\
&= \sum_{\substack{i_1 < \ldots < i_m \\ \delta_1,\ldots,\delta_m}} \sum_{\sigma \in S_m} Z^{\,\rightarrow}(|\alpha|,|\delta_1|,\ldots,|\delta_m|) (-1)^{\binom{m}{2}} \theta_{i_{\sigma 1}} \cdots \theta_{i_{\sigma m}} \prod_{r=1}^m \Big\{ \partial_{t_{i_{\sigma r}}}(t^{\delta_r}) d_{\mathcal{A}}^{\,(\delta_r)} \Big\} (\alpha)\\
&= \sum_{i_1 < \ldots < i_m} \theta_{\bold{i}} \sum_{\sigma \in S_m} (-1)^{\binom{m}{2} + |\sigma|} \sum_{\delta_1,\ldots,\delta_m} Z^{\,\rightarrow}(|\alpha|,|\delta_1|,\ldots,|\delta_m|) \prod_{r=1}^m \Big\{ \partial_{t_{i_{\sigma r}}}(t^{\delta_r}) d_{\mathcal{A}}^{\,(\delta_r)} \Big\} (\alpha)\,.
\end{align*}
Note that only sequences $\delta_1,\ldots,\delta_m \in \mathbb{N}^n$ with all $\delta_i \neq \bold{0}$ contribute to this sum, so by hypothesis all the operators $d_{\mathcal{A}}^{\,(\delta_r)}$ involved anti-commute
\begin{align*}
&= \sum_{i_1 < \ldots < i_m} \theta_{\bold{i}} \sum_{\sigma \in S_m} (-1)^{\binom{m}{2} + |\sigma|} \sum_{\delta_1,\ldots,\delta_m} Z^{\,\rightarrow}(|\alpha|,|\delta_{\sigma 1}|,\ldots,|\delta_{\sigma m}|) \prod_{r=1}^m \Big\{ \partial_{t_{i_{\sigma r}}}(t^{\delta_{\sigma r}}) d_{\mathcal{A}}^{\,(\delta_{\sigma r})} \Big\} (\alpha)\\
&= \sum_{i_1 < \ldots < i_m} \theta_{\bold{i}} \sum_{\sigma \in S_m} (-1)^{\binom{m}{2} + |\sigma|} \sum_{\delta_1,\ldots,\delta_m} Z^{\,\rightarrow}(|\alpha|,|\delta_{\sigma 1}|,\ldots,|\delta_{\sigma m}|) (-1)^{|\sigma|} \prod_{r=1}^m \Big\{ \partial_{t_{i_r}}(t^{\delta_{r}}) d_{\mathcal{A}}^{\,(\delta_{r})} \Big\} (\alpha)\\
&= \sum_{i_1 < \ldots < i_m} (-1)^{\binom{m}{2}} \theta_{\bold{i}} \sum_{\delta_1,\ldots,\delta_m} \sum_{\sigma \in S_m} Z^{\,\rightarrow}(|\alpha|,|\delta_{\sigma 1}|,\ldots,|\delta_{\sigma m}|) \prod_{r=1}^m \Big\{ \partial_{t_{i_r}}(t^{\delta_{r}}) d_{\mathcal{A}}^{\,(\delta_{r})} \Big\} (\alpha)\\
&= \sum_{i_1 < \ldots < i_m} (-1)^{\binom{m}{2}} \theta_{\bold{i}} \sum_{\delta_1,\ldots,\delta_m} Z(|\alpha|,|\delta_{1}|,\ldots,|\delta_{m}|) \prod_{r=1}^m \Big\{ \partial_{t_{i_r}}(t^{\delta_{r}}) d_{\mathcal{A}}^{\,(\delta_{r})} \Big\} (\alpha)
\end{align*}
as claimed.
\end{proof}
The lemma is sometimes useful in reducing the number of Feynman diagrams that one has to actually calculate, see Remark \ref{remark:symmetry_A_type}. While the hypotheses of Lemma \ref{lemma:technical_antic} are technical, in the typical cases they are easy to check:
\begin{example}\label{example:computing_min_kstab} Suppose that $X$ is a Koszul matrix factorisation as in \eqref{eq:defn_X_tilde} and that we use the isomorphism of Lemma \ref{lemma:iso_rho} to identify $\mathcal{H}_{\mathcal{A}'_\theta}(X,X)$ with
\[
\bigwedge \big( F_\theta \oplus F_\xi \oplus F_{\Bar{\xi}} \big) \otimes R/I \otimes k\llbracket \bold{t} \rrbracket
\]
on which space by Lemma \ref{lemma:transfer_rho} we have
\[
d_{\mathcal{A}}^{\,(\delta)} = \sum_{i=1}^r f_i^{\,(\delta)} \xi_i^* + \sum_{i=1}^r g_i^{\,(\delta)} \Bar{\xi}_i^*
\]
for some operators $f_i^{\,(\delta)},g_i^{\,(\delta)}$ on $R/I$ computed by Lemma \ref{lemma:rsharp_explicit}. Let $\mathcal{K}$ be the subspace
\[
\bigwedge \big( F_\theta \oplus F_{\Bar{\xi}} ) \otimes R/I \otimes k\llbracket \bold{t} \rrbracket
\]
with no $\xi_i$'s, then as an operator on $\mathcal{K}$
\[
d_{\mathcal{A}}^{\,(\delta)}\Big|_{\mathcal{K}} = \sum_{i=1}^r g_i^{\,(\delta)} \Bar{\xi}_i^*\,.
\]
These operators will all pair-wise anticommute, provided that $[ g_i^{\,(\delta)}, g_j^{\,(\varepsilon)}] = 0$ as operators on $R/I$ for all $1 \le i,j \le r$ and $\delta, \epsilon \neq \bold{0}$. This is true trivially when $R/I = k$, which means that the previous Lemma applies to calculating $(\zeta \vAt_{\mathcal{A}})^m$ everywhere in Feynman diagrams computing the minimal model of $\mathcal{A}(k^{\operatorname{stab}},k^{\operatorname{stab}})$, see Section \ref{section:generator}.
\end{example}
\begin{remark} Scalar factors like $Z$ occur in the context of infrared divergences involving soft virtual particles (such as soft virtual photons in quantum electrodynamics) see for instance \cite[Ch. 13]{weinberg} and \cite[p.204]{ps}. The operator $\zeta$ is part of a propagator \cite[\S 4.1.3]{lazaroiu_sft} which like $\frac{1}{p^2 - m^2 + i \varepsilon}$ in QFT has the effect generically of suppressing contributions from terms far off the mass-shell (the further off the mass-shell you are, the larger $p^2-m^2$ is). In our case, Feynman diagrams with large numbers of internal virtual particle lines ($\theta$ and $t$ lines) are suppressed with respect to the usual metric on $\mathbb{Q}$.
\end{remark}
The most commonly treated case of the soft amplitudes in textbooks is the case of an on-shell external electron line, which corresponds to taking $a = 0$. In this case there is a simple formula for $Z$, which is easily proved by induction:
\begin{lemma} Given a sequence $d_1,\ldots,d_m > 0$ of integers,
\begin{equation}
Z(0,d_1,\ldots,d_m) = \sum_{\sigma \in S_m} \frac{1}{d_{\sigma 1}} \frac{1}{d_{\sigma 1} + d_{\sigma 2}} \cdots \frac{1}{d_{\sigma 1} + \cdots + d_{\sigma m}} = \frac{1}{d_1 \cdots d_m}\,.
\end{equation}
\end{lemma}
We do not know any simple formula for $Z$ in general.
\section{Feynman diagrams}\label{section:feynman_diagram}
In quantum field theory, the calculus of Feynman diagrams provides algorithms for computing scattering amplitudes (with some caveats) and reasoning about physical processes. The role of Feynman diagrams in the theory of $A_\infty$-categories is similar: they provide an algorithmic method for computing the higher $A_\infty$-products on $\mathcal{B}$ as well as a set of tools for reasoning about these products. The connection between $A_\infty$-structures, homological perturbation and Feynman diagrams is well-known; see \cite{lazaroiu_sft,lazaroiu_roiban}, \cite[p.42]{lazaroiu} and \cite[\S 2.5]{gwilliam}. However, in this context nontrivial examples with fully explicit Feynman rules accounting for all signs and symmetry factors and \emph{actual diagrams} like Figure \ref{fig:feynman_1} below, are rare. For background in the physics of Feynman diagrams we recommend \cite[Ch. 6]{weinberg}, \cite[\S 4.4]{ps} and for a more mathematical treatment \cite{qftstring}.
The presentation of $A_\infty$-products in terms of Feynman diagrams is most useful when the objects of $\mathcal{A}$ are matrix factorisations of Koszul type, and so we will focus on this case below. Throughout we adopt the hypotheses of Setup \ref{setup:overall}, and we write
\begin{equation}\label{eq:hhalt_koszul}
\mathcal{H} = \mathcal{H}_{\mathcal{A}'_\theta}\,, \qquad \mathcal{H}(X,Y) = \bigwedge F_\theta \otimes R/I \otimes \operatorname{Hom}_k(\widetilde{X},\widetilde{Y}) \otimes k\llbracket \bold{t} \rrbracket\,.
\end{equation}
Our aim is give a diagrammatic interpretation of the operators $\rho_T$, as given for example in \eqref{eq:explicit_tree_operator}. Implicitly $\rho_T$ consists of many summands, obtained by expanding the $e^{\delta}, e^{-\delta}$ and $\sigma_\infty, \phi_\infty$ operators. Among the summands generated from \eqref{eq:explicit_tree_operator} is for example
\begin{equation}\label{eq:contrib_feynman}
\pi \delta^2 r_2\Big( (\zeta \vAt_{\mathcal{A}})^2 \sigma \otimes \delta^3 (\zeta \vAt_{\mathcal{A}})^3 \zeta \nabla \delta r_2\Big( \delta^5 (\zeta \vAt_{\mathcal{A}})^6 \sigma \otimes (\zeta \vAt_{\mathcal{A}}) \sigma \Big) \Big)\,.
\end{equation}
The aim is to
\begin{itemize}
\item represent the space $\mathcal{H}$ on which these operators act as a tensor product of exterior algebras and (completed) symmetric algebras, and
\item represent the operators as polynomials in creation and annihilation operators (that is, as multiplication with, or the derivative with respect to, even or odd generators of the relevant algebras).
\end{itemize}
Once this is done we can represent the operator \eqref{eq:contrib_feynman} as the contraction of a set of polynomials in creation and annihilation operators, with the pattern of contractions dictated by the structure of the original tree. The process of \emph{reducing this contraction to normal form} (with all annihilation operators on the right, and creation operators on the left) involves commuting creation and annihilation operators past one another, and their commutation relations generate many new terms. Feynman diagrams provide a calculus for organising these terms, and thus computing the normal form. There are three classes of operators making up \eqref{eq:contrib_feynman} which need to be given a diagrammatic interpretation:
\begin{itemize}
\item In Section \ref{section:koszul_mf} we represent $\mathcal{H}$ as suggested above.
\item In Section \ref{section:fenyman_diagram_1} we treat $\vAt_{\mathcal{A}}, \delta$.
\item In Section \ref{section:fenyman_diagram_2} we treat $\zeta$.
\item In Section \ref{section:feynman_diagram_3} we treat $\mu_2$.
\end{itemize}
Finally, in Section \ref{section:feynman_diagram_4} we give the Feynman rules and explain the whole process of computing with Feynman diagrams in an example.
\subsection{Koszul matrix factorisations}\label{section:koszul_mf}
Our Feynman diagrams will have vertices representing certain operators on $\mathcal{H}(X,Y)$ for a pair of matrix factorisations $X,Y$ of $W$ of Koszul type. This means that we suppose given collections of polynomials $\{ f_i, g_i \}_{i=1}^r$ and $\{ u_j, v_j \}_{j=1}^s$ in $R$ satisfying
\[
W = \sum_{i=1}^r f_i g_i = \sum_{j=1}^s u_j v_j\,.
\]
To these polynomials we may associate matrix factorisations $X,Y$ defined as follows: we take odd generators $\xi_1,\ldots,\xi_r,\eta_1,\ldots,\eta_s$, set $F_\xi = \bigoplus_{i=1}^r k \xi_i, F_\eta = \bigoplus_{j=1}^s k \eta_j$ and
\begin{align*}
\widetilde{X} &= \bigwedge F_\xi = \bigwedge( k \xi_1 \oplus \cdots \oplus k \xi_r )\\
\widetilde{Y} &= \bigwedge F_\eta = \bigwedge( k \eta_1 \oplus \cdots \oplus k \eta_s )
\end{align*}
and then define
\begin{align}
X &= \big( \widetilde{X} \otimes R, \sum_{i=1}^r f_i \xi_i^* + \sum_{i=1}^r g_i \xi_i \big)\,,\label{eq:defn_X_tilde}\\
Y &= \big( \widetilde{Y} \otimes R, \sum_{j=1}^s u_j \eta_j^* + \sum_{j=1}^s v_j \eta_j \big)\,.\label{eq:defn_Y_tilde}
\end{align}
We ultimately want to give a graphical representation of operators on the $k$-module \eqref{eq:hhalt_koszul}, for which relevant operators are polynomials in creation and annihilation operators. It is therefore convenient to rewrite $\operatorname{Hom}_k( \widetilde{X}, \widetilde{Y} )$ in the form of an exterior algebra.
\begin{lemma}\label{lemma:iso_nu} There is an isomorphism of $\mathds{Z}_2$-graded $k$-modules
\[
\nu: \operatorname{Hom}_k( \widetilde{X}, \widetilde{Y} ) \longrightarrow \bigwedge F_\eta \otimes \bigwedge F_\xi^*
\]
defined by
\[
\nu( \phi ) = \sum_{p \ge 0} \sum_{i_1 < \cdots < i_p} (-1)^{\binom{p}{2}} \phi( \xi_{i_1} \cdots \xi_{i_p} ) \xi_{i_1}^* \cdots \xi_{i_p}^*\,.
\]
\end{lemma}
The contraction operator which removes $\xi^*_i$ from a wedge product in $\bigwedge F_\xi^*$ can be written $(\xi_i^*)^* \lrcorner (-)$ or $(\xi_i^*)^*$ for short, but this is awkward. Even worse, the operation of wedge product $\xi_i^* \wedge (-)$ in this exterior algebra cannot be safely abbreviated to $\xi_i^*$ because some of our formulas will involve precisely the same notation to denote the contraction operator on $\bigwedge F_\xi$. So we introduce the following notational convention:
\begin{definition}\label{defn:bar_convention} We write $\Bar{\xi}_i$ for $\xi_i^*$ and $F_{\Bar{\xi}} = F_\xi^*$ so that as operators on $\bigwedge F_\xi^*$ we have
\[
\Bar{\xi}_i = \xi_i^* \wedge (-)\,, \qquad \Bar{\xi}_i^* = (\xi_i^*)^* \lrcorner (-)\,.
\]
The same conventions apply to $\eta$ and any other odd generators.
\end{definition}
Using $\nu$ we may identify $\mathcal{H}(X,Y)$ as a $\mathds{Z}_2$-graded $k$-module with
\begin{equation}\label{eq:presentationHHalt}
\bigwedge \big( F_\theta \oplus F_\eta \oplus F_{\Bar{\xi}} \big) \otimes R/I \otimes k\llbracket \bold{t} \rrbracket
\end{equation}
which we view as the tensor product of a $k$-module of coefficients $R/I$ with the (completed) bosonic Fock space $k\llbracket \bold{t} \rrbracket$ with creation and annihilation operators $t_i, \partial_{t_i}$ and fermionic Fock spaces $\bigwedge F_{\Bar{\xi}}, \bigwedge F_\eta, \bigwedge F_\theta$ with creation operators $\Bar{\xi}_i, \eta_j, \theta_k$ and annihilation operators $\Bar{\xi}_i^*, \eta_j^*, \theta_k^*$ respectively.
\subsection{Diagrams for $\vAt_{\mathcal{A}}, \delta$}\label{section:fenyman_diagram_1}
In the notation of the previous section we now elaborate on the explicit formulas for $\vAt_{\mathcal{A}},\delta$ in terms of creation and annihilation operators.
\begin{lemma}\label{lemma:transfer} The operator $(*)$ which makes the diagram
\[
\xymatrix@C+3pc@R+2pc{
\operatorname{Hom}_k( \widetilde{X}, \widetilde{Y} ) \otimes R \ar[d]_-{d_{\mathcal{A}}} \ar[r]_-{\cong}^-{ \nu \otimes 1} & \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\xi}} \otimes R \ar[d]^-{(*)}\\
\operatorname{Hom}_k( \widetilde{X}, \widetilde{Y} ) \otimes R \ar[r]^-{\cong}_-{ \nu \otimes 1 } & \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\xi}} \otimes R
}
\]
is given by the formula
\[
\sum_{j=1}^s u_j \eta_j^* + \sum_{j=1}^s v_j \eta_j - \sum_{i=1}^r f_i \Bar{\xi}_i + \sum_{i=1}^r g_i \Bar{\xi}_i^*\,.
\]
\end{lemma}
\begin{proof}
By direct calculation.
\end{proof}
With this notation, the operator $\vAt_{\mathcal{A}}$ on $\mathcal{H}(X,Y)$ corresponds to an operator on \eqref{eq:presentationHHalt} given by the following formula (we use the superscript $\nu$ to record that this isomorphism is used to transfer $\vAt_{\mathcal{A}}$)
\begin{align}
\vAt^{\nu}_{\mathcal{A}} &= [ d_{\mathcal{A}}, \nabla ] = \sum_{k=1}^n \theta_k [ \partial_{t_k}, d_{\mathcal{A}} ]\nonumber\\
&= \sum_{k=1}^n \sum_{j=1}^s \theta_k \partial_{t_k} (u_j) \eta_j^* + \sum_{k=1}^n \sum_{j=1}^s \theta_k \partial_{t_k}( v_j ) \eta_j\label{eq:vat_formula}\\
& \qquad - \sum_{k=1}^n \sum_{i=1}^r \theta_k \partial_{t_k}(f_i) \Bar{\xi}_i + \sum_{k=1}^n \sum_{i=1}^r \theta_k \partial_{t_k}(g_i) \Bar{\xi}_i^*\nonumber
\end{align}
where for an element $r \in R$ what we mean by $\partial_{t_k}(r)$ is the $k$-linear operator on $R/I \otimes k\llbracket \bold{t} \rrbracket$ which is the commutator of $\partial_{t_k}$ with the operator $r^{\#}$ of Definition \ref{defn:rsharp}. By Lemma \ref{lemma:rsharp_explicit} we can write this explicitly in terms of the multiplication tensor $\Gamma$ of Definition \ref{defn_gamma} as
\begin{equation}
\partial_{t_k}(r)(z_h \otimes t^{\tau}) = \sum_{l=1}^\mu \sum_{\delta \in \mathbb{N}^n} \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu r_{(m,\alpha)} \Gamma^{mh}_{l\beta} \Big] z_l \otimes \partial_{t_k}(t^\delta) t^{\tau}
\end{equation}
where the coefficients $r_{(m,\alpha)} \in k$ are as in Definition \ref{defn:rsharp}, $\{ z_h \}_{h=1}^\mu$ is our chosen $k$-basis of $R/I$ and $\Gamma$ is the multiplication tensor. Reading $\partial_{t_k}(t^\delta)$ as the operator of left multiplication by this monomial, we may write
\begin{equation}\label{eq:partial_derivative_op}
\partial_{t_k}(r) = \sum_{h=1}^\mu \sum_{l=1}^\mu \sum_{\delta \in \mathbb{N}^n} \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu r_{(m,\alpha)} \Gamma^{mh}_{l\beta} \Big] z_l \partial_{t_k}(t^\delta) z_h^*\,.
\end{equation}
Next we describe the operators $\delta = \sum_{k=1}^n \lambda^\bullet_k \theta_k^*$ and for this we need to choose a particular homotopy $\lambda^Y_k: Y \longrightarrow Y$ with $[ \lambda^Y_k, d_Y ] = t_k \cdot 1_Y$. Recall that $t_1,\ldots,t_n$ is a quasi-regular sequence satisfying some hypotheses satisfied in particular by the partial derivatives of the potential $W$, but other choices are possible. We assume here that our homotopies $\lambda^Y_k$ are chosen to be of the form
\begin{equation}\label{eq:formula_lambdak_koszul}
\lambda^Y_k = \sum_{j=1}^s F_{kj} \eta_j^* + \sum_{j=1}^s G_{kj} \eta_j
\end{equation}
for polynomials $\{ F_{kj}, G_{kj} \}_{j=1}^s$ in $R$ which satisfy the equations
\[
\sum_{j=1}^s( F_{kj} v_j + G_{kj} u_j ) = t_k \qquad 1 \le k \le n\,.
\]
\begin{remark}\label{remark:default_homotopies}
If $\bold{t} = (\partial_{x_1} W, \ldots, \partial_{x_n} W)$ then $F_{kj} = \partial_{x_k}(u_j), G_{kj} = \partial_{x_k}(v_j)$ satisfy these requirements and hence define a valid sequence of homotopies $\lambda^Y_1,\ldots,\lambda^Y_n$.
\end{remark}
The operator $\lambda_k^\bullet$ of Definition \ref{defn:important_operators} acts by post-composition with $\lambda^Y_k$, and so the corresponding operator on \eqref{eq:presentationHHalt} under $\nu$ is by the same calculation as Lemma \ref{lemma:transfer} given by the formula \eqref{eq:formula_lambdak_koszul}. In this notation (again using a superscript $\nu$ to indicate the transfer)
\begin{equation}\label{eq:delta_formula}
\delta^\nu = \sum_{k=1}^n \lambda^\bullet_k \theta_k^* = \sum_{k=1}^n \sum_{j=1}^s F_{kj} \eta_j^* \theta_k^* + \sum_{k=1}^n \sum_{j=1}^s G_{kj} \eta_j \theta_k^*\,.
\end{equation}
Combining \eqref{eq:vat_formula} and \eqref{eq:partial_derivative_op} yields:
\begin{lemma}
The critical Atiyah class $\vAt_{\mathcal{A}}$ may be presented using $\nu$ as an operator on \eqref{eq:presentationHHalt} given by the sum of the four terms given below, each of which is itself summed over the indices $1 \le h,l \le \mu, 1 \le k \le n$ and $\delta \in \mathbb{N}^n$:
\begin{align}
& \sum_{j=1}^s \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (u_j)_{(m,\alpha)} \Gamma^{m h}_{l \beta} \Big] \theta_k z_{l} \partial_{t_k}(t^\delta) z_h^* \eta_j^* \qquad (\textup{A}.1)^{\nu} \label{eq:a1vertex}\\
& \sum_{j=1}^s \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (v_j)_{(m,\alpha)} \Gamma^{mh}_{l\beta} \Big] \theta_k z_l \partial_{t_k}(t^\delta) \eta_j z_h^* \qquad (\textup{A}.2)^{\nu} \label{eq:a2vertex}\\
-&\sum_{i=1}^r \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (f_i)_{(m,\alpha)} \Gamma^{mh}_{l\beta} \Big] \theta_k z_l \partial_{t_k}(t^\delta) \Bar{\xi}_i z_h^* \qquad (\textup{A}.3)^{\nu} \label{eq:a3vertex}\\
&\sum_{i=1}^r \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (g_i)_{(m,\alpha)} \Gamma^{mh}_{l\beta} \Big] \theta_k z_l \partial_{t_k}(t^\delta) z_h^* \Bar{\xi}_i^* \qquad (\textup{A}.4)^{\nu}\,. \label{eq:a4vertex}
\end{align}
Schematically, we can write
\begin{equation}
\vAt^\nu_{\mathcal{A}} = \sum_{h,l=1}^\mu \sum_{k=1}^n\sum_{\delta \in \mathbb{N}^n} \Big[ (\textup{A}.1)^{\nu} + (\textup{A}.2)^{\nu} + (\textup{A}.3)^{\nu} + (\textup{A}.4)^{\nu} \Big]\,.
\end{equation}
\end{lemma}
Combining \eqref{eq:delta_formula} and \eqref{eq:partial_derivative_op} yields:
\begin{lemma}
The operator $\delta$ may be presented using $\nu$ as an operator on \eqref{eq:presentationHHalt} given by the sum of the two terms given below, each of which is itself summed over the indices $1 \le h,l \le \mu, 1 \le k \le n$ and $\delta \in \mathbb{N}^n$:
\begin{align}
& \sum_{j=1}^s \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (F_{kj})_{(m,\alpha)} \Gamma^{m h}_{l \beta} \Big] z_{l} t^\delta \eta_j^* z_h^* \theta_k^* \qquad (\textup{C}.1)^{\nu}\label{eq:c1vertex}\\
& \sum_{j=1}^s \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (G_{kj})_{(m,\alpha)} \Gamma^{m h}_{l \beta} \Big] z_{l} t^\delta \eta_j z_h^* \theta_k^* \qquad (\textup{C}.2)^{\nu}\,.\label{eq:c2vertex}
\end{align}
Schematically, we can write
\begin{equation}
\delta^\nu = \sum_{h,l=1}^\mu \sum_{k=1}^n\sum_{\delta \in \mathbb{N}^n} \Big[ (\textup{C}.1)^{\nu} + (\textup{C}.2)^{\nu} \Big]\,.
\end{equation}
\end{lemma}
Each of these monomials in creation and annihilation operators is associated with its own type of \emph{interaction vertex} in our Feynman diagrams. Eventually these vertices will be drawn on the same trees used to define the $A_\infty$-products $\rho_k$ and they will be given a formal interpretation by the Feynman rules, but for the moment they are just pictures. In our description we tend to imagine time evolving from from top of the page (the input) to the bottom (the output). At an interaction vertex associated with a monomial, each annihilation operator becomes an \emph{incoming line} (entering the vertex from above) and each creation operator becomes an \emph{outgoing line} (leaving the vertex downward). Different line styles are used to distinguish creation and annihilation operators of different ``types''.
\begin{center}
\begin{tabular}{ >{\centering}m{8cm} >{\centering}m{6cm} }
\[
\xymatrix@C+2pc@R+3pc{
& \ar@{.}[d]^-{h} & \ar[dl]^-{\eta_j}\\
& \bullet \ar@{=}[dl]^-{\theta_k} \ar@{.}[d]^-{l} \ar@{~}[dr]^-{\partial_{t_k}(t^\delta)}\\
& &
}
\]
&
\textbf{(A.1)${}^\nu$}
\vspace{1cm}
\[\sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (u_j)_{(m,\alpha)} \Gamma^{m h}_{l \beta}\]
\vspace{0.5cm}
$\theta_k z_{l} \partial_{t_k}(t^\delta) z_h^* \eta_j^*$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{8cm} >{\centering}m{6cm} }
\[
\xymatrix@C+1pc@R+3pc{
& & \ar@{.}[d]^-{h}\\
& & \bullet \ar@{=}[dll]_-{\theta_k} \ar@{.}[dl]^-{l} \ar@{~}[dr]_-{\partial_{t_k}(t^\delta)} \ar[drr]^-{\eta_j}\\
& & & &
}
\]
&
\textbf{(A.2)${}^\nu$}
\vspace{1cm}
\[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (v_j)_{(m,\alpha)} \Gamma^{mh}_{l\beta}\]
\vspace{0.5cm}
$\theta_k z_l \partial_{t_k}(t^\delta) \eta_j z_h^*$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{8cm} >{\centering}m{6cm} }
\[
\xymatrix@C+1pc@R+3pc{
& & \ar@{.}[d]^-{h}\\
& & \bullet \ar@{=}[dll]_-{\theta_k} \ar@{.}[dl]^-{l} \ar@{~}[dr]_-{\partial_{t_k}(t^\delta)}\\
& & & & \ar[ull]_-{\xi_i}
}
\]
&
\textbf{(A.3)${}^\nu$}
\vspace{1cm}
\[- \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (f_i)_{(m,\alpha)} \Gamma^{mh}_{l\beta}\]
\vspace{0.5cm}
$\theta_k z_l \partial_{t_k}(t^\delta) \Bar{\xi}_i z_h^*$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{8cm} >{\centering}m{6cm} }
\[
\xymatrix@C+2pc@R+3pc{
& \ar@{.}[d]^-{h} &\\
& \bullet \ar[ur]_-{\xi_i} \ar@{=}[dl]^-{\theta_k} \ar@{.}[d]^-{l} \ar@{~}[dr]^-{\partial_{t_k}(t^\delta)}\\
& &
}
\]
&
\textbf{(A.4)${}^\nu$}
\vspace{1cm}
\[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (g_i)_{(m,\alpha)} \Gamma^{mh}_{l\beta} \]
\vspace{0.5cm}
$\theta_k z_l \partial_{t_k}(t^\delta) z_h^* \Bar{\xi}_i^*$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{8cm} >{\centering}m{6cm} }
\[
\xymatrix@R+3pc{
\ar@{~}[d]^-{t_k}\\
\bullet \ar@{=}[d]^-{\theta_k}\\
\;
}
\]
&
\textbf{(B)}
\vspace{1cm}
\[\theta_k \partial_{t_k}\]
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{8cm} >{\centering}m{6cm} }
\[
\xymatrix@C+2pc@R+3pc{
\ar[dr]_-{\eta_j} & \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta_k}\\
& \bullet \ar@{.}[d]^-{l} \ar@{~}[dl]^-{t^\delta}\\
& &
}
\]
&
\textbf{(C.1)${}^\nu$}
\vspace{1cm}
\[\sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (F_{kj})_{(m,\alpha)} \Gamma^{m h}_{l \beta}\]
\vspace{0.5cm}
$z_{l} t^\delta \eta_j^* z_h^* \theta_k^*$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{8cm} >{\centering}m{6cm} }
\[
\xymatrix@C+2pc@R+3pc{
& \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta_k} \\
& \bullet \ar[dl]^-{\eta_j}\ar@{.}[d]^-{l} \ar@{~}[dr]^-{t^\delta}\\
& &
}
\]
&
\textbf{(C.2)${}^\nu$}
\vspace{1cm}
\[\sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (G_{kj})_{(m,\alpha)} \Gamma^{m h}_{l \beta}\]
\vspace{0.5cm}
$z_{l} t^\delta \eta_j z_h^* \theta_k^*$
\end{tabular}
\end{center}
We refer to the vertices \eqref{eq:a1vertex}-\eqref{eq:a4vertex} arising from the Atiyah class respectively as \emph{A-type vertices} (A.1, A.2, A.3, A.4) and the vertices \eqref{eq:c1vertex},\eqref{eq:c2vertex} arising from $\delta$ as \emph{C-type vertices} (C.1, C.2). There are also \emph{B-type vertices} given by the operator $\nabla = \sum_{k=1}^n \theta_k \partial_{t_k}$.
\begin{remark} Some remarks on these diagrams:
\begin{itemize}
\item We do not think of the vectors in $\mathcal{H}$ as states of literal particles (this word is generally reserved for state spaces transforming as representations of the inhomogeneous Lorentz group \cite{weinberg}) but the physics terminology is convenient and we sometimes refer to \emph{bosons} (the $t_i$) and \emph{fermions} (the $\xi_i, \eta_j, \theta_k$). Another useful concept is that of \emph{virtual particles} which is a term used to refer to lines propagating in the interior of Feynman diagrams. In our diagrams this role is played by the bosons $t_i$ and fermions $\theta_i$ (hence the virtual degree of Definition \ref{definition:zeta}) which represent the degrees of freedom that are being ``integrated out'' by the process of computing the $A_\infty$-products.
\item Following standard conventions bosons (commuting generators) are denoted by wiggly or dashed lines, and fermions (anticommuting generators) by solid lines \cite[\S 4.7]{ps} (perhaps doubled). For simplicity we distinguish the $\xi$ and $\eta$ lines only by their labels and we write $h$ for a line labelled $z_h$. Strictly speaking a squiggly line labelled $t^\delta$ should be interpreted as $\delta_i$ lines labelled $t_i$ for $1 \le i \le n$. We use the orientation on a fermion line to determine whether it should be read as a creation or annihilation operator for $\xi$ (downward) or for $\Bar{\xi}$ (upward).
\item Each vertex above actually represents a family indexed by possible choices of indices. If we wish to speak about a specific instance we use subscripts, for example $(\textup{B})_{k=2}$ is the interaction vertex with an incoming $t_2$ and outgoing $\theta_2$.\footnote{Depending on the matrix factorisations involved, some families of A or C-type interaction vertex may be \emph{infinite}. For example, if $(f_i)_{(m, \alpha)}$ is nonzero for infinitely many $\alpha$ there may be infinitely many $(A.3)^{\nu}$ vertices with nonzero coefficients. However only finitely many distinct types of interaction vertices can contribute for any particular tree $T$. If $T$ has $k$ leaves there are $k - 2$ internal edges and hence $k - 2$ occurrences of $\nabla$ in the associated operator. In terms of Feynman diagrams, that means there are precisely $k - 2$ B-type interactions. Since each $t_i$ that is generated in a Feynman diagram must eventually annihilate with a $\partial_{t_i}$ at a B-type interaction vertex, and each interaction vertex with indices $\alpha, \beta$ generates either $|\delta|$ or $|\delta| - 1$ copies of the $t_i$'s, only coefficients with $|\alpha|, |\beta| \le k - 2$ contribute. In short, larger trees can support more ``virtual bosons'' and more complex interactions.}
\end{itemize}
\end{remark}
\subsection{Alternative isomorphism $\rho$}\label{section:altrho}
We have used the isomorphism $\nu$ to present operators on $\mathcal{H}(X,Y)$ as creation and annihilation operators. In the case $X = Y$ there is an alternative isomorphism $\rho$ which leads to less interaction vertices.
\begin{lemma}\label{lemma:iso_rho} There is an isomorphism of $\mathds{Z}_2$-graded $k$-modules
\begin{gather*}
\rho: \bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \longrightarrow \End_k\big( \bigwedge F_\xi \big)\\
\rho\big( \xi_{i_1} \wedge \cdots \wedge \xi_{i_a} \otimes \Bar{\xi}_{j_1} \wedge \cdots \wedge \Bar{\xi}_{j_b} \big) = \xi_{i_1} \circ \cdots \circ \xi_{i_a} \circ \xi_{j_1}^* \circ \cdots \circ \xi_{j_b}^*
\end{gather*}
where on the right hand $\xi_i, \xi_j^*$ denote the usual operators $\xi_i = \xi_i \wedge (-)$ and $\xi_j^* = \xi_j^* \lrcorner (-)$.
\end{lemma}
\begin{remark}\label{remark_rhoisoalg} Let $C$ be the $\mathds{Z}_2$-graded algebra generated by odd $\xi_i, \Bar{\xi}_i$ for $1 \le i \le r$ subject to the relations $[ \xi_i, \xi_j ] = [ \Bar{\xi}_i, \Bar{\xi}_j ] = 0$ and $[ \xi_i, \Bar{\xi}_j ] = \delta_{ij}$. We denote multiplication in this Clifford algebra by $\bullet$. There is an isomorphism of $\mathds{Z}_2$-graded $k$-modules
\begin{gather*}
\bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \longrightarrow C\\
\xi_{i_1} \wedge \cdots \wedge \xi_{i_a} \otimes \Bar{\xi}_{j_1} \wedge \cdots \wedge \Bar{\xi}_{j_b} \mapsto \xi_{i_1} \bullet \cdots \bullet \xi_{i_a} \bullet \Bar{\xi}_{j_1} \bullet \cdots \bullet \Bar{\xi}_{j_b}\,.
\end{gather*}
Making this identification, $\rho$ is the linear map underlying an isomorphism of the Clifford algebra $C$ with the endomorphism algebra of $\bigwedge F_\xi$. In particular, the diagram
\begin{equation}
\xymatrix@C+2pc{
\big( \bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \big) \otimes \big( \bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \big) \ar[r]^-{\rho \otimes \rho}\ar[d]_-\bullet & \End_k\big( \bigwedge F_\xi\big) \otimes \End_k\big( \bigwedge F_\xi \big) \ar[d]^-{- \circ -}\\
\bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \ar[r]_-\rho & \End_k\big( \bigwedge F_\xi \big)
}
\end{equation}
commutes, where on the left $\bullet$ is the multiplication in the Clifford algebra.
\end{remark}
\begin{lemma}\label{lemma:commutators_on_rho} The diagrams
\[
\xymatrix@C+2pc{
\bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \ar[d]_-{\Bar{\xi}_i^*} \ar[r]^-{\rho} & \End_k\big( \bigwedge F_\xi \big) \ar[d]^-{[\xi_i, -]}\\
\bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \ar[r]_-{\rho} & \End_k\big( \bigwedge F_\xi \big)
}
\qquad
\xymatrix@C+2pc{
\bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \ar[d]_-{\xi_i^*} \ar[r]^-{\rho} & \End_k\big( \bigwedge F_\xi \big) \ar[d]^-{[ \xi_i^*, - ]}\\
\bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \ar[r]_-{\rho} & \End_k\big( \bigwedge F_\xi \big)
}
\]
commute, where graded commutators are in the algebra $\End_k(\bigwedge F_\xi)$.
\end{lemma}
\begin{proof}
By direct calculation.
\end{proof}
\begin{lemma}\label{lemma:transfer_rho} The operator $(*)$ which makes the diagram
\[
\xymatrix@C+3pc@R+2pc{
\operatorname{Hom}_k( \widetilde{X}, \widetilde{X} ) \otimes R \ar[d]_-{d_{\mathcal{A}}} \ar[r]_-{\cong}^-{ \rho \otimes 1} & \bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \otimes R \ar[d]^-{(*)}\\
\operatorname{Hom}_k( \widetilde{X}, \widetilde{X} ) \otimes R \ar[r]^-{\cong}_-{ \rho \otimes 1 } & \bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \otimes R
}
\]
is given by the formula
\[
\sum_{i=1}^r f_i \xi_i^* + \sum_{i=1}^r g_i \Bar{\xi}_i^*\,.
\]
\end{lemma}
\begin{proof}
By definition the differential is
\[
d_{\mathcal{A}} = \sum_{i=1}^r f_i [ \xi_i^*, - ] + \sum_{i=1}^r g_i [ \xi_i, - ]
\]
so this is immediate from Lemma \ref{lemma:commutators_on_rho}.
\end{proof}
Using $\rho$ we may identify $\mathcal{H}(X,X)$ as a $\mathds{Z}_2$-graded $k$-module with
\begin{equation}\label{eq:presentationHHalt_rho}
\xymatrix@C+2pc{
\bigwedge \big( F_\theta \oplus F_\xi \oplus F_{\Bar{\xi}} \big) \otimes R/I \otimes k\llbracket \bold{t} \rrbracket
}\,.
\end{equation}
With this notation, the operator $\vAt_{\mathcal{A}}$ on $\mathcal{H}(X,X)$ corresponds to
\begin{align*}
\vAt^\rho_{\mathcal{A}} &= [ d_{\mathcal{A}}, \nabla ] = \sum_{k=1}^n \theta_k [ \partial_{t_k}, d_{\mathcal{A}} ]\\
&= \sum_{k=1}^n \sum_{i=1}^r \theta_k \partial_{t_k}(f_i) \xi_i^* + \sum_{k=1}^n \sum_{i=1}^r \theta_k \partial_{t_k}(g_i) \Bar{\xi}_i^*\,.
\end{align*}
With the same conventions about $\delta = \sum_{k=1}^n \lambda^\bullet_k \theta_k^*$ as above, we have
\begin{equation}
\lambda^X_k = \sum_{i=1}^r F_{ki} \xi_i^* + \sum_{i=1}^r G_{ki} \xi_i\,.
\end{equation}
The operator $\lambda_k^\bullet$ acts on $\End_k( \bigwedge F_\xi ) \otimes R$ by post-composition with $\lambda^X_k$, that is to say, by left multiplication in the endomorphism ring, and since $\rho$ is an isomorphism of algebras the corresponding operator on \eqref{eq:presentationHHalt_rho} using $\rho$ is
\begin{equation}
\delta^\rho = \sum_{k=1}^n \lambda^\bullet_k \theta_k^* = \sum_{k=1}^n \sum_{i=1}^r F_{ki} [\Bar{\xi}_i \bullet (-)] \theta_k^* + \sum_{k=1}^n \sum_{i=1}^r G_{ki} [\xi_i \bullet (-)] \theta_k^*
\end{equation}
where $\bullet$ means multiplication in the Clifford algebra structure on $\bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}}$. It is easily checked that as operators on this tensor product, we have
\[
\xi_i \bullet (-) = \xi_i \otimes 1\,, \qquad \Bar{\xi}_i \bullet (-) = \xi_i^* \otimes 1 + 1 \otimes \Bar{\xi}_i
\]
with the usual convention that $\xi_i$ means $\xi_i \wedge (-)$ and $\xi_i^*$ means $\xi_i^* \lrcorner (-)$. So finally
\begin{equation}
\delta = \sum_{k=1}^n \sum_{i=1}^r F_{ki} \xi_i^* \theta_k^* + \sum_{k=1}^n \sum_{i=1}^r F_{ki} \Bar{\xi}_i \theta_k^* + \sum_{k=1}^n \sum_{i=1}^r G_{ki} \xi_i \theta_k^*\,.
\end{equation}
\begin{lemma}
The critical Atiyah class $\vAt_{\mathcal{A}}$ may be presented using $\rho$ as an operator on \eqref{eq:presentationHHalt_rho}, given by the sum of the four terms below, each of which is itself summed over the indices $1 \le h,l \le \mu, 1 \le k \le n$ and $\delta \in \mathbb{N}^n$:
\begin{align}
&\sum_{i=1}^r \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (f_i)_{(m,\alpha)} \Gamma^{mh}_{l\beta} \Big] \theta_k z_l \partial_{t_k}(t^\delta) \xi_i^* z_h^* \qquad (\textup{A}.1)^\rho \label{eq:a3vertex_rho}\\
&\sum_{i=1}^r \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (g_i)_{(m,\alpha)} \Gamma^{mh}_{l\beta} \Big] \theta_k z_l \partial_{t_k}(t^\delta) z_h^* \Bar{\xi}_i^* \qquad (\textup{A}.4)^\rho \label{eq:a4vertex_rho}
\end{align}
\end{lemma}
\begin{lemma}
The operator $\delta$ may be presented using $\rho$ as an operator on \eqref{eq:presentationHHalt_rho} given by the sum of the two terms given below, each of which is itself summed over the indices $1 \le h,l \le \mu, 1 \le k \le n$ and $\delta \in \mathbb{N}^n$:
\begin{align}
& \sum_{i=1}^r \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (F_{ki})_{(m,\alpha)} \Gamma^{m h}_{l \beta} \Big] z_{l} t^\delta \xi_i^* z_h^* \theta_k^* \qquad (\textup{C}.1)^\rho\label{eq:c1vertex_rho}\\
& \sum_{i=1}^r \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (G_{ki})_{(m,\alpha)} \Gamma^{m h}_{l \beta} \Big] z_{l} t^\delta \xi_i z_h^* \theta_k^* \qquad (\textup{C}.2)^\rho\label{eq:c2vertex_rho}\\
& \sum_{i=1}^r \Big[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (F_{ki})_{(m,\alpha)} \Gamma^{m h}_{l \beta} \Big] z_{l} t^\delta \Bar{\xi}_i z_h^* \theta_k^* \qquad (\textup{C}.3)^\rho\label{eq:c3vertex_rho}
\end{align}
\end{lemma}
The B-type interaction is as before. The other interactions are
\begin{center}
\begin{tabular}{ >{\centering}m{8cm} >{\centering}m{6cm} }
\[
\xymatrix@C+2pc@R+3pc{
& & \ar@{.}[d]^-{z_h} & \ar[dl]^-{\xi_i}\\
& & \bullet \ar@{=}[dl]_-{\theta_k} \ar@{.}[d]^-{z_l} \ar@{~}[dr]^-{\partial_{t_k}(t^\delta)}\\
& & & &
}
\]
&
\textbf{(A.1)${}^\rho$}
\vspace{1cm}
\[\sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (f_i)_{(m,\alpha)} \Gamma^{mh}_{l\beta}\]
\vspace{0.5cm}
$\theta_k z_l \partial_{t_k}(t^\delta) \xi_i^* z_h^*$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{8cm} >{\centering}m{6cm} }
\[
\xymatrix@C+2pc@R+3pc{
& \ar@{.}[d]^-{z_h} &\\
& \bullet \ar[ur]_-{\xi_i} \ar@{=}[dl]^-{\theta_k} \ar@{.}[d]^-{z_l} \ar@{~}[dr]^-{\partial_{t_k}(t^\delta)}\\
& &
}
\]
&
\textbf{(A.4)${}^\rho$}
\vspace{1cm}
\[ \sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (g_i)_{(m,\alpha)} \Gamma^{mh}_{l\beta} \]
\vspace{0.5cm}
$\theta_k z_l \partial_{t_k}(t^\delta) z_h^* \Bar{\xi}_i^*$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{8cm} >{\centering}m{6cm} }
\[
\xymatrix@C+2pc@R+3pc{
\ar[dr]_-{\xi_i} & \ar@{.}[d]^-{z_h} & \ar@{=}[dl]^-{\theta_k}\\
& \bullet \ar@{.}[d]^-{z_l} \ar@{~}[dl]^-{t^\delta}\\
& &
}
\]
&
\textbf{(C.1)${}^\rho$}
\vspace{1cm}
\[\sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (F_{ki})_{(m,\alpha)} \Gamma^{m h}_{l \beta}\]
\vspace{0.5cm}
$z_{l} t^\delta \xi_i^* z_h^* \theta_k^*$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{8cm} >{\centering}m{6cm} }
\[
\xymatrix@C+2pc@R+3pc{
& \ar@{.}[d]^-{z_h} & \ar@{=}[dl]^-{\theta_k} \\
& \bullet \ar[dl]^-{\xi_i}\ar@{.}[d]^-{z_l} \ar@{~}[dr]^-{t^\delta}\\
& &
}
\]
&
\textbf{(C.2)${}^\rho$}
\vspace{1cm}
\[\sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (G_{ki})_{(m,\alpha)} \Gamma^{m h}_{l \beta}\]
\vspace{0.5cm}
$z_{l} t^\delta \xi_i z_h^* \theta_k^*$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{8cm} >{\centering}m{6cm} }
\[
\xymatrix@C+2pc@R+3pc{
& \ar@{.}[d]^-{z_h} & \ar@{=}[dl]^-{\theta_k} \\
& \bullet \ar@{.}[d]^-{z_l} \ar@{~}[dr]^-{t^\delta}\\
\ar[ur]^-{\xi_i} & &
}
\]
&
\textbf{(C.3)${}^\rho$}
\vspace{1cm}
\[\sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (F_{ki})_{(m,\alpha)} \Gamma^{m h}_{l \beta}\]
\vspace{0.5cm}
$z_{l} t^\delta \Bar{\xi}_i z_h^* \theta_k^*$
\end{tabular}
\end{center}
\subsection{Diagrams for $\zeta$}\label{section:fenyman_diagram_2}
A power of $\zeta \vAt_{\mathcal{A}}$ will contribute a sequence of A-type interactions together with the scalar factors analysed in Section \ref{section:propagator}. To put these factors in the context of Feynman diagrams, consider a power $(\zeta \vAt)^m$ contributing $m$ A-type vertices, each of which emits a $\theta$, some monomial $t^\delta$, and acts in some way on the rest of $\mathcal{H}$ which we ignore. Such a process is depicted generically in Figure \ref{fig:tapesofU} and the factor contributed by this diagram is
\begin{equation}\label{eq:diagram_for_zeta_eq}
Z^{\,\rightarrow}(a,|\delta_1|,\ldots,|\delta_m|) = \frac{1}{a + |\delta_1|} \frac{1}{a + |\delta_1| + |\delta_3|} \cdots \frac{1}{a + |\delta_1| + \cdots + |\delta_m|}\,.
\end{equation}
\begin{figure}
\begin{tikzpicture}[scale=0.8,auto]
\draw[line width=2pt] (-8,0) -- (5,0);
\draw[snake=coil,segment aspect=0, segment amplitude=2pt, segment length=7pt] (-6.7,0) -- (-5.7,4);
\draw[double] (-7,0) -- (-6,4);
\node (1t) at (-5.5,4.5) {$\theta_1 \partial_{t_{i_1}}(t^{\delta_1})$};
\draw[snake=coil,segment aspect=0, segment amplitude=2pt, segment length=7pt] (-3.7,0) -- (-2.7,4);
\draw[double] (-4,0) -- (-3,4);
\node (2t) at (-2.5,4.5) {$\theta_2 \partial_{t_{i_2}}(t^{\delta_2})$};
\node (dots) at (0,3) {$\cdots$};
\draw[snake=coil,segment aspect=0, segment amplitude=2pt, segment length=7pt] (2.3,0) -- (3.3,4);
\draw[double] (2,0) -- (3,4);
\node (nt) at (3.7,4.5) {$\theta_m \partial_{t_{i_m}}(t^{\delta_m})$};
\node (1) at (-7.6,-0.5) {$a$};
\node (2) at (-5.5,-0.5) {$a + |\delta_1|$};
\node (m) at (4,-0.5) {$a + \sum_{i=1}^m |\delta_i|$};
\end{tikzpicture}
\centering
\caption{Depiction of a process contributing $\zeta$ factors. The labels indicate the virtual weight of the tensor obtained by cutting the diagram vertically at that position. Moving left to right in this diagram is to be read as moving down the tree.}\label{fig:tapesofU}
\end{figure}
There is also an occurrence of $\zeta$ as $\zeta \nabla$ in $\phi_\infty$, which contributes a scalar factor immediately after every B-type vertex. To be precise, $(\zeta \vAt)^m \zeta$ will contribute
\begin{equation}\label{eq:diagram_for_zeta_eq_b}
\frac{1}{a} \frac{1}{a + |\delta_1|} \frac{1}{a + |\delta_1| + |\delta_3|} \cdots \frac{1}{a + |\delta_1| + \cdots + |\delta_m|}\,.
\end{equation}
\subsection{Diagrams for $\mu_2$}\label{section:feynman_diagram_3}
Finally, we require a diagrammatic representation for the composition
\[
\mu_2: \mathcal{H}(Y, Z) \otimes \mathcal{H}(X, Y) \longrightarrow \mathcal{H}(X,Z)\,.
\]
In addition to the spaces $\widetilde{X} = \bigwedge F_\xi$ and $\widetilde{Y} = \bigwedge F_\eta$ underlying the matrix factorisations $X,Y$ we now introduce odd generators $\varepsilon_1,\ldots,\varepsilon_t$, $F_\varepsilon = \bigoplus_{i=1}^t k \varepsilon_i$ and set
\[
\widetilde{Z} = \bigwedge F_\varepsilon = \bigwedge\big( k \varepsilon_1 \oplus \cdots \oplus k \varepsilon_t \big)
\]
underlying a Koszul matrix factorisation $Z = \widetilde{X} \otimes R$.
\begin{lemma}\label{lemma_mu2presentation} For $\omega, \omega' \in \bigwedge F_\theta$ and $\alpha \in \operatorname{Hom}_k(\widetilde{Y}, \widetilde{Z}), \beta \in \operatorname{Hom}_k(\widetilde{X}, \widetilde{Y})$
\begin{gather*}
\mu_2\Big( [ \omega \otimes z_h \otimes \alpha ] \otimes [ \omega' \otimes z_l \otimes \beta ] \Big) \\
= (-1)^{|\alpha||\omega'|} \sum_{k=1}^\mu \sum_{\delta} \Gamma^{hl}_{k \delta} \cdot \omega \wedge \omega' \otimes z_k \otimes \alpha \circ \beta \otimes t^{\delta}\,.
\end{gather*}
\end{lemma}
First we explain how to represent $\alpha \circ \beta$ diagrammatically. The following lemmas are proven by straightforward direct calculations, which we omit.
\begin{lemma}\label{lemma:mixedr2_0} The diagram
\begin{equation}
\xymatrix@C+2pc@R+2pc{
\operatorname{Hom}_k( \widetilde{Y}, \widetilde{Z} ) \otimes \operatorname{Hom}_k( \widetilde{X}, \widetilde{Y} ) \ar[d]_-{\nu \otimes \nu} \ar[r]^-{- \circ -} & \operatorname{Hom}_k( \widetilde{X}, \widetilde{Z} ) \ar[d]^-\nu\\
\big( \bigwedge F_\varepsilon \otimes \bigwedge F_{\Bar{\eta}} \big) \otimes \big( \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\xi}} \big) \ar[d]_-{\exp( \sum_i \eta_i^* \Bar{\eta}_i^* )} & \bigwedge F_\varepsilon \otimes \bigwedge F_{\Bar{\xi}} \ar@{=}[d]\\
\big( \bigwedge F_\varepsilon \otimes \bigwedge F_{\Bar{\eta}} \big) \otimes \big( \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\xi}} \big) \ar[r]_-P & \bigwedge F_\varepsilon \otimes \bigwedge F_{\Bar{\xi}}
}
\end{equation}
commutes, where $P$ denotes the projection onto the subspace with no $\Bar{\eta}$'s or $\eta$'s.
\end{lemma}
\begin{lemma}\label{lemma:mixedr2_1} The diagram
\begin{equation}
\xymatrix@C+2pc@R+2pc{
\operatorname{Hom}_k( \widetilde{X}, \widetilde{Y} ) \otimes \operatorname{Hom}_k( \widetilde{X}, \widetilde{X} ) \ar[d]_-{\nu \otimes \rho^{-1}} \ar[r]^-{- \circ -} & \operatorname{Hom}_k( \widetilde{X}, \widetilde{Y} ) \ar[d]^-\nu\\
\big( \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\xi}} \big) \otimes \big( \bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \big) \ar[d]_-{\exp( \sum_i \xi_i^* \Bar{\xi}_i^* )} & \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\xi}} \\
\big( \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\xi}} \big) \otimes \big( \bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \big) \ar[r]_-P & \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\xi}} \otimes \bigwedge F_{\Bar{\xi}} \ar[u]_-{1 \otimes m}
}
\end{equation}
commutes, where $P$ denotes the projection onto the subspace with no $\xi$'s and $m$ is multiplication in the exterior algebra.
\end{lemma}
\begin{lemma}\label{lemma:mixedr2_2} The diagram
\begin{equation}
\xymatrix@C+2pc@R+2pc{
\operatorname{Hom}_k( \widetilde{Y}, \widetilde{Y} ) \otimes \operatorname{Hom}_k( \widetilde{X}, \widetilde{Y} ) \ar[d]_-{\rho^{-1} \otimes \nu} \ar[r]^-{- \circ -} & \operatorname{Hom}_k( \widetilde{X}, \widetilde{Y} ) \ar[d]^-\nu\\
\big( \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\eta}} \big) \otimes \big( \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\xi}} \big) \ar[d]_-{\exp( \sum_i \eta_i^* \Bar{\eta}_i^* )} & \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\xi}} \\
\big( \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\eta}} \big) \otimes \big( \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\xi}} \big) \ar[r]_-P & \bigwedge F_\eta \otimes \bigwedge F_\eta \otimes \bigwedge F_{\Bar{\xi}} \ar[u]_-{m \otimes 1}
}
\end{equation}
commutes, where $P$ projects onto the subspace with no $\Bar{\eta}$'s and $m$ is multiplication in the exterior algebra. The operator $\eta_i^*$ in the exponential acts on the third tensor factor.
\end{lemma}
These lemmas allow us to represent the $\alpha \circ \beta$ part of $\mu_2$ as a boundary condition $P$ together with new types of interaction vertices. From Lemma \ref{lemma:mixedr2_0} we obtain the (D.1)-type vertex, in which $\eta_i^*\Bar{\eta}_i^*$ couples an incoming $\eta_i$ in the right branch with an incoming $\Bar{\eta}_i$ (which we view as an $\eta_i$ travelling upward) in the left branch. From Lemma \ref{lemma:mixedr2_1} we obtain the (D.2)-type vertex, which has a similar description. To these interaction vertices we add the (D.3)-type vertex, which represents the $\Gamma^{hl}_{k\delta}z_k( z_h^* \otimes z_l^* )$ part of the $\mu_2$ operator in Lemma \ref{lemma_mu2presentation} (keeping in mind that the incoming $z_h$ and $z_l$ are on the left and right branch at an internal vertex of the tree, respectively):
\begin{center}
\begin{tabular}{ >{\centering}m{5cm} >{\centering}m{5cm} >{\centering}m{5cm} }
\textbf{(D.1) $+1$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& & \ar[dl]^-{\eta_i} \\
& \bullet \ar[ul]^-{\eta_i}\\
& & &
}
\]
&
\textbf{(D.2) $+1$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& & \ar[dl]^-{\xi_i} \\
& \bullet \ar[ul]^-{\xi_i}\\
& & &
}
\]
&
\textbf{(D.3) $\Gamma^{hl}_{k\delta}$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
\ar@{.}[dr]^-{h} & & \ar@{.}[dl]^-{l}\\
& \bullet \ar@{.}[d]_-{k} \ar@{~}[dr]^-{t^\delta}\\
& & &
}
\]
\end{tabular}
\end{center}
Note that in the ``mixed'' cases of Lemma \ref{lemma:mixedr2_1} and Lemma \ref{lemma:mixedr2_2} there are still multiplication operators $m$ in the final presentation. For example the $1 \otimes m$ in Lemma \ref{lemma:mixedr2_1} means that an upward travelling $\xi$ entering the vertex can either continue upwards into the left branch, or into the right branch. More precisely, since $\Bar{\xi}_i^*$ is a graded derivation $\Bar{\xi}_i^* m = m (\Bar{\xi}_i^* \otimes 1) + m( 1 \otimes \Bar{\xi}_i^* )$. A similar description applies to the $m \otimes 1$ in Lemma \ref{lemma:mixedr2_2}.
\subsection{The Feynman rules}\label{section:feynman_diagram_4}
We now integrate the previous sections into a method for reasoning about higher operations $\rho_k$ using Feynman diagrams. We fix matrix factorisations $X_0,\ldots,X_k \in \mathcal{A}$ which we assume to be Koszul with underlying graded $k$-modules $\widetilde{X}_i = \bigwedge F^{(i)}_\xi$. Once we choose for each pair $(i,i+1)$ and $(0,k)$ either $\nu$ or $\rho$ to present the mapping spaces $\mathcal{B}(X_i, X_{i+1})$ as a tensor product of exterior algebras, the higher operation $\rho_k$ is a $k$-linear map
\begin{equation}
\rho_k: \bigotimes_{i=0}^{k-1}\Big[ R/I \otimes \bigwedge F_{\xi}^{(i+1)} \otimes \bigwedge F_{\Bar{\xi}}^{(i)} \Big][1] \longrightarrow \Big[ R/I \otimes \bigwedge F_{\xi}^{(k)} \otimes \bigwedge F_{\Bar{\xi}}^{(0)} \Big] [1]
\end{equation}
which is a (signed) sum of operators $\rho_T$ for binary plane trees $T$ with $k$ inputs. According to Lemma \ref{prop:replacer2} evaluating $\rho_T( \beta_1, \ldots, \beta_k )$ involves applying $\operatorname{eval}_{D'}$ to the input $\beta_k \otimes \cdots \otimes \beta_1$. We develop a diagrammatic understanding of the evaluation of $\operatorname{eval}_{D'}$ on this tensor, via Feynman diagrams embedded in a thickening of the mirror tree $T'$. Given a basis vector
\[
\tau \in R/I \otimes \bigwedge F_\xi^{(k)} \otimes \bigwedge F_{\Bar{\xi}}^{(0)}
\]
we wish to know the coefficient of $\tau$ in the evaluation $\operatorname{ev}_{D'}( \beta_k, \ldots, \beta_1 )$, which we denote
\begin{equation}\label{eq:defn_ctau}
C_\tau := \tau^* \operatorname{ev}_{D'}( \beta_k, \ldots, \beta_1 ) \in k\,.
\end{equation}
The description of diagrams contributing to $C_\tau$ is reached in several stages, which are summarised by the Feynman rules in Definition \ref{defn:feynman_rules}. To explain the algorithm it will be helpful to keep in mind the data structure
\begin{equation}
\cat{D} = \{ ( 1, \tau, \operatorname{eval}_{D'}, \beta_k \otimes \cdots \otimes \beta_1 ) \}\,.
\end{equation}
This data structure will be modified as we proceed, but it will always be a sequence of tuples $(\lambda, \alpha, \psi, \beta)$ consisting of a scalar $\lambda \in k$, an output basis vector $\alpha$, a $k$-linear operator $\psi$ with the same domain and codomain as $\operatorname{eval}_{D'}$, and an input basis vector $\beta$. Each time we modify $\cat{D}$ the sum will remain invariant, that is, we will always have
\begin{equation}\label{eq:ctauconstraint}
C_\tau = \sum_{(\lambda, \alpha, \psi, \beta) \in \cat{D}} \lambda \alpha^* \psi( \beta )\,.
\end{equation}
In the following ``the tree'' means $T'$ unless specified otherwise.
\\
\textbf{Stage one: expansion.} Recall $\operatorname{eval}_{D'}$ is defined as a composition of operators
\[
\sigma_\infty, \phi_\infty, e^{\delta}, e^{-\delta}, \mu_2, \pi
\]
which may be written, with some signs and factorials, in terms of the operators\footnote{For the reader's convenience, here is a cheatsheet: for $\zeta$ see Section \ref{section:propagator}, $\vAt_{\mathcal{A}}$ is the Atiyah class of Definition \ref{defn:atiyah_class}, $\nabla$ the chosen connection from Corollary \ref{corollary:nabla_sigma}, $\sigma$ the chosen section of the quotient map $\pi: R \longrightarrow R/I$, for $\delta$ see Definition \ref{defn:important_operators}, $\mu_2$ is ordinary composition in $\mathcal{A}'_\theta$.}
\[
\zeta, \vAt_{\mathcal{A}}, \nabla, \sigma, \delta, \mu_2, \pi\,.
\]
The signs and factorials involved are accounted for carefully in Definition \ref{defn:feynman_rules} below; for clarity we omit them in the present discussion. Choose a presentation $\nu$ or $\rho$ for each edge in $T'$. We expand occurrences of $\vAt_{\mathcal{A}}, \nabla, \delta$ using \eqref{eq:a1vertex}-\eqref{eq:a4vertex},\eqref{eq:c1vertex},\eqref{eq:c2vertex} on the edges of the tree for which $\nu$ has been chosen and \eqref{eq:a3vertex_rho},\eqref{eq:a4vertex_rho},\eqref{eq:c1vertex_rho},\eqref{eq:c2vertex_rho} on the edges for which $\rho$ is chosen. We replace the occurrences of $\mu_2$ using the lemmas of Section \ref{section:fenyman_diagram_2} with additional exponentials, occurrences of $m$ and $\Gamma$ contributions from (D.3) vertices. Next we absorb the $\tau^*$ and $\beta_k \otimes \cdots \otimes \beta_1$ from \eqref{eq:defn_ctau} into the operator decorated tree, by writing the input tensors $\beta_i$ as products of creation operators and the projection $\tau^*$ as a product of annihilation operators followed by the projection $1^*$ (here we assume for simplicity that $z_1 = 1$ in the chosen basis for $R/I$). In the resulting expression the remaining terms that are \emph{not} creation and annihilation operators are occurrences of $\zeta$ and the multiplications $m$ on exterior algebras from Lemma \ref{lemma:mixedr2_1} and Lemma \ref{lemma:mixedr2_2}. Since the relevant virtual degrees are now all fixed, we can calculate the scalar contribution from the $\zeta$ operators and absorb them into the $\lambda$ coefficients. Hence we can replace $\cat{D}$ by a sequence of tuples
\[
(\lambda, 1^*, \psi, 1)
\]
in which every $\psi$ is the denotation of an operator decorated tree (in the sense of Definition \ref{defn:evaluation_tree}) with operators taken from the list (here $\xi, \Bar{\xi}$ stand for any fermionic generator coming from the matrix factorisations themselves, so we do not separately list $\eta, \Bar{\eta}$)
\begin{equation}\label{eq:possible_operators}
\theta, \theta^*, z, z^*, t, \partial_t, \xi, \xi^*, \Bar{\xi}, \Bar{\xi}^*, m, \pi\,.
\end{equation}
This completes the expansion stage.
\\
\textbf{Stage two: reduction to normal form.} The operators $\psi$ in $\cat{D}$ are denotations of trees decorated by monomials in creation and annihilation operators. Such an operator (or more precisely the decoration from which it arises) is said to be in \emph{normal form} if any path from an annihilation operator in the tree to an input leaf encounters no creation operators (roughly, annihilation operators appear to the right of creation operators). After stage one the operators $\psi$ are not in normal form, and we now explain a rewrite process which transforms $\cat{D}$ such that after each step \eqref{eq:ctauconstraint} holds, and the process terminates with the operator $\psi$ in every tuple of $\cat{D}$ in normal form. During some steps of the rewrite process a tuple $(\lambda, 1^*, \psi, 1) \in \cat{D}$ is replaced by a pair $\{ (\lambda_i, 1^*, \psi_i, 1) \}_{i=1}^2$, because the rewriting is ``nondeterministic'' in the sense that it involves binary choices. A Feynman diagram is a \emph{graphical representation of such binary choices made during rewriting.}
Here is the informal algorithm for the rewrite process: take the first tuple $(\lambda, 1^*, \psi, 1)$ in $\cat{D}$ which contains a fermionic annihilation operator, and let $\omega^*$ be the one occurring closest to the root (so $\omega$ is $\theta, \xi$ or $\Bar{\xi}$) and use the available (anti)commutation relations to move it up the tree past the other operators in \eqref{eq:possible_operators}, changing the sign of $\lambda$ as appropriate. The only nontrivial anticommutators that we encounter are:
\begin{itemize}
\item[(a)] $\omega^*$ meets $m$ and generates two additional terms
\[
\omega^* m = m( 1 \otimes \omega^* ) + m( \omega^* \otimes 1)
\]
\begin{center}
\includegraphics[scale=0.6]{dia20}
\end{center}
\item[(b)] $\omega^*$ meets $\omega$ and generates two additional terms
\[
\omega^* \omega = - \omega \omega^* + 1
\]
\begin{center}
\includegraphics[scale=0.6]{dia21}
\end{center}
\item[(c)] $\omega^*$ meets the input $1$.
\end{itemize}
In (a), (b) the tuple $(\lambda, 1^*, \psi, 1)$ is replaced in $\cat{D}$ by two new tuples, in which the decoration of the tree differs from the one determining $\psi$ only in the indicated way (changing the coefficient $\lambda$ by a sign if $\omega^* \omega$ is replaced by $\omega \omega^*$). In (c) we remove the tuple $(\lambda, 1^*, \psi, 1)$ from $\cat{D}$, since it contributes zero to $C_\tau$. The meaning of the pictures will become clear later. We say the occurrence of $\omega^*$ in these new tuples is \emph{descended} from the original $\omega^*$ and we continue the process of commuting these descendents upwards until in $\cat{D}$ there are no tuples containing operators $\omega^*$ descended from our original annihilation operator. Once this is done we return to the beginning of the loop, choosing a new fermionic annihilation operator in $\cat{D}$ as our $\omega^*$. This part of the algorithm terminates when there are no fermionic annihilation operators remaining in $\cat{D}$. It is possible that $\cat{D}$ is now empty, so that $C_\tau = 0$ and the overall algorithm terminates.
We next treat the occurrences of the bosonic annihilation operators $\partial_{t_i}$ in the same way, with the only nontrivial commutation relation being $\partial_{t_i} t_i = t_i \partial_{t_i} + 1$ which generates two new tuples in $\cat{D}$. The $z_h^*$ operators act on the next $z_l$ to give scalar factors $\delta_{h=l}$. This part of the algorithm terminates when there are no annihilation operators remaining, that is, we have replaced $\cat{D}$ by a sequence of tuples $(\lambda, 1^*, \psi, 1)$ in which every $\psi$ is the denotation of an operator decorated tree with operators taken from
\[
\theta, t, \xi, \Bar{\xi}, m, \pi\,.
\]
Since we apply $\pi$ and $1^*$ at the bottom of our diagrams, we do not change the coefficient \eqref{eq:ctauconstraint} if we delete from $\cat{D}$ any tuple in which $\psi$ contains a creation operator. After doing so, the remaining tuples all have the same decoration by $m$'s and $\pi$ and hence the same operator $\psi_{final}$, so at the completion of the second stage we have replaced $\cat{D}$ by a set
\[
\Big\{ (C_{\tau, F}, 1^*, \psi_{final}, 1) \Big\}_{F \in \cat{F}}
\]
where $\cat{F}$ is some index set. Hence $C_\tau = \sum_{F \in \cat{F}} C_{\tau, F}$.
\\
\textbf{Stage three: drawing the diagram.} An index $F \in \cat{F}$ contains the information of a sequence of binary choices made at each nontrivial step of the rewrite process: for example, the choice to replace $\omega^* \omega$ by either $\pm \omega \omega^*$ or $1$, where $\omega$ is one of $\theta, t, \xi, \Bar{\xi}$. The tuple $(\lambda, 1^*, \psi, 1)$ to which this choice refers has a unique ancestor among the tuples in $\cat{D}$ at the end of stage one. In that ancestor tuple, the creation operator $\omega$ is associated with a particular monomial $P$ inserted at a location on $T'$, and the annihilation operator $\omega^*$ is associated with a monomial $P'$ at some other location. By construction $P'$ occurs lower on the tree than $P$. These monomials are precisely the \emph{interaction vertices} to which we have assigned names and diagrams above.
To encode the information in $F$ diagrammatically, we draw all the interaction vertices (in the unique ancestor of the tuple $(C_{\tau, F}, 1^*, \psi_{final}, 1)$ among the tuples of stage one) at the location that they occur in the decoration of $T'$. Each incoming line labelled $\omega$ to such an interaction vertex is uniquely associated by the choices in $F$ with an outgoing line labelled $\omega$ at some vertex higher up the tree, namely, the first and only creation operator where $F$ chooses $1$ rather than $\pm \omega \omega^*$ for the annihilation operator $\omega^*$ whose ancestor is the chosen incoming line. We connect the two interaction vertices by joining them with a line labelled $\omega$. The resulting diagrammatic representation of $F$ is called a \emph{Feynman diagram}, see Figure \ref{fig:feynman_1} for an example.
\\
In summary, we have the following Feynman rules for enumerating Feynman diagrams $F$ and computing their coefficient $C_{\tau, F}$. See \cite[\S 6.1]{weinberg} for real Feynman rules in QFT.
\begin{definition}[(Feynman rules)]\label{defn:feynman_rules} The coefficient $C_\tau$ is a sum $\sum_{F \in \cat{F}} C_{\tau, F}$ of coefficients $C_{\tau, F}$ associated to Feynman diagrams $F$. To enumerate the possible Feynman diagrams, first draw $\beta_k,\ldots,\beta_1$ in order on the input leaves and $\tau$ on the root of $T'$, as sequences of incoming and outgoing particle lines. Then choose for each
\begin{itemize}
\item \textbf{input} a sequence of A-type vertices, then a sequence of C-type vertices (from $e^{\delta} \sigma_\infty$).
\item \textbf{internal vertex} a sequence of vertices of type (D.1),(D.2) and exactly one (D.3) vertex (this arising from multiplication in $R/I$).
\item \textbf{internal edge} a sequence of C-type vertices, then a single B-type vertex, then a sequence of A-type vertices and a sequence of C-type vertices (from $e^{\delta} \phi_\infty e^{-\delta}$).
\item and for the \textbf{outgoing edge} a sequence of C-type vertices (from $\pi e^{-\delta}$).
\end{itemize}
Draw the chosen interaction vertices on the thickened tree and choose a way of connecting outgoing lines at vertices to incoming lines of the same type lower down the tree. Since the only lines incident with the incoming and outgoing boundary of the tree are those arising from the $\beta_i$ and $\tau$ no ``virtual particles'' ($t$'s or $\theta$'s) may enter or leave the diagram. In each case the number of A, C, D-type vertices chosen may be zero, but every internal edge has precisely one B-type vertex. These choices parametrise a finite set of Feynman diagrams $\cat{F}$. The coefficient $C_{\tau, F}$ contributed by the Feynman diagram $F$ is the product of the following five contributing factors:
\begin{itemize}
\item[(i)] The coefficients associated to each A,C-type interaction vertex as given above (for example the vertex $(A.1)^{\nu}$ has coefficient $\sum_{\alpha + \beta = \delta } \sum_{m=1}^\mu (u_j)_{(m,\alpha)} \Gamma^{m h}_{l \beta}$ multiplied by the scalar arising from the partial derivative $\partial_{t_k}(t^\delta)$).
\item[(ii)] For each sequence of $m$ C-type vertices a factorial $\frac{1}{m!}$ and if the sequence immediately preceedes a B-type vertex or the root, a sign $(-1)^m$ (from $e^{-\delta}$ versus $e^\delta$).
\item[(iii)] $Z^{\,\rightarrow}$ factors from $\zeta$ operators (see Section \ref{section:fenyman_diagram_2}).
\item[(iv)] For each time two fermion lines cross, a factor of $-1$.
\item[(v)] For each A-type vertex a factor of $-1$ (from the $(-1)^m$ in $\sigma_\infty,\phi_\infty$).
\end{itemize}
\end{definition}
\begin{remark}\label{remark:boson_symmetry} Recall that we draw an interaction vertex $v$ with an outgoing line labelled $t^\delta = t_1^{\delta_1} \cdots t_n^{\delta_n}$ in our pictures, the convention is that such a line stands for $|\delta|$ separate lines labelled $t_i$ for some $i$. The above description of the Feynman rules involves choosing, for each such $v$, a series of B-type vertices to pair with each of these lines.
This leads to $\delta_1! \cdots \delta_n!$ otherwise identical diagrams, in which the only difference is \emph{which} of the $\delta_i$ lines labelled $t_i$ is paired with which $(B)_i$ vertex. The usual convention is to draw just one such diagram, counted with a symmetry factor $\delta_1! \cdots \delta_n!$.
\end{remark}
\begin{remark}\label{remark:symmetry_A_type} The rules involve \emph{sequences} of A and C-type interactions: different orderings are different Feynman diagrams. In general the Feynman diagrams $F,F'$ associated to different orderings have $C_{\tau, F} \neq C_{\tau, F'}$ because the interaction vertices are operators that do not necessarily commute. In practice, however, one can often infer that the incoming state to a particular edge is constrained to lie in a subspace $\mathcal{K} \subseteq \mathcal{H}$ on which all the relevant operators \emph{do} commute, in which case the different orderings can be grouped together and counted as a single diagram with an appropriate symmetry factor (cancelling, in the C-type case, the scalar factor of (ii) in the Feynman rules).
The situation for a length $m$ sequence of A-type vertices is more subtle, because these arise from powers of $\zeta \vAt_{\mathcal{A}}$ and the operators arising from $\vAt_{\mathcal{A}}$ do not commute with $\zeta$. However, when the incoming state lies in a subspace $\mathcal{K}$ to which Lemma \ref{lemma:technical_antic} applies, we can group together permutations of the $m$ vertices and the $Z^{\,\rightarrow}$ factors become a symmetrised $Z$. This applies in the context of Section \ref{section:generator}.
\end{remark}
\begin{example} Consider the potential $W = \frac{1}{5} x^5$ and matrix factorisations
\begin{align}
X &= \big( \bigwedge(k \xi) \otimes k[x], x^2 \xi^* + \frac{1}{5} x^3 \xi \big)\\
Y &= \big( \bigwedge(k \eta) \otimes k[x], x^3 \eta^* + \frac{1}{5} x^2 \eta \big)
\end{align}
so $f = x^2, u = x^3, g = \frac{1}{5}x^3, v = \frac{1}{5} x^2$. We set $t = \partial_x W = x^4$ and choose our connection $\nabla$ and operators $\partial_t$ as in Example \ref{example:dt_xd} with $d = 4$. For the homotopies $\lambda^Y, \lambda^X$ we use the default choices of Remark \ref{remark:default_homotopies}, so that
\begin{align*}
\lambda^X &= \partial_x( d_X ) = 2 x \xi^* + \frac{3}{5} x^2 \xi && F^X = 2x, \quad G^X = \frac{3}{5} x^2\\
\lambda^Y &= \partial_x( d_Y ) = 3 x \eta^* + \frac{2}{5} x \eta && F^Y = 3x, \quad G^Y = \frac{2}{5} x\,.
\end{align*}
To compute the coefficients associated to all the interaction vertices, we need to compute for various $r \in R$ the coefficients $r_{(m,\alpha)}$ (see Definition \ref{defn:rsharp}) as well as the tensor $\Gamma$. This involves fixing the $k$-basis $R/I = k[x]/x^4 = k1 \oplus kx \oplus kx^2 \oplus kx^3$ that is, $z_h = x^h$ for $0 \le h \le 3$, and the section $\sigma(x^i) = x^i$. Then for example
\[
x^3 = 1 \cdot \sigma( x^3 ) t^0
\]
and in general $(x^a)_{(m,\alpha)} = \delta_{m = a} \delta_{\alpha = 0}$ for $0 \le a \le 3$. Since all the polynomials occurring in $f,g,u,v,F,G$ have degree $\le 3$ these coefficients are all easily calculated as delta functions in this way. The tensor $\Gamma$ encodes the multiplication in $R/I$ and is given by Definition \ref{defn_gamma} for $0 \le m,h \le 3$ by
\[
\Gamma^{mh}_{l \beta} = \delta_{m+h \le 3}\delta_{l = m+h}\delta_{\beta = 0} + \delta_{m+h > 3} \delta_{l = m+h-4} \delta_{\beta = 1}\,.
\]
To present interaction vertices as creation and annihilation operators we use $\rho$ on all edges of the tree involving pairs $(Y,Y)$ and $(X,X)$ and $\nu$ on edges involving $(X,Y)$. Such edges involve $\eta$'s travelling downward and $\xi$'s travelling upward, and the interactions with their coefficients are (recall our convention is to write $h$ for $z_h$ in these diagrams):
\begin{center}
\begin{tabular}{ >{\centering}m{5cm} >{\centering}m{5cm} >{\centering}m{5cm} }
\textbf{(A.1)${}^\nu_{h > 0}$ $+1$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar[dl]^-{\eta}\\
& \bullet \ar@{=}[dl]^-{\theta} \ar@{.}[d]^-{h-1}\\
& &
}
\]
&
\textbf{(A.2)${}^\nu_{h > 1}$ $+\frac{1}{5}$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& \ar@{.}[d]^-{h}\\
& \bullet \ar@{=}[dl]_-{\theta} \ar@{.}[d]^-{h-2} \ar[dr]^-{\eta}\\
& &
}
\]
&
\textbf{(A.3)${}^\nu_{h>1}$ $-1$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& \ar@{.}[d]^-{h}\\
& \bullet \ar@{=}[dl]_-{\theta} \ar@{.}[d]^-{h-2} \\
& & \ar[ul]_-{\xi}
}
\]
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{5cm} >{\centering}m{5cm} >{\centering}m{5cm} }
\textbf{(A.4)${}^\nu_{h>0}$ $+\frac{1}{5}$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& \ar@{.}[d]^-{h} &\\
& \bullet \ar[ur]_-{\xi} \ar@{=}[dl]^-{\theta} \ar@{.}[d]^-{h-1} \\
& &
}
\]
&
\textbf{(B)}
\vspace{0.1cm}
\[
\xymatrix@R+1.5pc{
\ar@{~}[d]^-{t}\\
\bullet \ar@{=}[d]^-{\theta}\\
\;
}
\]
&
\textbf{(C.1)${}^\nu_{h\le2}$ $+3$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
\ar[dr]_-{\eta} & \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta}\\
& \bullet \ar@{.}[d]^-{h+1} \\
& &
}
\]
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{5cm} >{\centering}m{5cm} >{\centering}m{5cm} }
\textbf{(C.1)${}^\nu_{h>2}$ $+3$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
\ar[dr]_-{\eta} & \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta}\\
& \bullet \ar@{.}[d]^-{h-3} \ar@{~}[dl]^-{t}\\
& &
}
\]
&
\textbf{(C.2)$^\nu_{h \le 2}$ $+\frac{2}{5}$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta} \\
& \bullet \ar[dl]^-{\eta}\ar@{.}[d]^-{h+1} \\
& &
}
\
&
\textbf{(C.2)$^\nu_{h > 2}$ $+\frac{2}{5}$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta} \\
& \bullet \ar[dl]^-{\eta}\ar@{.}[d]^-{h-3} \ar@{~}[dr]^-{t}\\
& &
}
\]
\end{tabular}
\end{center}
On the edges involving $X$ purely, where we are using the $\rho$ presentation, we have the following interaction vertices (we omit the B-type which is as above)
\begin{center}
\begin{tabular}{ >{\centering}m{5cm} >{\centering}m{5cm} >{\centering}m{5cm} }
\textbf{(A.1)${}^{X,\rho}_{h>1}$ $+1$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar[dl]^-{\xi}\\
& \bullet \ar@{=}[dl]_-{\theta} \ar@{.}[d]^-{h-2} \\
& & &
}
\]
&
\textbf{(A.4)${}^{X,\rho}_{h>0}$ $+\frac{1}{5}$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& \ar@{.}[d]^-{h} &\\
& \bullet \ar[ur]_-{\xi} \ar@{=}[dl]^-{\theta} \ar@{.}[d]^-{h-1}\\
& &
}
\]
&
\textbf{(C.1)${}^{X,\rho}_{h \le 2}$ $+2$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
\ar[dr]_-{\xi} & \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta}\\
& \bullet \ar@{.}[d]^-{h+1}\\
& &
}
\]
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{5cm} >{\centering}m{5cm} >{\centering}m{5cm} }
\textbf{(C.1)${}^{X,\rho}_{h > 2}$ $+2$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
\ar[dr]_-{\xi} & \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta}\\
& \bullet \ar@{.}[d]^-{h-3} \ar@{~}[dl]^-{t}\\
& &
}
\]
&
\textbf{(C.2)${}^{X,\rho}_{h \le 1}$ $+\frac{3}{5}$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta} \\
& \bullet \ar[dl]^-{\xi}\ar@{.}[d]^-{h+2}\\
& &
}
\]
&
\textbf{(C.2)${}^{X,\rho}_{h > 1}$ $+\frac{3}{5}$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta} \\
& \bullet \ar[dl]^-{\xi}\ar@{.}[d]^-{h-2} \ar@{~}[dr]^-{t}\\
& &
}
\]
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{6cm} >{\centering}m{6cm} }
\textbf{(C.3)${}^{X,\rho}_{h \le 2}$ $+2$}
\vspace{0.1cm}
\[
\xymatrix@C+2pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta} \\
& \bullet \ar@{.}[d]^-{h+1} \\
\ar[ur]^-{\xi} & &
}
\]
&
\textbf{(C.3)${}^{X,\rho}_{h > 2}$ $+2$}
\vspace{0.1cm}
\[
\xymatrix@C+2pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta} \\
& \bullet \ar@{.}[d]^-{h-3} \ar@{~}[dr]^-{t}\\
\ar[ur]^-{\xi} & &
}
\]
\end{tabular}
\end{center}
On edges involving $Y$ purely, where again we use the $\rho$ presentation, we have:
\begin{center}
\begin{tabular}{ >{\centering}m{5cm} >{\centering}m{5cm} >{\centering}m{5cm} }
\textbf{(A.1)${}^{Y,\rho}_{h > 0}$ $+1$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar[dl]^-{\eta}\\
& \bullet \ar@{=}[dl]_-{\theta} \ar@{.}[d]^-{h-1} \\
& &
}
\]
&
\textbf{(A.4)${}^{Y,\rho}_{h>1}$ $+\frac{1}{5}$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
& \ar@{.}[d]^-{h} &\\
& \bullet \ar[ur]_-{\eta} \ar@{=}[dl]^-{\theta} \ar@{.}[d]^-{h-2} \\
& &
}
\]
&
\textbf{(C.1)${}^{Y,\rho}_{h \le 2}$ $+3$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
\ar[dr]_-{\eta} & \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta}\\
& \bullet \ar@{.}[d]^-{h+1} \\
& &
}
\]
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{5cm} >{\centering}m{5cm} >{\centering}m{5cm} }
\textbf{(C.1)${}^{Y,\rho}_{h>2}$ $+3$}
\vspace{0.1cm}
\[
\xymatrix@C+1pc@R+1.5pc{
\ar[dr]_-{\eta} & \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta}\\
& \bullet \ar@{.}[d]^-{h-3} \ar@{~}[dl]^-{t}\\
& &
}
\]
&
\textbf{(C.2)${}^{Y,\rho}_{h \le 2}$ $+\frac{2}{5}$}
\vspace{0.1cm}
\[
\xymatrix@C+2pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta} \\
& \bullet \ar[dl]^-{\eta}\ar@{.}[d]^-{h+1}\\
& &
}
\]
&
\textbf{(C.2)${}^{Y,\rho}_{h > 2}$ $+\frac{2}{5}$}
\vspace{0.1cm}
\[
\xymatrix@C+2pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta} \\
& \bullet \ar[dl]^-{\eta}\ar@{.}[d]^-{h-3} \ar@{~}[dr]^-{t}\\
& &
}
\]
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ >{\centering}m{6cm} >{\centering}m{6cm} }
\textbf{(C.3)${}^{Y,\rho}_{h \le 2}$ $+3$}
\vspace{0.1cm}
\[
\xymatrix@C+2pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta} \\
& \bullet \ar@{.}[d]^-{h+1}\\
\ar[ur]^-{\eta} & &
}
\]
&
\textbf{(C.3)${}^{Y,\rho}_{h > 2}$ $+3$}
\vspace{0.1cm}
\[
\xymatrix@C+2pc@R+1.5pc{
& \ar@{.}[d]^-{h} & \ar@{=}[dl]^-{\theta} \\
& \bullet \ar@{.}[d]^-{h-3} \ar@{~}[dr]^-{t}\\
\ar[ur]^-{\eta} & &
}
\]
\end{tabular}
\end{center}
Suppose we want to evaluate the forward suspended product
\[
\rho_3: \mathcal{H}(X,X)[1] \otimes \mathcal{H}(X,Y)[1] \otimes \mathcal{H}(Y,Y)[1] \longrightarrow \mathcal{H}(X,Y)[1]
\]
on an input tensor
\[
x^2 \xi \otimes x \eta \Bar{\xi} \otimes x^3 \Bar{\eta} = z_2 \xi \otimes z_1 \eta \Bar{\xi} \otimes z_3 \Bar{\eta}\,.
\]
We examine one Feynman diagram contributed by our standard tree $T$ of Figure \ref{eq:explicit_tree_operator}. We are in the situation examined in the proof of Lemma \ref{prop:replacer2}, with
\[
\rho_T( z_2 \xi, z_1 \eta \Bar{\xi}, z_3 \Bar{\eta} ) = \pi e^{-\delta} \mu_2\Big\{ e^{\delta} \phi_\infty e^{-\delta} \mu_2\Big( e^\delta \sigma_\infty( z_3 \Bar{\eta} ) \otimes e^\delta \sigma_\infty( z_1 \eta \Bar{\xi} ) \Big) \otimes e^\delta\sigma_\infty( z_2 \xi ) \Big\}\,.
\]
Next we expand the exponentials and $\phi_\infty, \sigma_\infty$, among the summands is
\begin{align*}
\pi \mu_2\Big\{ \delta \zeta \nabla (-\delta) \mu_2\Big( z_3 \Bar{\eta} \otimes (-\zeta \vAt_{\mathcal{A}})( z_1\eta \Bar{\xi} ) \Big) \otimes \delta (-\zeta \vAt_{\mathcal{A}})( z_2 \xi ) \Big\}\,.
\end{align*}
If we now further expand $\delta, \vAt_{\mathcal{A}}, \nabla$ among the summands is
\begin{gather*}
- \pi \mu_2\Big\{ [ 3 z_1 \eta^* \theta^* ] \zeta \theta \partial_t [ \tfrac{2}{5} \eta t z_3^* \theta^* ] \mu_2\Big( z_3 \Bar{\eta} \otimes \zeta [\tfrac{1}{5} \theta z_1^* \Bar{\xi}^*] z_1 \eta \Bar{\xi}\Big) \otimes [2z_1 \Bar{\xi} \theta^*] \zeta [\theta \xi^* z_2^*] z_2 \xi \Big\}\\
= - \tfrac{3 \cdot 2 \cdot 1 \cdot 2}{5^2} \pi \mu_2\Big\{ [ z_1 \eta^* \theta^* ] \zeta \theta \partial_t [ \eta t z_3^* \theta^* ] \mu_2\Big( z_3 \Bar{\eta} \otimes [ \theta z_1^* \Bar{\xi}^*] z_1 \eta \Bar{\xi}\Big) \otimes [z_1 \Bar{\xi} \theta^*] [\theta \xi^* z_2^*] z_2 \xi \Big\}
\end{gather*}
which are, reading from left to right, the vertices $(\textup{C}.1)^\nu_{h=0}, (B), (\textup{C}.2)^{\nu}_{h=3}, (\textup{A}.4)^\nu_{h=1}, (\textup{C}.3)^{X,\rho}_{h=0}$ and $(\textup{A}.1)^{X,\rho}_{h=2}$. Next we replace all occurrences of $\mu_2$ according to Section \ref{section:feynman_diagram_3}. Among the summands are (we omit the boundary condition operators $P$ for legibility)
\begin{gather*}
- \tfrac{3 \cdot 2 \cdot 1 \cdot 2}{5^2} \pi m\Big\{ [ z_1 \eta^* \theta^* ] \zeta \theta \partial_t [ \eta t z_3^* \theta^* ] m ( \eta^* \Bar{\eta}^* ) \Big( z_3 \Bar{\eta} \otimes [ \theta z_1^* \Bar{\xi}^*] z_1 \eta \Bar{\xi}\Big) \otimes [z_1 \Bar{\xi} \theta^*] [\theta \xi^* z_2^*] z_2 \xi \Big\}\,.
\end{gather*}
Now we commute the leftmost fermionic annihilation operator $\eta^*$ to the right. The only nonzero contribution is when this pairs with the next $\eta$. The leftmost $\theta^*$ has to annihilate with the closest $\theta$, the $\partial_t$ has to annihilate with the $t$, and so on. There is only one pattern of contractions which has a nonzero coefficient, and the pairings of fermionic creation and annihilation operators is shown in the diagram
\begin{center}
\includegraphics[scale=0.45]{dia22}
\end{center}
The corresponding Feynman diagram $F$ with outgoing state $\tau = z_2 \Bar{\xi}$ is shown in Figure \ref{fig:feynman_1}. If we had taken $\tau$ as our outgoing state in the Feynman rules, then this Feynman diagram $F$ would be one of the contributors to $C_{\tau}$ and its contribution is $C_{\tau, F} = - \tfrac{12}{25}$.
\begin{figure}
\begin{center}
\includegraphics[scale=1.0]{dia16}
\end{center}
\centering
\caption{Example of a Feynman diagram, where blue lines denote $\eta$ and green lines $\xi$.}\label{fig:feynman_1}
\end{figure}
\end{example}
\section{The stabilised residue field}\label{section:generator}
In this section we sketch how our approach recovers the usual $A_\infty$-minimal model \cite{seidel_hms, d0904.4713, efimov, sheridan} of $\mathcal{A}(k^{\operatorname{stab}},k^{\operatorname{stab}})$ when $k$ is a field, as an illustration of an example where $E$ may be split by hand. Let $k$ be a characteristic zero field, $W \in k[x_1,\ldots,x_n]$ a potential and $\mathcal{A}$ the DG-category with the single object
\[
k^{\operatorname{stab}} = \big( \bigwedge F_\xi \otimes R, \sum_{i=1}^n x_i \xi_i^* + \sum_{i=1}^n W^i \xi_i \big)
\]
for some chosen decomposition $W = \sum_{i=1}^n x_i W^i$ where $F_\xi = \bigoplus_{i=1}^n k \xi_i$. Hence
\begin{equation}
\mathcal{A}(k^{\operatorname{stab}}, k^{\operatorname{stab}}) = \Big( \End_k\big(\bigwedge F_\xi\big) \otimes R, \sum_{i=1}^n x_i [\xi_i^*,-] + \sum_{i=1}^n W^i [\xi_i,-] \Big)\,.
\end{equation}
This Koszul matrix factorisation is a classical generator of the homotopy category of matrix factorisations over $k\llbracket \bold{x} \rrbracket$ as was shown first by Schoutens in the setting of maximal Cohen-Macaulay modules \cite{schoutens} and then rediscovered by Orlov \cite{orlov} (see \cite[Lemma 12.1]{seidel_hms}) Dyckerhoff \cite[Corollary 5.3]{d0904.4713} and others \cite[Proposition A.2]{keller}. In Setup \ref{setup:overall} we take:
\begin{itemize}
\item $\bold{t} = (x_1,\ldots,x_n)$ so $R/I = k$.
\item $\lambda = \xi_i$ meaning as usual $\xi_i \wedge (-)$.
\item $\sigma: k \longrightarrow R$ is the inclusion of scalars, and $\nabla = \sum_{i=1}^n \partial_{x_i} \theta_i$.
\end{itemize}
It can be shown that
\[
e^{-\delta} r_2 = r_2 e^{\Xi} ( e^{-\delta} \otimes e^{-\delta} )
\]
where $\Xi = \sum_{i=1}^n \theta_i^* \otimes [\xi_i, -]$, and hence in the operator decorated trees computing the higher operations we can remove all the occurrences of $e^{\delta}, e^{-\delta}$ at the cost of replacing $r_2$ at each internal vertex by $r_2 e^{\Xi}$. Under the isomorphism $\rho$ of Lemma \ref{lemma:iso_rho}
\begin{equation}\label{eq:rho_now}
\xymatrix@C+2pc{
\bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}} \otimes R \ar[r]^-{\rho}_-{\cong} & \End_k\big( \bigwedge F_\xi \big) \otimes R
}
\end{equation}
the differential $d_{\mathcal{A}}$ on the right hand corresponds on the left hand side to
\begin{equation}\label{eq:daa_stab}
d^{\rho}_{\mathcal{A}} = \sum_{i=1}^n x_i \xi_i^* + \sum_{i=1}^n W^i \Bar{\xi}_i^*
\end{equation}
by Lemma \ref{lemma:commutators_on_rho}. Moreover under $\rho$ the operator $\Xi$ corresponds to $\theta_i^* \otimes \Bar{\xi}_i^*$, so $e^{\Xi}$ is a family of interaction vertices that allows a downward travelling $\theta_i$ on the left branch to convert into an upward travelling $\xi$ on the right. We identify $\mathcal{B}(k^{\operatorname{stab}}, k^{\operatorname{stab}})$ with $\bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}}$ with zero differential. The Atiyah classes $\At_i$ as operators on $\mathcal{B}$ are
\[
\gamma_j = \At_j = [d_{\mathcal{A}}^{\rho},\partial_{x_j}] = -\xi_j^* - \sum_{i=1}^n \partial_{x_j}(W^i)\Big|_{\bold{x} = \bold{0}} \Bar{\xi}_i^*\,.
\]
Together with $\gamma_j^\dagger = - \xi_j$ these Atiyah classes form a representation of the Clifford algebra, which determines a subspace $F_W$ such that
\[
\operatorname{Im}(E_1) = \bigcap_{i=1}^n \operatorname{Ker}(\gamma_i) = \bigwedge F_W \subseteq \bigwedge F_\xi \otimes \bigwedge F_{\Bar{\xi}}\,.
\]
For example, if $W \in (x_1,\ldots,x_n)^3$ so that $\gamma_j = - \xi_j^*$ then $\bigwedge F_W = \bigwedge F_{\Bar{\xi}}$. This subspace is closed under the higher operations on $\mathcal{B}$ since neither $\vAt_{\mathcal{A}}$ nor $\Xi$ can introduce a $\xi_i$. It follows that $\bigwedge F_{\Bar{\xi}}$ equipped with the restricted operations is a minimal $A_\infty$-category which splits the idempotent $E$.
| 54,596 |
\section{Introduction} \label{intro}
High Performance Computing (HPC) is increasingly relying on more heterogeneity, e.g., by combining Central Processing Units (CPUs) with accelerators in the form of Graphics Processing Units (GPUs) programmed with languages such as OpenCL~\cite{khronosopenclworkinggroup2015opencl} or CUDA~\cite{Nickolls:2008:SPP:1365490.1365500}.
For example, the Swiss National Supercomputing Centre's Piz Daint achieves a Linpack performance of 19.6 PFlop/s with a combination of Intel Xeon E5 V3 CPUs and NVIDIA P100 GPUs~\cite{pizdaint}.
Heterogeneous systems are used to achieve energy efficiency and/or performance levels that are not achievable on a single computing device/architecture.
For example, matrix multiplication is much faster on GPUs than CPUs for the same power/energy budget~\cite{gpuenergyefficiency}.
These accelerators offer a large number of specialized cores that CPUs can use to offload computation that exhibits data-parallelism and often other types of parallelism as well (e.g., task-level parallelism).
This adds an extra layer of complexity if one wants to target these systems efficiently, which in the case of HPC systems such as supercomputers is of utmost importance.
An inefficient use of the hardware is amplified by the magnitude of such systems (hundreds/thousands of CPU cores and accelerators), potentially resulting in large increases in the utilization/power bill and/or exacerbating cooling challenges as a consequence.
Ensuring that the hardware is efficiently used is in part the responsibility of the compiler used (e.g., GCC~\cite{gcc} and Clang/LLVM~\cite{clang,llvm}) to target the code to the computing devices in such systems and also the responsibility of the compiler users.
The programmer(s) and the compiler(s) have to be able to target different computing devices, e.g., CPU, GPU, and/or Field Programmable Gate Array (FPGA), and/or architectures, e.g., systems with ARM and x86 CPUs, in a manner that achieves suitable results for certain metrics, such as execution time and energy efficiency.
Compiler users tend to rely on the standard compiler optimization levels, typically represented by flags such as GCC's \texttt{-O2} or \texttt{-O3}, when using compilers such as GCC and Clang/LLVM.
These flags represent fixed sequences of analysis and transformation compiler passes, also referred to as compiler phase orders.
Programs compiled with these flags tend to outperform the unoptimized equivalent.
However, there are often other assembly/binary representations of the source application in the solution space with higher performance than the ones achieved through the use of the standard optimization levels~\cite{Kulkarni:2012:MCO:2384616.2384628,Purini:2013:FGO:2400682.2400715,Martins:2016:CSE:2899032.2883614,Nobre:2015:UPA:2764967.2764978,Nobre:2016:GIC:2907950.2907959}.
There are a number of scenarios that benefit from specialized compiler sequences, as there is potential to achieve considerable performance, energy or power improvements in comparison with what is achieved with the standard optimization levels.
Domains such as embedded systems or HPC tend to prioritize metrics such as energy efficiency that typically receive less attention from the compiler developers, so these domains can benefit further from these specialized sequences~\cite{Nobre:2016:CPC}.
Heterogeneous systems are composed of multiple devices with different architectures, each one needing different optimization strategies.
With phase selection/ordering, one can achieve higher optimization for these devices, by specifying different compiler sequences for each of them.
In addition, the use of compiler phase selection/ordering specialization can reduce engineering costs, as
in a number of cases the same source code can be used when targeting architecturally different computing devices and/or different metrics through the use of different compiler phase orders.
This can reduce or mitigate the need to develop, manually optimize, and maintain multiple versions of the same function/application.
Automatic approaches have been proposed by a number of researchers in the context of compilation for CPUs and FPGAs.
These approaches can rely on iterative optimization algorithms such as sequential insertion~\cite{huangfpga}, simulated annealing (SA)~\cite{nobresa} and genetic algorithms (GAs)~\cite{Martins:2016:CSE:2899032.2883614}, where new compiler phase orders are iteratively generated and evaluated.
Other approaches rely on machine learning techniques (see, e.g.,~\cite{amir2016,amir,Sher:2014:PRN:2568326.2568328}), mapping a number of static and/or dynamic code features to compiler phase selections and/or phase orders.
This paper presents our work on identification of phase orderings when targeting OpenCL kernels to GPUs using LLVM.
The contributions of this paper are the following:
\begin{enumerate}
\item Assess the performance improvement that can be achieved when targeting OpenCL kernels to an NVIDIA GPU (Pascal architecture) using compiler pass phase ordering specialization with the LLVM compiler toolchain, in comparison with both the use of LLVM without the use of phase ordering specialization and the default OpenCL and CUDA kernel compilation strategies to NVIDIA GPUs.
\item Compare performance between OpenCL and CUDA kernels implementing the same freely available and representative benchmarks using recent NVIDIA drivers and CUDA toolchain, on an NVIDIA GPU with an up-to-date architecture.
\item Further motivate the importance of compiler phase ordering specialization with three experiments. These experiments demonstrate that compiler sequences that lead to performance improvements on some benchmarks do not necessarily do the same for others, and that the order of the passes on compiler sequences have a significant impact on performance.
\item Propose and evaluate a simple feature-based scheme to suggest phase orders that is able to achieve significant speedups with a low number of evaluations.
\end{enumerate}
This paper is an extended version of work published in~\cite{nobreheteropar}.
Contributions 3 and 4 are content exclusive to this paper.
The rest of this paper is organized as follows.
Section~\ref{section-experiments} describes the methodology for the experiments presented in this paper.
Section~\ref{section:gpu_results} presents experiments performed to assess the performance improvement that can be achieved with phase ordering specialization when targeting OpenCL kernels to a NVIDIA GPU (Pascal architecture).
This section includes a set of additional experiments, targeting the NVIDIA GPU, performed to further motivate the use of efficient compiler phase ordering specialization approaches.
Finally, it also includes an explanation at the NVIDIA PTX assembly level (for OpenCL compilation with LLVM and for CUDA compilation with the NVIDIA CUDA Compiler (NVCC)), for each kernel, of the causes of the performance improvements achieved with specialized compiler sequences.
Section~\ref{section:features} presents results, targeting the same NVIDIA GPU, for the use of a simple approach that relies on OpenCL code features to suggest the use of specific compiler sequences for the compilation of a new unseen OpenCL kernel.
Section~\ref{section:relatedwork} presents related work in the context of compiler phase ordering.
Final remarks about the presented work and ongoing and future work are presented in Section \ref{section-conclusions}.
\section{Experimental Setup} \label{section-experiments}
We extended our compiler phase selection/ordering Design Space Exploration (DSE) system, previously used in~\cite{Nobre:2016:GIC:2907950.2907959,Nobre:2016:CPC},
to support exploring compiler sequences targeting NVIDIA GPUs using Clang/LLVM 3.9~\cite{clang} and the libclc OpenCL library 0.2.0~\cite{libclc}.
\subsection{Target systems and system configurations}
The NVIDIA GPU used for the experiments is a variant of the NVIDIA GP104 GPU in the form of an EVGA NVIDIA GeForce GTX 1070~\cite{gtx1070} graphics card (08G-P4-6276-KR) with a 1607/1797 MHz base/boost graphics clock and 8GB of 256 bit GDDR5 memory with a transfer rate of 8008 MHz (256.3 GB/s memory bandwidth).
The GPU is connected to a PCI-Express 3.0 16x interface on its respective workstation.
For the experiments targeting a NVIDIA GPU, we used a workstation with an Intel Xeon E5-1650 v4 CPU, running at 3.6 GHz (4.0 GHz Turbo) and 64 GB of Quad-channel ECC DDR4 @2133 MHz.
We relied on Ubuntu 16.04 64-bit with the NVIDIA CUDA 8.0 toolchain (released in Sept. 28, 2016) and the NVIDIA 378.13 Linux Display Driver (released in Feb. 14, 2017).
The NVIDIA GPU is set to persistence mode with the command \texttt{nvidia-smi -i <target gpu> -pm ENABLED}.
This forces the kernel mode driver to keep the GPU initialized at all instances, avoiding the overhead caused by triggering GPU initialization at application start.
The preferred performance mode is set to \emph{Prefer Maximum Performance} under the \emph{PowerMizer settings} tab in the \emph{NVIDIA X Server Settings}, in order to reduce the occurrence of extreme GPU and memory frequency variation during execution of the GPU kernels.
In order to reduce DSE overhead, and given the fact that we found experimentally that multiple executions of the same compiled kernel had a small standard deviation in respect to measured wall time, each generated code is only executed a single time during DSE.
Only in a final phase on the DSE process are the top solutions executed $30$ times and averaged in order to select a single compiler phase order.
All execution time metrics reported (baseline CUDA/OpenCL and OpenCL optimized with phase ordering) in this paper correspond to the average over $30$ executions.
\subsection{Kernels and objective metric}
In this paper, we use kernels from the PolyBench/GPU benchmark~\cite{polybenchgpu} suite to assess the potential for improvement with phase ordering specialization when targeting a NVIDIA GPU.
This benchmark suite includes kernels from 15 benchmarks from different domains which represent computations that would be performed on GPUs in the context of HPC, including convolution kernels (\texttt{2DCONV}, \texttt{3DCONV}), linear algebra (\texttt{2MM}, \texttt{3MM}, \texttt{ATAX}, \texttt{BICG}, \texttt{GEMM}, \texttt{GESUMMV}, \texttt{GRAMSCH}, \texttt{MVT}, \texttt{SYR2K}, \texttt{SYRK}), data mining (\texttt{CORR}, \texttt{COVAR}), and stencil computations (\texttt{FDTD-2D}).
PolyBench/GPU includes implementations of these kernels in CUDA, OpenCL, and HMPP.
Finally, this benchmark suite is freely available and thus contributes for making the results presented in this paper reproducible.
We experimented with both the CUDA and the OpenCL implementations available for each PolyBench/GPU benchmark.
We rely on the default dataset shape so that reproducibility of the performance metrics reported in this paper is more straightforward.
We performed the minimum of changes to ensure a fair comparison between the OpenCL and the CUDA versions.
This included modifying the benchmarks to ensure that both use the same floating-point precision.
In addition, we also modified the initialization/validation code of some kernels to detect errors produced by some phase orders, by initializing data to non-zero values and adding missing checks.
\subsection{Compilation and execution flow with specialized phase ordering}
Figure~\ref{fig:compilationflow} depicts the compilation flow we use for targeting the host CPU and the GPU, with the use of specialized phase orders when compiling for the latter.
We use Clang compiler's OpenCL frontend with the libclc library to generate an LLVM assembly representation of a given input OpenCL kernel.
The libclc library is an open source library with support for OpenCL functions on AMD and NVIDIA GPUs.
\begin{figure}[t]
\centering
\includegraphics[scale=0.4]{compilationflow.eps}
\caption{Compilation targeting a host CPU and an OpenCL-compatible GPU using Clang/LLVM and libclc. Phase orders are generated/selected by our DSE system.}
\label{fig:compilationflow}
\end{figure}
We use the LLVM Optimizer tool (\texttt{opt}
to optimize the ~LLVM intermediate representation (IR) using a specific optimization strategy represented by a compiler phase order, and we link this optimized IR with the libclc OpenCL functions for our target using \texttt{llvm-link}.
Finally, using Clang, we generate the NVIDIA PTX~\cite{nvidiaptx} from the LLVM bytecode resulting from the previous step, using the \texttt{nvptx64-nvidia-nvcl} target.
PTX is NVIDIA's IR for GPU computations, and is used by NVIDIA's OpenCL and CUDA implementations.
Although PTX is not the final assembly executed on the GPU, it is typically the closest representation one is able to access without direct access to the internals of NVIDIA's drivers.
In the code that executes in the host, each PTX representation generated from the optimized LLVM IR is used to create an OpenCL program object instead of loading the OpenCL kernel source file that comes with the specific PolyBench/GPU benchmark.
To do this, we used offline compilation. That is, we load the compiled NVIDIA PTX code with the \texttt{clCreateProgramWithBinary} function instead of passing the OpenCL source code to \texttt{clCreateProgramWithSource}, which is the most commonly used mechanism in OpenCL.
A compiler phase order represents not only the compiler passes to execute in the compiler optimizer, which can be in the order of the hundreds, but also their order of execution. For instance, as of LLVM 3.9, the phase order for --O3 includes the execution of a total of 172 instances of compiler passes, from which only 72 represent the execution of distinct passes.
The fact that compiler passes are interdependent and interfere with each other's execution in ways that are difficult to predict can make it extremely hard to manually generate suitable compiler sequences.
\subsection{Validation of the code generated after phase ordering}
Each PolyBench/GPU benchmark has verification embedded in its code that consists in executing the OpenCL GPU kernel(s) followed by the execution of a functionally equivalent sequential C version on the CPU, and comparing the two.
This alone poses a challenge, as CPU execution using the same parameters as the ones used for GPU execution takes a long time for a considerable number of PolyBench/GPU benchmarks.
This would have an unreasonable impact on the phase ordering exploration time.
During DSE, we separate validation from the measurement phases to reduce the time for each DSE iteration.
We validate the code generated by compilation with each particular compiler phase order by executing the unoptimized serial version on the CPU (as in the original PolyBench/GPU code) and the optimized version on the GPU with inputs that can be processed quickly.
However, we also execute the same GPU code using the original inputs in order to measure the execution time.
We further reduce exploration time by checking whether an identical optimized version (i.e., NVIDIA PTX) was previously generated. If so, we reuse the results (i.e., correctness and measured performance) from that previous execution.
At the end of phase ordering exploration, all compiler pass sequences that were iteratively evaluated during DSE are ordered by their resulting fitness regarding the objective metric of interest.
With performance as objective metric, sequence/fitness pairs are ordered from the one resulting in fastest execution time to the one resulting in least performance.
Then, as a final step, the optimized version that resulted in highest performance is executed with the original inputs on both the non-optimized CPU version and the optimized GPU version, with $30$ randomly generated inputs that result in the same number of each kind of GPU instructions being executed.
We choose the fastest version that passes validation.
This is performed to eliminate possible situations where a compiled OpenCL code gives correct results using a small input set but gives wrong results with the original input set.
This is just a precaution, and in our tests we did not encounter any case where this was a problem.
The PolyBench/GPU kernels are mostly composed of floating-point operations and the result of floating-point operations can be affected by reordering operations and rounding.
Because of this, we allow for up to 1\% difference between the outputs of CPU and GPU executions when checking if a given compiler phase order results in code that generates valid output.
\section{Impact of Phase Selection and Ordering} \label{section:gpu_results}
We present in this section results of experiments consisting in compiling/evaluating the OpenCL kernels from each of the PolyBench/GPU benchmarks with a set of $10,000$ randomly generated compiler phase orders (given a compiler/target pair, the same set of phase orders was used with all OpenCL codes) composed of up to $256$ pass instances.
We allow any given sequence of LLVM passes to include repeated calls to the same pass.
Passes were selected from a list with all LLVM passes, except passes with names starting with \texttt{-view-*} and the ones that resulted in compilation and/or execution problems when used individually to compile the PolyBench/GPU OpenCL kernels.
For each of the benchmarks, we measured the execution times for the CUDA version, the original OpenCL version compiled from source, an offline compiled OpenCL without optimization, an offline compiled OpenCL with standard LLVM optimization levels (i.e., the best of \texttt{-O1}, \texttt{-O2}, \texttt{-O3} and \texttt{-Os} for each benchmark, which we will refer to as \texttt{-OX}) and an offline compiled OpenCL with our custom compiler optimization phase orders.
\subsection{Performance evaluation}
We compared the results for the various versions of the benchmarks (offline OpenCL versions, OpenCL from source and CUDA) to determine how they perform. Using custom phase orders found by iterative compilation produced code that consistently outperforms the other OpenCL variants, and nearly always outperforms the CUDA version.
Figure \ref{fig:phase_ordering_gtx1070} depicts the speedups obtained with phase ordering over the standard OpenCL (compiled from source), CUDA and the other OpenCL (offline compilation with LLVM) baselines.
With phase ordering specialization we were able to achieve a geometric mean speedup of 1.54$\times$ over the CUDA version and a speedup of 1.65$\times$ over the execution of the OpenCL kernels compiled from source.
Additionally, code compiled with specialized phase ordering can be up to 5.48$\times$ and up to 5.7$\times$ faster than the respective CUDA implementation and the OpenCL compiled from source.
Table \ref{tab_phase_orders} depicts the LLVM 3.9 compiler phase orders that result in the performance improvement factors presented in Figure~\ref{fig:phase_ordering_gtx1070}.
\begin{figure}[t]
\centering
\includegraphics[scale=0.69]{phase_ordering_gtx1070.eps}
\caption{Performance improvements from phase ordering with LLVM over CUDA (\emph{Over CUDA}) and OpenCL using the default compilation method (\emph{Over OpenCL}) for the NVIDIA GP104 GPU and over OpenCL to PTX compilation using Clang/LLVM without (\emph{Over OpenCL w/LLVM}) and with the standard optimization levels (\emph{Over OpenCL w/LLVM --OX}). Data labels are shown for the CUDA and OpenCL using the default compilation method.}
\label{fig:phase_ordering_gtx1070}
\end{figure}
\begin{table}[t]
\caption{LLVM 3.9 compiler phase orders that resulted in compiled kernels with highest performance when targeting an NVIDIA GP104 GPU. Compiler passes that resulted in no performance improvement were eliminated from the compiler phase orders. No compiler phase orders resulted in improving the performance of 2DCONV, 3DCONV or FDTD-2D, when targeting the NVIDIA GPU.}
\label{tab_phase_orders}
\begin{tabularx}{\textwidth}{ l X }
\toprule
\textbf{Benchmark} & \textbf{Compiler Phase Order} \\
\midrule
2MM & \texttt{-cfl-anders-aa -dse -loop-reduce -licm -instcombine} \\
3MM & \texttt{-loop-reduce -gvn-hoist -reg2mem -cfl-anders-aa -sroa -licm} \\
ATAX & \texttt{-bb-vectorize -loop-reduce -licm -cfl-anders-aa} \\
BICG & \texttt{-gvn -loop-reduce -cfl-anders-aa -licm -loop-reduce} \\
CORR & \texttt{-cfl-anders-aa -loop-reduce -gvn -sink -loop-extract-single -loop-unswitch -loop-unswitch -ipsccp -reg2mem -licm -nvptx-lower-alloca} \\
COVAR & \texttt{-cfl-anders-aa -loop-unswitch -reassociate -jump-threading -loop-reduce -gvn -loop-unswitch -reassociate -sink -loop-unswitch -loop-reduce -jump-threading -reg2mem -licm -nvptx-lower-alloca} \\
GEMM & \texttt{-cfl-anders-aa -print-memdeps -loop-reduce -licm} \\
GESUMMV & \texttt{-instcombine -reg2mem -mem2reg} \\
GRAMSCHM & \texttt{-sink -reg2mem -licm -cfl-anders-aa -sroa} \\
MVT & \texttt{-gvn -loop-reduce -cfl-anders-aa -licm} \\
SYR2K & \texttt{-loop-reduce -loop-unroll -instcombine -loop-reduce -licm -cfl-anders-aa} \\
SYRK & \texttt{-licm -cfl-anders-aa -reg2mem -licm -sroa} \\
\bottomrule
\end{tabularx}
\end{table}
There were no significant performance difference between the offline compilation model using Clang/LLVM without custom phase ordering and the OpenCL versions from source.
Likewise, using the LLVM standard optimization level flags did not result in noticeable improvements in terms of the performance of the generated code for most benchmarks.
For \texttt{2DCONV}, \texttt{3DCONV}, \texttt{FDTD-2D} and \texttt{SYR2K} none of the standard optimization level flags resulted in different code being generated.
For benchmarks \texttt{2MM}, \texttt{3DCONV}, \texttt{3MM}, \texttt{ATAX}, \texttt{BICG}, \texttt{GEMM}, \texttt{GESUMMV}, \texttt{GRAMSCHM}, \texttt{MVT} and \texttt{SYRK}, the generated code using the optimization level flags differs from the generated code without optimizations. However, the different optimization levels (i.e., \texttt{-O1}, \texttt{-O2}, \texttt{-O3}, \texttt{-Os}) all produced the same code.
\texttt{CORR} and \texttt{COVAR} are the only benchmarks for which different optimization level flags produce different code. However, even in these benchmarks, the performance impact was usually minimal (within 1\%).
The only exceptions were \texttt{GESUMMV} and \texttt{GRAMSCHM}.
In the case of \texttt{GESUMMV}, the use of all tested optimization levels resulted in 1.07$\times$ performance improvement over the non-optimized version.
For \texttt{GRAMSCHM}, the non-optimized version was 1.04$\times$ faster than all the versions produced by the optimization level flags.
The difference between the OpenCL baselines is that one represents the de facto OpenCL compilation flow (with OpenCL program created from source) and the others represent the compilation using LLVM (with OpenCL program created from binary) using the standard optimization level that results in the generation of code with highest performance on a kernel-by-kernel basis, and compilation using LLVM but with no optimization.
Finally, on these benchmarks, performance with CUDA tends to be better than with OpenCL, if no specialized phase ordering is considered.
The geometric mean (considering all 15 PolyBench/GPU benchmarks) of the performance improvement with CUDA (over OpenCL from source) is 1.07$\times$.
The \texttt{2DCONV}, \texttt{3MM}, \texttt{ATAX}, \texttt{BICG} and \texttt{SYRK} benchmarks are at least 1.1$\times$ faster in CUDA than in OpenCL.
All other benchmarks with exception for \texttt{3DCONV} and \texttt{GESUMMV} are still faster in CUDA than in OpenCL, although by a smaller margin.
We also evaluated the same PolyBench/GPU kernels with compiler phase ordering using LLVM when targeting an AMD Fiji GPU using the ROCm 1.6 drivers, compared with the code generated by online compilation and with the use of LLVM without phase ordering, and were able to achieve speedups both over compilation from source ($1.65\times$) and compilation with LLVM using the standard optimization levels ($1.73\times$).
These results can be of special interest, as the compilation approach for AMD and NVIDIA are significantly different.
When targeting AMD GPUs, LLVM generates the final AMD GPU ISA code, whereas the LLVM backend targeting NVIDIA GPUs generates NVIDIA's PTX, which is an intermediate representation that is further optimized by the CUDA compiler.
The speedups using phase ordering differ substantially from the ones obtained when targeting the NVIDIA GPU, with benchmarks \texttt{2MM}, \texttt{3DCONV}, \texttt{3MM}, \texttt{GEMM}, \texttt{GESUMMV}, \texttt{GRAMSCHM} and \texttt{SYRK} improving substantially more with phase ordering specialization when targeting the AMD Fiji GPU, while \texttt{CORR}, \texttt{COVAR}, \texttt{MVT} and \texttt{SYR2K} benefited more when targeting the NVIDIA GPU.
This reinforces the notion that the efficiency of custom phase orders is highly device-dependent, so target specialization is important.
\subsection{Problematic phase orders}
Considering the evaluation of all $10,000$ sequences with all PolyBench/GPU benchmarks (\texttt{TOTAL} column), the most common problem is the report being non-existent or broken ($17\%$), the second is the generation of incorrect output by the compiled OpenCL kernels ($13\%$), and the third (and last) is the optimized LLVM IR not being generated ($3\%$).
Typically, the non-existence of optimized LLVM IR after calling the LLVM Optimizer is caused by a compiler crash.
In some cases, the execution of the optimized/compiled kernels does not terminate.
This can happen because of problems in the kernel themselves or because the compiled kernels are not given enough time to finish execution.
Our phase ordering exploration system has a timeout parameter for limiting the overhead of exploration allowed to the execution of the OpenCL kernels compiled after phase ordering.
The fact there is a timeout is not detrimental in the sense that it does not result in discarding any given suitable compiler phase orders in the context of performance maximization, because if a compiled OpenCL kernel takes too much time to execute, it means that it was not compiled with a phase order suitable to maximize performance.
Although compiler developers typically make an effort to assure that any given compiler pass that transforms an IR will have as output other IR that is functionally equivalent to the original, this is difficult to implement in practice,
especially given the fact that compilers can have tens or hundreds of compiler passes and it is difficult to predict all possible interactions between passes.
For instance, Eide and Regehr evaluated thirteen production-quality C compilers and, for each, were able to find cases where incorrect code to access volatile variables was generated~\cite{eide}.
Moreover, given the fact that during our phase ordering exploration we are compiling with sequences that were possibly never tested/evaluated by the compiler developers, there is a greater potential for generating code that does not conform with the same functionality of the original code and that will generate outputs that are different than expected.
\subsection{Additional experiments}
We performed the following experiments in order to better assess the importance of specialization of compiler sequences when using LLVM to target NVIDIA GPUs:
\begin{enumerate}
\item Evaluate the use of the sequences found for any given PolyBench/GPU OpenCL benchmark (see Table~\ref{tab_phase_orders}) in the remaining benchmarks.
\item Show how the performance of the different PolyBench/GPU OpenCL benchmarks is impacted by different compiler sequences, comparing the performance differences registered for different kernels using the same set of compiler sequences.
\item Show the effect of the different phase orders, on each benchmark, constructed with permutations of the sequences found for the same benchmark (see Table~\ref{tab_phase_orders}).
\end{enumerate}
Figure~\ref{fig:sequences_in_other_kernels} shows the matrix resulting from evaluating the sequences from Table~\ref{tab_phase_orders} for compilation of each of the PolyBench/GPU OpenCL benchmarks.
Values, between $0$ and $1$, represent performance factors resulting from comparing with the performance obtained with the sequence found to be the best for each benchmark.
\begin{figure}[t]
\centering
\includegraphics[scale=0.65]{sequences_in_other_kernels.eps}
\caption{Performance ratios for using sequences found for each of the benchmark in all benchmarks. The $X$ axis represents the benchmarks and the $Y$ axis represents the sequences (i.e., the best found for each benchmark). Performance ratios are represented with a precision of $5\%$, so values represented as $1.0$ can be as low as $0.95$. Values represented as $1.0$ but that are closer to $0.95$ are represented with a slightly lighter shade.
}
\label{fig:sequences_in_other_kernels}
\end{figure}
The phase orders found individually for each benchmark result in a very wide performance factor range when used to compile the OpenCL code of the remaining $14$ benchmarks.
Moreover, some benchmark/sequence pairs (\texttt{CORR}/\texttt{2MM}, \texttt{GESUMMV}/\texttt{COVAR}, \texttt{GESUMMV}/\texttt{GRAMSCHM}, \texttt{GESUMMV}/\texttt{SYRK}) did not pass validation, resulting in incorrect outputs when executing the generated code.
Each individual compiler sequence has a different impact on different OpenCL kernels.
Figure \ref{fig:100_seqs} presents the performance impact (i.e., speedup) of the first $100$ compiler sequences evaluated during the initial DSE process (from the set of $10,000$ sequences evaluated) on each of the OpenCL benchmarks.
The performance baseline is offline compilation with LLVM with no optimization, as it was experimentally determined that the use of the standard optimization levels in LLVM only very rarely improves the performance of the generated NVIDIA PTX assembly code (see Figure \ref{fig:phase_ordering_gtx1070}).
The performance of the best phase order (see Table \ref{tab_phase_orders}) found for each OpenCL benchmark is represented as a horizontal reference line.
Speedups above this reference line are entirely caused by random performance variations.
For the \texttt{2MM}, \texttt{3MM}, \texttt{CORR} and \texttt{COVAR} OpenCL benchmarks, the first $100$ DSE iterations could not achieve the speedup achieved with the $10,000$ iterations.
It is also interesting to notice that there is a cluster of points from the compiler sequence solution space close to the baseline (i.e., close to horizontal line $y=1$), and that for some kernels the concentration of points close to the highest achieved speedup (i.e., close to the reference line) is very scarce (e.g., \texttt{GEMM}, \texttt{GRAMSCHM}, \texttt{SYR2K}), while for other kernels there are several sequences close to the best found (e.g., \texttt{SYRK}).
Benchmark/sequence pairs that result in generation of incorrect output, compiler crashes or execution timeout expiration, are represented on top of the $X$ axis (i.e., $y=0$).
These results show that: 1) given a compiler sequence randomly selected from the compiler sequence's space, it is most likely that it will generate code without performance improvements in relation to the code achieved without compiler phase ordering specialization; and 2) for some kernels, it is considerably less likely that a given compiler sequence will result in generated code that performs close to the performance achieved with a considerable number of iterations of iterative compilation.
\begin{figure}[t]
\centering
\includegraphics[scale=0.605]{100_seqs.eps}
\caption{Speedup for the same sequences on different kernels. Each point represents a phase order, with its $X$ and $Y$ position representing the speedup for the 2DCONV benchmark and for the benchmark indicated on top of each chart, respectively.
The horizontal reference line represents the speedup for the best found sequence for each benchmark.
The same baseline was used for all benchmarks so that comparisons between charts possible.}
\label{fig:100_seqs}
\end{figure}
Figure~\ref{fig:permutations} shows the results obtained for experiments where permutations of the phase orders from Table~\ref{tab_phase_orders} are evaluated.
Up to $1,000$ randomly generated permutations were evaluated for compilation of each kernel.
Each permutation includes all the compiler passes that are present in the phase order from Table~\ref{tab_phase_orders} (including the number of pass instances that are repeated in the sequence).
The \texttt{2DCONV}, \texttt{3DCONV} and \texttt{FDTD-2D} OpenCL benchmarks are not included because the initial DSE process could not find a LLVM phase order that resulted in improved performance.
The execution performance after compilation with a large number of permutations resulted in considerable performance degradation.
Some of these permutations were only able to achieve $10\%$ or less of the execution performance achieved with the initial phase order (e.g., \texttt{3MM}, \texttt{CORR}, \texttt{COVAR}).
We believe these results strongly motivate, at least under some circumstances, the use of compiler phase ordering when targeting an NVIDIA GPU.
\begin{figure}[t]
\centering
\includegraphics[scale=0.62]{permutations.eps}
\caption{Impact of the order of compiler passes in the best-found sequence for each benchmark. This is demonstrated in the form of distribution of speedups over the best order ($X$ axis). The $Y$ axis indicates the percentage of permutations that achieve that range of speedups.}
\label{fig:permutations}
\end{figure}
\subsection{Insights about performance improvements}
In this subsection, we explain, for each PolyBench/GPU benchmark, some of the key reasons behind the performance improvement achieved with phase ordering, compared with the performance achieved by the OpenCL baselines and the baseline CUDA versions.
More specifically, we compare the PTX output resulting from OpenCL offline compilation with specialized phase ordering with PTX generated from OpenCL offline compilation without phase ordering and with PTX generated from the CUDA versions.
For \texttt{2DCONV}, CUDA is $1.26\times$ faster than all versions compiled from OpenCL.
The compiler pass phase ordering DSE process was not able to find an LLVM sequence capable of optimizing this benchmark.
The main improvement regarding the CUDA version over the OpenCL versions seems to be the generation of more efficient code for loads from global memory. Figure~\ref{fig:ptx-load-comparison} shows the difference between the two approaches. Whereas CUDA load operations typically result in a single PTX instruction, the equivalent for OpenCL typically results in 5 PTX instructions. We believe that this difference in load instructions is the primary reason for CUDA's advantage over OpenCL for the \texttt{2DCONV} OpenCL code.
\begin{figure}[t]
\centering
\begin{minipage}{.45\textwidth}
\scriptsize
\begin{lstlisting}[language=ptx]
ld.global.f32
\end{lstlisting}
\textbf{(a)} PTX load code generated from CUDA.
\end{minipage}
\centering
\begin{minipage}{.45\textwidth}
\scriptsize
\begin{lstlisting}[language=ptx]
add.s32
cvt.s64.s32
shl.b64
add.s64
ld.global.f32
\end{lstlisting}
\textbf{(b)} PTX load code generated from OpenCL (\texttt{-O3}).
\end{minipage}
\caption{PTX code for equivalent load operations, for CUDA and OpenCL (2DCONV benchmark).}
\label{fig:ptx-load-comparison}
\end{figure}
For \texttt{2MM}, the OpenCL version optimized with phase ordering is $1.63\times$ and $1.56\times$ faster than the OpenCL (compiled from source) and CUDA baselines, respectively.
The main reason for these speedups is the removal of store operations within the kernel loop.
Both the OpenCL and the CUDA baseline versions of this kernel repeatedly overwrite the same element, to the detriment of program performance.
The phase ordered version instead uses an accumulator register and performs the store only after all the loop computations are complete, which substantially reduces the number of costly memory accesses.
It is unclear why the baseline OpenCL and CUDA versions do not perform this optimization.
One possibility is that the NVIDIA OpenCL/CUDA compiler and LLVM w/o the use of special phase orders are unable to determine that there are no aliasing issues.
In the context of the \texttt{2MM} benchmark, it is correct to assume that there is no aliasing, as any aliasing would result in a data race (in OpenCL 2.0), which is undefined behavior~\cite{khronosopenclworkinggroup2015opencl}.
We do not know if the optimization was applied because LLVM correctly discovered this fact, or if there is a bug that happened to result in correct code by accident.
Even if the optimization turns out to be the result of a bug, we believe this speedup represents an opportunity for approaches based on \emph{Loop Versioning} transformations~\cite{loopversioning}.
Although this benchmark uses two kernels, both are equivalent (the only difference being kernel and variable names), and thus the same analysis applies to both.
There are two differences between the CUDA version and OpenCL versions compiled offline that can explain the different execution times.
The first being the aforementioned issue with load instructions (see Figure~\ref{fig:ptx-load-comparison}), the second being a different loop unroll factor as the phase ordered version based on OpenCL uses efficient load instructions, but uses a loop unroll factor of $2$ (while the CUDA version uses an unrolling factor of $8$).
For \texttt{3DCONV}, we were unable to achieve a speedup using any of the compiler phase orders evaluated, when compared with LLVM w/ or w/o the optimization level flags.
We believe this happens because most of the time spent on the benchmark is due to global memory loads that are not removed or improved by any LLVM pass.
Any optimization will only modify the rest of the code, which takes a negligible amount of time compared to the memory operations.
Interestingly, there is a speedup from the use of the LLVM PTX backend compared with the OpenCL from source compilation path ($1.05\times$) and the compilation from CUDA ($1.06\times$).
On the \texttt{3MM} benchmark, we were able to achieve speedups of $1.55\times$ and $1.82\times$ over the baseline CUDA and OpenCL version compiled from source, respectively. The main reason for the performance improvement is the removal of the memory store operation from the computation loop.
The OpenCL version of \texttt{ATAX} optimized with phase ordering achieves a speedup of $1.47\times$ and $1.25\times$ over the baseline OpenCL and CUDA versions, respectively.
Once again, the phase ordered version is able to move memory stores out of the innermost loops of the kernels, which explains the speedups.
Additionally, the difference between the CUDA and the baseline OpenCL versions can be explained by a different loop unroll factor ($2$ for OpenCL, $8$ for CUDA).
The CUDA version uses the previously described simpler code pattern for memory loads compared to the baseline OpenCL version, but the phase ordering version also uses an efficient memory load pattern.
On the \texttt{BICG} benchmark, we achieved a speedup of $1.48\times$ over OpenCL compiled from source, and $1.28\times$ over CUDA. Main differences between the versions are the memory stores in the kernel loop, the unroll factor and the inefficient memory access patterns in the baseline OpenCL version.
The \texttt{CORR} benchmark is one of the benchmarks that benefit the most from phase ordering ($5.36\times$ and $5.14\times$ over baseline OpenCL from source and CUDA versions, respectively).
The version generated by phase ordering contains several memory accesses to a local memory storage buffer (named \verb|__local_depot|) that serves a purpose similar to the stack of the CPU.
We believe this buffer is inserted due to the use of the \verb|reg2mem| pass without a corresponding \verb|mem2reg|.
These instructions do not seem to have a significant impact, either because they are eliminated in the compilation of the PTX to device-specific code by the NVIDIA GPU driver, or because the accesses to local memory are too fast to affect non-negligible performance variations.
Phase ordering is also capable of moving global memory stores out of loops, which neither the CUDA version nor the baseline OpenCL versions do.
In general, for this benchmark, the CUDA version tends to produce more compact load instructions and use higher loop unroll factors than the OpenCL versions.
The \texttt{COVAR} and \texttt{CORR} benchmarks both rely on the same \verb|mean_kernel| and \verb|reduce_kernel| functions. However, these functions represent only a fragment of the total execution code.
Regardless, the same conclusions from \texttt{CORR} apply to \texttt{COVAR}: phase ordering removes global stores from the loop, but introduces several new registers and local memory accesses.
The functions of the \texttt{FDTD-2D} benchmark are very straightforward, with little potential for optimization. As such, phase ordering had no impact.
The performance differences for the \texttt{GEMM} benchmark ($1.67\times$ and $1.73\times$ over the OpenCL from source and the CUDA baselines) can be explained by the removal of the memory store operation from the kernel loop, the different unroll factor and the different pattern of memory load instructions.
There was only a small performance improvement for the \texttt{GESUMMV} benchmark ($1.07\times$ over CUDA and $1.02\times$ over the baseline OpenCL from source).
Phase ordering specialization is able to extract the memory stores out of the main computation loop, but uses a smaller loop unroll factor ($2$) than the baseline OpenCL and CUDA versions ($4$ and $16$, respectively).
We were able to obtain speedups of $1.49\times$ and $1.52\times$ over the baseline CUDA and OpenCL versions on the \texttt{GRAMSCHM}, respectively.
Phase ordering is able to move the memory storage operations out of the loop. Aside from that, it uses the same load from memory instruction pattern and unroll factor as the baseline OpenCL version.
The \texttt{MVT} benchmark benefits from phase ordering by a factor of $1.32\times$ and $1.44\times$ over the baseline CUDA and OpenCL versions. The main reason for this improvement is the extraction of the store operation from the computation loop.
The \texttt{SYR2K} benchmarks benefits from phase ordering by a factor of $1.99\times$ and $2.05\times$ over the baseline CUDA version and baseline OpenCL compiled from source, respectively.
In general, the same memory load pattern, loop unroll factor and loop invariant memory storage code motion conclusions apply to this benchmark.
Phase ordering also seems to outline the segment of the code containing the kernel loop, but this does not seem to be the reason for the performance difference.
For the \texttt{SYRK} benchmark, phase ordering improves performance by $1.14\times$ over the OpenCL baselines.
Once again, the main reason for this improvement is the extraction of the store from the loop.
We could not achieve significant speedups over the CUDA version.
\section{Feature-based Phase Ordering Suggestions} \label{section:features}
It is intuitive to think that the set of compiler passes and compiler sequences that are most suited to compile a given function/program given the target computing platform/device and metrics (e.g., performance, energy, code size) are, at least to some extent, related to static and/or dynamic features of the function/program.
Taking into account such information when deciding what compiler sequences to evaluate during DSE for a given program/function can have the potential to reduce exploration time, or alternatively, achieve better solutions with the same number of compile/execute DSE iterations.
Given the large exploration space for phase selection and ordering, in this paper we propose a simple feature-based approach for suggesting compiler sequences when targeting OpenCL code to GPUs. The experiments show the efficiency of the particular approach and the potential of feature-based approaches in general for compilation targeting GPUs, particularly for use cases demanding fast DSE of compiler phase orders. We believe that the results of these experiments can be of importance in guiding the development of future approaches.
Given a new OpenCL benchmark, we select $K$ sequences from Table \ref{tab_phase_orders} that are assigned to the $K$ functions/programs most similar with it (k-nearest neighbors).
Those sequences are then used for compilation/evaluation with the new function/program, and the compiled code resulting in the highest performance is selected.
This method allows to suggest compiler sequences very efficiently, and with an overhead that grows in proportion to the value $K$ selected by the compiler user (i.e., number of compilations/evaluations that the user affords).
Phase ordering exploration can have additional termination conditions. For instance, in a real-world use case, instead of terminating only when all sequences from this set having been evaluated or the maximum number of evaluations tolerable by the user being achieved, the DSE could end as soon as the compiled code complies with the non-functional requirements for the function/program.
\subsection{OpenCL code features}
The code-features used in the work presented in this paper are extracted with the MILEPOST GCC toolchain~\cite{fursin:inria-00294704}, which includes the Interactive Compilation Interface (ICI) 2.0 and Machine-Learning (ML) feature extractor version 2.0.
Although the MILEPOST feature extractor was originally developed to extract static C code features, given OpenCL C (the language used to program OpenCL compute kernels) is based on C99, we are using the same flow to extract features from the OpenCL PolyBench/GPU kernels.
MILEPOST features represent an absolute count (e.g., number of basic blocks in the method, number of basic blocks with a single successor) or an average of some count (e.g., average of number of instructions in basic blocks, average of number of phi-nodes at the beginning of a basic block).
Pairs of feature vectors with MILEPOST code features (e.g., feature vectors for function/program from the reference set of functions and the feature vector of a given new function/program) are used to compute a similarity metric that determines the compiler sequences that are suggested for evaluation.
We did not perform feature selection, thus all $55$ code features extracted by MILEPOST GCC are represented in the feature vectors.
The host code is not taken into account when extracting code features.
Only the OpenCL kernels execute on the GPU, therefore only they are being optimized through compiler sequence specialization.
\subsection{Experiments using code features}
In the experiments presented here we use the cosine distance between feature vectors associated with the OpenCL codes as metric of similarity between pairs of OpenCL codes.
The evaluation of the impact on the results of other similarity metrics is planned for future work.
Figure~\ref{fig:cosinedistance_vs_random_1_14} shows the performance improvement achieved over compilation from source for different numbers of evaluations of compiler sequences from Table~\ref{tab_phase_orders}.
LLVM without any optimization is used as fallback in case no additional sequence evaluated results in better performance.
The approach that relies on evaluating $K$ (a number given by the programmer/user) sequences associated with the $K$ functions/programs more similar with the new OpenCL kernel is compared with the random selection of PolyBench/GPU kernels and using the sequences previously found for them.
Each random selection is performed $1,000$ times, and the geometric mean of the resulting speedups from using the sequences associated with the selected benchmarks is reported.
We also compare with the IterGraph approach (see~\cite{Nobre:2016:GIC:2907950.2907959}), where a graph representing favorable compiler pass subsequences was generated from the sequences from Table~\ref{tab_phase_orders}, using a leave-one-out approach for validation (i.e., when compiling a given OpenCL kernel the sequence associated with that kernel is not used to build the graph).
In fact, all experiments presented in this section are performed with a leave-one-out approach.
When a given PolyBench/GPU kernel is the input of exploration, only the other $14$ PolyBench/GPU kernels and their sequences are considered.
\begin{figure}[t]
\centering
\includegraphics[scale=0.685]{cosinedistance_vs_random_1_14.eps}
\caption{Performance improvement on a NVIDIA GP104 GPU for using MILEPOST static code features in the PolyBench OpenCL kernels to select a number of most similar OpenCL kernels and using their sequences. The reference line represents the geometric mean speedup of using the best order for each benchmark, as seen in Figure~\ref{fig:phase_ordering_gtx1070}.}
\label{fig:cosinedistance_vs_random_1_14}
\end{figure}
The results show that the selection of the $K$ most similar OpenCL kernels using the cosine distance and using the sequences from Table~\ref{tab_phase_orders} associated with them, results in considerably higher geometric mean performance improvements, for all $K$, when compared with random selection of $K$ OpenCL kernels (and using their sequences from Table \ref{tab_phase_orders}) and also when compared with the use of the IterGraph approach.
With only $2$ additional sequence evaluations, the method based on the cosine distance results in a $1.56\times$ performance improvement compared with compilation w/o phase ordering specialization, while random selection of OpenCL kernels and using the compiler sequences associated with them required $9$ evaluations to achieve the same performance improvement.
For $14$ sequence evaluations, both methods result in the same performance improvement of $1.6\times$, as at that point all sequences from Table~\ref{tab_phase_orders} have been evaluated by both methods.
The IterGraph approach beats random selection of sequences from Table~\ref{tab_phase_orders} for evaluation
when considering up to only $3$ sequence evaluations (on top of LLVM w/o optimization), and after that, only for $8$ and $9$ evaluations.
When considering only up to $14$ sequence evaluations, the IterGraph approach tends to result in compiled code with reduced performance when compared with the simple code-feature based approach presented here.
This may change on new OpenCL kernels from a different benchmark suite or from a different domain.
An advantage of the IterGraph approach in comparison with methods that rely on evaluating predetermined sequences is that new sequences can be generated from the graph (i.e., different from the ones used to build it), which may be more suitable to optimize a given unseen kernel than any of the sequences previously determined to be best for compiling the kernels from a reference set.
The results of our experiments combining the MILEPOST code features extracted from OpenCL C kernels with the use of the cosine similarity as metric for identifying similar OpenCL kernels based on the feature vectors are promising.
Using only the sequence associated with the most similar OpenCL kernel results in a geomean speedup of $1.49\times$ and $1.03\times$, for selection based on cosine similarity and for random selection, respectively.
For $3$ sequence evaluations, the speedups improve to $1.56\times$ and $1.22\times$ respectively.
Finally, for $5$ evaluations, the speedups are $1.59\times$ and $1.5\times$ respectively.
The speedup using $5$ evaluations of compiler sequences suggested using code features is very close to the highest speedup that is achieved when testing the $14$ sequences of all other OpenCL kernels ($1.6\times$).
The performance of the GPU code generated from offline compilation without phase ordering
is close to the performance obtained with online compilation (see Figure ~\ref{fig:phase_ordering_gtx1070}).
For some OpenCL codes, if the number of evaluated sequences is too small, phase ordering can lead to slowdowns over a given baseline (i.e., one of the standard optimization levels, including \texttt{-O0}).
To prevent these situations, it is important to use a safe fallback, such as \texttt{-O3}.
However, when using the similarity-based approach these situations are quite rare.
For instance, considering only the evaluation of a single compiler sequence, \texttt{GESUMV}, \texttt{GRAMCHM} and \texttt{SYR2K} are the only OpenCL benchmarks where the similarity-based selection approach would result in generating code that is worse than the code generated from online compilation.
Two evaluations of sequences are enough to reduce this to only a single benchmark (\texttt{GESUMV}).
In contrast, when randomly selecting sequences to evaluate from Table~\ref{tab_phase_orders}, $8$ OpenCL benchmarks would result in worse performance than online compilation if not using the fallback GPU code for these benchmarks; $2$ evaluations would only reduce that number to $3$.
While the IterGraph approach does not tend to perform very well in comparison with the other two approaches for a small number of evaluations (e.g., up to 14 evaluations in these experiments), it could generate, for any number of sequence evaluations, compiler optimization sequences that result in higher performance for the \texttt{SYR2K} OpenCL kernel than the two other approaches.
We believe that when we validate these approaches using OpenCL kernels from a different benchmark (instead of relying on leave-one-out validation), there will be more cases where the IterGraph approach outperforms the approach for feature-based suggestion of sequences from Table~\ref{tab_phase_orders} evaluated here, and even more cases where it surpasses the efficiency of randomly selecting sequences from Table~\ref{tab_phase_orders}.
Notice that in these experiments, given an OpenCL kernel, the number of compilations and executions of compiled kernels caused by the selection of $K$ kernels from the reference set can be lower than $K$, even for small $K$ values.
Each sequence is only used for compilation once, even if the same sequence is chosen for two or more OpenCL reference codes.
When evaluating the impact of a given selected set of $K$ sequences, subsequent evaluations of identical sequences result in no additional overhead, as the DSE process simply recalls the fitness value (i.e., execution performance) when a previously seen compiler sequence is evaluated for use with the same kernel.
The DSE system also recalls the fitness value for the metric of interest in case a given generated GPU code is identical to the code generated with the use of other compiler sequence previously evaluated, but in this case there is an overhead that incurs because of compilation, which in our experiences is not negligible; especially because we perform validation, during the execution of the main DSE loop, using inputs that result in faster execution time.
\section{Related Work}\label{section:relatedwork}
There have been attempts to identify a small universal set of promising phase orders that could then be used by compiler users to extend the portfolio of the standard optimization levels provided by a compiler (e.g., --O3).
Purini and Jain~\cite{Purini:2013:FGO:2400682.2400715} presented and evaluated approaches for devising a universal set of compiler sequences that is able to cover the program space of a reference set of programs. Given a new program, the sequences from this set are evaluated in a predefined order.
Other authors use features to identify similarities between an unseen code and the codes from a set of reference functions/programs in order to reduce the design space.
For example, the approach presented by Martins et al.~\cite{Martins:2016:CSE:2899032.2883614} relies on special code fingerprints to reduce the number of compiler passes to consider for exploration, reducing DSE overhead considerably; and Amir et al.\cite{amir2016} relies on dynamic code features to guide the generation of specialized compiler phase orders.
The work presented in this paper has key differences to those approaches.
For instance, while other approaches rely on program-features (either static or dynamic) for focus exploration space, our approach uses code-features to select from a set of previously generated sequences.
Unlike sequences generated online using optimization algorithms, these sequences can be strenuously validated by the compiler developers, in a similar way to the validation of the sequences represented by the compiler's default optimization levels (e.g., --O3).
Compiler bugs are a serious hindrance when considering the use of compiler sequence specialization in use-cases where the correct execution of an application is of utmost importance (e.g., safe-critical systems), therefore any approach that drastically reduces the likelihood of the expression of compiler bugs is important.
In addition, the fact that the sequences can be previously validated for correctness allows compiler users to use them with more confidence, reducing the need for extra validation (e.g., testing many pairs of inputs/outputs) than when using the traditional flags such as --O3.
In comparison with the approach as presented by Purini and Jain.~\cite{Purini:2013:FGO:2400682.2400715}, although we also evaluate sequences from a small set of sequences previously generated with iterative compilation, our approach relies on code-features to help efficiently identifying suitable compiler sequences from those sequences.
In this work we identify promising phase orders for a given input function using similarity and code features.
However, an approach such as the one provided by Martins et al.~\cite{Martins:2016:CSE:2899032.2883614}, or any other approach (e.g.,~\cite{Martins:2016:CSE:2899032.2883614,Nobre:2014:ESD:2556863.2556870,Nobre:2016:GIC:2907950.2907959}) that can be used to accelerate iterative DSE can be used to make the initial iterative DSE process required to create the model correlating code-features with suitable compiler sequences faster.
Other authors have also performed a performance comparison between applications written in OpenCL and CUDA. For instance, Fang et al.~\cite{fang} compared benchmarks written in CUDA and OpenCL, and found that the performance differences can be attributed to 4 main factors: programming model differences (e.g., use of texture memory in only one of the versions), different optimizations on the kernels (e.g., one version takes care to coalesce global memory accesses and the other does not), architecture-related differences (e.g., benchmarks that have been tuned for one specific device may perform poorly on others) and compiler/run-time differences. In the benchmarks we tested, we made sure that the code versions are equivalent across the two languages (incl. manual source code optimizations), so the first 3 differences do not apply here. The performance differences can then be explained only by toolchain differences, which explains why our compiler improvements let us bridge the difference between OpenCL and CUDA.
Komatsu et al.~\cite{komatsu} also studied the performance differences between OpenCL and CUDA and found that the CUDA compiler performed significantly more optimizations than the OpenCL code, notably loop unrolling, and that by manually performing the same optimizations, the OpenCL implementation could perform competitively.
Note that in both of these cases, the authors tested the default (\emph{online}) OpenCL implementation, whereas our phase ordering approach is based on an offline compiler.
\subsection{Enumeration-based approaches}
To the best of our knowledge, our work is the first focusing on compiler sequence exploration including phase ordering when targeting GPUs.
However, phase selection and ordering has been focused on by many authors in the context of CPUs, and to a lesser extent FPGAs.
We present next enumeration-based and ML-based approaches for exploring phase orders.
Cooper et al.~\cite{Cooper:1999:ORC:314403.314414} were to the best of our knowledge the first to propose iterative compilation as a means to find phase orders to improve the quality of the compiled code with respect to a given metric.
They used iterative compilation in the form of a GA as a way to minimize the executable footprint.
Cooper et al.~\cite{Cooper2006} explore compiler optimization phase ordering testing different randomized search algorithms based on genetic algorithms, hill climbers and randomized sampling.
Almagor et al.~\cite{Almagor:2004:FEC:997163.997196} rely on genetic algorithms, hill climbers, and greedy constructive algorithms to explore compiler phase ordering at program-level to a simulated SPARC processor.
Huang et al.~\cite{huangfpga} propose insertion-based iterative approaches for compiler optimization phase ordering in the context of hardware compilation targeting an Altera Cyclone II FPGA, to improve circuit area, execution cycles, maximum operating clock frequency, and wall-clock time.
Purini and Jain~\cite{Purini:2013:FGO:2400682.2400715} propose an approach that relies on a list of compiler sequences previously found to be suitable, relying on iterative algorithms, for a reference set of benchmarks representative of all classes of programs.
Their approach circumvents program classification by relying on a small set of sequences which has the particularity of including an optimization sequence for each possible program class.
Given a new program, each of these compiler sequences is tested and the one leading to better binary execution performance after compilation of the new program is selected.
Nobre~\cite{nobrefpl2013} presents results for the use of a SA-based approach to specialize compiler sequences in the context of software and hardware compilation.
More recently, Nobre et al.~\cite{Nobre:2016:GIC:2907950.2907959} presented an approach based on sampling over a graph representing transitions between compiler passes, targeting the LEON3 microarchitecture.
\subsection{ML-based approaches}
Agakov et al.~\cite{agakov2006} present a methodology to reduce the number of evaluations of the program being compiled with iterative approaches.
Models are generated taking into account program features and the shapes of compiler sequence spaces generated from iteratively evaluating a reference set of programs.
These models are used to focus the iterative exploration for a new program, targeting the TI C6713 and AMD Au1500 embedded processors.
Kulkarni and Cavazos~\cite{Kulkarni:2012:MCO:2384616.2384628} proposed an approach that formulates the phase ordering challenge as a Markov process where the current state of a function being optimized conforms to the Markov property (i.e., the current state must have all the information to decide what to do next).
Instead of suggesting complete compiler sequences, these authors use a neural network to propose the next compiler pass based on current code features.
Sher et al.~\cite{Sher:2014:PRN:2568326.2568328} describe a compilation system that relies on evolutionary neural networks for phase ordering.
Neural networks constructed with reinforcement learning output a set of probabilities of use for each compiler pass, which is then sampled to generate compiler sequences based on the input program/function features.
Martins et al.~\cite{Martins:2016:CSE:2899032.2883614} propose the use of a clustering method to reduce the exploration space in the context of compiler pass phase order exploration.
Amir et al.~\cite{amir2016} present an for compiler phase ordering that relies on predictive modeling, using dynamic features to suggest the next compiler phase to execute to maximize execution performance given the current status.
\section{Conclusions}\label{section-conclusions}
This paper showed that compiler pass phase ordering specialization allows achieving considerable performance improvements when compiling OpenCL kernels to GPUs.
In our first set of experiments we explored the performance impact of specialized phase orders obtained by iterative compilation.
In our second set of experiments we explored the use of a feature-based approach to identify specialized phase orders.
Targeting an NVIDIA GP104 GPU using Clang, we were able to improve the performance of code compiled from PolyBench/GPU OpenCL kernels $1.65\times$ on average (up to $5.70\times$) over the default compilation flow.
The use of phase ordering on top of the OpenCL versions of the kernels resulted in a geometric mean speedup of $1.54\times$ (up to $5.48\times$) when compared with the performance of the equivalent CUDA kernels compiled with NVCC.
We gave insights explaining why using specialized phase orders tends to result in speedups. We found that due to phase ordering, the compiler was able to extract memory writes from loops by using an accumulator register, reducing the number of expensive global memory writes.
We presented results that give confidence that static features in OpenCL kernels can be used in the context of suggesting suitable compiler sequences more efficiently when targeting GPUs.
Evaluating sequences, previously found in an initial exploration phase to be more suitable for compiling OpenCL code from a reference set, ordered by similarity to the new unseen OpenCL code, results in higher optimization with the same number of compiler sequence evaluations on an NVIDIA GPU; compared both with randomly evaluating sequences from this set of sequences and with evaluating sequences generated by the IterGraph approach.
Using a leave-one-out validation approach, geometric mean performance improvements of $1.49\times$, $1.56\times$ and $1.59\times$, versus the same OpenCL baseline, were achieved while evaluating only $1$, $3$ and $5$ compiler sequences from this set, respectively. For comparison, our best phase orders found by our initial experiments lead to improvements of $1.65\times$.
We are currently evaluating how to better extend our compiler phase ordering exploration framework to allow exploration targeting the CUDA kernels instead of only OpenCL kernels.
We are also evaluating the potential of compiler phase ordering for GPU energy consumption reduction, and accessing how it correlates with execution performance, as GPUs are used in domains with energy (and power) concerns (e.g., HPC, embedded), so there may exist scenarios were it is acceptable to sacrifice performance for less total energy use.
As future work, we plan to analyze experimental data about phase orders that result in problematic or incorrect situations, to create a model to evaluate the likelihood that a new compiler sequence will be unsuitable and reduce the number of evaluations for these sequences.
\subsubsection*{Acknowledgments}
This work was partially supported by the TEC4Growth project, NORTE-01-0145-FEDER-000020, financed by the North Portugal Regional Operational Programme (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement, and through the European Regional Development Fund (ERDF). Reis acknowledges the support by Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia (FCT) through PD/BD/105804/2014.
In addition, we acknowledge Tiago Carvalho for his work on the LARA framework, used to develop the DSE tool used for this paper.
\bibliographystyle{splncs04}
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\section{Introduction}
In many (if not most) countries/regions, the prehospital resources available, e.g.~ambulances, are scarce, and this clearly affects the availability and general quality of the prehospital service in question. For example, a small decrease in the median response time, which could be the effect of the addition of an ambulance, could save a significant number of lives annually. However, simply adding additional ambulances to a given region might not necessarily yield the expected outcome, i.e.~drastically decreased response times, but instead the main challenge might be one of resource optimisation. More specifically, it may very well be the case that the existing resources are not utilised optimally; note that, given an expected performance outcome in terms of the response time distribution, one might have to both add and optimise prehospital resources. Moreover, in most/all cases, the addition of a new ambulance comes at a relatively high cost, so simply adding ambulances until the expected outcome has been achieved is essentially never a viable option. Such optimisation can be quite challenging and extensive; in 2018, in Sweden, approximately 660 ambulances responded to roughly 1.2 million {\em ambulance/emergency alarm calls}, i.e.~events defined by the deployment of an ambulance. In addition, the deployment of ambulances in 2018 cost more than 4 billion Swedish krona. \\
Carrying out resource optimisation within a prehospital care organisation/system, which to a large degree boils down to placing ambulance stations at locations where, e.g.~the median response time is minimised annually, requires knowledge/understanding of where (and when) calls tend to occur spatially (and temporally). In essence, minimising the response times of ambulances is crucial for obtaining the desired clinical outcomes following ambulance calls \citep{blackwell2002response,pell2001effect,o2011role}. The call occurrence risk can be influenced by various underlying factors, such as demographic factors. More specifically, within the spatial context, if one is able to exploit different covariates/predictors to model the expected number of calls occurring within a given region, within an arbitrary period of time, then one can obtain an understanding of which spatial population characteristics (e.g.~age distribution in a given subregion) that are driving the risk of an ambulance call occurring at a given spatial location. In addition, such a model may be extended to be exploited for predictive purposes, most notably within a spatio-temporal context. One can easily imagine that the distribution of calls in Sweden is both complex, dynamical, and heterogeneously distributed within the spatial study region. \\
This work is aimed at describing the spatial dynamics of ambulance calls in the municipality of Skellefte{\aa}, Sweden. The main focus of our study is to generate a risk-map for the ambulance calls, which is modelled by means of different spatial covariates, and to identify hotspot regions in Skellefte{\aa}, which can play a key role in designing optimal dispatching strategies for prehospital resources. It also focuses on identifying spatial covariates that tend to influence the occurrence of call events. Hence, we aim for a model with good predictive performance. \\
Note that, a priori, we do not know how many calls there will be within the spatial region studied and within the timeframe considered, which here is given by the years 2014--2018. In addition, we have access to the exact (GPS) locations of the events, as well as additional information, so called marks, attached to each event. Such datasets are commonly referred to as {\em (marked) point patterns}, and their natural modelling framework is that of {\em (marked) spatial point processes} \citep{baddeley2015spatial, moller2003statistical, diggle2013statistical,van2000markov}, which may be thought of as generalised random samples with the properties that the number of points/elements in the sample is allowed to be random and the points are allowed to be dependent. When the marks are discrete, one commonly speaks of a {\em multivariate/multitype} point process. Here, in our specific dataset, each event has two mark components attached to each spatial location: the priority/risk level (1 or 2) of the call and the gender (male or female) of the patient associated with the call.\\
The properties of a univariate point process are most commonly characterised through its intensity function, which essentially reflects the probability of the point process having a point at a given location in the study region. Formally, it is defined as the density function of the first-order expectation measure over the spatial domain. In most applications, it is unrealistic to assume that the underlying point process is homogeneous, i.e.~the intensity function is constant, and there are various approaches available to deal with the spatial inhomogeneity of the points/events \citep{baddeley2015spatial,diggle2013statistical}. Quite often, one has access to a collection of spatial covariates, which may be used to model the spatially varying intensity function. In particular, it is commonly assumed that the intensity has a log-linear form, i.e.~the log-intensity is a linear combination of the covariates. Here, the common practice is to first model the intensity function using a Poisson process log-likelihood function, which is a closed-form function of only the (assumed) intensity function. In the case of a general point process, this is commonly referred to as a composite likelihood estimation and, although a Poisson process is a point process with independent points/events and Poisson distributed counts within any subregion, the indicated intensity estimation approach still has good large sample properties for general point processes \citep{coeurjolly2019understanding}. Once an intensity function estimate is obtained, one would then proceed with analysing and modelling possible spatial interaction, i.e.~dependence between the events \citep{baddeley2000non, baddeley2015spatial, van2011aj, cronie2016summary}. It should here be emphasised that observed clumps of points in a point pattern may be the result of either inhomogeneity, spatial interaction (clustering/aggregation/attraction) or both. Since our main interest here is generating a spatial risk map for the observed events, i.e.~the ambulance calls, which can be exploited for predictive purposes and also used as input in the problem of designing an optimal dispatching strategy of ambulances, we will solely focus on the former, i.e. modelling the intensity function as a log-linear function of a collection of spatial covariates. This allows us to address one of the main objectives of this work, which is to identify hotspot regions of the events. A further main objective of this work is to select covariates governing the intensity function. Here the list of covariates includes population density, shortest distance to road networks, line density of road networks, line density of densely populated regions, bench mark estimated intensity, proportion of population by age category, proportion of population by gender, proportion of population (aged 20 + years) by income status, proportion of household (aged 20 + years) by economic standard, proportion of population (aged 25-64 years) by education level, proportion of population by Swedish or foreign background, and proportion of population (aged 20-64 years) by employment status. Aside from including the individual covariates in our model framework, we also include the cross-terms given by the products of the individual covariates, since there may potentially be interactive effects of the covariates. This results in a high-dimensional data setting (a total of nine hundred eighty-nine covariates), with possibly hard to interpret cross-terms, and it is clearly a challenge to identify which covariates sensibly explain the actual intensity of the events, i.e.~which covariates should actually be included in the final model. As a solution, one may apply regularisation when fitting the model, which, aside from carrying out variable selection, also reduces variance inflation from overfitting and bias from underfitting. In addition, we want to adjust for the fact that the demographic covariates we have access to only reflect where different demographic groups live, and not how they move around. Since we do not have access to any human mobility covariates, we have to be pragmatic and use some proxy for such covariates. Our solution is to additionally include a non-parametric intensity estimate of the spatial locations of the events as an additional covariate. In other words, these added covariates would (to some degree) represent the portion of the intensity function that the original spatial covariates could not explain. Here we have another benefit of the regularisation: having accounted for the other covariates, if this added covariate has little/no influence on the intensity function, then the regularisation would indicate this. \\
Since many of the spatial covariates which we deal with are most likely strongly correlated, variable selection in the modelling of the spatially varying intensity function is most likely necessary to overcome the use of correlated covariates in the modelling of ambulance call events. \citet{tibshirani1996regression} introduced a penalised likelihood procedure, which has been a cornerstone in the development of variable selection methods via regularisation (or penalisation). The idea of \citet{tibshirani1996regression} is to add a least absolute shrinkage and selection operator (lasso) penalty to the loss function, often the likelihood function, to shrink small coefficients of covariates to zero while retaining covariates with large coefficients in the model, thus leaving us with a sparse model with highly influential covariates. Hence, the approach simultaneously performs variable selection and parameter estimation. A plethora of regularisation methods, such as elastic-net \citep{zou2005regularization} and adaptive lasso \citep{zou2006adaptive}, have been developed subsequent to the work of \citet{tibshirani1996regression}. A predecessor of the lasso penalty, the ridge penalty, which is an $\ell_2$-penalty, may be combined linearly with the lasso $\ell_1$-penalty to obtain elastic-net regularisation, which is used to select variables and shrinks the coefficients of correlated variables to each other. Moreover, in adaptive lasso regularisation, adaptive weights are used in the penalisation of the coefficients of the variables. It should be emphasised that these shrinkage/regularisation methods have the effect of balancing estimation bias and variance, which is an additional motivation for their employment. Turning to the point process context, also here variable selection has been developed to reduce variance inflation from overfitting and bias from underfitting. Considering Poisson process likelihood estimation together with a lasso penalty, \citet{renner2013equivalence} has introduced a maximum entropy approach for modelling the spatial distribution of a specie. \citet{thurman2014variable} considered an adaptive lasso penalty to select variables in Poisson point process modelling. Regularisation methods such as lasso, adaptive lasso, and elastic-net have also been considered in the context of a wide range of inhomogeneous spatial point processes models by \citet{yue2015variable}. \citet{choiruddin2018convex} further extended the above works to include a larger range of models and penalties. We here essentially follow the track of \citet{yue2015variable}. \\
With regard to the estimation of the regularised models, \citet{efron2004least} have developed the least angle regression to estimate the entire lasso regularisation paths. According to \cite{friedman2007pathwise} and \cite{friedman2010regularization}, in comparison with the least angle regression algorithm, the cyclical coordinate descent algorithm computes the regularisation paths of different regularisation methods with lower computational costs. In this study, the cyclical coordinate descent method has been used to estimate the entire regularisation paths since it is computationally fast on large datasets \citep{fercoq2016optimization}. The general idea of the coordinate descent algorithm is that the objective function is optimised with respect to a single parameter at a time, and the optimisation of the objective function is iteratively carried out for all parameters until a convergence criterion is fulfilled. In this work, the objective function represents a regularised quadratic approximation to the log-likelihood function of an inhomogeneous spatial Poisson process. \\
Two approaches have been used to evaluate the performance of the proposed model. The first approach involves training the proposed model on the whole dataset. We treat the trained (or estimated) intensity function on the whole dataset as the true intensity function of the call events. Based on the estimated intensity function, undersampling i.e. 70\% of the whole dataset, has been used to generate a hundred datasets, which are then used to generate a hundred intensity images. Using quantiles and mean absolute errors between pixel-by-pixel differences of the estimated image (i.e. image based on the whole dataset) and the hundred estimated intensity images (i.e. the intensity images based on the hundred undersampled datasets) can be used to evaluate the stability of the proposed model in estimating the intensity function of the emergency alarm call events. The second approach is to visually evaluate the performance of the proposed model. We train the proposed model on 70\% and on the remaining 30\% of the whole dataset. Then, we compare the patterns of hotspot regions in the estimated intensity images obtained from the two datasets. If the patterns of hotspot regions and the spatial locations in the respective datasets seemingly coincide with each other, then the proposed model is more likely applicable for the modelling of the call events. \\
In summary, the aim of this article is to explore the space-dynamics of ambulance calls and to identify spatial covariates associated with the call events. The result of this work will be used as input in designing optimal dispatching strategies for prehospital resources such as ambulances. \\
The structure of the article is as follows. Section \ref{data} and \ref{Statisticalmethod} provide an overview of the data and statistical methods used in this work. Section \ref{s:CreatedCovariates} and \ref{Results} present the created spatial features for modelling emergency alarm call events and the results of the study. We evaluate the fitted model in section \ref{Evaluation} and discuss the implication of the results and provide a precise summary of the work in section \ref{Discn}.
\section{Data}\label{data}
Given a spatial/geographical region $W$, which we assume to be a subset of $\mathbb R^2$, by an {\em event} we will mean a location (GPS position) in $W$ to which an ambulance is dispatched during a given time period and the total collection of events will be referred to as a set of {\em ambulance/emergency call dataset}. Here, the available information associated to an ambulance call dataset is the collection of locations $\{x_1,\ldots,x_n\}\subseteq W$, $n\geq0$, as well a {\em mark} $m_i$ which is attached to each location $x_i$, $i=1,\ldots,n$. As we a priori do not know $n$, i.e.~how many calls there will be within $W$ during the period in question, such a dataset $\mathbf y=\{(x_1,m_1),\ldots,(x_n,m_n)\}$ is most naturally classified as a {\em marked point pattern} \citep{baddeley2015spatial}. Note that when the marks are discrete, we usually refer to the point pattern as {\em multivariate/multitype} rather than as {\em marked}.
\subsection{Spatial ambulance call dataset}
Turning to the dataset at hand, $W$ will represent the Swedish municipality of Skellefte{\aa}, and the time period under consideration is given by the years 2014--2018. Our dataset consists of 14,919 events where the mark structure is such that for each event, and thereby the person associated with the ambulance call in question, there is a recording of the event's priority label (1 indicates the highest severity, implying turned on sirens, and 2 is a lower severity level) and a recording of the sex/gender of the person (female or male). Among the 14,919 events, 7,204 and 7,715 of them have priority labels 1 and 2, respectively. Unfortunately, a missing data issue is present here: 5,236 and 5,238 of the events are recorded as male and female, respectively, and the remaining events do not have any sex recording for the person related to the event, so they are labelled as "missing". Since our interest in the sex label lies mostly in highlighting possible structural differences related to the different covariates at hand, we have decided to proceed by studying the data as two separate marked point patterns: one where the marks are given by the priority labels and one where the marks are given by the sex labels. It should be acknowledged that there are other ways of dealing with this issue. \\
Figure \ref{Or213} demonstrates the locations of all calls as well as the main road network of Skellefte{\aa}, Sweden. The figure highlights that the call locations are unevenly distributed over the study region and they tend to lie along the road network, which will make the statistical modelling quite challenging. For data privacy reasons, the axis labels of the figure have been scaled (i.e.~divided) by a factor of 1000.
\begin{figure}[!htpb]
\centering
\includegraphics[width=0.4\textwidth, height=0.35\linewidth]{Dataplot}
\caption{Ambulance emergency alarm call locations and the main road network of Skellefte{\aa}.}
\label{Or213}
\end{figure}
\subsection{Spatial covariates}
In order to properly model the ambulance call risk, we also need a range of spatial covariates. To begin with, a closer look at Figure \ref{Or213} and the relation between the road network and the call locations justifies the inclusion of road network related covariates. In addition, demographic spatial covariates should also play a role here, given that different demographic zones have different behavioural patterns. The demographic spatial covariates considered have been supplied by Statistics Sweden (SCB), and the road network related covariates considered have been provided by the Swedish Transport Administration (Trafikverket). \\
Here, we will distinguish between two categories of covariates. The first is the collection of 'original' covariates, which are raw covariates retrieved from SOS alarm, Statistics Sweden (SCB), and Trafikverket. These covariates will, in turn, be used to generate a collection of new covariates, which we will refer to as 'created' covariates. In Section \ref{s:CreatedCovariates}, we outline the construction of the created covariates in detail.\\
All covariates considered have been based on the year 2018 and in their final forms they are given as functions of the form $z_i(s)$, $s\in W$. Moreover, in addition to including each individual covariate $z_i$ in the analysis, we also include each interaction term $z_{ij}(s)=z_i(s)z_j(s)$, $s\in W$, which makes sense because we suspect that many of the covariates interact in one way or another. We have a total of 43 individual spatial covariates and, all in all, we consider a total of 989 $(={43 \choose 2} + 2\times 43)$ covariates as candidates to be included in the modelling of the ambulance call risk.
\subsubsection{Demographic spatial covariates}
Each demographic spatial covariate $z_i(s),$ $s\in W,$ which has been sampled on the 31st of December 2018, is piecewise constant and its value changes depending on which DeSO zone the location $s$ belongs to. The DeSO zones, which we use to define different demographic zones, partition Sweden into 5984 smaller spatial sub-regions, which do not overlap with the borders of any of the country's 290 municipalities and are encoded based on e.g.~how rural a DeSO zone is. Below follows a short description of these covariates, where some have been graphically illustrated in Figure \ref{Covariates}:
\begin{itemize}
\item {\bf Population density:} Per DeSO, this gives us the ratio of the total population size of the DeSO to the area of the DeSO.
\item {\bf Population by age (counts):} The number of individuals of a given age category living in a given DeSO zone. There are a total of 17 such covariates, reflecting the following age categories (ages in years): 0-5, 6-9, 10-15, 16-19, 20-24, 25-29, 30-34, 35-39, 40-44, 45-49, 50-54, 55-59, 60-64, 65-69, 70-74, 75-79 and 80+.
\item {\bf Population by sex (counts):} The number of individuals of a given sex classification living in a given DeSO zone. There are 2 sex related covariates: women and men.
\item {\bf Population by Swedish/non-Swedish background (counts):} The number of individuals of a given Swedish/non-Swedish classification living in a given DeSO zone. There are 2 such covariates: Swedish background and non-Swedish background (the latter includes people born outside Sweden as well as people born in Sweden with both parents born outside Sweden).
\item {\bf Population 20-64 years of age by occupation (counts):} The number of individuals of a given occupational status living in a given DeSO zone. There are a total of 2 such covariates: (gainfully) employed and unemployed.
\item {\bf Population 25-64 years of age by education level (counts):} The number of individuals of a given maximal education level category living in a given DeSO zone. There are a total of 4 such covariates: no secondary education, secondary education, post-secondary education of at most three years, and post-secondary education of more than three years (including doctoral degrees).
\item {\bf Population 20+ years of age by accumulated income (counts):} The number of individuals of a given income level category living in a given DeSO zone. There are a total of 2 such covariates: below the national median income and above the national median income.
\item {\bf Population 20+ years of age by economical standard (counts):} The number of individuals of a given economical standard category living in a given DeSO zone. There are a total of 2 such covariates: below the national median economical standard and above the national median economical standard.
\end{itemize}
\subsubsection{Road network related covariates}
The road network data at hand have been broken down into sub-networks, which reflect the complete road network, the main road network, and the densely populated areas; a graphical illustration of these can be found in Figure \ref{BandWidthLineDensity}. Each network is included in the analysis with the aim of indicating a different level of call activity: The complete road network indicates the overall spatial region where calls tend to occur, the main road network indicates the parts of the complete network where there is a reasonable amount of activity, and the densely populated area indicates on which parts of the complete road network most people live. \\
We would further like to adjust for the fact that the covariates above mainly reflect where different people (of different demographic groups) live, and not how they move about. Unfortunately, we do not have access to explicit mobility covariates such as aggregated movement patterns in the population, and, as one may guess, people do not only need access to ambulances when they are at home. A partial solution to this, as we see it, is to consider an additional covariate, namely a spatial point pattern of bus stop locations. The idea is that bus stops reflect where there is a large amount of human mobility/activity. \\
It may be noted that the above original covariates are not functions of the form $z_i(s)$, $s\in W$, but rather line segment patterns and point patterns \citep{baddeley2015spatial}. Our approach to including them in the analysis is to let them give rise to a set of created covariates, which are described in detail in Section \ref{s:LineDensity}.
\subsubsection{The spatial domain}
We finally emphasis that the $x$- and $y$-coordinates of the spatial domain $W\subseteq\mathbb R^2$ will be included as covariates. These are intended to explain residual and explicit spatial variation, having adjusted for the full range of spatial covariates.
\section{Statistical methods}\label{Statisticalmethod}
As emphasised in Section \ref{data}, in particular through Figure \ref{Or213}, the ambulance calls are mainly located on/close to the underlying road network. One way to deal with the analysis is to project all events to a linear network representation of the road network and proceed with a linear network point pattern analysis \citep{baddeley2015spatial}. However, we here avoid such projections and instead choose to treat the spatial domain, and thereby the data point pattern, as Euclidean. As a result, we will instead introduce road network generated covariates which control that the fitted model generates events close to the underlying road network; details on the construction of such covariates can be found in Section \ref{s:LineDensity}.
\subsection{Point process preliminaries}\label{Ppp}
In application areas such as environmental science, epidemiology, ecology, etc., aside from the spatial locations of the events, additional information about the events may be available, which can be associated to the locations of the events. Such information pieces are referred to as marks, and by including them in the analysis, we can often obtain more realistic spatial point process models for the events -- note that, in contrast, a covariate reflects information which is known throughout the spatial domain before realisation of the events. For instance, as we saw above, it is both common and practically appropriate in emergency medical services to document the priority levels, the gender, the incident time, etc.~for a call/patient; recall that by spatial points of the events we mean the set of spatial locations of the patients to which ambulances have been dispatched or the set of spatial locations of patients from which calls have been made to the dispatcher/emergency alarm center. \\
As previously indicated, marked point patterns are modelled by {\em marked point processes} \citep{baddeley2015spatial, moller2003statistical, diggle2013statistical,van2000markov}. Given a spatial domain $W\subseteq\mathbb R^2$ and a mark space\footnote{Formally, $\mathcal{M}$ is assumed to be a Polish space.} $\mathcal{M}$, a point process:
$$
Y=\{(x_i,m_i)\}_{i=1}^{N}\subseteq W\times \mathcal{M},
\qquad
0\leq N \leq\infty,
$$
is a random subset such that, with probability 1, $Y\cap A\times B$ has finite cardinality $Y(A\times B)$ for any\footnote{Throughout, any set under consideration is a Borel set.} $A\times B\subseteq W\times \mathcal{M}$; this is referred to as local finiteness. If we additionally have that $Y_g=\{x_i\}_{i=1}^{N}\subseteq W$ is a well-defined point process on $W$ (locally finite) in its own right, then we say that $Y$ is a marked point process. Note that each $(x_i,m_i)$ is a random variable and note in particular that if $W$ is bounded then $Y$ is automatically a marked point process.
When the mark space is discrete, e.g.~$\mathcal{M}= \left\lbrace j: j = 1, 2, \ldots,k,\; k>1\right\rbrace $, we say that $Y$ is multitype and we note that we may split $Y$ into the marginal (purely spatial) point processes:
$$
Y_j=\{x_i:(x_i,m_i)\in Y, m_i=j\},
\qquad
j=1, 2, \ldots,k.
$$
This collection may formally be represented by the vector $(Y_1, Y_2, \ldots,Y_k)$, which is referred to be a {\em multivariate} point process, and one commonly uses the two notions interchangeably.
\subsection{Spatial intensity functions}
Our main interest here is to create a set of "heat maps" which describe the risk of a call occurring at a given spatial location $x\in W$, given an associated mark. This is accomplished by assuming that our data are generated by a multivariate point process $Y\subseteq W\times\{1, 2, \ldots,k\}$, $k>1$ and then modelling the {\em spatial intensity function} of each of the component $Y_j, j = 1, 2, \ldots, k.$ Formally, the spatial intensity function of $Y$ is defined as the function $\rho_Y$ satisfying\footnote{We assume that $\rho$ is the Radon-Nikodym derivative of the Borel measure $(A\times B)\mapsto E[Y(A\times B)]$ w.r.t.~the product measure given by the product of a Lebesgue measure and a counting measure.}
$$
E[Y(A\times B)]= \sum_{j=1}^k \mathbf 1\{j\in B\}\int_{A} \rho_Y(x,j)dx,
\quad
A\times B\subseteq W\times \{1, 2, \ldots,k\},
$$
where $\mathbf 1\{\cdot\}$ denotes the indicator function. Heuristically, $\rho_Y(x,j)dx$ may be interpreted as the probability that $Y$ has an event with mark $j$ in an infinitesimal neighbourhood of $x$ with size $dx$. Since $E[Y(A\times\{j\})]=E[Y_j(A)]$, it follows that the spatial intensity function of $Y_j$ satisfies
$$
\rho_j(x)=\rho_Y(x,j),
\quad
x\in W, j=1, 2, \ldots,k.
$$
Hence, by modelling the marginal processes $\rho_j$ separately, we obtain a model for $\rho_Y$. Letting $X$ denote an arbitrary $Y_j$, $j=1, 2, \ldots,k$, we see that its spatial intensity function $\rho$ satisfies
$$
E[X(A)] = \int_{A} \rho(x)dx,
\quad
A\subseteq W,
$$
or equivalently,
$
\rho(x) = \lim_{\abs{dx}\to 0} E\left[X\left(dx\right)\right]/\abs{dx}.
$
If a point process has constant intensity function then we say that it is homogeneous, otherwise we refer to it as inhomogeneous. In a similar fashion, we may define higher-order intensity functions $\rho_n(x_1, x_2, \ldots,x_n)$, $x_1, x_2\ldots,x_n\in W$, $n\geq2$ as $\rho_n(x_1, x_2, \ldots,x_n)=E[X(dx_1)X(dx_2)\cdots X(dx_n)]=P(X(dx_1)=1, X(dx_2)=1, \ldots,X(dx_n)=1)$ for disjoint infinitesimal neighbourhoods $dx_1, dx_2, \ldots,dx_n\subseteq W$ of $x_1, x_1, \ldots,x_n$. If $x_i=x_j$ for any $i\neq j$, then $\rho_n(x_1, x_2, \ldots,x_n)=0$.
\subsection{Poisson processes}
Poisson processes, for which $\rho_n(x_1, x_2, \ldots,x_n)=\rho(x_1)\rho(x_2)\cdots\rho(x_n)$, $n\geq1$ and $X(A)$ is Poisson distributed with mean $\int_A\rho(x)dx$ for any $A\subseteq W$, are used as baseline models for the case of complete spatial randomness, i.e.,~ the case where there is no spatial interaction/dependence present. Poisson processes are completely governed by their spatially varying intensity functions and may be viewed as generalisations of random samples to the case where the size of the sample is random.
\subsection{Parametric spatial intensity function modelling}
We have observed a high degree of inhomogeneity in the ambulance call locations and we believe that a large portion of this inhomogeneity can be attributed to (some of) the spatial covariates which we have access to. Hence, as a starting point, for a given mark, we will consider parametric modelling of the call locations.\\
Consider a family of spatial intensity functions $\rho_{\boldsymbol\theta}(x)$, $x\in W$, which depends on spatial covariates through a parameter vector $\boldsymbol\theta\in\mathbb{R}^{K+1}$. A common and convenient approach when modelling the intensity function of an arbitrary point process is to proceed as if we are considering a Poisson process, which is commonly referred to as composite likelihood estimation and is often motivated by good large sample properties \citep{coeurjolly2019understanding}. Accordingly, suppose that a point pattern $\mathbf x = \left\lbrace x_{1}, x_{2}, \cdots, x_{n} \right\rbrace \subseteq W$ represents a realisation of a spatial point process $X$ which is observed within a bounded study region $W$. If $X$ is an inhomogeneous Poisson point process, then the associated log-likelihood function is given by
\begin{eqnarray}
\label{ogeetti3}
\log \mathcal{L}\left(\boldsymbol\theta\mid \mathbf{x} \right) = \sum_{i=1}^{n}\log\rho_{\boldsymbol\theta}\left(x_{i}\right) - \int_{W}\rho_{\boldsymbol\theta}\left(x\right)dx .
\end{eqnarray}
Using any quadrature rule, the integral in \eqref{ogeetti3} can be approximated by a finite sum
\begin{eqnarray}\label{Ogeettikoo}
\int_{W}\rho_{\boldsymbol\theta}\left(x\right)dx
\approx
\sum_{j=1}^{m} \rho_{\boldsymbol\theta}(s_{j})w_{j},
\end{eqnarray}
where the positive numbers $w_{j}$, $j = 1, 2, \ldots, m$, are quadrature weights summing to the area $\abs{W}$ and $s_{j}\in W$, $j = 1, 2, \ldots, m,$ are quadrature points. Following this approximation, the approximated log-likelihood function may be expressed as
\begin{eqnarray}\label{logl}
\log \mathcal{L}\left(\boldsymbol\theta\mid \mathbf{x} \right)
\approx
\sum_{i=1}^{n}\log\rho_{\boldsymbol\theta}\left(x_{i}\right) - \sum_{j=1}^{m} \rho_{\boldsymbol\theta}(s_{j})w_{j},
\end{eqnarray}
where $\textbf{s} = \{ s_1,\ldots, s_m\} \subseteq W$ represents the union of the observed spatial locations of events $\mathbf{x} = \{ x_1,\ldots, x_n\}$ and the set of dummy points $\textbf{s}\setminus\mathbf{x} = \{v_1,\ldots,v_q\}$, $q = m-n$. Here we assume that $m$ is much larger than $n$ for a better approximation of the log-likelihood function $\log \mathcal{L}\left(\boldsymbol\theta\mid \mathbf{x} \right)$. The approximated log-likelihood function in expression \eqref{logl} can then be rewritten as follows:
\begin{eqnarray}
\label{Ogeettikoo2}
\log \mathcal{L}\left(\boldsymbol\theta\mid \mathbf{x} \right)
\approx
\sum_{j=1}^{m}\left( y_{j}\log\rho_{\boldsymbol\theta}(s_{j}) - \rho_{\boldsymbol\theta}(s_{j})\right) w_{j},
\end{eqnarray}
where
\begin{eqnarray}
\label{indicator}
y_{j} = \frac{a_{j}}{ w_{j}}
\quad
\text{and}
\quad
a_{j} = \mathbf 1\{s_{j}\in \mathbf{x}\}.
\end{eqnarray}
Note that $a_j=0$ means that $s_{j}$ is a dummy point. Exploiting the approximation in equation \eqref{Ogeettikoo}, a large number of dummy points are required to obtain accurate parameter estimation using equation \eqref{Ogeettikoo2}. \cite{waagepetersen2008estimating} proposed two ways of obtaining dummy points and quadrature weights. With regard to the quadrature weights, the first method is a grid approach in which the observation window $W$ is partitioned into a collection of rectangular tiles. The quadrature weight for a quadrature point $s\in \textbf{s}$ falling in a tile $R$ is the area of $R$ divided by the total number of quadrature points falling in $R$. This approach is advantageous since the computation of the quadrature weights is easy. The second approach is the Dirichlet approach \citep{okabe2009spatial} in which the quadrature weights are the areas of the tiles of the Dirichlet/Voronoi tessellation generated by the quadrature points in $\textbf{s}$. With regard to the dummy points, \cite{waagepetersen2008estimating} proposed two ways of generating dummy points. The first approach is to use stratified dummy points combined with grid-type weights while the second approach is to exploit binomial dummy points with the Dirichlet-type weights. \\
According to \cite{baddeley2000practical} and \cite{thurman2015regularized}, a computationally cheaper approach to generate dummy points and compute quadrature weights is to partition the study region $W$ into tiles $R$ of equal area. To generate dummy points, we place one dummy point exactly in each tile either systematically or randomly. It follows that the quadrature weights for quadrature points $s_{j}$ can be set to $w_{j} = \Delta/E_{j}$, where $\Delta$ is the area of each tile and $E_{j}$ is the number of events and dummy points in the same tile as point $s_{j}$. \\
Modelling spatial intensity functions parametrically, in particular modelling based on spatial covariates, we often assume that $\rho_{\boldsymbol\theta}$ has a log-linear form. More specifically, letting $\boldsymbol\beta = \left(\beta_{1}, \beta_{2}, \ldots, \beta_{K}\right)$ and $\boldsymbol{\theta} = \left(\beta_{0}, \boldsymbol\beta\right)$, we assume that
\begin{eqnarray}\label{candidate}
\rho_{\boldsymbol\theta}(x) = \exp\left\lbrace \beta_{0} + \mathbf{z}(x)\boldsymbol\beta'\right\rbrace,
\end{eqnarray}
where $\mathbf{z}(x)= (z_{1}(x), z_{2}(x), \cdots, z_{K}(x))$ is a vector of spatial covariates at location $x\in W$. Combining equations \eqref{Ogeettikoo2} and \eqref{candidate}, we thus find that the log-likelihood function $\log \mathcal{L}\left(\boldsymbol\theta\mid \mathbf{x}\right)$ can be approximated by
\begin{eqnarray}
\label{werqqq}
\log \mathcal{L}\left(\boldsymbol\theta\mid \mathbf{x}\right) \approx \sum_{j=1}^{m}\left( y_{j}\left(\beta_{0} + \mathbf{z}(s_{j})\boldsymbol\beta'\right) - \exp\left\lbrace \beta_{0} + \mathbf{z}(s_{j})\boldsymbol\beta'\right\rbrace\right) w_{j}.
\end{eqnarray}
It follows that the expression on the right-hand side of the approximation sign in equation \eqref{werqqq} is a weighted log-likelihood function of independent Poisson random variables $Y_{j}, j = 1, 2, \cdots, m$. That is, for $j = 1, 2, \cdots, m$, $y_{j}$ are the observations, $\rho_{\boldsymbol\theta}(s_{j}) = \exp\{\beta_{0} + \mathbf{z}(s_{j})\boldsymbol\beta'\}$ are the intensities of the Poisson distributions, and $w_{j}$ are the weights.
Thus, the weighted log-likelihood function in equation \eqref{werqqq} can be maximised using standard software for fitting generalised linear models \citep{McCullagh1989generalised}.
\subsection{Variable selection: Elastic-net regularisation}\label{s:Regularisation}
Incorporating regularisation into the log-likelihood function in equation \eqref{werqqq} can help to simultaneously select variables and estimate the parameters of the model. A penalised log-likelihood function based on equation \eqref{werqqq} may be given by
\begin{eqnarray}
\label{WaaqayyoItalyEegi}
\mathcal{L}_{p}\left(\boldsymbol\theta\right)
\approx
\frac{1}{m}\log \mathcal{L}\left(\boldsymbol\theta\mid \mathbf{x}\right) + \lambda R(\boldsymbol\beta),
\end{eqnarray}
where $R(\boldsymbol\beta)$ is a regularisation method or penalty function, and $\lambda\ge 0$ is a tuning or smoothing parameter determining the strength of the penalty, or the amount of shrinkage. Several penalties such as garrote \citep{breiman1995better}, least absolute shrinkage and selection operator (lasso), elastic-net and fused lasso \citep{tibshirani2005sparsity}, group lasso \citep{yuan2006model}, Berhu penalty \citep{owen2007robust}, adaptive lasso \citep{zou2006adaptive}, and LAD-lasso \citep{wang2007robust} have been developed for penalised regression modelling. The most common regularisation methods are lasso, ridge regression, elastic-net, and adaptive lasso. In short, lasso has a tendency to shrink several coefficients to zero, leaving only the most influential ones in the model, while ridge regression shrinks the coefficients of correlated covariates towards each other to borrow strength from each other \citep{friedman2010regularization}. The elastic-net penalty provides a mix between the ridge and the lasso penalty, and it is useful in cases where there are many correlated covariates or when the number of covariates exceeds the size of observations. The elastic-net regularisation penalty has the form
\begin{align}\label{Regu}
R(\boldsymbol\beta) = \sum_{k=1}^{K}\left\lbrace \frac{1}{2}\left(1-\alpha\right) \beta^{2}_{k} + \alpha\abs{\beta_{k}}\right\rbrace,
\end{align}
where the elastic-net parameter $\alpha \in [0, 1]$ turns \eqref{Regu} into a ridge penalty if $\alpha = 0$ and a lasso penalty if $\alpha = 1$. If $\alpha = 1-\epsilon$ for small $\epsilon>0$, then elastic-net performs like lasso but it avoids unstable behaviour due to extreme correlation \citep{yue2015variable};
empirical studies have indicated that elastic-net technique tends to outperform lasso on data with highly correlated features \citep{comber2018geographically, friedman2010regularization}.
\cite{zou2006adaptive} proposed an adaptive lasso to address the shortcomings of lasso, such as biased estimates of large coefficients and conflict between optimal prediction and consistent variable selection. According to \cite{kramer2009regularized}, however, the performance of adaptive lasso is poor in the presence of highly correlated variables.
Hence, our way forward here is to consider elastic-net penalisation and substituting equation \eqref{Regu} into equation \eqref{WaaqayyoItalyEegi}, the optimization problem of the elastic-net penalization of the log-likelihood function can be summarized as
\begin{align}\label{waaqayyoo234}
\operatorname*{argmin}_{\boldsymbol\theta\in \mathbb{R}^{K+1} }
\mathcal{L}_{p}\left(\boldsymbol\theta\right)
\approx
\operatorname*{argmin}_{\boldsymbol\theta\in \mathbb{R}^{K+1} }
\left\lbrace
- \frac{1}{m}\log\mathcal{L}\left(\boldsymbol\theta\right) + \lambda \sum_{k=1}^{K}\left\lbrace \frac{1}{2}\left(1-\alpha\right) \beta^{2}_{k} + \alpha\abs{\beta_{k}}\right\rbrace\right\rbrace.
\end{align}
\subsubsection{Optimisation methods}\label{Optimisation}
To deal with the optimisation problem in \eqref{waaqayyoo234}, we carry out the optimisation using a cyclical coordinate descent method, which optimises a target function/optimisation problem with respect to a single parameter at a time and iteratively cycles through all parameters until a convergence criterion is reached. Here, we present the coordinate descent algorithm for solving the regularized log-likelihood function with the elastic-net penalty. \\
Let $f\left(\boldsymbol\theta\right)$ be the approximated log-likelihood function \eqref{werqqq}, i.e.,
\begin{eqnarray*}
f(\boldsymbol\theta)
=
f(\beta_{0}, \boldsymbol\beta)
= \sum_{j=1}^{m} w_{j}\left(y_{j}\left(\beta_{0} + \mathbf{z}(s_{j})\boldsymbol\beta'\right) - \exp\left\lbrace \beta_{0} + \mathbf{z}(s_{j})\boldsymbol\beta'\right\rbrace\right).
\end{eqnarray*}
Let $r$ denote the step number in the optimisation algorithm, and $\boldsymbol\theta^{(r-1)} = (\beta_{0}^{(r-1)}, \boldsymbol\beta^{(r-1)})$ represent the current estimates of the parameters.
A quadratic approximation of $f\left(\boldsymbol\theta\right)$ at the point $\boldsymbol\theta^{(r-1)}$ is given by
\begin{eqnarray}\label{werqqqqweq}
f(\boldsymbol\theta)
\approx
f\left(\boldsymbol\theta^{(r-1)}\right) + \frac{d f\left(\boldsymbol\theta^{(r-1)}\right) }{d\boldsymbol\theta}\left( \boldsymbol\theta - \boldsymbol\theta^{(r-1)}\right)' + \frac{1}{2}\left( \boldsymbol\theta - \boldsymbol\theta^{(r-1)}\right)\frac{d^{2}f\left(\boldsymbol\theta^{(r-1)}\right) }{d\boldsymbol\theta d\boldsymbol\theta'}\left( \boldsymbol\theta - \boldsymbol\theta^{(r-1)}\right)',
\end{eqnarray}
where the first and the second-order derivatives of the function $f$ with respect to $\boldsymbol\theta$ are given by
\begin{eqnarray}
\frac{d f(\boldsymbol\theta) }{d\boldsymbol\theta}
&=&
\sum_{j=1}^{m} w_{j}\left( y_{j}\mathbf{\bar{z}}(s_{j}) - \exp\left\lbrace \beta_{0} + \mathbf{z}(s_{j})\boldsymbol\beta'\right\rbrace\mathbf{\bar{z}}(s_{j})\right),
\label{werqqq2341}
\\
\frac{d^{2} f(\boldsymbol\theta) }{d\boldsymbol\theta d\boldsymbol\theta'}
&=&
- \sum_{j=1}^{m} w_{j} \exp\left\lbrace \beta_{0} + \mathbf{z}(s_{j})\boldsymbol\beta'\right\rbrace\mathbf{\bar{z}}(s_{j})'\mathbf{\bar{z}}(s_{j}),
\label{werqqq2342}
\end{eqnarray}
and $\mathbf{\bar{z}}(s_{j}) = (1, \mathbf{z}(s_{j}))$.
Hence, a quadratic approximation of the approximated log-likelihood function in \eqref{werqqq} can be obtained through equations \eqref{werqqq2341}, \eqref{werqqq2342}, and \eqref{werqqqqweq}, i.e.,
\begin{eqnarray}
\label{addafacaane2}
\mathcal{L}_{q}\left(\boldsymbol\theta\right) = -\frac{1}{2} \sum_{j=1}^{m}u_{j} \left(y^{*}_{j} - \beta_{0} - \mathbf{z}(s_{j})\boldsymbol\beta'\right)^{2} + C\left(\boldsymbol\theta^{(r-1)}\right),
\end{eqnarray}
where $C(\boldsymbol\theta^{(r-1)})$ is a constant function of $\boldsymbol\theta$ and the remaining variables in equation \eqref{addafacaane2} are given by
\begin{eqnarray}\label{addafacaane3452}
y^{*}_{j} = \beta^{(r-1)}_{0} + \mathbf{z}(s_{j}){\boldsymbol\beta^{(r-1)}}'+ \frac{y_{j} }{\exp\left\lbrace \beta^{(r-1)}_{0} + \mathbf{z}(s_{j}){\boldsymbol\beta^{(r-1)}}'\right\rbrace} - 1
\quad \text{and} \quad
u_{j} = w_{j}\exp\left\lbrace \beta^{(r-1)}_{0} + \mathbf{z}(s_{j}){\boldsymbol\beta^{(r-1)}}'\right\rbrace.
\end{eqnarray}
As can be seen from equations \eqref{addafacaane2} and \eqref{addafacaane3452}, the variable $y^{*}_{j}$ is the working response variable while $u_{j}$ is the updated weight. Replacing the log-likelihood function $\log\mathcal{L}\left(\boldsymbol\theta\right)$ in equation \eqref{waaqayyoo234} by the quadratic approximation $\mathcal{L}_{q}\left(\boldsymbol\theta\right)$,
the optimisation problem of the regularised quadratic approximation of the log-likelihood function becomes
\begin{eqnarray}
\label{addafacaane3}
\displaystyle\operatorname*{argmin}_{\boldsymbol\theta\in \mathbb{R}^{K+1} } \mathcal{L}_{qp}\left(\boldsymbol\theta\right) = \displaystyle\operatorname*{argmin}_{\boldsymbol\theta\in \mathbb{R}^{K+1} } \left\lbrace -\frac{1}{m}\mathcal{L}_{q}\left(\boldsymbol\theta\right) + \lambda \displaystyle\sum_{k=1}^{K}\left\lbrace \frac{1}{2}\left(1-\alpha\right) \beta^{2}_{k} + \alpha\abs{\beta_{k}}\right\rbrace\right\rbrace.
\end{eqnarray}
The optimisation problem in equation \eqref{addafacaane3} can be solved by the coordinate descent algorithm.
More specifically, for any pre-specified value of the tuning parameter $\lambda$, in iteration $r = 1, 2, \ldots$, the coordinate descent algorithm partially optimises the optimisation problem with respect to $\beta_{k}$, given the estimates $\beta_{0}^{(r-1)}$ and $\beta_{h}^{(r-1)}$, $h\in\{1,\ldots,K\}\setminus\{k\}$.
Explicitly, the optimisation can be described by
\begin{eqnarray}\label{addafacaane4}
\displaystyle\operatorname*{argmin}_{\boldsymbol\theta\in \mathbb{R}^{K+1} } \mathcal{L}_{qp}\left(\boldsymbol\theta\right) \approx \displaystyle\operatorname*{argmin}_{\beta_{k}\in \mathbb{R}} \mathcal{L}_{qp}\left(\beta^{(r-1)}_{0}, \beta^{(r-1)}_{1}, \cdots, \beta^{(r-1)}_{k-1}, \beta_{k}, \beta^{(r-1)}_{k+1}, \cdots, \beta^{(r-1)}_{K}\right).
\end{eqnarray}
According to \cite{friedman2007pathwise}, there are closed form coordinate-wise updates to estimate the parameters of the optimisation problem. Letting $\beta_{k}\geq 0$, the first-order derivative of $\mathcal{L}_{qp}\left(\boldsymbol\theta\right)$ in equation \eqref{addafacaane4} with respect to $\beta_{k}$ is given by
\begin{eqnarray}\label{addafww2}
\frac{d\mathcal{L}_{qp}\left(\boldsymbol\theta\right)}{d\beta_{k}} = -\frac{1}{m} \sum_{j=1}^{m}u_{j}z_{k}(s_{j}) \left(y^{*}_{j} - \tilde{y}^{(k)}_{j}\right) + \frac{1}{m}\sum_{j=1}^{m}u_{j}z^{2}_{k}(s_{j})\beta_{k} + \lambda(1-\alpha)\beta_{k} + \lambda\alpha,
\end{eqnarray}
where $\tilde{y}^{(k)}_{j} = \beta^{(r-1)}_{0} + \sum_{h\ne k}^{K}\beta_{h}^{(r-1)}z_{h}(s_{j})$ is the fitted value excluding the covariate $z_{k}(s_{j})$. Similarly, the first-order derivative of $\mathcal{L}_{qp}\left(\boldsymbol\theta\right)$ for the case $\beta_{k} < 0$ can easily be obtained.
It follows that the coordinate-wise updates for parameter estimation in the elastic-net penalisation can be obtained by
\begin{eqnarray}\label{addafacaane5}
\beta^{(r)}_{k} =
\frac{S\left(\frac{1}{m}\displaystyle\sum_{j=1}^{m} u_{j} z_{k}(s_{j})\left(y^{*}_{j}-\tilde{y}^{(k)}_{j} \right), \lambda\alpha\right) }{\frac{1}{m}\displaystyle\sum_{j=1}^{m} u_{j} z^{2}_{k}(s_{j})+\lambda\left(1-\alpha\right)}, \quad r = 1, 2, \ldots, \quad k = 1, 2, \ldots, K,
\end{eqnarray}
where $S(z, \vartheta)$ is the soft-thresholding operator given by
\begin{eqnarray*}
S(z, \vartheta) = \text{sign}\left(z\right) \left(\abs{z}-\vartheta \right)_{+} =
\begin{cases}
z-\vartheta, & \text{if } z > 0 \text{ and } \vartheta < \abs{z} ,\\
0, & \text{if } \vartheta \ge \abs{z},\\
z + \vartheta, & \text{if } z < 0 \text{ and }\vartheta < \abs{z}.
\end{cases}
\end{eqnarray*}
The intercept parameter need not be penalised as it has no role in the variable selection. The estimate of the intercept term can be obtained by
\begin{align*}
\beta^{(r)}_{0} =
\frac{1}{\sum_{j=1}^{m} u_{j}}
\sum_{j=1}^{m} u_{j}\left(y^{*}_{j}-\mathbf{z}(s_{j}) \boldsymbol\beta'^{(r-1)}\right)
, \quad r = 1, 2, \ldots.
\end{align*}
The parameter estimates are updated until the algorithm converges.
With regard to the tuning parameter $\lambda\in \left[\lambda_{max}, \lambda_{min}\right]$, we start with the smallest value $\lambda_{max}$ of the tuning parameter for which the entire vector is zero. That is, we begin with an $\lambda_{max}$ for which $\widehat{\boldsymbol\beta} = 0$ to obtain solutions for a decreasing sequence of $\lambda$ values. Using a prediction performance measure, e.g.~cross-validation, for the estimated model, the user can select a particular value of $\lambda$ from the candidate sequence of $\lambda$ values. Since the parameter estimation updating equation \eqref{addafacaane5} is obtained for elastic-net penalisation, we may set $\alpha = 0$ to implement ridge regression and $\alpha = 1$ to use the lasso approach; other elastic-net regularisation can be implemented by picking $\alpha\in(0, 1)$. Recall that elastic-net is useful when there are many correlated covariates in the statistical model and the data are high-dimensional, i.e.,~data with the property that $K\gg m$. Cyclical coordinate descent methods are a natural approach to solving convex problems and each coordinate-descent step of the algorithm is fast with an explicit formula for each coordinate-wise optimisation. It also exploits the sparsity of the model and it has better computational speed both for high dimensional data and big data \citep{friedman2010regularization}.
\begin{algorithm}[H]
\caption{\small Parameter estimation algorithm for regularized log-likelihood function of inhomogeneous Poisson point process.}
\label{alg:urgooftuuko}
\begin{algorithmic}[1]
\State Identify the spatial domain $W$,
\State Generate a set of dummy points $\textbf{v} = \lbrace v_{1}, v_{2}, \ldots, v_{q} \rbrace$ in $W$,
\State Combine the dummy points $\textbf{v} = \lbrace v_{1}, v_{2}, \ldots, v_{q}\rbrace$ with the data points $\mathbf{x} = \lbrace x_{1}, x_{2}, \ldots, x_{n}\rbrace $ to form a set of quadrature points $\textbf{s} = \lbrace s_{j}\mid j = 1, 2, \ldots, m \rbrace$,
\State Compute the quadrature weights $w_{j}$,
\State Following equation \eqref{indicator}, determine the indicator $a_{j}$ and compute the variable $y_{j} = a_{j}/w_{j} $,
\State Obtain the vector of spatial covariates $\mathbf{z}(s_{j}) = (z_{1}(s_{j}), \ldots, z_{K}(s_{j}))$ at each quadrature point $s_{j}$,
\State Use existing model-fitting software such as \texttt{glmnet} \citep{JSSv033i01}, specifying that the model is a log-linear Poisson regression model, $\log\rho_{\boldsymbol\theta}(s_{j}) = \beta_{0} + \mathbf{z}(s_{j})\boldsymbol\beta'$,
in order to fit the responses $y_{j}$ and vector of covariate values $\mathbf{z}(s_{j})$ with weights $w_{j}$,
\State The coefficient estimates returned by the software give the approximate maximum log-likelihood estimate of $\boldsymbol\theta$,
\end{algorithmic}
\end{algorithm}
The optimisation problem in equation \eqref{addafacaane4} can be implemented using Algorithm \ref{alg:urgooftuuko}. The approximated log-likelihood function in equation \eqref{werqqq} and the log-likelihood function of the weighted generalised linear model (Poisson distribution) have the same deviance function $\mathcal{D}\left(\boldsymbol\theta\right) = 2\lbrace \log \mathcal{L}(\mathbf{y}\mid \mathbf{y})-\log \mathcal{L}(\boldsymbol\theta\mid \mathbf{y})\rbrace$. Hereby, the deviance $\mathcal{D}$ of the regularised weighted generalised linear model (Poisson distribution) obtained by the model-fitting software, e.g.~\texttt{glmnet}, can be exploited to select an optimal tuning parameter in the optimisation of the regularised quadratic approximation of the log-likelihood function of the inhomogeneous Poisson process in equation \eqref{addafacaane3}. Choosing an optimal tuning parameter value can be done using K-fold cross-validation where, in short, we set a sequence of tuning parameter candidate values $\lambda_{1}>\lambda_{2}>\cdots>\lambda_{T}$ and split the data into K-folds. Then, for each $\lambda$-candidate value, we leave out a data fold/piece and perform parameter estimation on all the remaining $K-1$ data folds, thus obtaining a deviance for the left-out data fold. We repeat the parameter estimation and deviance computation for the remaining $K-1$ possible folds to be left out. This means that we obtain $K$ out-of-sample deviances for each $\lambda$ value. Among the sequenced $\lambda$ values, the one giving the smallest mean deviance can be picked as an optimal estimate of the tuning parameter $\lambda$ of the regularised quadratic approximation of the log-likelihood function. We then use the selected optimal $\lambda$ to again carry out the regularized fitting, this time using the full dataset, in order to obtain a final estimate of the model parameter $\boldsymbol\theta$. Finally, the stopping criterion for the cyclic coordinate descent algorithm is generally based on the change of the fitted quadratic approximation of the log-likelihood function value.
\subsection{Semi-parametric intensity function modelling}\label{Semiparametric}
We propose the approach outlined below for the setting where i) the main goal is a prediction/predictive model, i.e.,~one wants to predict a collection of further/future events as precisely as possible, ii) one believes that the observed covariates can only describe a part of the spatial intensity variation, and iii) added spatial flexibility is warranted in the modelling. In the case of our ambulance data, as our end goal is to build optimal dispatching strategies, we mainly want a predictive model. We want to mention again that the demographic covariates we have at hand only reflect where different demographic groups live but not how they move about. Explicitly, we do not have access to explicit mobility covariates such as aggregated movement patterns in the population, and, as one may guess, people do not only need access to ambulances when they are at home.\\
Our solution is quite pragmatic and simple. We simply add a further spatial covariate to the existing collection of covariates, which is given by a non-parametric spatial intensity estimate $\widetilde\rho(x)$, $x\in W$ and will be referred to as the {\em benchmark spatial intensity}. Hence, we include the spatial intensity estimate $\widetilde\rho$ as a covariate in the approximated log-likelihood expression in \eqref{werqqq} and the inclusion has the effect that the modelling steps away from a purely parametric setting to a semi-parametric approach. This added covariate should pick up on regions where there is an increased intensity due to human mobility, which cannot be explained by the existing list of covariates. Note that this is similar in nature to the semi-parametric (spatio-temporal) log-Gaussian Cox process modelling approach advocated for in e.g.~\citet{diggle2013statistical}. To be able to discern whether this added covariate is in fact necessary/useful in the presence of the other covariates, we carry out elastic-net regularisation-based variable selection (see Section \ref{s:Regularisation}) to indicate whether the benchmark spatial intensity has any added value in terms of describing the true intensity function.\\
A natural question here is what kind of non-parametric intensity estimator one should use to generate the benchmark spatial intensity. There are different candidates for this, and the main distinction one usually makes is between global and adaptive/local smoothers. Adaptive smoothing techniques include adaptive kernel intensity estimation \citep{davies2018tutorial} and (resample-smoothed) Voronoi intensity estimation \citep{ogata2003modelling, ogata2004space,moradi2019resample}. These have some clear benefits (in particular the latter, \citet{moradi2019resample}), but here we do not want to put too much weight on the local features since we may run the risk of overfitting. Instead, we here consider (global) kernel intensity estimation \citep{diggle1985kernel, baddeley2015spatial}, which is arguably the most prominent approach to global smoothing and is defined as
$$
\widetilde\rho(x)=\sum_{y\in \mathbf x}\kappa_h(x-y)/w_h(x,y), \quad x\in W,
$$
where $\kappa_h(\cdot)=h^{-1}\kappa(\cdot/h)$, $\kappa$ is a symmetric density function and the smoothing parameter $h>0$ is the bandwidth.
The function $w_h(x,y)$ is a suitable edge correction factor which adjusts the effect of unobserved events outside $W$ on the intensity of the observed events \citep{baddeley2015spatial}; we here use the local corrector $w_h(x,y)=\int_W\kappa_h(u-y) du$ which ensures mass preservation, i.e.~that $\int_W\widetilde\rho(x)dx=n$, the number of observed events. Practically, to carry out kernel intensity estimation we make use of the function \texttt{density.ppp} in the \textsf{R} package \textsf{spatstat} \citep{baddeley2015spatial}. \\
Although the choice of kernel may play a certain role, the choice of bandwidth is absolutely the main determinant for the quality of the intensity estimate; recall that the bandwidth governs how much we smooth the data. However, optimal bandwidth selection is a well-studied and challenging problem. Concerning the state of the art, the bandwidth selection criterion of \citet{cronie2018non} is generally the most stable with respect to accounting for spatial interaction; observed clusters of points in a point pattern may be the effect of aggregation/clustering (dependence) or intensity peaks, or a combination of the two. However, there is one scenario where it tends to not perform too well, and that is when there are large regions in $W$ where there are no points present, which is the case for our ambulance dataset. Other standard methods for bandwidth selection include the state estimation approach of \citet{diggle1985kernel} (called \texttt{bw.diggle} in \textsf{spatstat}), the Poisson process likelihood leave-one-out cross-validation approach in \citet{baddeley2015spatial} and \citet{loader1999local} (called \texttt{bw.ppl} in \textsf{spatstat}), and the recent machine learning-based approach of \citet{bayisa2020large} (see Algorithm \ref{alg:Oroq}).
\subsubsection{New heuristic algorithm for bandwidth selection}
In K-means clustering, the dataset is partitioned into a number of clusters, and each cluster consists of data points whose intra-point distances, i.e.,~ distances between points with in a cluster, are smaller than their inter-point distances, i.e.,~ their distances to points outside of the cluster. In a recent study, \cite{bayisa2020large} proposed a K-means clustering-based bandwidth selection approach for kernel intensity estimation, where the average of the standard deviations of the clusters is used as an optimally selected bandwidth. Although the approach performed well in terms of non-parametrically describing the current ambulance call dataset, it has some limitations/issues. Firstly, the number of clusters used in the K-means algorithm has been selected visually, and thereby subjectively. Secondly, clusters with high point densities and clusters with widely dispersed data points tend to uniformly determine the resulting bandwidth. Evidently, a cluster with widely dispersed data points has a larger standard deviation, which can be an outlier, and hence, it distances the estimated bandwidth away from the standard deviations of clusters with highly clustered data points. As a result, the selected bandwidth leads to oversmoothing of the spatial intensity.
\begin{algorithm}[!htpb]
\caption{$K$--means clustering-based heuristic algorithm for bandwidth selection}
\label{alg:Oroq}
\begin{algorithmic}[1]
\State Consider the observed spatial location data: $\mathbf x = \left\lbrace x_{1}, x_{2}, \cdots, x_{n} \right\rbrace \subseteq W\subset\mathbb R^{2}$,
\State Candidates for the number of clusters: $K = 2, 3, \cdots, K_{max}$,
\State Let $\tr\left(\cdot\right)$ denote the trace of a square matrix,
\State Let $d$ denote the number of variables,
\State Set while condition determining parameter: $\Delta = 1$,
\State Set the maximum number of iterations for the while loop: $max.iter$,
\State Let $\varpi_{q}$, $q = 1, 2, \cdots, P$ represent the centroid of a cluster $q$,
\State Let $C_{q}$ denote the collection of observations in cluster $q$,
\State Let $n_{q}$ is the number of data points in $C_{q}$,
\State Set a parameter determining the convergence of the algorithm: $\varepsilon$,
\State Let $D_{P} = \displaystyle\sum_{q = 1}^{P}\sum_{x\in C_{q}} \left(x - \varpi_{q}\right)' \left(x -\varpi_{q}\right)$ denote within-cluster dispersion matrix for $P$ clusters, \label{withincluster}
\State Let $KL_{P} = \displaystyle\left\lvert\frac{\left(P-1\right)^{2/d}\tr\left(D_{P-1}\right) - P^{2/d}\tr\left(D_{P}\right)}{P^{2/d}\tr\left(D_{P}\right) - \left(P+1\right)^{2/d}\tr\left(D_{P+1}\right)}\right\rvert$ represent $KL$ index, \label{KLIndex}
\Statex \For {$K \gets 2, 3, \cdots, K_{max}$} \label{DD1}
\For {$P \gets K+1$} \label{DD2}
\Statex \hspace{0.35in} Initialize the centroids: $\varpi^{\left(v\right) }_{q}$, $v = 0$ and $q = 1, 2, \ldots, P$,
\Statex \hspace{0.35in} Call algorithm \ref{alg:Subalgorithm},
\EndFor \label{For1}
\If {$K = 2$,}
\For {$P \gets K$}
\Statex \hspace{0.35in} Initialize the centroids: $\varpi^{\left(v\right) }_{q}$, $v = 0$ and $q = 1, 2, \ldots, P$,
\Statex \hspace{0.35in} Call algorithm \ref{alg:Subalgorithm},
\EndFor \label{For}
\For {$P \gets K-1 = 1$}
\Statex \hspace{0.35in} $\varpi_{1} = \left( \displaystyle\displaystyle\sum_{i = 1}^{n}x_{i}\Biggm/n\right)$,
\EndFor
\EndIf
\If {$K > 2$,}
\For {$P \gets K-1, K$}
\Statex \hspace{0.3in} Initialize the centroids: $\varpi^{\left(v\right) }_{q}$, $v = 0$ and $q = 1, 2, \ldots, P$,
\Statex \hspace{0.3in} Call algorithm \ref{alg:Subalgorithm},
\EndFor
\EndIf\label{DD3}
\State \hspace{0.1in} Obtain the optimal centroids $\left\lbrace \varpi_{q}\right\rbrace_{q = 1}^{P} $ for $P = K-1$, $K$, and $K+1$ from step \ref{DD2} to \ref{DD3}, \label{ObtainCentroids2}
\State \hspace{0.1in} Using the result from step \ref{ObtainCentroids2}, compute the expression $D_{P}$ in step \ref{withincluster} for $P = K-1$, $K$, and $K+1$, \label{ObtainCentroidskl2}
\State \hspace{0.1in} Based on the result from step \ref{ObtainCentroidskl2}, compute $KL_{P}$ in step \ref{KLIndex} for $P = K$.
\EndFor \label{DD4}
\State From step \ref{DD1} to \ref{DD4}, obtain the optimal number of clusters: $P_{0} = \displaystyle\argmax_{P\in\left\lbrace 2, 3, \ldots, K_{max}\right\rbrace}\Big\{ KL_{P}\Big\}$,
\State Initialize the centroids for the optimal number of clusters $P_{0}$: $\varpi^{\left(v\right) }_{q}$, $v = 0$ and $q = 1, 2, \ldots, P = P_{0}$, \label{InitializeD}
\State Based on step \ref{InitializeD}, call algorithm \ref{alg:Subalgorithm} and obtain the optimal centroids $\left\lbrace \varpi_{q}\right\rbrace_{q = 1}^{P} $ and classes of data points $\left\lbrace k_{i}\right\rbrace_{i = 1}^{n}$,
\State Compute cluster dispersion measure: $\sigma_{q}^{2} = \displaystyle\frac{1}{2n_{q}}\sum_{x_{i}\in C_{q}} \left\|x_{i} - \varpi_{q} \right\|^{2}$, $q = 1, 2, \ldots, P$,
\State Compute the weight: $w_{q} = \displaystyle\frac{1}{g_{q}\left(x, \varpi_{q}\right)}$, $q = 1, 2, \ldots, P$ and $g_{q}\left(x, \varpi_{q}\right) = \displaystyle\frac{1}{n}\sum_{i=1}^{n}\left\|x_{i} - \varpi_{q} \right\|^{2}$,
\State Obtain an optimal bandwidth estimate: $h = \sqrt{\displaystyle\sum_{q=1}^{P}w_{q} \Biggm/ \displaystyle\sum_{q=1}^{P}\frac{w_{q}}{\sigma_{q}^{2}}}$.
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[H]
\caption{Subalgorithm of the main algorithm}
\label{alg:Subalgorithm}
\begin{algorithmic}[1]
\While{$\Delta > \varepsilon$ and $v\le max.iter$}
\State Obtain optimal classes $k_{i} $ and $1$-of-$P$ class indicator variables $\iota_{iq}$ for data points $ x_{i}$: \label{CLUSTER1}
\begin{align*}
k^{(v)}_{i} = \argmin_{q\in\left\lbrace 1, 2, \ldots, P\right\rbrace } \big \| x_{i} - \varpi^{\left(v\right)}_{q} \big \|^{2}, \quad \iota^{(v)}_{iq} =
\mathbf 1_{\left\{q = k^{(v)}_{i}\right\}}, \quad q = 1, 2, \ldots, P; i = 1, 2, \ldots, n,
\end{align*}
\State Update the centroids of the clusters (the mean locations of the clusters) $\varpi_{q}$: \label{CLUSTER2}
\begin{align*}
\varpi^{\left(v+1\right)}_{q} = \left\lbrace \sum_{i = 1}^{n} \iota^{\left(v\right)}_{iq}x_{i}\right\rbrace
\Bigg/
\sum_{i = 1}^{n}\iota^{\left(v\right)}_{iq},
\quad q = 1, 2, \ldots,P,
\end{align*}
\State $\Delta = \displaystyle\sum_{q=1}^{P}\left\| \varpi^{(v+1)}_{q} - \varpi^{(v)}_{q} \right\|^{2}$. \label{CLUSTER3}
\EndWhile
\end{algorithmic}
\end{algorithm}
To overcome these limitations, we propose a new heuristic algorithm, which is outlined in Algorithm \ref{alg:Oroq}, to establish the ideal number of clusters and thereby to obtain an optimal estimate of the bandwidth. The main algorithm, which is Algorithm \ref{alg:Oroq}, consists of two crucial steps. It continually invokes Algorithm \ref{alg:Subalgorithm}, which is a K-means algorithm, to establish the optimal number of clusters using the $KL$ index of \citet{krzanowski1988criterion}. Once an optimal number of clusters has been obtained, the main algorithm determines an optimal bandwidth by calling the K-means algorithm and using a weighted harmonic mean of dispersion measures for the clusters, where the weight for each cluster is given by the inverse of the average of the squares of the distances from the centroid of the cluster to each observation. When establishing the bandwidth, the weights help in balancing the contributions of clusters with closely spaced spatial points and clusters with widely spaced spatial points. In Algorithm \ref{alg:Subalgorithm}, the spatial data points are re-assigned to clusters (see step \ref{CLUSTER1} in Algorithm \ref{alg:Subalgorithm}), the cluster means are re-computed (see step \ref{CLUSTER2} in Algorithm \ref{alg:Subalgorithm}), and these steps are repeated until the sum composed of the squared Euclidean distances between all successive centroids is smaller than a user-specified value (see step \ref {CLUSTER3} in Algorithm \ref{alg:Subalgorithm}). Alternatively, one may control the convergence of Algorithm \ref{alg:Subalgorithm} by repeating steps \ref{CLUSTER1} and \ref{CLUSTER2} until there is either no further change in the assignments of data points to clusters or until some maximum number of iterations has been reached. In our case, we have used $\varepsilon = 10^{-5}$ and $max.iter = 100$.
\section{Created covariates}\label{s:CreatedCovariates}
Recall the notions of 'original' and 'created' covariates from Section \ref{data}. We illustrate some of the selected 'original' and 'created' covariates considered in this study in Figure \ref{Covariates}. Below, we provide a description of the construction of all created covariates.
\begin{figure}[!htpb]
\centering
\includegraphics[height=22cm, width=18cm]{spatialcovariates}
\caption{Spatial demographic and road network-related covariates. Population 20+ years old income status: a) below and b) above median income. Population 20-64 years old employment status: c) employed and d) unemployed. Population by age: e) 35-39 years old, f) 55-59 years old, and g) 80+ years old. Population by sex: h) male and i) female. j) complete road network line density. k) main road network line density. l) population density. Households 20+ years old economic status: m) below and n) above median income. o) Densely populated line density. }
\label{Covariates}
\end{figure}
\subsection{Benchmark intensity covariate for the ambulance data}
Recall that our semi-parametric approach is based on the idea of including a non-parametric intensity estimate, the so-called benchmark intensity, as a covariate in our log-linear intensity form. To generate the benchmark intensity for the ambulance data, we employ our new algorithm to obtain the intensity estimate found in panel (e) of Figure \ref{BandWidth}. We further compare the result with the aforementioned approaches, namely the state estimation approach, the Poisson process likelihood leave-one-out cross-validation approach, and the approach of \citet{bayisa2020large}, which is based on 5 clusters (this number has been obtained through visual inspection). The resulting intensity estimates for the ambulance data can be found in panels (b)-(d) in Figure \ref{BandWidth}. Note that throughout we have used a Gaussian kernel in combination with the aforementioned local edge correction factor. We argue that the state estimation approach and the Poisson process likelihood leave-one-out cross-validation approach tend to under-smooth the data and thereby do not reflect the general overall variations of the data, whereas the K-means clustering based bandwidth selection of \citet{bayisa2020large} instead tends to over-smooth the ambulance data. Recall that the number of clusters, which is a necessary input in the K-means clustering based bandwidth selection of \citet{bayisa2020large}, has to be selected through visual inspection. Looking at panel (e) of Figure \ref{BandWidth}, we see that by employing our proposed $KL$ index to automatically select the number of clusters, we obtain a total of 16 clusters. We argue that, compared to the different panels in Figure \ref{BandWidth}, the new heuristic algorithm performs the best in terms of balancing over- and under-smoothing of the ambulance call events.
\begin{figure}[!htpb]
\centering
\includegraphics[width= 0.8\textwidth, height=0.6\linewidth]{BandwidthSelection4}
\caption{The roles of different bandwidth selection methods using Gaussian kernel density estimation for the unmarked ambulance call data in panel (a). b) State estimation \citep{diggle1985kernel}. c) Poisson likelihood cross-validation \citep{loader1999local}. d) K-means clustering \citep{bayisa2020large}. e) The new heuristic algorithm developed in this work. Note that what interests us here are the relative scales rather than the raw scales; the values in the plots above have been multiplied by 1000 for ease of visualisation.}
\label{BandWidth}
\end{figure}
\subsection{Creation of (road) network-related covariates}\label{s:LineDensity}
The role of the road network-related covariates is to control that the fitted model generates events close to the underlying road network. But, as previously indicated, the original road network-related covariates are not of the form $z_i(s)$, $s\in W$.
\begin{itemize}
\item Complete road networks line density. It is a spatial pattern of line segments, which is converted to a pixel image. The value of each pixel in the image is measured as the total length of intersection between the pixel and the line segments [1].
\item Main road networks line density. It is a spatial pattern of line segments, which is converted to a pixel image. The value of each pixel in the image is measured as the total length of intersection between the pixel and the line segments [1].
\item Densely populated line density. It is a spatial pattern of line segments, which is converted to a pixel image. The value of each pixel in the image is measured as the total length of intersection between the pixel and the line segments [1].
\item Bus stops density. It is a spatial point pattern, which is converted to a pixel image. The value of each pixel is an intensity, which is measured as "points per unit area" [1].
\item Shortest distance to bus stops. It is the shortest distance in meters from the ambulance location data to the bus stops [1].
\item Shortest distance to densely populated areas. It is the shortest distance in meters from the ambulance location data to the densely populated areas [1].
\item Shortest distance to main road networks. It is the shortest distance in meters from the ambulance location data to the main road networks [1].
\item Shortest distance to complete road networks. It is the shortest distance in meters from the ambulance location data to the complete road networks [1].
\end{itemize}
We here propose to treat the road networks under consideration as line segment patterns \citep{baddeley2015spatial}, which essentially means that each road network considered is a realisation of a point process in the space of line segments in $\mathbb R^2$.
The spatial covariate corresponding to a given line segment pattern is then given by the estimated line segment intensity, which is obtained as the convolution of an isotropic Gaussian kernel with the line segments of the pattern in question. Practically, such an estimate may be obtained through the function \texttt{density.psp} in the \textsf{R} package \textsf{spatstat}, and the standard deviation of the Gaussian kernel, the bandwidth, determines the degree of smoothing. The default bandwidth choice in \texttt{density.psp} is given by the diameter of the observation window multiplied by 0.1. As an alternative, we propose to use our new heuristic algorithm for bandwidth selection, which is achieved by letting $W = [0,\infty)$, and letting the observations $\mathbf x = \left\lbrace x_{1}, x_{2}, \cdots, x_{n} \right\rbrace \subseteq W$ considered in Algorithm \ref{alg:Oroq} represent the lengths of the line segments. Figure \ref{BandWidthLineDensity} compares the default bandwidth choice of \texttt{density.psp} to our heuristic algorithm, and it clearly suggests that the line segment intensities generated using the heuristic algorithm bandwidth selection have captured the spatial pattern of the line segments in the observed data better than the default bandwidth choice. Note that what interests us here are the relative scales rather than the raw scales; the values in the plots have been multiplied by 1000 for ease of visualisation.
\begin{figure}[H]
\centering
\includegraphics[width= 1\textwidth, height=0.9\linewidth]{CovariatesFromLineSegmentPatterns}
\caption{Creation of covariates from line segment patterns. The first column represents a) the complete road network, d) the main road network and g) the densely populated area road network, respectively. Created road network covariates using the \textsf{spatstat} function \texttt{density.psp}: (b), (e) and (h) have been obtained using the default bandwidth of \texttt{density.psp}, whereas (c), (f) and (i) have been obtained using the new heuristic bandwidth selection algorithm. The values in the plots have been multiplied by 1000 for ease of visualisation.}
\label{BandWidthLineDensity}
\end{figure}
\section{Data analysis}\label{Results}
Having presented the semi-parametric regularised fitting procedure to be employed, we next turn to its actual modelling of the ambulance call data; recall the structure of the data (including the missing sex/gender label issue) and that we may model the intensity of each marginal process $Y_j$ separately.\\
As carefully emphasised and laid out above, we have chosen to follow \citet{yue2015variable} and thus chosen to employ elastic-net regularisation of a Poisson process log-likelihood function, where one of the covariates in fact is a non-parametric intensity estimate of the data (the so-called benchmark intensity). Our motivation for elastic-net regularisation is mainly that we have a large number of spatial covariates, which may be highly correlated. The choice of $\alpha$ is subjective, and we usually pick an $\alpha$ between 0 and 1, and then proceed to the estimation of the proposed model \citep{friedman2010regularization}. Recall further that elastic-net performs like a lasso but avoids unstable behaviour due to extreme correlation if the elastic-net parameter $\alpha$ is large enough but less than one \citep{yue2015variable}; $\alpha=1$ yields the lasso penalty and $\alpha=0$ yields the ridge penalty. Since we here are interested in carrying out lasso-like elastic-net regularization, we let $\alpha = 0.95$, following a recommendation of \cite{friedman2010regularization} and \citet{yue2015variable}. Such lasso-like elastic-net regularisation results in variable selection (coefficients of less determining covariates are set to zero), but the penalty also forces highly correlated features to have similar coefficients.
\subsection{Modelling the ambulance call intensity function} \label{SpatialCovariates}
Figure \ref{Orgr2LassoElastic} shows the estimated coefficients of the spatial covariates, using both lasso regularisation and lasso-like elastic-net regularisation, for the spatial point pattern constituting the events with priority label 1. Note that the numbers at the top of each panel in Figure \ref{Orgr2LassoElastic} indicate the number of spatial covariates with non-zero coefficients, i.e.~the number of covariates that are associated with the fitted spatial intensity functions for the indicated value of $\lambda$. We clearly see how a large number of covariates are quickly excluded as we increase $\log\lambda$.\\
Using the aforementioned covariates and their first-order interaction terms, the trace plots of the estimated parameters for lasso regularization and lasso-like elastic-net regularisation with $\alpha = 0.95$ are shown in Figure \ref{Orgr2LassoElastic}. At their corresponding optimal tuning parameters, which are shown by blue-coloured vertical lines in Figure \ref{Orgr2LassoElastic}, lasso and lasso-like elastic-net regularisations have selected 185 (18.69\%) and 207 (20.91\%) of the covariates that are associated with the intensity of the events. As it is expected, even though the lasso-like elastic-net regularisation selects more variables than the lasso regularization, it forces the highly correlated covariates to have similar coefficients \citep{friedman2010regularization, yue2015variable}. As a result, the correlated covariates have similar roles on the intensity of the events, and their presence in the fitted model can be advantageous for interpretation. Therefore, hereafter, we use lasso-like elastic-net regularisation with $\alpha = 0.95$.
\begin{figure}[!htpb]
\centering
\includegraphics[width=0.8\textwidth, height=0.4\linewidth]{spatialtraceplotLasso}\\
\includegraphics[width=0.8\textwidth, height=0.4\linewidth]{spatialtraceplotElasticNet}
\caption{A trace plot of the estimated coefficients for lasso (the first row) and lasso-like elastic-net (the second row) regularisations. The numbers at the top of each panel indicate the number of spatial covariates with non-zero estimated coefficients. The plots in the right panel of the left plots represent the zoomed in portion of the region marked by the rectangular region. For the purpose of visualization, the estimated coefficients are scaled (divided) by $10^{10}$. }
\label{Orgr2LassoElastic}
\end{figure}
A grid of $\lambda$ values has been exploited to train the proposed model, and among the candidate $\lambda$ values, the one which gives the smaller deviance, $\mathcal{D}$, has been selected as an optimal estimate of $\lambda$. The optimal elastic-net regularisation parameter $\lambda$ has been selected using ten-fold cross-validation as shown in Figure \ref{Orgr2}.
\begin{figure}[!htpb]
\centering
\includegraphics[width=0.7\textwidth, height=0.5\linewidth]{Deviancespatial}
\caption{An optimal tuning parameter selection via ten-fold cross-validation: location of the logarithm of the optimal estimate of the tuning parameter (blue) and the location of the logarithm of the estimate of the tuning parameter that is one standard error away from the optimal estimate (red).}
\label{Orgr2}
\end{figure}
\noindent The two vertical lines in the figure have been drawn to show the location of the logarithm of the optimal estimate of the tuning parameter (blue) and the location of the logarithm of the estimate of the tuning parameter that is one standard error away from the optimal estimate (red); the one-standard-error-rule \citep{hastie2017elements} says that one should go with the simplest model, which is no more than one standard error worse than the best model.\\
The spatial intensity function of the events has been obtained at each observed spatial location using the estimated model, and it has been obtained at the desired spatial locations in the study area using kernel-smoothed spatial interpolation of the estimated intensity function, which is used as the mark of the observed point pattern. The estimated spatial intensities of the marginal spatial point patterns are shown in Figure \ref{Orgr3}; note that we have scaled the intensity estimates to range between 0 and 1 so that we may compare them more easily. For the priority level 1 and 2 events, about 20.91\% and 36.06\% of the spatial covariates have been associated with/included in the final intensity function estimates, respectively. The lasso-like elastic-net has discerned about 17.68\% and 35.76\% of the spatial covariates which determine the spatial intensities of the point patterns corresponding to male and female, respectively.
\begin{figure}[H]
\centering
\includegraphics[width=0.25\textwidth, height=0.2\linewidth]{SpatialIntensityplotp1}
\includegraphics[width=0.2\textwidth, height=0.2\linewidth]{Dataplotp1}
\includegraphics[width=0.25\textwidth, height=0.2\linewidth]{SpatialIntensityplotm}
\includegraphics[width=0.2\textwidth, height=0.2\linewidth]{Dataplotm}\\
\includegraphics[width=0.25\textwidth, height=0.2\linewidth]{SpatialIntensityplotp2}
\includegraphics[width=0.2\textwidth, height=0.2\linewidth]{Dataplotp2}
\includegraphics[width=0.25\textwidth, height=0.2\linewidth]{SpatialIntensityplotf}
\includegraphics[width=0.2\textwidth, height=0.2\linewidth]{Dataplotf}
\caption{The estimated spatial intensities of the emergency alarm calls and their corresponding observed marginal spatial point patterns. The first column presents estimated intensities for priority levels 1 and 2, respectively, while the second column demonstrates their corresponding observed marginal spatial point patterns. In a similar manner, the third column presents the estimated intensities for genders, male and female, while the fourth column shows their corresponding observed marginal spatial point patterns. Note that we have scaled the intensity estimates to range between 0 and 1 so that we may compare them more easily.}
\label{Orgr3}
\end{figure}
The aforementioned analyses have utilised the spatial covariates $z_i(s)$, $s\in W$, and the interaction terms of the form $z_i(s)z_j(s)$, which holds for the cases $i = j$ and $i \ne j$. However, with this approach, it is not easy to interpret the results and present the estimated models. Therefore, we are interested in further exploring the estimation of the intensity of the emergency alarm call events using only the original spatial covariates $z_i(s)$, $s\in W$, i.e.,~no first-order and higher-order interactions of spatial covariates in the modelling of the spatial intensity. Modelling the spatial intensity of the emergency alarm call events using only the original spatial covariates can help to identify the spatial covariates that play a key role in determining the spatial distribution of the emergency alarm call events, and it also ease the interpretation and presentation of the results. Moreover, we can also compare the estimated intensities that are obtained by the two approaches (i.e.,~modelling with only the original spatial covariates and modelling with the original spatial covariates and the first-order interaction terms) to discern the approach that can be practicable. Using only the original spatial covariates, a lasso-like elastic-net ($\alpha = 0.95$) with an optimal tuning parameter estimate does not provide sparse solutions, i.e.,~most of the coefficients of the spatial covariates are non-zero. That is to say, the lasso-like elastic-net provides dense solutions, i.e.,~it has a low rate of variable exclusion, see Table \ref{tab:EstimatedModels1}.
\begin{center}
\begin{longtable}{|l|l|ccccc}
\caption{\normalsize The estimated dense models for the marginal and unmarked emergency alarm call events/patterns. The dots in the table represent small coefficients of covariates that are shrunk to zero.}
\label{tab:EstimatedModels1}\\
\hline \multicolumn{1}{|c|}{\textbf{No.}} & \multicolumn{1}{c|}{\textbf{Covariates}} & \multicolumn{1}{c}{\textbf{Priority 1}} & \multicolumn{1}{c}{\textbf{Priority 2}} & \multicolumn{1}{c}{\textbf{Male}} & \multicolumn{1}{c}{\textbf{Female}} & \multicolumn{1}{c}{\textbf{Unmarked}}\\ \hline
\endfirsthead
\multicolumn{3}{c}%
{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
\hline \multicolumn{1}{|c|}{\textbf{No.}} & \multicolumn{1}{c|}{\textbf{Covariates}} & \multicolumn{1}{c}{\textbf{Priority 1}} & \multicolumn{1}{c}{\textbf{Priority 2}} & \multicolumn{1}{c}{\textbf{Male}} & \multicolumn{1}{c}{\textbf{Female}} & \multicolumn{1}{c}{\textbf{Unmarked}} \\ \hline
\endhead
\hline \multicolumn{3}{l}{{Continued on next page}} \\ \hline
\endfoot
\hline \hline
\endlastfoot
&\textbf{Ambulance location data}: \\\cline{1-2}
1\footnote[1]{\label{1sttablefoot}The parameter estimates are divided by $10^{-5}$ for better visualisation of the estimates}&\hspace{2cm}x & 0.823 &0.826 &0.951 &0.522&0.725\\
2\footref{1sttablefoot}&\hspace{2cm}y &-0.144 &-0.225&-0.175 &-0.463&-0.243\\\hline
&\textbf{Shortest distance to}: \\\cline{1-2}
3\footref{1sttablefoot}& Bus stops&-42.810 &-52.120&-53.820&-49.810& -44.776\\
4\footref{1sttablefoot}&Densely populated area &-3.685 &-4.380&-2.344& -4.246&-3.674\\
5\footref{1sttablefoot}&Main road networks &-32.030 &-16.740&-16.890&-20.800&-24.907\\
6\footref{1sttablefoot}&Complete road networks &-206.300&-238.600&-203.100&-259.200&-229.287\\\hline
7&Population density &18.991 &-141.524&68.728&$\cdot$& -31.185\\
8&Benchmark intensity & 949.209 &3802.066&-15119.642 &$\cdot$& 276.511\\\hline
&\textbf{Line densities}:\\\cline{1-2}
9&Complete road networks & 243.727&280.890 &308.540 &304.280& 275.264\\
10&Main road networks &42.073 &-54.408 &-78.472 & -57.196&25.836\\
11&Densely populated & -158.792 &-155.344 &-175.181 &-157.708& -161.800\\\hline
12&Bus stops density &548072.243&-163337.873 &604558.360 & 81888.976&494114.346\\\hline
&\textbf{Population by}: \\\cline{1-2}
&a) \textbf{employment status:} \\
13&Employed & $\cdot$ & $\cdot$ & $\cdot$ &$\cdot$& $\cdot$\\
14&Unemployed& $\cdot$ & $\cdot$ & $\cdot$ &$\cdot$& $\cdot$\\\hline
&b) \textbf{income status:} \\
15&Below national median income & 4.747 &0.387 &0.365 &0.866&4.446\\
16&Above national median income& -0.116 &-0.001 &-0.001 & -0.010&-0.079\\\hline
&c) \textbf{educational level}: \\
17&Pre-high school & -0.355 &-2.112 & -1.996 & -2.260& -2.150 \\
18&High school & 1.808 & 0.962 & 2.001 & 1.365& 1.190 \\
19&Post-secondary -- less than 3 years & 3.105 & 1.196&2.534 &$\cdot$& 1.034\\
20&Post-secondary -- 3 years or longer& $\cdot$ & $\cdot$ &$\cdot$ &$\cdot$& $\cdot$\\\hline
&d) \textbf{age:} \\
21&0 -- 5 & 7.999 & 5.926 &4.683 & 6.189&6.000\\
22&6 -- 9& -1.115 &-12.052 &-3.768 &-12.189&-4.575\\
23&10 -- 15 &4.505 &1.310 &5.901 & 3.828&7.542\\
24&16 -- 19 & $\cdot$ &4.433 &2.148 &0.247&-3.813\\
25&20 -- 24 & $\cdot$ &-3.429 &$\cdot$ &$\cdot$& 0.063\\
26&25 -- 29 & -0.179 &-0.764 &-1.902 & -4.533&-0.163\\
27&30 -- 34 & -3.306 & -3.821 &$\cdot$ & $\cdot$&-0.599\\
28&35 -- 39 &1.486 & 12.824 &4.757 & 15.668&5.583\\
29&40 -- 44 &-3.386 &0.112 &-8.503 & -3.040& 0.737\\
30&45 -- 49 & -1.064 &-3.946 &-4.763 & -4.098& -4.794\\
31&50 -- 54 & -10.938 &-8.441 &-7.539 & -5.996&-10.981\\
32&55 -- 59 & 3.340 & 4.501 &6.950 & 7.621&6.450\\
33&60 -- 64 &-1.003 &0 & 1.370 & -0.171&-0.422\\
34&65 -- 69 & -13.011 &-5.694 &-6.573 & -8.789& -7.743\\
35&70 -- 74 & 0.131 &0.025 &$\cdot$ & 0.098&-4.211\\
36&75 -- 79 & 3.020 & $\cdot$ &$\cdot$ & 3.359& 2.524\\
37&80 + & $\cdot$ &0.294 &$\cdot$ & $\cdot$&2.507\\\hline
&e) \textbf{gender:}\\
38&Female &-0.816 &2.586 & 1.431 & 2.145&-1.302\\
39&Male &0.058 & -0.017 &-0.004 & -0.043& 0.106\\\hline
&f) \textbf{background:}\\
40&Swedish background & -0.177 & -1.460 &-1.715 & $\cdot$&-0.671\\
41&Swedish foreign background&0.046 & 0.028 &0.062 & $\cdot$&0.044\\\hline
&g) \textbf{household economic standard:} \\
42&Below the national median & -3.741 & $\cdot$ &$\cdot$ & $\cdot$ &-3.195\\
43&Above the national median & 0.131 & $\cdot$& $\cdot$ & $\cdot$&0.079\\\hline
&\textbf{Intercept parameter estimate}: & 2.410 &8.409 & 3.646 &25.929&10.886\\
\end{longtable}
\end{center}
\noindent On the other hand, exploiting the one-standard-error rule in \cite{JSSv033i01}, where we pick the most parsimonious estimated model within one standard error of the minimum, in the context of regularisation, we present the result of the estimation for each of the intensity functions of the marginal point processes/patterns in Table \ref{tab:EstimatedModels2}.
\begin{center}
\begin{longtable}{|l|l|ccccc}
\caption{\normalsize The estimated parsimonious/sparse models for the marginal and unmarked emergency alarm call events/patterns. The dots in the table represent small coefficients of covariates that are shrunk to zero.}
\label{tab:EstimatedModels2}\\
\hline \multicolumn{1}{|c|}{\textbf{No.}} & \multicolumn{1}{c|}{\textbf{Covariates}} & \multicolumn{1}{c}{\textbf{Priority 1}} & \multicolumn{1}{c}{\textbf{Priority 2}} & \multicolumn{1}{c}{\textbf{Male}} & \multicolumn{1}{c}{\textbf{Female}} & \multicolumn{1}{c}{\textbf{Unmarked}}\\ \hline
\endfirsthead
\multicolumn{3}{c}
{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
\hline \multicolumn{1}{|c|}{\textbf{No.}} & \multicolumn{1}{c|}{\textbf{Covariates}} & \multicolumn{1}{c}{\textbf{Priority 1}} & \multicolumn{1}{c}{\textbf{Priority 2}} & \multicolumn{1}{c}{\textbf{Male}} & \multicolumn{1}{c}{\textbf{Female}} & \multicolumn{1}{c}{\textbf{Unmarked}} \\ \hline
\endhead
\hline \multicolumn{3}{l}{{Continued on next page}} \\ \hline
\endfoot
\hline \hline
\endlastfoot
&\textbf{Ambulance location data}: \\\cline{1-2}
1\footnote[1]{\label{1sttablefoot}The parameter estimates are divided by $10^{-5}$ for better visualisation of the estimates}&\hspace{1.75cm}x & 0.098 &0.324&0.390&0.119& 0.128\\
2\footref{1sttablefoot}&\hspace{1.75cm}y &-0.077 &$\cdot$ &-0.002 &-0.015& -0.133\\\hline
&\textbf{Shortest distance to}: \\\cline{1-2}
3\footref{1sttablefoot}&Bus stops&-38.910 &-45.250 &-48.560&-44.100 &-41.486\\
4\footref{1sttablefoot}&Densely populated area &-2.633 &-2.882 &-1.567 & -2.899&-2.898\\
5\footref{1sttablefoot}&Main road networks &-22.740 &-8.121&-8.359 & -11.310& -19.045\\
6\footref{1sttablefoot}&Complete road networks &-65.710&-54.770 &-57.790 & -64.720&-101.270\\\hline
7&Population density &$\cdot$ &$\cdot$ &$\cdot$ &$\cdot$& $\cdot$\\
8&Benchmark intensity & $\cdot$ &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$\\\hline
&\textbf{Line densities}: \\\cline{1-2}
9&Complete road networks & 273.826&315.707 &319.707 & 335.114&309.844\\
10&Main road networks &74.692 &$\cdot$ &$\cdot$ & $\cdot$ & 34.869\\
11&Densely populated&-141.438 &-129.585 &-152.325 & -141.335& -159.128\\\hline
12&Bus stops density &601370.222&52605.907 &76587.453 & $\cdot$& 604564.297\\\hline
&\textbf{Population by}:\\\cline{1-2}
&a) \textbf{employment status:}\\
13&Employed & $\cdot$ &$\cdot$ &$\cdot$ & $\cdot$ &$\cdot$\\
14&Unemployed& $\cdot$ &$\cdot$ &$\cdot$ & $\cdot$&$\cdot$\\
&b) \textbf{income status:}\\
15&Below national median income &$\cdot$ &$\cdot$ &$\cdot$ &$\cdot$ &$\cdot$\\
16&Above national median income&$\cdot$ & $\cdot$ &$\cdot$ & $\cdot$ &$\cdot$\\\hline
&c) \textbf{educational level:} \\
17&Pre-high school &$\cdot$ &$\cdot$ &$\cdot$ &$\cdot$ &$\cdot$\\
18&High school &$\cdot$ &$\cdot$ &$\cdot$ &$\cdot$ &$\cdot$\\
19&Post-secondary -- less than 3 years &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$ &$\cdot$\\
20&Post-secondary -- 3 years or longer& $\cdot$ & $\cdot$ &$\cdot$ & $\cdot$ &$\cdot$\\\hline
&d) \textbf{age:}\\
21&0 -- 5 &1.917 &2.120 & 2.780 & 3.782&3.786\\
22&6 -- 9& 0.045 & $\cdot$ &1.197 & $\cdot$&$\cdot$\\
23&10 -- 15 &2.123 &2.552 &3.126 & 0.937&0.924\\
24&16 -- 19 &$\cdot$ &$\cdot$ &$\cdot$ &$\cdot$ &$\cdot$\\
25&20 -- 24 &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$ &$\cdot$\\
26&25 -- 29 &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$ &$\cdot$\\
27&30 -- 34 &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$&$\cdot$\\
28&35 -- 39 &$\cdot$ &4.098 &$\cdot$ & 9.585&1.289\\
29&40 -- 44 &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$ &$\cdot$\\
30&45 -- 49 &$\cdot$ &$\cdot$ & $\cdot$ & $\cdot$ &$\cdot$\\
31&50 -- 54 &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$&-1.541\\
32&55 -- 59 &$\cdot$ &-0.901 & $\cdot$ & $\cdot$&$\cdot$\\
33&60 -- 64 &-6.267 &-2.451 &-3.532 & $\cdot$ &-3.204\\
34&65 -- 69 &-0.466 &$\cdot$ &$\cdot$ & $\cdot$ &-0.144\\
35&70 -- 74 &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$&$\cdot$\\
36&75 -- 79 &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$ &$\cdot$\\
37&80 + & $\cdot$ &$\cdot$ &$\cdot$ &$\cdot$&$\cdot$\\\hline
&e) \textbf{gender:} \\
38&Female &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$&$\cdot$\\
39&Male &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$&$\cdot$\\\hline
&f) \textbf{background}: \\
40&Swedish background &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$&$\cdot$\\
41&Swedish foreign background&$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$&$\cdot$\\\hline
&g) \textbf{household economic standard}: \\
42&Below the national median &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$ &$\cdot$\\
43&Above the national median &$\cdot$ &$\cdot$ &$\cdot$ & $\cdot$ &$\cdot$\\\hline
&\textbf{Intercept parameter estimate}: &3.255 &-4.821 &-4.872 & -2.757&6.651\\
\end{longtable}
\end{center}
The three approaches, which are the estimated spatial intensity based on the original spatial covariates and the first-order interaction terms, and the estimated spatial intensities that are obtained using the estimated dense model and parsimonious/sparse model based on only the original spatial covariates, can all be compared for how well they capture the spatial variation of the events. Here, we need to remark that when we say dense model and parsimonious/sparse model, we are referring to the estimated models obtained based on only the original spatial covariates, i.e.,~no interaction term in the model setting. Figure \ref{Orgr3} indicates that the spatial distributions of the events are well captured by their corresponding estimated spatial intensities, with the exception that the distribution of the events with priority level 2 is not well captured by its corresponding estimated spatial intensity. This can be viewed as the shortcoming of employing the original covariates and the first-order interaction terms in modelling the ambulance call events. Following this, we are interested in comparing the performance of the estimated model based on the original spatial covariates and the first-order interaction terms with that of the estimated dense and sparse models based on only the original covariates on the marginal point pattern corresponding to priority level 2. In a broad sense, the three approaches have similar performance for the marginal point patterns corresponding to the marks priority level 1, male, and female. Here we present the results of the three approaches, i.e.,~the estimated spatial intensity using the estimated model based on the original spatial covariates and the first-order interaction terms; the estimated spatial intensities using the estimated dense and the parsimonious/sparse models based on only the original spatial covariates. Figure \ref{OrgrgGd3Compare} presents the estimated spatial intensities using the three approaches for the marginal point pattern with priority level 2.
\begin{figure}[H]
\centering
\includegraphics[width=0.23\textwidth, height=0.2\linewidth]{SpatialIntensityplotp2}
\includegraphics[width=0.23\textwidth, height=0.2\linewidth]{SpatialIntensityDensep2}
\includegraphics[width=0.23\textwidth, height=0.2\linewidth]{SpatialIntensitySparsep2}
\includegraphics[width=0.18\textwidth, height=0.2\linewidth]{Dataplotp2}
\caption{The estimated spatial intensities of the emergency alarm call events with priority level 2 using the estimated model based on the original spatial covariates and the first-order interaction terms, the estimated dense and sparse models, which are based on only the original covariates, respectively. The performance of the estimated models can be evaluated based on the pattern of the events in the observed marginal spatial point pattern. Note that we have scaled the intensity estimates to range between 0 and 1 so that we may compare them more easily.}
\label{OrgrgGd3Compare}
\end{figure}
\noindent The figure clearly demonstrates that the dense and sparse models capture the spatial distribution of the events more accurately than the estimated model based on the original spatial covariates and the first-order interaction terms. Those spatial covariates in the dense model with non-zero estimated coefficients that continue to exist in the sparse model, i.e.,~those spatial covariates with non-zero estimated coefficients, have a strong association with the spatial intensity of the events. As the figure shows, the estimated dense and sparse models perform well in capturing the spatial variation of the events, and, relatively speaking, we can loosely and strongly interpret the results from the dense and sparse models, respectively. \\
Next, we examine the modelling of the spatial intensity function for unmarked ambulance call data, i.e.,~we ignore the marks of the ambulance call data, using only the original spatial covariates. Here, we are interested in recognising how the marks influence the inclusion of different covariates. As in the modelling of each marked spatial point pattern, the spatial intensity function modelling of the unmarked spatial point pattern based solely on the original spatial covariates is expected to better capture the spatial distribution of the emergency alarm call events than the corresponding model setting including both the original spatial covariates and the first-order interaction terms. Then, using only the original spatial covariates, we focus on modelling the spatial intensity function of the unmarked ambulance call data. The last column in Table \ref{tab:EstimatedModels1} presents the estimated dense model for the unmarked spatial point pattern, while the last column in Table \ref{tab:EstimatedModels2} displays the corresponding sparse model for the unmarked spatial point pattern that is obtained by utilising the one-standard-error rule. Figure \ref{UnmarkedOrgrg3} shows the estimated spatial intensities of the unmarked ambulance call data using the estimated dense and sparse models.
\begin{figure}[H]
\centering
\includegraphics[width=0.23\textwidth, height=0.2\linewidth]{SpatialIntensityDenseUnmarked}
\includegraphics[width=0.23\textwidth, height=0.2\linewidth]{SpatialIntensitySparseUnmarked}
\includegraphics[width=0.17\textwidth, height=0.2\linewidth]{DataplotUnmarked}
\caption{The estimated spatial intensities of the unmarked ambulance call data using the estimated dense model (left) and sparse model (middle). The pattern of the events in the observed unmarked spatial point pattern (right) can be used to assess the performance of the estimated models. To make it easier to compare the intensity estimates, we scaled them to have a range between 0 and 1.}
\label{UnmarkedOrgrg3}
\end{figure}
\section{Evaluation of the fitted model}\label{Evaluation}
The evaluation of the fitted spatial models is generally challenging, and here we use two approaches. Firstly, we evaluate the stability of the estimated models. We employ undersampling of the observed point patterns, i.e.,~70\% for the unmarked and marginal point patterns, to compare the stability of the fitted models. The estimates of the spatial intensities obtained for the unmarked and marginal point patterns using the fitted dense and sparse models are treated as the true spatial intensities of the corresponding point patterns. In this context, hereafter, we refer to the estimated spatial intensities based on the marginal and unmarked point patterns as the true spatial intensities. We generated one hundred undersampled random realisations, i.e.,~70\% for each of the unmarked and marginal spatial point patterns, in order to produce one hundred estimated spatial intensities for each of the unmarked and marginal spatial point patterns. Pixel-wise mean absolute errors, the 5\% and 95\% quantiles of pixel-wise absolute errors of the true intensities, and each of the one hundred estimated spatial intensities are used to assess the stabilities of the fitted dense and sparse models for each of the unmarked and marginal point patterns. Figure \ref{Densemae} shows the evaluation of the stabilities of the estimated dense models in estimating the spatial intensities of the emergency alarm call events. The overall means and standard deviations of the pixel-wise mean absolute errors for the dense models are 0.098 and 0.107, 0.099 and 0.123, 0.057 and 0.052, 0.060 and 0.052, and 0.052 and 0.051, respectively, for the point patterns with priority level 1, priority level 2, male, and female, as well as for the unmarked point pattern. Figure \ref{Sparsemae} demonstrates the evaluation of the stabilities of the estimated sparse models in estimating the spatial intensities of the emergency alarm call events. The overall means and standard deviations of the pixel-wise mean absolute errors for the sparse models for the priority level 1, priority level 2, male, and female point patterns, as well as for the unmarked point pattern, are 0.103 and 0.128, 0.103 and 0.140, 0.056 and 0.058, 0.062 and 0.066, and 0.054 and 0.061, respectively. Based on a comparison of the plots in Figures \ref{Densemae} and \ref{Sparsemae}, the estimated dense models seem to be more stable than the sparse models. Even though the 95\% and 5\% quantiles of pixel-wise absolute errors for the estimated dense models do not entirely lie below the corresponding 95\% and 5\% quantiles of pixel-wise absolute errors for the estimated sparse models, the maximum 95\% and 5\% quantiles can be used to compare the stabilities of the estimated dense and sparse models. An estimated model can be unstable if it has maximum 95\% and 5\% quantiles and has a broader band between the two quantiles. Despite the fact that the overall means and standard deviations of the pixel-wise mean absolute errors for the estimated dense and sparse models do not differ much as the aforementioned results suggest, we still believe that the estimated dense models are more stable than the estimated sparse models. Here, we want to emphasize that the spatial covariates that continue to exist in the sparse model and have non-zero estimated coefficients have a strong association with the emergency alarm call events, and as a result, we may draw a strong interpretation based on the estimated sparse model.
\begin{figure}[!htpb]
\centering
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Dense0595p1}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Dense0595p2}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Dense0595m}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Dense0595f}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Dense0595un}
\caption{The smoothed plots of pixel-wise mean absolute errors, the 5\% and 95\% quantiles of pixel-wise absolute errors of the true intensities and the estimated spatial intensities based on the undersampled point patterns. Row-wise, the plots show the evaluation of the estimated dense models for the point patterns with priority level 1, priority level 2, male, and female, as well as for the unmarked point pattern. The top, middle, and bottom curves in each plot represent the 95\% quantiles, the mean absolute errors, and the 5\% quantiles.}
\label{Densemae}
\end{figure}
\begin{figure}[!htpb]
\centering
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Sparse0595p1}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Sparse0595p2}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Sparse0595m}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Sparse0595f}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Sparse0595un}
\caption{The smoothed plots of pixel-wise mean absolute errors, the 5\% and 95\% quantiles of pixel-wise absolute errors of the true intensities and the estimated spatial intensities based on the undersampled point patterns. Row-wise, the plots show the evaluation of the estimated sparse models for the point patterns with priority level 1, priority level 2, male, and female, as well as for the unmarked point pattern. The top, middle, and bottom curves in each plot represent the 95\% quantiles, the mean absolute errors, and the 5\% quantiles.}
\label{Sparsemae}
\end{figure}
We can also visually assess how well the estimated models capture the spatial distribution of the emergency alarm call events. In order to do this, each of the datasets corresponding to the unmarked and marginal point patterns can be randomly divided into two parts. That is, we estimate the proposed model using 70\% of each dataset corresponding to the unmarked and marginal point patterns, and we use the remaining 30\% of each dataset to assess or validate the performance of the estimated models. We refer to the 30\% of each dataset that is retained for model validation in this context as a test point pattern. The estimated intensities for the test point patterns and their corresponding test point patterns can then be used to visually evaluate the performance of the estimated models. As can be seen in Figure \ref{CVDenseSparseData}, the hotspot regions in the test point patterns are well captured by their respective estimated spatial intensities. The plots in the figure also suggest that the estimated dense models seemingly perform better than their corresponding estimated sparse models, which are taken to be submodels of the estimated dense models.
\begin{figure}[H]
\centering
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Densep1CV}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Densep2CV}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{DensemCV}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{DensefCV}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{DenseunCV}\\
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Sparsep1CV}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Sparsep2CV}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{SparsemCV}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{SparsefCV}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{SparseunCV}\\
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Validationp1CV}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{Validationp2CV}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{ValidationmCV}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{ValidationfCV}
\includegraphics[width=0.18\textwidth, height=0.17\linewidth]{ValidationunCV}
\caption{A cross-validation based evaluation of the performance of the estimated dense and sparse models in estimating the spatial intensities of the emergency alarm call events. The estimated spatial intensities are shown in the first and second rows using the estimated dense and sparse models based on the corresponding test datasets in the third row. From left to right, the plots in the figure present the estimated spatial intensities for the test point patterns with priority level 1, priority level 2, male, and female, as well as for the unmarked test point pattern. Note that we have scaled the intensity estimates to range between 0 and 1 so that we may compare them more easily. We also multiplied the normalized versions by 1000 for ease of visualisation.}
\label{CVDenseSparseData}
\end{figure}
\section{Discussion}\label{Discn}
The purpose of this study is to model the spatial distribution of ambulance/medical emergency alarm call events in order to establish the framework for developing optimal ambulance dispatching strategies/algorithms. In order to develop the optimal dispatching strategies, it is essential to consider the response times and operational costs of prehospital resources such as ambulances. The optimal design can assist the concerned body in providing emergency medical assistance to life-threatening emergencies as quickly as possible. The locations of hotspot areas within the study region are crucial components of such designs. This means that identifying study area subregions with a high risk of events plays a crucial role in developing the optimal dispatching strategies. This work focuses on discerning such hotspot areas as well as selecting/exploring spatial covariates that are associated with the spatial distribution of the call events. \\
The data at hand consists of the locations of the events, which are the medical emergency alarm calls from patients in Skellefte{\aa}, Sweden, during the years 2014 -- 2018. The events have marks associated with them in addition to their spatial locations. These marks take the form of priority levels (1 or 2, where 1 denotes the highest priority) and the genders (male, female, or missing) of the patients linked to the calls. We have 14,919 events in total, with 7,204 and 7,715 locations having priority level 1 and priority level 2 marks, respectively. Of the events, 5,236 and 5,238 have male and female marks, respectively. In this case, the marks of some events are missing. We want to emphasise that we have chosen to treat the spatial locations associated with males, females, priority level 1s, and priority level 2s as individual point patterns, and the main reason for this is that we are interested in identifying spatial covariates associated with the intensity function of a spatial point process corresponding to each mark. This approach can also help to pinpoint subregions of the study area that have a high risk of events of a given mark. The other reason is that we have a large number of events with missing sex or gender labels. It should be emphasised that we could have, e.g.,~treated the missing entries as an independent thinning (a $p$-thinning) of the complete data set and thus scaled the intensity accordingly.\\
An exploratory analysis of the data shows that the observations tend to be along road networks, making the statistical modelling difficult (see Figure \ref{Or213}). Although a non-parametric estimate such as an (adaptive) kernel intensity estimate \citep{cronie2018non,davies2018tutorial,diggle1985kernel, baddeley2015spatial} or a (resample-smoothed) Voronoi intensity estimate \citep{ogata2003modelling, ogata2004space,moradi2019resample} does shed some light on the hotspots, we have observed that it does not do so to the fullest extent. More specifically, non-adaptive kernel estimators seem to over-smooth the data, whereas the other estimators, which are all adaptive estimators, tend to under-smooth the data (cf.~\citet{moradi2019resample}). Kernel intensity estimation, where a bandwidth was selected using a machine learning technique, i.e., K-means clustering, attempts to balance the over-and under-smoothing roles of the aforementioned intensity estimation methods, see \cite{bayisa2020large}, which selected the number of clusters through visual evaluation. In this work, a new heuristic algorithm is developed to obtain an optimal number of clusters and, thus, optimal bandwidth. The spatial distribution of the events or hotspot regions is well captured using the kernel intensity estimation based on the new bandwidth selection algorithm, as shown in \ref{BandWidth}. In addition, and more importantly, since we have access to various spatial covariates, we have shed some light on the spatial covariates in Subsection \ref{SpatialCovariates} that may influence the spatial variation of the intensity of the events. Here, it is important to emphasise that the spatial covariates we employ to model the spatial intensity function of the events are either based on demographics or the structure of the road networks.\\
Our modeling strategy/approach is to treat the data as a realization of a spatial point process and model its intensity function to quantify the spatially varying call risk. We assume that the intensity function is a log-linear function of the various spatial covariates under consideration, and we fit the intensity function by means of a Poisson process log-likelihood function. To carry out variable selection and adjust for over-/under-fitting, a regularisation term is added to the (approximated) log-likelihood function. Following \citet{yue2015variable}, we have chosen to employ an elastic-net penalty (a convex combination of the ridge and lasso penalties), which is useful when the number of covariates considered in the modeling exceeds the total point count or when the model contains several correlated spatial covariates. The elastic-net penalty is governed by a parameter $\alpha$, which controls how much weight we put on either the lasso or the ridge penalty, and by setting $\alpha = 1-\epsilon$ for some small $\epsilon > 0$, it performs much like the lasso, but avoids any unstable behaviour caused by extreme correlations. We used $\alpha = 0.95$, i.e.,~a lasso-like elastic-net, and ten-fold cross-validation to select an optimal estimate of the tuning parameter $\lambda.$\\
We considered two scenarios for the spatial covariates in this study. The first scenario is that the individual spatial covariates (we have also called them the original spatial covariates) and their products/pairwise interaction terms have been used in the modelling of the call events. In this case, we have nine hundred eighty-nine spatial covariates, which have been used in a regularised modelling of medical emergency alarm call events. The regularisation can select the original and pairwise interaction variables, making the results difficult to interpret. The second scenario involves only using the original spatial covariates in the modelling of the emergency alarm call events. In this work, we have also considered the cases of marked/marginal and unmarked spatial point patterns. Furthermore, we have investigated the dense model (we may call it the full model) and the sparse model (we may call it the submodel) for each of the spatial point patterns. A cyclical coordinate descent algorithm has been used for fast estimation of the parameters of the model. The intensity function of the events at the observed locations has been estimated using the estimated model parameters. To obtain the estimated intensities at the desired spatial locations in the study area, we have smoothed the estimated intensities at the observed locations, which are irregular spatial locations in the study area, using a Gaussian kernel.\\
Using the original spatial covariates and the first-order interaction terms, the estimated model parameters/coefficients for spatial point pattern with priority level 1 under lasso and lasso-like elastic-net regularisations have been shown in Figure \ref{Orgr2LassoElastic}. According to the figure, lasso and lasso-like elastic-net have chosen approximately 185 (18.69\%) and 207 (20.91\%) of the spatial covariates as important covariates in determining the spatial distribution of emergency alarm call events, respectively. Following the suggestion by \cite{yue2015variable}, we have used elastic-net regularisation, in particular, lasso-like elastic-net with $\alpha = 0.95$. Figure \ref{Orgr2} shows a ten-fold cross-validation method for selecting an optimal estimate of the tuning parameter for the spatial point pattern with priority level 1. The tuning parameter values corresponding to the blue and red vertical lines in the figure show an optimal and one standard error away from the optimal tuning parameter estimate. We may call the estimated model corresponding to the optimal estimate of the tuning parameter a dense model (or full model), while the estimated model corresponding to one standard error away from the optimal estimate of the tuning parameter may be referred to as a sparse model (or submodel). The optimal tuning parameter that was chosen has been used to obtain optimal parameter estimates, which have been used to generate the spatial intensities of the emergency alarm call events. The lasso-like elastic-net identified approximately 20.91\% and 36.06\% of the spatial covariates that are associated with the emergency alarm call events with priority levels 1 and 2, respectively. Furthermore, it has identified approximately 17.68\% and 35.76\% of the spatial covariates for the male and female marks of the alarm call events, respectively. The estimated intensity models based on the original spatial covariates and first-order interaction terms are depicted in Figure \ref{Orgr3}. The spatial variations of emergency alarm call events are not well captured using the estimated models based on the original spatial covariates and first-order interaction terms, as shown in the figure, particularly for emergency alarm call events with priority level 2. This finding prompted us to look into estimating the spatial intensities of emergency alarm call events using only the original covariates. \\
Using only the original spatial covariates, the lasso-like elastic-net provides dense solutions at the optimal tuning parameter estimates. That is, the majority of the spatial covariate coefficients are non-zero, see Table \ref{tab:EstimatedModels1}. Table \ref{tab:EstimatedModels2} depicts the sparse models that correspond to the estimated dense models. We would like to point out here that the spatial covariates that remain in the sparse models have a strong association with emergency alarm call events. Those spatial covariates that exist in the estimated dense models but not in the estimated sparse models may have weak associations with the events in comparison. To keep things simple, we will interpret the results based on the estimated sparse models identified by one-standard-error rule in \cite{hastie2017elements}. Accordingly, the one-standard-error rule identified event locations, population age categories (such as 0--5, 6--9, 10--15, 60--64, 65--69), and spatial covariates related to bus stops, main road networks, complete road networks, and densely populated areas as having strong associations with the spatial intensities of emergency alarm call events with priority level 1. With regard to a spatial point pattern with priority level 2, population age categories (such as 0--5, 10--15, 35--39, 55--59, 60--64), as well as spatial covariates related to bus stops, main road networks, complete road networks, and densely populated areas, have played an important role in determining the spatial variation of emergency alarm call events. Population age categories (such as 0--5, 6--9, 10--15, 60--64), event locations, and spatial covariates related to bus stops, main road networks, complete road networks, and densely populated areas are strongly associated with the spatial variation of emergency alarm call events with mark male. The one-standard-error rule has identified spatial covariates such as population age categories (such as 0--5, 10--5, 35--39), event locations, and spatial covariates related to bus stops, main road networks, complete road networks, and densely populated areas as key features in determining the spatial distribution of emergency alarm call events with mark female. In the case of an unmarked spatial point pattern, spatial covariates such as population age categories (such as 0--5, 10--15, 35--39), event locations, and spatial covariates related to bus stops, main road networks, complete road networks, and densely populated areas are strongly related to the spatial variation of emergency alarm call events. In the presence of the remaining spatial covariates, the benchmark intensity is not strongly associated with the emergency alarm call events for each of the marginal spatial point patterns, which may be due to the fact that the spatial locations of the events/event locations are more informative about human mobility than the benchmark intensity. Table \ref{tab:EstimatedModels2} also shows that, in the presence of other spatial covariates, population density is not strongly associated with the spatial variation of events. It could be due to the presence of a densely populated area in the model, which may provide more information about the distribution of the events than population density. Figure \ref{OrgrgGd3Compare} compares the estimated spatial intensity models based on the original spatial covariates and the first-order interaction terms to the estimated spatial intensity models based only on the original spatial covariates. The figure clearly shows that the estimated spatial intensity models based solely on the original covariates performed well in capturing the spatial distribution of the events.\\
We evaluated the performance of the estimated dense and sparse models for the unmarked and each of the marginal spatial point patterns in two ways. Firstly, we assessed the performance of the estimated dense and sparse models on undersampled data. To accomplish this, spatial intensity estimates of the unmarked and each of the marginal emergency alarm call events are generated and treated as true spatial intensities of the emergency alarm call events. Each dataset corresponding to the unmarked point pattern and the marginal spatial point patterns is undersampled. That is, from each spatial point pattern, we created one hundred undersampled datasets, accounting for 70\% of each dataset. Using the undersampled datasets, we estimated one hundred dense and sparse models and thus one hundred spatial intensities. The pixel-wise mean absolute errors of the estimated spatial intensity that was treated as the true spatial intensity of the emergency alarm call events and each of the hundred estimated spatial intensities are used to evaluate the stabilities of the estimated dense and sparse models. Furthermore, we assessed the performance of the estimated dense and sparse models using the 5\% and 95 \% quantiles of the pixel-wise absolute errors of the estimated spatial intensity that was treated as the true spatial intensity and each of the hundred estimated spatial intensities. Figures \ref{Densemae} and \ref{Sparsemae} show the evaluations of the stabilities of the estimated dense and sparse models in estimating the spatial intensities of emergency alarm call events. Secondly, we evaluated the performance of the estimated dense and sparse models for the unmarked point pattern and each of the marginal spatial point patterns, using 70\% of the total data for model fitting and 30\% of the total data for validating or testing the model. The performance of the estimated dense and sparse models in estimating the spatial intensities of emergency alarm call events can be evaluated visually by comparing the estimated spatial intensities and their corresponding patterns of spatial locations in the validation/test spatial point patterns, as shown in Figure \ref{CVDenseSparseData}. The plots in the figure show that the hotspot regions in the estimated spatial intensities of the events and the patterns of the spatial locations in the test spatial point patterns appear to coincide. That is, both the estimated dense and sparse models capture the spatial variations of emergency alarm call events well. Even though the estimated dense models appear to outperform the sparse models, the results show that both models are feasible.\\
The primary challenge in this work is the nature of the spatial data; that is, the locations of the events tend to be on road networks. Traditional kernel smoothing methods are incapable of displaying the spatial variation of emergency alarm call events. To address this issue, semi-parametric modelling of the spatial intensity function of the events has been developed, and it appears that the method is feasible for obtaining the estimated spatial intensity of the events, which will be used to design optimal ambulance dispatching strategies to provide life-threatening emergency medical services as quickly as possible.
Even though the semi-parametric modelling of the spatial intensity function of the events captures the spatial distribution of the emergency alarm call events well, the effect of the nature of the spatial data persists. To address this issue, we will continue to work on modelling the spatial intensity function of road network events. The proposed statistical method is tested with data gathered from Skellefte{\aa}, a municipality in northern Sweden. The spatial intensity function of emergency alarm call events will be modelled for each municipality in northern four counties (Norrbotten, V{\aa}sterbotten, V{\aa}sternorrland, and J{\aa}mtland) of Sweden using the proposed statistical model.\\
Finally, the study developed a new heuristic algorithm for bandwidth selection and discovered that spatial covariates such as population age categories, spatial location of events, and spatial covariates related to bus stops, main road networks, complete road networks, and densely populated areas play an important role in determining the spatial distribution of emergency alarm call events. The study also demonstrated that semi-parametric modelling of the spatial intensity function of the inhomogeneous Poisson process can handle the spatial variation of emergency alarm call events, and the estimated spatial intensity function of the events can be used as an input in designing optimal ambulance dispatching strategies to provide better emergency medical services for life-threatening health conditions.
\section*{Acknowledgments}
This work was supported by Vinnova [Grant No. 2018-00422] and the regions: V{\aa}sterbotten, Norrbotten, V{\aa}sternorrland and J{\aa}mtland-H{\aa}rjedalen. We are also grateful to our project partners SOS Alarm and the regions: V{\aa}sterbotten Norrbotten, V{\aa}sternorrland and J{\aa}mtland-H{\aa}rjedalen for fruitful discussions on the Swedish prehospital care.
\section*{Disclosure of Conflicts of Interest}
The authors have no relevant conflicts of interest to disclose.
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\section{INTRODUCTION}
\label{intro}
In recent times the virtual element method has been successfully applied to a
variety of problems\cite{da2013virtual,brezzi2013virtual}. The basic principle of virtual
element method has been discussed in \cite{de2014nonconforming,beirao2013basic}. A mimetic
discretization method with arbitrary polynomial order is presented recently
in \cite{brezzi2009mimetic}, but the classical finite element framework with
arbitrary polynomial is still making the
presentation cumbersome \cite{ciarlet2002finite}. The idea of the virtual element
methods is very similar to mimetic finite difference
methods \cite{droniou2010unified,da2008higher}. Virtual element space is a
unisolvent space of smooth functions containing a polynomial subspace. In other way
we can say non conforming virtual element method is a generalization of classical
nonconforming finite element methods. Very recently it has been clearly
understood that the degrees of freedom associated to trial/test functions is enough to
construct finite element framework, which lead to the study of virtual element method. Unlike
classical nonconforming FEM \cite{ciarlet2002finite,gao2011note,brenner2008mathematical,fortin1983non}
VEM has the advantage that virtual element space together with polynomial consistency
property allows us to approximate the bilinear form without explicit knowledge of basis
function.
Stability analysis of virtual element method irrespective of conforming or
non-conforming is quite different from classical finite element method. If the bilinear form
is symmetric then we divide it into two parts
\cite{ahmad2013equivalent,de2014nonconforming}, one is responsible for polynomial consistency
property and other one for stability analysis. The framework for stability analysis for
conforming and nonconforming virtual element is almost same. A pioneering work using
elliptic projection operator has been introduced to approximate symmetric bilinear form
by Brezzi et al in their papers
\cite{da2013virtual,beirao2013basic}. If the bilinear form is not symmetric like
convection diffusion reaction form, we can not extend this idea directly which may be
considered as the drawback of using elliptic projection operator
\cite{ahmad2013equivalent,brezzi2013virtual}. In this case we may use $L^{2}$ projection
operator for the modified approximation of bilinear form. The name virtual comes from the
fact that the local approximation space in each mesh either polygon or polyhedra contains
the space of polynomials together with some non-polynomial smooth function satisfying the
weak formulation of model problem. The novelty of this method is to take the spaces and the
degrees of freedom in such a way that the elementary stiffness matrix can be computed without
actually computing non-polynomial functions, but just using the degrees of freedom.
In this paper we have approximated non-symmetric bilinear form using $L^{2}$ projection
operator. We did not approximate the bilinear form same as \cite{de2014nonconforming} to
avoid difficulty. Conforming virtual element method using $L^{2}$ projection operator has
been already discussed for convection diffusion reaction equation with variable
coefficient. We have considered the nonconforming discretization same as defined
in \cite{knobloch2003p}. In two dimension, the design of schemes of order of
accuracy $k\geq 1$ was guided by the patch test \cite{irons1972experience,fortin1983non}
which enforces continuity at $k$ Gauss-Legendre points on edge. Over the last few
years, further generalization of non-conforming elements have been still considered by
several authors, one such generalization to stokes problem is considered
in \cite{stoyan2006crouzeix,arnold1984stable,comodi1989hellan}.
This article is organized as follows. In Section \ref{sec1} we discuss continuous setting
of the model problem (\ref{temp2}). In Section \ref{nconf} we have discussed construction of non-conforming virtual element
briefly since it has been discussed in the following paper\cite{de2014nonconforming}. Also, the
global and local setting of discrete bilinear formulation has been presented explicitly. In
Section \ref{step2} we have discussed the construction of bilinear form, stability analysis
and polynomial consistency property of the discrete bilinear form. Section \ref{step3},
discusses the construction of source term, boundary term and Section \ref{step5}
provides well-posedness and convergence analysis. Stability
analysis discussed in this paper is applicable only for the diffusion dominated problem. Finally in the last section we have
shown the optimal convergence rate in broken norm $\parallel.\parallel_{1,h}$.
\subsection{Continuous Problem}
\label{sec1}
In this section we present the basic setting and describe the continuous problem. Throughout
the paper, we use the standard notation of Sobolev spaces \cite{ciarlet2002finite}. Moreover, for
any integer $l \geq 0$ and a domain $D\in \mathbb{R}^{m} $ with
$m \leq d,d=2,3 $, $\mathbb{P}^{l}(D) $ is the space of polynomials of degree at most $l$
defined on $D$. We also assume the convention that $\mathbb{P}^{-1}=\{0\}$. Let the domain
$\Omega $ in $\mathbb{R}^{d} $ with $d=2,3$ be a bounded open polygonal domain with straight
boundary edges for $d=2$ or a polyhedral domain with flat boundary faces for $d=3$. Let
us consider the model problem:
\begin{flalign}
-& \nabla \cdot \left(\mathbf{K}(x) \, \nabla(u) \right)+\boldsymbol{\beta}(x) \cdot \nabla u +c(x) \, u = f(x) \quad \text{in} \quad \Omega, \nonumber \\
& u = g \quad \text{on} \quad \partial \Omega,
\label{temp2}
\end{flalign}
where $\mathbf{K} \in (C^{1}(\Omega))^{d\times d}$ is the diffusive
tensor, $\boldsymbol{\beta}(x) \in (C(\Omega))^{d} $ is the convection
field, $c\in C(\Omega)$ is the reaction field and $f\in L^{2}(\Omega)$. We assume
that $(c(x)-\frac{1}{2} \nabla \cdot \boldsymbol{\beta}(x))\geq c_{0}$, where $c_{0}$ is a
positive constant. This assumption guarantees that (\ref{temp2}) admits a unique
solution. The diffusive tensor is a full symmetric $d\times d$ sized matrix and strongly
elliptic, i.e. there exists two strictly positive real constants $\xi$ and $ \eta$ such that
\begin{center}
$\eta \, |\textbf{v}|^{2} \leq \textbf{v} \, \textbf{K} \, \textbf{v} \leq \xi \, |\textbf{v}|^{2} $
\end{center}
for almost every $ x \in \Omega$ and for any sufficiently smooth vector field
$v$ defined on $\Omega$, where $|\cdot|$ denotes the standard euclidean norm on
$\mathbb{R}^{d}$. $\mathbf{K},\boldsymbol{\beta},c$ are chosen to be
polynomials for the present problem.
The weak formulation of the model problem (\ref{temp2}) reads:
Find $u \in V_{g} \, \, \text{such} \, \, \text{that}$
\begin{equation*} A(u,v)=<f,v> \quad \forall \ v \in V \end{equation*}
where the bilinear form $A(\cdot,\cdot):V_{g} \times V \rightarrow \mathbb{R}$ is given by
\begin{eqnarray}
A(u,v) & = &\int_{\Omega} \textbf{K} \, \nabla u \cdot \nabla v +\int_{\Omega}(\boldsymbol{\beta} \cdot \nabla u)v+\int_{\Omega}c \, u \, v \nonumber \\
<f,v> & = &\int_{\Omega} fv
\label{temp3}
\end{eqnarray}
$<\cdot,\cdot>$ denotes the duality product between the functional space $V^{\prime}$ and $V$, where the space $V_{g}$ and $V$ are defined by
\begin{eqnarray}
V_{g} & = &\{ v \in H^{1}(\Omega):v|_{\partial \Omega }=g\} \nonumber \\
V & = &H^{1}_{0}(\Omega)
\label{tempb5}
\end{eqnarray}
We define the elemental contributions of the bilinear form $A(\cdot,\cdot)$ by
\begin{eqnarray}
a(u,v) & = &\int_{\Omega} \textbf{K} \, \nabla u \cdot \nabla v \ d\Omega \\
b(u,v) & = &\int_{\Omega}(\boldsymbol{\beta} \cdot \nabla u)v\ d\Omega \\
c(u,v) & = &\int_{\Omega}c \, u \, v \ d\Omega
\end{eqnarray}
Now we can estimate
\begin{eqnarray}
A(v,v) & = &\int_{\Omega}\textbf{K} \, \nabla v \cdot \nabla v+\int_{\Omega}(\boldsymbol{\beta} \cdot \nabla v) \, v+\int_{\Omega}c \, v^{2} \nonumber\\
& \geq &\eta \, \| \nabla v \|^2_0+\int(c-\frac{1}{2}\nabla \cdot \boldsymbol{\beta}) \, v^{2} \nonumber \\
& \geq & C \, \| v \|^2_1
\label{temp4}
\end{eqnarray}
Combining inequality (\ref{temp4}) with the continuity of $A(\cdot, \cdot)$ it follows that there
exists a unique solution to the variational form of equation (\ref{temp2}).
\subsection{Basic Setting}
We describe now the basic assumptions of the mesh partitioning and related
function spaces. Let $\{ \tau_{h}\}$ be a family of decompositions of $\Omega $ into
elements, $T$ and $ \varepsilon_{h}$ denote a single element and set of edges of a
particular partition respectively. By $\varepsilon^{0}_{h}$ and
$\varepsilon^{\partial}_{h}$ we refer to the set of interior and boundary
edges/faces respectively. We will follow the same assumptions on the family of partitions
as \cite{de2014nonconforming}.
\subsection{Assumptions on the family of partitions $\{\tau_{h} \}$}
There exists a positive $\rho >0 $ \ such that
\begin{itemize}
\item[(A1)] for every element $ T$ and for every edge/face $e\subset \partial T$, we
have $h_{e}\geq \rho \, h_{T} $.
\item[(A2)] every element $T$ is star-shaped with respect to all the points of a sphere of
radius $\geq \rho \, h_{T}$.
\item[(A3)] for $d=3$, every face $e \in \varepsilon_{h}$ is star-shaped with respect to all the
points of a disk having radius $\geq \rho \, h_{e} $.
\end{itemize}
The maximum of the diameters of the elements $T \in \tau_{h} $ will be denoted by $h$. For
every $h>0$, the partition $\tau_{h}$ is made of a finite number of polygons or polyhedra.
We introduce the broken sobolev space for any $s>0$
\begin{equation}
H^{s}(\tau_{h})=\epi_{T \in \tau_{h}} H^{s}(T)=\left\{v \in L^{2}(\Omega):v|_{T} \in H^{s}(T) \right\}
\label{temp5}
\end{equation}
and define the broken $ H^{s}$-norm
\begin{equation}
\| v \|^{2}_{s,\tau_{h}}=\sum_{T \in \tau_{h}} \| v \|^{2}_{s,T} \quad \forall v \in H^{s}(\tau_{h})
\label{temp6}
\end{equation}
In particular for $s=1$
\begin{equation}
\| v \|^{2}_{1,\tau_{h}}=\sum_{T \in \tau_{h}} \| v \|^{2}_{1,T} \quad \forall v \in H^{1}(\tau_{h}) \nonumber
\end{equation}
Let $e \in \varepsilon^{0}_{h}$ be an interior edges and let $T^{+}, T^{-} $ be two
triangles which share $e$ as a common edge. We denote the unit normal on $e$ in the
outward direction with respect to $T^{\pm}$ by $ n_{e}^{\pm}$. We then define the
jump operator as:
\begin{center}
$[\vert v \vert]$ := \, $v^{+} n_{e}^{+} + v^{-}n_{e}^{-}$ \quad on \quad $ e \in \varepsilon^{0}_{h}$
\end{center}
\begin{center}
and $[\vert v \vert]$ := \, $v \, n_{e}$ \quad on \quad $e \in \varepsilon^{\partial}_{h}$
\end{center}
\subsection{Discrete space}
In this section we will introduce discrete space same as \cite{de2014nonconforming}. For
an integer $k \geq 1$ we define
\begin{equation}
H^{1,nc}(\tau_{h};k)=\left\{ v \in H^{1}(\tau_{h}): \int_{e}[| v |] \cdot n_{e} \, q \, ds=0 \ \forall \ q \in P^{k-1}(e), \ \forall \ e \in \varepsilon_{h} \right\}
\label{temp7}
\end{equation}
We mention that if $v \in H^{1,nc}(\tau_{h};k) $ then $v $ in said to satisfy patch
test \cite{knobloch2003p} of order $k$. To approximate second order problems we must satisfy the patch test. The
space $H^{1,nc}(\tau_{h};1)$ is the space with minimal required order of patch test
to ensure convergence analysis.
\subsection{Nonconforming virtual element methods}
\label{nconf}
In this section we explain discrete nonconforming VEM framework for the equation
(\ref{temp2}). Before passing from the weak formulation to discrete problem, we first
apply integration by parts to the convective term $(\boldsymbol{\beta} \cdot \nabla u,v)$ to
obtain
\begin{equation}
\int_{\Omega} (\boldsymbol{\beta} \cdot \nabla u) v=\frac{1}{2} \left[\int_{\Omega} (\boldsymbol{\beta} \cdot \nabla u) v -\int_{\Omega} (\boldsymbol{\beta} \cdot \nabla v) u -\int_{\Omega}\nabla \cdot \boldsymbol{\beta} \, u \, v \right] \nonumber
\end{equation}
for $u \in H^{1}(\Omega),v\in H^{1}_{0}(\Omega)$
Bilinear form (\ref{temp3}) can be written as
\begin{equation}
A(u,v)=\sum_{T \in \tau_{h}} A^{T}(u,v)
\label{temp8}
\end{equation}
where
\begin{flalign}
A^{T}(u,v) & =\int_{T} \textbf{K} \, \nabla u \cdot \nabla v +b^{\text{conv}}_{T}(u,v)+\int_{T} c \, u \, v \nonumber \\
\text{and} \quad b^{\text{conv}}_{T}(u,v) & = \frac{1}{2} \left[\int_T (\boldsymbol{\beta}
\cdot \nabla u) v -\int_T (\boldsymbol{\beta} \cdot \nabla v) u -
\int_{T} \nabla \cdot \boldsymbol{\beta} \, u \, v \right]
\label{temp9}
\end{flalign}
We want to construct a finite dimensional space $ V^{k}_{h} \subset H^{1,nc}(\tau_{h};k) $, a
bilinear form $A_{h}(\cdot , \cdot):V_{h, g}^{k} \times V_{h}^{k} \rightarrow \mathbb{R}$, and
an element $f_{h} \in (V^{k}_{h})^{\prime}$ such that the discrete problem:
\linebreak
Find $u_{h} \in V_{h,g}^{k}$ such that
\begin{equation}
A_{h}(u_{h},v_{h})=<f_{h},v_{h}> \ \ \forall \ v_{h}\in V_{h}^{k}
\label{temp10}
\end{equation}
\quad has a unique solution $u_{h}$.
\subsection{Local nonconforming virtual element space:}
We define for $k\geq 1$ the finite dimensional space $V^{k}_{h}(T)$ on $T$ as
\begin{equation}
V^{k}_{h}(T)=\left\{v \in H^{1}(T):\frac{\partial v}{\partial \mathbf{n}} \in \mathbb{P}^{k-1}(e),v \mid_{e}\in \mathbb{P}^{k}(e) \ \forall \ e \subset \partial T, \Delta v \in \mathbb{P}^{k-2}(T) \right\}
\label{temp11}
\end{equation}
with the usual convention that $\mathbb{P}^{-1}(T)=\{0\} .$ \\
From the definition, it immediately follows that $\mathbb{P}^{k}(T) \subset
V^{k}_{h}(T)$. The non-conforming VEM is formulated through the $L^2$ projection
operators,
\begin{equation}
\Pi_{k}:V^{k}_{h}(T)\rightarrow \mathbb{P}^{k}(T) \nonumber
\end{equation}
\begin{equation}
\Pi_{k-1}:\nabla\left(V^{k}_{h}(T)\right) \rightarrow \left(\mathbb{P}_{k-1}(T)\right)^{d} \nonumber.
\end{equation}
For $k=1$, degrees of freedom are defined same as Crouzeix-Raviart
element \cite{crouzeix1973conforming,brenner2008mathematical}. In general we can say
normal derivative $\frac{\partial u}{\partial n}$ of an arbitrary element of
$V^{1}_{h}$ is constant on each edge $e\subset \partial T$
(and different on each edges)and inside $T$ are harmonic(i.e., $\Delta v=0$). It can
be easily concluded that the total no of degrees of freedom of a particular element $T$
with $n$ edges/faces is $n$ which is explained in detail in \cite{de2014nonconforming}.
For $k=2$, the space $V^{2}_{h}(T)$ consists of functions whose normal
derivative $\frac{\partial u}{\partial n}$ is a polynomial of degree $1$ on each
edge/face, i.e. $\frac{\partial u}{\partial n} \subset \mathbb{P}^{1}(e)$ and is a
polynomial of degree $0$ on interior region, i.e. a constant function. The dimension
of $V^{2}_{h}(T)$ is $d \, n+1$ where $n, d$ denote number of edges/face associated with
an element $T$ and spatial dimension of $T$ respectively.
For each element $T$,the dimension of $V^{k}_{h}(T)$ is given by
\begin{equation*}
N_{T} =
\begin{cases}
\displaystyle nk+\frac{(k-1)k}{2} & \text{for} \quad d=2, \\
\displaystyle \frac{nk(k+1)}{2}+\frac{(k-1)k(k+1)}{6} & \text{for} \quad d=3
\end{cases}
\end{equation*}
which is explained in detail in \cite{de2014nonconforming,brezzi2013virtual}. We
need to introduce some further notation to define degrees of freedom same
as \cite{de2014nonconforming}. Let us define space of scaled monomials
$M^{l}(e)$ and $M^{l}(T)$ on $e$ and $T$ as
\begin{equation}
M^{l}(e)=\left\{\left(\frac{(\mathbf{x}-\mathbf{x}_{e})}{h_{e}}\right)^{s}, \vert s \vert \leq l \right\} \nonumber
\end{equation}
\begin{equation}
M^{l}(T)=\left\{\left(\frac{(\mathbf{x}-\mathbf{x}_{T})}{h_{T}}\right)^{s}, \vert s \vert \leq l \right\} \nonumber
\end{equation}
where $s=(s_{1},s_{2}, \cdots, s_{d})$ be a $d$-dimensional multi index notation with
$\vert s \vert =\sum_{i=1}^{d}s_{i}$ and $\displaystyle \mathbf{x}^{s}=\Pi_{i=1}^{d} x_{i}^{s_{i}}$
where $\mathbf{x}=(x_{1},\dot{...}, x_{d}) \in \mathbb{R}^{d} $ and $l\geq 0$ be an integer.
In $V^{k}_{h}(T)$ we will choose same degrees of freedom as defined in \cite{de2014nonconforming}.
On each edge $e\subset \partial T$
\begin{equation}
\mu^{k-1}_{e}(v_{h})=\left\{ \frac{1}{e} \int_{e} v_{h} \, q \, ds,\quad \forall q \in M^{k-1}(e)\right\}
\label{temp12}
\end{equation}
on each element $T$
\begin{equation}
\mu^{k-2}_{T}(v_{h})= \left\{ \frac{1}{\vert T \vert} \int_{T} v_{h} \, q, \quad \forall q \in M^{k-2}(T) \right\}
\label{temp13}
\end{equation}
The set of functional defined in (\ref{temp12}) and (\ref{temp13}) are unisolvent for
the space $V^{k}_{h}(T)$
\begin{lem}
Let T be a simple polygon/polyedra with $n$ edges/faces, and let $V^{k}_{h}(T)$ be the space
defined in (\ref{temp11}) for any integer $k \geq 1$.The degrees of freedom (\ref{temp12})
and(\ref{temp13}) are unisolvent for $V^{k}_{h}(T)$.
\end{lem}
\begin{proof}
See the details in \cite{de2014nonconforming}.
\end{proof}
The degrees of freedom equation (\ref{temp12}) and (\ref{temp13}) are defined by using
the monomials in $\mu^{k-1}_{e}$ and $\mu^{k-2}_{T}$ as basis functions for the polynomial
spaces $\mathbb{P}^{k-1}(e)$ and $\mathbb{P}^{k-2}(T)$. This special chioce of the basis
functions gives advantages to implement the nonconforming VEM on arbitrary polygonal
domain. Implementation part is described explicitly in articles
\cite{ahmad2013equivalent,da2013virtual}.
\subsection{Global nonconforming virtual element space }
We now introduce the nonconforming(global) virtual element space $V^{k}_{h}$ of order $k$. We
have already defined local nonconforming virtual element space $V^{k}_{h}$(T) on each
element $T$ of partition $\tau_{h}$. The global nonconformi
ng virtual element space
$V^{k}_{h}$ of order $k$ is defined by
\begin{equation}
V^{k}_{h}=\left\{ v_{h}\in H^{1,nc}(\tau_{h};k):v_{h}|_{T} \in V^{k}_{h}(T)\quad \forall \, T \in \tau_{h} \right\}
\label{temp14}
\end{equation}
\subsection{Interpolation error}
We can define an interpolation operator in $V^{k}_{h}$ having optimal approximation
properties using same idea as described in
\cite{brezzi2013virtual,ciarlet1991basic,ciarlet2002finite,de2014nonconforming}. We can
define an operator $\chi_{i}$ which associates each function $\phi$ to the $i^{th}$ degree
of freedom and virtual basis functions $\psi_{i} $ of global virtual element space satisfies
the condition $\chi_{i}(\psi_{i})=\delta_{ij}$ for $i,j=1,2,\cdots,N $ where $N$ denotes
the number of degrees of freedom of global space. Then for
any $ v \in H^{1,nc}(\tau_{h},k)$, there exists unique $ v_{I} \in V^{k}_{h}$ such that
\begin{equation}
\chi_{i}(v-v_{I})=0 \quad \forall \ i=1,2,\cdots,N \nonumber
\end{equation}
Using all these properties we can claim that there exists a constant $C>0$, independent
of $h$ such that for every $h>0$, every $K \in \tau_{h}$, every $s$ with $2\leq s\leq k+1 $
and every $v \in H^{s}(K)$ the interpolant
$v_{I} \in V^{k}_{h}$ satisfies:
\begin{equation}
\| v-v_{I} \|_{0,T}+h_{T} \| v-v_{I} \|_{1,K} \leq C h^{s}_{T} \| v \|_{s,T}
\label{temp15}
\end{equation}
Technical detail of the above approximation is described in \cite{de2014nonconforming}.
\section{Construction of $A_{h}$}
\label{step2}
The goal of this section is to define the nonconforming virtual element discretization
(\ref{temp10}). If the discretized bilinear form is symmetric then we can easily prove the
good stability and nice approximation property and ensure the computability of the defined
bilinear form $A_{h}(\cdot,\cdot)$ over functions in $V^{k}_{h}$. But since the bilinear
form of the convection diffusion reaction equation is not symmetric we will accept certain
assumptions on the model problem (\ref{temp2}). Diffusive and reactive part are symmetric
and therefore we will split these two terms as a sum of polynomial part or consistency
part and stability part. Convective part is not symmetric therefore we will take only
polynomial approximation of this part. The present framework is only applicable to
the diffusion-dominated case when the Peclet number is sufficiently small.
\begin{equation}
A_{h}(u_{h},v_{h})=\sum_{T \in \tau_{h}} A^{T}_{h}(u_{h},v_{h}) \quad \forall\ u_{h},v_{h} \in V^{k}_{h},
\label{temp16}
\end{equation}
where $A^{T}_{h}:V^{k}_{h}\times V^{k}_{h} \rightarrow \mathbb{R} $ denoting the restriction
to the local space $V^{k}_{h}(T)$. The bilinear form $ A^{T}_{h} $ can be decomposed into
sum of element terms. Thus defining the approximate bilinear form
\begin{equation}
A^{T}_{h}(u_{h},v_{h}):=a^{T}_{h}(u_{h},v_{h})+b^{T}_{h}(u_{h},v_{h})+c^{T}_{h}(u_{h},v_{h})
\label{temp16}
\end{equation}
for each element $T$, we define the element contributions to $A^{T}_{h}$ by
\begin{eqnarray}
a^{T}_{h}(u_{h},v_{h}) & := & \int_{T} \mathbf{K} \, \Pi_{k-1}(\nabla u_{h}) \cdot \Pi_{k-1}(\nabla v_{h}) \, dT \nonumber \\
& + &\mathbf{S}^{T}_{a}\left((I-\Pi_{k})u_{h},(I-\Pi_{k})v_{h} \right) \nonumber\\
b^{T}_{h}(u_{h},v_{h}) & := & \frac{1}{2} \Big( \int_{T}\boldsymbol{\beta} \, \Pi_{k-1}(\nabla u_{h}) \, \Pi_{k}(v_{h}) \, dT \nonumber \\
&- &\int \boldsymbol{\beta} \cdot \Pi_{k-1}(\nabla v_{h}) \, \Pi_{k}(u_{h}) \, dT \nonumber \\
& - &\int_{T}(\nabla \cdot \boldsymbol{\beta}) \, \Pi_{k}(u_{h}) \, \Pi_{k}(v_{h}) \, dT \Big) \nonumber \\
c^{T}_{h}(u_{h},v_{h}) & := & \int_{T} {c} \, \Pi_{k}(u_{h}) \, \Pi_{k}(v_{h}) dT \nonumber \\
&+ & \mathbf{S}^{T}_{c}\left((I-\Pi_{k})u_{h},(I-\Pi_{k})v_{h}\right)
\label{temp17}
\end{eqnarray}
where $\mathbf{S}^{T}_{a}$ and $\mathbf{S}^{T}_{c}$ are the stabilising terms. These terms
are symmetric and positive definite on the quotient space
$V^{k}_{h}(T)\diagup \mathbb{P}_{k}(T) $ and satisfy the stability property:
\begin{center}
$\alpha_{\ast} a^{T}(v_{h},v_{h})\leq \mathbf{S}^{T}_{a}(v_{h},v_{h}) \leq \alpha^{\ast}a^{T}(v_{h},v_{h}),$
\label{temp18}
\end{center}
\begin{center}
$\gamma_{\ast}c^{T}(v_{h},v_{h}) \leq \mathbf{S}^{T}_{c}(v_{h},v_{h}) \leq \gamma^{\ast}c^{T}(v_{h},v_{h}), $
\label{temp19}
\end{center}
for all $v_{h}\in V^{k}_{h} $ with $\Pi_{k}(v_{h})=0 $. The first term ensures polynomial
consistency property and second term ensures stability property of the corresponding bilinear
form $ a^{T}_{h}(u_h, v_h)$ and $c^{T}_{h}(u_h,v_h)$.
\subsection{Consistency}
\begin{lem}
Let ${u_{h}} |_{T} \in \mathbb{P}^k(T)$ and $v_{h} |_{T} \in H^2(T)$, then the
bilinear forms $ a^{T}_{h}, b^{T}_{h}, c^{T}_{h}$ defined in equation (\ref{temp17})
satisfy the following consistency property for all $h>0$ and for all $T \in \tau_{h}$ .
\end{lem}
\begin{proof}
Whenever either $u_{h}$ or $ v_{h}$ or both are elements of the polynomial space
$\mathbb{P}_{k}(T)$, the following consistency property satisfy
\begin{eqnarray}
a^{T}_{h}(u_{h},v_{h})& = & a^{T}(u_{h},v_{h}) \nonumber\\
b^{T}_{h}(u_{h},v_{h}) & = & b^{T}(u_{h},v_{h}) \nonumber \\
c^{T}_{h}(u_{h},v_{h}) &= & c^{T}(u_{h},v_{h})
\label{temp20}
\end{eqnarray}
The consistency property in (\ref{temp20}) follows immediately since
$\Pi_{k}( \mathbb{P}^{k}(T))=\mathbb{P}^{k}(T) $ which implies
$\mathbf{S}^{T}_{a}\left(p-\Pi_{k}(p),v_{h}-\Pi_{k}(v_{h})\right)=0$ and
$\mathbf{S}^{T}_{c}\left(p-\Pi_{k}(p),v_{h}-\Pi_{k}(v_{h})\right)=0$. Now we will prove
$a^{T}_{h}(p,v_{h}) = a^{T}(p,v_{h}), \, b^{T}_{h}(p,v_{h}) = b^{T}(p,v_{h}), \, c^{T}_{h}(p,v_{h}) = c^{T}(p,v_{h})$
for all $p \in \mathbb{P}^{k}(T)$ and for all $v_{h}\in V^{k}_{h}(T)$.
\begin{eqnarray}
a^{T}_{h}(p,v_{h}) &= & \int_{T} \mathbf{K} \, \nabla p \cdot \Pi_{k-1}(\nabla v_{h}) \, dT \nonumber \\ \nonumber
& = & \int_{T} \left(\Pi_{k-1}(\nabla v_{h})-\nabla v_{h} \right) \, \mathbf{K} \, \nabla p \, dT \\
& + & \int_{T} \nabla v_{h} \, \mathbf{K} \, \nabla p \, \nonumber \, dT \\
& = & \int_{T} \mathbf{K} \, \nabla p \, \nabla v_{h} \nonumber \, dT \\
& = & a^{T}(p,v_{h})
\label{temp21}
\end{eqnarray}
\begin{eqnarray}
b^{T}_{h}(p,v_{h}) & = & \frac{1}{2} \Big(\int_{T} \boldsymbol{\beta} \cdot \nabla p \, \Pi_{k}(v_{h}) dT \nonumber \\
& - & \int_{T} \boldsymbol{\beta} \cdot \Pi_{k-1}(\nabla v_{h}) \, p \, dT-\int_{T}(\nabla. \boldsymbol{\beta}) \, p \, \Pi_{k}(v_{h}) \, dT \Big)
\label{tempc1}
\end{eqnarray}
\begin{eqnarray}
\int_{T} \boldsymbol{\beta} \cdot \nabla p \, \Pi_{k}(v_{h}) \, dT & = & \int_{T} (\Pi_{k}(v_{h})-v_{h}) \, \nabla p \cdot \boldsymbol{\beta} \, dT \nonumber\\
& + & \int_{T} \boldsymbol{\beta} \cdot \, \nabla p \, v_{h} \nonumber \\
& = & \int_{T} \boldsymbol{\beta} \cdot \nabla p \, v_{h}
\label{tempc2}
\end{eqnarray}
\begin{eqnarray}
\int_{T} \boldsymbol{\beta} \cdot \Pi_{k-1}(\nabla v_{h}) \, p &=& \int_{T} (\Pi_{k-1}(\nabla v_{h})-\nabla v_{h}) \cdot \boldsymbol{\beta} \, p \nonumber \\
& + &\int_{T} \boldsymbol{\beta} \cdot \nabla v_{h} \, p \nonumber \\
& \leq & \parallel \boldsymbol{\beta} \parallel_{\infty,T} \parallel p \parallel_{0,T} \parallel \nabla v_{h}-\Pi_{k-1}(\nabla v_{h}) \parallel +\int_{T} \boldsymbol{\beta} \cdot \nabla v_{h} \, p \nonumber \\
& \leq & C \parallel \boldsymbol{\beta} \parallel_{\infty,T} h_{T} \vert \nabla v_{h} \vert_{1,T} \parallel p \parallel_{0,T} \nonumber \\
& +& \int_{T} \boldsymbol{\beta} \cdot \nabla v_{h} \, p \nonumber \\
& \approx & \int_{T} \boldsymbol{\beta} \cdot \nabla v_{h} \, p \quad (\, \text{for small values of} \, \, h_{T})
\label{tempc3}
\end{eqnarray}
\begin{eqnarray}
\int_{T} (\nabla \cdot \boldsymbol{\beta}) \, p \, \Pi_{k}(v_{h}) \, dT &=& \int_{T} (\nabla \cdot \boldsymbol{\beta})\left(\Pi_{k}(v_{h})-v_{h}\right) \, p +\int_{T} (\nabla \cdot \boldsymbol{\beta}) p \, v_{h} \nonumber\\
&=& \int_{T} (\nabla \cdot \beta) \, p \, v_{h}
\label{tempc4}
\end{eqnarray}
Putting estimations (\ref{tempc2}), (\ref{tempc3}) and (\ref{tempc4}) in (\ref{tempc1})
we get $b^{T}_{h}(p,v_{h})=b^{T}(p,v_{h})$
Similarly,
\begin{equation}
c^{T}_{h}(p,v_{h})=c^{T}(p,v_{h})
\label{tempc5}
\end{equation}
Hence we proved required polynomial consistency of local discrete bilinear
form $A^{T}_{h}(u_{h},v_{h})$, i.e.
\begin{equation}
A^{T}_{h}(p,v_{h})=A^{T}(p,v_{h})
\label{tempc6}
\end{equation}
for $p \in \mathbb{P}^{k}$ and $ v_{h} \in V^{k}_{h}$
\end{proof}
\subsection{Discrete stability}
Before discussing stability property of the discrete bilinear form
$A^{T}_{h}(u_{h},v_{h})$ we reveal that the following framework is applicable for
diffusion dominated case. The stabilizing part $\mathbf{S}^{T}_{a}$ and $\mathbf{S}^{T}_{c}$
of the discrete bilinear form(\ref{temp17}) ensure the stability of the bilinear
form, precisely we can conclude that there exist two pairs of positive
constants $\alpha_{\ast} $, $\alpha^{\ast}$ and $\gamma^{\ast}, \gamma_{\ast} $ that are
independent of $h$ such that
\begin{equation}
\alpha_{\ast}a^{T}(v_{h},v_{h})\leq a^{T}_{h}(v_{h},v_{h}) \leq \alpha^{\ast}a^{T}(v_{h},v_{h})
\label{temp22}
\end{equation}
\begin{equation}
\gamma_{\ast}c^{T}(v_{h},v_{h})\leq c^{T}_{h}(v_{h},v_{h}) \leq \gamma^{\ast}c^{T}(v_{h},v_{h})
\label{temp23}
\end{equation}
for all $v_{h} \in V^{k}_{h}(T)$ and mesh elements $T$.
\section{Construction of right hand side term}
\label{step3}
In order to build the the right hand side $ <f_{h},v_{h}>$ for $v_{h}\in V^{k}_{h}$ we need
polynomial approximation of degree $(k-2)\geq 0$,that is $f_{h}=\textbf{P}^{T}_{k-2}f$ on
each $T \in \tau_{h}$, where $\textbf{P}^{T}_{k-2}$ is $L^{2}(T)$ projection operator
on $\mathbb{P}^{k-2}(T)$ for each element $T \in \tau_{h}$. We define $f_{h}$ locally by:
\begin{equation*}
(f_{h})|_{T}:=
\begin{cases}
\textbf{P}^{T}_{0}(f) & \text{for $k=1 $} \\
\textbf{P}^{T}_{k-2}(f)& \text{ for $k\geq 2 $}
\label{temp24}
\end{cases}
\end{equation*}
The projection operator is orthogonal to the polynomial space $\mathbb{P}^{k}(T)$. Therefore
we can write as
\begin{equation}
<f_{h},v_{h}> :=\sum_{T} \int_{T} \mathbf{P}^{T}_{k-2}(f) \, v_{h} \, dT=\sum_{T}\int_{T}f \, \, \mathbf{P}^{T}_{k-2}(v_{h}) \, dT
\end{equation}
Now we can prove the error estimates using orthogonality property of
projection operator, Cauchy-Schwarz inequality and standard
approximates \cite{ciarlet2002finite,ciarlet1991basic} for $k\geq2$, $s \geq 1.$
\begin{eqnarray}
\left| <f,v_{h}>-<f_{h},v_{h}> \right| & = & \left| \sum_{T} \int_{T}\left(f-\mathbf{P}^{T}_{k-2}(f)\right) \, v_{h} \, dT \right| \nonumber\\
& = &\left| \sum_{T} \int_{T}(f-\mathbf{P}^{T}_{k-2}(f))(v_{h}-\mathbf{P}_{0}^{T}(v_{h})) \, dT \right| \nonumber \\
& \leq & \parallel (f-\mathbf{P}^{T}_{k-2}(f)) \parallel_{0,\tau_{h}} \parallel (v_{h}-\mathbf{P}^{T}_{0}(v_{h}))\parallel_{0,\tau_{h}} \nonumber \\
& \leq & C h^{\text{min}(k,s)} \vert f \vert_{s-1,h} \vert v_{h} \vert_{1,h}
\label{temp25}
\end{eqnarray}
For $k=1$ the above analysis is not applicable, so we do the following
\begin{equation}
\tilde{v}_{h}|_{K}:=\frac{1}{n} \sum_{e \in \partial K} \frac{1}{\vert e \vert} \int_{e} v_{h} \, ds \approx \mathbf{P}^{T}_{0}(v_{h}),
\label{temp26}
\end{equation}
\begin{equation}
<f_{h},\tilde{v}_{h}>:=\sum_{T}\int_{T} \mathbf{P}^{T}_{0}(f) \, \tilde{v}_{h} \approx \sum_{T} \vert T \vert \, \mathbf{P}^{T}_{0}(f) \, \mathbf{P}^{T}_{0}(v_{h})
\end{equation}
\begin{eqnarray}
\left|<f,v_{h}>-<f_{h},\tilde{v}_{h}> \right| &= & \left| \sum_{T} \int_{T}(f v_{h}-\mathbf{P}^{T}_{0}(f)\tilde{v}_{h}) \right| \nonumber \\
&\leq & \left| \sum_{T} \int_{T}(f-\mathbf{P}_{0}^{T}(f))v_{h} \right|+ \left| \sum_{T} \int_{K} \mathbf{P}_{0}^{T}(f)(v_{h}-\tilde{v}_{h}) \right| \nonumber \\
&= & \left| \sum_{T} \int_{T}(f-\mathbf{P}_{0}^{T}(f))v_{h} \right| \nonumber \\
& \leq & C h \vert f \vert_{0,h} \vert v_{h} \vert_{1,h}
\label{temp27}
\end{eqnarray}
\subsection{Construction of the boundary term. }
We define $g_{h}:=\mathbf{P}^{e}_{k-1}(g)$ where $g$ is non-homogeneous Dirichlet
boundary value.
\begin{equation}
\int_{\varepsilon^{\partial}_{h}} g_{h} \, v_{h} \, ds:=\sum_{e \in {\varepsilon^{\partial}_{h}}} \int_{e} \mathbf{P}^{e}_{k-1}(g) \, v_{h} \, ds=\sum_{e \in {\varepsilon^{\partial}_{h}}} \int_{e} g \, \mathbf{P}^{e}_{k-1}(v_{h}) \, ds \quad \forall \ v_{h}\in V^{k}_{h}
\end{equation}
The above estimation guides us to compute the boundary terms easily using
degrees of freedom.
\subsection{Estimation of the jump term}
\begin{equation}
J_{h}(u,v_{h})=\sum_{T \in \tau_{h}} \int_{\partial T} \frac{\partial u}{\partial \mathbf{n}_{T}} \, v \, ds=\sum_{e \in \varepsilon_{h}} \int_{e} \nabla u \cdot [\vert v_{h}\vert ]
\label{temp28}
\end{equation}
\begin{lem}
Let (A1,A2,A3) be satisfied, $k\geq 1$ and $ u \in H^{s+1}(\Omega)$ with $ s\geq 1$ be the
solution of the model problem (\ref{temp2}). Let
$v_{h}\in H^{1,nc}(\tau_{h};1)$ be an arbitrary function. Then, there exists a
constant $C>0 $ independent of $h$ such that
\begin{equation}
\vert J_{h}(u,v_{h})\vert \leq C h^{\text{min}(s,k)} \parallel u \parallel_{s+1,\Omega} \vert v_{h} \vert_{1,h}
\label{tempb6}
\end{equation}
\end{lem}
\begin{proof}
Let $v_{h} \in H^{1,nc}(\tau_{h};k)$ be an arbitrary element. From the definition of the
finite element space $H^{1,nc}(\tau_{h};k)$, we can say $v_{h}$ satisfies patch test of
order $k$. Hence the following equality holds
\begin{equation}
\int_{e} [\vert v_{h}\vert ] \, q \, ds=0, \quad \forall q \in \mathbb{P}_{k-1}
\label{temp29}
\end{equation}
Let $\mathbf{P}^{e}_{k}:L^{2}(e)\rightarrow \mathbb{P}^{k}(e) $ is the $L^{2}$-orthogonal
projection operator onto the space $\mathbb{P}^{k}(e)$ for $k\geq 1$. Using patch test of
order $k$, Cauchy-Schwarz inequality and $L^{2}(e)$ orthogonal projection
operator $\mathbf{P}^{e}_{k}$ we can find
\begin{eqnarray}
J_{h}(u,v_{h}) & = &\sum_{T \in \tau_{h}} \int_{\partial T} \frac{\partial u}{\partial \mathbf{n}_{T}} v ds\nonumber\\
&= &\sum_{e \in \varepsilon_{h}} \int_{e} \nabla u.[\vert v_{h}\vert ]\nonumber \\
& = &\vert \sum_{e \in \varepsilon_{h}} \int_{e} (\nabla u-\mathbf{P}^{e}_{k-1}(\nabla u)).[\vert v_{h}\vert] ds \vert \nonumber \\
& = &\vert\sum_{e \in \varepsilon} \int_{e} (\nabla u-\mathbf{P}^{e}_{k-1}(\nabla u)).([\vert v_{h}\vert]-\mathbf{P}^{e}_{0}([\vert v_{h}\vert]))\vert \nonumber \\
& \leq & \sum_{e \in \varepsilon_{h}} \parallel \nabla u-\mathbf{P}^{e}_{k-1}(\nabla u)\parallel_{0,e} \parallel [\vert v_{h}\vert]-\mathbf{P}^{e}_{0}([\vert v_{h}\vert])\parallel_{0,e}
\label{temp29}
\end{eqnarray}
using standard polynomial approximation on edge $e$
\begin{align}
\parallel \nabla u-\mathbf{P}^{e}_{k-1}(\nabla u)\parallel_{0,e} & \leq C h^{min(m,k)-\frac{1}{2}} \parallel u \parallel_{m+1,T} \label{tempb7}\\
\parallel [\vert v_{h} \vert]-\mathbf{P}^{e}_{0}([\vert v_{h} \vert])\parallel_{0,e} & \leq C h^{\frac{1}{2}} \vert v_{h} \vert_{1,T}
\label{temp30}
\end{align}
we can easily bound the above two terms. Hence, putting the
estimation equation (\ref{tempb7}), (\ref{temp30}) in (\ref{temp29}) we obtain required result.
\end{proof}
\section{Well-posedness of nonconforming-virtual element methods}
\label{step4}
In this section we will discuss the well-posedness of nonconforming-virtual element
method. Let the assumption (A1,A2,A3), polynomial consistency, stability defined
in equation (\ref{temp22}), (\ref{temp23}) holds then the bilinear form $A_{h}$ is coercive
with respect to broken-norm $\parallel .\parallel_{1,h} $, i.e.
\begin{equation}
A_{h}(v_{h},v_{h}) \geq \alpha \parallel v_{h} \parallel_{1,h}^{2} \quad \forall v_{h} \in V_{h}
\label{temp31}
\end{equation}
where $\alpha$ is a positive constant.
Using the stability properties equation (\ref{temp22}),(\ref{temp23}) for diffusion and
reaction parts of discrete bilinear form we can bound
\begin{eqnarray}
A^{T}_{h}(v_{h},v_{h}) & \geq & \alpha_{\ast} \, a^{T}(v_{h},v_{h})+b_{h}^{T}(v_{h},v_{h})+\gamma_{\ast} \, c^{T}(v_{h},v_{h}) \nonumber \\
& \geq & \alpha_{\ast} \, \eta \, \vert v_{h}\vert^{2}_{1,T} +b^{T}(v_{h},v_{h})+\gamma_{\ast} \, c^{T}(v_{h},v_{h})+[b^{T}_{h}(v_{h},v_{h})-b^{T}(v_{h},v_{h})] \nonumber \\
&\geq & \alpha_{\ast} \, \eta \, \vert v_{h} \vert^{2}_{1,T}+\text{min}(1,\gamma_{\ast})(b^{T}(v_{h},v_{h}) \nonumber \\
&+ &c^{T}(v_{h},v_{h}))+[b^{T}_{h}(v_{h},v_{h})-b^{T}(v_{h},v_{h})] \nonumber \\
& \geq & \alpha_{\ast} \, \eta \, \vert v_{h} \vert^{2}_{1,T}+\text{min}(1,\gamma_{\ast}) \, c_{0} \parallel v_{h} \parallel^{2}_{0,T} \nonumber \\
& - & \vert b^{T}_{h}(v_{h},v_{h})-b^{T}(v_{h},v_{h})\vert \quad \forall v_{h} \in V^{k}_{h}(T)
\label{temp32}
\end{eqnarray}
\begin{eqnarray}
\vert b^{T}_{h}(v_{h},v_{h})-b^{T}(v_{h},v_{h}) \vert & =& \frac{1}{2} \left| \int_{T}(\nabla \cdot \boldsymbol{\beta})(\Pi_{k}(v_{h}))^{2} dT -\int_{T} (\nabla.\boldsymbol{\beta}) \, v_{h}^{2} \, dT \right| \nonumber \\
&=& \frac{1}{2} \Big| \int_{T}(\nabla \cdot \boldsymbol{\beta})(\Pi_{k}(v_{h}))^{2} -\int_{T} (\nabla \cdot \boldsymbol{\beta})\Pi_{k}(v_{h}) \, v_{h} \, dT \nonumber\\
&+& \int_{T}(\nabla \cdot \boldsymbol{\beta}) \, \Pi_{k}(v_{h}) v_{h} dT - \int_{T} (\nabla.\boldsymbol{\beta}) \, v_{h}^{2} \, dT \Big| \nonumber\\
& = & \frac{1}{2} \Big| \int_{T} (\nabla \cdot \boldsymbol{\beta}) \, \Pi_{k}(v_{h}) (\Pi_{k}(v_{h})-v_{h}) \, dT \nonumber\\
&+ & \int_{T} (\nabla \cdot \boldsymbol{\beta}) \cdot v_{h}(\Pi_{k}(v_{h})-v_{h}) \, dT \Big | \nonumber\\
& =& \frac{1}{2} \left| \int_{T} (\nabla \cdot \boldsymbol{\beta}) \, v_{h} (\Pi_{k}(v_{h})-v_{h}) \right| \nonumber\\
& \leq & C \parallel \nabla \cdot \boldsymbol{\beta} \parallel_{\infty,T} \parallel v_{h} \parallel_{0,T} \parallel v_{h}-\Pi_{k}(v_{h}) \parallel_{0,T} \nonumber \\
& \leq & C \parallel \nabla \cdot \boldsymbol{\beta} \parallel_{\infty,T} \parallel v_{h} \parallel_{0,T} \, h_{T} \, \vert v_{h} \vert_{1,T}
\label{tempc7}
\end{eqnarray}
Since $\|v_{h}\|_{0,T} \leq \| v_{h} \|_{1,T}$ and $\vert v_{h} \vert_{1,T} \leq \| v_{h} \|_{1,T} $ inequalities
hold \cite{ciarlet1991basic} we can estimate
\begin{equation}
-\vert b^{T}_{h}(v_{h},v_{h})-b^{T}(v_{h},v_{h}) \vert \geq - \, C \parallel \nabla.\boldsymbol{\beta} \parallel_{\infty,T} h_{T} \parallel v_{h} \parallel_{1,T}
\label{tempc8}
\end{equation}
Therefore
\begin{eqnarray}
A_{h}(v_{h},v_{h}) &= &\sum_{T}A^{T}_{h}(v_{h},v_{h}) \nonumber \\
& \geq & \sum_{T} \alpha_{\ast} \, \eta \, \vert v_{h} \vert_{1,T}^{2}+\sum_{T}\text{min}(1,\gamma_{\ast}) \, c_{0} \, \| v_{h} \|_{0,T}^{2} \nonumber\\
& - & \sum_{T} C \, \| \nabla \cdot \boldsymbol{\beta} \|_{\infty,T} \, h_{T} \, \| v_{h} \|_{1,T} \nonumber\\
&\geq & \sum_{T} \alpha_{T} \| v_{h} \|_{1}^{2} \nonumber \\
& \geq & \alpha\sum_{T} \| v_{h} \|_{1}^{2} \nonumber\\
& = & \alpha \| v_{h} \|_{1,h} \nonumber
\end{eqnarray}
where $\alpha=\text{min}(\alpha_{T})$, and
\begin{equation}
\alpha_{T}=\text{min} \{ \alpha_{\ast}\eta-C \parallel \nabla.\boldsymbol{\beta} \parallel_{\infty} h_{T} ,\text{min}(1,\gamma_{\ast})c_{0}-C \parallel \nabla.\boldsymbol{\beta} \parallel_{\infty}h_{T} \} \nonumber
\end{equation}
\subsection{Continuity of discrete bilinear form}
\begin{lem}
Under the assumption of the polynomial consistency and stability along with
the coefficients $\mathbf{K}, \boldsymbol{\beta}, c $ the
bilinear form $ A_{h} $ defined in equation (\ref{temp16}) is continuous.
\label{41}
\end{lem}
\begin{proof}
Diffusive part $a_{h}^{T}(u_{h},v_{h})$ and reactive part $c^{T}_{h}(u_{h},v_{h}) $ of the
bilinear form $A^{T}(u_{h},v_{h})$ are symmetric and hence they can be view as inner product
in VE space $V^{T}_{h}$ over each element $T$. Convective part $b^{T}_{h}(u_{h},v_{h})$ is
not symmetric and hence we cannot bound it like diffusive part and reactive
part, but using properties of the projection operator we can simply bound it. Hence
we conclude
\begin{eqnarray}
a^{T}_{h}(u_{h},v_{h}) & \leq &(a^{T}_{h}(u_{h},u_{h}))^{\frac{1}{2}} (a^{T}_{h}(v_{h},v_{h}))^{\frac{1}{2}} \nonumber\\
& \leq &\alpha^{\ast} (a^{T}(u_{h},u_{h}))^{\frac{1}{2}} (a^{T}(v_{h},v_{h}))^{\frac{1}{2}} \nonumber \\
& \leq & \alpha^{\ast} \| \mathbf{K} \|_{\infty} \| \nabla u_{h} \|_{0,T} \| \nabla v_{h} \|_{0,T}
\label{tempb10}
\end{eqnarray}
similarly
\begin{equation}
c^{T}_{h}(u_{h},v_{h}) \leq \gamma^{\ast} \| c \|_{\infty} \| u_{h} \|_{0,T} \| v_{h} \|_{0,T}
\end{equation}
\begin{eqnarray}
b^{T}_{h}(u_{h},v_{h}) & = & \frac{1}{2} \Big( \int_{T}\boldsymbol{\beta} \, \Pi_{k-1}(\nabla u_{h}) \, \Pi_{k}(v_{h}) \, dT \nonumber \\
&- &\int \boldsymbol{\beta} \cdot \Pi_{k-1}(\nabla v_{h}) \, \Pi_{k}(u_{h}) \, dT \nonumber\\
& - &\int_{T}(\nabla \cdot \boldsymbol{\beta}) \, \Pi_{k}(u_{h}) \, \Pi_{k}(v_{h}) \, dT \Big) \nonumber
\end{eqnarray}
Again
\begin{equation}
\int_{T}\boldsymbol{\beta} \, \Pi_{k-1}(\nabla u_{h}) \, \Pi_{k}(v_{h}) \, dT \leq C \, \|\boldsymbol{\beta}\| _{\infty} \|\nabla u_{h} \|_{0,T} \|v_{h} \|_{0,T}
\end{equation}
\begin{equation}
\int_{T} \boldsymbol{\beta} \cdot \Pi_{k-1}(\nabla v_{h}) \, \Pi_{k}(u_{h}) \, dT \leq C \, \|\boldsymbol{\beta} \| _{\infty} \| \nabla v_{h} \|_{0,T} \| u_{h} \|_{0,T}
\end{equation}
\begin{equation}
\int_{T}(\nabla \cdot \boldsymbol{\beta}) \, \Pi_{k}(u_{h}) \, \Pi_{k}(v_{h}) \, dT \leq C \, \| \boldsymbol{\beta} \|_{1,\infty} \| u_{h} \|_{0,T} \| v_{h} \|_{0,T}
\end{equation}
Thus,
\begin{eqnarray}
A^{T}_{h}(u_{h},v_{h}) & = & a^{T}_{h}(u_{h},v_{h}) +b^{T}_{h}(u_{h},v_{h})+c^{T}_{h}(u_{h},v_{h}) \nonumber\\
& \leq & \alpha^{\ast} \| \mathbf{K} \|_{\infty} \| \nabla u_{h} \|_{0,T} \| \nabla v_{h} \|_{0,T} \nonumber \\
& + & C \| \boldsymbol{\beta} \|_{1,\infty} \| u_{h} \|_{1,T} \| v_{h} \|_{0,T} \nonumber \\
& + &\gamma^{\ast} \| c \|_{\infty} \| u_{h} \|_{0,T} \| v_{h} \|_{0,T} \nonumber\\
& \leq & C_{T} \|v_{h} \|_{1,T} \| u_{h} \|_{1,T}
\end{eqnarray}
\begin{eqnarray}
A_{h}(u_{h},v_{h}) & = & \sum_{T} A^{T}_{h}(u_{h},v_{h}) \nonumber \\
& \leq & \sum_{T} C_{T} \|v_{h} \|_{1,T} \| u_{h} \|_{1,T} \nonumber \\
& \leq & C\left(\sum_{T} \| u_{T} \|^{2}_{1,T}\right)^{\frac{1}{2}} \left(\sum_{T} \| v_{h} \|^{2}_{1,T}\right)^{\frac{1}{2}} \nonumber \\
& = & C \, \| u_{h} \|_{1,h} \| v_{h} \|_{1,h}
\label{tempb11}
\end{eqnarray}
Therefore the bilinear form is continuous.
\end{proof}
The bilinear form $A^{T}_{h}$ is $\textit{discrete coercive}$ and
$\textit{bounded or continuous} $ in discrete norm $\| \cdot \|_{1,h}$ on
nonconforming virtual element space $ V^{k}_{h}$, defined in equation (\ref{temp14}). Hence the bilinear
form has unique solution in $V^{k}_{h,g}$ by Banach Necas Babuska(BNB)
theorem \cite{di2011mathematical}.
\section{Convergence analysis and apriori error analysis in $\parallel.\parallel_{1,h}$ norm }
\label{step5}
In this section we reveal nonconforming convergence analysis of discrete solution
$u_{h} \in V^{k}_{h}$ which satisfy the discrete bilinear form equation (\ref{temp10}). The basic
idea is same as non conforming error analysis of convection diffusion problem proposed by
Tobiska et al \cite{john1997nonconforming,knobloch2003p}. It is well known that
non conforming virtual element space
$V^{k}_{h} \subset H^{1,nc}(\tau_{h};k)\nsubseteq H^{1}(\Omega)$ and introduce
consistency error. The finite element solution $ v_{h} \in V^{k}_{h}$ is not continuous
along interior edge $e$ except certain points which implies an additional jump
term $ J(u,v_{h})$ defined in equation (\ref{temp28}). Nonconforming virtual
element space $V^{k}_{h}$ satisfy 'patch-test'\cite{irons1972experience} of order $k$. Using
this property we can easily bound the jump term.
\begin{thm}
Let $u$ be the exact solution problem (\ref{temp3}) with polynomial
coefficients $\mathbf{K}, \boldsymbol{\beta}, c$ . Let $u_{h} \in V^{k}_{h}$ be the solution
of the non conforming virtual element approximation(\ref{temp10}). Let $f \in L^2({\Omega})$
and $ u \in H^{k+1}(\Omega)$ \,($k \geq 1$) then
\begin{equation}
\| u-u_{h} \|_{1,h} \leq C h^{k} \| u \|_{k+1,h} +C \, h \, \vert f \vert_{0,h}
\end{equation}
where $\| \cdot \|_{1,h} $ denote broken norm in the space $\mathbf{H}^{1,NC}(\tau_{h},k)$.
\end{thm}
\begin{proof}
We consider $u_{I}$ be the approximation of $u$ in $V^{k}_{h}$ and $u_{\Pi}$ be
polynomial approximation of $u$ in $\mathbb{P}^{k}(\tau_{h})$.
Define $ \delta:=u_{h}-u_{I}$.
using coercivity of the virtual element form, we can write
\begin{eqnarray}
\alpha \| \delta \|^{2}_{1} & \leq & A_{h}(\delta,\delta) \nonumber \\
& = & A_{h}(u_{h},\delta)-A_{h}(u_{I},\delta) \nonumber \\
& = & <f_{h},\delta>-A_{h}(u_{I},\delta) \nonumber \\
& = & <f_{h},\delta>-\sum_{T}A^{T}_{h}(u_{I},\delta)
\label{temp42}
\end{eqnarray}
Now we shall analyse the local bilinear form $A^{T}_{h}(u_{I},\delta) $ term by term
\begin{eqnarray}
A^{T}_{h}(u_{I},\delta) & = & A^{T}_{h}(u_{I}-u_{\Pi},\delta)+A^{T}_{h}(u_{\Pi,\delta}) \nonumber \\
& = & A^{T}_{h}(u_{I}-u_{\Pi},\delta)+A^{T}_{h}(u_{\Pi},\delta)-A^{T}(u_{\Pi},\delta)+A^{T}(u_{\Pi},\delta) \nonumber \\
& = & A^{T}_{h}(u_{I}-u_{\Pi},\delta)+A^{T}_{h}(u_{\Pi},\delta)-A^{T}(u_{\Pi},\delta) \nonumber \\
& + & A^{T}(u_{\Pi}-u,\delta)+A^{T}(u,\delta)
\label{tempb12}
\end{eqnarray}
Discrete bilinear form $A^{T}_{h}(u_{h},v_{h}) $ is polynomial consistent, hence
$A^{T}_{h}(u_{\Pi},\delta)=A^{T}(u_{\Pi},\delta) $
Therefore
\begin{equation}
A^{T}_{h}(u_{I},\delta)=A^{T}_{h}(u_{I}-u_{\Pi},\delta)+A^{T}(u_{\Pi}-u,\delta)+A^{T}(u,\delta)
\label{tempd3}
\end{equation}
Now using Green's theorem on the triangle $T$, we get
\begin{eqnarray}
\int_{T} (\boldsymbol{\beta} \cdot \nabla u) \, \delta \, dT &= &\frac{1}{2} \Big(\int_{T}(\boldsymbol{\beta} \cdot \nabla u) \, \delta \, dT -\int_{T} (\boldsymbol{\beta} \cdot \nabla \delta )u \, dT \nonumber\\
&-&\int_{T}(\nabla \cdot \boldsymbol{\beta})u \, \delta \, dT+\int_{\partial T} (\boldsymbol{\beta} \cdot \mathbf{n}_{\partial T})u \, \delta \, ds \Big)
\label{tempd1}
\end{eqnarray}
Rearranging terms we can write
\begin{eqnarray}
\frac{1}{2} \left(\int_{T}(\boldsymbol{\beta} \cdot \nabla u) \, \delta \, dT -\int_{T} (\boldsymbol{\beta} \cdot \nabla \delta )u -\int_{T}(\nabla \cdot \boldsymbol{\beta}) \, u \, \delta \, dT \right) & =& \int_{T} (\boldsymbol{\beta} \cdot \nabla u)\delta \, dT \nonumber\\
&-&\frac{1}{2}\int_{\partial T} (\boldsymbol{\beta} \cdot \mathbf{n}_{\partial T})u \, \delta \, ds
\label{tempd2}
\end{eqnarray}
Taking sum over all element $ T \in \tau_{h}$ we get
\begin{eqnarray}
\sum_{T}A^{T}(u,\delta) & = & \sum_{T} (-\nabla(\mathbf{K} \, \nabla u)+(\boldsymbol{\beta} \cdot \nabla u)+cu) \, \delta) \nonumber\\
& + &\sum_{T} \int_{\partial T} \mathbf{K} \, \nabla u \cdot \mathbf{n}_{e} \, \delta \, ds -\sum_{T} \frac{1}{2}\int_{\partial T} (\boldsymbol{\beta} \cdot \mathbf{n}_{\partial T})u \, \delta \, ds \nonumber\\
& = & <f,\delta> +\sum_{e \in \varepsilon_{h}} \int_{e} (\mathbf{K} \, \nabla u \cdot \mathbf{n}_{e}) \, [\vert \delta \vert] \, ds-\frac{1}{2} \sum_{e \in \varepsilon_{h}} \int_{e} (\boldsymbol{\beta}.\mathbf{n}_{e}) \, u \, [\vert \delta \vert]
\label{temp44}
\end{eqnarray}
In the above equation we get jump term corresponding to $\delta $ only, since $\delta$ is
discontinuous along interior edge $ e \in \varepsilon^{0}_{h}$
Globally the equation (\ref{tempd3}) can be written as
\begin{eqnarray}
A_{h}(u_{I},\delta) & = & \sum_{T} A^{T}_{h}(u_{I},\delta) \nonumber \\
& = & \sum_{T} A^{T}_{h}(u_{I}-u_{\Pi},\delta) +\sum_{T}A^{T}(u_{\Pi}-u,\delta) \nonumber \\
& + & <f,\delta> +\sum_{e \in \varepsilon_{h}} \int_{e}(\mathbf{K} \, \nabla u \cdot \mathbf{n}_{e}) \, [\vert \delta \vert] \, ds -\frac{1}{2} \sum_{e \in \varepsilon_{h}} \int_{e} (\boldsymbol{\beta} \cdot \mathbf{n}_{e}) \, u \, [\vert \delta \vert]
\label{temp45}
\end{eqnarray}
Let us denote
\begin{eqnarray}
M_{1} & = & A^{T}_{h}(u_{I}-u_{\Pi},\delta) \nonumber \\
M_{2} & = & A^{T}(u_{\Pi}-u,\delta) \nonumber
\end{eqnarray}
\begin{equation}
\vert A^{T}_{h}(u_{I}-u_{\Pi},\delta) \vert \leq C \| u_{I}-u_{\Pi} \|_{1,T} \| \delta \|_{1,T}
\label{temp46}
\end{equation}
\begin{equation}
\vert A^{T}(u_{\Pi}-u,\delta) \vert \leq C \| u_{\Pi}-u \|_{1,T} \| \delta \|_{1,T}
\label{temp47}
\end{equation}
using (\ref{temp46}), (\ref{temp47}) and interpolation error estimation (\ref{temp15}) we
can bound
\begin{eqnarray}
\vert M_{1} \vert +\vert M_{2} \vert & \leq & C \, \|\delta \, \|_{1,T} (\| u_{I}-u_{\Pi} \|_{1,T}+ \|u_{\Pi}-u \|_{1,T} ) \nonumber\\
& \leq & C \, \| \delta \, \|_{1,T} \, h^{k} \, \| u \|_{k+1,T}
\label{temp48}
\end{eqnarray}
we use equation (\ref{temp25}) for $k\geq2$ and (\ref{temp27}) for $k=1$ to bound the
right hand side
\begin{eqnarray}
\vert <f_{h},\delta>-<f,\delta> \vert & = & \left| \sum_{T} \int_{T} (f_{h}-f) \, \delta \right| \nonumber\\
& \leq & C h^{k} \vert f \vert_{k-1,\tau_{h}} \vert \delta \vert_{1,h}
\label{temp49}
\end{eqnarray}
In particular for $k=1$
\begin{equation*}
\left| <f_{h},\delta> - <f,\delta> \right| \leq C \, h \, |f|_{0,\tau_{h}} \, \delta_{1,h}
\end{equation*}
using estimation (\ref{tempb6}) we bound consistency error
\begin{eqnarray}
\left| \int_{e} (\mathbf{K} \, \nabla u \cdot \mathbf{n}_{e}) [\vert\delta\vert] \right| & \leq & \| \mathbf{K} \|_{\infty} \vert \int_{e} (\nabla u \cdot \mathbf{n}_{e}) [\vert\delta\vert] \vert \nonumber \\
& \leq & C h^{k} \| u \|_{k+1,T^{+}\cup T^{-}} \vert \delta \vert_{1,T^{+}\cup T^{-}} \nonumber
\end{eqnarray}
Edge $e$ is an interior edge shared by triangles $T^{+}$ and $T^{-}$. Therefore
\begin{equation}
\left| \sum_{e \in \varepsilon_{h}} \int_{e}(\mathbf{K} \, \nabla u \cdot \mathbf{n}_{e}) [\vert\delta \vert] \right| \leq C \, h^{k} \, \| u \|_{k+1,h} \|\delta \|_{1,h}
\label{tempm1}
\end{equation}
again
\begin{eqnarray}
\left| \sum_{e\in \varepsilon_{h}} \int_{e} (\boldsymbol{\beta} \cdot \mathbf{n}_{e}) \, u \, [\vert \delta \vert]\right| & \leq &
\left| \sum_{e \in \varepsilon_{h}} \| \boldsymbol{\beta} \cdot \mathbf{n}_{e} \|_ {\infty} \int_{e} u \, [\vert \delta \vert] \ \right| \nonumber\\
& \leq & C \, \left| \sum_{e \in \varepsilon_{h}} \int_{e}(u-P^{k-1}(u)) \, [\vert \delta \vert] \ \right| \nonumber\\
& = & C \, \left| \sum_{e \in \varepsilon_{h}} \int_{e}(u-P^{k-1}(u)) \, ([\vert \delta \vert]-P^{0}([\vert \delta \vert])) \ \right| \nonumber\\
& \leq & C \sum_{e \in \varepsilon_{h}} \| u-P^{k-1}(u) \|_{0,e} \|[\vert \delta \vert]-P^{0}([\vert \delta \vert]) \|_{0,e} \nonumber
\end{eqnarray}
using standard approximation \cite{ciarlet1991basic}
\begin{eqnarray}
\| u-P^{k-1}(u) \|_{0,e} & \leq & C h^{\text{min}(k,s)-1/2} \| u \|_{s,T^{+}\cup T^{-}} \nonumber \\
\|[\vert \delta \vert]-P^{0}([\vert \delta \vert]) \|_{0,e} & \leq & C h^{1/2} \| \delta \|_{1,T^{+}\cup T^{-}} \nonumber
\end{eqnarray}
we can bound
\begin{eqnarray}
\left| \sum_{e\in \varepsilon_{h}} \int_{e} (\boldsymbol{\beta} \cdot \mathbf{n}_{e})u [\vert \delta \vert] \right| &\leq & C h^{\text{min}(k,s)}\sum_{T} \| u \|_{s,T} \| \delta \|_{1,T} \nonumber\\
& \leq & C h^{\text{min}(k,s)} \| u \|_{s+1,h} \| \delta \|_{1,h} \nonumber
\end{eqnarray}
In particular for $s=k$
\begin{equation}
\left| \sum_{e\in \varepsilon_{h}} \int_{e} (\boldsymbol{\beta} \cdot \mathbf{n}_{e}) \, u \, [\vert \delta \vert]\right| \leq C \, h^{k} \, \| u \|_{k+1,h} \| \delta \|_{1,h}
\label{tempn1}
\end{equation}
Using (\ref{temp48}), (\ref{temp49}), (\ref{tempm1}), (\ref{tempn1}) we bound
\begin{eqnarray}
\alpha \| \delta \|_{1,h}^{2} & \leq & (C h^{k} \|u \|_{k+1,h}+C \, h \, \vert f \vert_{0,h}) \| \delta \|_{1,h}\nonumber\\
\| \delta \|_{1,h} & \leq & C h^{k} \| u \|_{k+1,h} +C h \vert f \vert_{0,h} \label{tempn2}
\end{eqnarray}
We can write
\begin{equation}
\|{u-u_{h}} \|_{1,h}\leq \|(u-u^{I}) \|_{1,h}+ \|(u^{I}-u_{h}) \|_{1,h}
\end{equation}
The first term can be estimated using the standard approximation (\ref{temp15})
and second term can be estimated using (\ref{tempn2}). Hence we obtain
\begin{equation}
\| u-u_{h} \|_{1,h} \leq C h^{k} \| u \|_{k+1,h}+C \, h \, \vert f \vert_{0,h}
\end{equation}
\end{proof}
\section{Conclusions}
\label{last1}
In this work we presented the analysis of nonconforming virtual element
method for convection diffusion reaction equation with polynomial
coefficients using $L^2$ projection. $L^2$ projection can be partially computed using
degrees of freedom of the finite element. The external virtual element method using
$L^2$ projection operator is not fully computable which may be considered
as a drawback of this method. We have proved stability of the method
assuming that the model problem is diffusion dominated. If the model
problem is convection dominated then the present analysis is not applicable
and hence we require a new framework for convection dominated problem, which
may be carried out as a future work.
\bibliographystyle{plain}
| 25,388 |
\section{Introduction}
\noindent
So-called NOON states are an important resource in optical quantum information science \cite{Sanders,Downling1, Mitchell, Walther}.
They are bipartite entangled, $N$-photon two-mode states where the $N$ photons occupy either one of two optical modes, $\frac{1}{\sqrt{2}}(|N0\rangle+|0N\rangle)$.
NOON states have been widely used in quantum communication \cite{Gisin}, quantum metrology \cite{metrology} and quantum lithography \cite{lithography},
because they allow for super-sensitive measurements, e.g. in optical interferometry. This is related to the substandard quantum-limit behaviour of NOON states, i.e. a factor $\sqrt{N}$ improvement to the shot noise limit can be achieved \cite{Rohde}.
Due to their practical relevance, various schemes for NOON state generation based on strong non-linearities \cite{Gerry, Kapale}
or measurement and feed-forward \cite{Cerf, VanMeter, Cable} have been proposed. Unfortunately, NOON states are very fragile, which focused
recent research on their entanglement and phase properties in noisy environments \cite{Sperling} or on the enhancements of NOON state sensitivity by non-Gaussian operations \cite{Jeffers}.\\
Though very sensitive to losses, NOON states can be useful resources to build quantum error correcting codes, as will be shown in this paper. Optical quantum information and especially their use in long-distance quantum communication suffers from loss.
Here, the main mechanism of decoherence is photon loss which is theoretically described by the amplitude-damping (AD) channel that acts on each field mode. To protect quantum information from photon loss, various kinds of quantum error correction codes for AD were proposed \cite{Banaszek, bosonic_codes, Ralph}.
In this context, it was also observed that quantum error correction codes fall in one of two classes: exact or approximate.
The usual quantum error correction conditions are strictly fulfilled by exact quantum codes, whereas approximate codes only fulfil a set of relaxed conditions \cite{approximate_codes}. \\
Recently, quantum parity codes (QPCs) \cite{lutkenhaus, Ralph} as an example of exact codes for AD were employed in the context of long-distance quantum communication.
A QPC consumes $N^{2}$ photons distributed in $2N^{2}$ modes with at most one photon per mode. This code therefore requires a maximal number of modes. In contrast, an important class of exact AD codes, the so-called bosonic codes
using $N^{2}$ total photons, introduced in \cite{bosonic_codes}, may use no more than just two modes at the expense of having up to $N^{2}$ photons per mode. The codes to be developed
in this paper are intermediate between QPC and bosonic codes, because they use $N^{2}$ photons in $2N$ modes with at most $N$ photons per mode. Our code is a block code
like QPC and unlike the general bosonic code, with the same number of blocks as QPC, but with the $N^{2}$ photons distributed among a smaller number of modes in every block compared to QPC. Despite their structural differences,
all these loss codes, including our code, protect a logical qubit exactly against $N-1$ photon losses using $N^{2}$ photons. Thus, only in our scheme, both the total mode number and the maximal photon number per mode scale linearly with $N$ to achieve protection against $N-1$ losses.
Another crucial difference, compared to QPC and bosonic codes lies in the systematic accessibility of our codewords from NOON states and linear optics.
For the simplest special case of a one-photon-loss qubit code, even only one-photon Fock states are sufficient as resources for codeword generation, as the $N=2$ NOON
state corresponds to the well-known Hong-Ou-Mandel state. Our systematic approach can be also applied to qudit-code constructions, while certain examples of bosonic qudit codes were also given in \cite{bosonic_codes} (see also the recent work in \cite{Grassl}).\\
\noindent The structure of the paper is as follows: in the second section, the AD model and the basics of quantum error correction are introduced and some known photon loss codes are
reviewed. In the succeeding section, quantum codes for logical qubits are systematically developed.
The third section discusses the extension of this systematic scheme to logical qudits in a natural manner by switching from beam splitters to general $N$-port devices.
It is shown that the scaling of the fidelity only depends on the total photon number and, especially, that it is independent of the dimension of the logical qudit. Section 5 presents an in-principle method for the generation of an arbitrary logical qubit
state for the one-photon-loss qubit code based on linear optics and light-matter interactions. The last section, as an example of an application, describes the use of the qudit code in a one-way quantum communication scheme. This scheme
does not intrinsically provide an optimal rate between physical versus logical qubits like another recent approach \cite{Jiang2}, but nonetheless allows
for sending more quantum information at each time step with the same loss protection.
\section{Quantum error correction and photon loss}
\noindent Photon loss can be modelled by the AD channel. The non-unitary error operators $A_{k}$, specifying the loss of $k$ photons in a single mode, are given by \cite{bosonic_codes}
\begin{equation}
A_{k}=\sum\limits_{n=k}^{\infty}\sqrt{\binom{n}{k}}\sqrt{\gamma}^{n-k}\sqrt{1-\gamma}^{k}|n-k\rangle\langle n|,
\end{equation}
$\forall k\in \{0,1,\cdots,\infty\}$. Here, $\gamma$ is the damping parameter and $1-\gamma$ corresponds to the probability of losing one photon. The operators $A_{k}$ form a POVM, i.e. $A_{k}\geq 0$ and $\sum\limits_{k=0}^{\infty}A_{k}^{\dagger}A_{k}=1$.
The non-unitary evolution of an arbitrary single-mode density operator $\rho$ under the effect of
AD is
\begin{equation}\label{AD}
\rho\rightarrow\rho_{f}=\sum\limits_{k=0}^{\infty}A_{k}\rho A_{k}^{\dagger}.
\end{equation}
By employing a quantum code, one is partially able to reverse the dynamics implied by Eq. \eqref{AD} and recover the original state. A proper quantum code enables one to detect and correct a certain set of errors on the encoded state. A quantum code is a vector space
spanned by basis codewords, denoted by $|\bar{0}\rangle\equiv |c_{1}\rangle$ and $|\bar{1}\rangle\equiv |c_{2}\rangle$ for a qubit code, and a subspace of some higher-dimensional Hilbert space. Normalized elements of this vector space of the form
$\alpha|\bar{0}\rangle+\beta|\bar{1}\rangle$ are called logical qubits. This notion can be extended to qudit codes, where there are more than two codewords $|c_{1}\rangle,\cdots,|c_{d}\rangle$ to encode a logical $d$-level system.
To form a proper quantum code, the logical basis codewords have to fulfil certain conditions. We state the famous Knill-Laflamme conditions which are a set of necessary and sufficient condition for the existence of a recovery operation \cite{N-C}:
\begin{Theorem}{(Knill-Laflamme)}\\
Let $C=span\{|c_{1}\rangle,|c_{2}\rangle,\cdots,|c_{d}\rangle\}$ be a quantum code, $P$ be the projector onto $C$ and $\{E_{i}\}$ the set of error operators. There exists an error-correction operation $\mathcal{R}$ that corrects
the errors $\{E_{i}\}$ on C, iff
\begin{equation}\label{knill-laflamme}
PE_{i}^{\dagger}E_{j}P=\Lambda_{ij}P,~~~ \forall i,j
\end{equation}
for some semi-positive, Hermitian matrix $\Lambda$ with matrix elements $\Lambda_{ij}$.
\end{Theorem}
\noindent For photon loss codes (in particular, those exact codes with a fixed total photon number), the matrix $\Lambda$ is typically diagonal, i.e. $\Lambda_{ij}=g_{i}\delta_{ij}$. This defines non-degenerate codes with different
loss errors (especially different numbers of photons lost, but also different modes subject to loss) corresponding to orthogonal error spaces. Nonetheless, certain instances of our code do exhibit
degeneracy for a given number of lost photons.\\
The Knill-Laflamme (KL) conditions contain two basic notions. The first notion is the orthogonality of corrupted codewords, i.e.
\begin{equation}\label{ortho}
\langle c_{k}|E_{i}^{\dagger}E_{j}|c_{l}\rangle =0~~ \text{if}~~ k\neq l.
\end{equation}
The second one is the non-deformability condition, i.e.
\begin{equation}\label{deform}
\langle c_{l}|E_{i}^{\dagger}E_{i}|c_{l}\rangle =g_{i},~~ \forall l.
\end{equation}
This means that the norm of a corrupted codeword only depends on the error operator and not on the codeword itself.\\
Before proceeding with the code construction, we highlight some examples of existing photon loss codes.
In the first example, a logical qubit is encoded in a certain two-dimensional subspace of two bosonic modes. The basis codewords are chosen
in the following way \cite{bosonic_codes}:
\begin{equation}\label{eq:Leung}
\begin{aligned}
|\bar{0}\rangle=&\frac{1}{\sqrt{2}}(|40\rangle+|04\rangle),\\
|\bar{1}\rangle=&|22\rangle,
\end{aligned}
\end{equation}
i.e. any logical qubit has a total photon number $N^{2}=4$. This code corrects exactly $N-1=1$ photon losses. The worst-case fidelity, as defined further below, is found to be
$F=\gamma^{4}+4\gamma^{3}(1-\gamma)=1-6(1-\gamma)^{2}+8(1-\gamma)^{3}-3(1-\gamma)^{4}$.
In the same reference \cite{bosonic_codes}, the following code is given:
\begin{equation}
\begin{aligned}
|\bar{0}\rangle&=\frac{1}{\sqrt{2}}(|70\rangle+|16\rangle),\\
|\bar{1}\rangle&=\frac{1}{\sqrt{2}}(|52\rangle+|34\rangle).
\end{aligned}
\end{equation}
This code corrects also all one-photon losses and its worst-case fidelity is $\gamma^{7}+7\gamma^{6}(1-\gamma)=
1-21(1-\gamma)^{2}+70(1-\gamma)^{3}-105(1-\gamma)^{4}\linebreak +84(1-\gamma)^{5}-35(1-\gamma)^{6}+6(1-\gamma)^{7}$.
\\Another example that encodes a qubit in three optical modes with a total photon number of 3 was
proposed in \cite{Banaszek}. The basis codewords are
\begin{equation}
\begin{aligned}
|\bar{0}\rangle&=\frac{1}{\sqrt{3}}(|300\rangle+|030\rangle+|003\rangle),\\
|\bar{1}\rangle&=|111\rangle.\\
\end{aligned}
\end{equation}
The fidelity in this case is $\gamma^{3}+3\gamma^{2}(1-\gamma)=1-3(1-\gamma)^{2}+2(1-\gamma)^{3}$. Moreover, note that all three codes
given above are capable of exactly correcting only the loss of one photon, as can be easily seen by checking the KL conditions. An example
for a proper two-photon-loss code is \cite{bosonic_codes}
\begin{equation}
\begin{aligned}
|\bar{0}\rangle&=\frac{1}{2}|90\rangle+\frac{\sqrt{3}}{2}|36\rangle,\\
|\bar{1}\rangle&=\frac{1}{2}|09\rangle+\frac{\sqrt{3}}{2}|63\rangle,\\
\end{aligned}
\label{eq: nine}
\end{equation}
whose worst-case fidelity is found to be $F=\gamma^{9}+9\gamma^{8}(1-\gamma)+36\gamma^{7}(1-\gamma)^{2}\approx 1-84(1-\gamma)^{3}$.\\
What these codes also have in common is their small number of optical modes, at the expense of having rather large maximal photon numbers in each mode
(in order to obtain a sufficiently large Hilbert space). Conversely, a code that has at most one photon in any mode, but a correspondingly large total mode number,
is the QPC. The simplest non-trivial QPC, denoted as QPC(2,2), reads as follows \cite{Ralph}:
\begin{equation}
\begin{aligned}
|\bar{0}\rangle&=\frac{1}{\sqrt{2}}(|10101010\rangle+|01010101\rangle),\\
|\bar{1}\rangle&=\frac{1}{\sqrt{2}}(|10100101\rangle+|01011010\rangle).\\
\end{aligned}
\end{equation}
It also corrects exactly the loss of one photon. Different from all these codes that all consist of superpositions of states
with a fixed photon number is the following code \cite{approximate_codes}:
\begin{equation}
\begin{aligned}
|\bar{0}\rangle=\frac{1}{\sqrt{2}}(|0000\rangle+|1111\rangle),\\
|\bar{1}\rangle=\frac{1}{\sqrt{2}}(|0011\rangle+|1100\rangle).\\
\end{aligned}
\end{equation}
This code is conceptually distinct, because it does not satisfy the usual KL conditions. It satisfies certain relaxed conditions,
which leads, in a more general setting, to approximate quantum error correcting schemes \cite{approximate_codes}.
The above approximate code still satisfies the KL conditions up to linear order in $1-\gamma$, corresponding to one-photon-loss correction, while it requires
4 physical qubits (single-rail qubits encoded as vacuum $|0\rangle$ and single-photon $|1\rangle$) instead of 5 physical qubits for the minimal universal
one-qubit-error code. Note that for dual-rail physical qubits (i.e., the approximate Leung code \cite{approximate_codes} concatenated with standard optical dual-rail encoding), one
obtains QPC(2,2), which is then an exact one-photon-loss code.\\
After setting the stage, we will now start to discuss how to construct new quantum codes for AD to suppress the effect of photon losses.
\section{Qubit codes}
\label{sec: Qubit codes}
\noindent Let us consider the following qubit codewords defined in the three-dimensional Hilbert space of two photons distributed among
two modes,
\begin{equation}
\begin{aligned}
|\bar{0}\rangle&=\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle),\\
|\bar{1}\rangle&=|11\rangle.\\
\end{aligned}
\end{equation}
The action of the AD channels on the two modes of the logical qubit $|\bar{\Psi}\rangle=c_{0}|\bar{0}\rangle+c_{1} |\bar{1}\rangle$ is
\footnote{In the remainder of the paper $c_{0},c_{1},..,c_{d-1}\in \mathbb{C}$ are the coefficients of our logical qudits.}
\begin{equation}
\begin{aligned}
A_{0}\otimes A_{0}|\bar{\Psi}\rangle&=\sqrt{\gamma^{2}}|\bar{\Psi}\rangle,\\
A_{1}\otimes A_{0}|\bar{\Psi}\rangle&=\sqrt{\gamma(1-\gamma)}(c_{0} |10\rangle+c_{1}|01\rangle)),\\
A_{0}\otimes A_{1}|\bar{\Psi}\rangle&=\sqrt{\gamma(1-\gamma)}(c_{0} |01\rangle+c_{1}|10\rangle)),\\
\end{aligned}
\end{equation}
including the first three error operators $E_{1}=A_{0}\otimes A_{0}$, $E_{2}=A_{1}\otimes A_{0}$ and $E_{3}=A_{0}\otimes A_{1}$, of which the last
two describe the loss of a photon. Obviously, the one-photon-loss spaces are not orthogonal (they are even identical) and the qubit is subject to a random bit flip
for the one-photon-loss case. A different choice would be:
\begin{equation}
\begin{aligned}
|\bar{0}\rangle&=\frac{1}{2}|20\rangle+\frac{1}{2}|02\rangle+\frac{1}{\sqrt{2}}|11\rangle,\\
|\bar{1}\rangle&=\frac{1}{2}|20\rangle+\frac{1}{2}|02\rangle-\frac{1}{\sqrt{2}}|11\rangle.
\end{aligned}
\label{eq: fake}
\end{equation}
After AD, this becomes:
\begin{align*}
A_{0}\otimes A_{0}|\bar{\Psi}\rangle&=\sqrt{\gamma^{2}}|\bar{\Psi}\rangle,\numberthis \\
A_{1}\otimes A_{0}|\bar{\Psi}\rangle&=\sqrt{\gamma(1-\gamma)}\\
&\times(c_{0} \frac{1}{\sqrt{2}}(|10\rangle+|01\rangle)+c_{1}\frac{1}{\sqrt{2}}(|10\rangle-|01\rangle)),\\
A_{0}\otimes A_{1}|\bar{\Psi}\rangle&=\sqrt{\gamma(1-\gamma)}\\
&\times(c_{0} \frac{1}{\sqrt{2}}(|10\rangle+|01\rangle)-c_{1}\frac{1}{\sqrt{2}}(|10\rangle-|01\rangle)).
\end{align*}
\noindent
Here, the phase flip in the last line corresponds to a violation of the KL criteria,$\langle \bar{0}|E_{2}^{\dagger}E_{3}|\bar{0}\rangle\neq \langle\bar{1}|E_{2}^{\dagger}E_{3}|\bar{1}\rangle$,
preventing the encoding from being a proper quantum error correcting code. Indeed, again we have identical one-photon-loss spaces. One can easily verify that any choice of codewords
will either lead to overlapping one-photon-loss spaces or the qubit is completely lost. A possible remedy is to construct codes composed of blocks.\\
To demonstrate this, we first deal with the specific example for encoding a logical qubit. Define
\begin{equation}
\begin{aligned}
|t_{0}^{2,2}\rangle&=BS[|20\rangle]=\frac{1}{2}|20\rangle+\frac{1}{2}|02\rangle+\frac{1}{\sqrt{2}}|11\rangle,\\
|t_{1}^{2,2}\rangle&=BS[|02\rangle]=\frac{1}{2}|20\rangle+\frac{1}{2}|02\rangle-\frac{1}{\sqrt{2}}|11\rangle,
\end{aligned}
\end{equation}
as the "input states" for our encoding, where $BS[~]$ denotes a 50:50 beam splitter transformation.
Note that, in general, a beam splitter with reflectivity $r$ and transmittance $t$ acts on a two-mode Fock state as
\begin{align*}
|m,n\rangle\mapsto \sum\limits_{j,k=0}^{m,n}\sqrt{\frac{(j+k)!(m+n-j-k)!}{m!n!}}\binom{m}{j}\binom{n}{k}\\
\times(-1)^{k}t^{n+j-k}r^{m-j+k}|m+n-j-k,j+k\rangle.\numberthis
\end{align*}
\noindent
In the case of a 50:50 beam splitter ($t=r=\frac{1}{\sqrt{2}}$), this reduces to
\begin{equation}
\begin{aligned}
|m,n\rangle&\mapsto\sum\limits_{j,k=0}^{m,n}\sqrt{\frac{1}{2}}^{n+m}\sqrt{\frac{(j+k)!(m+n-j-k)!}{m!n!}}\binom{m}{j}\\
&\times\binom{n}{k}(-1)^{k}|m+n-j-k,j+k\rangle,
\end{aligned}
\end{equation}
\noindent
and we obtain in particular:
\begin{equation}
\begin{aligned}
BS[|N0\rangle]&=\sqrt{\frac{1}{2}}^{N}\sum\limits_{j=0}^{N}\sqrt{\binom{N}{j}}|N-j,j\rangle,\\
BS[|0N\rangle]&=\sqrt{\frac{1}{2}}^{N}\sum\limits_{j=0}^{N}(-1)^{j}\sqrt{\binom{N}{j}}|N-j,j\rangle.
\end{aligned}
\end{equation}
\noindent
Now by means of a Hadamard-type operation on $|t_{0}^{2,2}\rangle$ and $|t_{1}^{2,2}\rangle$, the following states are obtained:
\begin{equation}
\begin{aligned}
|\widetilde{0}\rangle&=\frac{1}{\sqrt{2}}(|t_{0}^{2,2}\rangle+|t_{1}^{2,2}\rangle)=\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle),\\
|\widetilde{1}\rangle&=\frac{1}{\sqrt{2}}(|t_{0}^{2,2}\rangle-|t_{1}^{2,2}\rangle)=|11\rangle.
\end{aligned}
\end{equation}
Note that $|\widetilde{1}\rangle$ equals $BS[\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle)]$, whereas $|\widetilde{0}\rangle$ is the two-photon NOON state which is invariant
under the beam splitter transformation. A logical qubit can now be encoded according to
\begin{equation}\label{twophotons}
|\bar{\Psi}\rangle=c_{0}|\widetilde{0}\rangle|\widetilde{0}\rangle+c_{1}|\widetilde{1}\rangle|\widetilde{1}\rangle\equiv c_{0}|\bar{0}\rangle+c_{1}|\bar{1}\rangle.
\end{equation}
\noindent
We prove in the following that the codewords
\begin{equation}
\begin{aligned}
|\bar{0}\rangle&=|\widetilde{0}\rangle|\widetilde{0}\rangle=\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)\\
&=\frac{1}{2}(|2020\rangle+|2002\rangle+|0220\rangle+|0202\rangle),\\
|\bar{1}\rangle&=|\widetilde{1}\rangle|\widetilde{1}\rangle=|1111\rangle,
\end{aligned}
\end{equation}
form a quantum error correcting code for the AD channel.
Calculating the action of AD on the basis codewords and checking the KL conditions, we obtain
\begin{align*}
&A_{0}\otimes A_{0}\otimes A_{0}\otimes A_{0}|\bar{\Psi}\rangle\\
&=\sqrt{\gamma^{4}}|\bar{\Psi}\rangle,\\
&A_{1}\otimes A_{0}\otimes A_{0}\otimes A_{0}|\bar{\Psi}\rangle\\
&=\sqrt{\gamma^{3}(1-\gamma)}\left(\frac{c_{0}}{\sqrt{2}}(|1020\rangle+|1002\rangle)+c_{1}|0111\rangle\right),\\
&A_{0}\otimes A_{1}\otimes A_{0}\otimes A_{0}|\bar{\Psi}\rangle\\
&=\sqrt{\gamma^{3}(1-\gamma)}\left(\frac{c_{0}}{\sqrt{2}}(|0120\rangle+|0102\rangle)+c_{1}|1011\rangle\right),\numberthis \\
&A_{0}\otimes A_{0}\otimes A_{1}\otimes A_{0}|\bar{\Psi}\rangle\\
&=\sqrt{\gamma^{3}(1-\gamma)}\left(\frac{c_{0}}{\sqrt{2}}(|2010\rangle+|0210\rangle)+c_{1}|1101\rangle\right),\\
&A_{0}\otimes A_{0}\otimes A_{0}\otimes A_{1}|\bar{\Psi}\rangle\\
&=\sqrt{\gamma^{3}(1-\gamma)}\left(\frac{c_{0}}{\sqrt{2}}(|2001\rangle+|0201\rangle)+c_{1}|1110\rangle\right).\\
\end{align*}
\noindent
The KL conditions are obviously fulfilled for one-photon-loss errors. Note that, after losing any two or more photons, the logical qubit cannot be recovered
anymore.\\
To be able to actively perform quantum error correction, it is a necessary task to determine the syndrome information, i.e. in our
case the location (the mode) where a photon loss occurred. To get this information, we first measure the total photon number per block. If
the result is "2" on each block, there is no photon missing and the logical qubit is unaffected. However, if for example a photon got lost on the first
mode, the result is "1" for the first block and "2" for the other. This result is not unique, because there are still two possible corrupted states with this
measurement pattern. In order to resolve this, inter-block photon number parity measurements with respect to modes 2+3 and 1+4 are suitable. The results
"even-odd" and "odd-even" uniquely determine the corrupted state which can then be accordingly recovered.
Note that all the measurements discussed here are assumed to be of QND-type such that also higher photon losses can be non-destructively detected.
But so far these cannot be corrected by means of the encoding.\\
A convenient measure for the quality of a quantum error correcting code is the worst-case fidelity, defined as \cite{N-C,bosonic_codes}
\begin{equation}
F=\min\limits_{|\bar{\Psi}\rangle\in C} \langle\bar{\Psi}|\mathcal{R}(\bar{\rho}_{f})|\bar{\Psi}\rangle,
\label{eq: fidelity}
\end{equation}
where $\bar{\rho}_{f}$ is the final mixed state after multi-mode amplitude damping (with the only assumption that each AD channel
acts independently on each mode) and $\mathcal{R}$ is the recovery operation. Note that the recovery operation
always exists if the KL conditions are fulfilled. The fidelity defined in Eq.\eqref{eq: fidelity} is a suitable figure of merit to assess
the performance of a quantum error correction code \footnote{The exact loss codes considered in this paper have identical worst-case and average fidelities.}. In particular, it also reveals if an encoding is not a proper code (see, e.g. the encoding
in Eq.\eqref{eq: fake}). In our case, the worst-case fidelity
is easily calculated as
\begin{equation}
F=\gamma^{4}+4\gamma^{3}(1-\gamma)\approx 1-6(1-\gamma)^{2}.
\end{equation}
Note that this code has the same scaling as the four-photon-code of \cite{bosonic_codes} described by Eq.\eqref{eq:Leung}.\\
For higher losses, we can use NOON states with higher photon number to encode a logical qubit. For this purpose, let us define the input states for the codewords
as
\begin{equation}
\begin{aligned}
|t_{0}^{2,3}\rangle=BS[|30\rangle]&=\frac{1}{2\sqrt{2}}|03\rangle+\frac{1}{2}\sqrt{\frac{3}{2}}|12\rangle+\frac{1}{2}\sqrt{\frac{3}{2}}|21\rangle\\
&+\frac{1}{2\sqrt{2}}|30\rangle,\\
|t_{1}^{2,3}\rangle=BS[|03\rangle]&=-\frac{1}{2\sqrt{2}}|03\rangle+\frac{1}{2}\sqrt{\frac{3}{2}}|12\rangle-\frac{1}{2}\sqrt{\frac{3}{2}}|21\rangle\\
&+\frac{1}{2\sqrt{2}}|30\rangle,
\end{aligned}
\end{equation}
such that this time
\begin{equation}
\begin{aligned}
|\widetilde{0}\rangle&=\frac{1}{\sqrt{2}}(|t_{0}^{2,3}\rangle+|t_{1}^{2,3}\rangle)=\frac{1}{2}|30\rangle+\frac{\sqrt{3}}{2}|12\rangle,\\
|\widetilde{1}\rangle&=\frac{1}{\sqrt{2}}(|t_{0}^{2,3}\rangle-|t_{1}^{2,3}\rangle)=\frac{1}{2}|03\rangle+\frac{\sqrt{3}}{2}|21\rangle,
\end{aligned}
\end{equation}
\noindent
become the states after the Hadamard-type gate. We could now again build a qubit like in Eq.\eqref{twophotons}.
However, we find that the resulting six-photon two-block (four-mode) code only corrects certain two-photon losses and therefore there is no significant enhancement compared to the $N=2$ code above. This can be
understood by looking at the corrupted logical qubit for losses of up to two photons. The details for this are presented in Appendix \ref{sec: inefficiency}.
The conclusion is that some of the orthogonality requirements are violated for certain two-photon losses which consequently cannot be corrected.
To overcome this problem and to improve the code, instead we take the following codewords for $N=3$ photons per block (with $N^{2}=9$ as the total number of photons):
\begin{equation}
\begin{aligned}
|\bar{0}\rangle=|\widetilde{0}\rangle|\widetilde{0}\rangle|\widetilde{0}\rangle,\\
|\bar{1}\rangle=|\widetilde{1}\rangle|\widetilde{1}\rangle|\widetilde{1}\rangle,
\end{aligned}
\end{equation}
which are now composed of three blocks for a total number of six modes. To verify that this code corrects all losses up to two photons, we can calculate the action of AD on the logical qubit. Due to symmetry reasons, it is sufficient to calculate the action of only certain
error operators on the codewords, because all other corrupted codewords with at most two lost photons can be obtained by permutations of the blocks. Therefore, if the KL conditions are fulfilled for the
following error operators, then they are also satisfied by the block-permuted corrupted states. The relevant error operators are
\begin{widetext}
\begin{align*}
A_{1}\otimes A_{0}\otimes A_{0}\otimes A_{0}\otimes A_{0}\otimes A_{0}|\bar{\Psi}\rangle
& =\sqrt{\frac{3}{2}\gamma^{8}(1-\gamma)}(\frac{c_{0}}{\sqrt{2}}(|20\rangle+|02\rangle)|\widetilde{0}\rangle|\widetilde{0}\rangle+c_{1}|11\rangle|\widetilde{1}\rangle|\widetilde{1}\rangle),\\
A_{0}\otimes A_{1}\otimes A_{0}\otimes A_{0}\otimes A_{0}\otimes A_{0}|\bar{\Psi}\rangle
& =\sqrt{\frac{3}{2}\gamma^{8}(1-\gamma)}(c_{0}|11\rangle|\widetilde{0}\rangle|\widetilde{0}\rangle+\frac{c_{1}}{\sqrt{2}}(|20\rangle+|02\rangle)|\widetilde{1}\rangle|\widetilde{1}\rangle),\\
A_{1}\otimes A_{1}\otimes A_{0}\otimes A_{0}\otimes A_{0}\otimes A_{0}|\bar{\Psi}\rangle
& =\sqrt{\frac{3}{2}\gamma^{7}(1-\gamma)^{2}}(c_{0}|01\rangle|\widetilde{0}\rangle|\widetilde{0}\rangle+c_{1}|10\rangle|\widetilde{1}\rangle|\widetilde{1}\rangle),\\
A_{2}\otimes A_{0}\otimes A_{0}\otimes A_{0}\otimes A_{0}\otimes A_{0}|\bar{\Psi}\rangle
& =\frac{\sqrt{3}}{2}\sqrt{\gamma^{7}(1-\gamma)^{2}}(c_{0}|10\rangle|\widetilde{0}\rangle|\widetilde{0}\rangle+c_{1}|01\rangle|\widetilde{1}\rangle|\widetilde{1}\rangle),\\
A_{0}\otimes A_{2}\otimes A_{0}\otimes A_{0}\otimes A_{0}\otimes A_{0}|\bar{\Psi}\rangle
& =\frac{\sqrt{3}}{2}\sqrt{\gamma^{7}(1-\gamma)^{2}}(c_{0}|10\rangle|\widetilde{0}\rangle|\widetilde{0}\rangle+c_{1}|01\rangle|\widetilde{1}\rangle|\widetilde{1}\rangle),\numberthis \\
A_{1}\otimes A_{0}\otimes A_{1}\otimes A_{0}\otimes A_{0}\otimes A_{0}|\bar{\Psi}\rangle
& =\frac{3}{2}\sqrt{\gamma^{7}(1-\gamma)^{2}}(c_{0}\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)|\widetilde{0}\rangle+c_{1}|11\rangle|11\rangle|\widetilde{1}\rangle),\\
A_{0}\otimes A_{1}\otimes A_{1}\otimes A_{0}\otimes A_{0}\otimes A_{0}|\bar{\Psi}\rangle
& =\frac{3}{2}\sqrt{\gamma^{7}(1-\gamma)^{2}}(c_{0}|11\rangle\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)|\widetilde{0}\rangle+c_{1}\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)|11\rangle|\widetilde{1}\rangle),\\
A_{0}\otimes A_{1}\otimes A_{0}\otimes A_{1}\otimes A_{0}\otimes A_{0}|\bar{\Psi}\rangle
& =\frac{3}{2}\sqrt{\gamma^{7}(1-\gamma)^{2}}(c_{0}|11\rangle|11\rangle|\widetilde{0}\rangle+c_{1}\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)|\widetilde{1}\rangle),\\
A_{1}\otimes A_{0}\otimes A_{0}\otimes A_{1}\otimes A_{0}\otimes A_{0}|\bar{\Psi}\rangle
& =\frac{3}{2}\sqrt{\gamma^{7}(1-\gamma)^{2}}(c_{0}\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)|11\rangle|\widetilde{0}\rangle+c_{1}|11\rangle\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)|\widetilde{1}\rangle).\\
\end{align*}
\end{widetext}
One can easily verify that a recovery of the logical qubit is, in principle, possible by again detecting the photon number for each block with additional
inter-block parity measurements. It is then also not difficult to see that the KL conditions are fulfilled for these operators, so indeed
the corresponding two-photon loss errors can be corrected with this encoding. Note that the code is degenerate, i.e. the effect of some non-identical loss
errors on the logical qubit is identical. For the loss of three or more photons, the code ceases to be a
complete loss code.
The corresponding worst-case fidelity is
\begin{equation}
\begin{aligned}
F&=\gamma^{9}+9\gamma^{8}(1-\gamma)+36\gamma^{7}(1-\gamma)^{2}\\
&\approx 1-84(1-\gamma)^{3}.
\end{aligned}
\end{equation}
\noindent
This is the same result as for the bosonic code in Eq.\eqref{eq: nine}. However, note that in order to promote the encoding from a one-photon-loss
to a two-photon-loss code, in our scheme the maximal photon number per mode only needs to go up from two to three photons (as opposed
to four versus nine photons in Eq.\eqref{eq:Leung} and Eq.\eqref{eq: nine}, respectively). Similarly, the two-photon-loss code QPC(3,3) requires as many as 18 optical modes
compared to a modest number of six modes in our case.\\
Our procedure can be generalised for arbitrary $N$ (i.e., $N$ photons per block
and $N^{2}$ total number of photons), setting
\newpage
\begin{equation}
\begin{aligned}
|t_{0}^{2,N}\rangle&=BS[|N0\rangle],\\
|t_{1}^{2,N}\rangle&=BS[|0N\rangle],
\end{aligned}
\end{equation}
applying the Hadamard-type gate,
\begin{equation}
\begin{aligned}
|\widetilde{0}\rangle&=\frac{1}{\sqrt{2}}\left(|t_{0}^{2,N}\rangle+|t_{1}^{2,N}\rangle\right),\\
|\widetilde{1}\rangle&=\frac{1}{\sqrt{2}}\left(|t_{0}^{2,N}\rangle-|t_{1}^{2,N}\rangle\right),\\
\end{aligned}
\end{equation}
and finally introducing the $N$-block structure,
\begin{align*}
|\bar{0}\rangle&=|\widetilde{0}\rangle^{\otimes N}=\left(BS\left[\frac{1}{\sqrt{2}}(|N0\rangle+|0N\rangle)\right]\right)^{\otimes N}\\
&=\left(\frac{1}{\sqrt{2}^{N-1}}\sum\limits_{j=0}^{N}\sqrt{\binom{N}{2j}}|N-2j,2j\rangle\right)^{\otimes N}, \\
\numberthis \\
|\bar{1}\rangle&=|\widetilde{1}\rangle^{\otimes N}=\left(BS\left[\frac{1}{\sqrt{2}}(|N0\rangle-|0N\rangle)\right]\right)^{\otimes N} \\
&=\left(\frac{1}{\sqrt{2}^{N-1}}\sum\limits_{j=0}^{N}\sqrt{\binom{N}{2j+1}}|N-2j-1,2j+1\rangle\right)^{\otimes N}.
\end{align*}
\noindent
By construction (for more details, see the next section), this code corrects the loss of up to $N-1$ photons using $N^{2}$ photons. The worst-case fidelities
of different qubit codes are compared in Fig.~\ref{fig: qubit-codes} .
\begin{figure}[t!]
\centering
\includegraphics[width=0.51\textwidth]{qubit-codes2.png}
\caption{Worst-case fidelities for different qubit loss NOON codes as a function of $\gamma$:
$N^{2}=4$ (orange), $N^{2}=9$ (green), $N^{2}=16$ (blue) and $N^{2}=25$ (magenta), each correcting $N-1$ photon losses. Notice
the change of ordering with higher-order codes beating the lower-order codes for small losses and the converse for larger losses [see inset].
The small-loss regime $\gamma \in [0.95,1]$ would correspond to a communication channel length of $\sim 1~\text{km}$ (see Section \ref{sec: communication}~).}
\label{fig: qubit-codes}
\end{figure}
\noindent
One interesting feature of our qubit code construction is the interchangeability of the beam splitter transformation, Hadamard operation, and block building.
For example, consider the $N^{2}=4$ case. In order to produce the codewords, we first apply the symmetric beam splitter transformation
on $|20\rangle$ and $|02\rangle$, followed by the Hadamard gate, and finally build the blocks. The logical basis codewords obtained in this way are
\begin{equation}
\begin{aligned}
|\bar{0}\rangle&=\left[\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)\right]^{\otimes 2}\\
&=\frac{1}{2}(|2020\rangle+|2002\rangle+|0220\rangle+|0202\rangle),\\
|\bar{1}\rangle&=\left[\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle\right]^{\otimes 2}\\
&=\frac{1}{2}(|2020\rangle-|2002\rangle-|0220\rangle+|0202\rangle),
\end{aligned}
\end{equation}
which correspond to the codewords obtained as before up to a beam splitter transformation on each block. The details to verify that this encoding is also a proper code
as well as its extension to qudits can be found in Appendix \ref{sec: alternative noon}~.
\newpage
\section{Generalisation to qudit codes}
\noindent Our method can be directly generalised to logical qudits. Let us again illustrate the idea by a specific example, namely that for a
qutrit code ($d=3$). Define the states
\begin{align*}
|t_{0}^{3,2}\rangle&=T[|200\rangle]\\
&=\frac{1}{3}|200\rangle+\frac{1}{3}|020\rangle+\frac{1}{3}|002\rangle+\frac{\sqrt{2}}{3}|101\rangle\\
&~~+\frac{\sqrt{2}}{3}|011\rangle+\frac{\sqrt{2}}{3}|110\rangle,\numberthis\\
|t_{1}^{3,2}\rangle&=T[|020\rangle]\\
&=\frac{1}{3}|200\rangle+\frac{1}{3}\exp(4\pi i/3)|020\rangle+\frac{1}{3}\exp(-4\pi i/3)|002\rangle\\
&~~+\frac{\sqrt{2}}{3}\exp(-2\pi i/3)|101\rangle+\frac{\sqrt{2}}{3}|011\rangle\\
&~~+\frac{\sqrt{2}}{3}\exp(2\pi i/3)|110\rangle,\\
|t_{2}^{3,2}\rangle&=T[|002\rangle]\\
&=\frac{1}{3}|200\rangle+\frac{1}{3}\exp(-4\pi i/3)|020\rangle+\frac{1}{3}\exp(4\pi i/3)|002\rangle\\
&~~+\frac{\sqrt{2}}{3}\exp(2\pi i/3)|101\rangle+\frac{\sqrt{2}}{3}|011\rangle\\
&~~+\frac{\sqrt{2}}{3}\exp(-2\pi i/3)|110\rangle,
\end{align*}
\noindent
where $T$ now represents a "tritter" transformation, i.e. a symmetric 3-splitter.
The encoding works via a qutrit Hadamard-type gate:
\begin{align*}
|\widetilde{0}\rangle&=\frac{1}{\sqrt{3}}(|t_{0}^{3,2}\rangle+|t_{1}^{3,2}\rangle+|t_{2}^{3,2}\rangle) \numberthis\\
&=\frac{1}{\sqrt{3}}|200\rangle+\sqrt{\frac{2}{3}}|011\rangle,\\
|\widetilde{1} \rangle&=\frac{1}{\sqrt{3}}(|t_{0}^{3,2}\rangle+\exp(2\pi i/3)|t_{1}^{3,2}\rangle+\exp(-2\pi i/3)|t_{2}^{3,2}\rangle)\\
&=\frac{1}{\sqrt{3}}|020\rangle+\sqrt{\frac{2}{3}}|101\rangle,\\
|\widetilde{2} \rangle&=\frac{1}{\sqrt{3}}(|t_{0}^{3,2}\rangle+\exp(-2\pi i/3)|t_{1}^{3,2}\rangle+\exp(2\pi i/3)|t_{2}^{3,2}\rangle)\\
&=\frac{1}{\sqrt{3}}|002\rangle+\sqrt{\frac{2}{3}}|110\rangle.\\
\end{align*}
The logical qutrit state is then defined as
\begin{equation}
\bar{\Psi}\rangle=c_{0}|\widetilde{0}\rangle|\widetilde{0}\rangle+c_{1}|\widetilde{1} \rangle|\widetilde{1} \rangle+ c_{2} |\widetilde{2} \rangle| \widetilde{2} \rangle.
\end{equation}
\noindent
The states obtained from the logical qutrit after the loss of exactly one photon are:
\begin{align*}
&A_{1}\otimes A_{0}\otimes A_{0}\otimes A_{0} \otimes A_{0}\otimes A_{0} |\bar{\Psi}\rangle\\
&=\sqrt{\frac{2}{3}\gamma^{3}(1-\gamma)}(c_{0}|100\rangle|\widetilde{0}\rangle
+ c_{1}|001\rangle|\widetilde{1} \rangle+ c_{2}|010\rangle|\widetilde{2}\rangle),\\
&A_{0}\otimes A_{1}\otimes A_{0}\otimes A_{0} \otimes A_{0}\otimes A_{0} |\bar{\Psi}\rangle\\
&=\sqrt{\frac{2}{3}\gamma^{3}(1-\gamma)}(c_{0}|001\rangle|\widetilde{0}\rangle
+c_{1} |010\rangle|\widetilde{1} \rangle+c_{2} |100\rangle|\widetilde{2} \rangle),\\
&A_{0}\otimes A_{0}\otimes A_{1}\otimes A_{0} \otimes A_{0}\otimes A_{0} |\bar{\Psi}\rangle\\
&=\sqrt{\frac{2}{3}\gamma^{3}(1-\gamma)}(c_{0}|010\rangle|\widetilde{0}\rangle\numberthis
+c_{1} |100\rangle|\widetilde{1} \rangle +c_{2}|001\rangle |\widetilde{2} \rangle),\\
&A_{0}\otimes A_{0}\otimes A_{0}\otimes A_{1} \otimes A_{0}\otimes A_{0} |\bar{\Psi}\rangle\\
&=\sqrt{\frac{2}{3}\gamma^{3}(1-\gamma)}(c_{0}|\widetilde{0}\rangle|100\rangle
+c_{1} |\widetilde{1}\rangle|001\rangle+c_{2} |\widetilde{2} \rangle|010\rangle),\\
&A_{0}\otimes A_{0}\otimes A_{0}\otimes A_{0} \otimes A_{1}\otimes A_{0} |\bar{\Psi}\rangle\\
&=\sqrt{\frac{2}{3}\gamma^{3}(1-\gamma)}(c_{0}|\widetilde{0}\rangle|001\rangle
+c_{1}|\widetilde{1}\rangle|010\rangle+c_{2}|\widetilde{2} \rangle|100\rangle),\\
&A_{0}\otimes A_{0}\otimes A_{0}\otimes A_{0} \otimes A_{0}\otimes A_{1} |\bar{\Psi}\rangle\\
&=\sqrt{\frac{2}{3}\gamma^{3}(1-\gamma)}(c_{0}|\widetilde{0}\rangle|010\rangle
+ c_{1}|\widetilde{1}\rangle|100\rangle +c_{2}|\widetilde{2}\rangle|001\rangle).\\
\end{align*}
\noindent
Again, the KL conditions are obviously fulfilled, so the code can correct the loss of up to one single photon.
As before, the error spaces can be discriminated by identifying in which block the photon was lost and by measuring global inter-block observables (while simple
inter-block parities no longer work). An extension to higher photon numbers and to higher dimensional quantum systems is natural,
\begin{equation}
|t_{0}^{d,N}\rangle=S_{d}[|N00...0\rangle],..., |t_{d-1}^{d,N}\rangle=S_{d}[|000... 0N\rangle].
\end{equation}
Here, $S_{d}$ represents a $d$-splitter, i.e. a symmetric $d$-port device where $d$ is the number of modes. It is the multi-mode generalisation of a symmetric beam splitter
and the tritter as discussed above (thus $S_{2}=BS$ and $S_{3}=T$). As a linear optical device, it is defined by the linear relation between the annihilation operators of the input modes
$a_{i}, i=1,..., d$ and the annihilation operators of the output modes $b_{i}$:
\begin{equation}
b_{i}=\sum\limits_{j=1}^{d}U_{ij}a_{j}.
\end{equation}
Here, the unitary matrix $U$, connecting the input and output modes and ensuring photon number preservation, is given by
\begin{equation}
U_{kl}=\frac{1}{\sqrt{d}}\exp\left(i\frac{2\pi kl}{d}\right).
\end{equation}
\noindent
Then we define the following states:
\begin{equation}
|\widetilde{k}\rangle=\frac{1}{\sqrt{d}}\sum\limits_{j=0}^{d-1}\exp(2\pi i kj/d)|t_{j}^{d,N}\rangle,
\end{equation}
\noindent
for $k=0,..., d-1$. A general logical qudit is then expressed by the $dN$-mode, $N^{2}$-photon state
\begin{equation}
|\bar{\Psi}\rangle=c_{0}|\widetilde{0}\rangle^{\otimes N}+c_{1}|\widetilde{1}\rangle^{\otimes N}
+...+c_{d-1}|\widetilde{d-1}\rangle^{\otimes N}.
\end{equation}
By construction, this code can correct up to $N-1$ photon losses. The orthogonality of corrupted codewords, required by the KL conditions,
is easy to check, because the codewords are built blockwise. The non-deformation criterion, however, requires a more rigorous check. Let us first calculate the input
state $|t_{0}^{d,N}\rangle$ for general $N$ and $d$,
\begin{align*}
&|N00\cdots\rangle= \frac{a_{1}^{\dagger N}}{\sqrt{N!}}|000...\rangle
\rightarrow S_{d}[|N000...\rangle]\\
&=\frac{1}{\sqrt{N!}}\sqrt{\frac{1}{d}}^{N}(a_{1}^{\dagger}+a_{2}^{\dagger}+\cdots+a_{d}^{\dagger})^{N}|000\cdots\rangle \\
&=\frac{1}{\sqrt{N!}}\sqrt{\frac{1}{d}}^{N}\sum\limits_{\vec{k}\in \mathcal{A}} \begin{pmatrix}
N\\
k_{1},k_{2},\cdots, k_{d}
\end{pmatrix}
a_{1}^{\dagger k_{1}} a_{2}^{\dagger k_{2}}\cdots a_{d}^{\dagger k_{d}}|000\rangle\numberthis \\
&=\frac{1}{\sqrt{N!}}\sqrt{\frac{1}{d}}^{N}\sum\limits_{\vec{k}\in \mathcal{A}} \begin{pmatrix}
N\\
k_{1},k_{2},\cdots, k_{d}
\end{pmatrix}\\
&\times \sqrt{k_{1}!}\sqrt{k_{2}!}\cdots\sqrt{k_{d}!}|k_{1}, k_{2}, \cdots, k_{d}\rangle\\
&=\frac{1}{\sqrt{N!}}\sqrt{\frac{1}{d}}^{N}\sum\limits_{\vec{k}\in \mathcal{A}} \frac{N!}{\sqrt{k_{1}!k_{2}!\cdots k_{d}!}}
|k_{1},k_{2},\cdots, k_{d}\rangle.
\end{align*}
\noindent
In the third line, we used the multinomial theorem, bearing in mind that all the creation operators commute with each other.
Furthermore, we defined the multinomial coefficient,
\begin{equation}
\begin{pmatrix}
N\\
k_{1},k_{2},\cdots, k_{d}
\end{pmatrix}=\frac{N!}{k_{1}!k_{2}!\cdots k_{d}!},
\end{equation}
as the number of arrangements of $N$ objects in which there are $k_{j}$ objects of type j, $k_{q}$ objects of type q and so on.
We also introduced the set of $d$-dimensional vectors with fixed column sum, i.e. $\mathcal{A}\equiv \{\vec{k}\in \mathbb{N}_{0}^{d}|\sum\limits_{i=1}^{d}k_{i}=N\}$,
to parametrise the set of all $d$-mode Fock states with fixed photon number $N$. Furthermore, we define
$\mathcal{A}^{\prime}\equiv \{\vec{k}\in \mathbb{N}_{0}^{d}|\sum\limits_{i=1}^{d}k_{i}=N~ \text{and}~ k_{1}\geq 1\}$
and $\mathcal{A}^{\prime\prime}\equiv \{\vec{k}\in \mathbb{N}_{0}^{d}|\sum\limits_{i=1}^{d}k_{i}=N-1\}$.\\
We consider the loss of exactly one photon in the first mode, i.e. we apply the operator $A_{1}\otimes A_{0}^{\otimes d-1}$:
\begin{align*}
&A_{1}\otimes A_{0}^{\otimes d-1}S_{d}[|N00...\rangle]\\
&=\frac{1}{\sqrt{N!}}\sqrt{\frac{1}{d}}^{N}\sum\limits_{\vec{k}\in \mathcal{A}} \frac{N!}{\sqrt{k_{1}!}\sqrt{k_{2}!}\cdots\sqrt{k_{d}!}}\\
&\times A_{1}\otimes A_{0}^{\otimes d-1}|k_{1},k_{2},\cdots, k_{d}\rangle \\
&=\frac{1}{\sqrt{N!}}\sqrt{\frac{1}{d}}^{N}\sum\limits_{\vec{k}\in \mathcal{A^{\prime}}} \frac{N!}{\sqrt{k_{1}!}\sqrt{k_{2}!}\cdots\sqrt{k_{d}!}}\\
&\times\sqrt{\gamma}^{N-1}\sqrt{1-\gamma}\sqrt{k_{1}}|k_{1}-1, k_{2},\cdots, k_{d}\rangle\\
&=\frac{1}{\sqrt{N!}}\sqrt{\frac{1}{d}}^{N}\sqrt{\gamma}^{N-1}\sqrt{1-\gamma}\\
&\times\sum\limits_{\vec{k}\in \mathcal{A^{\prime}}} \frac{N!}{\sqrt{(k_{1}-1)!}\sqrt{k_{2}!}\cdots\sqrt{k_{d}!}}|k_{1}-1, k_{2},\cdots, k_{d}\rangle\\
&=\frac{1}{\sqrt{N!}}\sqrt{\frac{1}{d}}^{N}\sqrt{\gamma}^{N-1}\sqrt{1-\gamma}\\
&\times\sum\limits_{\vec{q}\in \mathcal{A^{\prime \prime}}} \frac{N!}{\sqrt{q_{1}!}\sqrt{q_{2}!}\cdots\sqrt{q_{d}!}}|q_{1}, q_{2},\cdots, q_{d}\rangle\\
&=\sqrt{N}\sqrt{\frac{1}{d}}\sqrt{\gamma}^{N-1}\sqrt{1-\gamma}S_{d}(|N-1,0,0,\cdots\rangle)\numberthis
\end{align*}
\noindent
For symmetry reasons, the loss of a photon in a different mode acts identically. The same is true for the other input states, i.e. $S_{d}[|0,0,\cdots,N,0,0,\cdots\rangle]$
decays into $S_{d}[|0,0,\cdots,N-1,0,0,\cdots\rangle]$ after losing one photon. Higher losses can be treated by induction. Because the blocks of the basis codewords are exactly
superpositions of these states, no deformation can take place after photon loss. Together with the orthogonality of corrupted codewords, this proves our qudit encoding to be a quantum
error correction code.
\section{Physical implementation}
\noindent In order to substantiate the importance of the encodings, we describe a scheme how to generate an arbitrary logical qubit for the simplest
code with just two photons per block ($N=2$). We assume that the states $\frac{1}{\sqrt{2}}(|20\rangle\pm|02\rangle)$ are experimentally accessible
from two single-photon states $|1\rangle\otimes |1\rangle$ with a phase-free and an appropriately phase-inducing, 50:50 beam splitter.
In addition, we need one auxiliary photon in two ancilla modes to produce the following states:
\begin{equation}
\begin{aligned}
|\psi_{1}\rangle=|0\rangle\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)|1\rangle,\\
|\psi_{2}\rangle=|1\rangle\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle)|0\rangle.\\
\end{aligned}
\end{equation}
As pointed out in \cite{sharypov}, by employing an ancilla ion-trap system, the generation of a symmetric entangled state,$\frac{1}{\sqrt{2}}(|\phi_{1}\rangle|\phi_{2}\rangle+|\phi_{2}\rangle|\phi_{1}\rangle),$
is, in principle, possible for arbitrary photonic input states $|\phi_{1}\rangle$ and $|\phi_{2}\rangle$. Applied to $|\psi_{1}\rangle$
and $|\psi_{2}\rangle$, one obtains
\begin{align*}
&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle)|0110\rangle\right.\numberthis \\
&\left.+\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)|1001\rangle\right),
\end{align*}
\noindent
where we already reordered the modes. The next step is to apply a general beam splitter with transmittance $t$, with the coefficients in the desired superposition determined
later, to the first and second pair of the ancilla modes. This leads to
\begin{align*}
&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle)\right.\\
&\times(\sqrt{1-t}|10\rangle-\sqrt{t}|01\rangle)(\sqrt{t}|10\rangle+\sqrt{1-t}|01\rangle)\numberthis \\
&+\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle) \\
&\left. (\sqrt{1-t}|01\rangle+\sqrt{t}|10\rangle)(-\sqrt{t}|01\rangle+\sqrt{1-t}|10\rangle)\right).
\end{align*}
\noindent
Measuring the photons after the beam splitter and detecting $'1001'$ projects the state onto
\begin{equation}
\begin{aligned}
&\frac{1-t}{\sqrt{t^{2}+(1-t)^{2}}}\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle))\\
&-\frac{t}{\sqrt{t^{2}+(1-t)^{2}}}\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)).
\end{aligned}
\end{equation}
Finally, a phase shift of $\pi/2$ on the last mode gives the logical qubit (, i.e applying $\exp\left(\frac{i\pi \hat{n}}{2}\right)$ to it)
\begin{align*}
|\bar{\Psi}\rangle&=\frac{1-t}{\sqrt{t^{2}+(1-t)^{2}}}\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle))\\
&-\frac{t}{\sqrt{t^{2}+(1-t)^{2}}}\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle)) \\
&=c_{0}(t)\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle+|02\rangle))\numberthis \label{eq:alternative} \\
&c_{1}(t)\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle)\frac{1}{\sqrt{2}}(|20\rangle-|02\rangle)),
\end{align*}
\noindent
similar to Eq.\eqref{twophotons}. This means that with an appropriated choice of $t$ and a final symmetric beam splitter transformation on the blocks,
any superposition of the logical codewords can be generated. Note that the logical qubit in Eq.\eqref{eq:alternative}
(without the final symmetric beam splitter) corresponds to the four-photon, alternative NOON code qubit [see Section \ref{sec: Qubit codes} and Appendix \ref{sec: alternative noon}~].
\section{Application in a one-way communication scheme}
\label{sec: communication}
\noindent In practice, especially for quantum communication, the direct transmission of a photonic state is performed through a noisy quantum channel which leads to an exponential decay of the success rate with the total distance due to photon loss.
To overcome this problem, besides the standard quantum repeater \cite{Briegel1, Briegel2}, a one-way quantum communication scheme can be applied \cite{lutkenhaus}. Here, an encoded quantum state is sent from a sending station directly through an optical
fibre of length $L_{0}$ to reach the first repeater station while suffering from a moderate amount of photon loss for sufficiently small $L_{0}$. In each intermediate station, teleportation-based error correction (TEC)\cite{Knill} is performed
before the corrected state is sent to the next repeater station. For logical qubits, TEC is realised by Bell-state preparation
and Bell measurements at the encoded level, which requires encoded Pauli operations as well as encoded Hadamard and CNOT gates.
As pointed out in \cite{Jiang2}, TEC can be generalised to logical qudits using qudit Pauli and SUM gates together with qudit Hadamard gates.\\
Based on the results of the last sections, the success probability for one-way communication over a total distance $L$
with repeater spacings $L_{0}$ of an $(N^{2},d)$ encoded qudit is \footnote{The success probability corresponds to a multiple of the fidelity
calculated in the last sections, $P_{succ}=[F(L_{0})]^{L/L_{0}}$.} \\
\begin{equation}
P_{succ}=\left(\sum\limits_{k=0}^{N-1}\binom{N^{2}}{k}\gamma^{N^{2}-k}(1-\gamma)^{k}\right)^{L/L_{0}}.
\end{equation}
Here, the damping parameter is given by $\gamma=\exp\left(-\frac{L_{0}}{L_{att}}\right)$ with the attenuation length $L_{att}=22~\text{km}$ for telecom fibres
and photons at telecom wavelengths. Note that $P_{succ}$ only depends on $N$ and especially not on $d$. The success probability
for the one-way scheme over a total distance of 1000 km using various codes is shown in Fig.~\ref{fig: compare} .\\
To assess the resources needed in a scheme with our qudit codes, we furthermore define a (spatial) cost function as \cite{lutkenhaus}
\begin{equation}
C(N,d)=\frac{N^{2}}{P_{succ}\log_{2}(d)L_{0}},
\end{equation}
which depends on the photon number $N$ per block and the dimension of the qudit $d$ \footnote{Compared to \cite{lutkenhaus}, here
we shall only consider the cost for transmitting logical qubits over a total distance $L$, instead of secure classical bits eventually obtained via quantum
communication.}. Fixing the total
photon number, the cost is obviously suppressed by the inverse of the binary logarithm of the qudit dimension (corresponding
to the effective number of encoded logical qubits) such that qudit encodings
make the one-way scheme more efficient. More interesting is the comparison of different qudit encodings with different total photon numbers, as
shown in Fig.~\ref{fig: cost}~. The plot shows the cost functions of various codes. The cost decreases with $N$ as $P_{succ}$ is increasing
at the same time for a suitably chosen $L_{0}$.
\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{one-way-codes.png}
\caption{Success probabilities for the one-way scheme with different encodings:
$N^{2}=4$ (orange), $N^{2}=9$ (green), $N^{2}=16$ (blue) and $N^{2}=100$ (magenta) total photons for $L=1000~ \text{km}$.}
\label{fig: compare}
\end{figure}
\begin{figure}[t!]
\centering
\subfigure[$N^{2}=1$]{\includegraphics[width=0.22\textwidth]{cost_1.png}}\quad
\subfigure[$N^{2}=9$]{\includegraphics[width=0.22\textwidth]{cost9.png}}\quad
\subfigure[$N^{2}=16$]{\includegraphics[width=0.22\textwidth]{cost16.png}}\quad
\subfigure[$N^{2}=25$]{\includegraphics[width=0.22\textwidth]{cost25.png}}
\caption{Cost function for codes with different photon numbers and dimensions for $L=1000~ \text{km}$:
$d=2$ (red), $d=3$ (green), $d=4$ (blue) (from top to bottom).}
\label{fig: cost}
\end{figure}
\noindent
Note that also $N^{2}=1$ can be realised in the so-called multiple-rail qudit encoding, where a single photon occupies
one of $d$ modes, i.e. $|\bar{0}\rangle=|1000..\rangle, |\bar{1}\rangle=|0100..\rangle,..., |\overline{d-1}\rangle=|00...01\rangle$. Since the scaling
of the transmission probability with the loss parameter $\gamma$ only depends on the total photon number (and especially not on the qudit dimension $d$),
a cost reduction can be achieved already in this case by increasing $d$. However, the multiple-rail encoding is not a quantum error correction code; it is
only a quantum error detection code \cite{N-C} that can detect but not correct loss errors.
\section{Conclusions}
\noindent
We presented a systematic approach for constructing a class of exact quantum error correcting codes for the amplitude-damping channel. Based on
quantum optical NOON state resources, logical qubits can be encoded in a block code consuming a total of $N^{2}$ photons in $N$ blocks.
These codes are capable of correcting $N-1$ photon losses, which is the same scaling obtainable with existing exact loss codes for the same fixed
total photon number. Nonetheless, only our codes have a total mode number and a maximal photon number per mode that both scale linearly with $N$.\\
All our codes have logical codewords that can be built from NOON states with linear optics. A method for the experimental
generation of the $N^{2}=4$ qubit code including arbitrary logical qubits was also proposed. This method relies on the presence of an ion-trap ancilla system.
Furthermore, the NOON code approach can be generalised to logical qudits of arbitrary dimension by increasing the mode number per block without
losing the loss robustness, i.e. the fidelity always only depends on the total photon number $N^{2}$ and not on the dimension of the logical qudit.\\
As for an application, this feature is exploited in a one-way communication scheme where general qudit codes turn out to be beneficial in terms of
the spatial resource cost. Limitations of our codes are that there is no simple and efficient method known for the experimental generation of qubit codes
with higher loss resistance and for that of arbitrary qudit codes (including arbitrary logical quantum states).
This is, however, necessary for the presented one-way scheme, because for achieving a useful success probability at moderate intermediate distances
$L_{0} \thicksim 1~ \text{km}$, $N^{2}=\mathcal{O}(100)$ will be required.
In addition, the proposed QND-type measurement for syndrome identification and the corresponding recovery operation, possibly implemented via encoded
qudit quantum teleportation, are experimentally hard to achieve.
\section{Acknowledgements}
\noindent We acknowledge support from Q.com (BMBF) and Hipercom (ERA-NET CHISTERA).
\bibliographystyle{apsrev4-1}
| 20,071 |
\section{Introduction}
\large
\noindent Generally used techniques for multivariate approximation were polynomial interpolation and piece-wise polynomial splines till 1971, Rolland Hardy proposed a new method using a variable kernel for each interpolation point, assembled as a function, depending only on the radial distance from the origin or any specific reference point termed as `\textit{center}', \citep{Hardy1971}. Such a kernel is known as radial kernel or radial basis functions (RBFs). In 1979, Richard Franke studied many available approaches for scattered data interpolation and established the utility of RBFs over many other schemes {\color{blue}\citep{franke1979}}. Since RBF approximation methods do not require to be interpolated over tensor grids using rectangular domains, they have been proven to work effectively, where the polynomial approximation could not be precisely applied {\color{blue}\citep{Sarra201168}}.
Global interpolation methods based on radial basis functions have been found to be efficient for surface fitting of scattered data sampled at $s$-dimensional scattered nodes. For a large number of samples, however, such interpolations lead to the solution of ill-conditioned system of equations {\color{blue}\citep{Fass2009,Forn2011,Fass2012,Lin20122,Chen2014}}. This ill-conditioning occurs due to the global nature of the RBF interpolation, where the interpolated value at each node is influenced by all the node points in the domain providing full matrices that tend to become more ill-conditioned as the shape parameter gets smaller \citep{Driscoll2002}.
Several approaches have been proposed to deal with ill-conditioning in global interpolation using RBFs: Kansa and Hon performed a series of numerical tests using various schemes like replacement of global solvers by block partitioning or LU decomposition, use of matrix preconditioners, variable shape parameters based on the function curvature, multizone methods, and node adaptivity which minimizes the required number of nodes for the problem {\color{blue}\citep{KansaHon2002}}. Some other approaches to deal with the ill-conditioning in the global RBF interpolation are: accelerated iterated approximate moving least squares {\color{blue}\citep{Fass2009}}, random variable shape parameters {\color{blue}\citep{Sarra20091239}}, Contour-Pad\'e and RBF-QR algorithms {\color{blue}\citep{Forn2011}}, series expansion of Gaussian RBF {\color{blue}\citep{Fass2012}}, and regularized symmetric positive definite matrix factorization \citep{Sarra2014}. The Contour-Pad\'e approach is limited to few degrees of freedom only. According to Fornberg and Flyer {\color{blue}\citep{Fornberg2013627,Forn2015}}, the current best (in terms of computation time) is the RBF-GA algorithm which is limited to the Gaussian RBFs only. RBF-GA is a variant of the RBF-QR technique developed by Fornberg and Piret {\color{blue}\citep{Fornberg200760}}. There are other modern variants: the Hilbert-Schmidt SVD approach developed by Fasshauer and McCourt which can be shown to be equivalent to RBF-QR, but approaches the problem from the perspective of Mercer's theorem and eigenfunction expansions {\color{blue}\citep{Fass2015}}. Another approach has been proposed by DeMarchi what he termed as the Weighted SVD method, which works with any RBF, but requires a quadrature/cubature rule, and only partially offsets ill-conditioning {\color{blue}\citep{DeMarchi20131}}. Recently Kindelan et al. proposed an algorithm to study RBF interpolation with small parameters, based on Laurent series of the inverse of the RBF interpolation matrix {\color{blue} \citep{Gonzale2015,Kindelan2016}}. Another parallel approach to deal with the ill-conditioning problem is to use the local approximations like RBF-FD; however in this paper, we consider only the global approximations to focus on the effect of the proposed approach.
In this paper, we propose a hybrid radial basis function (HRBF) using the Gaussian and a cubic kernels, which significantly improves the condition of the system matrix avoiding the above mentioned ill-conditioning in RBF interpolation schemes. In order to examine the effect of various possible hybridization with these two kernels, we use the global particle swarm optimization for deciding the optimal weights for the both kernels as well as the optimal value of the shape parameter of the Gaussian kernel. Through a number of numerical tests, we discuss the advantages of the proposed kernel over the Gaussian and the cubic kernels, when used individually. We also also test the proposed kernel for the interpolation of a synthetic topographical data near a normal fault, at a relatively large number of desired locations, which could not be interpolated with only the Gaussian kernel, due to the ill-conditioning issue.
The rest of the paper is structured as follows. In section 2, we give a brief introduction of the radial basis functions and the fundamental interpolation problem. In section 3, after briefly arguing over the need of RBFs for interpolation problems and its advantage over geostatistical approaches for the same, we introduce our proposed hybrid kernel along with the motivation behind such a hybridization. Section 4 contains the description of polynomial augmentation in the hybrid kernel and additional constrains to ensure the non-singularity of the system. In order to get the optimal combination of the parameters introduced in section 3, we have used global particle swarm optimization algorithm, which has been briefly explained in section 5. Through numerical tests, we check the linear reproduction property of the proposed hybrid kernel and application of this kernel for 2-D interpolation problems in sections 6.1 and 6.2 respectively. We analyze the eigenvalues of the interpolation matrices obtained by using the proposed hybrid kernel in section 6.3. In section 6.4, a comparative study of two different cost functions has been shown. In order to bring in some complexity in the analysis, in section 6.5, the proposed hybrid radial basis function has been used for interpolation of a 2-D geophysical data and its comparison with some other interpolation methods like linear, cubic, ordinary kriging and the Gaussian radial basis interpolation. Finally we discuss the computational cost of the present method, followed by the conclusion.
\section{Radial basis functions and spatial interpolation}
RBF and geostatistical approach for scattered data interpolation are quite similar and provide almost equally good results, in general; however, for the data set which has very small number of measurements at locations making the spatial correlation very difficult, the application of geostatistical methods offers several challenges.
In such cases, variographic study is quite difficult to perform, which is a primary requirement in understanding the data for proper application of statistical tools. In 2009, Cristian and Virginia performed several interpolation tests with real data sets and discussed some of the advantages of using RBF for scattered data interpolation {\color{blue}\citep{Rusu2006}}. For the completeness, we briefly define the RBF and the fundamental interpolation problem.
\noindent \textbf{Definition 2.1.} A function $\bm{\Phi}: \mathbb{R}^s \rightarrow \mathbb{R}$ is said to be radial if there exists a univariate function $\phi : [0, \infty) \rightarrow \mathbb{R}$ such that
\begin{equation}
\bm{\Phi} \left(\bm{x} \right) = \phi(r), \qquad r=\parallel\bm{x}\parallel.
\end{equation}
\noindent$\parallel\cdot\parallel $ here, represents Euclidean norm. Some commonly used RBFs have been listed in Table \ref{tab:rbflist}. The constant $\epsilon$ is termed as the shape parameter of the corresponding RBF.
\begin{table*}
\large
\begin{tabular*}{250pt}{ll}
\hline
RBF Name & Mathematical Expression \\
\hline
Multiquadric & $\phi(r)= (1+(\epsilon r)^2)^{1/2}$ \\
Inverse multiquadric & $\phi(r)= (1+(\epsilon r)^2)^{-1/2} $\\
Gaussian & $\phi(r)=e^{-(\epsilon r)^2}$\\
Thin plate spline & $\phi(r)=r^2 \log(r) $\\
Cubic & $\phi(r)=r^3$\\
Wendland's & $\phi(r)=(1-\epsilon r)^{4}_{+}(4\epsilon r+1)$\\
\hline
\end{tabular*}
\caption{Typical RBFs and their expressions}
\label{tab:rbflist}
\end{table*}
\noindent For completeness, we briefly define a general interpolation problem here. Let us assume that we have some measurements $y_j\in \mathbb{R}$ at some scattered locations $\bm{x}_j \in \mathbb{R}^s$. For most of the benchmark tests, we assume that these measurements $\left(y_j\right)$ are obtained by sampling some test function $\bm{f}$ at locations $\bm{x}_j$. \\
\noindent \textbf{Problem 2.1.} Given the data $y_j \in \mathbb{R}$ at locations $\bm{x}_j \in \mathbb{R}^s$ $(j=1,2,..N)$, find a continuous function $\mathcal{F}$ such that,
\[\mathcal{F}(\bm{x}_j) = y_j \qquad j=1,2,...N.\]
\noindent Given the set of scattered centers $\bm{x}_j$ an approximation of the Franke's test function can be written as,
\begin{eqnarray}{\label{eq:one}}
\mathcal{F}(\bm{x}) = \sum_{j=1}^{N} c_j \phi (\parallel \bm{x}-\bm{x_j}\parallel).
\end{eqnarray}
where $\phi (\parallel \bm{x}-\bm{x_i}\parallel)$ is the value of the radial kernel, $\parallel \bm{x}-\bm{x_i}\parallel$ is the Euclidean distance between the observational point and the center, and $c_j =\{ c_1, ..., c_N \}$ are the unknown coefficients which are determined by solving a linear system of equations depending on the interpolation conditions. The system of linear equations for above representation can be written as, \\
\begin{eqnarray}
\begin{bmatrix}
\phi (\parallel \bm{x}_1-\bm{x}_1\parallel) & \cdots & \phi (\parallel \bm{x}_1-\bm{x}_N\parallel) \\
\phi (\parallel \bm{x}_2-\bm{x}_1\parallel) & \cdots & \phi (\parallel \bm{x_2}-\bm{x}_N\parallel) \\
\vdots & \vdots & \ddots & \vdots \\
\vdots & \vdots & & \vdots \\
\phi (\parallel \bm{x_N}-\bm{x}_1\parallel) & \cdots &\phi (\parallel \bm{x}_N-\bm{x}_N\parallel)
\end{bmatrix}
\begin{bmatrix}
c_1 \\
c_2\\
\vdots\\
\vdots \\
c_N
\end{bmatrix} =
\begin{bmatrix}
f(\bm{x_1}) \\
f(\bm{x_2})\\
\vdots\\
\vdots \\
f(\bm{x_N})
\end{bmatrix}
\end{eqnarray}
\section{Hybrid Gaussian-cubic kernels}
\noindent The stability and accuracy of the RBF interpolation depends on certain aspects of the algorithm, and the data involved. For example, if the scattered data comes from a sufficiently smooth function, the accuracy of the interpolation using the Gaussian RBF will increase relatively more as we increase the number of data points used, than those obtained with other kernels. Use of positive definite kernels like the Gaussian radial basis function assures the uniqueness in the interpolation. This means that the system matrix of interpolation is non-singular even if the input data points are very few and poorly distributed making the Gaussian RBF a popular choice for interpolation and numerical solution of PDEs. Since the accuracy and stability of Gaussian radial basis function interpolation mostly depends on the solution of the system of linear equations, the condition number of the system matrix plays an important role. It has been seen that the system of linear equations is severely ill-conditioned either for large number of data points in the domain or for very small shape parameter i.e., flat radial basis. A ``small" shape parameter becomes typical for sparse data. Also, to deal with coarse sampling and a low-level information content, the RBFs must be relatively flat. The Gaussian radial basis function leads to ill-conditioned system when the shape parameter is small.
Cubic radial basis function $(\phi (r) = r^3)$, on the other hand, is an example of finitely smooth radial basis functions. Unlike the Gaussian RBF, it is free of shape parameter which excludes the possibility of ill-conditioning due to small shape parameters. However, using a cubic RBF only in the interpolation might be problematic since the resulting linear system may become singular for certain point locations. Another difference between cubic RBFs and Gaussians lies in their approximation power. This means that assuming the scattered data comes from a sufficiently smooth function, for a Gaussian RBF the accuracy of an interpolant will increase much more rapidly than the cubic RBF, as the number of used data points are increased. On the other hand, cubic RBFs can provide more stable and better converging interpolations for some specific data set but the risk of singularity will always be there associated with a typical node arrangement depending on the data type.
From what we have discussed above, it is certain that both the Gaussian and cubic RBFs have their own advantage for scattered data interpolation. In order to make the interpolation more flexible, we propose a hybrid basis function using a combination of both the Gaussian and the cubic radial basis function as given by,
\begin{eqnarray}
\phi(r) = \alpha e^{ -(\epsilon r)^2} +\beta r^3.
\end{eqnarray}
\noindent Here, $\epsilon$ is the usual shape parameter associated with the RBFs. We have introduced two weights $\alpha$ and $\beta$, which control the contribution of the Gaussian and the cubic part in the hybrid kernel, depending upon the type of problem and the input data points, to ensure the optimum accuracy and stability. Such a combination would seem appealing for different reasons, such as (1) the involvement of the cubic kernel in the hybridization helps conditioning when using low values of shape parameter, which likely is good for accuracy and (2) supplementary polynomials (of low order) can help cubics at boundaries and also improve (derivative) approximations of very smooth functions (such as constants). Figure \ref{1dkernel} contains the 1-D plots of the proposed kernel with some arbitrarily chosen parameters. In order to find the best combination of $\epsilon$, $\alpha$, and $\beta$, corresponding to maximum accuracy, we use the global particle swarm optimization algorithm.
\begin{figure}[hbtp]
\centering
\includegraphics[scale=0.35]{kernelplot1D.pdf}
\caption{1D Plots of the hybrid kernels with different combinations of the parameters.}
\label{1dkernel}
\end{figure}
\section{Polynomial augmentation and linear reproduction}
Application of radial basis functions for multivariate data interpolation is strongly supported by the limitation of polynomials for the same. In general, the well-posedness of multivariate polynomial interpolation can only be assured if the data sites are in certain special locations. However, polynomial augmentation in radial basis function has been reported to improve the convergence. A desirable property of a typical interpolant is linear reproduction,\emph{i.e.}, if the data is sampled from a linear function, the interpolation should reproduce it exactly. The interpolation using the Gaussian radial basis function does not reproduce the simple linear polynomials. Such polynomial reproduction is recommended especially when the interpolation is intended to be used in numerical solution of partial differential equations. The ability to reproduce the linear function is termed as the ``\emph{patch test}''. The Gaussian, and other radial basis functions, which are integrable, do not reproduce linear polynomials {\color{blue}\citep{Sback1994,Fassbook2007}}. However, a polynomial augmentation in the Gaussian radial basis function reproduces the linear polynomial to the machine precision {\color{blue}\cite{Fassbook2007}}. Fornberg and Flyer have also shown the improvement due to polynomial augmentation in cubic and quintic radial basis function {\color{blue}\cite{Fornberg2002,Barnett2015}}.
It is well known that the Gaussian kernel is positive definite. This implies that the Gaussian kernel is conditionally positive definite of any order too. Any non-negative linear combination of conditionally positive definite kernels (of the same order) is again conditionally positive definite (of the same order). Therefore, adding a linear polynomial to the hybrid kernel expansion ensures that the resulting augmented interpolation matrix is non-singular; just as it does for the cubic kernel.
In order to reproduce linear functions, we add low order polynomial function, \emph{i.e.}, $\bm{x}\mapsto 1$,$\bm{x}\mapsto x$, and $\bm{x}\mapsto y$ to the proposed kernel. By doing so, we have added three more unknowns to the interpolation problem, which makes the total number of unknowns equal to $N+3$. The polynomial augmentation, explained here, is for the 2D case only. Linear polynomial augmentation in other dimensions works analogously. The approximation now becomes,
\begin{eqnarray}
\mathcal{F}(\bm{x}) = \sum_{k=1}^{N} c_k \phi(\parallel \bm{x}-\bm{x_k}\parallel)+c_{N+1}+c_{N+2}x+c_{N+3}y \qquad \bm{x}=(x,y)\in \mathbb{R}^2,
\end{eqnarray}
At this point, we have $N$ interpolation conditions, i.e.,
\begin{eqnarray}
\mathcal{F}(\bm{x}_j) = \bm{f}(\bm{x}_j) \qquad j=1,2,...N.
\end{eqnarray}
In accordance with \cite{Fassbook2007,Sback1994,Fornberg2002}, we add three additional conditions to ensure the non-singular system as follows.
\begin{eqnarray}
\sum_{k=1}^N c_k =0, \qquad \sum_{k=1}^N c_k x_k =0, \qquad \sum_{k=1}^N c_k y_k =0.
\end{eqnarray}
Now we need to solve a system of the following form.
\begin{eqnarray}
\begin{bmatrix} \mathbf{A} & \mathbf{P} \\ \mathbf{P}^T & O \end{bmatrix}
\begin{bmatrix} \mathbf{c} \\ \mathbf{d}\end{bmatrix}
=
\begin{bmatrix} \mathbf{y}\\ \mathbf{0}\end{bmatrix},
\end{eqnarray}
where,
\[\mathbf{A}_{j,k}= \phi(\parallel \bm{x}_j-\bm{x}_k\parallel) \qquad j,k=1,...,N, \]
\[\mathbf{c}=[c_1,...,c_N]^T,\]
\[\mathbf{y}=[f(\bm{x}_1)....,f(\bm{x}_N)]^T,\]
\[\mathbf{P}_{j,k} =p_k(\bm{x}_k), \qquad p =[1\qquad x \qquad y], \]
$ \mathbf{0}$ is a zero vector of length 3, and $O$ is a zero matrix of size $3\times 3$.
\section{Parameter Optimization}
\noindent The performance of the hybrid kernel varies with different combination of the parameters $\alpha$, $\beta$, and $\epsilon$. Therefore, there must be an optimum combination of these parameters for which the performance of the hybrid kernel is the ``best'' for a typical problem. Ever since the RBFs have come into use, the selection of a good shape parameter has been a prime concern. In the context of dealing with ill-conditioned system in RBF interpolation and application, the work done by Fornberg and his colleagues established that there exists an optimal value of the shape parameter which corresponds to the optimal accuracy and stability. We have used global particle swarm optimization approach to determine the optimal shape parameter and weights in the proposed hybrid kernel. We have considered two objective functions; the RMS error in the interpolation and \emph{leave-one-out-cross-validation} criterion, to be minimized subjected to the given constrains.
\subsection{Particle swarm optimization}
\noindent The term optimization refers to the process of finding a set of parameters corresponding to a given criterion among many possible sets of parameters. One such optimization algorithm is particle swarm optimization (PSO), proposed by James Kennedy and Russell Eberhart in 1995 \citep{Everhart1995,Everhart2001}. PSO is known as an algorithm which is inspired by the exercise of living organisms like bird flocking and fish schooling. In PSO, the system is initiated with many possible random solutions and it finds optima in the given search space by updating the solutions over the specified number of generations. The possible solutions corresponding to a user defined criterion are termed as \emph{particles}. At each generation, the algorithm decides optimum particle towards which all the particles fly in the problem space. The rate of change in the position of a particle in the problem space is termed as \emph{particle velocity}. In each generation, all the particles are given two variables which are known as \emph{pbest} and \emph{gbest}. The first variable (\emph{pbest}) stores the best solution by a particle after a typical number of iteration. The second variable (\emph{gbest}) stores the global best solution, obtained so far by any particle in the search space \cite{Shaw2007,Singh2015}. Once the algorithm finds these two parameters, it updates the velocity and the position of all the particles according to the following pseudo-codes,
\begin{eqnarray}
\nonumber
v[.]= v[.]+c_1*rand(.)*(pbest[.] - present[.]) + c_2 * rand(.) * (gbest[.] - present[.]),
\end{eqnarray}
\begin{eqnarray}
\nonumber
present[.] = present[.] + v[.]
\end{eqnarray}
Where, $v[.]$ is the particle velocity, present[.] is the particle at current generation, and $c_1$ and $c_2$ are learning factors. According to the studies of Perez and Behdinan \cite{Perez2007}, the particle swarm algorithm is stable only if the following conditions are fulfilled;
\[0< c_1+c_2 < 4\]
\[\left( \frac{c_1+c_2}{2}\right)-1 < w<1\]
The optimization of the parameters of hybrid Gaussian-cubic kernel using particle swarm optimization is summarized in the flowchart given in the appendix A.
We consider two approaches for the construction of the objective function for the interpolation problem. If we know the exact solution of the interpolation problem, which works mostly for theoretical purpose, the formula for the RMS error can be taken as the objective function which is minimized for the best parameters by the optimization algorithm. On the other hand, if the exact solution is not known (which is most likely the case), leave-one-out-cross-validation (LOOCV), can be used as an objective function in the optimization algorithm. In the next two subsections we explain these two objective functions.
\subsection{RMS Error as the objective function}
\noindent The optimum combination of $\epsilon$, $\alpha$, and $\beta$ will depend on the type of the problem to which we are applying the proposed hybrid radial basis function. Such problem is called the \emph{forward problem} in the context of an optimization algorithm. The root mean square error over $M$ evaluation points is computed according to the formula given by,
\begin{eqnarray}
\label{error}
E_{rms} = \sqrt{\frac{1}{M}\sum_{j=1}^{M} \left[\mathcal{F}(\xi_j) - f(\xi_j) \right]^2}
\end{eqnarray}
Where $\xi_j, (j=1,...,M)$ are the evaluation points and $E_{rms}$ is the error function which is to be optimized for the minimum values for a set of $\epsilon$, $\alpha$, and $\beta$.
The optimization problem here, can be written in the mathematical form as following,
\[ Minimize \rightarrow E_{rms}(\epsilon, \alpha, \beta,), \]
subject to the following constraints,
\[\epsilon \geq 0,\]
\[0 \leq \alpha \leq 1,\]
\[0 \leq \beta \leq 1,\]
\subsection{Leave-one-out-cross-validation (LOOCV) as the objective function}
\noindent In practical problems of scattered data interpolation, the exact solution is most likely to be unknown. In such situations, it is not possible to calculate the exact RMS error. Cross-validation is a statistical approach. The basic idea behind cross-validation is to separate the available data into two or more parts and test the accuracy of the involved algorithm. Leave-one-out-cross-validation (LOOCV) in particular, is a special case of an exhaustive cross-validation, \textit{i.e.}, leave-$p$-out-cross-validation with $p=1$. The LOOCV algorithm separates one datum from the whole dataset of size $N$ and calculates the error of the involved method using the rest of the data. This is done repeatedly, each time excluding different datum and the corresponding errors are stored in an array. Unlike the regular LOOCV approach which performs relative error analysis \cite{Who90}, we consider this LOOCV error as the objective function in the particle swarm optimization algorithm. A detailed explanation of LOOCV can be found in \citep{Rippa}, however, we briefly explain the LOOCV used in this paper as follows.
\noindent Following the notations used in problem 2.1, let us write the datasites without the $k^{th}$ data as,
\[\bm{x}^{[k]} = [\bm{x}_1,...,\bm{x}_{k-1}, \bm{x}_{k+1},...,\bm{x}_N ]^T.\]
The removed point has been indicated by the superscript $[k]$. This superscript will differentiate the quantities computed with ``full" dataset and partial data set without the $k^{th}$ point. Hence, the partial RBF interpolant $\mathcal{F}(\bm{x})$ of the given data $\bm{f}(\bm{x})$ can be written as,
\[\mathcal{F}(\bm{x}) = \sum_{j=1}^{N-1} c^{[k]}_{j} \phi (\parallel \bm{x}-\bm{x}^{[k]}_{j}\parallel). \]
The error estimator can, therefore, be written as,
\[e_k = \bm{f}(\bm{x}_k) - \mathcal{F}^{[k]}(\bm{x}_k).\]
The norm of the error vector $\bm{e} = [ e_1,..., e_N]^T$, obtained by removing each one point and comparing the interpolant to the known value at the excluded point determines the quality of the interpolation. This norm serves as the ``cost function" which is the function of the kernel parameters $\epsilon$, $\alpha$, and $\beta$. We consider $l_2$ norm of the error vectors for our purpose. The algorithm for constructing the ``cost function" for RBF interpolation via LOOCV has been summarized in Algorithm 1. We recommend \cite{Fass2009,Fass2012} for some more insights of the application of LOOCV in radial basis interpolation problems. Here $c_k$ is the $k^{th}$ coefficient for the interpolant on ``full data" set and $\mathbf{A}^{-1}_{kk}$ is the $k^{th}$ diagonal element in the inverse of the interpolation matrix for ``full data".
\begin{algorithm}[!htbp]
\begin{algorithmic}[1]
\STATE Fix a set of parameters $[\epsilon, \alpha, \beta]$
\STATE For $k=1,...,N$
\STATE Compute the interpolant by excluding the $k^{th}$ point as,
\begin{eqnarray}
\mathcal{F}(\bm{x}) = \sum_{j=1}^{N-1} c^{[k]}_{j} \phi (\parallel \bm{x}-\bm{x}^{[k]}_{j}\parallel).
\end{eqnarray}
\STATE Computer the $k^{th}$ element of the error vector $e_k$
\begin{eqnarray}
e_k = \mid\bm{f}(\bm{x}_k) - \mathcal{F}^{[k]}(\bm{x}_k) \mid,
\end{eqnarray}
\STATE As proposed by Rippa \citep{Rippa}, a simplified alternative approach to compute $e_k$ is,
\begin{eqnarray}
e_k = \frac{c_k}{\mathbf{A}^{-1}_{kk}}.
\end{eqnarray}
\STATE end
\STATE Assemble the ``cost vector'' as $\bm{e} = [ e_1,..., e_N]^T$.
\STATE \noindent The optimization problem here, can be written in the mathematical form as,
\[ Minimize \rightarrow \parallel\bm{e}\parallel (\epsilon, \alpha, \beta), \]
subject to the following constraints,
\[\epsilon \geq 0,\]
\[0 \leq \alpha \leq 1,\]
\[0 \leq \beta \leq 1.\]
\end{algorithmic}
\caption{Leave-one-out-cross-validation for radial basis interpolation schemes.}
\label{alg:test}
\end{algorithm}
\section{Numerical Tests}
\subsection{Linear Reproduction}
\noindent In this test, we examine the linear reproduction of the hybrid kernel and compare it with that of the Gaussian kernel. We consider the following linear function to sample the test data,
\[f(\mathbf{x})= \frac{x+y}{2}.\]
\noindent Table \ref{tab:LRtest} contains the results of particle swarm optimization for the aforementioned linear polynomial reproduction. The objective function, which is frequently referred as the ``cost function" in the RBF literature, is RMS error. In order to compare the convergence of linear reproduction test using the Gaussian kernel, the hybrid kernel and the hybrid kernel with polynomial augmentation, parameter optimization has been performed for various degrees of freedoms, i.e.,$ [ 25, 49, 144, 196, 625, 1296, 2401, 4096]$. The RMS errors for various degrees of freedom using the hybrid kernel, the Gaussian kernel and the hybrid kernel with polynomial augmentation has been denoted as $E_{H}$, $E_{G}$, and $E_{H+P}$ respectively. Although we have used PSO for the optimization of the parameters here, it may not play a relevant role here for the fact that there are many (near) optimal results which reproduce the linear polynomial. It is well known that the Gaussian kernel alone does not reproduce any order of polynomial and needs polynomial augmentation for the same. This limitation is continued with the proposed hybrid kernel too. However, if we notice the convergence comparison for this test in Figure \ref{fig:LRtest2}(a), the proposed hybrid kernel offers better convergence as compared to the Gaussian kernel, which actually shows no convergence. The reason for such better convergence is the reduced condition number due to the proposed hybridization which could be seen in Figure \ref{fig:LRtest2}(b). Although the hybrid kernel does not reproduce the linear polynomial exactly, the condition number of the interpolation matrix is almost similar to those obtained by hybrid kernel with polynomial augmentation. Interestingly, a very small doping ($\beta$ values in Table 2) of the cubic kernel into the Gaussian kernel reduces the condition number of the interpolation, which is a solution to the ill-conditioned problem of the Gaussian kernel at large degrees of freedom.
\begin{table*}
\small
\centering
\begin{tabular*}{\textwidth}{l@{\extracolsep\fill}cccccc}
\hline
N & $\epsilon $ & $\alpha $ & $\beta$ & $E_{H}$ & $E_{G}$ & $E_{H+P}$ \\
\hline
25 & 0.1600 &0.9592 &$1.73e-09$ &$4.14e-07$ &$4.27e-07$ &$0.00e-00$ \\
49 &0.2151 &0.9216 &$2.03e-08$ &$4.15e-07$ &$1.29e-07$ &$6.00e-17$ \\
81 &0.5580 &0.5790 &$3.29e-08$ &$2.83e-07$ &$8.14e-09$ &$0.00e-00$ \\
144 &0.8633 &0.6994 &$4.58e-08$ &$1.36e-07$ &$3.18e-08$ &$4.81e-17$ \\
196 &0.5911 &0.9277 &$1.34e-07$ &$8.57e-08$ &$2.63e-08$ &$4.96e-17$ \\
625 &1.5290 &0.4350 &$8.89e-08$ &$2.20e-08$ &$3.33e-08$ &$4.40e-17$ \\
1296 &0.9397 &0.2791 &$8.46e-08$ &$1.85e-09$ &$2.31e-07$ &$4.18e-17$ \\
2401 &0.9450 &0.7403 &$6.27e-07$ &$1.50e-09$ &$7.14e-08$ &$7.43e-17$ \\
4096 &1.1183 &0.7590 &$2.34e-07$ &$5.37e-10$ &$4.24e-08$ &$4.40e-17$ \\
\hline
\end{tabular*}
\caption{Linear reproduction with parameter optimization, using hybrid kernel with polynomial augmentation.}
\label{tab:LRtest}
\end{table*}
Another interesting observation in this numerical test is the small values of the shape parameter. This observation exhibits the efficiency of the proposed hybridization at small shape parameters, which explains the higher accuracy of linear reproduction. The flat kernels made possible by the hybridization are closer to polynomials than what is possible with the Gaussian kernel alone due to the more severe ill-conditioning.
\begin{figure}[hbtp]
\centering
\includegraphics[scale=0.5]{Ntest01.pdf}
\caption{Numerical test to check the linear reproduction property of the proposed hybrid Gaussian-cubic kernel. The RMS error convergence and the condition number variation with increasing degrees of freedom using the Gaussian, the hybrid and the hybrid kernel with polynomial augmentation have been compared. }
\label{fig:LRtest2}
\end{figure}
\subsection{Franke's test}
\noindent Here, we perform 2D interpolation test using the proposed hybrid kernel and its comparative study with the Gaussian and cubic spline. A benchmark test \citep{franke1979} for 2D interpolation is to sample and reconstruct Franke's test function which is given by,
\begin{eqnarray}
\label{Franke}
f(x,y) = f_1+f_2+f_3-f_4,
\end{eqnarray}
where,
\[f_1=0.75 e^{\left( -\frac{1}{4} \left( (9x-2)^2 + (9y-2)^2 \right)\right)},\]
\[ f_2=0.75 e^{\left( -\frac{1}{49} (9x+1)^2 + \frac{1}{4}(9y+1)^2 \right)}, \]
\[f_3=0.50 e^{\left( -\frac{1}{4} \left( (9x-7)^2 + (9y-3)^2 \right)\right)},\]
\[f_4=0.20 e^{\left( (9x-4)^2 - (9y-7)^2 \right)},\]
The RMS error was kept as the objective function in the particle swarm optimization. We sample the Franke's test function at various number of data points, \textit{i.e.} $ [ 25, 49, 144, 196, 625, 1296, 2401, 4096]$ and try to reconstruct it while finding the optimal parameter combination for the hybrid kernel. Figure \ref{fig:PSOconv} shows a typical particle swarm optimization procedure for this test when we try to reconstruct the Franke's test function using $625$ data points. The convergence of the $pbest$ and $gbest$ values of $\epsilon$, $\alpha$, and $\beta$ over the generations, have been shown in Figures \ref{fig:PSOconv}(a)-(c). The swarm size for this test was $40$ and the optimization has been performed for 5 iterations. Figures \ref{fig:PSOconv}(d)-(f) show the histogram of the parameters' values acquired by each swarm over 5 iterations ($40\times 5=200$).
Table \ref{tab:PSO2DFranke} contains the results of this numerical test. It has been observed that the hybrid kernel performs very similar with or without the polynomial augmentation. Figure \ref{fig:Franke2D}(a) shows the RMS error variation with different shape parameters in this test with $625$ data points using hybrid kernel, hybrid kernel with polynomial augmentation, and only the Gaussian kernel. There are two interesting observations in Figure \ref{fig:Franke2D}(a). First, the optimal value of the shape parameter $\epsilon$ is almost the same for all the three kernels, which infers that the hybridization or even an additional polynomial augmentation does not affect the original optimal shape of the Gaussian part in the hybrid kernel. Also, Figure \ref{fig:Franke2D}(d) shows the interdependence of $\alpha$ and $\beta$ for which the corresponding $\epsilon$ does not change significantly. The observation here is that the optimization algorithm suggests various values of $\alpha$ in the solution inferring that the weight of the Gaussian kernel could have any values between $0.2-0.9$. The second interesting observation in Figure \ref{fig:Franke2D}(a) is the performance of the hybrid kernel for very small values of shape parameters. The hybrid kernel with or without the polynomial augmentation is quite stable in this range too. The corresponding condition number variations have been shown in Figure \ref{fig:Franke2D}(b). Like the previous numerical test of linear reproduction, the condition number here is also significantly reduced for the hybrid kernel, with or without polynomial augmentation, which is not surprising since the condition number of the matrix does not depend on the data values, only on the kernel chosen and the data locations. Figure \ref{fig:Franke2D}(c) shows the RMS error convergence with increasing degrees of freedom for the aforementioned kernels and an additional cubic kernel. The convergence of hybrid kernel with or without polynomial augmentation is better than that for the Gaussian kernel and far better than that of cubic kernel. The cubic kernel, however, has the least values of the condition numbers as shown in the Figure \ref{fig:Franke2D}(d). The overall observation in numerical test is that the hybrid kernel works better than the Gaussian and the cubic kernel as far as the RMS error convergence is concerned. Also, the polynomial augmentation is not required in the hybrid kernel unless the data comes from a linear function.
\begin{table*}
\centering \small
\label{test_kernelp2}
\begin{tabular*}{\textwidth}{c@{\extracolsep\fill}cccccccc}
\hline
N & $\epsilon $ & $\alpha $ & $\beta$ & $E_{rms}$ & $\epsilon^p$ & $\alpha^p$ & $\beta^p$ & $E^{p}_{rms}$ \\
\hline
25 & 2.9432 & $3.161e-01$ & $4.661e-01$ & $2.724e-02$ & $3.7378$ & $8.253e-01$ & $ 2.544e-06$ & $ 2.552e-02$\\
49 & 4.8600 & $1.138e-01 $ & $ 8.603e-01$ & $ 1.070e-02$ & $5.0242$ & $1.633e-01$ & $ 5.817e-01$ & $ 1.029e-02$\\
81 & 5.1345 & $4.462e-02$ &$ 9.316e-01$ & $ 4.044e-03$ & $5.2688$ &$4.532e-02$ & $9.582e-01$ & $ 3.900e-03 $ \\
144 & 6.2931 & $1.700e-02$ &$ 8.494e-01$ & $ 9.054e-04$ & $6.6797$ &$1.500e-02$ &$9.970e-01$ & $ 8.287e-04 $\\
196 & 5.5800 & $7.087e-02 $ & $9.445e-01$ & $1.658e-04$ &$5.3149$ & $4.134e-01$ & $4.094e-01$ & $ 2.912e-04 $ \\
400 & 5.5683 & $4.500e-01$ &$ 4.649e-05$ & $2.311e-05$ & $5.7856$ & $6.531e-01$ & $4.275e-07$ & $ 1.718e-05$\\
625 & 5.5434 & $6.749e-01$ & $4.915e-07 $ & $1.400e-06$ & $5.7530$ & $7.584e-01$ & $1.342e-06$ & $ 1.716e-05$\\
1296 & 6.2474 & $7.880e-01$ & $9.109e-09$ & $ 8.582e-09$ & $5.9265$ & $8.790e-01$ & $1.832e-09$ & $ 5.073e-09 $\\
2401 & 6.0249 & $5.600e-01$ & $2.503e-08$ & $2.106e-09$ & $6.3070$ & $9.520e-01$ &$5.704e-09$ & $9.006e-10 $\\
4096 & 5.7700& $9.107e-01$ & $7.090e-08 $ & $1.150e-09$ & $5.9397$ &$6.548e-01$ & $1.756e-08$ & $ 7.730e-10$\\
\hline
\end{tabular*}
\caption{Results of the parameter optimization test for 2-D interpolation using hybrid kernel. The superscript $p$ means that the parameter has been optimized with linear polynomial augmentation in the hybrid kernel.}
\label{tab:PSO2DFranke}
\end{table*}
\begin{figure}[hbtp]
\centering
\includegraphics[scale=0.4]{PSOpdata4096.pdf}
\caption{(a)-(c) Convergence of $pbest$ and $gbest$ over generations for optimization of $\epsilon$, $\alpha$, and $\beta$ respectively. (d)-(f) frequency histogram of the optimized solution for 40 swarms over 5 iterations for $\epsilon$, $\alpha$, and $\beta$ respectively. (g) shows the
interdependence of $\alpha$ and $\beta$ for fixed value of shape parameter $\epsilon = 5.5434$.}
\label{fig:PSOconv}
\end{figure}
\begin{figure}[hbtp]
\centering
\includegraphics[scale=0.58]{Ntest02_1_FrankeRMS.pdf}
\caption{Interpolation test with Franke's Test function using RMS Error as the objective function; (a)-(b) RMS error variation with the shape parameter for various kernel and (c)-(d) RMS error convergence for various kernels.}
\label{fig:Franke2D}
\end{figure}
\newpage
\subsection{Eigenvalue Analysis of Interpolation Matrices}
\noindent In this numerical test, we visualize the eigenvalue spectra of the interpolation matrices for the hybrid kernel with and without polynomial augmentation. Figure \ref{fig:eigentest} shows the eigenvalue analysis of the interpolation matrix in 2-D interpolation using Franke's test function and corresponding optimal values of $\epsilon$, $\alpha$, and $\beta$. The objective function for the optimization was RMS Error. The eigenvalue spectra of the interpolation matrices are shown for the hybrid kernel (top) and the hybrid kernel with polynomial augmentation (bottom). We have used the data for this numerical test from Table \ref{tab:PSO2DFranke}. The eigenvalues for various degrees of freedom, i.e., $N= [ 25, 49, 144, 196, 625, 1296, 2401, 4096]$ have been plotted together. For the hybrid kernel, the spectra has some negative eigenvalues for $N = [25, 49, 144, 196]$. The reason for this is the significantly large values of $\beta$ (see Table 3) representing the dominance of the cubic kernel in the hybridization. It has been observed that similar (slightly smaller) RMS errors can also be achieved if we force the values of $\beta$ to be smaller. For other larger degrees of freedoms (excluding the extra data points for polynomial augmentation for the augmented system), i.e., $N=[625,1296,2401,4096]$, all the eigenvalues are positive. The observation here is that if the cubic part is significantly small, the eigenvalues of the proposed hybrid kernel are positive making the kernel positive definite. On the other hand, if we include polynomial terms in the hybrid kernel, the spectra has some negative part for all degrees of freedom, even with the very small cubic part in the kernel.
\begin{figure}[hbtp]
\centering
\includegraphics[scale=0.38]{eigenvalues.pdf}
\caption{Eigenvalue analysis of the interpolation matrix in 2-D interpolation using Franke's test function and corresponding optimal values of $\epsilon$, $\alpha$, and $\beta$. The objective function for the optimization was RMS Error. The eigenvalue spectra of the interpolation matrix has been shown for the hybrid kernel (top) and the hybrid kernel with polynomial augmentation (bottom).}
\label{fig:eigentest}
\end{figure}
\subsection{The objective functions}
\noindent In this test we compare the optimization for two different objective functions, i.e., the RMS error and the cost function through leave-one-out-cross-validation. We take the same Franke's test function for this interpolation test. The optimization has been performed for various degrees of freedom, i.e., $N= [ 25, 49, 144, 196, 625, 1296, 2401, 4096]$. The results have been tabulated in Table \ref{tab:rmsvsloocv}. $\epsilon$,$\alpha$, $\beta$, and $E_{max}$ are the shape parameter and weight coefficients for the Gaussian and the cubic kernel when the objective function is RMS error, whereas $\epsilon^{'}$,$\alpha^{'}$, $\beta^{'}$ are the similar quantities when the objective function is the cost function given by LOOCV. Figure \ref{fig:rmsvsloocv} shows the convergence pattern of both the optimizations. The values in Table \ref{fig:rmsvsloocv} suggest that the global minima (as determined by PSO) are sometimes considerably different, depending on whether one uses RMS error or LOOCV as the objective function.
\begin{table*}
\centering \small
\label{test_kernelp3}
\begin{tabular*}{\textwidth}{l@{\extracolsep\fill}cccccc}
\hline
N & $\epsilon $ & $\alpha $ & $\beta$ & $\epsilon^{'}$ & $\alpha^{'}$ & $\beta^{'}$ \\
\hline
25 &2.9432 &0.3161 &0.4660 &2.4150 &0.3694 &0.7226 \\
49 &4.8584 &0.0770 &0.5830 &2.7318 &0.6733 &0.6690 \\
81 &5.1344 &0.0407 &0.8518 &3.6403 &0.7150 &0.06492 \\
144 &6.2932 &0.0170 &0.8494 &4.1900 &0.7419 &0.2815\\
196 &5.2291 & 0.8033 &$3.40e-03$ &4.3400 & 0.7400 &$2.00e-03$\\
400 &5.5683 & 0.4500 &$4.65e-05$ &5.3350 &0.9203 &$4.56e-08$ \\
625 &5.5434 & 0.6749 &$4.91e-07$ &5.1700 &0.6690 &$1.41e-07$ \\
1296&6.2474 & 0.7880 &$9.11e-09$ &4.2420 &0.9300 &$4.00e-03$ \\
2401&5.8711 & 0.6318 &$1.28e-07$ &5.1105 &0.7427 &$1.90e-05$ \\
\hline
\hline
\end{tabular*}
\caption{The optimized parameters using two different objective functions (OF),\textit{ i.e.}, RMS error and Leave-one-out-cross-validation.}
\label{tab:rmsvsloocv}
\end{table*}
\begin{figure}[hbtp]
\centering
\includegraphics[scale=0.5]{RMSvsLOOCVD.pdf}
\caption{Convergence of interpolation errors using two different optimization criteria,\textit{i.e.}, RMS error and LOOCV.}
\label{fig:rmsvsloocv}
\end{figure}
\subsection{Interpolation of normal fault data}
\noindent Next, an interpolation test has been performed for synthetic geophysical data. For this, a synthetic dataset representing the vertical distance of the surface of a stratigraphic horizon from a reference surface has been considered \citep{Martin2010}. The data set contains a lithological unit displaced by a normal fault. The foot wall has very small variations in elevations whereas the elevation in the hanging wall is significantly variable representing two large sedimentary basins. This data contains the surface information at irregularly spaced $78$ locations in the 2500 $km^2$ domain as shown in Figure \ref{fig:compareinterp}. This data has been reconstructed at $501\times501$ i.e. $251001$ regularly spaced new locations using hybrid radial basis interpolation. Although the used data is synthetic, the exact solution is not known. Optimization has been performed for this case, using the cost function, generated by the LOOCV scheme, which has been explained earlier. The optimal value of the shape parameter are $\epsilon=0.4318$, $\alpha = 0.7265$, and $\beta=0.4440$. Since we have already observed that the optimum value of $\epsilon$ is similar for the Gaussian kernel and the hybrid kernel, we compare the interpolation outputs with the Gaussian kernel using the same shape parameter as the hybrid one.
Figure \ref{fig:compareinterp} shows the interpolation of this fault data with various interpolation techniques like, MATLAB's linear and cubic interpolation, ordinary kriging, the Gaussian RBF and the hybrid RBF interpolation. It can be seen that the conventional Gaussian kernel does not interpolate this data properly, and the interpolation using hybrid kernel performs nice interpolation in agreement with the other approaches.
\begin{figure}[hbtp]
\centering
\includegraphics[scale=0.4]{compinterp.pdf}
\caption{Interpolation of the normal fault data using various techniques. The integer part of the data values have been displayed with the scattered data.}
\label{fig:compareinterp}
\end{figure}
\subsection{Computational Cost}
The computational cost study has been performed using a MATLAB implementation of the 2D interpolation algorithm using Franke's test function. Figure \ref{fig:cputime} illustrates the computational cost of 2D interpolation with the proposed hybrid kernel including the cost of parameter optimization. The cost of global RBF interpolation with the hybrid kernels varies approximately as $N^3$, which is similar to the RBF-QR approach without the optimization of kernel parameters \cite{Forn2011}. Moreover, the cost of RBF-QR significantly increases with the optimal value of the shape parameter, which is not the case here. The computational cost of the presented approach is likely to be reduced when used in local approximation form such as radial basis-finite difference (RBF-FD)--- as the ``matrices in the RBF-FD methodology go from being completely full to $99\%$ empty" \citep{Flyer2020}.
\begin{figure}[hbtp]
\centering
\includegraphics[scale=0.6]{CPUTime.pdf}
\caption{Elapsed CPU time 2D RBF interpolation with hybrid Gaussian-cubic kernels, including the time taken in the optimization process. The PSO algorithm in this test used the swarm size of 20 for a single iteration.}
\label{fig:cputime}
\end{figure}
\section{Conclusions}
\noindent We have proposed a hybrid kernel for radial basis interpolation and its applications. This hybrid kernel utilizes the advantages of the Gaussian and the cubic kernel according to the problem type. Based on the numerical tests performed in this work, we draw the following conclusions:
\begin{enumerate}
\item Combination of a small part of the cubic kernel in the Gaussian kernel reduces the condition number significantly, making the involved algorithm well-posed.
\item The optimal value of the shape parameter remains nearly the same for the Gaussian and hybrid Gaussian-cubic kernel.
\item The interpolation using the proposed hybrid kernel remains stable under the low shape parameter paradigm unlike the one with only the Gaussian RBF.
\item The hybrid kernel was used to interpolate real type geophysical data, which had close observational points and large changes, due to which, the Gaussian kernel could not interpolate this data. However, the interpolation using the hybrid kernel exhibits convincing results in agreement with the ordinary kriging approach.
\item When used in the global form, the computational cost of the proposed approach was found to vary as $N^3$, which is similar to the RBF-QR approach. However, unlike RBF-QR, the cost of the present approach is not parameter dependent. Also, the computational cost of the present approach can be further minimized by normalizing the hybrid kernel---therefore reducing the number of kernel parameters: from three to two.
\item Future work might involve the application of the proposed hybrid kernels in local RBF interpolations such as RBF-FD and for stable meshless schemes for numerical solution of PDEs.
\end{enumerate}
\section*{Acknowledgment}
\noindent Authors would like to acknowledge Dr. Wolfgang Schwanghart for providing source code for geostatistical kriging interpolation at MATLAB's file exchange. In this paper, this code (Ordinary kriging) has been used in the comparative study for normal fault interpolation. This code can be found at\\ \mcode{http://www.mathworks.com/matlabcentral/fileexchange/29025-ordinary-kriging}.
\section*{Appendix A:}
Figure \ref{fig:psoflowchart} shows a flowchart for parameter optimization for the RBF interpolation scheme.
\begin{figure}[hbtp]
\centering
\includegraphics[scale=0.6]{flowchart.pdf}
\caption{Flowchart of particle swarm optimization in the context of numerical test.}
\label{fig:psoflowchart}
\end{figure}
\bibliographystyle{spmpsci}
| 14,271 |
\section*{\refname}}
\begin{document}
\title{Terahertz Nonlinearity in Graphene Plasmons}
\author{Mohammad M. Jadidi}
\email{mmjadidi@umd.edu}
\affiliation{Institute for Research in Electronics \& Applied Physics, University of Maryland, College Park, MD 20742, USA}
\author{Jacob C. K\"onig-Otto}
\email{j.koenig-otto@hzdr.de}
\affiliation{Helmholtz-Zentrum Dresden-Rossendorf, PO Box 510119, D-01314 Dresden, Germany}
\affiliation{Technische Universit\"at Dresden, 01069 Dresden, Germany}
\author{Stephan Winnerl}
\email{s.winnerl@hzdr.de}
\affiliation{Helmholtz-Zentrum Dresden-Rossendorf, PO Box 510119, D-01314 Dresden, Germany}
\author{Andrei B. Sushkov}
\email{sushkov@umd.edu}
\affiliation{Center for Nanophysics and Advanced Materials, University of Maryland, College Park, Maryland 20742, USA}
\author{H. Dennis Drew}
\email{hdrew@umd.edu}
\affiliation{Center for Nanophysics and Advanced Materials, University of Maryland, College Park, Maryland 20742, USA}
\author{Thomas E. Murphy}
\email{tem@umd.edu}
\affiliation{Institute for Research in Electronics \& Applied Physics, University of Maryland, College Park, MD 20742, USA}
\author{Martin Mittendorff}
\email{martin@mittendorff.email}
\affiliation{Institute for Research in Electronics \& Applied Physics, University of Maryland, College Park, MD 20742, USA}
\begin{abstract}
Sub-wavelength graphene structures support localized plasmonic resonances in the terahertz and mid-infrared spectral regimes\cite{Ju2011,Brar2013,Yan2013}. The strong field confinement at the resonant frequency is predicted to significantly enhance the light-graphene interaction\cite{Mikhailov2008,Koppens2011}, which could enable nonlinear optics at low intensity \cite{Gullans2013,Manjavacas2011,Jablan2015} in atomically thin, sub-wavelength devices \cite{Kauranen2012}. To date, the nonlinear response of graphene plasmons and their energy loss dynamics have not been experimentally studied. We measure and theoretically model the terahertz nonlinear response and energy relaxation dynamics of plasmons in graphene nanoribbons. We employ a THz pump-THz probe technique at the plasmon frequency and observe a strong saturation of plasmon absorption followed by a 10 ps relaxation time. The observed nonlinearity is enhanced by two orders of magnitude compared to unpatterned graphene with no plasmon resonance. We further present a thermal model for the nonlinear plasmonic absorption that supports the experimental results.
\end{abstract}%
\keywords{graphene, plasmons, nonlinear, pump-probe, terahertz}
\maketitle
Graphene exhibits a broadband intrinsic nonlinear optical response \cite{Mikhailov2008,Hendry2010} that has been used in mode-locking \cite{Zhang2009} and harmonic generation \cite{Hong2013}. In the optical and near-infrared regime, the nonlinear response of graphene is primarily attributed to transient Pauli blocking, which leads to an ultrafast saturable absorption and nonlinear refraction\cite{Zhang2012}. In the terahertz regime\cite{Jnawali2013,Shi2014}, however, the nonlinear response is primarily caused by fast thermal heating and cooling of the electron population, which effects the intraband absorption\cite{Kadi2014,Mics2015}.
In the terahertz and mid-IR regime, the light-graphene interaction can be greatly increased by exploiting plasmon resonances, where the field is strongly localized and resonantly enhanced in a sub-wavelength graphene region\cite{Koppens2011}. A dramatic enhancement of the linear absorption has been experimentally observed in isolated subwavelength graphene elements\cite{Ju2011,Brar2013,Yan2013}, and graphene-filled metallic apertures \cite{Jadidi2015} at resonant frequencies that can be controlled through the graphene dimensions and carrier concentration. Significant enhancement in the nonlinear response of graphene can be expected and has been theoretically predicted\cite{Koppens2011,Manjavacas2011,Gullans2013,Yao2014,Jablan2015}. To date, there have been no experimental demonstrations to study this effect, or to explore the energy loss dynamics of these collective plasmonic excitations.
In this letter, we measure the nonlinear response of plasmon resonances in an array of graphene nanoribbons using THz pump-THz probe measurements with a free-electron laser that is tuned to the plasmon resonance (9.4 THz.) We observe a resonantly-enhanced pump-induced nonlinearity in the transmission that is orders of magnitude stronger than that of unpatterned graphene. The pump-probe measurements reveal an energy relaxation time of approximately 10 ps (measured at 20K). We present a thermal model of the nonlinear plasmonic response that includes scattering through LA phonons and disorder-assisted supercooling, which matches both the observed timescale and power-scaling of the nonlinear response. With the model we show that the strong pump-induced transmission change is caused by an unexpected red-shift and broadening of the resonance. Furthermore the model predicts that even greater resonant enhancement of the nonlinear response can be expected in high-mobility graphene, suggesting that nonlinear graphene plasmonic devices could be promising candidates for classical and quantum nonlinear optical processing.
Fig.~\ref{fig:1}a-b shows the structure, dimensions, and scanning-electron micrograph of the graphene plasmonic resonant structure considered here, and Fig.~\ref{fig:1}c shows the measured room-temperature linear transmission spectrum of the sample, which exhibits a strong dip in transmission centered at 9.4 THz that is associated with plasmonic absorption of the nanoribbons.
The plasmon resonance can be approximated by assuming an equivalent sheet conductivity of the graphene ribbon array \cite{Chen2013,Cai2015} (Supplementary Equation S9),
\begin{equation}\label{eq:1}
\sigma(\omega) =
\frac{w}{\Lambda}\frac{D}{\pi[\Gamma-i(\omega^2-\omega_p^2)/\omega]}
\end{equation}
where $\Gamma$ is the scattering rate and $D \simeq \sqrt{\pi n}e^2v_F/\hbar$ is the Drude weight of graphene with a carrier concentration of $n$ and Fermi velocity $v_F$. The plasmon resonant frequency is related to the Drude weight by $\omega_p^2 \equiv D w/\left[2 \Lambda^2 \epsilon_0\bar\epsilon \ln\left(\sec\left(\pi w/2\Lambda\right)\right)\right]$, where $\bar\epsilon = (\epsilon_1 + \epsilon_2)/2$ is the average of the substrate and incident dielectric constants \cite{Chen2013}.
The relative power transmission through such a conductive sheet is given by $\tau(\omega)/\tau_0 = \left|1+\sigma(\omega)/(Y_1+Y_2)\right|^{-2}$, where $\tau_0$ denotes the transmission with the graphene film absent, and $Y_j \equiv (\epsilon_0\epsilon_j/\mu_0)^{1/2}$ is the admittance of the incident ($j=1$) or substrate ($j=2$) region (see Supplementary Equation S4.) The green curve in Fig.~\ref{fig:1}b shows the best-fit transmission spectrum calculated using this model, from which we determined the carrier concentration and graphene scattering rate to be $n = 9\times 10^{12}$ cm$^{-2}$ and $\Gamma = 23$ rad/ps, respectively at room temperature, which corresponds to a Fermi energy of 0.35 eV and carrier mobility of 1,250 cm$^2$V$^{-1}$s$^{-1}$.
The pump-induced transmission change $\Delta \tau/\tau$ at the center of the plasmonic resonance was measured with spectrally narrow radiation (cf. Fig.~\ref{fig:1}c) in a setup that is depicted in Fig.~\ref{fig:2}. This signal, recorded as a function of the pump-probe delay $\Delta t$, is depicted in Fig.~\ref{fig:3}a for several pump fluences. In all cases, the pump causes a transient increase in transmission that is accompanied by a decrease in absorption. The observed nonlinear response decays in the wake of the pump pulse with a time constant of $\sim$ 10 ps, which is close to the previously reported hot electron-phonon relaxation time in graphene at the measurement temperature (20 K) \cite{Winnerl2011}.
The electron temperature $T$ in the graphene evolves in response to the terahertz pump pulse with intensity $I(t)$ at the center frequency $\omega_0$ according to \cite{Song2012}
\begin{equation}\label{eq:2}
\alpha T \frac{dT}{dt} + \beta (T^3-T_L^3) = A(\omega_0;T) I(t)
\end{equation}
where $\alpha=2\pi k_B^2 \varepsilon_F/(3\hbar^2 v_F^2)$ is the specific heat of graphene, $\beta=\zeta(3) V_D^2 \varepsilon_F k_B^3/(\pi^2 \rho \hbar^4 v_F^3 s^2 l)$ is the cooling coefficient, $T_L$ is the lattice temperature, $A(\omega_0;T)$ is the fractional absorption in the graphene, which itself depends on temperature. $k_B$ is the Boltzmann constant, $\rho$ is the areal mass density, $s$ is the speed of sound in graphene, $\zeta$ is the Riemann zeta function, $l$ is the electron-disorder mean free path, and $V_D$ is the acoustic deformation potential. We assume that the temperature relaxation is dominated by disorder-assisted supercollision cooling $\propto T^3$ \cite{Viljas2010,Song2012}, rather than momentum-conserving cooling\cite{Kar2014}.
The fractional absorption appearing in \eqref{eq:2} can be derived from the equivalent conductivity \eqref{eq:1} (Supplementary Equation S3),
\begin{equation}\label{eq:3}
A(\omega_0;T) = \frac{4Y_1\mathbf{Re}\left\{\sigma(\omega_0)\right\}}{|Y_1+Y_2+\sigma(\omega_0)|^2}
\end{equation}
where $\omega_0$ denotes the carrier frequency of the quasi-CW pump and probe pulses.
The basis of the thermal model is that the Drude weight $D$, scattering rate $\Gamma$, and plasmon frequency appearing in \eqref{eq:1} implicitly depend upon the electron temperature $T$, which increases when the incident pump pulse is absorbed in the graphene layer.
The temperature-dependent Drude weight \cite{DasSarma2011,Frenzel2014} and plasmon frequency (supplementary Section S3) are calculated as
\begin{align}
\label{eq:5}
D(T) &= \frac{2e^2}{\hbar^2}k_BT\ln\left[2\cosh\left(\frac{\mu(T)}{k_BT}\right)\right] \\
\label{eq:6}
\omega_p^2(T) &= \frac{D(T)w}{2 \epsilon_0\bar\epsilon \Lambda^2 \ln\left(\sec\left(\pi w/2\Lambda\right)\right)}
\end{align}
The scattering rate $\Gamma$ also varies with temperature, both because of temperature-dependent scattering from long-range Coulomb impurities and longitudinal acoustic (LA) phonons \cite{Chen2008}.
\begin{equation}\label{eq:7}
\Gamma(T) = \frac{\Gamma_0 \varepsilon_F}{\mu(T)} + \frac{k_B T \varepsilon_F V_D^2}{4\hbar^3v_F^2 \rho s^2} \\
\end{equation}
The second term in \eqref{eq:7} describes the temperature dependent LA phonon scattering, which was essential in order to match the observed fluence dependence of the nonlinear response, shown in Fig.~\ref{fig:3}c (see Supplementary Section S4.)
The results from the thermal model (Fig.~\ref{fig:3}b) are in close agreement with the experimental data (Fig.~\ref{fig:2}a), and correctly predict the 10 ps response time. The increased transmission is a result of a decreased plasmon frequency, which is caused by a reduced value of the chemical potential at elevated electron temperatures (cf. Eq. \eqref{eq:4} and \eqref{eq:5}), and a broadening of the resonance caused by a faster scattering rate. Fig.~\ref{fig:3}c plots the peak value of the (observed and calculated) transient response as a function of the incident pump fluence $F$, which shows an approximate $F^{1/2}$ dependence. Along the right axis, we plot the corresponding simulated peak electron temperature as a function of fluence, showing the expected $F^{1/3}$ dependence. The observed power scaling was best matched by assuming supercollision cooling as the single dominant cooling mechanism, together with temperature-dependent momentum scattering through LA phonons\cite{DasSarma2011,Chen2008}.
Only two free parameters were used in the numerical simulations: the acoustic deformation potential $V_D$ and the electron disorder mean free path $l$, which together control the strength of dominant cooling and scattering mechanisms. The observed $F^{1/2}$ power scaling seen in Fig.~\ref{fig:3}c was matched by choosing $V_D = 11$ eV, which is consistent with values reported in the literature for similar graphene \cite{Kar2014,McKitterick2015}. The mean free path $l$ was adjusted to match the overall magnitude of the nonlinearity, from which we obtained $l=2$ nm -- which is smaller than that expected from the scattering rate, but consistent with other recent experimental measurements of cooling in large-area graphene\cite{McKitterick2015}. The origin of this discrepancy remains to be explained.
To confirm the plasmonic enhancement of the nonlinearity, we repeated the pump-probe measurements with the pump and probe co-polarized in the direction parallel to the graphene ribbons, thus ensuring that the plasmons are not excited. Fig.~\ref{fig:4}a compares the measurements from the two polarization cases for the same incident pump fluence and frequency. The measured nonlinearity is far stronger when the plasmons are excited than for the opposite polarization, consistent with the thermal predictions. Fig.~\ref{fig:4}b shows the electric field profile at the plasmon resonance calcuated using (linear) finite element simulations, showing the dramatic field enhancement that occurs near the graphene sheet, which is responsible for the enhanced nonlinearity.
Fig.~\ref{fig:5} presents a calculation of how this nonlinearity would be further enhanced by employing higher quality graphene nanoribbons with a mobility of 25,100 cm$^2$V$^{-1}$s$^{-1}$\cite{Wang2013}. The calculated power transmission is shown as a function of frequency (in the vertical direction) and time (in the horizontal direction), assuming an input fluence of 1.27 $\mu$J/cm$^2$. The nonlinear transmission is caused by a transient red-shift of the plasmon frequency and broadening of the plasmon linewidth, causing a pump-induced change in transmission of order unity.
To conclude, the temperature dependent absorption, cooling, and scattering of hot electrons in graphene causes a nonlinear response to terahertz waves. Using terahertz pump-probe measurements, we show that when graphene is patterned into sub-wavelength structures that exhibit a plasmon resonance, this nonlinearity is greatly enhanced at the resonant frequency. This enhanced nonlinearity is caused by a stronger on-resonance absorption, followed by a spectral red-shift and broadening of the plasmon resonance with electron temperature. We provide a thermal model that explains the observed nonlinear enhancement, and sheds light on the dominant cooling and scattering mechanisms for hot electrons collectively excited in a graphene plasmon. The theory predicts that in higher-mobility graphene the nonlinearity in transmission could approach unity, enabling high-speed terahertz-induced switching or modulation.
\begin{figure}
\centering
\includegraphics[scale=1.75]{fig1}
\caption{\textbf{Structure and transmission spectrum of graphene nano ribbons.} (a) False color scanning electron micrograph of fabricated graphene ribbons. (b) Cross sectional diagram of device. (c) Measured (blue) and best fit (green) linear transmission spectrum of device, showing a decreased transmission at the plasmon frequency of 9.4 THz. The superposed red curve shows the measured spectrum of the free electron laser pulse source that was used to observe the nonlinear response.}
\label{fig:1}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=1.75]{fig2}
\caption{\textbf{Terahertz pump-probe measurement system.} The free-electron laser (FEL) was tuned to generate 5.5 ps pulses at a carrier frequency of 9.4 THz and repetition rate of 13 MHz. An optional reflective polarization rotation system orients the polarization perpendicular to the graphene ribbons. The pulses were separated into parallel, co-polarized pump and probe pulses that were focused onto the graphene sample inside of a cryostat. The transmitted probe power was measured as a function of the relative pump-probe delay $\Delta t$, which was controlled through a mechanical delay stage.}
\label{fig:2}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.95]{fig3}
\caption{\textbf{Pump-probe measurements and model.} (a) Measured relative change in transmission of the probe signal for different pump fluences as a function of pump-probe time delay $\Delta t$. The positive signal indicates a decrease in absorption that becomes stronger at higher pump fluences. (b) Calculated relative change in transmission based on a nonlinear thermal model for plasmonic absorption in graphene nanoribbons that includes supercollision cooling and LA phonon scattering. (c) Measured and simulated peak of relative transmission change (left) and peak electron temperature (right) as a function of pump fluence $F$.}
\label{fig:3}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=1.75]{fig4}
\caption{\textbf{Plasmonic enhancement of nonlinearity.} (a) Comparison of normalized change in transmission for two different polarizations. The blue curves show the measured and simulated pump-probe response when the pump and probe were polarized perpendicular to the graphene nanoribbons, thereby exciting the plasmon. The red curves show the measured and simulated response for the same incident pump fluence, but opposite polarization, where there is no plasmonic excitation, and the nonlinear response is correspondingly much lower. (b) Electric field profile at the resonant frequency, calculated using a (linear) finite element time domain method with a normally incident wave from above, showing the field-enhancement at the graphene surface. The color indicates the electric field intensity $|\mathbf{E}|^2$, relative to that of the incident plane wave, showing a nearly 9-fold intensity enhancement at the graphene surface. }
\label{fig:4}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=1.75]{fig5}
\caption{\textbf{Transient change in plasmon transmission spectrum.} Numerically predicted change in transmission as a function of frequency $f$ and time $\Delta t$, calculated assuming a higher graphene mobility of 25,100 cm$^2$V$^{-1}$s$^{-1}$. The pump pulse causes a transient red-shift and broadening of the plasmon resonance, as shown by the two vertical sections plotted on the right. The dashed curves indicated in the right panel show the calculated Drude response for an unpatterned graphene sheet, which shows no plasmon resonance, and very little pump-induced change in transmission. A signal tuned to the resonant frequency would experience a corresponding transient increase in transmission, as shown in the horizontal section plotted on the top.}
\label{fig:5}
\end{figure}
\bibliographystyle{naturemag}
| 5,083 |
\section{Introduction}
Nonparametric estimation under shape constraints is currently a very active research area in statistics.
A frequently encountered problem in this field is the estimation of the hazard rate, which, in survival analysis, is defined as the probability that an individual will experience an event within a small time interval given that the subject has survived until the beginning of this interval. In this context, monotonicity constraints arise naturally reflecting the property of aging or becoming more reliable as the survival time increases.
Beside this, it might be also of interest to characterize the distribution of the event times in terms of the density assuming that it is monotone.
Popular estimators, such as the nonparametric maximum likelihood estimator (NPMLE) or
Grenander type estimators, are typically piecewise constant and converge at rate~$n^{1/3}$.
However, at the price of additional smoothness assumptions on the hazard or density function, the cube-root-$n$ rate of convergence can be improved.
Smooth estimation has received considerable attention in the literature, because it is needed to prove that a bootstrap method works
(see for instance, \citet{kos2008}; \citet{SBW}).
Moreover, it provides a straightforward estimate of the derivative of the function of interest, which is of help when constructing confidence intervals (see for instance, \citet{nane15}).
Various approaches can be used to obtain smooth shape constrained estimators.
It essentially depends on the methods of both isotonization and smoothing and on the order of operations (see for instance, \citet{mammen1991} or \citet{GJW10}).
Chapter~8 in \citet{GJ14} gives an overview of such methods.
In this paper, we focus on kernel smoothed Grenander-type estimator (SG)
of the hazard function and the probability density in the presence of randomly right censored data.
\citet{huang-wellner1995} consider the random censorship model and
Grenander estimators of a monotone hazard and density,
obtained by taking slopes of the greatest convex minorant (lowest concave majorant) of the Nelson-Aalen
or Kaplan-Meier estimator.
Consistency and asymptotic distribution are established, together with the asymptotic equivalence with the maximum likelihood estimator.
The same model and the $L_p$-error of this type of estimators was investigated in~\citet{durot2007}.
However, both these papers do not take in consideration smoothing options. On the other hand, the kernel smoothed Grenander-type estimator of a monotone hazard in the context of the Cox model, which is a generalization of the right censoring that takes into account covariates, was introduced in \citet{Nane}, but without further development of its asymptotic distribution. However, on the basis of Theorem 3.1 in \citet{GJ13}, where no censoring takes place, our main result (Theorem~\ref{theo:distr}) was conjectured by \citet{Nane}. Afterwards, Theorem 11.8 in \citet{GJ14} states the limit distribution of the smoothed maximum likelihood estimator (SMLE) of a monotone hazard function using a more delicate argument.
Hence, it seems quite natural to address the problem of the smoothed Grenander-type estimator.
The present paper, aims at giving a rather short and direct proof of its limit distribution, relying on the method developed in \citet{GJ13} together with a Kiefer-Wolfowitz type of result derived in~\citet{DL14}.
Both Theorem~\ref{theo:distr} and Theorem~\ref{theo:distr-dens}, highlight the fact that also after applying smoothing techniques, the NPMLE and the Grenander estimator remain asymptotically equivalent.
Furthermore, we study inconsistency problems at the boundaries of the support.
In order to prevent those, different approaches have been tried, including penalization (see for instance, \citet{GJ13}) and boundary corrections
(see for instance, \citet{Albers}).
However, no method performs strictly better than the others.
We choose to use boundary kernels, but we discover that still the inconsistency at the right boundary can not be avoided.
The main reason for this is that a bound on the distance between the cumulative hazard (cumulative distribution) function and the Nelson-Aalen
(Kaplan-Meier) estimator is only available on intervals strictly smaller than the end point of the support.
The paper is organized as follows.
In Section~\ref{sec:model} we briefly introduce the Grenander estimator in the random censorship model and recall some results needed in the sequel.
The smoothed estimator of a monotone hazard function is described in Section~\ref{sec:asymptotics} and it is shown to be asymptotically normally distributed.
Moreover, a smooth estimator based on boundary kernels is studied and uniform consistency is derived.
Using the same approach, in Section~\ref{sec:dens} we deal with the problem of estimating a smooth monotone density function.
Section~\ref{sec:conf-int} is devoted to numerical results on pointwise confidence intervals.
Finally, we end with a short discussion on how these results relate to a more general picture.
\section{The random censorship model}
\label{sec:model}
Suppose we have an i.i.d.~sample $X_1,\dots,X_n$ with distribution function $F$ and density~$f$, representing the survival times.
Let $C_1,\dots,C_n$ be the i.i.d.~censoring variables with a distribution function $G$ and density~$g$.
Under the random censorship model, we assume that the survival time $X$ and the censoring time $C$ are independent and the observed data consists of
i.i.d.~pairs
of random variables $(T_1,\Delta_1),\dots,(T_n,\Delta_n)$, where $T$ denotes the follow-up time~$T=\min(X,C)$
and $\Delta=\mathds{1}_{\{X\leq C\}}$ is the censoring indicator.
Let $H$ and $H^{uc}$ denote the distribution function of the follow-up time and the sub-distribution function of the uncensored observations, respectively, i.e.,
$H^{uc}(x)=\mathbb{P}(T\leq x,\Delta=1)$.
Note that $H^{uc}(x)$ and $H(x)$ are differentiable with derivatives
\[
h^{uc}(x)=f(x)\left(1-G(x)\right)
\]
and
\[
h(x)=f(x)(1-G(x))+g(x)(1-F(x))
\]
respectively.
We also assume that $\tau_H=\tau_G<\tau_F\leq\infty$, where $\tau_F,\,\tau_G$ and $\tau_H$ are the end points of the support of $F,\,G$ and $H$.
The hazard rate $\lambda$ is characterized by the following relation
\[
\lambda(t)=\frac{f(t)}{1-F(t)}
\]
and we refer to the quantity
\[
\Lambda(t)=\int_0^t\lambda(u)\,\mathrm{d}u,
\]
as the cumulative hazard function.
First, we aim at estimating $\lambda$, subject to the constraint that it is increasing (the case of a decreasing hazard is analogous), on the basis of $n$ observations $(T_1,\Delta_1),\dots,(T_n,\Delta_n).$
The Grenander-type estimator $\tilde{\lambda}_n$ of $\lambda$ is defined as the left-hand slope of the greatest convex minorant $\tilde{\Lambda}_n$ of the Nelson-Aalen estimator $\Lambda_n$
of the cumulative hazard function $\Lambda$, where
\[
\Lambda_n(t)=
\sum_{i=1}^n
\frac{\mathds{1}_{\{T_i\leq t\}}\Delta_i}{\sum_{j=1}^n \mathds{1}_{\{T_j\geq T_i\}}}.
\]
Figure~\ref{fig:NA} shows the Nelson-Aalen estimator and its greatest convex minorant for a sample of $n=500$ from
a Weibull distribution with shape parameter $3$ and scale parameter $1$ for the event times and the uniform distribution on~$(0,1.3)$ for the censoring times.
We consider only the data up to the last observed time before the $90\%$ quantile of $H$.
The resulting Grenander-type estimator can be seen in Figure~\ref{fig:gren}.
\begin{figure}[t]
\includegraphics[width=.7\textwidth]{NA}
\caption{The Nelson-Aalen estimator (piecewise constant solid line) of the cumulative hazard (dotted line) and its greatest convex minorant (solid line).}
\label{fig:NA}
\end{figure}%
In \citet{huang-wellner1995} it is shown that the Grenander estimator of a nondecreasing hazard rate satisfies the following pointwise consistency result
\begin{equation}
\label{eqn:consG}
\lambda(t-)\leq \liminf_{n\to \infty} \tilde{\lambda}_n(t)\leq \limsup_{n\to \infty} \tilde{\lambda}_n(t)\leq \lambda(t+),
\end{equation}
with probability one and for all $0<t<\tau_H$, where $\lambda(t-)$ and $\lambda(t+)$ denote the left and right limit at $t$.
Moreover, we will also make use of the fact that for any $0<M<\tau_H$,
\begin{equation}
\label{eqn:chbound}
\sqrt{n}
\sup_{u\in[0,M]}
\left|\tilde{\Lambda}_n(u)-\Lambda(u)\right|
=
O_P(1),
\end{equation}
(see for instance, \citet{LopuhaaNane2013}, Theorem 5, in the case $\beta=0$, or \citet{VW96}, Example~3.9.19).
It becomes useful to introduce
\begin{equation}
\label{def:Phi}
\Phi(x)=
\int \mathds{1}_{[x,\infty)}(y)\,\mathrm{d}\mathbb{P}(y,\delta)=1-H(x)
\end{equation}
and
\begin{equation}
\label{def:Phin}
\Phi_n(x)=\int \mathds{1}_{[x,\infty)}(y)\,\mathrm{d}\mathbb{P}_n(y,\delta),
\end{equation}
where $\mathbb{P}$ is the probability distribution of $(T,\Delta)$ and $\mathbb{P}_n$ is the empirical measure of the pairs
$(T_i,\Delta_i)$, $i=1,\dots,n$. From Lemma~4 in~\citet{LopuhaaNane2013} we have,
\begin{equation}
\label{eqn:phi}
\sup_{x\in [0,\tau_H]}
|\Phi_n(x)-\Phi(x)|
\to0, \text{ a.s., and }
\sqrt{n}\sup_{x\in[0,\tau_H]}|\Phi_n(x)-\Phi(x)|=O_P(1).
\end{equation}
Let us notice that, with these notations, we can also write
\begin{equation}
\label{eqn:cum-haz}
\Lambda_n(t)
=
\int\frac{\delta\mathds{1}_{\{u\leq t\}}}{\Phi_n(u)}\,\mathrm{d}\mathbb{P}_n(u,\delta),
\qquad
\Lambda(t)
=
\int\frac{\delta\mathds{1}_{\{u\leq t\}}}{\Phi(u)}\,\mathrm{d}\mathbb{P}(u,\delta).
\end{equation}
Our second objective is to estimate a monotone (e.g., increasing) density function $f$ .
In this case the Grenander-type estimator $\tilde{f}_n$ of $f$ is defined as the left-hand slope of the greatest convex minorant $\tilde{F}_n$ of the Kaplan-Meier estimator $F_n$ of the cumulative distribution function~$F$.
Pointwise consistency of the Grenander estimator of a nondecreasing density:
\begin{equation}
\label{eqn:consGdens}
f(t-)\leq \liminf_{n\to \infty} \tilde{f}_n(t)\leq \limsup_{n\to \infty} \tilde{f}_n(t)\leq f(t+),
\end{equation}
with probability one, for all $0<t<\tau_H$, where $f(t-)$ and $f(t+)$ denote the left and right limit at $t$, is proved in \citet{huang-wellner1995}.
Moreover, for any $0<M<\tau_H$, it holds
\begin{equation}
\label{eqn:cdbound}
\sqrt{n}
\sup_{u\in[0,M]}
\left|\tilde{F}_n(u)-F(u)\right|
=
O_P(1),
\end{equation}
(see for instance, \citet{BC74}, Theorem 5). By Theorem 2 in~\citet{MR88}, for each $0<M<\tau_H$ and $x\geq 0$, we have the following strong approximation
\begin{equation}
\label{eqn:approx}
\mathbb{P}\left\{\sup_{t\in[0,M]}n\left|F_n(t)-F(t)-n^{-1/2}(1-F(t))W\circ L(t)\right|>x+K_1\log n\right\}\leq K_2\mathrm{e}^{-K_3x},
\end{equation}
where $K_1,\,K_2,\,K_3$ are positive constants, $W$ is a Brownian motion and
\begin{equation}
\label{eqn:L}
L(t)=\int_0^t \frac{\lambda(u)}{1-H(u)}\,\mathrm{d}u.
\end{equation}
\section{Smoothed Grenander-type estimator of a monotone hazard}
\label{sec:asymptotics}
Next, we introduce the smoothed Grenander-type estimator $\tilde{\lambda}_n^{SG}$ of an increasing hazard.
Kernel smoothing is a rather simple and broadly used method.
Let $k$ be a standard kernel, i.e.,
\begin{equation}
\label{def:kernel}
\text{$k$ is a symmetric probability density with support $[-1,1]$.}
\end{equation}
We will consider the scaled version
\[
k_b(u)=\frac{1}{b}k\left(\frac{u}{b}\right)
\]
of the kernel function $k$, where $b=b_n$ is a bandwidth that depends on the sample size, such that
\begin{equation}
\label{eqn:band}
0<b_n\to 0\quad\text{and}\quad nb_n\to\infty, \quad\text{as }n\to\infty.
\end{equation}
From now on we will use the notation $b$ instead of $b_n$.
For a fixed $x\in[0,\tau_H]$, the smoothed Grenander-type estimator $\tilde{\lambda}_n^{SG}$ is defined by
\begin{equation}
\label{eqn:SG}
\tilde{\lambda}_n^{SG}(x)=\int_{(x-b)\vee 0}^{(x+b)\wedge\tau_H} k_b(x-u)\,\tilde{\lambda}_n(u)\,\mathrm{d}u=\int k_b(x-u)\,\mathrm{d}\tilde{\Lambda}_n(u).
\end{equation}
Figure~\ref{fig:gren} shows the Grenander-type estimator together with the kernel smoothed version for the same sample as in Figure~\ref{fig:NA}.
We used the triweight kernel function
\[
k(u)=\frac{35}{32}(1-u^2)^3\mathds{1}_{\{|u|\leq 1\}}
\]
and the bandwidth $b=c_{opt}\,n^{-1/5}$, where $c_{opt}$ is the asymptotically MSE-optimal constant (see \eqref{eqn:c_opt}) calculated in the point $x_0=0.5$.
\begin{figure}[t]
\includegraphics[width=.7\textwidth]{SG-hazard}
\caption{The Grenander-type estimator (dashed line) of the hazard rate (dotted line) and the kernel smoothed version (solid line).}
\label{fig:gren}
\end{figure}%
The following result is rather standard when dealing with kernel smoothed isotonic estimators
(see for instance, \citet{Nane},Chapter 5).
For completeness, we provide a rigorous proof.
\begin{theo}
\label{theo:cons}
Let $k$ be a kernel function satisfying~\eqref{def:kernel} and let
$\tilde{\lambda}^{SG}$ be the smoothed Grenander-type estimator defined in~\eqref{eqn:SG}. Suppose that the hazard function $\lambda$ is nondecreasing and continuous. Then for each $0<\epsilon<\tau_H$, it holds
\[
\sup_{x\in[\epsilon,\tau_H-\epsilon]}|\tilde{\lambda}_n^{SG}(x)-\lambda(x)|\to 0
\]
with probability one.
\end{theo}
\begin{proof}
First, note that for a fixed $x\in(0,\tau_H)$ and sufficiently large $n$, we have $0<x-b< x+b<\tau_H.$
We start by writing
\[
\tilde{\lambda}_n^{SG}(x)-\lambda(x)
=
\tilde{\lambda}_n^{SG}(x)-\tilde{\lambda}_n^s(x)
+
\tilde{\lambda}_n^{s}(x)-\lambda(x),
\]
where $\tilde{\lambda}^s_n(x)=\int k_b(x-u)\,\lambda(u)\,\mathrm{d}u$.
Then, a change of variable yields
\[
\tilde{\lambda}^{s}_n(x)-\lambda(x)
=
\int_{-1}^1 k(y)
\left\{\lambda(x-by)-\lambda(x)\right\}\,\mathrm{d}y.
\]
Using the continuity of $\lambda$ and applying the dominated convergence theorem, we obtain that,
for each $x\in (0,\tau_H)$,
\begin{equation}
\label{eqn:partcons}
\tilde{\lambda}^s_n(x)\to\lambda(x),
\quad
\text{as }n\to\infty.
\end{equation}
On the other hand,
\[
\tilde{\lambda}^{SG}_n(x)-\tilde{\lambda}^s_n(x)
=
\int_{-1}^1 k(y)
\left\{
\tilde{\lambda}_n(x-by)-\lambda(x-by)
\right\}\,\mathrm{d}y.
\]
Choose $\epsilon>0$.
Then by continuity of $\lambda$, we can find $\delta>0$,
such that $0<x-\delta<x+\delta<\tau_H$ and $|\lambda(x+\delta)-\lambda(x-\delta)|<\epsilon$.
Then, there exists $N$ such that, for all $n\geq N$ and for all $y\in[-1,1]$, it holds $|by|<\delta$.
Hence, by the monotonicity of the hazard rate, we get
\[
\tilde{\lambda}_n(x-\delta)-\lambda(x+\delta)\leq \tilde{\lambda}_n(x-by)-\lambda(x-by)\leq \tilde{\lambda}_n(x+\delta)-\lambda(x-\delta).
\]
It follows from~\eqref{eqn:consG} and~\eqref{def:kernel} that
\[
-\epsilon
\leq
\liminf_{n\to\infty}
\tilde{\lambda}^{SG}_n(x)-\tilde{\lambda}^s_n(x)
\leq
\limsup_{n\to\infty}
\tilde{\lambda}^{SG}_n(x)-\tilde{\lambda}^s_n(x)
\leq
\epsilon,
\]
with probability one.
Since $\epsilon>0$ is arbitrary, together with~\eqref{eqn:partcons}, this proves the strong pointwise consistency
at each fixed $x\in(0,\tau_H)$.
Finally, uniform consistency in $[\epsilon,\tau_H-\epsilon]$ follows from the fact that we have a sequence of monotone functions converging pointwise to a continuous, monotone function on a compact interval.
\end{proof}
It is worth noticing that, if one is willing to assume that $\lambda$ is twice differentiable with uniformly bounded first and second derivatives,
and that $k$ is differentiable with a bounded derivative, then we get a more precise result on the order of convergence
\[
\sup_{x\in[\epsilon,\tau_H-\epsilon]}|\tilde{\lambda}_n^{SG}(x)-\lambda(x)|=O_P(b^{-1}n^{-1/2}+b^2).
\]
Such extra assumptions are considered in Theorem 5.2 in \citet{Nane} for the Cox model and the right censoring model is just a particular case with regression parameters $\beta=0$. Furthermore, in a similar way, it can be proved that also the estimator for the derivative of the hazard is uniformly consistent in $[\epsilon,\tau_H-\epsilon]$, provided that $\lambda$ is continuously differentiable and the kernel is differentiable with bounded derivative.
The pointwise asymptotic normality of the smoothed Grenander estimator is stated in the next theorem.
Its proof is inspired by the approach used in \citet{GJ13}.
The key step consists in using a Kiefer-Wolfowitz type of result for the Nelson-Aalen estimator,
which has recently been obtained by~\citet{DL14}.
\begin{theo}
\label{theo:distr}
Let $\lambda$ be a nondecreasing and twice continuously differentiable hazard such that $\lambda$ and~$\lambda'$ are strictly positive.
Let $k$ satisfy~\eqref{def:kernel} and suppose that it is
differentiable with a uniformly bounded derivative.
If $bn^{1/5}\to c\in(0,\infty)$, then for each $x\in(0,\tau_h)$,
\[
n^{2/5}
\big(
\tilde{\lambda}_n^{SG}(x)-\lambda(x)
\big)
\xrightarrow{d}
N(\mu,\sigma^2),
\]
where
\begin{equation}
\label{eqn:mu-sigma}
\mu=\frac{1}{2}c^2\lambda''(x)\int u^2k(u)\,\mathrm{d}u
\quad\text{ and }\quad
\sigma^2=
\frac{\lambda(x)}{c\,(1-H(x))}\int k^2(u)\,\mathrm{d}u.
\end{equation}
For a fixed $x\in(0,\tau_h)$, the asymptotically MSE-optimal bandwidth $b$ for $\tilde{\lambda}^{SG}$ is given by
$c_{opt}(x)n^{-1/5}$, where
\begin{equation}
\label{eqn:c_opt}
c_{opt}(x)=\left\{\lambda(x)\int k^2(u)\,\mathrm{d}u \right\}^{1/5}\left\{(1-H(x))\lambda''(x)^2 \left(\int u^2\,k(u)\,\mathrm{d}u\right)^2\right\}^{-1/5}.
\end{equation}
\end{theo}
\begin{proof}
Once again we fix $x\in(0,\tau_H)$. Then, for sufficiently large $n$, we have $0<x-b< x+b\leq M<\tau_H$.
We write
\begin{equation}
\label{eqn:distr1}
\begin{split}
\tilde{\lambda}^{SG}_n(x)
&
=\int k_b(x-u)\,\mathrm{d}\Lambda(u)+\int k_b(x-u)\,\mathrm{d}(\Lambda_n-\Lambda)(u)\\
&\quad+\int k_b(x-u)\,\mathrm{d}(\tilde{\Lambda}_n-\Lambda_n)(u).
\end{split}
\end{equation}
The first (deterministic) term on the right hand side of~\eqref{eqn:distr1} gives us
the asymptotic bias by using a change of variables, a Taylor expansion, and the properties of the kernel:
\[
\begin{split}
n^{2/5}
\left\{
\int_{x-b}^{x+b} k_b(x-u)\,\lambda(u)\,\mathrm{d}u-\lambda(x)
\right\}
&=
n^{2/5}
\int_{-1}^1 k(y)\,
\left\{
\lambda(x-by)-\lambda(x)
\right\}\,\mathrm{d}y\\
&=
n^{2/5}
\int_{-1}^1 k(y)
\left\{
-\lambda'(x)by+\frac12\lambda''(\xi_n)b^2y^2
\right\}\,\mathrm{d}y,\\
&\to
\frac{1}{2}\,c^2\,\lambda''(x)
\int_{-1}^1 y^2\,k(y)\,\mathrm{d}y.
\end{split}
\]
where $|\xi_n-x|<b|y|\leq b\to0$.
On the other hand, the last term on the right hand side of~\eqref{eqn:distr1} converges to $0$ in probability.
Indeed, integration by parts formula enables us to write
\[
\begin{split}
n^{2/5}\int_{x-b}^{x+b}k_b(x-u)\,\mathrm{d}(\tilde{\Lambda}_n-\Lambda_n)(u)
&=
n^{2/5}
\int_{x-b}^{x+b}
\left\{
\tilde{\Lambda}_n(u)-\Lambda_n(u)
\right\}
\frac{1}{b^2}\,k'\left(\frac{x-u}{b}\right)\,\mathrm{d}u\\
&=
\frac{n^{2/5}}{b}\int_{-1}^1
\left\{
\tilde{\Lambda}_n(x-by)-\Lambda_n(x-by)
\right\}
k'(y)\,\mathrm{d}y,
\end{split}
\]
and then we use $\sup_{t\in[0,\tau_H]}|\tilde{\Lambda}_n(t)-\Lambda_n(t)|=O_p(n^{-2/3}(\log n)^{2/3})$ (see \citet{DL14}, Corollary 3.4) together with the boundedness of $k'$.
What remains is to prove that
\[
n^{2/5}\,\int k_b(x-u)\,\mathrm{d}(\Lambda_n-\Lambda)(u)\xrightarrow{d} N(0,\sigma^2),
\]
where $\sigma^2$ is defined in~\eqref{eqn:mu-sigma}. Let us start by writing
\[
n^{2/5}\int_{x-b}^{x+b} k_b(x-u)\,\mathrm{d}(\Lambda_n-\Lambda)(u)=\frac{1}{\sqrt{bn^{1/5}}}\int_{-1}^1 k(y)\,\mathrm{d}\hat{W}_n(y),
\]
where, for each $y\in[-1,1]$, we define
\begin{equation}
\label{eqn:distr2}
\begin{split}
\hat{W}_n(y)
&=
\sqrt{\frac{n}{b}}
\left\{
\Lambda_n(x-by)-\Lambda_n(x)-\Lambda(x-by)+\Lambda(x)
\right\}\\
&=
\sqrt{\frac{n}{b}}
\int
\frac{\delta}{\Phi_n(u)}\left\{\mathds{1}_{[0,x-by]}(u)-\mathds{1}_{[0,x]}(u)\right\}\,\mathrm{d}\mathbb{P}_n(u,\delta)\\
&\qquad-
\sqrt{\frac{n}{b}}
\int\frac{\delta}{\Phi(u)}\left\{\mathds{1}_{[0,x-by]}(u)-\mathds{1}_{[0,x]}(u)\right\}\,\mathrm{d}\mathbb{P}(u,\delta)\\
&=
b^{-1/2}
\int
\frac{\delta}{\Phi(u)}\left\{\mathds{1}_{[0,x-by]}(u)-\mathds{1}_{[0,x]}(u)\right\}\,\mathrm{d}\sqrt{n}(\mathbb{P}_n-\mathbb{P})(u,\delta)\\
&\qquad+
\sqrt{\frac{n}{b}}
\int \delta\left\{\mathds{1}_{[0,x-by]}(u)-\mathds{1}_{[0,x]}(u)\right\}
\left\{
\frac{1}{\Phi_n(u)}-\frac{1}{\Phi(u)}
\right\}\,\mathrm{d}\mathbb{P}_n(u,\delta).
\end{split}
\end{equation}
Here we took advantage of the representations in~\eqref{eqn:cum-haz}.
The last term in the right hand side of~\eqref{eqn:distr2} is bounded in absolute value by
\[
\sqrt{\frac{n}{b}}
\frac{1}{\Phi_n(M)\,\Phi(M)}
\int \delta
\left|\mathds{1}_{[0,x-by]}(u)-\mathds{1}_{[0,x]}(u)\right|
\left|\Phi(u)-\Phi_n(u)\right|\,
\mathrm{d}\mathbb{P}_n(u,\delta)=o_P(1).
\]
Indeed, by using~\eqref{eqn:phi}, we obtain that $1/\Phi_n(M)=O_P(1)$ and then it suffices to prove that
\[
b^{-1/2}
\int
\delta\left|\mathds{1}_{[0,x-by]}(u)-\mathds{1}_{[0,x]}(u)\right|
\,\mathrm{d}\mathbb{P}_n(u,\delta)=o_P(1).
\]
To do so, we write the left hand side as
\begin{equation}
\label{eqn:decompose}
\begin{split}
&
b^{-1/2}\,\int \delta\left|\mathds{1}_{[0,x-by]}(u)-\mathds{1}_{[0,x]}(u)\right|\,\mathrm{d}\mathbb{P}(u,\delta)\\
&\quad+
b^{-1/2}\,\int \delta\left|\mathds{1}_{[0,x-by]}(u)-\mathds{1}_{[0,x]}(u)\right|\,\mathrm{d}(\mathbb{P}_n-\mathbb{P})(u,\delta)\\
&=
b^{-1/2}\,\big|H^{uc}(x-by)-H^{uc}(x)\big|+O_p(b^{-1/2}n^{-1/2})
=
o_P(1).
\end{split}
\end{equation}
Here we have used that $H^{uc}$ is continuously differentiable
and that the class of indicators of intervals forms a VC-class, and is therefore Donsker
(see \citet{VW96}, Theorem 2.6.7 and Theorem 2.5.2).
The last step consists in showing that
\[
b^{-1/2}
\int
\frac{\delta}{\Phi(u)}\left\{\mathds{1}_{[0,x-by]}(u)-\mathds{1}_{[0,x]}(u)\right\}\,\mathrm{d}\sqrt{n}(\mathbb{P}_n-\mathbb{P})(u,\delta)
\xrightarrow{d}
\sqrt{\frac{\lambda(x)}{1-H(x)}}\,W(y),
\]
where $W$ is a two sided Brownian motion. This follows from Theorem~2.11.23 in \citet{VW96}.
Indeed, we can consider the functions
\[
f_{n,y}(u,\delta)
=
b^{-1/2}\frac{\delta}{\Phi(u)}
\left\{
\mathds{1}_{[0,x-by]}(u)-\mathds{1}_{[0,x]}(u)
\right\},
\qquad y\in[-1,1].
\]
with envelopes $F_n(u,\delta)=b^{-1/2}\Phi(M)^{-1}\delta\mathds{1}_{[x-b,x+b]}(u)$.
It can be easily checked that
\[
\Vert F_n\Vert_{L_2(\mathbb{P})}^2=\frac{1}{b\,\Phi^2(M)}\int_{x-b}^{x+b} f(u)(1-G(u))\,\mathrm{d}u=O(1),
\]
and that for all $\eta>0$,
\[
\frac{1}{b\Phi^2(M)}
\int_{\left\{u:b^{-1/2}\Phi(M)^{-1}\mathds{1}_{[x-b,x+b]}(u)>\eta\sqrt{n}\right\}}
f(u)(1-G(u))\,\mathrm{d}u
\to0.
\]
Moreover, for every sequence $\delta_n\downarrow0$, we have
\[
\frac{1}{b\Phi^2(M)}
\sup_{|s-t|<\delta_n}
\int_{x-b(s\vee t)}^{x-b(s\wedge t)}
f(u)(1-G(u))\,\mathrm{d}u
\to0.
\]
Since $f_{n,y}$ are sums and products of bounded monotone functions, the bracketing number is bounded
(see \citet{VW96}, Theorem 2.7.5)
\[
\log N_{[\,]}\left(\epsilon\Vert F_n\Vert_{L_2(\mathbb{P})},\mathcal{F}_n, \Vert \cdot\Vert_{L_2(\mathbb{P})}\right)
\lesssim
\log\left(1/\epsilon\,\Vert F_n\Vert_{L_2(\mathbb{P})}\right).
\]
Hence, since $\Vert F_n\Vert_{L_2(\mathbb{P})}$ is bounded we obtain
\[
\int_0^{\delta_n}\sqrt{\log N_{[\,]}\left(\epsilon\Vert F_n\Vert_{L_2(\mathbb{P})},\mathcal{F}_n, \Vert \cdot\Vert_{L_2(\mathbb{P})}\right)}\,\mathrm{d}\epsilon
\lesssim
\delta_n + \int_0^{\delta_n}\sqrt{\log(1/\epsilon)}\,\mathrm{d}\epsilon
\to0.
\]
Finally, as in~\eqref{eqn:decompose}, for any $s\in[-1,1]$,
\[
\mathbb{P} f_{n,s}
=
b^{-1/2}
\left\{
H^{uc}(x-bs)-H(x)
\right\}
\to0.
\]
Furthermore, for $0\leq s\leq t$,
\[
\mathbb{P} f_{n,s}f_{n,t}
=
b^{-1}
\int_{x-bs}^x\frac{f(u)(1-G(u))}{\Phi^2(u)}\,\mathrm{d}u
=
b^{-1}
\int_{x-bs}^x\frac{\lambda(u)}{1-H(u)}\,\mathrm{d}u
\to
\frac{\lambda(x)}{1-H(x)}s.
\]
Similarly, for $t\leq s\leq 0$, $\mathbb{P} f_{n,s}f_{n,t}\to -s\lambda(x)/(1-H(x))$,
whereas $\mathbb{P} f_{n,s}f_{n,t}=0$, for $st<0$.
It follows that
\begin{equation}
\label{def:limiting cov}
\mathbb{P} f_{n,s}f_{n,t}-\mathbb{P} f_{n,s}\mathbb{P} f_{n,t}
\to
\begin{cases}
\dfrac{\lambda(x)}{1-H(x)}(|s|\wedge|t|) & \text{, if }st\geq 0;\\
0& \text{, if }st<0.
\end{cases}
\end{equation}
Consequently, according to~Theorem~2.11.23 in \citet{VW96}, the sequence of stochastic processes
$\sqrt{n}(\mathbb{P}_n-\mathbb{P})f_{n,y}$ converges in distribution to a tight Gaussian process $\mathbb{G}$ with mean zero
and covariance given on the right hand side of~\eqref{def:limiting cov}.
Note that this is the covariance function of $\sqrt{\lambda(x)/[1-H(x)]}W$, where $W$ is a two sided Brownian motion.
We conclude that
\[
\begin{split}
&
n^{2/5}\int_{x-b}^{x+b} k_b(x-u)\,\mathrm{d}(\Lambda_n-\Lambda)(u)\\
&\quad=
\frac{1}{\sqrt{bn^{1/5}}} \int_{-1}^1 k(y)\,\mathrm{d}\hat{W}_n(y)\\
&\quad\stackrel{d}{\to}
\left(\frac{\lambda(x)}{c(1-H(x))}\right)^{1/2}
\int_{-1}^1 k(y)\,\mathrm{d}W(y)
\stackrel{d}{=}
N\left(0,\frac{\lambda(x)}{c\,(1-H(x))}\int_{-1}^1 k^2(y)\,\mathrm{d}y\right).
\end{split}
\]
This proves the first part of the theorem.
The optimal $c$ is then obtained by minimizing
\[
\mathrm{AMSE}(\tilde{\lambda}^{SG} ,c)=\frac{1}{4}\,c^4\,\lambda''(x)^2\, \left(\int u^2\,k(u)\,\mathrm{d}u\right)^2+\frac{\lambda(x)}{c\,(1-H(x))}\,\int k^2(u)\,\mathrm{d}u
\]
with respect to $c$.
\end{proof}
This result is in line with Theorem 11.8 in~\citet{GJ14} on the asymptotic distribution of the SMLE under the same model,
which highlights the fact that even after applying a smoothing technique the MLE and the Grenander-type estimator remain asymptotically equivalent.
Standard kernel density estimators lead to inconsistency problems at the boundary.
In order to prevent those, different methods of boundary corrections can be used.
Here we consider boundary kernels and one possibility is to construct linear combinations of $k(u)$ and $uk(u)$ with coefficients depending on the value near the boundary (see~\citet{DGL13}; \citet{ZK98}).
To be precise, we define the smoothed Grenander-type estimator $\hat{\lambda}_n^{SG}$ by
\begin{equation}
\label{eqn:sg}
\hat{\lambda}_n^{SG}(x)=\int_{(x-b)\vee 0}^{(x+b)\wedge\tau_H} k_b^{(x)}(x-u)\,\tilde{\lambda}_n(u)\,\mathrm{d}u=\int_{(x-b)\vee 0}^{(x+b)\wedge\tau_H} k_b^{(x)}(x-u)\,\mathrm{d}\tilde{\Lambda}_n(u),\quad x\in[0,\tau_H],
\end{equation}
with $k_b^{(x)}(u)$ denoting the rescaled kernel $b^{-1}k^{(x)}(u/b)$ and
\begin{equation}
\label{eqn:bound-kernel}
k_{b}^{(x)}(u)=
\begin{cases}
\phi(\frac{x}{b})k(u)+\psi(\frac{x}{b})uk(u) & x\in[0,b],\\
k(u) & x\in(b,\tau_H-b)\\
\phi(\frac{\tau_H-x}{b})k(u)-\psi(\frac{\tau_H-x}{b})uk(u) & x\in[\tau_H-b,\tau_H],
\end{cases}
\end{equation}
where $k(u)$ is a standard kernel satisfying~\eqref{def:kernel}.
For $s\in[-1,1]$, the coefficients $\phi(s)$, $\psi(s)$ are determined by
\begin{equation}
\label{eqn:coefficient}
\begin{aligned}
&\phi(s)\int_{-1}^s k(u)\,\mathrm{d}u+\psi(s)\int_{-1}^s uk(u)\,\mathrm{d}u=1,\\
&\phi(s)\int_{-1}^s uk(u)\,\mathrm{d}u+\psi(s)\int_{-1}^s u^2k(u)\,\mathrm{d}u=0.
\end{aligned}
\end{equation}
Note that $\phi$ and $\psi$ are not only well defined, but they are also continuously differentiable if the kernel $k$ is assumed to be continuous (see \citet{DGL13}).
Furthermore, it can be easily seen that, for each $x\in[0,b]$, equations~\eqref{eqn:coefficient} lead to
\begin{equation}
\label{eqn:integral}
\int_{-1}^{x/b}k^{(x)}(u)\,\mathrm{d}u=1\qquad\text{and}\qquad\int_{-1}^{x/b}uk^{(x)}(u)\,\mathrm{d}u=0.
\end{equation}
In this case, we obtain a stronger uniform consistency result which is stated in the next theorem.
\begin{theo}
\label{theo:boundcorr}
Let $\hat{\lambda}_n^{SG}$ be defined by~\eqref{eqn:sg} and suppose that $\lambda$ is nondecreasing and uniformly continuous.
Assume that $k$ satisfies~\eqref{def:kernel} and is differentiable with a uniformly bounded derivative
and that $bn^{\alpha}\to c\in(0,\infty)$.
If $0<\alpha<1/2$, then for any $0<M<\tau_H$,
\[
\sup_{x\in[0,M]}
\left|\hat{\lambda}_n^{SG}(x)-\lambda(x)\right|
\to
0
\]
in probability.
\end{theo}
\begin{proof}
For $x\in[0,M]$, write
\[
\hat{\lambda}_n^{SG}(x)-\lambda(x)
=
\left(\hat{\lambda}_n^{SG}(x)-\lambda_n^s(x)\right)
+
\Big(\lambda_n^{s}(x)-\lambda(x)\Big),
\]
where
\[
\lambda^s_n(x)=\int_{(x-b)\vee 0}^{(x+b)\wedge\tau_H} k_b^{(x)}(x-u)\,\lambda(u)\,\mathrm{d}u.
\]
We have to distinguish between two cases.
First, we consider the case $x\in[0,b]$.
By means of~\eqref{eqn:integral} and the fact that $\lambda$ is uniformly continuous, a change of variable yields
\begin{equation}
\label{eqn:cons2}
\sup_{x\in[0,b]}|\lambda^{s}_n(x)-\lambda(x)|
\leq
\sup_{x\in[0,b]}
\int_{-1}^{x/b} k^{(x)}(y)
\big|\lambda(x-by)-\lambda(x)\big|\,\mathrm{d}y
\to0.
\end{equation}
On the other hand, integration by parts and a change of variable give
\begin{equation}
\label{eqn:partcons2}
\begin{split}
\hat{\lambda}^{SG}_n(x)-\lambda^s_n(x)
&=
\int_{0}^{x+b} k_b^{(x)}(x-u)\,\mathrm{d}\big(\tilde{\Lambda}_n-\Lambda\big)(u)\\
&=
-\int_0^{x+b}\frac{\partial}{\partial u}k_b^{(x)}(x-u)
\big(\tilde{\Lambda}_n(u)-\Lambda(u)\big)\,\mathrm{d}u\\
&=
\frac{1}{b}
\int_{-1}^{x/b}
\frac{\partial}{\partial y}k_b^{(x)}(y)
\big(\tilde{\Lambda}_n(x-by)-\Lambda(x-by)\big)\,\mathrm{d}y.
\end{split}
\end{equation}
Consequently, since for $n$ sufficiently large $x+b\leq M$, we obtain
\[
\sup_{x\in[0,b]}
\left|\hat{\lambda}^{SG}_n(x)-\lambda^s_n(x)\right|
\lesssim
\frac{1}{b}
\sup_{u\in[0,M]}
\left|
\tilde{\Lambda}_n(u)-\Lambda(u)
\right|
=O_p(n^{-1/2+\alpha}),
\]
because of~\eqref{eqn:chbound} and the boundedness of the coefficients $\phi$, $\psi$ and of $k(u)$ and $k'(u)$.
Together with~\eqref{eqn:cons2} and since $0<\alpha<1/2$, this proves that,
\[
\sup_{x\in[0,b]}
\left|\hat{\lambda}^{SG}_n(x)-\lambda(x)\right|
=o_P(1).
\]
When $x\in(b,M]$, for sufficiently large $n$, we have $0<x-b<x+b<\tau_H$, so that
by a change of variable and uniform continuity of $\lambda$,
it follows that
\begin{equation}
\label{eqn:cons1}
\sup_{x\in(b,M]}
\left|\lambda^{s}_n(x)-\lambda(x)\right|
\leq
\int_{-1}^{1} k(y)
|\lambda(x-by)-\lambda(x)|\,\mathrm{d}y
\to0.
\end{equation}
Furthermore,
\[
\hat{\lambda}^{SG}_n(x)
=
\int_{x-b}^{x+b} k_b(x-u)\,\tilde{\lambda}_n(u)\,\mathrm{d}u,
\]
so that, arguing as in~\eqref{eqn:partcons2}, we find that
\[
\sup_{x\in(b,M]}
\left|\hat{\lambda}^{SG}_n(x)-\lambda^s_n(x)\right|
=O_p(n^{-1/2+\alpha}),
\]
which, together with~\eqref{eqn:cons1}, proves that
\[
\sup_{x\in(b,M]}\left|\hat{\lambda}^{SG}_n(x)-\lambda(x)\right|
=o_P(1).
\]
This proves the theorem.
\end{proof}
The previous result illustrates that even if we use boundary kernels, we can not avoid inconsistency problems at the end point of the support.
Although a bit surprising, this is to be expected because we can only control the distance between the Nelson-Aalen estimator and the cumulative hazard on intervals that stay away from the right boundary (see~\eqref{eqn:chbound}).
\begin{figure}
\includegraphics[width=.6\textwidth]{SGC-hazard}
\caption{The kernel smoothed versions (standard-dashed line and boundary corrected-solid line) of the hazard rate (dotted line).}
\label{fig:bound}
\end{figure}
Figure~\ref{fig:bound} illustrates that boundary corrections improve the performance of the smooth estimator constructed with the standard kernel.
\section{Smoothed Grenander-type estimator of a monotone density}\label{sec:dens}
This section is devoted to the smoothed Grenander-type estimator $\tilde{f}_n^{SG}$ of an increasing density $f$. Let $k$ be a kernel function satisfying~\eqref{def:kernel}. For a fixed $x\in[0,\tau_H]$, $\tilde{f}_n^{SG}$ is defined by
\begin{equation}
\label{eqn:SGd}
\tilde{f}_n^{SG}(x)=\int_{(x-b)\vee 0}^{(x+b)\wedge\tau_H} k_b(x-u)\,\tilde{f}_n(u)\,\mathrm{d}u=\int k_b(x-u)\,\mathrm{d}\tilde{F}_n(u).
\end{equation}
We also consider the boundary corrected version $\hat{f}_n^{SG}$ of the smoothed Grenander-type estimator, defined by
\begin{equation}
\label{eqn:sgd}
\hat{f}_n^{SG}(x)=\int_{(x-b)\vee 0}^{(x+b)\wedge\tau_H} k_b^{(x)}(x-u)\,\tilde{f}_n(u)\,\mathrm{d}u=\int_{(x-b)\vee 0}^{(x+b)\wedge\tau_H} k_b^{(x)}(x-u)\,\mathrm{d}\tilde{F}_n(u),\quad x\in[0,\tau_H],
\end{equation}
with $k_b^{(x)}(u)$ as in~\eqref{eqn:bound-kernel}.
The following results can be proved in exactly the same way as Theorem~\ref{theo:cons} and Theorem~\ref{theo:boundcorr}.
\begin{theo}
\label{theo:cons-dens}
Let $k$ be a kernel function satisfying~\eqref{def:kernel} and let
$\tilde{f}^{SG}$ be the smoothed Grenander-type estimator defined in~\eqref{eqn:SGd}. Suppose that the density function $f$ is nondecreasing and continuous. Then for each $0<\epsilon<\tau_H$, it holds
\[
\sup_{x\in[\epsilon,\tau_H-\epsilon]}|\tilde{f}_n^{SG}(x)-f(x)|\to 0
\]
with probability one.
\end{theo}
\begin{theo}
\label{theo:boundcorr-dens}
Let $\hat{f}_n^{SG}$ be defined by~\eqref{eqn:sg} and suppose that $f$ is nondecreasing and uniformly continuous.
Assume that $k$ satisfies~\eqref{def:kernel} and is differentiable with uniformly bounded derivatives
and that $b\,n^{\alpha}\to c\in(0,\infty)$.
If $0<\alpha<1/2$, then for any $0<M<\tau_H$,
\[
\sup_{x\in[0,M]}
\left|\hat{f}_n^{SG}(x)-f(x)\right|
\to
0
\]
in probability.
\end{theo}
\begin{figure}
\includegraphics[width=.6\textwidth]{SGC-density}
\caption{The kernel smoothed versions (standard-dashed line and boundary corrected-solid line) of the density function (dotted line).}
\label{fig:bound-dens}
\end{figure}
Figure~\ref{fig:bound-dens} shows the smooth isotonic estimators of a decreasing density for a sample of size $n=500$.
We choose an exponential distribution with mean $1$ truncated to $[0,5]$ for the event times and
an exponential distribution with mean $2$ truncated to $[0,5]$ for the censoring times.
The bandwidth used is $b=c_{opt}\,n^{-1/5}$, where $c_{opt}=5.14$ is the asymptotically MSE-optimal constant (see~\eqref{eqn:c_opt-dens})
corresponding to $x_0=2.5$.
In order to derive the asymptotic normality of the smoothed Grenander-type estimator $\tilde{f}_n^{SG}$ we first provide a Kiefer-Wolfowitz type of result for the Kaplan-Meier estimator.
\begin{lemma}
\label{le:KW}
Let $0<M<\tau_H$. Let $f$ be a nondecreasing and continuously differentiable density such that $f$ and $f'$ are strictly positive.
Then we have
\[
\sup_{x\in[0,M]}|\tilde{F}_n(x)-F_n(x)|=O_P\left(\frac{\log n}{n}\right)^{2/3},
\]
where $F_n$ is the Kaplan-Meier estimator and $\tilde{F}_n$ is the greatest convex minorant of $F_n$.
\end{lemma}
\begin{proof}
We consider $f$ on the interval $[0,M]$ and apply Theorem~2.2 in~\citet{DL14}.
The density $f$ satisfies condition (A1) of this theorem with $[a,b]=[0,M]$.
Condition (2) of Theorem~2.2 in~\citet{DL14} is provided by
the strong approximation~\eqref{eqn:approx}, with $L$ defined in~\eqref{eqn:L}, $\gamma_n=O(n^{-1}\log n)^{2/3}$, and
\[
B(t)=\left(1-F(L^{-1}(t))\right)W(t),
\quad
t\in[L(0),L(M)]
\]
where $W$ is a Brownian motion.
It remains to show that $B$ satisfies conditions (A2)-(A3) of Theorem~2.2 in~\citet{DL14} with $\tau=1$.
In order to check these conditions for the process $B$, let
$x\in[L(0),L(M)]=[0,L(M)]$, $u\in(0,1]$ and $v>0$.
Then
\begin{equation}
\label{eq:bound A2}
\begin{split}
&
\mathbb{P}\left(
\sup_{|x-y|\leq u} |B(x)-B(y)|>v\right)\\
&\leq
\mathbb{P}\left(\sup_{|x-y|\leq u,\, y\in[0,L(M)]} |B(x)-B(y)|>v\right)\\
&\leq
2\mathbb{P}\left( \sup_{|x-y|\leq u} |W(x)-W(y)|>\frac{v}{3}\right)\\
&\qquad+
\mathbb{P}\left( \sup_{|x-y|\leq u,\, y\in[0,L(M)]} |F(L^{-1}(x))-F(L^{-1}(y)) |\,|W(x)|>\frac{v}{3}\right).
\end{split}
\end{equation}
Note that from the proof of Corollary 3.1 in~\citet{DL14} it follows that $W$ satisfies condition (A2) in~\citet{DL14}.
This means that there exist $K_1,K_2>0$, such that the first probability on the right hand side of~\eqref{eq:bound A2} is bounded by
$K_1\exp(-K_2v^2u^{-1})$.
Furthermore, since $u\in(0,1]$ and
\[
|F(L^{-1}(x))-F(L^{-1}(y)) |\leq \frac{\sup_{u\in[0,M]}f(u)}{\inf_{u\in[0,M]}L'(u)}\,|x-y|\leq \frac{\sup_{u\in[0,M]}f(u)}{\inf_{u\in[0,M]}\lambda(u)}\,|x-y|,
\]
the second probability on the right hand side of~\eqref{eq:bound A2} is bounded by
\[
\mathbb{P}\left(|W(x)|>\frac{v}{K_3\sqrt{u}}\right)
\leq
\mathbb{P}\left(\sup_{t\in[0,L(M)]}|W(t)|>\frac{v}{K_3\sqrt{u}}\right),
\]
for some $K_3>0$.
Hence, by the maximal inequality for Brownian motion, we conclude that
there exist $K_1',K_2'>0$ such that
\[
\mathbb{P}\left(
\sup_{|x-y|\leq u} |B(x)-B(y)|>v\right)
\leq
K_1'\exp\left(-K_2'v^2u^{-1}\right)
\]
which proves condition (A2) in~\citet{DL14}.
Let us now consider (A3).
For all $x\in[0,L(M)]$, $u\in(0,1]$, and $v>0$, we obtain
\begin{equation}
\label{eq:bound A3}
\begin{split}
&
\mathbb{P}\left( \sup_{u\leq z\leq x}
\left\{B(x-z)-B(x)-vz^2\right\}>0\right)\\
&\leq
\mathbb{P}\left( \sup_{u\leq z\leq x}
\left\{W(x-z)-W(x)-\frac{vz^2}{3}\right\}>0\right)\\
&\quad+
\mathbb{P}\left( \sup_{u\leq z\leq x} \left\{F(L^{-1}(x-z))[W(x)-W(x-z)]-\frac{vz^2}{3}\right\}>0\right)\\
&\qquad+
\mathbb{P}\left(\sup_{u\leq z\leq x}
\left\{
\left(F(L^{-1}(x))-F(L^{-1}(x-z))\right)W(x)-\frac{vz^2}{3}
\right\}>0\right).
\end{split}
\end{equation}
Again, from the proof of Corollary 3.1 in~\citet{DL14} it follows that $W$ satisfies condition (A3) in~\citet{DL14},
which means that there exist $K_1,K_2>0$, such that the first probability on the right hand side of~\eqref{eq:bound A3} is bounded by
$K_1\exp(-K_2v^2u^{3})$.
We establish the same bound for the remaining tow probabilities.
By the time reversal of the Brownian motion, the process $W'(z)=W(x)-W(x-z)$ is also a Brownian motion on the interval $[u,x]$.
Then, using the change of variable $u/z=t$ and the fact that $\widetilde{W}(t)=tu^{-1/2}W'(u/t)$, for $t>0$, is again a Brownian motion,
the second probability on the right hand side of~\eqref{eq:bound A3} is bounded by
\begin{equation}
\label{eq:bound A3 term2}
\mathbb{P}\left(\sup_{t\in(0,1]} \frac{t}{\sqrt{u}}\left|W'\left(\frac{u}{t}\right)\right|>\frac{vu^{3/2}}{3t}\right)
=
\mathbb{P}
\left(
\sup_{t\in[0,1]}
|\widetilde{W}(t)|>\frac{vu^{3/2}}{3}
\right).
\end{equation}
Finally,
\[
\sup_{u\leq z\leq x} \left|\frac{F(L^{-1}(x))-F(L^{-1}(x-z))}{z}\right|\leq \frac{\sup_{u\in[0,M]}f(u)}{\inf_{u\in[0,M]}\lambda(u)},
\]
so that the third probability on the right hand side of~\eqref{eq:bound A3} is bounded by
\begin{equation}
\label{eq:bound A3 term3}
\begin{split}
&
\mathbb{P}\left(\sup_{u\leq z\leq x} \left|\frac{F(L^{-1}(x))-F(L^{-1}(x-z))}{z}\right|\,|W(x)|>\frac{vu}{3}\right)\\
&\qquad\leq
\mathbb{P}\left(|W(x)|>\frac{vu^{3/2}}{K_3}\right)
\leq
\mathbb{P}\left(\sup_{t\in[0,L(M)]}|W(t)|>\frac{vu^{3/2}}{K_3}\right),
\end{split}
\end{equation}
for some $K_3>0$.
By applying the maximal inequality for Brownian motion to the right hand sides of~\eqref{eq:bound A3 term2} and~\eqref{eq:bound A3 term3},
we conclude that there exist $K_1',K_2'>0$, such that
\[
\mathbb{P}\left( \sup_{u\leq z\leq x} \{B(x-z)-B(x)-vz^2\}>0\right)
\leq K'_1\exp(-K'_2v^2u^{3}),
\]
which proves condition (A3) in~\citet{DL14}.
\end{proof}
\begin{theo}
\label{theo:distr-dens}
Let $f$ be a nondecreasing and twice continuously differentiable density such that $f$ and $f'$ are strictly positive. Let $k$ satisfy~\eqref{def:kernel} and suppose that it is differentiable with a uniformly bounded derivative. If $bn^{1/5}\to c\in(0,\infty)$, then for each $x\in(0,\tau_h)$, it holds
\[
n^{2/5}\,\big(\,\tilde{f}_n^{SG}(x)-f(x)\,\big)\xrightarrow{d}N(\mu,\sigma^2),
\]
where
\begin{equation}
\label{eqn:mu-sigma-dens}
\mu=\frac{1}{2}\,c^2\,f''(x)\, \int u^2\,k(u)\,\mathrm{d}u\quad\text{ and }\quad \sigma^2= \frac{f(x)}{c\,(1-G(x))}\,\int k^2(u)\,\mathrm{d}u.
\end{equation}
For a fixed $x\in(0,\tau_h)$, the asymptotically MSE-optimal bandwidth $b$ for $\tilde{\lambda}^{SG}$ is given by
$c_{opt}(x)n^{-1/5}$, where
\begin{equation}
\label{eqn:c_opt-dens}
c_{opt}=\left\{f(x)\int k^2(u)\,\mathrm{d}u \right\}^{1/5}\left\{(1-G(x))f''(x)^2 \left(\int u^2\,k(u)\,\mathrm{d}u\right)^2\right\}^{-1/5}.
\end{equation}
\end{theo}
\begin{proof}
Fix $x\in(0,\tau_H)$. Then, for sufficiently large $n$, we have $0<x-b< x+b\leq M<\tau_H.$ Following the proof of~\ref{theo:distr}, we write
\begin{equation}
\label{eqn:distr1-dens}
\tilde{f}^{SG}_n(x)=\int k_b(x-u)\,\mathrm{d}F(u)+\int k_b(x-u)\,\mathrm{d}(F_n-F)(u)+\int k_b(x-u)\,\mathrm{d}(\tilde{F}_n-F_n)(u).
\end{equation}
Again the first (deterministic) term on the right hand side of~\eqref{eqn:distr1-dens} gives us the asymptotic bias:
\[
n^{2/5}
\left\{
\int_{x-b}^{x+b} k_b(x-u)f(u)\,\mathrm{d}u-f(x)
\right\}
\to
\frac{1}{2}c^2f''(x)
\int_{-1}^1 y^2 k(y)\,\mathrm{d}y,
\]
and by the Kiefer-Wolfowitz type of result in Lemma~\ref{le:KW}, the last term on the right hand side of~\eqref{eqn:distr1-dens} converges to $0$ in probability. What remains is to prove that
\[
n^{2/5}\,\int k_b(x-u)\,\mathrm{d}(F_n-F)(u)\xrightarrow{d} N(0,\sigma^2),
\]
where $\sigma^2$ is defined in~\eqref{eqn:mu-sigma-dens}. We write
\[
n^{2/5}\int_{x-b}^{x+b} k_b(x-u)\,\mathrm{d}(F_n-F)(u)=\frac{1}{\sqrt{b\,n^{1/5}}} \int_{-1}^1 k(y)\,\mathrm{d}\hat{W}_n(y),
\]
where, for each $y\in[-1,1]$, we define
\begin{equation}
\label{eqn:distr2-dens}
\begin{split}
\hat{W}_n(y)
&=
\sqrt{\frac{n}{b}}
\Big\{F_n(x-by)-F_n(x)-F(x-by)+F(x)\Big\}\\
&=
\sqrt{\frac{n}{b}}
\Big\{F_n(x-by)-F(x-by)-n^{-1/2}(1-F(x-by))W\circ L(x-by)\Big\}\\
&\qquad-
\sqrt{\frac{n}{b}}
\left\{F_n(x)-F(x)-n^{-1/2}(1-F(x))W\circ L(x)\right\}\\
&\qquad+
\frac{1}{\sqrt{b}}
\Big(1-F(x)\Big)\Big\{W\circ L(x-by)-W\circ L(x)\Big\}\\
&\qquad+
\frac{1}{\sqrt{b}}\Big(F(x)-F(x-by)\Big)W\circ L(x-by).
\end{split}
\end{equation}
Using the strong approximation~\eqref{eqn:approx}, we obtain
\[
\begin{split}
&
\mathbb{P}\left(\sup_{u\in[0,M]}\sqrt{\frac{n}{b}}\left|F_n(u)-F(u)-n^{-1/2}(1-F(u))W\circ L(u)\right|>\epsilon\right)\\
&\qquad\leq
K_1\exp\left\{-K_2(\epsilon\sqrt{nb}-K_3\log n)\right\}\to 0,
\end{split}
\]
and it then follows that the first two terms on the right hand side of~\eqref{eqn:distr2-dens} converge to $0$ in probability uniformly in $y$.
For the last term, we get
\[
\begin{split}
&
\mathbb{P}\left(\sup_{y\in[-1,1]}
\left|\frac{1}{\sqrt{b}}\Big(F(x)-F(x-by)\Big)W\circ L(x-by)\right|>\epsilon\right)\\
&\qquad\leq
\mathbb{P}\left(\sqrt{b}\sup_{u\in[0,M]}f(u)\sup_{u\in [0,\Vert L\Vert_\infty]}|W(u)|>\epsilon\right)
\leq K_1\exp\left(-\frac{K_2\epsilon^2}{b}\right)\to 0.
\end{split}
\]
For the third term on the right hand side of~\eqref{eqn:distr2-dens}, note
that $y\mapsto b^{-1/2}(W\circ L(x-by)-W\circ L(x))$, for $y\in[-1,1]$, has the same distribution as the process
\begin{equation}
\label{eqn:process}
y\mapsto \widetilde{W}\left(\frac{L(x)-L(x-by)}{b}\right),
\quad
\text{for }y\in[-1,1],
\end{equation}
where $\widetilde{W}$ is a two-sided Brownian motion.
By uniform continuity of the two-sided Brownian motion on compact intervals,
the sequence of stochastic processes in~\eqref{eqn:process} converges to the process
$\{\widetilde{W}\left(L'(x)y\right):y\in[-1,1]\}$:
\[
\sup_{y\in[-1,1]}\left|\widetilde{W}\left(\frac{L(x)-L(x-by)}{b}\right)-\widetilde{W}\left(L'(x)y\right)\right|\xrightarrow{\mathbb{P}} 0.
\]
As a result
\[
\begin{split}
\frac{1}{\sqrt{bn^{1/5}}}
\int_{-1}^1 k(y)\,\mathrm{d}\hat{W}_n(y)
&\xrightarrow{d}
\sqrt{\frac{f(x)}{c(1-G(x))}}
\int_{-1}^1 k(y)\,\mathrm{d}\widetilde{W}(y)\\
&\sim
N\left(0,\frac{f(x)}{c(1-G(x))}\int_{-1}^1 k^2(y)\,\mathrm{d}y \right).
\end{split}
\]
The optimal $c$ is then obtained by minimizing
\[
\mathrm{AMSE}(\tilde{f}^{SG} ,c)=\frac{1}{4}c^4f''(x)^2
\left(\int u^2\,k(u)\,\mathrm{d}u\right)^2
+
\frac{f(x)}{c(1-G(x))}
\int k^2(u)\,\mathrm{d}u,
\]
with respect to $c$.
\end{proof}
\section{Pointwise confidence intervals}
\label{sec:conf-int}
In this section we construct pointwise confidence intervals for the hazard rate and the density based on the asymptotic distributions derived in
Theorem~\ref{theo:distr} and Theorem~\ref{theo:distr-dens} and compare them to confidence intervals constructed using Grenander-type estimators without smoothing.
According to Theorem~2.1 and Theorem~2.2 in~\citet{huang-wellner1995}, for a fixed $x_0\in(0,\tau_H)$,
\[
n^{1/3}\left|\frac{1-G(x_0)}{4f(x_0)f'(x_0)}\right|^{1/3}\left(\tilde{f}_n(x_0)-f(x_0)\right)
\xrightarrow{d}
\mathbb{Z},
\]
and
\[
n^{1/3}\left|\frac{1-H(x_0)}{4\lambda(x_0)\lambda'(x_0)}\right|^{1/3}\left(\tilde{\lambda}_n(x_0)-\lambda(x_0)\right)
\xrightarrow{d}
\mathbb{Z},
\]
where $W$ is a two-sided Brownian motion starting from zero
and $\mathbb{Z}=\mathop{\text{argmin}}_{t\in\mathbb{R}}\left\{W(t)+t^2\right\}$.
This yields $100(1-\alpha)\%$-confidence intervals for $f(x_0)$ and $\lambda(x_0)$ of the following form
\[
C^1_{n,\alpha}
=
\tilde{f}_n(x_0)\pm n^{-1/3}\hat{c}_{n,1}(x_0)q(\mathbb{Z},1-\alpha/2),
\]
and
\[
C^2_{n,\alpha}
=
\tilde{\lambda}_n(x_0)\pm n^{-1/3}\hat{c}_{n,2}(x_0)q(\mathbb{Z},1-\alpha/2),
\]
where $q(\mathbb{Z},1-\alpha/2)$ is the $(1-\alpha/2)$ quantile of the distribution $\mathbb{Z}$ and
\[
\hat{c}_{n,1}(x_0)
=
\left|\frac{4\tilde{f}_n(x_0)\tilde{f}_n'(x_0)}{1-G_n(x_0)}\right|^{1/3},
\qquad
\hat{c}_{n,2}(x_0)
=
\left|\frac{4\tilde{\lambda}_n(x_0)\tilde{\lambda}'_n(x_0)}{1-H_n(x_0)}\right|^{1/3}.
\]
Here, $H_n$ is the empirical distribution function of $T$ and in order to avoid the denominator taking the value zero,
instead of the natural estimator of $G$, we consider a slightly different version as in~\citet{MP87}:
\begin{equation}
\label{eqn:G_n}
G_n(t)
=
\begin{cases}
0 & \text{ if } 0\leq t< T_{(1)},\\
\displaystyle{1-\prod_{i=1}^{k-1}\left(\frac{n-i+1}{n-i+2} \right)^{1-\Delta_i}} & \text{ if } T_{(k-1)}\leq t< T_{(k)},\quad k=2,\dots,n,\\
\displaystyle{1-\prod_{i=1}^{n}\left(\frac{n-i+1}{n-i+2} \right)^{1-\Delta_i}} & \text{ if } t\geq T_{(n)}.
\end{cases}
\end{equation}
Furthermore, as an estimate for $\tilde{f}_n'(x_0)$ we choose $(\tilde{f}_n(\tau_m)-\tilde{f}_n(\tau_{m-1}))/(\tau_m-\tau_{m-1})$,
where $\tau_{m-1}$ and $\tau_m$ are two succeeding points of jump of $\tilde{f}_n$ such that $x_0\in(\tau_{m-1},\tau_m]$,
and~$\tilde{\lambda}_n'(x_0)$ is estimated similarly.
The quantiles of the distribution $\mathbb{Z}$ have been computed in~\citet{GW01} and we will use $q(\mathbb{Z},0.975)=0.998181$.
The pointwise confidence intervals based on the smoothed Grenander-type estimators are constructed from
Theorem~\ref{theo:distr} and Theorem~\ref{theo:distr-dens}.
We find
\[
\widetilde{C}^1_{n,\alpha}
=
\tilde{f}^{SG}_n(x_0)\pm n^{-2/5}
\left(
\hat{\sigma}_{n,1}(x_0)q_{1-\alpha/2}+\hat{\mu}_{n,1}(x_0)
\right),
\]
and
\[
\widetilde{C}^2_{n,\alpha}
=
\tilde{\lambda}^{SG}_n(x_0)\pm
n^{-2/5}(\hat{\sigma}_{n,2}(x_0)q_{1-\alpha/2}+\hat{\mu}_{n,2}(x_0)),
\]
where $q_{1-\alpha/2}$ is the $(1-\alpha/2)$ quantile of the standard normal distribution.
The estimators $\hat{\sigma}_{n,1}(x_0)$ and $\hat{\mu}_{n,1}(x_0)$ are obtained by plugging $\tilde{f}^{SG}_n$ and its second derivative for $f$ and $f''$, respectively,
and $G_n$ and $c_{opt}(x_0)$ for $G$ and $c$, respectively, in~\eqref{eqn:mu-sigma-dens}, and similarly $\hat{\sigma}_{n,2}(x_0)$ and $\hat{\mu}_{n,2}(x_0)$
are obtained from~\eqref{eqn:mu-sigma}.
Estimating the bias seems to be a hard problem because it depends on the second derivative of the function of interest.
As discussed, for example in~\citet{Hall92}, one can estimate the bias by using a bandwidth of a different order for estimating the second derivative or one
can use undersmoothing (in that case the bias is zero and we do not need to estimate the second derivative).
We tried both methods and it seems that undersmoothing performs better, which is in line with other results available in the literature (see for instance,~\citet{Hall92};~\citet{GJ15};~\citet{CHT06}).
When estimating the hazard rate,
we choose a Weibull distribution with shape parameter $3$ and scale parameter $1$ for the event times and the uniform distribution on~$(0,1.3)$ for the censoring times.
Confidence intervals are calculated at the point $x_0=0.5$ using 1000 sets of data and the bandwidth in the case of undersmoothing is $b=c_{opt}(x_0)n^{-1/4}$,
where $c_{opt}(x_0)=1.2$.
In the case of bias estimation we use $b=c_{opt}(x_0)n^{-5/17}$ to estimate the hazard and $b_1=c_{opt}(x_0) n^{-1/17}$ to estimate its second derivative (as suggested in~\citet{Hall92}).
Table~\ref{tab:1} shows the performance, for various sample sizes, of the confidence intervals based on the asymptotic distribution (AD) of the Grenander-type estimator and of the smoothed Grenander estimator (for both undersmoothing and bias estimation).
\begin{table}
\begin{tabular}{ccccccc}
\toprule
& \multicolumn{2}{c}{Grenander} & \multicolumn{2}{c}{SG-undersmoothing} & \multicolumn{2}{c}{SG-bias estimation} \\
n & AL & CP & AL & CP & AL & CP \\
100 & 0.930 & 0.840 & 0.648 & 0.912 & 0.689 & 0.955 \\
500 & 0.560 & 0.848 & 0.366 & 0.948 & 0.383 & 0.975 \\
1000 & 0.447 & 0.847 & 0.283 & 0.954 & 0.295 & 0.977 \\
5000 & 0.255 & 0.841 & 0.155 & 0.953 & 0.157 & 0.978 \\
\bottomrule
\end{tabular}
\caption{The average length (AL) and the coverage probabilities (CP) for $95\%$ pointwise confidence intervals of the hazard rate at the point $x_0=0.5$ based on the asymptotic distribution. }
\label{tab:1}
\end{table}
The poor performance of the Grenander-type estimator seems to be related to the crude estimate of the derivative of the hazard with the slope of the correspondent segment.
On the other hand, it is obvious that smoothing leads to significantly better results in terms of both average length and coverage probabilities.
As expected, when using undersmoothing, as the sample size increases we get shorter confidence intervals and coverage probabilities that are closer to the nominal level of $95\%$.
By estimating the bias, we obtain coverage probabilities that are higher than $95\%$,
because the confidence intervals are bigger compared to the average length when using undersmoothing.
\begin{figure}
\centering
\subfloat[][]
{\includegraphics[width=.48\textwidth]{conf-int-haz-x}} \quad
\subfloat[][]
{\includegraphics[width=.48\textwidth]{conf-int-haz}}
\caption{Left panel: Actual coverage of confidence intervals with nominal coverage $95\%$ for the hazard rate based on the asymptotic distribution. Dashed line-nominal level; dotdashed line-Grenander-type; solid line-SG undersmoothing; dotted line-SG bias estimation. Right panel: $95\%$ confidence intervals based on the asymptotic distribution for the hazard rate using undersmoothing.}
\label{fig:subfig4}
\end{figure}
Another way to compare the performance of the different methods is to take
a fixed sample size $n=500$ and different points of the support of the hazard function.
Figure~\ref{fig:subfig4} shows that confidence intervals based on undersmoothing behave well also at the boundaries in terms of coverage probabilities,
but the length increases as we move to the left end point of the support.
In order to maintain good visibility of the performance of the smooth estimators, we left out the poor performance of the Grenander estimator at point $x=0.1$.
Finally, we consider estimation of the density.
We simulate the event times and the censoring times from exponential distributions with means $1$ and $2$, truncated to the interval $[0,5]$.
Confidence intervals are calculated at the point $x_0=1$ using 1000 sets of data and the bandwidth in the case of undersmoothing is $b=c_{opt}(2.5)n^{-1/4}$,
where $c_{opt}(2.5)=5.14$.
When estimating the bias we use $b=c_{opt}(2.5)n^{-5/17}$ to estimate the hazard and $b_1=c_{opt}(2.5) n^{-1/17}$ to estimate its second derivative (as suggested in~\citet{Hall92}).
Table~\ref{tab:2} shows the performance, for various sample sizes, of the confidence intervals based on the asymptotic distribution (AD)
of the Grenander-type estimator and of the smoothed Grenander estimator (for both undersmoothing and bias estimation).
\begin{table}
\begin{tabular}{ccccccc}
\toprule
& \multicolumn{2}{c}{Grenander} & \multicolumn{2}{c}{SG-undersmoothing} & \multicolumn{2}{c}{SG-bias estimation} \\
n & AL & CP & AL & CP & AL & CP \\
100 & 0.364 & 0.867 & 0.23 & 0.999 & 0.24 & 0.997 \\
500 & 0.210 & 0.870 & 0.126 & 0.980 & 0.134 & 0.992 \\
1000 & 0.160 & 0.865 & 0.096 & 0.974 & 0.105 & 0.985 \\
5000 & 0.090 & 0.845 & 0.052 & 0.962 & 0.056 & 0.976 \\
10000 & 0.070 & 0.855 & 0.040 & 0.959 & 0.041 & 0.970 \\
\bottomrule
\end{tabular}
\caption{The average length (AL) and the coverage probabilities (CP) for $95\%$ pointwise confidence intervals of the density function at the point $x_0=1$ based on the asymptotic distribution. }
\label{tab:2}
\end{table}
Again, confidence intervals based on the Grenander-type estimator have a poor coverage.
On the other hand, by considering the smoothed version,
we usually obtain very high coverage probabilities which tend to get closer to the nominal level as the sample size increases.
Again, undersmoothing behaves slightly better.
The performance of these three methods for a fixed sample size $n=500$ and different points of the support of the density is illustrated in Figure~\ref{fig:subfig5}.
\begin{figure}
\centering
\subfloat[][]
{\includegraphics[width=.48\textwidth]{conf-int-dens-x}} \quad
\subfloat[][]
{\includegraphics[width=.48\textwidth]{conf-int-dens}}
\caption{Left panel: Actual coverage of confidence intervals with nominal coverage $95\%$ for the density function based on the asymptotic distribution. Dashed line-nominal level; dotdashed line-Grenander-type; solid line-SG undersmoothing; dotted line-SG bias estimation. Right panel: $95\%$ confidence intervals based on the asymptotic distribution for the density function using undersmoothing.}
\label{fig:subfig5}
\end{figure}
\section{Discussion}\label{sec:discussion}
In the present paper, we have considered kernel smoothed Grenander estimators for a monotone hazard and a monotone density under right censoring.
We have established uniform strong convergence of the estimators in the interior of the support of distribution of the follow-up times
and asymptotic normality at a rate of convergence of $n^{2/5}$.
The behavior of the estimators has been illustrated in a small simulation study, where it can be seen that the smoothed versions perform better than the ordinary Grenander estimators.
The proof of asymptotic normality is more or less straightforward thanks to a Kiefer-Wolfowitz type of result provided in \citet{DL14}.
The right censoring model is a special case of the Cox regression model, where in addition one can consider various covariates.
A natural question then is whether the previous approach for proving the asymptotic normality of a smoothed Grenander-type estimator, for example, for the hazard rate can be extended to such a more general setting.
Unfortunately, no Kiefer-Wolfowitz nor an embedding into the Brownian motion is available for the Breslow estimator, being the natural naive estimator for the cumulative hazard.
Recently, \citet{GJ14} developed different approach to establish asymptotic normality of smoothed isotonic estimators,
which is mainly based on uniform $L_2$-bounds for the distance between the non-smoothed estimator and the true function.
This approach seems to have more potential for generalizing our results to the Cox model.
However, things will not go smoothly, because the presence of covariates makes it more difficult to obtain bounds on the tail probabilities
for the inverse process involved and in the Cox model one has to deal with a rather complicated martingale associated with the Breslow estimator.
This is beyond the scope of this paper, but will be the topic of future research.
\printbibliography
\end{document}
| 23,327 |
\section{Introduction}
The aim of this paper is to establish a connection between the
topology of the automorphism group of a symplectic manifold
$(M,\o)$ and the quantum product on its homology. More precisely,
we assume that $M$ is closed and connected, and consider the group $\mathrm{Ham}(M,\o)$
of Hamiltonian automorphisms with the $C^\infty$-topology.
$\mathrm{Ham}(M,\o)$ is a path-connected subgroup of the symplectic automorphism
group $\mathrm{Aut}(M,\o)$; if $H^1(M,\mathbb{R}) = 0$, it is the connected component
of the identity in $\mathrm{Aut}(M,\o)$. We introduce a homomorphism $q$
from a certain extension of the fundamental group $\pi_1(\mathrm{Ham}(M,\o))$ to the
group of invertibles in the quantum homology ring of $M$.
This invariant can be used to detect nontrivial elements
in $\pi_1(\mathrm{Ham}(M,\o))$. For example, consider
$M = S^2 \times S^2$ with the family of product structures
$\o_\lambda = \lambda(\o_{S^2} \times 1) + 1 \times {\o_{S^2}}$,
$\lambda > 0$. This example has been studied by Gromov \cite{gromov85},
McDuff \cite{mcduff87} and Abreu \cite{abreu97}. McDuff
showed that for $\lambda \neq 1$, $\pi_1(\mathrm{Ham}(M,\omega_\lambda))$ contains
an element of infinite order. This result can be recovered
by our methods. In a less direct way, the existence of $q$
imposes topological restrictions on all
elements of $\pi_1(\mathrm{Ham}(M,\o))$. An example
of this kind of reasoning can be found in section
\ref{sec:an-application}; a more important one will appear
in forthcoming work by Lalonde, McDuff and Polterovich.
To define $q$, we will use the general relationship
between loops in a topological group and bundles over
$S^2$ with this group as structure group. In our case,
a smooth map $g: S^1 \longrightarrow \mathrm{Aut}(M,\o)$ determines
a smooth fibre bundle $E_g$ over $S^2$ with a family
$\O_g = (\O_{g,z})_{z \in S^2}$ of symplectic structures
on its fibres. If $g(S^1)$ lies in $\mathrm{Ham}(M,\o)$, the
$\O_{g,z}$ are restrictions of a closed $2$-form on
$E_g$. We will call a pair $(E,\O)$ with this property a
{\em Hamiltonian fibre bundle}.
Witten \cite{witten88} proposed to define an invariant of such a bundle
$(E,\O)$ over $S^2$ with fibre $(M,\o)$ in the
following way: choose a positively oriented complex structure $j$ on $S^2$
and an almost complex structure $\hat{J}$ on $E$ (compatible
with the symplectic structure on each fibre) such that
the projection $\pi: E \longrightarrow S^2$ is $(\hat{J},j)$-linear.
For generic $\hat{J}$, the space $\sec(j,\hat{J})$ of pseudoholomorphic sections
of $\pi$ is a smooth finite-dimensional manifold, with a natural evaluation
map $\mathrm{ev}_z: \sec(j,\hat{J}) \longrightarrow E_z$ for $z \in S^2$.
After fixing a symplectic isomorphism
$i : (M,\o) \longrightarrow (E_z,\O_z)$ for some
$z$, $\mathrm{ev}_z$ should define a homology class in $M$.
This rough description does not take into account the lack of
compactness of $\sec(j,\hat{J})$. Part of this problem is due to
`bubbling': since we deal with it
using the method of Ruan and Tian in \cite{ruan-tian94},
we have to make the following
\begin{assumption}[$\mathrm{\mathbf{W^+}}$]
$(M,\o)$ satisfies one of the following
conditions:
\renewcommand{\theenumi}{(\alph{enumi})}
\renewcommand{\labelenumi}{\theenumi}
\begin{enumerate}
\item there is a $\lambda \geq 0$ such that $\o(A) = \lambda c_1(A)$
for all $A \in \pi_2(M)$;
\item $c_1 | \pi_2(M) = 0$;
\item the minimal Chern number $N \geq 0$ (defined by
$c_1(\pi_2(M)) = N\mathbb{Z}$) is at least $n-1$, where $2n = \dim M$.
\end{enumerate}
\end{assumption}
Here $c_1$ stands for $c_1(TM,\o)$. An equivalent assumption is
\begin{equation} \label{eq:wplus-equivalent}
A \in \pi_2(M), 2-n \leq c_1(A) < 0 \Longrightarrow \o(A) \leq 0.
\end{equation}
Therefore {$(W^+)$} is more restrictive than the notion of weak
monotonicity (see e.g. \cite{hofer-salamon95}) in which $2-n$ is
replaced by $3-n$. In particular, the Floer homology $HF_*(M,\o)$
\cite{hofer-salamon95} and the quantum cup-product \cite{ruan-tian94}
are well-defined for manifolds which satisfy {$(W^+)$}. Note that
all symplectic four-manifolds belong to this class. Recent work on
Gromov-Witten invariants seems to indicate that the restriction {$(W^+)$}
might be removed at the cost of introducing rational coefficients,
but this will not be pursued further here.
The other aspect of the non-compactness of $\sec(j,\hat{J})$ is
that it may have infinitely many components corresponding
to different homotopy classes of sections of $E$. By separating
these components, one obtains infinitely many invariants
of $(E,\O)$. These can be arranged
into a single element of the quantum homology group $QH_*\mo$, which is
the homology of $M$ with coefficients in the Novikov ring
$\Lambda$. This element is normalized by the choice of a
section of $E$; it depends
on this section only up to a certain equivalence relation, which
we call {\em $\Gamma$-equivalence}. If $S$ is a
$\Gamma$-equivalence class of sections, we denote the invariant obtained
in this way by $Q(E,\O,S) \in QH_*\mo$.
The bundles $(E_g,\O_g)$ for a Hamiltonian loop $g$ do not
come with a naturally preferred $\Gamma$-equivalence class of sections.
Therefore we introduce an additional piece of data:
$g$ acts on the free loop space $\Lambda M = C^\infty(S^1,M)$ by
\begin{equation} \label{eq:action}
(g \cdot x)(t) = g_t(x(t)).
\end{equation}
Let $\mathcal{L}M$ be the connected component of $\Lambda M$ containing
the constant loops. There is an abelian covering $p: \widetilde{\loops} \longrightarrow
\mathcal{L}M$ such that $\Gamma$-equivalence classes of sections of $E_g$
correspond naturally to lifts of the action of $g$ to $\widetilde{\loops}$.
Let $S_{\tilde{g}}$ be the equivalence class corresponding to a
lift $\tilde{g}: \widetilde{\loops} \longrightarrow \widetilde{\loops}$. We define
\[
q(g,\tilde{g}) = Q(E_g,\O_g,S_{\tilde{g}}).
\]
Let $G$ be the group of smooth loops $g: S^1 \longrightarrow \mathrm{Ham}(M,\o)$
such that $g(0) = \mathit{Id}$, and $\widetilde{G}$ the group of pairs $(g,\tilde{g})$.
We will use the $C^\infty$-topology on $G$, and a topology on $\widetilde{G}$
such that the homomorphism $\widetilde{G} \longrightarrow G$ which
`forgets' $\tilde{g}$ is continuous with discrete kernel.
The covering $\widetilde{\loops}$ was originally introduced by Hofer and
Salamon in their definition of $HF_*(M,\o)$ \cite{hofer-salamon95}.
They set up Floer homology as a formal analogue of Novikov homology
for this covering. Now the $\widetilde{G}$-action on $\widetilde{\loops}$ commutes
with the covering transformations.
Pursuing the analogy, one would expect an induced action of
$\widetilde{G}$ on Floer homology. If we assume that {$(W^+)$} holds,
this picture is correct: there are induced maps
\[
HF_*(g,\tilde{g}): HF_*\mo \longrightarrow HF_*\mo
\]
for $(g,\tilde{g}) \in \tilde{G}$. These maps are closely related
to $q(g,\tilde{g})$. The relationship involves
the `pair-of-pants' product
\[
\qp_{\mathrm{PP}}: HF_*\mo \times HF_*\mo \longrightarrow HF_*\mo
\]
and the canonical isomorphism $\Psi^+:
QH_*\mo \longrightarrow HF_*\mo$ of Piunikhin, Salamon
and Schwarz \cite{piunikhin-salamon-schwarz94},\cite{schwarz97}.
\begin{bigtheorem} \label{bigth:comparison}
For any $(g,\tilde{g}) \in \widetilde{G}$ and $b \in HF_*\mo$,
\[
HF_*(g,\tilde{g})(b) = \Psi^+(q(g,\tilde{g})) \qp_{\mathrm{PP}} b.
\] \end{bigtheorem}
This formula provides an alternative approach to
$q$ and $Q$, and we will use it to derive
several properties of these invariants. Let $\ast$ be the
product on $QH_*\mo$ obtained from the quantum cup-product
by Poincar{\'e} duality. We will call $\ast$ the
quantum intersection product. The quantum homology ring
$(QH_*\mo,\ast)$ is a ring with unit and commutative in the
usual graded sense.
\begin{bigcorollary} \label{bigth:invertible}
For any Hamiltonian fibre
bundle $(E,\O)$ over $S^2$ with fibre $(M,\o)$
and any $\Gamma$-equivalence class $S$ of sections
of $E$, $Q(E,\O,S)$ is an invertible element of
$(QH_*\mo,\ast)$. \end{bigcorollary}
By definition, $Q(E,\O,S)$ is homogeneous and even-dimensional,
that is, it lies in $QH_{2i}(M,\o)$ for some $i \in \mathbb{Z}$. We will
denote the group (with respect to $\ast$) of
homogeneous even-dimensional invertible elements of
$QH_*\mo$ by $QH_*\mo^{\times}$.
\begin{bigcorollary} \label{bigth:homomorphism}
$q(g,\tilde{g})$ depends only
on $[g,\tilde{g}] \in \pi_0(\widetilde{G})$, and
\[
q: \pi_0(\tilde{G}) \longrightarrow QH_*\mo^{\times}
\]
is a group homomorphism. \end{bigcorollary}
Let $\Gamma$ be the group of covering transformations of
$\widetilde{\loops}$. Since the kernel of the homomorphism
$\widetilde{G} \longrightarrow G$ consists of the pairs
$(\mathit{Id},\gamma)$ with $\gamma \in \Gamma$, there is an exact
sequence
\begin{equation} \label{eq:pizero}
\cdots \longrightarrow
\Gamma \longrightarrow \pi_0(\tilde{G}) \longrightarrow
\pi_0(G) \longrightarrow 1.
\end{equation}
On the other hand, for every $\gamma \in \Gamma$ there is an element
$[M] \otimes \gen{\gamma} \in QH_*(M,\o)$ which is easily seen to be invertible.
The map $\tau: \Gamma \longrightarrow QH_*\mo^{\times}$ defined in this way is an
injective homomorphism.
\begin{bigproposition} \label{bigth:trivial}
For $\gamma \in \Gamma$, $q(\mathit{Id},\gamma) = \tau(\gamma)$.
\end{bigproposition}
It follows that there is a unique homomorphism $\bar{q}$
such that the diagram
\begin{equation} \label{diag:basic}
\xymatrix{
\Gamma \ar[r] \ar@{=}[d] & \pi_0(\widetilde{G}) \ar[r] \ar[d]^{q}
& \pi_0(G) \ar[d]^{\bar{q}}\\
\Gamma \ar[r]^-{\tau} & QH_*\mo^{\times} \ar[r] & QH_*\mo^{\times}/\tau(\Gamma)
}
\end{equation}
commutes. $\bar{q}$ is more interesting for applications
to symplectic geometry than $q$ itself, since
$\pi_0(G) \iso \pi_1(\mathrm{Ham}(M,\o))$.
In the body of the paper, we proceed in a different order than
that described above. The next section contains the
definition of $\widetilde{G}$ and of Hamiltonian fibre bundles.
Section \ref{sec:floer-homology} is a review (without proofs)
of Floer homology. Our definition is a variant of
that in \cite{hofer-salamon95}, the difference being that we
allow time-dependent almost complex structures. This makes
it necessary to replace weak monotonicity by the condition {$(W^+)$}.
The following three sections contain the definition and basic properties
of the maps $HF_*(g,\tilde{g})$. An argument similar to the use of
`homotopies of homotopies' in the definition of Floer homology
shows that $HF_*(g,\tilde{g})$ depends only on $[g,\tilde{g}] \in
\pi_0(\widetilde{G})$. Moreover, we prove that $HF_*(g,\tilde{g})$ is an automorphism
of $HF_*(M,\o)$ as a module over itself with the pair-of-pants product.
This second property is considerably simpler: after a slight modification
of the definition of $\qp_{\mathrm{PP}}$, the corresponding equation holds
on the level on chain complexes. $Q(E,\O,S)$ is defined in
section \ref{sec:holomorphic-sections}, and section
\ref{sec:gluing} describes the gluing argument which
establishes the connection between this invariant and the
maps $HF_*(g,\tilde{g})$. After that, we prove the results stated
above. The grading of $q(g,\tilde{g})$ is determined by a
simple invariant $I(g,\tilde{g})$ derived from the
first Chern class. This relationship is exploited in section
\ref{sec:an-application} to obtain some vanishing results for this
invariant. The final section contains two explicit computations of
$q(g,\tilde{g})$, one of which is the case of $S^2 \times S^2$
mentioned at the beginning.
\subsubsection*{Acknowledgments}
Discussions with S. Ag\-ni\-ho\-tri,
S. Do\-nald\-son and V. Mu\-noz were helpful in preparing
this paper. I have profited from the referee's comments
on an earlier version. I am particularly indebted to
L. Polterovich for explaining to me the point of view
taken in section \ref{sec:the-action} and for his encouragement.
Part of this paper was written during a stay at the Universit{\'e}
de Paris-Sud (Orsay) and the Ecole Polytechnique.
\renewcommand{\theenumi}{(\roman{enumi})}
\renewcommand{\labelenumi}{\theenumi}
\section{\label{sec:elementary} Hamiltonian loops and fibre bundles}
Throughout this paper, $(M,\o)$ is a closed connected symplectic
manifold of dimension $2n$. We will write $c_1$ for $c_1(TM,\o)$.
As in the Introduction, $\Lambda M = C^\infty(S^1,M)$ denotes the
free loop space of $M$, $\mathcal{L}M \subset \Lambda M$ the
subspace of contractible loops, $\mathrm{Ham}(M,\o)$ the group of Hamiltonian
automorphisms and $G$ the group of smooth
based loops in $\mathrm{Ham}(M,\o)$. We use the $C^\infty$-topology
on $\mathrm{Ham}(M,\o)$ and $G$.
\begin{lemma} \label{th:smooth-loops} The canonical homomorphism
$\pi_0(G) \longrightarrow \pi_1(\mathrm{Ham}(M,\o))$
is an isomorphism. \end{lemma}
This follows from the fact that any continuous loop in
$\mathrm{Ham}(M,\o)$ can be approximated by a smooth loop. Note that it
is unknown whether $\mathrm{Ham}(M,\o)$ with the $C^\infty$-topology is
locally contractible for all $(M,\o)$ (see the discussion on p. 321 of
\cite{mcduff-salamon}). However, there is a neighbourhood
$U \subset \mathrm{Ham}(M,\o)$ of the identity such that each path component
of $U$ is contractible. This weaker property is sufficient to
prove Lemma \ref{th:smooth-loops}.
$G$ acts on $\Lambda M$ by \eqref{eq:action}.
\begin{lemma} \label{th:contractible-component}
$g(\mathcal{L}M) = \mathcal{L}M$ for every $g \in G$. \end{lemma}
The proof uses an idea which is also the
starting point for the definition of induced maps on Floer homology.
Let $\H = C^\infty(S^1 \times M, \mathbb{R})$
be the space of periodic Hamiltonians\footnote{We
identify $S^1 = \mathbb{R}/\mathbb{Z}$ throughout.}. The perturbed action
one-form $\alpha_H$ on $\Lambda M$ associated to $H \in \H$ is
\[
\alpha_H(x)\xi = \int_{S^1} \omega(\dot{x}(t) - X_H(t,x(t)),\xi(t)) dt
\]
where $X_H$ is the time-dependent Hamiltonian vector field
of $H$. The zero set of $\alpha_H$ consists of the $1$-periodic solutions of
\begin{equation} \label{eq:hamilton}
\dot{x}(t) = X_H(t,x(t)).
\end{equation}
We say that a Hamiltonian $K_g \in \H$ generates $g \in G$ if
\begin{equation} \label{eq:generates}
\frac{\partial g_t}{\partial t}(y) = X_{K_g}(t,g_t(y)).
\end{equation}
\begin{lemma} \label{th:hg} Let $K_g$ be a Hamiltonian
which generates $g \in G$. For every $H \in \H$, define $H^g \in \H$ by
\[
H^g(t,y) = H(t,g_t(y)) - K_g(t,g_t(y)).
\]
Then $g^*\alpha_H = \alpha_{H^g}$. \end{lemma}
The proof is straightforward.
\proof[Proof of Lemma \ref{th:contractible-component}]
Assume that $g(\mathcal{L}M)$ is a connected component of $\Lambda M$
distinct from $\mathcal{L}M$. If $H$ is small, $\alpha_H$ has no
zeros in $g(\mathcal{L}M)$ and by Lemma \ref{th:hg}, $\alpha_{H^g}$
has no zeros in $\mathcal{L}M$. This contradicts the Arnol'd
conjecture (now a theorem)
which guarantees the existence of at least one contractible
$1$-periodic solution of \eqref{eq:hamilton} for every
Hamiltonian. \qed
Since we will only use this for manifolds satisfying
{$(W^+)$}, we do not really need the Arnol'd conjecture in full
generality, only the versions established in \cite{hofer-salamon95} and
\cite{ono94}.
The space $\widetilde{\loops}$ introduced in \cite{hofer-salamon95} is
defined as follows: consider pairs
$(v,x) \in C^\infty(D^2,M) \times \mathcal{L}M$ such that
$x = v|\partial D^2$. $\widetilde{\loops}$ is the set of equivalence classes
of such pairs with respect to the following relation:
$(v_0,x_0) \sim (v_1,x_1)$ if $x_0 = x_1$ and
$\o(v_0 \# \overline{v_1}) = 0$, $c_1(v_0 \# \overline{v_1}) = 0$.
Here $v_0 \# \overline{v_1}: S^2 \longrightarrow M$ is the map
obtained by gluing together $v_0,v_1$ along the boundaries.
$p: \widetilde{\loops} \longrightarrow \mathcal{L}M$, $p(v,x) = x$, is a covering
projection for the obvious choice of topology on $\widetilde{\loops}$.
Define $\Gamma = \pi_2(M)/\pi_2(M)_0$, where $\pi_2(M)_0$ is the
subgroup of classes $a$ such that $\o(a) = 0$, $c_1(a) = 0$.
$\Gamma$ can also be defined as a quotient of
$\mathrm{im}(\pi_2(M) \longrightarrow H_2(M;\mathbb{Z}))$.
Therefore the choice of base point for $\pi_2(M)$ is irrelevant,
and any $A: S^2 \longrightarrow M$ determines a class $[A] \in \Gamma$.
Clearly $\o(A)$ and $c_1(A)$ depend only on $[A]$. By an abuse
of notation, we will write $\o(\gamma)$ and $c_1(\gamma)$ for
$\gamma \in \Gamma$. $\Gamma$ is the covering group of $p$.
It acts on $\widetilde{\loops}$ by `gluing in spheres':
$[A] \cdot [v,x] = [A \# v,x]$
(see \cite{hofer-salamon95} for a precise description).
\begin{lemma} \label{th:lift-action}
The action of any $g \in G$ on $\mathcal{L}M$ can
be lifted to a homeomorphism of $\widetilde{\loops}$. \end{lemma}
\proof Since $\widetilde{\loops}$ is a connected covering, it
is sufficient to show that the action of $g$ on $\mathcal{L}M$
preserves the set of smooth maps $S^1 \longrightarrow \mathcal{L}M$ which can
be lifted to $\widetilde{\loops}$. Such a map is given by a
$B \in C^\infty(S^1 \times S^1, M)$ with $\o(B) = 0$, $c_1(B) = 0$.
Its image under $g$ is given by $B'(s,t) = g_t(B(s,t))$. Now
$\o(B') = \o(B)$ because $(B')^*\o = B^*\o + d\theta$, where
$\theta(s,t) = K_g(t,g_t(B(s,t))) dt$ for a Hamiltonian $K_g$
as in \eqref{eq:generates}. Similarly, $c_1(B') = c_1(B)$ because
there is an isomorphism $D: B^*TM \longrightarrow (B')^*TM$ of
symplectic vector bundles, given by $D(s,t) = Dg_t(B(s,t))$. \qed
\begin{definition} \label{def:group} $\widetilde{G} \subset
G \times \mathrm{Homeo}(\widetilde{\loops})$ is the subgroup of pairs
$(g,\tilde{g})$ such that $\tilde{g}$ is a lift of the $g$-action
on $\mathcal{L}M$. \end{definition}
We give $\widetilde{G}$ the topology induced from the $C^\infty$-topology
on $G$ and the topology of pointwise convergence on
$\mathrm{Homeo}(\widetilde{\loops})$. This makes $\widetilde{G}$ into a topological
group, essentially because a lift $\tilde{g}$ of a given $g$ is
determined by the image of a single point. The projection
$\widetilde{G} \longrightarrow G$ is onto by Lemma \ref{th:lift-action},
and since its kernel consists of the pairs $(\mathit{Id},\gamma)$ with
$\gamma \in \Gamma$, there is an exact sequence
\[
1 \longrightarrow \Gamma \longrightarrow \widetilde{G}
\longrightarrow G \longrightarrow 1
\]
of topological groups, with $\Gamma$ discrete.
A point $c = [v,x] \in \widetilde{\loops}$ determines a preferred homotopy
class of trivializations of the symplectic vector bundle
$x^*(TM,\o)$. This homotopy class consists of the maps
$\tau_c: x^*TM \longrightarrow S^1 \times (\mathbb{R}^{2n},\o_0)$ which
can be extended over $v^*TM$, and it is independent of the choice
of the representative $(v,x)$ of $c$. For $(g,\tilde{g}) \in \widetilde{G}$,
\[
l(t) = \tau_{\tilde{g}(c)}(t) Dg_t(x(t)) \tau_c(t)^{-1} \qquad (t \in S^1)
\]
is a loop in $\mathrm{Sp}(2n,\mathbb{R})$. Up to homotopy, it does not depend on $c$ and
on the trivializations. We define the `Maslov index' $I(g,\tilde{g}) \in \mathbb{Z}$ by
$I(g,\tilde{g}) = \mathrm{deg}(l)$, where
$\mathrm{deg}: H_1(\mathrm{Sp}(2n,\mathbb{R})) \longrightarrow \mathbb{Z}$ is the
standard isomorphism induced by the determinant on
$U(n) \subset \mathrm{Sp}(2n,\mathbb{R})$.
\begin{lemma} \label{th:easy-maslov} $I(g,\tilde{g})$ depends only
on $[g,\tilde{g}] \in \pi_0(\widetilde{G})$. The map
$I: \pi_0(\widetilde{G}) \longrightarrow \mathbb{Z}$ is a homomorphism,
and $I(\mathit{Id},\gamma) = c_1(\gamma)$ for all $\gamma \in \Gamma$.
\end{lemma}
We omit the proof. Because of \eqref{eq:pizero}, it follows
that $I(g,\tilde{g}) \text{ mod } N$ depends only on $g$.
In this way, we recover a familiar (see e.g. \cite[p. 80]{weinstein89})
invariant
\[
\bar{I}: \pi_0(G) = \pi_1(\mathrm{Ham}(M,\o)) \longrightarrow \mathbb{Z}/N\mathbb{Z}.
\]
Now we turn to symplectic fibre bundles. A smooth fibre
bundle $\pi: E \longrightarrow B$ together with a smooth family
$\O = (\O_b)_{b \in B}$ of symplectic forms on its fibres
is called a symplectic fibration;
a symplectic fibre bundle is a symplectic fibration which is
locally trivial. Note that $\O$ defines a symplectic structure
on the vector bundle $TE^v = \ker(D\pi) \subset TE$.
Here we consider only the case $B = S^2$.
It is convenient to think of $S^2$ as $D^+ \cup_{S^1} D^-$,
where $D^+$, $D^-$ are closed discs. Fix a point $z_0 \in D^-$.
A symplectic fibre bundle $(E,\O)$ over $S^2$ with a fixed isomorphism
$i: (M,\o) \longrightarrow (E_{z_0},\O_{z_0})$ will be called
a symplectic fibre bundle {\em with fibre $(M,\o)$}; we will
frequently omit $i$ from the notation.
Let $g$ be a smooth loop in $\mathrm{Aut}(M,\o)$. The `clutching' construction
produces a symplectic fibre bundle $(E_g,\O_g)$ over
$S^2$ by gluing together the trivial fibre
bundles $D^{\pm} \times (M,\omega)$ using
\[ \begin{gathered}
\phi_g: \partial D^+ \times M \longrightarrow
\partial D^- \times M,\\
\phi_g(t,y) = (t,g_t(y)).
\end{gathered} \]
In an obvious way, $(E_g,\O_g)$ is a symplectic fibre bundle
with fibre $(M,\o)$.
\begin{convention} \label{th:orientation-convention}
We have identified $\partial D^+,\partial D^-$
with $S^1$ and used these identifications to glue $D^+$ and
$D^-$ together. We orient $D^+,D^-$ in such a way
that the map $S^1 \longrightarrow \partial
D^+$ preserves orientation while the one $S^1 \longrightarrow
\partial D^-$ reverses it; this induces an orientation
of $S^2$.
\end{convention}
We will use this construction only for loops with $g(0) = \mathit{Id}$.
By a standard argument, it provides
a bijection between elements of $\pi_1(\mathrm{Aut}(M,\o))$ and isomorphism classes
of symplectic fibre bundles over $S^2$ with fibre $(M,\o)$.
\begin{defn} A symplectic fibre bundle $(E,\O)$ over a
surface $B$ is called {\em Hamiltonian} if there
is a closed two-form $\widetilde{\O}$ on $E$ such that
$\widetilde{\O}|E_b = \O_b$ for all $b \in B$. \end{defn}
For $B = S^2$, these are precisely
the bundles corresponding to Hamiltonian loops:
\begin{proposition} \label{th:hamiltonian-bundles}
$(E_g,\O_g)$ is Hamiltonian iff $[g]$ lies in the subgroup\\
$\pi_1(\mathrm{Ham}(M,\o)) \subset \pi_1(\mathrm{Aut}(M,\o))$. \end{proposition}
\proof Let $g \in G$ be a loop generated by $K_g \in \H$.
We denote the pullback of $\o$ to $D^\pm \times M$ by $\o^{\pm}$.
Choose a $1$-form $\theta$ on $D^+ \times M$ such that
$\theta(t,y) = K_g(t,g_t(y)) dt$ for $t \in S^1$ and
$\theta | \{z\} \times M = 0$ for all $z \in D^+$. There is
a continuous $2$-form
$\widetilde{\O}$ on $E_g$ such that $\widetilde{\O} | D^+ \times M =
\o^+ + d\theta$ and $\widetilde{\O} | D^- \times M = \o^-$,
and for a suitable choice of $\theta$ it is smooth.
$\widetilde{\O}$ is closed and extends the family
$\O_g$, which proves that $(E_g,\O_g)$ is Hamiltonian.
Conversely, let $g$ be a based loop in $\mathrm{Aut}(M,\o)$ such that
$(E_g,\O_g)$ is Hamiltonian with $\widetilde{\O} \in \Omega^2(E_g)$.
We will use the exact sequence
\begin{equation} \label{eq:banyaga}
1 \longrightarrow \pi_1(\mathrm{Ham}(M,\o)) \longrightarrow
\pi_1(\mathrm{Aut}(M,\o)) \stackrel{F}{\longrightarrow} H^1(M,\mathbb{R})
\end{equation}
where $F$ is the flux homomorphism (see \cite{banyaga78} or
\cite[Corollary 10.18]{mcduff-salamon96}). Choose a point
$z_+ \in D^+$. For $\lambda \in C^\infty(S^1,M)$, consider the maps
\begin{align*}
T_{\lambda}: S^1 \times S^1 &\longrightarrow M, \quad
T_{\lambda}(r,t) = g_t(\lambda(r)),\\
T_{\lambda}^-: S^1 \times S^1 &\longrightarrow D^- \times M, \quad
T_{\lambda}^-(r,t) = (z_0,g_t(\lambda(r))) \text{ and } \\
T_{\lambda}^+: S^1 \times S^1 &\longrightarrow D^+ \times M, \quad
T_{\lambda}^+(r,t) = (z_+,\lambda(r)).
\end{align*}
By definition
$\left<F(g),[\lambda]\right> = \o(T_\lambda) = \widetilde{\O}(T_{\lambda}^-)$.
Since $T_{\lambda}^-$ and $T_{\lambda}^+$ are homotopic in $E_g$,
$\widetilde{\O}(T_{\lambda}^-) = \widetilde{\O}(T_{\lambda}^+)$; but
clearly $\widetilde{\O}(T_{\lambda}^+) = 0$. Therefore $F(g) = 0$, and
by \eqref{eq:banyaga}, $[g] \in \pi_1(\mathrm{Ham}(M,\o))$. \qed
Let $(E,\O)$ be a Hamiltonian fibre bundle over $S^2$,
with $\widetilde{\O} \in \Omega^2(E)$ as above.
We say that two continuous sections $s_0$, $s_1$ of $E$ are
{\em $\Gamma$-equivalent} if $\widetilde{\O}(s_0) = \widetilde{\O}(s_1)$ and
$c_1(TE^v,\O)(s_0) = c_1(TE^v,\O)(s_1)$. Using the exact sequence
\begin{equation} \label{eq:homotopy-sequence}
\cdots \longrightarrow \pi_2(E_{z_0})
\longrightarrow \pi_2(E) \longrightarrow
\pi_2(S^2) \longrightarrow \cdots,
\end{equation}
it is easy to see that this equivalence relation is independent of the
choice of $\widetilde{\O}$. Let $S$ be the $\Gamma$-equivalence class of
a section $s$. By an abuse of notation, we write $\widetilde{\O}(S) =
\widetilde{\O}(s)$ and $c_1(TE^v,\O)(S) = c_1(TE^v,\O)(s)$.
\begin{lemma} \label{th:equivalence-classes}
Let $(E,\O)$ be a Hamiltonian fibre bundle over $S^2$ with fibre $(M,\o)$
and $\widetilde{\O} \in \Omega^2(E)$ a closed extension of $\O$.
\begin{enumerate} \item \label{item:cont-sec}
$E$ admits a continuous section.
\item \label{item:gamma-one}
For two $\Gamma$-equivalence classes $S_0$, $S_1$ of sections of $E$,
there is a unique $\gamma \in \Gamma$ such that
\begin{equation} \label{eq:oc-difference}
\begin{aligned}
\widetilde{\O}(S_1) &= \widetilde{\O}(S_0) + \o(\gamma),\\
c_1(TE^v,\O)(S_1) &= c_1(TE^v,\O)(S_0) + c_1(\gamma).
\end{aligned}
\end{equation}
$\gamma$ is independent of the choice of $\widetilde{\O}$.
\item \label{item:gamma-two}
Conversely, given a $\Gamma$-equivalence class $S_0$ and
$\gamma \in \Gamma$, there is a unique $\Gamma$-equivalence class
$S_1$ such that \eqref{eq:oc-difference} holds.
\end{enumerate} \end{lemma}
\proof \ref{item:cont-sec} $(E,\O)$ is isomorphic to $(E_g,\O_g)$
for some $g: S^1 \longrightarrow \mathrm{Aut}(M,\o)$. Since it is Hamiltonian,
we can assume that $g \in G$. Choose a point $y \in M$; by Lemma
\ref{th:contractible-component}, there
is a $v \in C^\infty(D^-,M)$ such that $v(t) = g_t(y)$ for
$t \in S^1 = \partial D^-$. To obtain a section of $E_g$, glue together
$s^+: D^+ \longrightarrow D^+ \times M$, $s^+(z) = (z,y)$ and
$s^-: D^- \longrightarrow D^- \times M$, $s^-(z) = (z,v(z))$. \\
\ref{item:gamma-one} is an easy consequence of
\eqref{eq:homotopy-sequence}. \\
\ref{item:gamma-two} Let $i: (M,\o) \longrightarrow (E_{z_0},\O_{z_0})$ be the
preferred symplectic isomorphism. Take $s_0 \in S_0$ and
$A: S^2 \longrightarrow M$ such that $[A] = \gamma$ and
$i(A(z)) = s_0(z_0)$ for some $z \in S^2$. A section $s_1$ with the
desired properties can be obtained by gluing together $s_0$ and $i(A)$.
The uniqueness of $S_1$ is clear from the definition. \qed
In the situation of Lemma \ref{th:equivalence-classes}, we will
write $\gamma = S_1 - S_0$ and $S_1 = \gamma + S_0$.
\begin{defn} Let $(E,\O)$ be a Hamiltonian fibre bundle over $S^2$
and $S$ a $\Gamma$-equivalence class of sections of
$E$. We will call $(E,\O,S)$ a {\em normalized Hamiltonian fibre bundle}.
\end{defn}
This notion is closely related to the group $\widetilde{G}$:
for $(g,\tilde{g}) \in \widetilde{G}$, choose a point $c \in \widetilde{\loops}$
and representatives $(v,x)$ of $c$ and $(v',x')$ of $\tilde{g}(c)$.
The maps
$s^+_{\tilde{g}}: D^+ \longrightarrow D^+ \times M$,
$s^+_{\tilde{g}}(z) = (z,v(z))$ and
$s^-_{\tilde{g}}: D^- \longrightarrow D^- \times M$,
$s^-_{\tilde{g}}(z) = (z,v'(z))$
define a section $s_{\tilde{g}}$ of $E_g$. To be precise,
$s^+_{\tilde{g}}$ is obtained from $v$ by identifying $D^+$ with the
standard disc $D^2$ such that the orientation (see
\ref{th:orientation-convention}) is
preserved; in the case of $D^-$, one uses an orientation-reversing
diffeomorphism. By comparing this with the definition of $I(g,\tilde{g})$,
one obtains
\begin{equation} \label{eq:maslov-and-chern}
c_1(TE_g^v,\O_g)(s_{\tilde{g}}) = -I(g,\tilde{g}).
\end{equation}
\begin{lemma} \label{th:equivalence-class}
The $\Gamma$-equivalence class of $s_{\tilde{g}}$ is independent of the
choice of $c$ and of $v,v'$. \end{lemma}
We omit the proof. It follows that $(g,\tilde{g})$ determines a normalized
Hamiltonian fibre bundle $(E_g,\O_g,S_{\tilde{g}})$, where $S_{\tilde{g}}$
is the $\Gamma$-equivalence class of $s_{\tilde{g}}$.
This defines a one-to-one correspondence between $\pi_0(\widetilde{G})$
and isomorphism classes of normalized Hamiltonian fibre bundles
with fibre $(M,\o)$. We will only use the easier half of this correspondence:
\begin{lemma} \label{th:completeness}
Every normalized Hamiltonian fibre bundle with fibre $(M,\o)$ is isomorphic
to $(E_g,\O_g,S_{\tilde{g}})$ for some
$(g,\tilde{g}) \in \widetilde{G}$. \qed \end{lemma}
\newcommand{\smooth_{\epsilon}}{C^\infty_{\epsilon}}
\section{\label{sec:floer-homology} Floer homology}
From now on, we always assume that $(M,\o)$ satisfies {$(W^+)$}.
In \cite{hofer-salamon95}, weak monotonicity is used
to show that a generic $\o$-compatible almost complex structure
admits no pseudo-holomorphic spheres with negative Chern
number. Under the more restrictive condition
{$(W^+)$}, this non-existence result can be extended
to families of almost complex structures depending on
$\leq\!3$ parameters. To make this precise, we need to introduce
some notation. Let $\mathbf{J} = (J_b)_{b \in B}$ be a smooth family
of almost complex structures on $M$ parametrized by a manifold
$B$. $\mathbf{J}$ is called $\o$-compatible if every $J_b$ is
$\o$-compatible in the usual sense. We denote the space of
$\o$-compatible families by $\modJ{B}$.
The parametrized moduli space
$\mathcal{M}^s(\mathbf{J}) \subset B \times C^\infty(\CP{1},M)$ associated to
$\mathbf{J} \in \modJ{B}$ is the space of pairs
$(b,w)$ such that $w$ is $J_b$-holomorphic and simple (not multiply
covered). In the case of a single almost complex structure $J$,
a $J$-holomorphic sphere $w$ is called regular if the linearization
of the equation $\bar{\partial}_J(u) = 0$ at $w$, which is
given by an operator
\[
D_J(w): C^\infty(w^*TM) \longrightarrow \Omega^{0,1}(w^*(TM,J)),
\]
is onto (see \cite[Chapter 3]{mcduff-salamon}). Similarly,
$(b,w) \in \mathcal{M}^s(\mathbf{J})$ is called regular if the extended operator
\begin{gather*}
\hat{D}_{\mathbf{J}}(b,w): T_bB \times
C^\infty(w^*TM) \longrightarrow \Omega^{0,1}(w^*(TM,J_b)), \\
\hat{D}_{\mathbf{J}}(b,w)(\beta,W) =
D_{J_b}(w)W + \frac{1}{2} \left( D\mathbf{J}(b)\beta \right) \circ dw \circ i
\end{gather*}
is onto. Here $i$ is the complex structure on $\CP{1}$
and $D\mathbf{J}(b): T_bB \longrightarrow C^\infty(\mathrm{End}(TM))$
is the derivative of the family $(J_b)$ at $b$.
$\mathbf{J}$ itself is called regular if all $(b,w) \in \mathcal{M}^s(\mathbf{J})$ are regular,
and the set of regular families is denoted by
$\modJreg{B} \subset \modJ{B}$.
$\hat{D}_{\mathbf{J}}(b,w)$ is a Fredholm operator of index
$2n + 2c_1(w) + \dim B$. For $k \in \mathbb{Z}$, let $\mathcal{M}^s_k(\mathbf{J})
\subset \mathcal{M}^s(\mathbf{J})$ be the subspace of pairs $(b,w)$ with
$c_1(w) = k$. By applying the implicit function theorem, one
obtains
\begin{lemma} \label{lemma:implicit-function}
If $\mathbf{J} \in \modJreg{B}$, $\mathcal{M}^s_k(\mathbf{J})$ is a smooth manifold of dimension
$2n + 2k + \dim B$ for all $k$. \end{lemma}
Assume that $B$ is compact and choose $\mathbf{J}_0 \in
\modJ{B}$. Let $U_\delta(\mathbf{J}_0) \subset \modJ{B}$ be a
$\delta$-ball around $\mathbf{J}_0$ with respect to Floer's
$\smooth_{\epsilon}$-norm (see \cite{hofer-salamon95}
or \cite[p. 101--103]{schwarz95}).
\begin{theorem} \label{th:para-trans-theorem}
For sufficiently small $\delta>0$,
$\modJreg{B} \cap U_\delta(\mathbf{J}_0) \subset U_\delta(\mathbf{J}_0)$
has second category; in particular, it is $\smooth_{\epsilon}$-dense.
\end{theorem}
This is a well-known result (see e.g. \cite[Theorem 3.1.3]{mcduff-salamon}).
It is useful to compare its proof with that of the basic transversality
theorem for pseudoholomorphic curves, which is the special case
$B = \{pt\}$: in that case, one shows first that the `universal moduli space'
\[
\mathcal{M}^{\mathrm{univ}} \stackrel{\pi}{\longrightarrow} \modJ{pt}
\]
is smooth. $\modJreg{pt}$ is the set of regular values of $\pi$,
which is shown to be dense by applying the Sard-Smale theorem
(we omit the details which arise from the use of $\smooth_{\epsilon}$-spaces).
In the general case, a family $(J_b)_{b \in B}$ is regular iff the
corresponding map $B \longrightarrow \modJ{pt}$ is transverse to $\pi$.
Therefore the part of the proof which uses specific properties of
pseudoholomorphic curves (the smoothness of $\mathcal{M}^{\mathrm{univ}}$)
remains the same, but a different general result
has to be used to show that generic maps $B \longrightarrow \modJ{pt}$
are transverse to $\pi$.
A family $\mathbf{J}$ is called semi-positive if $\mathcal{M}^s_k(\mathbf{J}) = \emptyset$
for all $k < 0$; the semi-positive families form a subset
$\modJplus{B} \subset \modJ{B}$.
\begin{lemma} If $\dim B \leq 3$, $\modJreg{B} \subset \modJplus{B}$.
\end{lemma}
\proof $PSL(2,\C)$ acts freely on $\mathcal{M}_k^s(\mathbf{J})$ for all $k$.
If $\mathbf{J} \in \modJreg{B}$, the quotient is a smooth manifold and
\[
\dim \mathcal{M}_k^s(\mathbf{J})/PSL(2,\C) \leq 2n + 2k - 3.
\]
Therefore $\mathcal{M}_k^s(\mathbf{J}) = \emptyset$ for $k \leq 1-n$.
But for $2-n \leq k < 0$, $\mathcal{M}_k^s(\mathbf{J}) = \emptyset$
by \eqref{eq:wplus-equivalent}. \qed
\begin{corollary} \label{cor:no-negative-spheres}
If $B$ is compact and $\dim B \leq 3$, $\modJplus{B}$
is $C^\infty$-dense in $\modJ{B}$. \end{corollary}
Later, we will also use a `relative'
version of Corollary \ref{cor:no-negative-spheres}, in which $B$ is
non-compact and one considers families $\mathbf{J}$ with a fixed behaviour
outside a relatively compact open subset of $B$. We omit the
precise statement.
In contrast with the case of negative Chern number, holomorphic
spheres with Chern number $0$ or $1$ can occur in a family
$\mathbf{J} = (J_b)_{b \in B}$ even if $\dim B$ is small. When defining
Floer homology, special attention must be paid to them. For any
$k \geq 0$, let
$V_k(\mathbf{J}) \subset B \times M$ be the set of pairs $(b,y)$ such
that $y \in \mathrm{im}(w)$ for a non-constant $J_b$-holomorphic sphere $w$ with
$c_1(w) \leq k$. $V_k(\mathbf{J})$ is the
union of the images of the evaluation maps
\[
\mathcal{M}_j^s(\mathbf{J}) \times_{PSL(2,\C)} \CP{1} \longrightarrow B \times M
\]
for $j \leq k$. For regular $\mathbf{J}$ and $j \leq 0$,
the dimension of $\mathcal{M}_j^s(\mathbf{J}) \times_{PSL(2,\C)} \CP{1}$ is
$\leq 2n + \dim B - 4$. Therefore $V_0(\mathbf{J})$
is (loosely speaking) a codimension-$4$ subset of
$B \times M$. Similarly, $V_1(\mathbf{J})$ has codimension $2$.
For any $H \in \H$, the pullback of $\alpha_H$ to $\widetilde{\loops}$
is exact: $p^*\alpha_H = da_H$ with
\[
a_H(v,x) = -\int_{D^2} v^*\o + \int_{S^1} H(t,x(t)) dt.
\]
Let $Z(\alpha_H) \subset \mathcal{L}M$ be the zero set of
$\alpha_H$ and $\mathrm{Crit}(a_H) = p^{-1}(Z(\alpha_H))$ the
set of critical points of $a_H$. $c = [v,x] \in \mathrm{Crit}(a_H)$
is nondegenerate iff $x$ is a regular zero of $\alpha_H$,
that is, if the linearization of \eqref{eq:hamilton} at
$x$ has no nontrivial solutions. We denote the Conley-Zehnder
index of such a critical point by $\mu_H(c) \in \mathbb{Z}$.
\begin{convention} Let $H$ be a time-independent Hamiltonian,
$y \in M$ a nondegenerate critical point of $H$ and $c = [v,x] \in \widetilde{\loops}$
the critical point of $a_H$ represented by the constant maps $v,x \equiv y$.
Our convention for $\mu_H$ is that for small $H$, $\mu_H(c)$ equals
the Morse index of $y$. This differs from the convention in
\cite{salamon-zehnder92} by a constant $n$. \end{convention}
Assume that all critical points of $a_H$ are nondegenerate, and
let $\mathrm{Crit}_k(a_H)$ be the set of critical points
of index $k$. The Floer chain group $CF_k(H)$ is the group of formal sums
\[
\sum_{c \in \mathrm{Crit}_k(a_H)} m_c \gen{c}
\]
with coefficients $m_c \in \mathbb{Z}/2$ such that
$
\{ c \in \mathrm{Crit}_k(a_H) \, | \, m_c \neq 0, a_H(c) \geq C \}
$
is finite for all $C \in \mathbb{R}$.
A family $\mathbf{J} = (J_t)_{t \in S^1} \in \modJ{S^1}$ defines a
Riemannian metric
\[
(\xi,\eta)_{\mathbf{J}} = \int_{S^1} \o(\xi(t),J_t \eta(t)) dt
\]
on $\mathcal{L}M$. Let $\nabla_{\mathbf{J}} a_H$ be the gradient of $a_H$ with respect
to the pullback of $(\cdot,\cdot)_{\mathbf{J}}$ to $\widetilde{\loops}$. A smooth path
$\tilde{u}: \mathbb{R} \longrightarrow \widetilde{\loops}$ is a flow line of
$-\nabla_{\mathbf{J}} a_H$ iff its projection to $\mathcal{L}M$
is given by a map $u \in C^\infty(\mathbb{R} \times S^1,M)$ such that
\begin{equation} \label{eq:floer}
\partials{u} + J_t(u(s,t))\left(
\partialt{u} - X_H(t,u(s,t))\right) = 0
\quad \text{ for all } (s,t) \in \mathbb{R} \times S^1.
\end{equation}
If $u$ is a solution of \eqref{eq:floer} whose
energy $E(u) = \int {\left|\partials{u}\right|}^2$ is
finite, there are $c_-,c_+ \in \mathrm{Crit}(a_H)$ such that
\[
\lim_{s \rightarrow \pm \infty} \tilde{u}(s) = c_\pm.
\]
Moreover, if the limits are nondegenerate, the linearization
of \eqref{eq:floer} at $u$ is a differential operator
$D_{H,\mathbf{J}}(u): C^\infty(u^*TM) \longrightarrow C^\infty(u^*TM)$
whose Sobolev completion
\[
D_{H,\mathbf{J}}(u): W^{1,p}(u^*TM) \longrightarrow L^p(u^*TM)
\]
$(p>2)$ is a Fredholm operator with
index $\mathrm{ind }(u) = \mu_H(c_-) - \mu_H(c_+)$.
For $c_-,c_+ \in \mathrm{Crit}(a_H)$, let $\mathcal{M}(c_-,c_+;H,\mathbf{J})$
be the space of solutions of \eqref{eq:floer} which have
a lift $\tilde{u}: \mathbb{R} \longrightarrow \widetilde{\loops}$ with limits
$c_-,c_+$. The Floer differential
\[
\partial_k(H,\mathbf{J}): CF_k(H) \longrightarrow CF_{k-1}(H)
\]
is defined by the formula
\begin{equation} \label{eq:differential}
\partial_k(H,\mathbf{J})(\gen{c_-}) = \sum_{c_+ \in \mathrm{Crit}_{k-1}(a_H)}
\#(\mathcal{M}(c_-,c_+;H,\mathbf{J})/\mathbb{R}) \gen{c_+},
\end{equation}
extended in the obvious way to infinite linear combinations
of the $\gen{c_-}$ (from now on, this is to be understood
for all similar formulae). $\mathbb{R}$ acts on $\mathcal{M}(c_-,c_+;H,\mathbf{J})$
by $(s_0 \cdot u)(s,t) = u(s-s_0,t)$, and
$\#$ denotes counting the number
of points in a set mod $2$. In order for \eqref{eq:differential}
to make sense and define the `right' homomorphism, $(H,\mathbf{J})$
has to satisfy certain conditions.
\begin{defn} \label{def:regular-pair}
$(H,\mathbf{J}) \in \H \times \modJreg{S^1}$ is called a
{\em regular pair} if
\begin{enumerate}
\item \label{item:one} every $[v,x] \in \mathrm{Crit}(a_H)$ is nondegenerate and
satisfies $(t,x(t)) \notin V_1(\mathbf{J})$ for all $t \in S^1$;
\item \label{item:regul}
every solution $u: \mathbb{R} \longrightarrow \mathcal{L}M$
of \eqref{eq:floer} with finite energy
is regular (that is, the linearization of \eqref{eq:floer} at $u$ is onto);
\item \label{item:three}
if in addition $\mathrm{ind }(u) \leq 2$, $(t,u(s,t)) \notin V_0(\mathbf{J})$
for all $(s,t) \in \mathbb{R} \times S^1$.
\end{enumerate} \end{defn}
These are essentially the conditions defining
$\mathcal{H}_{\mathrm{reg}}(J)$ in \cite{hofer-salamon95},
adapted to the case of time-dependent almost complex
structures. \ref{item:regul} implies that $\mathcal{M}(c_-,c_+;H,\mathbf{J})$
is a manifold of dimension $\mu_H(c_-) - \mu_H(c_+)$ for all
$c_-,c_+$. The proof of the main compactness result
\cite[Theorem 3.3]{hofer-salamon95} carries over to our
situation using Corollary \ref{cor:no-negative-spheres}.
It follows that if $(H,\mathbf{J})$ is a regular pair, the r.h.s. of
\eqref{eq:differential} is meaningful and defines differentials
$\partial_k(H,\mathbf{J})$ such that $\partial_{k-1}(H,\mathbf{J}) \partial_k(H,\mathbf{J}) = 0$.
\begin{definition} The Floer homology $HF_*(H,\mathbf{J})$ of a regular
pair $(H,\mathbf{J})$ is the homology of $(CF_*(H),\partial_*(H,\mathbf{J}))$. \end{definition}
The existence of regular pairs is ensured by the following Theorem,
which is a variant of \cite[Theorems 3.1 and 3.2]{hofer-salamon95}.
\begin{thm} \label{th:transv-floer}
The set of regular pairs is $C^\infty$-dense
in $\H \times \modJ{S^1}$. \end{thm}
So far, we have only used families of almost complex structures
parametrized by $S^1$. For such families Corollary
\ref{cor:no-negative-spheres} holds whenever $(M,\o)$ is weakly
monotone; therefore, the stronger assumption {$(W^+)$} has
not been necessary up to now. Two-parameter
families of almost complex structures occur first in the next step,
the definition of `continuation maps'.
A homotopy between regular pairs $(H^-,\mathbf{J}^-)$, $(H^+,\mathbf{J}^+)$ consists
of an $H \in C^\infty(\mathbb{R} \times S^1 \times M,\mathbb{R})$ and a
$\mathbf{J} \in \modJreg{\mathbb{R} \times S^1}$, such that $(H(s,t,\cdot), J_{s,t})$
is equal to $(H^-(t,\cdot),J^-_t)$ for $s \leq -1$ and to
$(H^+(t,\cdot),J^+_t)$ for $s \geq 1$. For $c_- \in \mathrm{Crit}(a_{H^-})$
and $c_+ \in \mathrm{Crit}(a_{H^+})$, let $\mathcal{M}^{\Phi}(c_-,c_+;H,\mathbf{J})$ be the
space of solutions $u \in C^\infty(\mathbb{R} \times S^1, M)$ of
\begin{equation} \label{eq:continuation}
\partials{u} + J_{s,t}(u)\left(\partialt{u} - X_H(s,t,u)\right) = 0
\end{equation}
which can be lifted to paths $\tilde{u}: \mathbb{R} \longrightarrow \widetilde{\loops}$
with limits $c_-,c_+$. A homotopy $(H,\mathbf{J})$ is regular if every
solution of \eqref{eq:continuation} is regular (the linearization
is onto) and if for those solutions with index $\leq 1$,
\[
(s,t,u(s,t)) \notin V_0(\mathbf{J}) \text{ for all } (s,t) \in \mathbb{R} \times S^1.
\]
For a regular homotopy, the `continuation homomorphisms'
\[
\Phi_k(H,\mathbf{J}): CF_k(H^-,\mathbf{J}^-) \longrightarrow CF_k(H^+,\mathbf{J}^+)
\]
are defined by
\[
\Phi_k(H,\mathbf{J})(\gen{c_-}) = \sum_{c_+ \in \mathrm{Crit}_k(a_{H^+})}
\#\mathcal{M}^{\Phi}(c_-,c_+;H,\mathbf{J}) \gen{c_+}.
\]
This is a homomorphism of chain complexes and induces an isomorphism of
Floer homology groups, which we denote equally by $\Phi_*(H,\mathbf{J})$.
For fixed $(H^{\pm},\mathbf{J}^{\pm})$, these isomorphisms are
independent of the choice of $(H,\mathbf{J})$. Moreover, they
are functorial with respect to composition of homotopies.
Therefore we can speak of a well-defined Floer homology
$HF_*(M,\o)$ independent of the choice of a regular pair
(see \cite{salamon-zehnder92} for a detailed discussion,
which carries over to our case with minor modifications).
The proof that the `continuation maps' are independent
of $(H,\mathbf{J})$ involves three-parameter families of almost
complex structures, and hence uses the full strength of
Corollary \ref{cor:no-negative-spheres}.
\begin{remark} The definition used here leads to a Floer homology
which is canonically isomorphic to that defined in \cite{hofer-salamon95}.
The only difference is that we have allowed a wider range
of perturbations. The reason for this will become clear in the next
section. \end{remark}
Since $a_H = p^*\alpha_H$, $\mathrm{Crit}(a_H)$ is $\Gamma$-invariant.
This induces an action of $\Gamma$ on $CF_*(H)$ for every regular
pair $(H,\mathbf{J})$. The action does not preserve the grading:
$\gamma$ maps $CF_k(H)$ to $CF_{k - 2c_1(\gamma)}(H)$, because
\begin{equation} \label{eq:grading-shift}
\mu_H(A \# v,x) = \mu_H(v,x) - 2c_1(A)
\end{equation}
for $[v,x] \in \mathrm{Crit}(a_H)$ and $A: S^2 \longrightarrow M$
(see \cite{hofer-salamon95} or \cite[Proposition 5]{floer-hofer93};
the difference in sign to \cite{hofer-salamon95} is due to the
fact that they consider Floer cohomology rather than homology).
To recover a proper grading, one should assign to $\gamma \in \Gamma$ the
`dimension' $-2c_1(\gamma)$. We denote the subset of elements
of `dimension' $k$ by $\Gamma_k \subset \Gamma$.
\begin{defn} \label{def:novikov-ring}
For $k \in \mathbb{Z}$, let $\Lambda_k$ be the group of formal sums
\begin{equation} \label{eq:element-of-novikov-ring}
\sum_{\gamma \in \Gamma_k} m_\gamma \gen{\gamma}
\end{equation}
with $m_\gamma \in \mathbb{Z}/2$, such that $\{\gamma \in \Gamma_k \,
| \, m_\gamma \neq 0, \o(\gamma) \leq C\}$ is finite for all
$C \in \mathbb{R}$. The multiplication
\[
\gen{\gamma} \cdot \gen{\gamma'} = \gen{\gamma + \gamma'}
\]
can be extended to the infinite linear combinations
\eqref{eq:element-of-novikov-ring}, and this makes
$\Lambda = \bigoplus_k \Lambda_k$ into a commutative graded ring
called the Novikov ring of $(M,\o)$. \end{defn}
For every $\gamma \in \Gamma$ and $c \in \widetilde{\loops}$,
$a_H(\gamma \cdot c) = a_H(c) - \o(\gamma)$.
Therefore the $\Gamma$-action on $CF_*(H)$ extends naturally to a graded
$\Lambda$-module structure. Since the differentials $\partial_*(H,\mathbf{J})$ and
the homomorphisms $\Phi_*(H,\mathbf{J})$ are $\Lambda$-linear,
this induces a $\Lambda$-module structure on $HF_*(H,\mathbf{J})$ and
on $HF_*(M,\o)$.
The {\em quantum homology} $QH_*\mo$ is the graded $\Lambda$-module
defined by
\[
QH_k(M,\o) = \bigoplus_{i + j = k} H_i(M;\mathbb{Z}/2) \otimes
\Lambda_j.
\]
\begin{theorem} \label{th:what-is-floer-homology}
$HF_*\mo$ is isomorphic to $QH_*\mo$ as a
graded $\Lambda$-module. \end{theorem}
This was proved by Piunikhin, Salamon and Schwarz {\cite{piunikhin-salamon-schwarz94}} (less general
versions had been obtained before by Floer \cite{floer88} and
Hofer-Salamon \cite{hofer-salamon95}); a detailed account of the
proof is in preparation \cite{schwarz97}. Their result is in fact
much stronger: it provides a canonical isomorphism
$\Psi^+: QH_*\mo \longrightarrow HF_*\mo$ which also relates
different product structures. We will use this construction
in section \ref{sec:gluing}.
\section{\label{sec:the-action} The $\widetilde{G}$-action on Floer homology}
We begin by considering the action of $G$ on $\modJ{S^1}$. For
$\mathbf{J} = (J_t)_{t \in S^1} \in \modJ{S^1}$ and $g \in G$, define $\mathbf{J}^g =
(J_t^g)_{t \in S^1}$ by $J_t^g = Dg_t^{-1} J_t Dg_t$.
\begin{lemma} \label{th:naturality-of-regularity}
If $\mathbf{J} \in \modJreg{S^1}$, $\mathbf{J}^g \in \modJreg{S^1}$ for
all $g \in G$. \end{lemma}
\proof $w \in C^\infty(\CP{1},M)$ is $J_t^g$-holomorphic iff
$w' = g_t(w)$ is $J_t$-holomorphic. Since
\begin{align*}
\hat{D}_{\mathbf{J}}(t,w')(1,\frac{\partial g_t}{\partial t}(w)) &=
\frac{\partial}{\partial t}
\left(\frac{1}{2} Dg_t \circ dw + \frac{1}{2} J_t \circ Dg_t \circ dw
\circ i\right)\\
&= \frac{1}{2} Dg_t \circ \frac{\partial J_t^g}{\partial t} \circ dw
\circ i,
\end{align*}
there is a commutative diagram
\begin{equation*} \xymatrix{
\mathbb{R} \times C^\infty(w^*TM) \ar[rr]^{\hat{D}_{\mathbf{J}^g}(t,w)} \ar[d]^{\delta}
&& \Omega^{0,1}(w^*(TM,J_t^g)) \ar[d]^{\delta'}\\
\mathbb{R} \times C^\infty((w')^*TM) \ar[rr]^{\hat{D}_{\mathbf{J}}(t,w')} &&
\Omega^{0,1}((w')^*(TM,J_t)).
}
\end{equation*}
Here $\delta(\tau,W) = (\tau, Dg_t(w)W + \tau \frac{\partial
g_t}{\partial t}(w))$, and $\delta'$ maps a homomorphism
$\sigma: T\CP{1} \longrightarrow w^*TM$ to $\delta'(\sigma) =
Dg_t(w) \circ \sigma$. If $w$ is simple, $\hat{D}_\mathbf{J}(t,w')$
is onto by assumption, and since $\delta,\delta'$ are isomorphisms,
$\hat{D}_{\mathbf{J}^g}(t,w)$ is onto as well. \qed
The same argument shows that $\hat{D}_{\mathbf{J}^g}(t,w)$ and
$\hat{D}_\mathbf{J}(t,w')$ have the same index; therefore
\begin{equation} \label{eq:g:v}
V_k(\mathbf{J}^g) = \{(t,y) \in S^1 \times M \; | \; (t,g_t(y)) \in V_k(\mathbf{J})\}
\end{equation}
for all $k \geq 0$. The metric on $\mathcal{L}M$ associated to $\mathbf{J}^g$ is
\begin{equation} \label{eq:g:metric}
(\cdot,\cdot)_{\mathbf{J}^g} = g^*(\cdot,\cdot)_\mathbf{J}.
\end{equation}
\begin{remark} The $G$-action does not preserve the subspace
of families $(J_t)$ such that $J_t = J_0$ for all $t$.
This is the reason why we have allowed $t$-dependent almost complex
structures in the definition of Floer homology. \end{remark}
For $(H,\mathbf{J}) \in \H \times \modJ{S^1}$ and $g \in G$, we call
the pair $(H^g,\mathbf{J}^g)$, where $H^g$ is as in Lemma \ref{th:hg},
the {\em pullback} of $(H,\mathbf{J})$ by $g$. Recall that $H^g$ depends
on the choice of a Hamiltonian which generates $g$. By Lemma
\ref{th:hg}, $\alpha_{H^g} = g^*\alpha_H$; therefore
\begin{equation} \label{eq:g:action}
a_{H^g} = \tilde{g}^*a_H + \mathrm{(constant)}
\end{equation}
and $\mathrm{Crit}(a_H) = \tilde{g}(\mathrm{Crit}(a_{H^g}))$ for every lift
$\tilde{g}: \widetilde{\loops} \longrightarrow \widetilde{\loops}$ of the action of $g$.
\begin{lemma} \label{th:functoriality}
For all $c_-,c_+ \in \mathrm{Crit}(a_{H^g})$, there
is a bijective map \[
\mathcal{M}(c_-,c_+;H^g,\mathbf{J}^g)/\mathbb{R} \longrightarrow
\mathcal{M}(\tilde{g}(c_-),\tilde{g}(c_+);H,\mathbf{J})/\mathbb{R}.
\] Moreover, if $(H,\mathbf{J})$ is a regular pair,
$(H^g,\mathbf{J}^g)$ is also regular. \end{lemma}
\proof Let $\tilde{u}: \mathbb{R} \longrightarrow \widetilde{\loops}$ be a smooth path
whose projection to $\mathcal{L}M$ is given by $u \in C^\infty(\mathbb{R} \times S^1,M)$.
Recall that $u$ satisfies \eqref{eq:floer} iff
\begin{equation} \label{eq:floer2}
\frac{d\tilde{u}}{ds} + \nabla_{\mathbf{J}}a_H(\tilde{u}(s)) = 0
\end{equation}
Let $\tilde{v}(s) = \tilde{g}^{-1}(\tilde{u}(s))$. By \eqref{eq:g:metric}
and \eqref{eq:g:action}, $\tilde{u}$ satisfies \eqref{eq:floer2}
iff $\tilde{v}$ is a solution of
\begin{equation} \label{eq:floer3}
\frac{d\tilde{v}}{ds} + \nabla_{\mathbf{J}^g}a_{H^g}(\tilde{v}(s)) = 0.
\end{equation}
The map of solution spaces defined in this way is clearly bijective
and $\mathbb{R}$-equivariant. Assuming that $(H,\mathbf{J})$ is regular, we now
check that its pullback $(H^g,\mathbf{J}^g)$ satisfies the conditions for
a regular pair. (i) Let $c = [x,v]$ be a critical point of $a_{H^g}$,
$c' = \tilde{g}(c)$ and $\xi \in C^\infty(x^*TM)$ with
$D^2a_{H^g}(c)(\xi,\cdot) = 0$. By \eqref{eq:g:action},
$D^2a_H(c')(Dg(x)\xi,\cdot) = 0$. Since
$c'$ is nondegenerate, $\xi = 0$, and therefore
$c$ is also nondegenerate. The last sentence in Definition
\ref{def:regular-pair}(i) follows from \eqref{eq:g:v}. (ii) can be
proved using the same method as in Lemma \ref{th:naturality-of-regularity},
which also shows that a solution of \eqref{eq:floer3} and
the corresponding solution of \eqref{eq:floer2} have the same index.
Together with \eqref{eq:g:v}, this implies that $(H^g,\mathbf{J}^g)$
satisfies condition \ref{def:regular-pair}(iii). \qed
\begin{prop} \label{th:shift}
If $c \in \mathrm{Crit}(a_{H^g})$ is nondegenerate,
$\mu_H(\tilde{g}(c)) = \mu_{H^g}(c) - 2I(g,\tilde{g})$. \end{prop}
Recall that $\mu_{H^g}(c)$ is defined in the following way
(see \cite{salamon-zehnder92}, \cite{hofer-salamon95}): choose
a representative $(v,x)$ of $c$ and a symplectic trivialization
$\tau_c: x^*TM \longrightarrow S^1 \times (\mathbb{R}^{2n},\o_0)$ which
can be extended over $v^*TM$. Let $\Psi_{H^g,c}: [0;1] \longrightarrow
\mathrm{Sp}(2n,\mathbb{R})$ be the path given by
\[
\Psi_{H^g,c}(t) = \tau_c(t) D\phi_{H^g}^t(x(0)) \tau_c(0)^{-1},
\]
where $(\phi_{H^g}^t)_{t \in \mathbb{R}}$ is the Hamiltonian flow of $H^g$.
$\Psi_{H^g,c}(0) = \mathit{Id}$, and since $c$ is nondegenerate,
$\mathrm{det}(\mathit{Id} - \Psi_{H^g,c}(1)) \neq 0$. A path $\Psi$ with these
properties has an index $\mu_1(\Psi) \in \mathbb{Z}$, and $\mu_{H^g}(c)$ is defined
by $\mu_{H^g}(c) = \mu_1(\Psi_{H^g,c})$. $\mu_1$ has the following property
\cite[Proposition 5]{floer-hofer93}:
\begin{lemma} \label{th:rescale} If $\Psi,\Psi'$ are related by
$\Psi'(t) = l(t) \Psi(t) l(0)^{-1}$ for some $l \in
C^\infty(S^1,\mathrm{Sp}(2n,\mathbb{R}))$, $\mu_1(\Psi') = \mu_1(\Psi) - 2 \deg(l)$. \qed
\end{lemma}
\proof[Proof of Proposition \ref{th:shift}]
Let $(v',x')$ be a representative of $\tilde{g}(c)$ and $\tau_{\tilde{g}(c)}$
a trivialization of $(x')^*TM$ which can be extended over $(v')^*TM$.
$\mu_H(\tilde{g}(c))$ is defined using the path
\[
\Psi_{H,\tilde{g}(c)}(t) = \tau_{\tilde{g}(c)}(t) D\phi^t_{H}(x'(0))
\tau_{\tilde{g}(c)}(0)^{-1}.
\]
$H^g$ is defined in such a way that
$\phi_H^t = g_t \, \phi_{H^g}^t \, g_0^{-1}$. Therefore
$
\Psi_{H,\tilde{g}(c)}(t) = l(t) \Psi_{H^g,c}(t) l(0)^{-1},
$
where $l(t) = \tau_{\tilde{g}(c)}(t) Dg_t(x(t)) \tau_c(t)^{-1}$. $l$ is
a loop in $\mathrm{Sp}(2n;\mathbb{R})$, and $\deg(l) = I(g,\tilde{g})$ by definition.
Proposition \ref{th:shift} now follows from Lemma \ref{th:rescale}. \qed
\begin{defn} \label{def:induced-maps}
Let $(H,\mathbf{J})$ be a regular pair, $(g,\tilde{g}) \in \widetilde{G}$
and $(H^g,\mathbf{J}^g)$ the pullback of $(H,\mathbf{J})$ by $g$.
For $k \in \mathbb{Z}$, define an isomorphism
\[
CF_k(g,\tilde{g};H,\mathbf{J},H^g): CF_k(H^g) \longrightarrow
CF_{k-2I(g,\tilde{g})}(H)
\]
by $CF_k(g,\tilde{g};H,\mathbf{J},H^g)(\gen{c}) = \gen{\tilde{g}(c)}$,
extended in the obvious way to infinite sums of the generators
$\gen{c}$. Because of \eqref{eq:g:action}, this respects the
finiteness condition for the formal sums in $CF_*(H)$. \end{defn}
In view of the definition of the
differential \eqref{eq:differential}, Lemma \ref{th:functoriality}
says that $CF_*(g,\tilde{g};H,\mathbf{J},H^g)$ is an isomorphism of
chain complexes. We denote the induced isomorphisms on Floer homology by
\[
HF_k(g,\tilde{g};H,\mathbf{J},H^g): HF_k(H^g,\mathbf{J}^g) \longrightarrow
HF_{k-2I(g,\tilde{g})}(H,\mathbf{J}).
\]
Consider two regular pairs $(H^-,\mathbf{J}^-)$, $(H^+,\mathbf{J}^+)$, two
Hamiltonians $K^-_g$, $K^+_g$ which generate $g$ and the
pullbacks $((H^-)^g,(\mathbf{J}^-)^g)$, $((H^+)^g,(\mathbf{J}^+)^g)$ using
$K^-_g$ and $K^+_g$, respectively. Let $(H,\mathbf{J})$ be a homotopy
between $(H^-,\mathbf{J}^-)$ and $(H^+,\mathbf{J}^+)$.
Choose $\psi \in C^\infty(\mathbb{R},\mathbb{R})$ with $\psi|(-\infty;-1] = 0$ and
$\psi|[1;\infty) = 1$, and define $H^g \in C^\infty(\mathbb{R} \times S^1
\times M,\mathbb{R})$ by
\[
H^g(s,t,y) = H(s,t,g_t(y)) - (1-\psi(s)) K_g^-(t,g_t(y))
- \psi(s) K_g^+(t,g_t(y)).
\]
Together with the family $\mathbf{J}^g$ given by $J^g_{s,t} = Dg_t^{-1}J_{s,t}Dg_t$,
this is a homotopy from $((H^-)^g,(\mathbf{J}^-)^g)$ to $((H^+)^g,(\mathbf{J}^+)^g)$.
\begin{lemma} \label{th:homotopy-is-natural}
For all $c_- \in \mathrm{Crit}(a_{(H^-)^g}), c_+ \in \mathrm{Crit}(a_{(H^+)^g})$,
there is a bijective map
\[
\mathcal{M}^{\Phi}(c_-,c_+;H^g,\mathbf{J}^g) \longrightarrow
\mathcal{M}^{\Phi}(\tilde{g}(c_-),\tilde{g}(c_+);H,\mathbf{J}).
\]
Moreover,
if $(H,\mathbf{J})$ is a regular homotopy, $(H^g,\mathbf{J}^g)$ is also regular. \end{lemma}
We omit the proof, which is similar to that of
Lemma \ref{th:functoriality}.
\begin{cor} If $(H,\mathbf{J})$ is a regular homotopy, the diagram
\begin{equation*}
\xymatrix{
CF_*((H^-)^g,(\mathbf{J}^-)^g) \ar[rrrr]^{CF_*(g,\tilde{g};H^-,\mathbf{J}^-,(H^-)^g)}
\ar[d]^{\Phi_*(H^g,\mathbf{J}^g)}
&&&& CF_*(H^-,\mathbf{J}^-) \ar[d]^{\Phi_*(H,\mathbf{J})}\\
CF_*((H^+)^g,(\mathbf{J}^+)^g) \ar[rrrr]^{CF_*(g,\tilde{g};H^+,\mathbf{J}^+,(H^+)^g)}
&&&& CF_*(H^+,\mathbf{J}^+)
}
\end{equation*}
commutes. \end{cor}
This is proved by a straightforward computation using Lemma
\ref{th:homotopy-is-natural}. It follows that for every
$(g,\tilde{g}) \in \widetilde{G}$, there is a unique automorphism
\[
HF_*(g,\tilde{g}): HF_*\mo \longrightarrow HF_*\mo
\]
independent of the choice of a regular pair.
We list some properties of these maps which are clear
from the definition.
\begin{proposition} \label{prop:further-properties}
\begin{enumerate}
\item
$HF_*(g,\tilde{g})$ is an automorphism of $HF_*\mo$ as a
$\Lambda$-module.
\item For $(g,\tilde{g}) = \mathit{Id}_{\widetilde{G}}$, $HF_*(g,\tilde{g}) = \mathit{Id}_{HF_*\mo}$.
\item \label{item:lambda-multiplication}
If $(g,\tilde{g}) = (\mathit{Id},\gamma)$ for some $\gamma \in \Gamma$, $HF_*(g,\tilde{g})$
is equal to the multiplication by $\gen{\gamma} \in \Lambda$.
\item \label{item:functor}
For $(g_1,\tilde{g}_1), (g_2,\tilde{g}_2) \in \widetilde{G}$,
$
HF_*(g_1 g_2,\tilde{g}_1 \tilde{g}_2) =
HF_*(g_1,\tilde{g}_1) HF_*(g_2,\tilde{g}_2)
$. \qed
\end{enumerate}
\end{proposition}
\newcommand{\modh}[2]{\mathcal{M}^h(#1,#2;\bar{H},\bar{\mathbf{J}})}
\newcommand{\cmodh}[2]{\overline{\mathcal{M}}^h(#1,#2;\bar{H},\bar{\mathbf{J}})}
\section{Homotopy invariance \label{sec:homotopy}}
Let $(g_{r,t})_{0 \leq r \leq 1, t \in S^1}$ be a smooth family of Hamiltonian
automorphisms of $(M,\o)$ with $g_{r,0} = \mathit{Id}$ for all $r$. This family
defines a path $(g_r)_{0 \leq r \leq 1}$ in $G$. Let
$(g_r,\tilde{g}_r)_{0 \leq r \leq 1}$ be a smooth lift of this path to
$\widetilde{G}$. The aim of this section is to prove the following
\begin{prop} \label{th:homotopy}
For $(g_r,\tilde{g}_r)_{0 \leq r \leq 1}$ as above,
\[
HF_*(g_0,\tilde{g}_0) = HF_*(g_1,\tilde{g}_1):
HF_*(M,\o) \longrightarrow HF_*(M,\o).
\]\end{prop}
Choose a smooth family $(K_r)_{0 \leq r \leq 1}$ of Hamiltonians such
that $K_r$ generates $g_r$. Let $(H,\mathbf{J})$ be a regular pair and
$(H^{g_r},\mathbf{J}^{g_r})$ its pullbacks (using $K_r$). The maps induced
by $(g_0,\tilde{g}_0)$ and $(g_1,\tilde{g}_1)$ are
\begin{align*}
HF_*(g_0,\tilde{g}_0;H,\mathbf{J},H^{g_0}): HF_*(H^{g_0},\mathbf{J}^{g_0}) &\longrightarrow
HF_*(H,\mathbf{J}),\\
HF_*(g_1,\tilde{g}_1;H,\mathbf{J},H^{g_1}): HF_*(H^{g_1},\mathbf{J}^{g_1}) &\longrightarrow
HF_*(H,\mathbf{J}).
\end{align*}
Proposition \ref{th:homotopy} says that if $(H',\mathbf{J}')$ is a regular
homotopy from $(H^{g_1},\mathbf{J}^{g_1})$ to $(H^{g_0},\mathbf{J}^{g_0})$,
\begin{equation} \label{eq:htp-one}
HF_*(g_1,\tilde{g}_1;H,\mathbf{J},H^{g_1}) = HF_*(g_0,\tilde{g}_0;H,\mathbf{J},H^{g_0})
\Phi_*(H',\mathbf{J}').
\end{equation}
Because of Proposition \ref{prop:further-properties}\ref{item:functor},
it is sufficient to consider the case $(g_0,\tilde{g}_0) = {\mathit{Id}}_{\widetilde{G}}$.
Choose $K_0 = 0$. Then $(H^{g_0},\mathbf{J}^{g_0}) = (H,\mathbf{J})$,
$HF_*(g_0,\tilde{g}_0; H,\mathbf{J},H) = \mathit{Id}_{HF_*(H,\mathbf{J})}$, and
\eqref{eq:htp-one} is reduced to
\begin{equation} \label{eq:htp-two}
\Phi_*(H',\mathbf{J}') HF_*(g_1,\tilde{g}_1;H,\mathbf{J},H^{g_1})^{-1} = \mathit{Id}_{HF_*(H,\mathbf{J})}.
\end{equation}
The rest of the section contains the proof of this equation.
\begin{defn} Let $(H,\mathbf{J})$, $(H^{g_r},\mathbf{J}^{g_r})$ and
$(H',\mathbf{J}')$ be as above. A {\em deformation of homotopies} compatible with
them consists of a function
$\bar{H} \in C^\infty([0;1] \times \mathbb{R} \times S^1 \times
M,\mathbb{R})$ and a family of almost complex structures $\bar{\mathbf{J}} =
(\bar{J}_{r,s,t}) \in \modJreg{[0;1] \times \mathbb{R} \times S^1}$, such that
\begin{align*}
& \bar{H}(r,s,t,y) = H^{g_r}(t,y), \; \bar{J}_{r,s,t} = J^{g_r}_t
&& \text{ for } s \leq -1, \\
& \bar{H}(r,s,t,y) = H(t,y), \; \bar{J}_{r,s,t} = J_t
&& \text{ for } s \geq 1, \\
& \bar{H}(0,s,t,y) = H(t,y), \; \bar{J}_{0,s,t} = J_t && \text{ and }\\
& \bar{H}(1,s,t,y) = H'(s,t,y), \; \bar{J}_{1,s,t} = J_{s,t}'. &&
\end{align*} \end{defn}
Let $(\bar{H},\bar{\mathbf{J}})$ be a deformation of homotopies. Consider a
pair $(r,u) \in [0;1] \times C^\infty(\mathbb{R} \times S^1,M)$ such that
\begin{equation} \label{eq:parametrized-floer}
\partials{u} + \bar{J}_{r,s,t}(u(s,t))\left(
\partialt{u} - X_{\bar{H}}(r,s,t,u(s,t))\right) = 0.
\end{equation}
We say that $u$ converges to
$c_-,c_+ \in \mathrm{Crit}(a_H)$ if there is a smooth path
$\tilde{u}: \mathbb{R} \longrightarrow \widetilde{\loops}$ with
\[
\lim_{s \rightarrow -\infty} \tilde{u}(s) = \tilde{g}_r^{-1}(c_-),
\lim_{s \rightarrow +\infty} \tilde{u}(s) = c_+
\]
such that $u(s,\cdot) = p(\tilde{u}(s))$ for all $s$.
We denote the space of such pairs $(r,u)$ with limits
$c_\pm$ by $\mathcal{M}^h(c_-,c_+;\bar{H},\bar{\mathbf{J}})$.
The linearization of \eqref{eq:parametrized-floer}
at a pair $(r,u) \in \mathcal{M}^h(c_-,c_+;\bar{H},\bar{\mathbf{J}})$
is given by an operator
\[
D_{\bar{H},\bar{\mathbf{J}}}^h(r,u): \mathbb{R} \times C^\infty(u^*TM)
\longrightarrow C^\infty(u^*TM).
\]
Since $c_-,c_+$ are nondegenerate critical points of $a_H$,
$\tilde{g}_r^{-1}(c_-)$ is a nondegenerate critical point
of $a_{H^{g_r}}$ for all $r$. It follows that the Sobolev
completion $(p>2)$
\begin{equation} \label{eq:h-operator}
D_{\bar{H},\bar{\mathbf{J}}}^h(r,u): \mathbb{R} \times W^{1,p}(u^*TM)
\longrightarrow L^p(u^*TM)
\end{equation}
is a Fredholm operator of index
$\mu_{H^{g_r}}(\tilde{g}_r^{-1}(c_-)) - \mu_H(c_+) + 1$.
By Lemma \ref{th:shift}, $\mu_H(c_-) =
\mu_{H^{g_r}}(\tilde{g}_r^{-1}(c_-)) - 2 I(g_r,\tilde{g}_r)$.
However, since $(g_r,\tilde{g}_r)$ is homotopic to the identity
in $\widetilde{G}$, $I(g_r,\tilde{g}_r) = 0$. It follows that
$\mathrm{ind } D_{\bar{H},\bar{\mathbf{J}}}^h(r,u) = \mu_H(c_-) - \mu_H(c_+) + 1$.
$(\bar{H},\bar{\mathbf{J}})$ is called regular if \eqref{eq:h-operator}
is onto for all $(r,u)$, and
$
(r,s,t,u(s,t)) \notin V_0(\bar{\mathbf{J}})
$
for all $(r,s,t) \in [0;1] \times \mathbb{R} \times S^1$ and
$u \in \mathcal{M}^h(c_-,c_+;\bar{H},\bar{\mathbf{J}})$ such that
$\mu_H(c_-) \leq \mu_H(c_+)$. An analogue of Theorem
\ref{th:transv-floer} ensures that regular deformations
of homotopies exists. Regularity implies that the spaces
$\mathcal{M}^h(c_-,c_+;\bar{H},\bar{\mathbf{J}})$ are smooth manifolds.
The boundary $\partial\mathcal{M}^h(c_-,c_+;\bar{H},\bar{\mathbf{J}})$ consists
of solutions of \eqref{eq:parametrized-floer} with
$r = 0$ or $1$. For $r = 0$, \eqref{eq:parametrized-floer} is
\begin{equation} \label{eq:r-is-zero}
\partials{u} + J_t(u)\left(\partialt{u} - X_H(t,u)\right) = 0
\end{equation}
and for $r = 1$, it is
\begin{equation} \label{eq:r-is-one}
\partials{u} + J_{s,t}'(u)\left(\partialt{u} - X_{H'}(s,t,u)\right) = 0.
\end{equation}
Since $(H,\mathbf{J})$ is a regular pair and $(H',\mathbf{J}')$ a regular homotopy,
this implies that $\partial\mathcal{M}^h(c_-,c_+;H,\mathbf{J}) = \emptyset$ if
$\mu_H(c_+) = \mu_H(c_-) + 1$.
\begin{lemma} \label{th:parametrized-compactness}
Let $(\bar{H},\bar{\mathbf{J}})$ be a regular deformation of homotopies
and $c_- \in \mathrm{Crit}_{k_-}(a_H)$, $c_+ \in \mathrm{Crit}_{k_+}(a_H)$.
\begin{enumerate}
\item \label{item:zero-d}
If $k_+ = k_- + 1$, $\modh{c_-}{c_+}$ is a finite set.
\item If $k_+ = k_-$, $\modh{c_-}{c_+}$ is one-dimensional,
and there is a smooth compactification $\cmodh{c_-}{c_+}$ whose boundary
$\partial\cmodh{c_-}{c_+}$ is the disjoint union of
$\partial\modh{c_-}{c_+}$,
\begin{equation} \label{eq:comp-component-1}
\modh{c_-}{c} \times (\mathcal{M}(c,c_+;H,\mathbf{J})/\mathbb{R})
\end{equation}
for all $c \in \mathrm{Crit}_{k_+ + 1}(a_H)$ and
\begin{equation} \label{eq:comp-component-2}
(\mathcal{M}(c_-,c';H,\mathbf{J})/\mathbb{R}) \times \modh{c'}{c_+}
\end{equation}
for all $c' \in \mathrm{Crit}_{k_+ - 1}(a_H)$. \end{enumerate}
\end{lemma}
This compactification is constructed in the same way as that used in
\cite[Lemma 6.3]{salamon-zehnder92}. We omit the proof.
The fact that the spaces $\modh{c_-}{c_+}$ of
dimension $\leq 1$ can be compactified without including limit points
with `bubbles' is a consequence of the regularity
condition (compare \cite[Theorem 5.2]{hofer-salamon95}).
The limits \eqref{eq:comp-component-2} arise in the following way:
let $(r_m,u_m)$ be a sequence in $\modh{c_-}{c_+}$ with
$\lim_m r_m = r$ and such that $u_m$ converges uniformly on compact
subsets to some $u_\infty \in C^\infty(\mathbb{R} \times S^1,M)$. Assume that
there are $s_m \in \mathbb{R}$, $\lim_m s_m = \infty$, such that the
translates $\hat{u}_m(s,t) = u_m(s-s_m,t)$
converge on compact subsets to a map $\hat{u}_\infty$ with
$\partial\hat{u}_\infty/\partial s \not\equiv 0$.
$\hat{u}_\infty$ is a solution of
\[
\partials{\hat{u}_\infty} + J_t^{g_r}(\hat{u}_\infty)\left(
\partialt{\hat{u}_\infty} - X_{H^{g_r}}(t,\hat{u}_\infty)\right) = 0.
\]
Therefore $\hat{v}_\infty(s,t) = g_{r,t}(\hat{u}_\infty(s,t))$ defines
a solution of \eqref{eq:r-is-zero}. $(\hat{v}_\infty,(r,u_\infty))$ is
the limit of the sequence $u_m$ in \eqref{eq:comp-component-2}.
The points \eqref{eq:comp-component-1} in the compactification
arise by a similar, but simpler process using translations with
$s_m \longrightarrow -\infty$.
\begin{defn} \label{def:chain-homotopy} Let $(\bar{H},\bar{\mathbf{J}})$ be
a regular deformation of homotopies. For $k \in \mathbb{Z}$, define
\begin{align*}
h_k(\bar{H},\bar{\mathbf{J}}): CF_k(H) &\longrightarrow CF_{k+1}(H),\\
h_k(\bar{H},\bar{\mathbf{J}})(\gen{c_-}) &=
\sum_{c_+} \#\modh{c_-}{c_+} \gen{c_+}.
\end{align*} \end{defn}
To show that this formal sum is indeed an element of $CF_*(H,\mathbf{J})$,
a slightly stronger version of Lemma
\ref{th:parametrized-compactness}\ref{item:zero-d}
is necessary; we omit the details.
\begin{lemma} \label{th:it-is-a-chain-homotopy} For all $k$,
\begin{multline*}
\partial_{k+1}(H,\mathbf{J}) h_k(\bar{H},\bar{\mathbf{J}}) + h_{k-1}(\bar{H},\bar{\mathbf{J}})
\partial_k(H,\mathbf{J}) =\\
= \Phi_k(H',\mathbf{J}')CF_k(g_1,\tilde{g}_1;H,\mathbf{J},H^{g_1})^{-1} - \mathit{Id}_{CF_k(H)}.
\end{multline*}
\end{lemma}
\proof By definition,
\begin{multline*}
\Phi_k(H',\mathbf{J}') CF_k(g_1,\tilde{g}_1;H,\mathbf{J},H^{g_1})^{-1} \gen{c_-} =\\
= \sum_{c_+} \#\mathcal{M}^{\Phi}(\tilde{g}_1^{-1}(c_-),c_+;H',\mathbf{J}')\gen{c_+}.
\end{multline*}
Comparing \eqref{eq:r-is-one} with \eqref{eq:continuation},
one sees that $u \in \mathcal{M}^\Phi(\tilde{g}_1^{-1}(c_-),c_+;H',\mathbf{J}')$
iff $(1,u) \in \modh{c_-}{c_+}$. The other part of
$\partial\modh{c_-}{c_+}$ is given by solutions of \eqref{eq:r-is-zero}
with limits $c_\pm$. If $c_-,c_+$ have the same Conley-Zehnder index,
any regular solution of \eqref{eq:r-is-zero} is stationary in the sense
that $\partial u/\partial s = 0$, and therefore
this part of $\partial\modh{c_-}{c_+}$ is empty unless
$c_- = c_+$, in which case it contains a single point. It follows that
\begin{multline*}
\left(\Phi_k(H',\mathbf{J}')CF_k(g_1,\tilde{g}_1;H,\mathbf{J},H^{g_1})^{-1} - \mathit{Id}\right)
\gen{c_-} = \\
= \sum_{c_+} \#\partial\modh{c_-}{c_+} \gen{c_+}
\end{multline*}
(the sign is irrelevant since we use $\mathbb{Z}/2$-coefficients).
Similarly, the coefficients of $\gen{c_+}$ in
$\partial_{k+1}(H,\mathbf{J}) h_k(\bar{H},\bar{\mathbf{J}})\gen{c_-}$ and
$h_{k-1}(\bar{H},\bar{\mathbf{J}}) \partial_k(H,\mathbf{J})\gen{c_-}$ are
given by the number of points in
\eqref{eq:comp-component-1} and \eqref{eq:comp-component-2}.
The statement now follows from the
fact that $\partial\cmodh{c_-}{c_+}$ contains an even number of points. \qed
Lemma \ref{th:it-is-a-chain-homotopy} shows that
$\Phi_*(H',\mathbf{J}') CF_*(g_1,\tilde{g}_1;H,\mathbf{J},H^{g_1})^{-1}$
is chain homotopic to the identity, which implies
\eqref{eq:htp-two}.
\newcommand{\moduli^{\mathrm{PP}}(c_-,c_0,c_+;H,\J)}{\mathcal{M}^{\mathrm{PP}}(c_-,c_0,c_+;H,\mathbf{J})}
\newcommand{\moduli^{\mathrm{PP}}(c_-,c_0,c_+;H^g,\J^g)}{\mathcal{M}^{\mathrm{PP}}(c_-,c_0,c_+;H^g,\mathbf{J}^g)}
\newcommand{\ggmodpp}{\mathcal{M}^{\mathrm{PP}}(\tilde{g}(c_-),c_0,
\tilde{g}(c_+);H,\mathbf{J})}
\newcommand{\perturb}[6]{\mathcal{P}(#1,#2,#3,#4,#5,#6)}
\newcommand{\perturb{H^-}{\J^-}{H^0}{\J^0}{H^+}{\J^+}}{\perturb{H^-}{\mathbf{J}^-}{H^0}{\mathbf{J}^0}{H^+}{\mathbf{J}^+}}
\newcommand{\frac{1}{4}}{\frac{1}{4}}
\section{Pair-of-pants product \label{sec:pair-of-pants}}
The main result of this section, Proposition \ref{th:pair-of-pants-linear},
describes the behaviour of the pair-of-pants product on Floer homology
under the maps $HF_*(g,\tilde{g})$. It turns out that $HF_*(g,\tilde{g})$ does
not preserve the ring structure; rather, it is an automorphism of
$HF_*\mo$ as a module over itself.
The definition of the pair-of-pants product for weakly monotone symplectic
manifolds can be found in {\cite{piunikhin-salamon-schwarz94}} or \cite[Chapter 10]{mcduff-salamon}
(a detailed construction in the case $[\o]|\pi_2(M) = 0$ is given in
\cite{schwarz95}). We need to modify this definition slightly by enlarging
the class of admissible perturbations. This would be necessary in any case
to adapt it to the use of time-dependent almost complex structures;
a further enlargement ensures that the relevant moduli spaces
transform nicely under (a subset of) $\widetilde{G}$.
We now sketch this modified definition.
Consider the punctured surface
$\Sigma = \mathbb{R} \times S^1 \setminus (0;0)$. It has a third tubular end
\[
e: \mathbb{R}^- \times S^1 \longrightarrow \Sigma, \quad
e(s,t) = (\frac{1}{4}e^{2\pi s}\cos(2\pi t),
\frac{1}{4}e^{2\pi s}\sin(2\pi t))
\]
around the puncture, and the surface $\widehat{\Sigma}$ obtained by
capping off this end with a disc can be identified with $\mathbb{R} \times S^1$.
For $u \in C^\infty(\Sigma,M)$ we will denote $u \circ e$ by $u^e$.
Two maps $u \in C^\infty(\Sigma,M)$ and $v_0 \in C^\infty(D^2,M)$ which
satisfy $\lim_{s \rightarrow -\infty} u^e(s,\cdot) = v_0|\partial D^2$
can be glued together to a map $u \# v_0: \widehat{\Sigma}
\longrightarrow M$. By identifying $\widehat{\Sigma} = \mathbb{R} \times S^1$,
this gives a path $\mathbb{R} \longrightarrow \Lambda M$. On the two
remaining ends, $u \# v_0$ has the same asymptotic behaviour as $u$.
In particular, if there are $x_-, x_+ \in \mathcal{L}M$ such that
$\lim_{s \rightarrow \pm\infty} u(s,\cdot) = x_\pm$, the path
$u \# v_0$ lies in $\mathcal{L}M$ and converges to $x_-$, $x_+$.
\begin{defn} We say that $u \in C^\infty(\Sigma,M)$ {\em converges
to} $c_-,c_0,c_+ \in \widetilde{\loops}$ if
\[
\lim_{s \rightarrow \pm\infty} u(s,\cdot) = p(c_\pm),
\quad \lim_{s \rightarrow -\infty} u^e(s,\cdot) = p(c_0)
\]
and if for $c_0 = [v_0,x_0]$, the path $u \# v_0: \mathbb{R} \longrightarrow
\mathcal{L}M$ has a lift $\widetilde{u \# v_0}: \mathbb{R} \longrightarrow \widetilde{\loops}$
with limits $c_-,c_+$. \end{defn}
This is independent of the choice of the representative $(v_0,x_0)$ for
$c_0$ and of the choices involved in the gluing.
Let $(H^-,\mathbf{J}^-)$, $(H^0,\mathbf{J}^0)$ and $(H^+,\mathbf{J}^+)$ be regular pairs.
Choose $\mathbf{J} = \! (J_z)_{z \in \Sigma} \in \modJreg{\Sigma}$ and
$H \in C^\infty(\Sigma \times M,\mathbb{R})$ with $H|e([-1;0] \times S^1) \times M
= 0$, such that $(H,\mathbf{J})$ agrees with one regular pair over each end
of $\Sigma$, more precisely:
\begin{align*}
& H(-s,t,y) = H^-(t,y), \; J_{(-s,t)} = J_t^-, \\
& H(s,t,y) = H^+(t,y), \; J_{(s,t)} = J_t^+, \\
& H(e(-s,t),y) = H^0(t,y), \; J_{e(-s,t)} = J_t^0
\end{align*}
for $s \geq 2$, $t \in S^1$ and $y \in M$. The space of such
$(H,\mathbf{J})$ will be denoted by $\perturb{H^-}{\mathbf{J}^-}{H^0}{\mathbf{J}^0}{H^+}{\mathbf{J}^+}$.
$u \in C^\infty(\Sigma,M)$ is called $(H,\mathbf{J})$-holomorphic if
\[
\partials{u}(s,t) + J_{s,t}(u(s,t))\left(
\partialt{u}(s,t) - X_H(s,t,u(s,t))\right)
= 0
\]
for $(s,t) \in \Sigma\setminus e((-\infty;-1] \times S^1)$ and
\[
\partials{u^e}(s,t) + J_{e(s,t)}(u^e(s,t)) \left(
\partialt{u^e}(s,t) - X_H(e(s,t),u^e(s,t))\right) = 0
\]
for $(s,t) \in \mathbb{R}^- \times S^1$.
For $z = (s,t) \in e([-1;0] \times S^1)$, the first equation be written
as $\bar{\partial}_{J_z}u(z) = 0$ using the complex structure
on $\Sigma \iso (\mathbb{C}/i\mathbb{Z}) \setminus \! (0;0)$. Similarly, the second equation
is $\bar{\partial}_{J_{e(s,t)}}u^e(s,t) = 0$
for $(s,t) \in [-1;0] \times S^1$. Since $e$ is holomorphic, this
implies that the two equations match up smoothly.
\begin{defn} For $c_- \in \mathrm{Crit}(a_{H^-}), c_0 \in \mathrm{Crit}(a_{H^0})$
and $c_+ \in \mathrm{Crit}(a_{H^+})$, $\moduli^{\mathrm{PP}}(c_-,c_0,c_+;H,\J)$ is the space of
$(H,\mathbf{J})$-holomorphic maps which converge to $c_-,c_0,c_+$
in the sense discussed above. \end{defn}
The linearization of the two equations for an $(H,\mathbf{J})$-holomorphic map at
$u \in \moduli^{\mathrm{PP}}(c_-,c_0,c_+;H,\J)$ is given by single differential operator
$D^\Sigma_{H,\mathbf{J}}(u)$ on $\Sigma$ which becomes a Fredholm operator
in suitable Sobolev spaces. Following the same method as in
the previous sections, we call $(H,\mathbf{J})$ regular if all these
operators are onto and if for all $(H,\mathbf{J})$-holomorphic maps $u$ with
$\mathrm{ind }(D^\Sigma_{H,\mathbf{J}}(u)) \leq 1$ and all $z \in \Sigma$,
$(z,u(z)) \notin V_0(\mathbf{J})$. A transversality result analogous to Theorem
\ref{th:transv-floer} shows that the set of regular $(H,\mathbf{J})$ is dense in
$\perturb{H^-}{\mathbf{J}^-}{H^0}{\mathbf{J}^0}{H^+}{\mathbf{J}^+}$. If $(H,\mathbf{J})$ is
regular and $\mu_{H^+}(c_+) = \mu_{H^-}(c_-) + \mu_{H^0}(c_0) - 2n$,
the space $\moduli^{\mathrm{PP}}(c_-,c_0,c_+;H,\J)$ is finite. The pair-of-pants product
\[
PP_{i,j}(H,\mathbf{J}): CF_{i}(H^-) \otimes CF_{j}(H^0)
\longrightarrow CF_{i+j-2n}(H^+)
\]
for regular $(H,\mathbf{J})$ is defined by
\[
PP_{i,j}(H,\mathbf{J})(\gen{c_-} \otimes \gen{c_0}) = \sum_{c_+} \#\moduli^{\mathrm{PP}}(c_-,c_0,c_+;H,\J) \gen{c_+}.
\]
A compactness theorem shows that this formal sum lies in $CF_*(H^+)$.
The grading is a consequence of our convention for
the Conley-Zehnder index.
The construction of the product is completed in the following
steps: using the spaces $\mathcal{M}^{\mathrm{PP}}$ of dimension $1$ and their
compactifications, one shows that $PP_*(H,\mathbf{J})$ is a chain homomorphism.
The induced maps
\begin{equation} \label{eq:ppp}
PP_*(H,\mathbf{J}): HF_*(H^-,\mathbf{J}^-) \times HF_*(H^0,\mathbf{J}^0) \longrightarrow
HF_*(H^+,\mathbf{J}^+)
\end{equation}
are independent of the choice of $(H,\mathbf{J})$ (the starting point for
the proof is that $\perturb{H^-}{\mathbf{J}^-}{H^0}{\mathbf{J}^0}{H^+}{\mathbf{J}^+}$
is contractible). Finally, a gluing argument proves that the
maps \eqref{eq:ppp} for different regular pairs are related by
continuation isomorphisms. Therefore they define a unique product
\[
\qp_{\mathrm{PP}}: HF_*(M,\o) \times HF_*(M,\o) \longrightarrow HF_*(M,\o).
\]
Using the assumption {$(W^+)$}, the proofs of these properties
for the product as defined in {\cite{piunikhin-salamon-schwarz94}} can be easily adapted to
our slightly different setup.
\begin{prop} \label{th:pair-of-pants-linear}
For all $(g,\tilde{g}) \in \widetilde{G}$ and $a,b \in HF_*(M,\o)$,
\[
HF_*(g,\tilde{g})(a \qp_{\mathrm{PP}} b) =
HF_*(g,\tilde{g})(a) \qp_{\mathrm{PP}} b.
\]
\end{prop}
Because of the homotopy invariance of $HF_*(g,\tilde{g})$
(Proposition \ref{th:homotopy}) it is sufficient to consider
the case where $g_t = \mathit{Id}_M$ for $t \in [-\frac{1}{4};\frac{1}{4}] \subset S^1$
(clearly, any path component of $\widetilde{G}$ contains a $(g,\tilde{g})$
with this property). Choose a Hamiltonian $K_g$ which generates $g$
such that $K_g(t,\cdot) = 0$ for $t \in [-\frac{1}{4};\frac{1}{4}]$.
Let $((H^{\pm})^g,(\mathbf{J}^{\pm})^g)$ be the pullback of $(H^\pm,\mathbf{J}^\pm)$ using
$K_g$. The proof of Proposition \ref{th:pair-of-pants-linear}
relies on the following analogue of the `pullback' of a regular
pair: for $(H,\mathbf{J}) \in \perturb{H^-}{\J^-}{H^0}{\J^0}{H^+}{\J^+}$, define $(H^g,\mathbf{J}^g)$ by
\[
J^g_{(s,t)} = Dg_t^{-1} J_{(s,t)} Dg_t, \quad
H^g(s,t,y) = H(s,t,g_t(y)) - K_g(t,g_t(y))
\]
for $(s,t) \in \Sigma$, $y \in M$. Because $g_t = \mathit{Id}_M$ for
$t \in [-\frac{1}{4};\frac{1}{4}]$ and
\[
\mathrm{im}(e) \subset (\mathbb{R} \times [-1/4;1/4]) \setminus (0;0)
\subset \Sigma,
\]
$(H^g,\mathbf{J}^g)$ satisfies
$H^g(e(s,t),\cdot) = H(e(s,t),\cdot)$ and $J^g_{e(s,t)} = J_{e(s,t)}$ for
all $(s,t) \in \mathbb{R}^- \times S^1$. In particular, $H^g$ vanishes on
$e([-1;0] \times S^1) \times M$. It follows that
$(H^g,\mathbf{J}^g) \in \perturb{(H^-)^g}{(\mathbf{J}^-)^g}{H^0}{\mathbf{J}^0}{(H^+)^g}{(\mathbf{J}^+)^g}$.
The next Lemma is the analogue of Lemma \ref{th:functoriality}:
\begin{lemma} \label{th:pp-functoriality} For all
$(c_-,c_0,c_+) \in \mathrm{Crit}(a_{(H^-)^g}) \times \mathrm{Crit}(a_{H^0})
\times \mathrm{Crit}(a_{(H^+)^g})$, there is a bijective map
\[
\moduli^{\mathrm{PP}}(c_-,c_0,c_+;H^g,\J^g) \longrightarrow \ggmodpp.
\]
Moreover, if $(H,\mathbf{J})$ is regular, so is $(H^g,\mathbf{J}^g)$.
\end{lemma}
\proof A straightforward computation shows that
if $u,v \in C^\infty(\Sigma,M)$ are related by $u(s,t) = g_t(v(s,t))$,
$u$ is $(H,\mathbf{J})$-holomorphic iff $v$ is $(H^g,\mathbf{J}^g)$-holomorphic
(note that $v^e = u^e$, and that the second equation
is the same in both cases). Now assume that $v$ converges
to $c_-,c_0,c_+$ as defined above, and choose a
representative $(v_0,x_0)$ of $c_0$. The maps
$u \# v_0, v \# v_0: \mathbb{R} \times S^1 \longrightarrow M$
can be chosen such that
\[
(u \# v_0)(s,t) = g_t((v \# v_0)(s,t)).
\]
By assumption, the path $\mathbb{R} \longrightarrow \mathcal{L}M$ given by
$v \# v_0$ has a lift $\widetilde{v \# v_0}: \mathbb{R} \longrightarrow
\widetilde{\loops}$ with limits $c_-,c_+$. Define
\[
\widetilde{u \# v_0}(s) = \tilde{g}(\widetilde{v \# v_0}(s)).
\]
Clearly, $\widetilde{u \# v_0}$ is a lift of $u \# v_0$ with
limits $\tilde{g}(c_-)$, $\tilde{g}(c_+)$. Therefore $u$ converges
to $\tilde{g}(c_-), c_0, \tilde{g}(c_+)$. The proof of regularity
is similar to that of Lemma \ref{th:naturality-of-regularity}; we omit
the details. \qed
From Lemma \ref{th:pp-functoriality} and the definition
of $PP_*(H,\mathbf{J})$, it follows that
\begin{multline*}
CF_*(g,\tilde{g};H^+,\mathbf{J}^+,(H^+)^g) PP_*(H^g,\mathbf{J}^g)(\gen{c_-}
\otimes \gen{c_0}) =\\
= PP_*(H,\mathbf{J})(\gen{\tilde{g}(c_-)} \otimes \gen{c_0})
\end{multline*}
for every regular $(H,\mathbf{J}) \in \perturb{H^-}{\J^-}{H^0}{\J^0}{H^+}{\J^+}$,
$c_- \in \mathrm{Crit}(a_{(H^-)^g})$ and $c_0 \in \mathrm{Crit}(a_{H^0})$.
Proposition \ref{th:pair-of-pants-linear} follows directly
from this.
\newcommand{\mathcal{J}(E,\O)}{\mathcal{J}(E,\O)}
\newcommand{\mathcal{J}^{\mathrm{reg}}(E,\O)}{\mathcal{J}^{\mathrm{reg}}(E,\O)}
\newcommand{\mathcal{J}^{\mathrm{reg},z_0}(E,\O)}{\mathcal{J}^{\mathrm{reg},z_0}(E,\O)}
\newcommand{\mathcal{C}}{\mathcal{C}}
\newcommand{\cu_{r,k}(J)}{\mathcal{C}_{r,k}(J)}
\newcommand{\widehat{\mathcal{J}}(j,\J)}{\widehat{\mathcal{J}}(j,\mathbf{J})}
\newcommand{\widehat{\mathcal{J}}^{\mathrm{reg}}(j,\J)}{\widehat{\mathcal{J}}^{\mathrm{reg}}(j,\mathbf{J})}
\newcommand{\mathcal{S}(j,\hatj)}{\mathcal{S}(j,\hat{J})}
\newcommand{D^{\mathrm{univ}}}{D^{\mathrm{univ}}}
\newcommand{\mathcal{U}}{\mathcal{U}}
\section{Pseudoholomorphic sections \label{sec:holomorphic-sections}}
Let $(E,\O)$ be a symplectic fibre bundle over $S^2$ with fibre
$(M,\o)$ and $\pi: E \longrightarrow S^2$ the projection.
We will denote by $\mathcal{J}(E,\O)$ the space of families
$\mathbf{J} = (J_z)_{z \in S^2}$ of almost complex
structures on the fibres of $E$ such that $J_z$ is $\O_z$-compatible
for all $z$. For $\mathbf{J} \in \mathcal{J}(E,\O)$ and $k \in \mathbb{Z}$, let $\mathcal{M}_k^s(\mathbf{J})$
be the space of pairs $(z,w) \in S^2 \times C^\infty(\CP{1},E)$ such that
$w$ is a simple $J_z$-holomorphic curve in $E_z$ with
$c_1(TE_z,\O_z)(w) = k$. Because $(E,\O)$ is locally trivial, these spaces
have the same properties as the spaces of holomorphic curves in $M$ with
respect to a two-parameter family of almost complex structures (see
section \ref{sec:floer-homology}). In particular, there is a dense subset
$\mathcal{J}^{\mathrm{reg}}(E,\O) \subset \mathcal{J}(E,\O)$ such that\footnote{This uses the
assumption {$(W^+)$}; again, weak monotonicity would not be sufficient.}
for $\mathbf{J} \in \mathcal{J}^{\mathrm{reg}}(E,\O)$, $\mathcal{M}_k^s(\mathbf{J}) = \emptyset$ for all $k<0$
and $\mathcal{M}_0^s(\mathbf{J})$ is a manifold of dimension
$\dim \mathcal{M}_0^s(\mathbf{J}) = \dim E$. It follows that the image of the
evaluation map
\[
\eta: \mathcal{M}_0^s(\mathbf{J}) \times_{PSL(2,\mathbb{C})} \CP{1} \longrightarrow E
\]
is a subset of codimension $4$.
Following the convention of section \ref{sec:elementary}, $(E,\O)$
is equipped with a preferred isomorphism $i: (M,\o) \longrightarrow
(E_{z_0},\O_{z_0})$ for the marked point $z_0 \in S^2$. We need
to recall the transversality theory of cusp-curves; a reference
is \cite[Chapter 6]{mcduff-salamon}.
\begin{defn} Let $J$ be an $\o$-compatible almost complex structure
on $M$. A simple $J$-holomorphic cusp-curve with $r \geq 1$ components is a
\[
v = (w_1,\dots w_r, t_1,\dots t_r, t_1',\dots t_r')
\in C^\infty(\CP{1},M)^r \times (\CP{1})^{2r}
\]
with the following properties:
\begin{enumerate} \item For $i = 1 \dots r$,
$w_i$ is a simple $J$-holomorphic curve;
\item $\mathrm{im}(w_i) \neq \mathrm{im}(w_j)$ for $i \neq j$;
\item $w_i(t_i) = w_{i+1}(t_{i+1}')$ for $i = 1 \dots r-1$.
\end{enumerate} \end{defn}
The Chern number of $v$ is defined by $c_1(v) = \sum_i c_1(w_i)$.
$PSL(2,\mathbb{C})^r$ acts freely on the space of simple cusp-curves
with $r$ components by
\begin{multline*}
(a_1 \dots a_r) \cdot (w_1,\dots w_r, t_1,\dots t_r, t_1',\dots t_r') =\\
(w_1 \circ a_1^{-1},\dots w_r \circ a_r^{-1}, a_1(t_1),\dots
a_r(t_r), a_1(t_1'),\dots a_r(t_r')).
\end{multline*}
Let $\cu_{r,k}(J)$ be the quotient of the space of cusp-curves with $r$
components and Chern number $k$ by this action, and
\[
\eta_1, \eta_2: \cu_{r,k}(J) \longrightarrow M
\]
the maps given by $\eta_1(v) = w_1(t_1')$, $\eta_2(v) = w_r(t_r)$ for
$v$ as above. For a generic $J$, $\cu_{r,k}(J)$ is a smooth manifold of
dimension $2n + 2k - 2r$ \cite[Theorem 5.2.1(ii)]{mcduff-salamon}.
Let $\mathcal{J}^{\mathrm{reg},z_0}(E,\O) \subset \mathcal{J}^{\mathrm{reg}}(E,\O)$ be the subset of families
$\mathbf{J} = (J_z)_{z \in S^2}$ such that $J = Di^{-1} J_{z_0} Di$ has this
regularity property; $\mathcal{J}^{\mathrm{reg},z_0}(E,\O)$ is dense in $\mathcal{J}(E,\O)$.
Let $j$ be a positively oriented complex structure on $S^2$ and
$\mathbf{J} \in \mathcal{J}(E,\O)$. We call an almost complex structure $\hatj$ on $E$
compatible with $j$ and $\mathbf{J}$ if $D\pi \circ \hat{J} = j \circ D\pi$ and
$\hat{J}|E_z = J_z$ for all $z \in S^2$. The space of such $\hat{J}$ will be
denoted by $\widehat{\mathcal{J}}(j,\J)$. If we fix a $\hat{J}_0 \in \widehat{\mathcal{J}}(j,\J)$, any other
such $\hat{J}$ is of the form $\hat{J}_0 + \theta \circ D\pi$,
where $\theta: \pi^*(TS^2,j) \longrightarrow (TE^v,\mathbf{J})$
is a smooth $\mathbb{C}$-antilinear vector bundle homomorphism
(recall that $TE^v = \ker D\pi$ is the tangent bundle along the fibres);
conversely, $\hat{J}_0 + \theta \circ D\pi \in \widehat{\mathcal{J}}(j,\J)$ for any such $\theta$.
For $j,\mathbf{J}$ as above and $\hat{J} \in \widehat{\mathcal{J}}(j,\J)$, we call a smooth section
$s: S^2 \longrightarrow E$ of $\pi$ {\em$(j,\hat{J})$-holomorphic} if
\begin{equation} \label{eq:holomorphic-section}
ds \circ j = \hatj \circ ds.
\end{equation}
If $E = S^2 \times M$ and $J_z$ is independent of
$z$, this is the `inhomogeneous Cauchy-Riemann equation' of
\cite{ruan-tian94} for maps $S^2 \longrightarrow M$
with an inhomogeneous term determined by the choice of $\hat{J}$.
Let $\mathcal{S}(j,\hatj)$ be the space of $(j,\hatj)$-holomorphic sections of $E$.
For every section $s$, $\mathbf{J}$ induces an almost complex structure
$s^*\mathbf{J}$ on $s^*TE^v$, given by $(s^*\mathbf{J})_z = (J_z)_{s(z)}$.
The linearization of \eqref{eq:holomorphic-section} at $s \in \mathcal{S}(j,\hatj)$
is a differential operator
\[
D_{\hatj}(s): C^\infty(s^*TE^v) \longrightarrow \Omega^{0,1}(s^*TE^v,s^*\mathbf{J})
\]
on $S^2$. $D_{\hatj}(s)$ differs
from the $\bar{\partial}$-operator of $(s^*TE^v,s^*\mathbf{J})$
by a term of order zero. Therefore it is a Fredholm operator with index
$d(s) = 2n + 2c_1(TE^v,\O)(s)$. If $D_{\hatj}(s)$ is onto,
$\mathcal{S}(j,\hatj)$ is a smooth $d(s)$-dimensional manifold near $s$.
\begin{defn} Let $j$ be a positively oriented complex structure on $S^2$,
$\mathbf{J} = (J_z)_{z \in S^2} \in \mathcal{J}^{\mathrm{reg},z_0}(E,\O)$ and $J$ the almost complex
structure on $M$ given by $J = Di^{-1} J_{z_0} Di$.
$\hat{J} \in \widehat{\mathcal{J}}(j,\J)$ is called regular if
$D_{\hatj}(s)$ is onto for all $s \in \mathcal{S}(j,\hatj)$, the map
\[
\mathrm{ev}: S^2 \times \mathcal{S}(j,\hatj) \longrightarrow E, \; \mathrm{ev}(z,s) = s(z)
\]
is transverse to $\eta$ and the map
\[
\mathrm{ev}_{z_0}: \mathcal{S}(j,\hatj) \longrightarrow M, \; \mathrm{ev}_{z_0}(s) = i^{-1}(s(z_0))
\]
is transverse to $\eta_1$. \end{defn}
\begin{prop} The subspace $\widehat{\mathcal{J}}^{\mathrm{reg}}(j,\J) \subset \widehat{\mathcal{J}}(j,\J)$
of regular $\hatj$ is $C^\infty$-dense.
\end{prop}
\proof[Proof (sketch)]
The strategy of the proof is familiar. The tangent space of
$\widehat{\mathcal{J}}(j,\J)$ at any point is a subspace of the space of bundle homomorphisms
$I: TE \longrightarrow TE$. As explained above, it contains precisely those
$I$ which can be written as $I = \theta \circ D\pi$, where
$\theta: \pi^*(TS^2,j) \longrightarrow (TE^v,\mathbf{J})$ is
a $\mathbb{C}$-antilinear homomorphism.
Let $\mathcal{U}$ be the space of such $\theta$ whose $\smooth_{\epsilon}$-norm is finite.
Consider first the condition that $D_{\hatj}(s)$ is onto. The main step
in the proof is to show that
\begin{align*}
D^{\mathrm{univ}}(s,\hatj): C^\infty(s^*TE^v) \times \mathcal{U} &\longrightarrow
\Omega^{0,1}(s^*TE^v,s^*\mathbf{J}),\\
D^{\mathrm{univ}}(s,\hatj)(S,\theta) &= D_{\hatj}(s)S + \frac12 (\theta \circ D\pi)
\circ ds \circ j
\end{align*}
is onto for all $\hatj \in \widehat{\mathcal{J}}(j,\J)$ and $s \in \mathcal{S}(j,\hatj)$.
$D\pi \circ ds = \mathit{Id}$ and therefore
$D^{\mathrm{univ}}(s,\hatj)(0,\theta)_z = \frac12 \theta(s(z)) \circ j$. Because
$s: S^2 \longrightarrow E$ is an embedding, this means that
$D^{\mathrm{univ}}(s,\hatj)(0 \times \mathcal{U})$ is dense in $\Omega^{0,1}(s^*TE^v,s^*\mathbf{J})$.
Since $D_{\hatj}(s)$ is a Fredholm operator, it follows that
$D^{\mathrm{univ}}(s,\hatj)$ is onto.
Now consider the condition that $\mathrm{ev}$ is transverse to $\eta$. The
proof does not use any specific properties of $\eta$; for any
smooth map $c: C \longrightarrow E$, the space of $\hatj$ such that
$\mathrm{ev}$ is transverse to $c$ is dense in $\widehat{\mathcal{J}}(j,\J)$. As in
\cite[Section 6.1]{mcduff-salamon}, this follows from the fact
that the evaluation map on a suitable `universal moduli space'
is a submersion. The main step is to prove that the operator
\begin{align*}
T_zS^2 \times C^\infty(s^*TE^v) \times \mathcal{U} &
\longrightarrow T_{s(z)}E \times \Omega^{0,1}(s^*TE^v,s^*\mathbf{J}),\\
(\xi, S, \theta) &\longmapsto (ds(z)\xi + S(z), D^{\mathrm{univ}}(s,\hatj)(S,\theta))
\end{align*}
is onto for all $\hatj \in \widehat{\mathcal{J}}(j,\J)$, $s \in \mathcal{S}(j,\hatj)$ and $z \in S^2$.
This is slightly stronger than the surjectivity of $D^{\mathrm{univ}}$,
but it can be proved by the same argument. The proof of the
transversality condition for $\mathrm{ev}_{z_0}$ is similar. \qed
From now on, we assume that $(E,\O)$ is Hamiltonian. Let
$\widetilde{\O} \in \Omega^2(E)$ be a closed form with $\widetilde{\O}|E_z = \O_z$
for all $z$.
\begin{lemma} \label{th:thurston} Let $\hat{J}$ be an almost complex
structure on $E$ such that $\hat{J} \in \widehat{\mathcal{J}}(j,\J)$ for some $j,\mathbf{J}$.
There is a two-form $\sigma$ on $S^2$ such that $\widetilde{\O} +
\pi^*\sigma$ is a symplectic form on $E$ which tames $\hat{J}$. \qed
\end{lemma}
This is a well-known method for constructing symplectic forms,
due to Thurston \cite{thurston76}. Lemma \ref{th:thurston} makes
it possible to apply Gromov's compactness theorem to $(j,\hat{J})$-holomorphic
sections, since any such section is a $\hat{J}$-holomorphic curve in $E$.
It is easy to see that any `bubble' in the limit of a sequence of
$(j,\hat{J})$-holomorphic sections must be a $J_z$-holomorphic curve in some
fibre $E_z$. As in \cite{ruan-tian94}, we can simplify the limits
by deleting some components and replacing multiply covered curves by
the underlying simple ones. If $\mathbf{J} \in \mathcal{J}^{\mathrm{reg}}(E,\O)$, every $J_z$ is
semi-positive, and the total Chern number does not increase
during the process. The outcome can be summarized as follows:
\begin{lemma} \label{th:bubbling} Let $j$ be a positively oriented
complex structure on $S^2$, $\hat{J}$ an almost complex structure
on $E$ such that $\hat{J} \in \widehat{\mathcal{J}}(j,\J)$ for some $\mathbf{J} \in \mathcal{J}^{\mathrm{reg}}(E,\O)$,
and $J$ the almost complex structure on $M$ which corresponds to $J_{z_0}$
through the isomorphism $i$.
Let $(s_m)_{m \in \N}$ be a sequence in $\mathcal{S}(j,\hatj)$ such that
$\widetilde{\O}(s_m) \leq C$ and $c_1(TE^v,\O)(s_m) \leq c$ for all $m$.
Assume that $(s_m)$ has no convergent subsequence, and that $s_m(z_0)$
converges to $y \in E_{z_0}$. Then one of the following
(not mutually exclusive) possibilities holds:
\begin{enumerate} \item \label{item:cancel-all-bubbles}
there is an $s \in \mathcal{S}(j,\hatj)$ with $c_1(TE^v,\O)(s) < c$ and $s(z_0) = y$;
\item \label{item:bubble-one}
there are $s \in \mathcal{S}(j,\hatj)$, $z \in S^2$ and
$w \in \mathcal{M}^s_0(\mathbf{J}) \times_{PSL(2,\mathbb{C})} \CP{1}$ such that
$c_1(TE^v,\O)(s) = c$, $s(z_0) = y$ and $s(z) = \eta(w)$;
\item \label{item:bubble-two}
there is an $s \in \mathcal{S}(j,\hatj)$ and a $v \in \cu_{r,k}(J)$ with
$k + c_1(TE^v,\O)(s) \leq c$, $i(\eta_1(v)) = s(z_0)$ and
$i(\eta_2(v)) = y$.
\end{enumerate} \end{lemma}
\ref{item:bubble-two} is the case where bubbling occurs at $z_0$
in such a way that $y$ does not lie in the image of the principal
component of the limiting cusp-curve.
We will denote by $\sec(j,\hat{J},S) \subset \mathcal{S}(j,\hatj)$ the space of
$(j,\hat{J})$-holomorphic sections of $E$ which lie in a given
$\Gamma$-equivalence class $S$.
\begin{lemma} \label{th:finite-non-empty}
For every $C \in \mathbb{R}$, there are only finitely
many $\Gamma$-equivalence classes $S$ with $\widetilde{\O}(S) \leq C$
and $\sec(j,\hat{J},S) \neq \emptyset$. \end{lemma}
\proof The stronger statements in which $\Gamma$-equivalence
is replaced by homological equivalence or even homotopy are well-known (see
\cite[Corollary 4.4.4]{mcduff-salamon}) consequences of
Gromov's compactness theorem. \qed
\begin{proposition} \label{th:pseudo-cycle}
If $\hat{J} \in \widehat{\mathcal{J}}^{\mathrm{reg}}(j,\J)$ for some $j$ and $\mathbf{J} \in \mathcal{J}^{\mathrm{reg},z_0}(E,\O)$, the map
\[
\mathrm{ev}_{z_0}: \sec(j,\hat{J},S) \longrightarrow M, \quad
\mathrm{ev}_{z_0}(s) = i^{-1}(s(z_0))
\]
is a pseudo-cycle (in the sense of \cite[Section 7.1]{mcduff-salamon})
of dimension $d(S) = 2n + 2c_1(TE^v,\O)(S)$ for any $\Gamma$-equivalence class
$S$. \end{proposition}
\proof Because $\hat{J}$ is regular, $\sec(j,\hat{J},S)$ is a smooth manifold.
Its dimension is given by the index of $D_{\hatj}(s)$ for any
$s \in \sec(j,\hat{J},S)$, which is $d(S)$. It remains to show that $\mathrm{ev}_{z_0}$
can be compactified by countably many
images of manifolds of dimension $\leq d(S) - 2$.
The compactification described in Lemma \ref{th:bubbling} has this
property: in case \ref{item:cancel-all-bubbles}, $y$ lies in the image of
\[
\mathrm{ev}_{z_0}: \sec(j,\hat{J},S') \longrightarrow M
\]
for some $S'$ with $c_1(TE^v,\O)(S') < c_1(TE^v,\O)(S)$.
Clearly $\dim \sec(j,\hat{J},S') \leq d(S) -2$. Now, consider case
\ref{item:bubble-one}. Let $\sec_0$ be the space of $(j,\hat{J})$-holomorphic
sections $s$ with $c_1(TE^v,\O)(s) = c_1(TE^v,\O)(S)$. The space
of $(s,z,w)$ as in Lemma \ref{th:bubbling}\ref{item:bubble-one}
is the inverse image of the diagonal $\Delta \subset E \times E$ by
the map
\[
\mathrm{ev} \times \eta: (S^2 \times \sec_0) \times
(\mathcal{M}^s_0(\mathbf{J}) \times_{PSL(2,\mathbb{C})} \CP{1}) \longrightarrow E \times E.
\]
$\dim \mathcal{M}^s_0(\mathbf{J}) \times_{PSL(2,\mathbb{C})} \CP{1} = \dim E - 4$,
and $S^2 \times \sec_0 $ is a manifold of dimension $d(S) + 2$.
Since $\hat{J}$ is regular, $\mathrm{ev}$ is transverse to $\eta$; it follows
that $(\mathrm{ev} \times \eta)^{-1}(\Delta)$ has dimension $d(S) - 2$.
Case \ref{item:bubble-two} is similar. \qed
The following way of assigning a homology class to a pseudo-cycle
is due to Schwarz \cite{schwarz96}.
\begin{lemma} \label{th:morse-homology}
Let $c: C \longrightarrow M$ be a $k$-dimensional
pseudo-cycle in $M$ which is compactified by
$c_\infty: C_\infty^{k-2} \longrightarrow M$. If $h$ is a Riemannian
metric on $M$ and $f \in C^\infty(M,\mathbb{R})$ is a Morse function for which
the Morse complex $(CM_*(f),\partial(f,h))$ is defined and
such that stable manifold $W^s(y;f,h)$ (for the negative gradient flow)
of any critical point $y$ of $f$ is transverse to $c,c_\infty$, the
sum
\[
\sum_y \#c^{-1}(W^s(y;f,h))\gen{y}
\]
over all critical points $y$ of Morse index $k$ is a cycle in $CM_*(f)$
(we use Morse complexes with $\mathbb{Z}/2$-coefficients). \qed \end{lemma}
Such a pair $(f,h)$ always exists, and the homology class of the
cycle in the `Morse homology' \cite{schwarz} of $M$ is independent
of all choices. Since Morse homology is canonically isomorphic to
singular homology \cite{schwarz97}, this defines a class in
$H_k(M;\mathbb{Z}/2)$ which we denote by $[c(C)]$. If $C$ is compact,
$[c(C)]$ is the fundamental class in the classical sense.
\begin{defn} \label{def:bigq}
Let $(E,\O,S)$ be a normalized Hamiltonian fibre bundle over
$S^2$ with fibre $(M,\o)$. Let $d = 2n + 2c_1(TE^v,\O)(S)$. We define
\[
Q(E,\O,S) = \sum_{\gamma \in \Gamma}
[\mathrm{ev}_{z_0}(\sec(j,\hat{J},\gamma + S))] \otimes \gen{\gamma}
\; \in QH_d(M,\o),
\]
where $j$ is a positively oriented complex structure on $S^2$
and $\hat{J} \in \widehat{\mathcal{J}}^{\mathrm{reg}}(j,\J)$ for some $\mathbf{J} \in \mathcal{J}^{\mathrm{reg},z_0}(E,\O)$. \end{defn}
The notation $\gamma + S$ was introduced in section \ref{sec:elementary}.
The formal sum defining $Q(E,\O,S)$ lies in $QH_*\mo$ because of Lemma
\ref{th:finite-non-empty}. It is of degree $d$ because the dimension
of $\sec(j,\hat{J},\gamma + S)$ is $d + 2c_1(\gamma)$ and $\gen{\gamma}$
has degree $-2c_1(\gamma)$.
\begin{prop} $Q(E,\O,S)$ is independent of the choice of $j$,
$\mathbf{J}$ and $\hat{J}$. \end{prop}
We omit the proof. It is based on the fact that cobordant pseudo-cycles
determine the same homology class, and is similar to
\cite[Proposition 7.2.1]{mcduff-salamon}.
Sometimes it is convenient to define $QH_*(M,\o)$ in terms of
Morse homology as the homology of the graded tensor
product $(CM_*(f) \otimes \Lambda, \partial(f,h) \otimes \mathit{Id})$.
An element of $CM_*(f) \otimes \Lambda$ is a (possibly infinite)
linear combination of $\gen{y} \otimes \gen{\gamma}$ for
$y \in \mathrm{Crit}(f)$, $\gamma \in \Gamma$.
Assume that $(f,h)$ satisfies the conditions of Lemma \ref{th:morse-homology}
with respect to the pseudo-cycles
$ev_{z_0}: \sec(j,\hat{J},S') \longrightarrow M$ for all $S'$, and define
\begin{equation*}
\sec(j,\hat{J},S',y) = \{ s \in \sec(j,\hat{J},S') \; | \;
\mathrm{ev}_{z_0}(s) \in W^s(y;f,h) \}
\end{equation*}
for $y \in \mathrm{Crit}(f)$. Then $Q(E,\O,S)$ is the homology class
of the cycle
\begin{equation} \label{eq:explicit}
\sum_{y,\gamma} \#\sec(j,\hat{J},\gamma + S,y) \gen{y} \otimes
\gen{\gamma} \in CM_*(f) \otimes \Lambda,
\end{equation}
where the sum is over all $(y,\gamma) \in \mathrm{Crit}(f) \times \Gamma$
such that $i_f(y) = d + 2c_1(\gamma)$ ($i_f$ denotes the Morse index).
It is often difficult to decide whether a given $\hat{J}$ and its
restriction $\mathbf{J} = \hat{J}|TE^v$ are regular, since this depends on
all holomorphic sections and the holomorphic curves in the fibres. Moreover,
in many examples the most natural choice of $\hat{J}$ is not regular.
However, almost complex structures which satisfy a weaker condition
can be used to determine $Q(E,\O,S)$ partially: to
compute $[\mathrm{ev}_{z_0}(\sec(j,\hat{J},\gamma + S))]$ for a single $\gamma$,
it is sufficient that $\sec(j,\hat{J},\gamma + S)$ and all its possible limits
listed in Lemma \ref{th:bubbling} are regular.
Since the contributions from those $\gamma$ with $d + 2c_1(\gamma) < 0$
or $d + 2c_1(\gamma) > 2n$ are zero, sometimes all of
$Q(E,\O,S)$ can be computed using an almost complex structure
which is not regular, as in the following case:
\begin{proposition} \label{th:computation}
Let $(E,J)$ be a compact complex manifold, $\pi: E
\longrightarrow \CP{1}$ a holomorphic map with no critical points,
$\widetilde{\O} \in \Omega^2(E)$ a closed form whose restrictions
$\O_z = \widetilde{\O}|E_z$ are K{\"a}hler forms,
and $i: \! (M,\o) \longrightarrow (E_{z_0},\O_{z_0})$ an
isomorphism for some $z_0 \in \CP{1}$. Assume that
\begin{enumerate}
\item the space $\sec$ of holomorphic sections $s$ of $\pi$
with $c_1(TE^v)(s) \leq 0$ is connected. In particular, all
of them lie in a single $\Gamma$-equivalence class $S_0$.
\item For any $s \in \sec$, $H^{0,1}(\CP{1},s^*TE^v) = 0$.
\item Any holomorphic map $w \in C^\infty(\CP{1},E)$ such that
$\mathrm{im}(w) \subset E_z$ for some $z \in \CP{1}$ satisfies $c_1(TE)(w) \geq 0$;
\item if $w$ is as before and not constant, and
$c_1(TE)(w) + c_1(TE^v)(S_0) \leq 0$,
then $s(z) \notin \mathrm{im}(w)$ for all $s \in \sec$.
\end{enumerate}
In that case, $\sec$ is a smooth compact manifold and
\[
Q(E,\O,S_0) = (\mathrm{ev}_{z_0})_*[\sec] \otimes \gen{0},
\]
where $\gen{0} \in \Lambda$ is the element corresponding to the
trivial class in $\Gamma$.
\end{proposition}
\newcommand{\Psi^+(e)}{\Psi^+(e)}
\newcommand{\mathcal{P}(\gamma)}{\mathcal{P}(\gamma)}
\newcommand{\choices}{\mathcal{C}(H^{\infty},\mathbf{J}^{\infty},
H^{-\infty},\mathbf{J}^{-\infty})}
\newcommand{\mathcal{S}'(H^+,\J^+,H^-,\J^-,\gamma,y)}{\mathcal{S}'(H^+,\mathbf{J}^+,H^-,\mathbf{J}^-,\gamma,y)}
\newcommand{\widehat{\#}_R}{\widehat{\#}_R}
\newcommand{\|}{\|}
\newcommand{?? \mathcal{N}}{?? \mathcal{N}}
\section{A gluing argument \label{sec:gluing}}
\begin{definition} For $(g,\tilde{g}) \in \widetilde{G}$, define
\[
q(g,\tilde{g}) = Q(E_g,\O_g,S_{\tilde{g}}) \in QH_*\mo.
\]
\end{definition}
By definition, $Q(E_g,\O_g,S_{\tilde{g}})$ is an element
of degree $2n + 2c_1(TE_g^v,\O_g)(S_{\tilde{g}})$. Using
\eqref{eq:maslov-and-chern} it follows that
\[
q(g,\tilde{g}) \in QH_{2n - 2I(g,\tilde{g})}(M,\o).
\]
Let $e = [M] \otimes \gen{0}$ be the `fundamental class' in
$QH_*\mo$. The connection between the
invariant $q$ and the $\widetilde{G}$-action on Floer homology is given by
\begin{theorem} \label{th:gluing}
For all $(g,\tilde{g}) \in \widetilde{G}$,
\[
q(g,\tilde{g}) = \Psi^- \, HF_*(g,\tilde{g}) \, \Psi^+(e).
\] \end{theorem}
Here
\[
\Psi^+: QH_*(M,\o) \longrightarrow HF_*(M,\o), \quad
\Psi^-: HF_*(M,\o) \longrightarrow QH_*(M,\o)
\]
are the canonical homomorphisms of Piunikhin, Salamon and Schwarz {\cite{piunikhin-salamon-schwarz94}}.
They proved that $\Psi^+,\Psi^-$ are isomorphisms and inverses of each
other, and a part of their proof will serve as a model for the
proof of Theorem \ref{th:gluing}. More precisely, we consider
the proof of $\mathit{Id}_{QH_*(M,\o)} = \Psi^- \Psi^+$ given in {\cite{piunikhin-salamon-schwarz94}}
and specialize it to the case
\[
e = \Psi^- \Psi^+(e).
\]
There are two steps in proving this: first, a gluing argument shows
that $\Psi^- \Psi^+(e)$ is equal to a certain Gromov-Witten invariant.
Then one computes that the value of this invariant is again $e$.
The Gromov-Witten invariant which arises can be described in our
terms as the $Q$-invariant of the trivial Hamiltonian fibre
bundle $(S^2 \times M,\o)$ with the $\Gamma$-equivalence class $S_0$
containing the constant sections. Therefore, what the gluing
argument in {\cite{piunikhin-salamon-schwarz94}} proves is
\begin{equation} \label{eq:pss}
Q(S^2 \times M, \omega, S_0) = \Psi^- \Psi^+(e).
\end{equation}
This is precisely the special case $(g,\tilde{g}) = \mathit{Id}_{\widetilde{G}}$ of
Theorem \ref{th:gluing}, since in that case $(E_g,\O_g,S_{\tilde{g}}) =
(S^2 \times M,\o,S_0)$ and $HF_*(g,\tilde{g})$
is the identity map. The analytical details
of the proof of Theorem \ref{th:gluing} are practically
the same as in the special case \eqref{eq:pss}; the
only difference is that the gluing procedure is modified
by `twisting' with $g$. Since a detailed account of the results
of {\cite{piunikhin-salamon-schwarz94}} based on the analysis in \cite{schwarz95} is in
preparation, we will only sketch the proof of Theorem \ref{th:gluing}.
The basic construction underlying the proof is familiar from
section \ref{sec:elementary}: a pair $(v^+,v^-) \in
C^\infty(D^+,M) \times C^\infty(D^-,M)$ such that $v^+|\partial D^+ = x
$ and $v^-|\partial D^- = g(x)$ for some $x \in \mathcal{L}M$
gives rise to a section of $E_g$. If
$[v^-,g(x)] = \tilde{g}([v^+,x]) \in \widetilde{\loops}$,
this section lies in the $\Gamma$-equivalence class
$S_{\tilde{g}}$. More generally, if $[v^-,g(x)] =
\gamma \cdot \tilde{g}([v^+,x])$ for some $\gamma \in \Gamma$,
the section lies in $(-\gamma) + S_{\tilde{g}}$. The sign
$-\gamma$ occurs for the following reason: to obtain the element
$[v^-,g(x)] \in \widetilde{\loops}$, one uses a diffeomorphism
from $D^-$ to the standard disc $D^2$. Following Convention
\ref{th:orientation-convention}, this diffeomorphism should be
orientation-reversing.
To adapt this construction to the framework of surfaces with
tubular ends, we replace $D^+,D^-$ by $\Sigma^+ = D^+ \cup_{S^1}
(\mathbb{R}^+ \times S^1)$ and $\Sigma^- = (\mathbb{R}^- \times S^1) \cup_{S^1}
D^-$, with the orientations induced from the standard ones on
$\mathbb{R}^{\pm} \times S^1$. A map $u \in C^\infty(\Sigma^+,M)$ such that
$\lim_{s \rightarrow \infty} u(s,\cdot) = x$ for some $x \in \mathcal{L}M$
can be extended continuously to the compactification $\overline{\Sigma}^+ =
\Sigma^+ \cup (\{\infty\} \times S^1)$. Using an orientation-preserving
diffeomorphism $D^2 \longrightarrow \overline{\Sigma}^+$, this defines
an element $c = [u,x] \in \widetilde{\loops}$. We say that $u$ converges to $c$.
There is a parallel notion for $u \in C^\infty(\Sigma^-,M)$, except
that in this case the identification of $\overline{\Sigma}^-$ with $D^2$
reverses the orientation.
\begin{defn} For $\gamma \in \Gamma$, let $\mathcal{P}(\gamma)$ be the space of
pairs $(u^+,u^-) \in C^\infty(\Sigma^+,M) \times C^\infty(\Sigma^-,M)$ such
that $u^+$ converges to $c$ and $u^-$ to $(-\gamma) \cdot \tilde{g}(c)$
for some $c \in \widetilde{\loops}$.
\end{defn}
Fix a point $z_0 \in D^- \subset \Sigma^-$. We will denote the trivial
symplectic fibre bundles $\Sigma^\pm \times (M,\o)$ by $(E^\pm,\O^\pm)$,
and the obvious isomorphism $(M,\o) \longrightarrow (E^-_{z_0},\O^-_{z_0})$
by $i$. For $R \geq 0$, consider $\Sigma_R^+ = D^+ \cup_{S^1}
([0;R] \times S^1) \subset \Sigma^+$, $\Sigma_R^- =
([-R;0] \times S^1) \cup D^- \subset \Sigma^-$ and $\Sigma_R =
\Sigma_R^+ \cup_{\partial\Sigma_R^{\pm}} \Sigma_R^-$, with the
induced orientation. With respect to Riemannian metrics
on $\Sigma^+,\Sigma^-$ whose restriction to $\mathbb{R}^{\pm} \times S^1$
is the standard tubular metric, $\Sigma_R$ is a surface with
an increasingly long `neck' as $R \rightarrow \infty$. Let
$(E_R,\O_R)$ be the symplectic fibre bundle over $\Sigma_R$
obtained by gluing together $E^+_R = E^+|\Sigma^+_R$ and
$E^-_R = E^-|\Sigma^-_R$ using $\phi_g:
E^+|\partial\Sigma^+_R \longrightarrow E^-|\partial\Sigma_R^-$,
$\phi_g(R,t,y) = (-R,t,g_t(y))$. The terminology introduced in section
\ref{sec:elementary} applies to bundles over $\Sigma_R$ since it is
homeomorphic to $S^2$. $(E_R,\O_R)$ is Hamiltonian;
it differs from $(E_g,\O_g)$ only because the base is parametrized
in a different way (more precisely, $(E_R,\O_R)$ is the pullback of
$(E_g,\O_g)$ by an oriented diffeomorphism
$\Sigma_R \longrightarrow S^2$). Let $S_R$ be the $\Gamma$-equivalence
class of sections of $E_R$ which corresponds to $S_{\tilde{g}}$.
For $(u^+,u^-) \in \mathcal{P}(\gamma)$, one can construct a section
$u^+ \widehat{\#}_R u^-$ of $E_R$ for large $R$ by
`approximate gluing': let $x = \lim_{s \rightarrow \infty} u^+(s,\cdot)
\in \mathcal{L}M$. There is an $R_0 \geq 0$ and a family
$(\hat{u}_R^+,\hat{u}_R^-)_{R \geq R_0}$ in $\mathcal{P}(\gamma)$
which converges uniformly to $(u^+,u^-)$ as $R \rightarrow \infty$
and such that $\hat{u}_R^+(s,t) = x(t)$ for $s \geq R$,
$\hat{u}_R^-(s,t) = g_t(x(t))$ for $s \leq -R$. Define
\[
(u^+ \widehat{\#}_R u^-)(z) = \begin{cases}
(z,\hat{u}_R^+(z)) & z \in \Sigma_R^+,\\
(z,\hat{u}_R^-(z)) & z \in \Sigma_R^-.
\end{cases}
\]
If $R$ is large, $u^+ \widehat{\#}_R u^-$ lies in the $\Gamma$-equivalence
class $\gamma + S_R$.
We will now introduce nonlinear $\bar{\partial}$-equations
on $\Sigma^+,\Sigma^-$ and state the gluing theorem for solutions
of them. Like the pair-of-pants product, these equations are part
of the formalism of `relative Donaldson type invariants' of {\cite{piunikhin-salamon-schwarz94}}.
Let $j^+$ be a complex structure on $\Sigma^+$ whose restriction
to $\mathbb{R}^+ \times S^1$ is the standard complex structure. Choose
a regular pair $(H^{\infty},\mathbf{J}^{\infty})$ and
$H^+ \in C^\infty(\mathbb{R}^+ \times S^1 \times M,\mathbb{R}),
\mathbf{J}^+ \in \modJ{\Sigma^+}$ such that $H^+|[0;1] \times S^1 \times M = 0$
and $H^+(s,t,\cdot) = H^{\infty}(t,\cdot), J^+_{(s,t)} = J^{\infty}_t$ for
$s \geq 2$, $t \in S^1$. For $c \in \mathrm{Crit}(a_{H^{\infty}}) \subset \widetilde{\loops}$,
let $\mathcal{M}^+(c;H^+,\mathbf{J}^+)$ be the space of $u \in C^\infty(\Sigma^+,M)$
which converge to $c$ and satisfy
\begin{equation} \label{eq:plus} \begin{aligned}
du(z) + J_z^+ \circ du(z) \circ j^+ &= 0 \quad \text{ for } z \in D^+,\\
\partials{u} + J_z^+(u)\left(\partialt{u} - X_{H^+}(s,t,u)\right) &= 0
\quad \text{ for } z = (s,t) \in \mathbb{R}^+ \times S^1.
\end{aligned} \end{equation}
For generic $(H^+,\mathbf{J}^+)$, $\mathcal{M}^+(c;H^+,\mathbf{J}^+)$ is a manifold
of dimension $2n - \mu_{H^{\infty}}(c)$ and the zero-dimensional
spaces are finite.
To write down the corresponding equations for $u \in C^\infty(\Sigma^-,M)$,
choose a complex structure $j^-$ on $\Sigma^-$,
a regular pair $(H^{-\infty},\mathbf{J}^{-\infty})$ and
$H^- \in C^\infty(\mathbb{R}^- \times S^1 \times M,\mathbb{R})$,
$\mathbf{J}^- \in \modJ{\Sigma^-}$ with properties symmetric to those above.
The equations are
\begin{equation} \label{eq:minus} \begin{aligned}
du(z) + J_z^- \circ du(z) \circ j^- &= 0 \quad \text{ for } z \in D^-,\\
\partials{u} + J_z^-(u)\left(\partialt{u} - X_{H^-}(s,t,u)\right) &= 0
\quad \text{ for } z = (s,t) \in \mathbb{R}^- \times S^1.
\end{aligned} \end{equation}
Let $h$ be a Riemannian metric on $M$ and $f \in C^\infty(M,\mathbb{R})$ a
Morse function. If $c$ is a critical point of $a_{H^{-\infty}}$ and $y$
a critical point of $f$, we denote by $\mathcal{M}^-(c,y;H^-,\mathbf{J}^-)$ the space of
solutions $u$ of \eqref{eq:minus} which converge to $c$ and with
$u(z_0) \in W^s(y;f,h)$. In the generic case, $\mathcal{M}^-(c,y;H^-,\mathbf{J}^-)$
is a manifold of dimension $\mu_{H^{-\infty}}(c) - i_f(y)$, and the
zero-dimensional spaces are again finite. For fixed
$(H^\infty,\mathbf{J}^\infty)$ and $(H^{-\infty},\mathbf{J}^{-\infty})$, we
will denote the space of all $(H^+,\mathbf{J}^+,H^-,\mathbf{J}^-,f,h)$ by
$\choices$.
\eqref{eq:plus} and \eqref{eq:minus} can be written in a different
way using an idea of Gromov. For $(z,y) \in \Sigma^+ \times M$, let
$\nu^+(z,y): T_z\Sigma^+ \longrightarrow T_yM$ be the
$(j^+,J_z^+)$-antilinear homomorphism given by
\[
\nu^+(z,y) = \begin{cases}
ds \otimes J_z^+ X_{H^+}(s,t,y) + dt \otimes X_{H^+}(s,t,y) &\text{for }
z = (s,t),\\
0 &\text{for } z \in D^+. \end{cases}
\]
$u$ is a solution of \eqref{eq:plus} iff
\begin{equation} \label{eq:plus-two}
du(z) + J_z^+ \circ du(z) \circ j^+ = \nu^+(z,u(z)).
\end{equation}
One can think of $\mathbf{J}^+ = (J^+_z)_{z \in \Sigma^+}$ as a family of almost
complex structures on the fibres of the trivial bundle $E^+$.
Consider the almost complex structure
\[
\hat{J}^+_{(z,y)}(Z,Y) = (j^+Z,J_z^+Y + \nu^+(z,y)(j^+Z))
\]
on $E^+$. $\hat{J}^+| \{z\} \times M = J^+_z$ for all $z \in \Sigma^+$, and
the projection $E^+ \longrightarrow \Sigma^+$ is $(\hat{J}^+,j^+)$-linear.
A straightforward computation shows that $u$ is a solution of
\eqref{eq:plus-two} iff the section $s(z) = (z,u(z))$ of $E^+$ is
$(j^+,\hat{J}^+)$-holomorphic. There is an almost
complex structure $\hat{J}^-$ on $E^-$ such that solutions
of \eqref{eq:minus} correspond to $(j^-,\hat{J}^-)$-holomorphic sections
in the same way.
From now on, we will assume that $(H^{\infty},\mathbf{J}^{\infty})$ is the pullback
of $(H^{-\infty},\mathbf{J}^{-\infty})$ by $g$. As a consequence,
$(J^+_z)_{z \in \Sigma^+_R}$ and $(J^-_z)_{z \in \Sigma^-_R}$ can
be pieced together to a smooth family $\mathbf{J}_R = (J_{R,z})_{z \in \Sigma_R}
\in \mathcal{J}(E_R,\O_R)$ for any $R \geq 2$. $j^+|\Sigma^+_R$ and
$j^-|\Sigma^-_R$ determine a complex structure $j_R$ on
$\Sigma_R$, and there is a unique $\hat{J}_R \in
\widehat{\mathcal{J}}(j_R,\mathbf{J}_R)$ such that $\hat{J}_R|E^{\pm}_R =
\hat{J}^{\pm}|E^{\pm}_R$. As in the previous section, we denote
the space of $(j_R,\hat{J}_R)$-holomorphic sections of $E_R$ in a
$\Gamma$-equivalence class $S$ by $\sec(j_R,\hat{J}_R,S)$, and by
$\sec(j_R,\hat{J}_R,S,y)$ for $y \in \mathrm{Crit}(f)$ the subspace
of those $s$ such that $i^{-1}(s(z_0)) \in W^s(y;f,h)$.
For $(\gamma,y) \in \Gamma \times \mathrm{Crit}(f)$ with
\begin{equation} \label{eq:dimension-condition}
i_f(y) = 2n - 2I(g,\tilde{g}) + 2c_1(\gamma),
\end{equation}
let $\mathcal{S}'(H^+,\J^+,H^-,\J^-,\gamma,y)$ be the disjoint union of
\begin{equation} \label{eq:the-product}
\mathcal{M}^+(c;H^+,\mathbf{J}^+) \times \mathcal{M}^-( (-\gamma) \cdot \tilde{g}(c),y;
H^-,\mathbf{J}^-)
\end{equation}
for all $c \in \mathrm{Crit}(a_{H^\infty})$. Equivalently, $\mathcal{S}'(H^+,\J^+,H^-,\J^-,\gamma,y)$ is the
space of $(u^+,u^-) \in \mathcal{P}(\gamma)$ such that $s^+(z) = (z,u^+(z))$ is
$(j^+,\hat{J}^+)$-holomorphic, $s^-(z) = (z,u^-(z))$ is
$(j^-,\hat{J}^-)$-holomorphic and $i^{-1}(s^-(z_0)) \in W^s(y;f,h)$.
For generic $(H^+,\mathbf{J}^+)$ and $(H^-,\mathbf{J}^-)$,
$\dim \mathcal{M}^+(c;H^+,\mathbf{J}^+) = 2n - \mu_{H^\infty}(c)$ and
$
\dim \mathcal{M}^-((-\gamma) \cdot \tilde{g}(c),y;H^-,\mathbf{J}^-)
= \mu_{H^{-\infty}}((-\gamma) \cdot \tilde{g}(c)) - i_f(y)
= \mu_{H^\infty}(c) - 2n
$
(the last equality uses \eqref{eq:dimension-condition},
\eqref{eq:grading-shift} and Lemma \ref{th:shift}). Therefore the product
\eqref{eq:the-product} is zero-dimensional
if $\mu_{H^\infty}(c) = 2n$, and is empty otherwise.
A compactness theorem shows that $\mathcal{S}'(H^+,\J^+,H^-,\J^-,\gamma,y)$ is a finite set.
We can now state the `gluing theorem' which is the main
step in the proof of Theorem \ref{th:gluing}.
\begin{theorem} \label{th:technical-gluing-theorem}
There is a subset of $\choices$ of second category such that
if $(H^+,\mathbf{J}^+,H^-,\mathbf{J}^-,f,h)$ lies in this subset, the following
holds: for $(\gamma,y) \in \Gamma \times \mathrm{Crit}(f)$ satisfying
\eqref{eq:dimension-condition}, there is an $R_0 > 2$ and a family
of bijective maps
\[
\#_R: \mathcal{S}'(H^+,\J^+,H^-,\J^-,\gamma,y) \longrightarrow \sec(j_R,\hat{J}_R,\gamma + S_R,y)
\]
for $R \geq R_0$. \end{theorem}
The construction of $\#_R$ relies on the following property
of the `approximate gluing' $\widehat{\#}_R$ for $(u^+,u^-) \in \mathcal{S}'(H^+,\J^+,H^-,\J^-,\gamma,y)$:
because $\hat{u}_R^+$ converges uniformly to $u^+$ as
$R \rightarrow \infty$, it approximately satisfies
\eqref{eq:plus} for large $R$. Therefore
$\hat{s}^+_R(z) = (z,\hat{u}_R^+(z))$
is an approximately $(j^+,\hat{J}^+)$-holomorphic section of $E^+$.
Using the same argument for $u^-$, it follows that
$\hat{s}_R = u^+ \widehat{\#}_R u^-$ is an approximately
$(j_R,\hat{J}_R)$-holomorphic section of $E_R$. More precisely,
\[
\| d\hat{s}_R + \hat{J}_R \circ d\hat{s}_R \circ
j_R \|_p \rightarrow 0 \quad \text{ as } R \rightarrow \infty.
\]
Here $\| \cdot \|_p$ $(p>2)$ is the $L^p$-norm on $\Sigma_R$ with
respect to a family of metrics with an
increasingly long `neck' as described above.
$\widehat{\#}_R$ can be set up in such a way that $i^{-1}(\hat{s}_R(z_0))
\in W^s(y;f,h)$ for all $R$. The section $u^+ \#_R u^-$
is obtained from $\hat{s}_R$ by an application of the implicit function
theorem in the space of $W^{1,p}$-sections $s$ of $E_R$ which satisfy
$i^{-1}(s(z_0)) \in W^s(y;f,h)$. The argument can be modelled on
\cite[Section 4.4]{schwarz95}. For large $R$,
$u^+ \#_R u^-$ and $\hat{s}_R$ are close in the $W^{1,p}$-metric
and therefore homotopic. It follows that $u^+ \#_R u^-$ lies
in the $\Gamma$-equivalence class $\gamma + S_R$.
By construction, $\#_R$ is injective for large $R$.
Surjectivity is proved by a limiting argument (`stretching the neck').
Let $s_m \in \sec(j_{R_m},\hat{J}_{R_m},\gamma + S_{R_m},y)$ be a
sequence of sections, with $R_m \rightarrow \infty$, such that
$s_m | \Sigma^+_{R_m}$ and $s_m | \Sigma^-_{R_m}$ converge on compact subsets
to sections $s^+(z) = (z,u^+(z))$ of $E^+$ and $s^-(z) = (z,u^-(z))$
of $E^-$. Then $s^\pm$ is $(j^\pm,\hat{J}^\pm)$-holomorphic.
If the pair $(s^+,s^-)$ describes the `geometric limit' of the
sequence $s_m$ completely, $(u^+,u^-) \in \mathcal{S}'(H^+,\J^+,H^-,\J^-,\gamma,y)$ and
$s_m = u^+ \#_{R_m} u^-$ for large $m$. Transversality and
dimension arguments are used to exclude more complicated limiting
behaviour.
It remains to explain how Theorem \ref{th:gluing} is derived
from the technical Theorem \ref{th:technical-gluing-theorem}.
The main point is that $\Psi^+,\Psi^-$ are the `relative
Donaldson type invariants' associated to $\Sigma^+,\Sigma^-$.
The element $\Psi^+(e) \in HF_{2n}(M,\o)$ has a particularly simple
description in terms of the moduli spaces $\mathcal{M}^+$: it is
the homology class of the cycle
\[
\Psi^+(e;H^+,\mathbf{J}^+) = \sum_{c \in \mathrm{Crit}_{2n}(a_{H^\infty})}
\#\mathcal{M}^+(c;H^+,\mathbf{J}^+) \gen{c} \in CF_{2n}(H^\infty)
\]
for generic $(H^+,\mathbf{J}^+)$. It follows that
$HF_*(g,\tilde{g}) \Psi^+(e) \in HF_{2n - 2I(g,\tilde{g})}(M,\o)$ is represented by
\[
\sum_c \#\mathcal{M}^+(c;H^+,\mathbf{J}^+)\gen{\tilde{g}(c)}
\in CF_{2n - 2I(g,\tilde{g})}(H^{-\infty}).
\]
To define $\Psi^-$, we consider $QH_*\mo$ as the homology
of $(CM_*(f) \otimes \Lambda, \partial(f,h) \otimes \mathit{Id})$.
For generic $(H^-,\mathbf{J}^-)$, the formula
\[
\Psi^-(H^-,\mathbf{J}^-,f,h)(\gen{c}) = \sum_{\gamma,y}
\#\mathcal{M}^-(\gamma \cdot c,y;H^-,\mathbf{J}^-) \gen{y} \otimes \gen{\!-\!\gamma},
\]
where the sum is over all $(\gamma,y) \in \Gamma \times \mathrm{Crit}(f)$ such that
$i_f(y) = \mu_{H^{-\infty}}(c) - 2c_1(\gamma)$, defines a $\Lambda$-linear
homomorphism of chain complexes
\[
\Psi^-(H^-,\mathbf{J}^-,f,h): CF_*(H^{-\infty}) \longrightarrow
CM_*(f) \otimes \Lambda.
\]
$\Psi^-$ is defined as the induced map on homology groups.
Here, as in the case of $\Psi^+$, we have omitted
several steps in the construction; we refer to {\cite{piunikhin-salamon-schwarz94}} for a more
complete description. As usual, our setup differs slightly from
that in {\cite{piunikhin-salamon-schwarz94}} because the almost complex structures may be $z$-dependent;
however, the proofs can be easily adapted, provided always that
{$(W^+)$} is satisfied.
By comparing the expressions for $HF_*(g,\tilde{g}) \Psi^+(e)$ and
$\Psi^-$ with the definition of $\mathcal{S}'(H^+,\J^+,H^-,\J^-,\gamma,y)$, one obtains
\[
\Psi^- HF_*(g,\tilde{g}) \Psi^+(e) =
\sum_{\gamma \in \Gamma} [c_\gamma] \otimes \gen{\gamma},
\]
where $[c_\gamma] \in H_*(M;\mathbb{Z}/2)$ is the homology class of the
cycle
\[
c_\gamma = \sum_y \#\mathcal{S}'(H^+,\J^+,H^-,\J^-,\gamma,y) \gen{y} \in CM_*(f);
\]
the sum runs over all $y \in \mathrm{Crit}(f)$ with
$i_f(y) = 2n - 2I(g,\tilde{g}) + 2c_1(\gamma)$.
Clearly, $Q(E_R,\O_R,S_R) = Q(E_g,\O_g,S_{\tilde{g}})$ for all $R \geq 0$.
To obtain an explicit expression for $Q(E_R,\O_R,S_R)$, choose
$\mathbf{J}_R' \in \mathcal{J}^{\mathrm{reg},z_0}(E_R,\O_R)$,
$\hat{J}_R' \in \widehat{\mathcal{J}}^{\mathrm{reg}}(j_R,\mathbf{J}_R')$, a Riemannian
metric $h'$ and a Morse function $f'$, all of which satisfy
appropriate regularity conditions. By \eqref{eq:explicit},
\[
Q(E_R,\O_R,S_R) = \sum_{\gamma \in \Gamma} [c_{\gamma,R}]
\otimes \gen{\gamma},
\]
where $c_{\gamma,R} \in CM_*(f)$ is given by
\[
c_{\gamma,R} = \sum_{y'} \#\sec(j_R,\hat{J}_R',\gamma + S_R,y') \gen{y'}.
\]
This time the sum is over those $y' \in \mathrm{Crit}(f')$ such that
$i_{f'}(y') = 2n + 2c_1(TE_R^v,\O_R)(S_R) + 2c_1(\gamma)$, but
by equation \eqref{eq:maslov-and-chern} the Morse indices
are the same as above.
The statement of Theorem \ref{th:gluing} is that
$[c_{\gamma,R}] = [c_\gamma]$ for all $\gamma$.
Since $[c_{\gamma,R}]$ is independent of $R$, it is sufficient
to show that for each $\gamma$ there is an $R$ such that
for a suitable choice of $\mathbf{J}_R',\hat{J}_R',f',h'$,
$[c_{\gamma,R}] = [c_\gamma]$. Choose some $\gamma_0 \in \Gamma$.
We can assume that $H^+,\mathbf{J}^+,H^-,\mathbf{J}^-,f,h$ have been chosen
as in Theorem \ref{th:technical-gluing-theorem}. Since $f$ has only
finitely many critical points, there is an $R$ such that
Theorem \ref{th:technical-gluing-theorem} holds for all $(\gamma_0,y)$.
Then
\[
c_{\gamma_0} = \sum_y \#\sec(j_R,\hat{J}_R,\gamma_0 + S_R,y) \gen{y}.
\]
and it follows that $c_{\gamma_0} = c_{\gamma_0,R}$ if we take
$\mathbf{J}_R' = \mathbf{J}_R$, $\hat{J}_R' = \hat{J}_R$ and $(f',h') = (f,h)$.
Note that $\mathbf{J}_R \in \mathcal{J}(E_R,\O_R)$ and
$\hat{J}_R \in \widehat{\mathcal{J}}(j_R,\mathbf{J}_R)$
are not `generic' choices because they are $s$-independent on the
`neck' of $\Sigma_R$. However, a final consideration shows that for
large $R$, they can be used to compute the coefficient
$[c_{\gamma_0,R}]$ of $Q(E_R,\O_R,S_R)$.
\newcommand{\tilde{\Phi}_{(A,\o)}}{\tilde{\Phi}_{(A,\o)}}
\section{Proof of the main results \label{sec:proofs}}
For $a_1, a_2, a_3 \in H_*(M;\mathbb{Z}/2)$ and $A \in H_2(M;\mathbb{Z})$, let
$\tilde{\Phi}_{(A,\o)}(a_1,a_2,a_3) \in \mathbb{Z}/2$ be the mod $2$ reduction of the
Gromov-Witten invariant of \cite[Section 8]{ruan-tian94}.
It is zero unless
\begin{equation} \label{eq:dimension-of-gw}
\dim(a_1) + \dim(a_2) + \dim(a_3) + 2c_1(A) = 4n.
\end{equation}
Intuitively, $\tilde{\Phi}_{(A,\o)}(a_1,a_2,a_3)$ is the number (modulo $2$) of
$J$-holomorphic spheres in the class $A$ which meet suitable
cycles representing $a_1,a_2$ and $a_3$.
Let $a_1 \ast_A a_2 \in H_*(M;\mathbb{Z}/2)$ be
the class defined by
\[
(a_1 \ast_A a_2) \cdot a_3 = \tilde{\Phi}_{(A,\o)}(a_1,a_2,a_3) \text{ for all } a_3,
\]
where $\cdot$ is the ordinary intersection product.
For $\gamma \in \Gamma$, let $a_1 \ast_{\gamma} a_2 \in H_*(M;\mathbb{Z}/2)$
be the sum of $a_1 \ast_A a_2$ over all classes $A$ which can
be represented by a smooth map $w: S^2 \longrightarrow M$ with
$[w] = \gamma$ (only finitely many terms of this sum are nonzero).
The quantum intersection product $\ast$ on $QH_*\mo$ is defined
by the formula
\[
(a_1 \otimes \gen{\gamma_1}) \ast (a_2 \otimes \gen{\gamma_2}) =
\sum_{\gamma \in \Gamma}
(a_1 \ast_\gamma a_2) \otimes \gen{\gamma_1 + \gamma_2 + \gamma},
\]
extended to infinite linear combinations in the obvious way.
$\ast$ is a bilinear $\Lambda$-module map with the following properties:
\begin{enumerate}
\item \label{item:grad} if $a_1 \in QH_i(M,\o)$ and $a_2 \in QH_j(M,\o)$,
$a_1 \ast a_2 \in QH_{i+j-2n}(M,\o)$.
\item \label{item:as} $\ast$ is associative.
\item \label{item:un}
$e = [M] \otimes \gen{0} \in QH_{2n}(M,\o)$ is the unit of $\ast$.
\item \label{item:pp}
$\Psi^+(a_1 \ast a_2) = \Psi^+(a_1) \qp_{\mathrm{PP}} \Psi^+(a_2) \in HF_*(M,\o).$
\end{enumerate}
\ref{item:grad} follows from \eqref{eq:dimension-of-gw}.
\ref{item:as} is due to Ruan and Tian \cite{ruan-tian94};
another proof is given in \cite{mcduff-salamon}.
\ref{item:un} is \cite[Proposition 8.1.4(iii)]{mcduff-salamon}
and \ref{item:pp} is the main result of {\cite{piunikhin-salamon-schwarz94}}. From the two last
items and the fact that $\Psi^+$ is an isomorphism, one obtains
\begin{lemma} \label{th:pp-unit}
$u = \Psi^+(e)$ is the unit of $(HF_*\mo,\qp_{\mathrm{PP}})$. \qed
\end{lemma}
\proof[Proof of Theorem \ref{bigth:comparison}] By Theorem \ref{th:gluing},
$q(g,\tilde{g}) = \Psi^- HF_*(g,\tilde{g})(u)$. Because $\Psi^-$ is the
inverse of $\Psi^+$, this can be written as $\Psi^+(q(g,\tilde{g})) =
HF_*(g,\tilde{g})(u)$. From Lemma \ref{th:pp-unit} and Proposition
\ref{th:pair-of-pants-linear}, it follows that
\[ \begin{split}
HF_*(g,\tilde{g})(b) &= HF_*(g,\tilde{g})(u \qp_{\mathrm{PP}} b)\\
&= HF_*(g,\tilde{g})(u) \qp_{\mathrm{PP}} b = \Psi^+(q(g,\tilde{g})) \qp_{\mathrm{PP}} b. \qed
\end{split} \]
\proof[Proof of Proposition \ref{bigth:trivial}] Proposition
\ref{prop:further-properties}\ref{item:lambda-multiplication}
says that $HF_*(\mathit{Id},\gamma)$ is given by multiplication with
$\gen{\gamma} \in \Lambda$. Using Theorem \ref{th:gluing},
$\Psi^-\Psi^+ = \mathit{Id}$ and the fact that
$\Psi^-$ is $\Lambda$-linear, we can compute
\begin{multline*}
q(\mathit{Id},\gamma) = \Psi^- HF_*(\mathit{Id},\gamma)(u) = \Psi^- (\gen{\gamma} \, u) = \\
= \gen{\gamma} \Psi^-(u) = \gen{\gamma} ([M] \otimes \gen{0})
= [M] \otimes \gen{\gamma}. \qed
\end{multline*}
\proof[Proof of Corollary \ref{bigth:homomorphism}] Essentially, this
is a consequence of the fact that the maps $HF_*(g,\tilde{g})$ define a
$\widetilde{G}$-action on Floer homology (Proposition
\ref{prop:further-properties}\ref{item:functor}). Take
$(g_1,\tilde{g}_1), (g_2,\tilde{g}_2) \in \widetilde{G}$. Using
Theorem \ref{th:gluing} twice, we obtain
\[ \begin{split}
q(g_1g_2,\tilde{g}_1\tilde{g}_2) &=
\Psi^- HF_*(g_1g_2,\tilde{g}_1\tilde{g}_2)(u) \\
& = \Psi^- HF_*(g_1,\tilde{g}_1) HF_*(g_2,\tilde{g}_2)(u)\\
& = \Psi^- HF_*(g_1,\tilde{g}_1) \Psi^+(q(g_2,\tilde{g}_2)).
\end{split} \]
Theorem \ref{bigth:comparison} with
$(g,\tilde{g}) = (g_1,\tilde{g}_1)$ and $b =
\Psi^+(q(g_2,\tilde{g}_2))$ says that
\[
HF_*(g_1,\tilde{g}_1) \Psi^+(q(g_2,\tilde{g}_2)) =
\Psi^+(q(g_1,\tilde{g}_1)) \qp_{\mathrm{PP}} \Psi^+(q(g_2,\tilde{g_2})).
\]
Using property \ref{item:pp} above, we conclude that
\[
q(g_1g_2,\tilde{g}_1\tilde{g}_2) = q(g_1,\tilde{g}_1) \ast
q(g_2,\tilde{g}_2).
\]
As a special case of Proposition \ref{bigth:trivial},
$q(\mathit{Id}_{\widetilde{G}}) = e$. The fact that $q(g,\tilde{g})$ depends on $(g,\tilde{g})$
only up to homotopy is a consequence of the homotopy invariance of
$HF_*(g,\tilde{g})$ (Proposition \ref{th:homotopy}) and Theorem \ref{th:gluing}.
\qed
\proof[Proof of Corollary \ref{bigth:invertible}] Because of Lemma
\ref{th:completeness}, this is an immediate consequence of Corollary
\ref{bigth:homomorphism}. \qed
\newcommand{\text{ mod }}{\text{ mod }}
\newcommand{\not\equiv}{\not\equiv}
\section{An application to the Maslov index\label{sec:an-application}}
Recall that $\bar{I}: \pi_1(\mathrm{Ham}(M,\o)) \longrightarrow \mathbb{Z}/N\mathbb{Z}$
($N$ denotes the minimal Chern number) is defined by
\[
\bar{I}(g) = I(g,\tilde{g}) \text{ mod } N,
\]
where $\tilde{g}: \widetilde{\loops} \longrightarrow \widetilde{\loops}$ is any lift of
$g \in G$. If $M$ is simply-connected, the definition of
$\bar{I}$ is elementary; in general, it uses Lemma
\ref{th:contractible-component}, which is based on the Arnol'd conjecture.
Our first result is a simple consequence of the existence of the
automorphisms $HF_*(g,\tilde{g})$.
\begin{proposition} \label{th:calabi-yau}
If $(M,\o)$ satisfies $c_1|\pi_2(M) = 0$,
$\bar{I}$ is the trivial homomorphism. \end{proposition}
\proof In this case, $\bar{I}$ is $\mathbb{Z}$-valued because
$I(g,\tilde{g}) \in \mathbb{Z}$ is independent of the choice of $\tilde{g}$.
Since $\bar{I}$ is a homomorphism, it is sufficient to show that
$\bar{I}(g) \leq 0$ for all $g$. The assumption on $c_1$
also implies that the grading of the Novikov ring $\Lambda$ is trivial.
It follows from Theorem \ref{th:what-is-floer-homology}
that $HF_0(M,\o) \iso H_0(M;\mathbb{Z}/2) \otimes \Lambda \iso \Lambda$ and
$HF_k(M,\o) = 0$ for $k<0$. By Proposition \ref{th:shift},
$HF_*(g,\tilde{g})$ maps $HF_0(M,\o)$ isomorphically to
$HF_{-2I(g,\tilde{g})}(M,\o)$, which is clearly impossible if
$I(g,\tilde{g}) > 0$. \qed
To obtain more general results, it is necessary to use
the multiplicative structure. Consider
\[
Q^+ = \bigoplus_{i < 2n} H_i(M;\mathbb{Z}/2) \otimes \Lambda \subset QH_*\mo.
\]
Since the degree of any element of $\Lambda$ is a multiple
of $2N$, $QH_k(M,\o) \subset Q^+$ for any $k$ such that
$k \not\equiv 2n \text{ mod } 2N$.
\begin{lemma} \label{th:qplus} If $Q^+ \ast Q^+ \subset Q^+$, every
homogeneous invertible element of $QH_*\mo$ has degree
$2n + 2iN$ for some $i \in \mathbb{Z}$. \end{lemma}
\proof Assume that $x \in QH_k(M,\o)$ is invertible and
$k \not\equiv 2n \text{ mod } 2N$; then $x \in Q^+$.
Since $\ast$ has degree $-2n$ and the unit $e$ lies in $QH_{2n}(M,\o)$,
the inverse $x^{-1}$ has degree $4n-k$. Clearly
$4n - k \not\equiv 2n \text{ mod } 2N$, and therefore
$x^{-1} \in Q^+$. $x^{-1} \ast x = e
\notin Q^+$, contrary to the assumption that $Q^+ \ast
Q^+ \subset Q^+$. \qed
\begin{proposition} If $(M,\o)$ satisfies {$(W^+)$} and
$Q^+ \ast Q^+ \subset Q^+$, $\bar{I}$ is trivial. \end{proposition}
\proof For all $(g,\tilde{g}) \in \widetilde{G}$, $q(g,\tilde{g}) \in QH_*\mo$ is
invertible by Corollary \ref{bigth:invertible}.
$q(g,\tilde{g})$ has degree $2n - 2I(g,\tilde{g})$. Using Lemma \ref{th:qplus}
it follows that $I(g,\tilde{g}) \equiv 0 \text{ mod } N$, and therefore
$\bar{I}(g) = 0$. \qed
By definition, $Q^+ \ast Q^+ \subset Q^+$ iff
$\tilde{\Phi}_{(A,\o)}(x_1,x_2,[pt]) = $ for all $A \in H_2(M;\mathbb{Z})$
and all $x_1, x_2 \in H_*(M;\mathbb{Z}/2)$ of dimension $<2n$. This is certainly true
if there is an $\o$-tame almost complex structure $J$ on $M$
and a point $y \in M$ such that no non-constant $J$-holomorphic
sphere passes through $y$. A particularly simple case is when there are no
non-constant $J$-holomorphic spheres at all. Then the quantum
intersection product reduces to the ordinary one, that is,
\begin{equation} \label{eq:undeformed}
(x_1 \otimes \gen{\gamma_1}) \ast (x_2 \otimes \gen{\gamma_2}) =
(x_1 \cdot x_2) \otimes \gen{\gamma_1 + \gamma_2}.
\end{equation}
For example, if
\begin{equation} \label{eq:ample-canonical}
(c_1 - \lambda[\o]) |\pi_2(M) = 0
\text{ for some } \lambda<0,
\end{equation}
$c_1(w) < 0$ for any non-constant pseudoholomorphic curve $w$.
If in addition $N \geq n-2$, there is
a dense set of $J$ such that any $J$-holomorphic sphere satisfies
$c_1(w) \geq 0$ and hence must be constant.
\begin{corollary} \label{cor:c-negative} If $(M^{2n},\omega)$
satisfies \eqref{eq:ample-canonical} and its
minimal Chern number is at least $n-1$, the homomorphism $\bar{I}$
is trivial. \qed \end{corollary}
Note that we have sharpened the condition on $N$ in order to
fulfil {$(W^+)$}. By a slightly different reasoning,
it seems likely that Corollary \ref{cor:c-negative}
remains true without any assumption on $N$;
this is one of the motivations for extending the theory
beyond the case where {$(W^+)$} holds.
As a final example, consider the case where $M$ is four-dimensional.
{$(W^+)$} holds for all symplectic four-manifolds.
For $N = 1$, $\bar{I}$ is vacuous. Assume
that $N \geq 2$ (this implies that $(M,\o)$ is minimal).
In that case, it is a result of McDuff
that \eqref{eq:undeformed} holds unless $(M,\o)$ is rational or ruled.
For convenience, we reproduce the proof from
\cite{mcduff-salamon96b}:
for generic $J$, there are no non-constant $J$-holomorphic spheres
$w$ with $c_1(w) \leq 0$ because the moduli space of such curves
has negative dimension. Therefore, if \eqref{eq:undeformed} does
not hold, there is a $w$ with $c_1(w) \geq 2$. After perturbing
$J$, we can assume that $w$ is immersed with transverse
self-intersections \cite[Proposition 1.2]{mcduff91}. Then $(M,\o)$
is rational or ruled by \cite[Theorem 1.4]{mcduff92}.
\begin{corollary} $\bar{I}$ is trivial for all $(M^4,\omega)$
which are not rational or ruled. \qed
\end{corollary}
\newcommand{\mathrm{Gr}}{\mathrm{Gr}}
\section{Examples \label{sec:examples}}
In the first example, $M$ is the Grassmannian $\mathrm{Gr}_k(\mathbb{C}^m)$
with the usual symplectic structure $\omega$ coming from the
Pl{\"u}cker embedding. $(M,\o)$ is a monotone symplectic manifold.
The $U(m)$-action on $\mathbb{C}^m$ induces a Hamiltonian $PU(m)$-action
$\rho$ on $M$. Let $g$ be the loop in $\mathrm{Ham}(M,\o)$ given by
$g_t = \rho(\mathrm{diag}(e^{2\pi i t}, 1, \dots, 1))$.
Obviously, $[g] \in \pi_1(\mathrm{Ham}(M,\o))$
lies in the image of $\rho_*: \pi_1(PU(m)) \longrightarrow
\pi_1(\mathrm{Ham}(M,\o))$, and since $\pi_1(PU(m)) \iso \mathbb{Z}/m$,
$[g]^m = 1$.
Let $H$ be the Hopf bundle over $\CP{1}$.
The fibre bundle $E = E_g$ is the bundle of Grassmannians
associated to the holomorphic vector bundle $H^{-1}
\oplus \mathbb{C}^{m-1}$:
\[
E = \mathrm{Gr}_k(H^{-1} \oplus \mathbb{C}^{m-1}) \stackrel{\pi}{\longrightarrow} \CP{1}.
\]
$H^{-1} \oplus \mathbb{C}^{m-1}$ is a subbundle of the trivial bundle
$\CP{1} \times \mathbb{C}^{m+1}$. This induces a holomorphic map
$f: E \longrightarrow \mathrm{Gr}_k(\mathbb{C}^{m+1})$ which is an embedding on
each fibre $E_z$. Let $\O_z \in \O^2(E_z)$ be the pullback of the
symplectic form on $\mathrm{Gr}_k(\mathbb{C}^{m+1})$ by $f|E_z$. $(E,\O)$ is a
Hamiltonian fibre bundle.
As for any bundle of Grassmannians associated to a vector bundle,
there is a canonical $k$-plane bundle $P_E \longrightarrow E$.
$P_E$ is a subbundle of $\pi^*(H^{-1} \oplus \mathbb{C}^{m-1})$ and
\begin{equation} \label{eq:vertical-tangent-bundle}
TE^v \iso \mathrm{Hom}(P_E,\pi^*(H^{-1} \oplus \mathbb{C}^{m-1})/P_E).
\end{equation}
Therefore $c_1(TE^v) = -k \, c_1(\pi^*H) - m \, c_1(P_E)$.
$P_E$ is isomorphic to $f^*\!P$, where $P$ is the canonical $k$-plane
bundle on $\mathrm{Gr}_k(\mathbb{C}^{m+1})$. It follows that
\begin{equation} \label{eq:c-one}
\deg(s^*TE^v) = -k -m \deg( s^*\!f^*\!P)
\end{equation}
for any section $s$ of $E$. If $s$ is holomorphic, $f(s)$ is a
holomorphic curve in $\mathrm{Gr}_k(\mathbb{C}^{m+1})$, and since
$c_1(P) \in H^2(\mathrm{Gr}_k(\mathbb{C}^{m+1}))$ is a negative multiple of the
symplectic class, either $\deg(s^*\!f^*\!P) \leq -1$ or $f(s)$ is constant.
In the first case, $\deg(s^*TE^v) > 0$ by \eqref{eq:c-one}. In the
second case, $s$ must be one of the `constant' sections
\[
s_W(z) = 0 \oplus W \in \mathrm{Gr}_k(H_z \oplus \mathbb{C}^{m-1})
\]
for $W \in \mathrm{Gr}_k(\mathbb{C}^{m-1})$.
We now check that $(E,\O)$ satisfies the conditions
of Proposition \ref{th:computation}.
\begin{enumerate} \item We have shown that any holomorphic section
with $\deg(s^*TE^v) \leq 0$ is a `constant' one. The space $\sec$
of such sections is certainly connected. Their $\Gamma$-equivalence
class $S_0$ satisfies $c_1(TE^v)(S_0) = -k$.
\item $s_W^*TE^v = \mathrm{Hom}(W, H^{-1} \oplus \mathbb{C}^{m-1}/W) \iso
\mathrm{Hom}(\mathbb{C}^k, H^{-1} \oplus \mathbb{C}^{m-k-1})$ is a sum of line bundles
of degree $0$ or $-1$, hence $H^{0,1}(\CP{1},s_W^*TE^v) = 0$.
\item and (iv) follow from the fact that $(M,\o)$ is monotone with minimal
Chern number $N = m > k$. \end{enumerate}
$(E_{z_0},\O_{z_0})$ is identified with $(M,\o)$ by choosing an
element of $H_{z_0}^{-1}$ which has length $1$ for the standard Hermitian
metric on $H^{-1}$. The evaluation map $\mathrm{ev}_{z_0}: \sec \longrightarrow M$
is an embedding whose image is $\mathrm{Gr}_k(\mathbb{C}^{m-1}) \subset \mathrm{Gr}_k(\mathbb{C}^m)$.
By Proposition \ref{th:computation},
$Q(E,\O,S_0) = [\mathrm{Gr}_k(\mathbb{C}^{m-1})] \otimes \gen{0}$.
Using the diagram \eqref{diag:basic}, we can derive from $[g]^m = 1$ that
\begin{equation} \label{eq:relation}
\left([\mathrm{Gr}_k(\mathbb{C}^{m-1})] \otimes \gen{0}\right)^m = [M] \otimes \gen{\gamma}
\end{equation}
for some $\gamma \in \Gamma$. Because of the
grading, $\gamma$ must be $k$ times the standard generator
of $\pi_2(M) \iso H_2(M;\mathbb{Z})$. \eqref{eq:relation} can be verified
by a direct computation, since $(QH_*(M,\o),\ast)$ is known
\cite{siebert-tian94} \cite{witten93}. \vspace{0.5em}
Our second example concerns the rational ruled surface
$M = \mathbb{F}_2$. Recall that $\mathbb{F}_r$ $(r>0)$
is the total space of the holomorphic fibre bundle
$p_r: \mathbb{P}(\mathbb{C} \oplus H^r) \longrightarrow \CP{1}$. We will use
two standard facts about $\mathbb{F}_r$.
\begin{lemma} \label{th:beauville} For any holomorphic
section $s$ of $p_r$,
$
\deg(s^*T\mathbb{F}_r^v) \geq -r.
$
There is a unique section such that equality holds, and all
others have $\deg(s^*T\mathbb{F}_r^v) \geq -r+2$. \end{lemma}
\begin{lemma} \label{th:beauville-b} For every
holomorphic map $w: \CP{1} \longrightarrow \mathbb{F}_2$,
$c_1(w) \geq 0$. All non-constant $w$ with $c_1(w) < 2$ are
of the form $w = s_- \circ u$, where $u: \CP{1} \longrightarrow \CP{1}$
is holomorphic and $s_-$ is the section
with $\deg(s_-^*T\mathbb{F}_2^v) = -2$. \end{lemma}
Lemma \ref{th:beauville} is a special case of a theorem on
irreducible curves in $\mathbb{F}_r$ \cite[Proposition IV.1]{beauville}. Lemma
\ref{th:beauville-b} can be derived from Lemma \ref{th:beauville}
by considering a map $w: \CP{1} \longrightarrow \mathbb{F}_2$
such that $p_2w$ is not constant as a
section of the pullback $(p_2w)^*\mathbb{F}_2$.
Since $\mathbb{P}(\mathbb{C} \oplus H^2) \iso \mathbb{P}(H^{-2} \oplus \mathbb{C})$, an embedding
of $H^{-2} \oplus \mathbb{C}$ into the trivial bundle $\CP{1} \times \mathbb{C}^3$ defines a
holomorphic map $f: M \longrightarrow \CP{2}$. Let $\tau_k$
be the standard integral K{\"a}hler form on $\CP{k}$. For all
$\lambda>1$, $\o_\lambda = (\lambda-1) p_2^*\tau_1 + f^*\tau_2$ is a
K{\"a}hler form on $M$.
The action of $S^1$ on $H^2$ by multiplication induces a Hamiltonian
circle action $g$ on $(M,\o_\lambda)$. $(E,\O) = (E_g,\O_g)$
can be constructed as follows: $E = \mathbb{P}(V)$ is the bundle of projective
spaces associated to the holomorphic vector bundle
\[
V = \mathbb{C} \oplus \mathrm{pr}_1^*H^2 \otimes \mathrm{pr}_2^*H^{-1}
\]
over $\CP{1} \times \CP{1}$ ($\mathrm{pr}_1,\mathrm{pr}_2$ are
the projections from $\CP{1} \times \CP{1}$ to $\CP{1}$).
$\pi: E \longrightarrow \CP{1}$ is obtained by composing
$\pi_V: \mathbb{P}(V) \longrightarrow \CP{1} \times \CP{1}$ with $\mathrm{pr}_2$.
The fibres $E_z = \pi^{-1}(z)$ are the ruled surfaces
$\mathbb{P}(\mathbb{C} \oplus H^2 \otimes H^{-1}_z)$. Any $\xi \in H^{-1}_z$
with unit length for the standard Hermitian metric determines a biholomorphic
map $E_z \longrightarrow M$. Since two such maps differ by
an isometry of $(M,\o_\lambda)$, the symplectic structure $\O_z$
on $E_z$ obtained in this way is independent of the choice of $\xi$.
$(E,\O)$ is a Hamiltonian fibre bundle.
A holomorphic section $s$ of $\pi$ can be decomposed into two pieces:
a section $s_1 = \pi_V \circ s: \CP{1} \longrightarrow \CP{1} \times
\CP{1}$ of $\mathrm{pr}_2$ and a section $s_2$ of
\[
F = \mathbb{P}(s_1^*V) \longrightarrow \CP{1}.
\]
In a sense, this second piece is given by $s$ itself, using the fact that
$s(z) \in \mathbb{P}(V_{s_1(z)})$ for all $z$. The decomposition leads to
an exact sequence
\begin{equation} \label{eq:vector-bundle-sequence}
0 \longrightarrow s_2^*(TF^v) \longrightarrow
s^*TE^v \stackrel{D\pi_V}{\longrightarrow} s_1^*(\ker(D\mathrm{pr}_2))
\longrightarrow 0
\end{equation}
of holomorphic vector bundles over $\CP{1}$. Since $s_1$ is a
section of $\mathrm{pr}_2$, it is given by $s_1(z) = (u(z),z)$ for some
$u: \CP{1} \longrightarrow \CP{1}$. Clearly,
$s_1^*(\ker(D\mathrm{pr}_2)) = u^*T\CP{1}$. Let $d$ be the degree of
$u$. The fibre bundle $F = \mathbb{P}(\mathbb{C} \oplus u^*H^2 \otimes
H^{-1})$ is isomorphic to $\mathbb{F}_1$ for $d = 0$ and to
$\mathbb{F}_{2d-1}$ for $d > 0$. If $d>0$,
$\deg(s_2^*TF^v) \geq 1-2d$ by Lemma \ref{th:beauville}. Since
$\deg(u^*T\CP{1}) = 2d$, it follows from
\eqref{eq:vector-bundle-sequence} that $\deg(s^*TE^v) > 0$.
If $d = 0$, $s_1(z) = (c,z)$ for some $c \in \CP{1}$. The same
argument as before shows that $\deg(s^*TE^v) > 0$ unless $s_2$
is the unique section of $F$ such that $\deg(s_2^*TF^v) = -1$.
It is easy to write down this section explicitly. We conclude
that any holomorphic section $s$ of $\pi$ with
$\deg(s^*TE^v) \leq 0$ belongs to the family
$\sec = \{s_c\}_{c \in \CP{1}}$, where
\begin{equation} \label{eq:explicit-section}
s_c(z) = [1:0] \in \mathbb{P}(\mathbb{C} \oplus H_c^2 \otimes H_z^{-1}) \subset E_z.
\end{equation}
These sections have $\deg(s_c^*TE^v) = -1$.
For any $z \in \CP{1}$, the evaluation map $\sec \longrightarrow E_z$
is an embedding. Its image is the curve of self-intersection $2$
which corresponds to $C^+ = \mathbb{P}(\mathbb{C} \,\oplus\, 0) \subset M$ under
an isomorphism $E_z \iso M$ as above.
We will now verify that the conditions of Proposition
\ref{th:computation} are satisfied.
\begin{enumerate} \item We have already shown that
$\sec$ is connected and that $c_1(TE^v)(S_0) = -1$ for
its $\Gamma$-equivalence class $S_0$.
\item For $s \in \sec$, \eqref{eq:vector-bundle-sequence} reduces to
\[
0 \longrightarrow H^{-1} \longrightarrow s^*TE^v \longrightarrow \mathbb{C}
\longrightarrow 0.
\]
By the exact sequence of cohomology groups,
$H^{0,1}(\CP{1},s^*TE^v) = 0$.
\item is part of Lemma \ref{th:beauville-b}.
\item Consider the curve $C^- = \mathbb{P}(0 \oplus H^2) \subset M$.
$C^-$ is the image of the unique section $s_-$ of $p_2$ with
$\deg(s_-^*T\mathbb{F}_2^v) = -2$. Let $w: \CP{1} \longrightarrow E_z$
be a non-constant holomorphic map with $c_1(TE)(w) < 2$.
By Lemma \ref{th:beauville-b}, the image of $w$ is mapped
to $C^-$ under an isomorphism $i: E_z \iso M$ chosen as above. We have
seen that $i(s(z)) \in C^+$ for all $s \in \sec$.
Since $C^+ \cap C^- = \emptyset$, it follows that $s(z) \notin \mathrm{im}(w)$.
\end{enumerate}
We obtain $Q(E,\O,S_0) = [C^+] \otimes \gen{0}$.
\begin{lemma} \label{th:quantum-relation}
Let $x^+, x^- \in \pi_2(M) \iso H_2(M;\mathbb{Z})$ be the classes of
$C^+, C^-$, and $\bar{x}^+,\bar{x}^- \in H_2(M;\mathbb{Z}/2)$ their
reductions mod $2$. Then
\[
(\bar{x}^+ \otimes \gen{0})^2 =
[M] \otimes \left(\gen{\tfrac{1}{2}(x^+ - x^-)} -
\gen{\tfrac{1}{2}(x^+ + x^-)}\right)
\]
in $(QH_*(M,\o_\lambda),\ast)$. \end{lemma}
\proof In \cite{mcduff87} McDuff showed that $(M,\o_\lambda)$ is
symplectically isomorphic to $\CP{1} \times \CP{1}$ with the
product structure $\lambda(\tau_1 \times 1) + 1 \times \tau_1$.
Such an isomorphism maps $x^{\pm}$ to
$a \pm b$, where $a = [\CP{1} \times \mathit{pt}]$ and
$b = [\mathit{pt} \times \CP{1}]$. Let $\bar{a}, \bar{b}$ the mod $2$
reductions of these classes. The
quantum intersection product on $\CP{1} \times \CP{1}$ is known
(see e.g. \cite[Proposition 8.2 and Example 8.5]{ruan-tian94}); it satisfies
\[
(\bar{a} \otimes \gen{0})^2 = [\CP{1} \times \CP{1}] \otimes \gen{b}, \;
(\bar{b} \otimes \gen{0})^2 = [\CP{1} \times \CP{1}] \otimes \gen{a}.
\]
Because of the $\mathbb{Z}/2$-coefficients, this implies the relation stated
above. \qed
This can be used to give a proof of the following result of McDuff.
\begin{cor} For all $\lambda > 1$,
$[g] \in \pi_1(\mathrm{Ham}(M,\omega_\lambda))$ has infinite order. \end{cor}
\proof Let $\tilde{g}: \widetilde{\loops} \rightarrow \widetilde{\loops}$ be the lift
of $g$ corresponding to the equivalence class $S_0$.
By Theorem \ref{bigth:homomorphism} and Lemma
\ref{th:quantum-relation},
\[
q(g^2,\tilde{g}^2) = Q(E,\O,S_0)^2 =
[M] \otimes \gen{\tfrac{1}{2}(x^+ - x^-)}(\gen{0} - \gen{x^-})
\]
and
\begin{align*}
q(g^{2m},\tilde{g}^{2m}) &= [M] \otimes \gen{\tfrac{m}{2}(x^+ - x^-)}
(\gen{0} - \gen{x^-})^m,\\
q(g^{2m+1},\tilde{g}^{2m+1}) &= \bar{x}^+ \otimes
\gen{\tfrac{m}{2}(x^+ - x^-)} (\gen{0} - \gen{x^-})^m
\end{align*}
for all $m \geq 0$. Since $x^-$ is represented by an algebraic curve,
$\o_\lambda(x^-) > 0$ for all $\lambda > 1$ (in fact,
$\o_\lambda(x^-) = \lambda - 1$). Therefore the class of
$x^-$ in $\Gamma$ has infinite order, and
\[
(\gen{0} - \gen{x^-})^m
\notin \{\gen{\gamma} \; | \; \gamma \in \Gamma\} \subset \Lambda
\]
for all $m \geq 1$. It follows that $q(g^k,\tilde{g}^k) \notin
\tau(\Gamma)$ for all $k>0$. Because of the diagram \eqref{diag:basic},
this implies that $[g^k] \in \pi_1(\mathrm{Ham}(M,\o))$ is nontrivial. \qed
\begin{rem} Using again Theorem \ref{bigth:homomorphism} and
Lemma \ref{th:quantum-relation}, one computes that
\begin{multline*}
q(g^{-1},\tilde{g}^{-1}) = (\bar{x}^+ \otimes \gen{0})^{-1} =\\
= \bar{x}^+ \otimes \gen{\tfrac{1}{2}(x^- - x^+)}
\left(\gen{0} + \gen{x^-} + \gen{2x^-} + \cdots \right).
\end{multline*}
In this case, a direct computation of the invariant seems to be more
difficult because infinitely many moduli spaces contribute to it.
\end{rem}
\bibliographystyle{amsplain}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
| 62,018 |
\section{COMPOSITE SYSTEMS: FROM MOLECULES TO QUARKS}
The first sign that there is an underlying structure at some level
of matter is the existence of an excitation spectrum. Thus molecules
exhibit a spectrum due to the quantised motions of their constituent atoms.
In turn atomic spectra are due to their electrons, those of nuclei are due to
their constituent protons and neutrons and, we now realise, those of the
hadrons are due to their constituent quarks. Qualitatively these look similar
but quantitatively they are radically different. Molecular excitations are on
the scale of $meV$, atoms $eV$, nuclei $MeV$ (note $M$ versus $m$ !) and the
hadrons are hundreds of $MeV$. These represent, inter alia, the different
length scales and binding energies of the respective structures. They also
are related to the different momentum or energy scales needed for probes to
resolve substructures directly.
The discovery of the atomic nucleus was achieved with probes ($\alpha$
particles) that are provided by
natural radioactive sources. These resolved the atom and revealed its
nucleus but saw the latter only as a point of charge; the inner structure
was not then apparent. If the beams are, say, electrons that have
been accelerated to energies of
some hundreds of $MeV$, the protons and neutrons become visible as
individuals. If the beams have energies of tens to hundreds of $GeV$, the
inner structure of the nucleons is resolved and their quarks are directly seen.
I do not wish here to enter into a debate on the detailed relation between
quarks as revealed in the latter ``deep inelastic" experiments and those
that drive spectroscop: it is the latter on which I shall concentrate.
In particular I shall motivate the exciting possibility that new varieties
of hadron are emerging which are associated with the
degrees of freedom available
to the force fields that bind quarks.
So first we need to ask what force holds hadrons together.
The quarks carry electrical charges but also carry an extra charge called
``colour". There are three varieties, let's call them red blue green, and they
attract and repel as do electrical charges: like colours repel
and unlike attract (technically when in
antisymmetric quantum states). Thus three different colours can mutually
attract
and form a baryon but a fourth is left neutral: attracted by two and repelled
by the third. The restriction of attractions to the antisymmetric state
causes the attraction and repulsion to counterbalance, (the importance
of symmetry will be discussed in section 2).
The analogy between colour charge and electrical charge goes further.
By analogy with QED one can form QCD, quantum chromo(or colour) dynamics.
Instead of photons as radiation and force carriers one has (coloured)
gluons. The gluons are in general coloured because a quark that is
coloured Red (say) can turn into a Green by radiating a gluon that is
coloured ``Red-Green"; (technically the $3$ colours form the basis
representation of an $SU(3)$ group and the gluons transform as the
regular representation, the octet - see section 2).
The fact that the gluons carry the colour, or charge, whereas
photons do not carry any charge, causes gluons to propagate differently from
photons. Whereas photons can voyage independently, gluons can mutually
attract en route (they ``shine in their own light"). Not only does this affect
the long range behaviour of the forces but it also suggests that bound states
of pure glue, known as glueballs, may exist. The existence of glueballs and
other
hadrons where glue is excited (``hybrids") will be the focus of these lectures.
First I shall discuss the ways that colour manifests itself in more familiar
systems such as baryons and nuclei.
To illustrate the similarities and differences between QED and QCD we can
list particles and clusters according to whether they feel the force or not.
Those that feel the force may do so because they manifestly carry the charge
(such as electrons or ions in QED or quarks and gluons in QCD) or because
the charge is hidden internally (such as atoms and molecules in QED or
nucleons and nuclei in QCD). Contrast these systems
with those states that do not feel the
force directly as they neither carry nor
contain the charge (such as neutrinos and photons
in QED or leptons in QCD). Note in particular that the force carriers, the
photon and gluon, are in different parts of this matrix; the gluons have
manifest colour charge and feel the QCD forces whereas the photons do not
have electrcial charge and do not directly feel the QED forces.
We can go further and examine the particular set of systems with ``hidden"
charge within them. There are three broad classes of these ``consequential"
forces. In QED the atoms and molecules feel covalent, van der Waals and ionic
forces; the former pair being due essentially to constituent exchange
and two-photon exchange between two separate pairs of constituents
respectively.
The analogues in QCD are quark exchange and two-gluon exchange; there is no
analogue of ionic forces at long range due to the property of confinement
of colour in QCD.
The confinement also breaks the naive similarity between QED and QCD forces
in that the quark exchange (``covalent" force) involves clusters or bags
of colourless combinations of
quark and antiquark known as mesons (of which the light
pion is the most obvious in nuclear forces). Nonetheless we can imagine
nuclei as being the QCD analogues of molecules. The van der Waal, two gluon
exchange, forces
will also be affected by confinement and, presumably, will involve
glueballs as effective exchange objects. However, as the lightest glueballs
are not expected to exist below 1GeV, their effective range is very restricted
and so they are unlikely to affect nuclear forces in any immediately
observable way. The search for colour analogues of van der Waals forces
is likely to be unfruitful in my opinion until someone comes up with a
smart idea.
I shall first show the important role that colour plays for quarks in baryons
and then I shall
contrast baryons, made of three constituent quarks, with the nuclei $^3H$
and $^3He$ which are made of three nucleons. Then I shall concentrate on
colour in mesons, extracting information about the colour forces from
spectroscopy
(section 3). I shall then discuss decays (section 4) in order to contrast with
glueball decays (section 5) before considering a ``realistic" picture where
glueball and quarkonia mix (section 6). The production of glueballs is
discussed in section 7, the phenomenology of hybrids in section 8 and some
attempts to test the hybrid interpretation are in the final section.
\section{COLOUR, THE PAULI PRINCIPLE AND SPIN-FLAVOUR CORRELATIONS}
\subsection{Colour}
The Pauli principle forbids fermions to coexist in the same
quantum state. Historically this created a paradox in that the baryon
$
\Omega^- (S^\uparrow S^\uparrow
S^\uparrow)$ appeared manifestly to violate this.
If quarks possess a property called colour, any quark being able to
carry any one of three colours (say red, green, blue), then the
$\Omega^-$ (and any baryon) can be built from distinguishable
quarks:
$$
\Omega^- (S^\uparrow_R S^\uparrow_G
S^\uparrow_B).
$$
This was how the idea of threefold colour first entered particle physics.
Subsequently the idea developed that colour is the source of a relativistic
quantum field theory, QCD or quantum chromodynamics, and is the source of
the strong forces that bind quarks in hadrons. I shall first discuss the idea
of colour and how, when combined with the Pauli principle, it determines
the properties of baryons. Then I shall develop the idea of it as source
of the interquark forces.
If quarks carry colour but leptons do not, then it is natural to
speculate that colour may be the property that is the source of the
strong interquark forces - absent for leptons.
Electric charges obey the rule ``like repel, unlike attract" and cluster
to net uncharged systems. Colours obey a similar rule: ``like
colours repel, unlike (can) attract". If the three colours form the
basis of an SU(3) group, then they cluster to form ``white" systems -
viz. the singlets of SU(3). Given a random soup of coloured
quarks, the attractions gather them into white clusters, at which
point the colour forces are saturated. Nuclear forces are then the
residual forces among these clusters.
If quark ($Q$) and antiquark ($\bar{Q}$) are the $\underline{3}$
and
$\underline{\bar{3}}$ of colour SU(3), then combining up to three
together gives SU(3) multiplets of dimensions as follows (see e.g.
ref.\cite{Closebook}):
\begin{eqnarray}
QQ = \mathop{3}_{\sim} \times
\mathop{3}_{\sim} = \mathop{6}_{\sim}
+ \mathop{\bar{3}}_{\sim}\nonumber \\
Q\bar{Q} = \mathop{3}_{\sim} \times \mathop{\bar{3}}_{\sim}
=
\mathop{8}_{\sim}+ \mathop{1}_{\sim}\nonumber
\end{eqnarray}
The $Q\bar{Q}$ contains a singlet - the physical mesons. Coloured
gluons belong to the $\mathop{8}$ representation and are confined.
Combining $QQ$ with
a third $Q$ gives \relax
$$
QQQ = \mathop{10}_{\sim}+ \mathop{8}_{\sim}+
\mathop{8}_{\sim}+\mathop{1}_{\sim}.
$$
where the $\mathop{1}\limits_{\sim}$ arose when $QQ$ pairs were in
$\mathop{\bar{3}}\limits_{\sim}$.
Note the singlet in $QQQ$ - the physical baryons.
For clusters of three or less, only $Q\bar{Q}$ and $QQQ$ contain
colour singlets and, moreover, these are the only states realized
physically. Thus are we led to hypothesize that only colour singlets
can exist free in the laboratory; in particular, the quarks will not
exist as free particles.
\subsection{Symmetries and correlations in baryons}
To have three quarks in colour singlet:
$$
1\equiv\frac{1}{\sqrt{6}} [(RB-BR)Y + (YR-RY)B + (BY-YB)R]
$$
any pair is in the $\mathop{\bar{3}}\limits_{\sim}$ and is
antisymmetric. Note that $\mathop{3}\limits_{\sim} \times
\mathop{3}\limits_{\sim} = \mathop{6}\limits_{\sim}
+\mathop{\bar{3}}\limits_{\sim}$. These are explicitly
\begin{eqnarray}
{\mathop{\bar{3}}\limits_{\sim}}_{anti}&
{\mathop{6}\limits_{\sim}}_{sym}\nonumber \\
RB-BR & RB+BR\nonumber \\
RY-YR & RY+YR\nonumber \\
BY-YB & BY+YB\nonumber \\
& RR\nonumber \\
&BB\nonumber \\
& YY
\end{eqnarray}
Note well: \underline{Any Pair is Colour Antisymmetric}\\
The Pauli principle requires total antisymmetry and therefore any
pair must be:
\underline{Symmetric in all else} (``else" means ``apart from colour").\\
This is an important difference from nuclear clusters where the
nucleons have no colour (hence are trivially \underline
{symmetric} in colour!). Hence for nucleons Pauli says
\begin{equation}
\mbox{\underline{Nucleons are Antisymmetric in Pairs}}
\end{equation}
and for quarks
\begin{equation}
\mbox{\underline{Quarks are Symmetric in Pairs}}
\end{equation}
(in all apart from colour).
If we forget about colour (colour has taken care of the
antisymmetry and won't affect us again), then
(i) Two quarks can couple their spins as follows
\begin{equation}
\left\{\begin{array}{rc}S = 1: & \mbox{symmetric}\\
S=0: &\mbox{antisymmetric}\end{array}\right\}
\end{equation}
(ii) Two $u,d$ quarks similarly form isospin states
\begin{equation}
\left\{\begin{array}{rc}I=1: &\mbox{symmetric} \\
I=0: &\mbox{antisymmetric}\end{array}\right\}
\end{equation}
(iii) In the ground state $L=0$ for all quarks; hence the orbital state
is trivially symmetric. Thus for pairs in $L=0$, we have due to Pauli that
\begin{equation}
\left\{\begin{array}{rcrc}S=1 & \mbox{and} & I=1 &
\mbox{correlate} \\
S=0& \mbox{and} & I=0 & \mbox{correlate}\end{array}\right\}
\end{equation}
Thus the $\Sigma^0$ and $\Lambda^0$ which are distinguished
by
their $u,d$ being $I=1$ or $0$ respectively also have the $u,d$ pair
in spin $=1$ or $0$ respectively:
\begin{equation}
\left\{\begin{array}{rcr}\Sigma^0[S(u,d)_{I=1}] &\leftrightarrow
&
S(u,d)_{S=1} \\
\Lambda^0[S(u,d)_{I=0}] &\leftrightarrow &
S(u,d)_{S=0}\end{array}\right\}
\end{equation}
Thus, the spin of the $\Lambda^0$ is carried entirely by the strange
quark.
\subsection{Colour, the Pauli principle and magnetic moments}
The electrical charge of a baryon is the sum of its constituent quark
charges. The magnetic moment is an intimate probe of the
correlations between the charges and spins of the constituents.
Being wise, today we can say that the neutron magnetic moment
was the first clue that the nucleons are not elementary particles.
Conversely the facts that quarks appear to have $g\simeq 2$
suggests that they {\it are} elementary (or that new dynamics is at
work if composite).
A very beautiful demonstration of symmetry at work is the
magnetic moment of two similar sets of systems of three, viz.
\begin{eqnarray}
\left\{\begin{array}{ccl}N ;& P \\
ddu ;& uud\end{array}\right\}
&\mu_p/\mu_N = -3/2\nonumber
\end{eqnarray}
and the nuclei
\begin{eqnarray}
\left\{\begin{array}{ccl}H^3;& He^3\\
NNP;& PPN\end{array}\right\}
&\mu_{He}/\mu_H = -2/3\nonumber
\end{eqnarray}
The Pauli principle for nucleons requires $He^4$ to have {\it no}
magnetic moment:
$$
\mu[He^4;P^\uparrow P^\downarrow N^\uparrow
N^\downarrow ] = 0.
$$
Then
\begin{eqnarray}
He^3 &\equiv He^4 - N\nonumber \\
H^3 &\equiv He^4 - P\nonumber
\end{eqnarray}
and so
$$
\frac{\mu_{He^3}}{\mu_{H^3}} =\frac{\mu_N}{\mu_p}
$$
To get at this result in a way that will bring best comparison with
the nucleon three-quark example, let's study the $He^3$ directly.
$He^3 = ppn: pp$ are flavour symmetric; hence Pauli requires that they be spin
antisymmetric; i.e., $S=0$.
Thus
\begin{equation}
[He^3]^\uparrow\equiv (pp)_0 n^\uparrow
\end{equation}
and so the $pp$ do not contribute to its magnetic moment. The
magnetic moment (up to mass scale factors) is
\begin{equation}
\mu_{He^3} = 0+\mu_n.
\end{equation}
Similarly,
\begin{equation}
\mu_{H^3} = 0+\mu_p.
\end{equation}
which, of course, gives the result that we got before, as it must. But
deriving it this way is instructive as we see when we
study the nucleons in an analogous manner.
The proton contains $u,u$ flavour symmetric and {\it colour
antisymmetric}; thus the spin of the ``like" pair is symmetric
$(S=1)$ in contrast to the nuclear example where this pair had
$S=0$. Thus coupling spin $1$ and spin $1/2$ together, the
Clebsches yield (where subscripts denote $S_z$)
\begin{equation}
p^\uparrow =\frac{1}{\sqrt{3}} (u,u)_0d^\uparrow
+\frac{2}{\sqrt{3}} (u,u)_1d^\downarrow
\end{equation}
(contrast Eq. (8)), and (up to mass factors)
\begin{equation}
\mu_p =\frac{1}{3} (0+d) +\frac{2}{3} (2u-d).
\end{equation}
Suppose that $\mu_{u,d} \propto e_{u,d}$, then
\begin{equation}
\mu_\mu = -2\mu_d
\end{equation}
so
\begin{equation}
\frac{\mu_p}{\mu_N} =\frac{4u-d}{4d-u} =-\frac{3}{2}
\end{equation}
(the neutron follows from proton by replacing $u\leftrightarrow
d$).
I cannot overstress the crucial, hidden role that colour played here
in getting the flavour-spin correlation right.
\section{THE POTENTIAL AND THE FORCE}
The following remarks are by no means rigorous and are intended only to
abstract some general suggestive features about the dynamics from the
spectroscopy of hadrons. They will also enable us to draw up some
empirical guidelines for identifying the nature of light hadrons.
We all know what the spectrum of a Coulomb potential looks like, with
the energy gap between the first two levels already being well on the way
to ionisation energies. The spectrum of hadrons is not like this in that
the gap between 1S and 2S is similar to (though slightly greater than) that
between 2S and 3S, and so on to 3S and 4S etc. The P states are found
slightly above the midway between the corresponding S states. This is
similar to that of a linear potential (which is near enough to a
harmonic that for many purposes the latter is often used for analytical
calculations). A comparison is shown in fig 1.
It is instructive to use the particle data tables\cite{PDG}
and to place the $b\bar{b}$
states on this spectrum, noting the relative energy gaps between the 1S,2S,
3S and the 1P,2P states. Now do the same for $c\bar{c}$ but rescaled
downwards by 6360MeV (so the $\psi(3097)$ and $\Upsilon(9460)$ start the
1S states at the same place). It is remarkable that where corresponding
levels have been identified in the two spectroscopies, there is a rather
similar pattern both qualitatively and even quantitatively (see figs 2a,b).
We shall consider the implications of this for light hadrons later but first
we can abstract the message that the potential between heavy flavours is
linear to a good approximation. This immediately tells us about the
spatial dynamics of the force fields. Let me show you how.
In the case of a U(1) charge, as in electrostatics, the force fields
spread out in space symmetrically in all three dimensions. Thus the
intensity crossing a sphere at distance R dies as the surface area, hence
as $\frac{1}{R^2}$. The potential is the integral of this, hence proportional
to $\frac{1}{R}$, the Coulomb form. We see that the Coulomb potential is
``natural" in a 3-D world.
Contrast this with the empirical message from the $Q\bar{Q}$ spectroscopy,
where $V(R) \sim R$. Here the intensity $\sim \frac{dV}{dR} \sim constant$.
The intensity does not spread at all; it is indeed ``linear". From this
empirical observation we have the picture that the gauge fields, the
gluons, transmit the force as if in a tube of colour flux. This is also
substantiated by computer simulations of QCD (``lattice QCD"). There is
some limited transverse spread but to a first approximation one is
encouraged towards models where a linear flux tube drives the dynamics.
This is what we find for the long range nature of the potential, where
the gluons have mutually interacted while transmitting the force. At
short range one expects there will be a significant perturbation
arising from single gluons travelling between the quarks independently;
this will be akin to the more familiar case of QED where independent
photon exchange generates the $\frac{1}{R}$ behaviour discussed above.
Hence our intuition is that the full potential in QCD will have a structure
along the lines of
\begin{equation}
V(R) \sim \frac{\alpha_s}{R} + aR
\end{equation}
where $\alpha_s$ is the strong coupling strength in QCD and $a$ is a
constant with dimensions of energy per unit length; this is in effect
the tension in the flux tube and empirically is about 1GeV/fermi.
This potential, when plotted on graph paper, looks similar to a log(R)
at the distance scales of hadrons. It is for this reason that the
absolute energy gap between 1S and 2S say is nearly independent of
the constituent mass: the solutions to the Schrodinger equation for
a log potential show that the energy gaps are independent of mass
(for $\frac{1}{R}$ they grow $\sim M$ whereas for $R$ they fall as
$M^{-\frac{1}{3}}$ and the competition ``accidentally" cancels.)
As an aside we can illustrate this.
Consider the Schrodinger equation
$$
(\nabla^2 + 2m_1 R^N) \psi_1(R) = 2m_1 E_1 \psi_1 (R)
$$
and similar form for $m_2$ with $E_2$ and $\psi_2 (R)$. Let $R\rightarrow
\lambda R$ where $\psi_2 (R) \equiv \psi_1 (\lambda R)$. Thus we
compare
$$
(\nabla^2 + 2m_1 \lambda^{N+2} R^N) \psi_1(\lambda R) = 2m_1 E_1
\lambda^2 \psi_1 (\lambda R)
$$
with
$$
(\nabla^2 + 2m_2 R^N) \psi_2( R) = 2m_2 E_2 \psi_2 ( R)
$$
Recognising that $\psi_2 (R) \equiv \psi_1 (\lambda R)$, on the L.H.S. we have
$$
\lambda^2 \equiv (\frac{m_2}{m_1})^{2/(N+2)}
$$
and on the R.H.S.
$$
\frac{E_2}{E_1} \equiv \frac{m_1}{m_2} \lambda^2
$$
Hence
$$
\frac{E_2}{E_1} = (\frac{m_2}{m_1})^{-\frac{N}{N+2}}
$$
shows how the energy levels scale with constituent mass in a potential $R^N$.
We can make a further analogy between QED and QCD via the magnetic
perturbations on the ground states. In hydrogen the magnetic interaction
between electron and proton causes a hyperfine splitting between the
$^3S_1$ and $^1S_0$ levels. This is inversely proportional to the
constituent masses and proportional to the expectation of the wavefunction
at the origin and to $\langle \vec{S_1} \cdot \vec{S_2} \rangle$.
For mesons one finds a similar splitting where
for $Q\bar{q}$ states the $^3S_1$ and $^1S_0$ levels are as follows
\vskip 0.1in
\begin{tabular}{l|lllll}
& K & D & $D_s$ & B & $B_s$ \\
$m(^3S_1)$ & 0.89 & 2.01 & 2.11 & 5.32 & 5.33\\
$m(^1S_0)$ & 0.49 & 1.87 & 1.97 & 5.27 & 5.28 \\
\end{tabular}
(the vector is raised by 1 unit and the pseudoscalar reduced
by three units relative to the unperturbed values; this follows from
$\langle 2\vec{S_1}\cdot \vec{S_2}\rangle \equiv
\langle(\vec{S_1}+\vec{S_2})^2
- 2\vec{S_i}^2\rangle \equiv S(S+1) -\frac{3}{2})$.
Qualitatively we see that the magnitude of the splitting is smaller as
one proceeds to heavier flavours, in line with the inverse mass property
of (chromo)magnetic interactions. Quantitavely the behaviour is interesting.
For a potential $V(R) \sim R^N$ the wavefunction at the origin behaves as
\begin{equation}
\psi(0)^2 \sim \mu_{ij}^{\frac{3}{2+N}}
\end{equation}
where $\mu$ is the reduced mass
\begin{equation}
\frac{1}{\mu_{ij}} = \frac{1}{m_i} + \frac{1}{m_j}
\end{equation}
Now if we assume that
\begin{equation}
\frac{(m_V + m_P)_{ij}}{2} \equiv m_i + m_j
\end{equation}
and note that, in hyperfine splitting
\begin{equation}
m_V - m_P \sim \frac{\psi(0)^2}{m_im_j}
\end{equation}
where $i,j$ are constituent quarks comprising Vector or Pseudoscalar mesons
($q_i\bar{q_j}$), we can find the best value of N in the potential.
If one forms $m_V^2 - m_P^2$ you will see that this is flavour independent to
a remarkable accuracy; then following the above hints you will immediately
see that $N=1$ is preferred; the wavefunctions of the linear potential are
those that fit best in the perturbation expression.
The mean mass of the ground states is nearer to the vector (spin triplet)
than the pseudoscalar (spin singlet). If we look at the
mass gap between the $^3S_1$ 1S and the $^3P_2$ 1P levels, we find
again a remarkable flavour independence, not just for the $b\bar{b}$ and
$c\bar{c}$ already mentioned but for the strange and nonstrange too.
\begin{tabular}{l|rrrrrr}
& $u\bar{d}$ & $u\bar{s}$ & $s\bar{s}$ &$ c\bar{u}$ & $c\bar{c}$
&$b\bar{b}$
\\
$m(^3P_2) $& 1320 & 1430 & 1525 & 2460 & 3550 & 9915 \\
$m(^3S_1)$ & 770 & 892 & 1020 & 2010 & 3100 & 9460\\
gap & 550 & 540 & 500 & 450 & 450 & 450\\
\end{tabular}
Thus although the splittings between $^3S_1$ and $^1S_0$ are strongly mass
dependent, as expected in QCD, the $S-P$ mass gaps are to good approximation
fairly similar across the flavours. Even though the light flavoured states are
above threshold for decays into hadrons, the memory of the underlying
potential remains and, at least empirically, we can produce an ouline
skeleton for the spectroscopic pattern anticipated for all flavours.
I illustrate this in fig 2. The absolute separations of 1S, 2S, 3S and those
of 1P,2P have been taken from the known heavy flavours and rescaled slightly
to make a best fit where the 1D,1F and even 1G are found by the high
spin states in each of these levels. A numerical solution of the
spectrum in a model where $Q\bar{Q}$ are connected by a linear flux-tube is
shown in fig.3; this is indeed very similar to the data and empirical spectrum
illustrated in fig 2c.
Unless certain $J^{PC}$ have strong energy shifts through coupling to
open channels, this should give a reliable guide to the energies of
light hadron multiplets.When we combine the $q\bar{q}$ spins to singlet or
triplet ($S=0,1$) and then combine in turn with the orbital angular
momentum we can construct a set of $^{2S+1}L_J$ states. We shall be interested
later in the possible discovery of a scalar glueball and so we shall also
need to be aware that scalar mesons can be formed in the quark model as $^3P_0$
states. From the figure we anticipate these to lie in the region around $1.2
(n\bar{n}) - 1.6 (s\bar{s})$GeV.
A list of the low lying quarkonium multiplets is given below
\begin{table}[htbp]
\begin{center}
\begin{tabular}{l|ll}
& S=1 triplet & S=0 singlet \\\hline
S & $^3S_1 \; 1^{--}$ & $^1S_0 \; 0^{-+} $\\\hline
P & $^3P_J \; 0^{++} 1^{++} 2^{++}$ & $^1P_1 \; 1^{+-} $\\
\hline
D & $^3D_J \; 1^{--} 2^{--} 3^{--}$ & $ ^1D_2 \; 2^{-+} $\\
\hline
F & $^3F_J \; 2^{++} 3^{++} 4^{++} $ & $ ^1F_3 \; 3^{+-}$\\
\end{tabular}
\end{center}
\end{table}
The spectroscopy of
baryons and mesons is now rather well understood, at least in outline,
to an extent that if there are ``strangers" lurking among the conventional
states, there is a strong likelihood that they can be smoked out.
Such a hope is now becoming important as new states are appearing
and may have a radical implication for our understanding of strong-QCD.
The reason has to do with the nature of gluons. As gluons carry colour charge
and can mutually attract, it is theoretically plausible that gluons can
form clusters that are overall colourless (like conventional hadrons)
but which contain only gluons. These are known as ``glueballs" and would
represent a new form of matter on the 1fm scale.
Glueballs are a missing link of the standard model.
Whereas the gluon degrees of freedom expressed
in $L_{QCD}$ have been established beyond doubt in high
momentum data, their dynamics in the strongly interacting limit
epitomised by hadron spectroscopy are quite obscure. This may be
about to change as a family of candidates for gluonic hadrons (glueballs and
hybrids) is now emerging \cite{amsler94,cafe95,cp94}. These contain both
hybrids around 1.9GeV and a scalar glueball candidate at $f_0(1500)$.
In advance of the most recent data, theoretical arguments
suggested that there may be gluonic activity manifested in the 1.5
GeV mass region. Lattice QCD is the best simulation of theory and
predicts the lightest ``primitive" (ie quenched approximation)
glueball to be $0^{++}$ with mass $1.55 \pm 0.05$ GeV
\cite{ukqcd}. Recent lattice computations place the glueball
slightly higher in mass at $1.74 \pm 0.07$ GeV \cite{weing}
with an optimised value for phenomenology proposed by Teper\cite{teper}
of $1.57 \pm 0.09$ GeV. That lattice QCD computations of the scalar glueball
mass are now concerned with such fine details represents considerable
advance in this field. Whatever the final concensus may be, these
results suggest that scalar mesons in
the 1.5 GeV region merit special attention.
Complementing this has been the growing realisation that there
are now too many $0^{++}$ mesons confirmed for them all to be
$Q\overline{Q}$ states \cite{PDG,amsler94,cafe95,close92}.
I will introduce some of my own prejudices about glueballs and how to
find them. I caution that we have no clear guide and so others may
have different suggestions. At this stage any of us, or none of us,
could be right. We have to do the best we can guided by experience.
It is indeed ironical that the lattice predicts that the lightest glueball
exists in the same region of mass as quarkonium states of the same
$J^{PC} =0^{++}$. If this is indeed the case in nature,
the phenomenology of glueballs may well be more subtle than
naive expectations currently predict.
We shall be interested later in the possible discovery of ``hybrid" states,
where
the gluonic fields are dynamically excited in presence of quarks. Among these
we shall be particularly interested in $0^{-+}, 1^{-+}, 2^{-+}, 1^{--}$ and
possibly $1^{++}$. Note that the $1^{-+}$ configuration does not occur for
$Q\bar{Q}$ and so discovery of such a resonant state would be direct evidence
for dynamics beyond the simple quark model. The other quantum numbers can
be
shared by hybrids and ordinary states. The mass of these lightest hybrids
is predicted to be around $1.9$GeV in a dynamical model where quarks are
connected by a flux tube. The numerical solution of the dynamics is
discussed in ref\cite{bcs} and endorses the earlier estimates by Isgur
and Paton\cite{paton85}.In fig.3 we see a comparison of the predicted
hybrid spectroscopy and that of the conventional states. The mass of the
$2^{-+}$ hybrid is predicted to be tantalisingly close to that of the
conventional
$^1D_2$ with which it shares the same overall $J^{PC}$ quantum numbers.
Comparison with fig.2 shows that this mass region is also near to that of
$3S$ states which include $0^{-+}$ and $1^{--}$, quantum numbers shared with
the lightest hybrids. Furthermore, the $1^{++}$ hybrid shares quantum numbers
with the $2P (^3P_1)$ quarkonium and, following fig.2, we may anticipate that
here too a similarlity in mass may ensue. Thus on mass grounds alone it may
be hard to disentangle hybrids and glueballs from conventional states. It will
be important to investigate both the production and decay patterns of these
various objects.
As regards the decays, we need to study both the flavour dependence in a
multiplet and also the spin and other intrinsic dynamical dependences
that may help to distinguish conventional quarkonia from states where
the gluonic degrees of freedom are excited. We shall therefore first look at
the flavour dependence.
\section{QUARKONIUM DECAY AMPLITUDES}
Let's review some basics of the flavour dependence of
two body decays for a $q\bar{q}$ state of arbitrary flavour.
This will be helpful in assigning meosns to nonets and will also
help us to understand some general features of glueball decays.
Consider a quarkonium state
\begin{equation}
|Q\overline{Q}\rangle = {\rm cos}\alpha |n\bar{n}\rangle -
{\rm sin}\alpha |s\bar{s}\rangle
\end{equation}
where
\begin{equation}
n\bar{n} \equiv (u\bar{u} + d\bar{d})/\sqrt{2}.
\end{equation}
The mixing angle $\alpha$ is related to the usual nonet mixing
angle $\theta$ \cite{PDG} by the relation
\begin{equation}
\alpha = 54.7^{\circ} + \theta.
\end{equation}
For $\theta=0$ the quarkonium state becomes pure SU(3)$_f$
octet, while for $\theta=\pm 90^{\circ}$ it becomes pure singlet.
Ideal mixing occurs for $\theta=35.3^{\circ}$ (-54.7$^{\circ}$)
for which the quarkonium state becomes pure $s\overline{s}$
($\overline{n}n$).
In general we define
\begin{equation}
\eta = {\rm cos}\phi |n\bar{n}\rangle - {\rm sin}\phi
|s\bar{s}\rangle
\end{equation}
and
\begin{equation}
\eta' = {\rm sin} \phi |n\bar{n}\rangle +
{\rm cos} \phi |s\bar{s}\rangle
\end{equation}
with $\phi = 54.7^{\circ} +\theta_{PS}$, where $\theta_{PS}$ is the
usual octet-singlet mixing angle in SU(3)$_f$ basis where
\begin{equation}
\eta = {\rm cos}(\theta_{PS}) |\eta_8\rangle -
{\rm sin}(\theta_{PS}) |\eta_1\rangle,
\end{equation}
\begin{equation}
\eta' = {\rm sin}(\theta_{PS}) |\eta_8\rangle +
{\rm cos}(\theta_{PS}) |\eta_1\rangle.
\end{equation}
The decay of quarkonium into a pair of mesons
$ Q\overline{Q} \rightarrow M(Q\bar{q_i}) M(q_i\bar{Q})$
involves the creation of $q_i\bar{q_i}$ from the vacuum.
If the ratio of the matrix elements for the creation
of $s\bar{s}$ versus $u\bar{u}$ or $d\bar{d}$ is denoted by
\begin{equation}
\rho \equiv \frac{\langle 0|V|s\bar{s}\rangle}{ \langle
0|V|d\bar{d}\rangle},
\end{equation}
then the decay amplitudes of an isoscalar $0^{++}$ (or $2^{++}$)
are proportional to
\begin{eqnarray}
\langleQ\overline{Q}|V|\pi\pi\rangle & = & {\rm cos} \alpha
\nonumber\\
\langleQ\overline{Q}|V|K\overline{K}\rangle & = & {\rm cos}
\alpha (\rho - \sqrt{2} {\rm tan}
\alpha)/2 \nonumber\\
\langleQ\overline{Q}|V|\eta\eta\rangle & = & {\rm cos}
\alpha (1 - \rho \sqrt{2} {\rm tan}
\alpha )/2 \nonumber\\
\langleQ\overline{Q}|V|\eta\eta^\prime\rangle & = & {\rm cos}
\alpha (1 + \rho \sqrt{2} {\rm
tan} \alpha )/2. \nonumber\\
\label{eq:quarkonium0}
\end{eqnarray}
The corresponding decay amplitudes of the isovector are
\begin{eqnarray}
\langleQ\overline{Q}|V|K\overline{K}\rangle & = & \rho/2 \nonumber\\
\langleQ\overline{Q}|V|\pi\eta\rangle & = & 1/\sqrt{2} \nonumber\\
\langleQ\overline{Q}|V|\pi\eta^\prime\rangle & = & 1/\sqrt{2},
\nonumber\\
\label{eq:quarkonium1}
\end{eqnarray}
and those for $K^*$ decay
\begin{eqnarray}
\langleQ\overline{Q}|V|K\pi\rangle & = & \sqrt{3}/2 \nonumber\\
\langleQ\overline{Q}|V|K\eta\rangle & =
& (\sqrt{2}\rho - 1)/\sqrt{8} \nonumber\\
\langleQ\overline{Q}|V|K\eta^\prime\rangle & = & (\sqrt{2}\rho + 1)
/\sqrt{8}. \nonumber\\
\label{eq:quarkonium2}
\end{eqnarray}
For clarity of presentation we have presented
eqn. \ref{eq:quarkonium0},\ref{eq:quarkonium1} and
\ref{eq:quarkonium2} in the approximation where
$\eta \equiv (n\bar{n} - s\bar{s}) / \sqrt{2}$ and
$\eta' \equiv (n\bar{n} + s\bar{s}) / \sqrt{2}$, i.e. for a
pseudoscalar mixing angle $\theta_{PS} \sim -10^{\circ}$ ($\phi =
45^{\circ}$). This is a useful mnemonic; the full expressions for
arbitrary $\eta, \eta'$ mixing angles $\theta_{PS}$ are given in
ref.\cite{cafe95}. Exact SU(3)$_f$ flavour symmetry corresponds to $\rho =
1$; empirically $\rho \geq 0.8$ for well established nonets such as
$1^{--}$ and $2^{++}$ \cite{dok95,Godfrey}.
The partial width into a particular meson pair $M_iM_j$ may be
written as
\begin{equation}
\Gamma_{ij} = c_{ij}|M_{ij}|^2\times |F_{ij} (\vec{q})|^2 \times p.s.
(\vec{q})
\equiv \gamma^2_{ij} \times |F_{ij} (\vec{q})|^2 \times p.s.
(\vec{q})
\label{Gamma}
\end{equation}
where $p.s.(\vec{q})$ denotes the phase-space, $F_{ij} (\vec{q})$
are model-dependent form factors which are discussed in detail in
ref.\cite{cafe95}, $M_{ij}$ is the relevant
amplitude (eqn. \ref{eq:quarkonium0},\ref{eq:quarkonium1} or
\ref{eq:quarkonium2}) and $c_{ij}$ is a weighting factor arising
from the sum over the various charge combinations, namely 4 for
$K\bar{K}$, 3 for $\pi\pi$, 2 for $\eta\eta^\prime$ and 1 for
$\eta\eta$ for isoscalar decay (eqn. \ref{eq:quarkonium0}), 4 for
$K\bar{K}$, 2 for $\pi\eta$ and 2 for $\pi\eta^\prime$ for
isovector decay (eqn. \ref{eq:quarkonium1}) and 2 for $K^*$
decays (eqn. \ref{eq:quarkonium2}). The dependence of
$\gamma^2_{ij} = c_{ij}|M_{ij}|^2$ upon the mixing angle $\alpha$
is shown in fig. \ref{alpha}a for the isoscalar decay in the case of
SU(3)$_f$ symmetry, $\rho = 1$.
The figure illustrates some general points.
An $s\bar{s}$ state corresponds to $\alpha = 90^0$ for which $\pi \pi$
vanishes. The $K\bar{K}$ vanishes when there is destructive interference
between $s\bar{s}$ and $n\bar{n}$; notice that the $\eta\eta$ tends to
vanish here also as it tends to be roughly $\frac{1}{4}$ of the $K\bar{K}$
independent of the mixing angle $\alpha$ (this would be an exact relation
for the ideal $\eta$ used in the text; the figure shows the results for
realistic $\eta$ flavour composition). This correlation between $\eta\eta$ and
$K\bar{K}$ is expected for any quarkonium state and a violation in data
will therefore be significant in helping identify ``strangers".
This pattern of decays is expected to hold true for any meson that
contains $q\bar{q}$ in its initial configuration. Thus it applies to
conventional or to hybrid multiplets and distinguishing between them will
depend on dynamical features associated with the gluonic excitation
or the spin states of the quarks. The case of glueballs is qualitatively
different in that there is no intrinsic flavour present initially and
so the pattern of decays will depend, inter alia, on the dynamics of
flavour creation.
The traditional assumption has been that as glueballs are flavour singlets,
their decays should be analogous to those of a flavour singlet quarkonium.
The case of a flavour singlet corresponds to $\alpha = -30^0$ (or $150^0$).
Here we see that $\eta \eta' \rightarrow 0$ and the other channels are
populated in proportion to their charge weighting (namely 4:3:1 for $K\bar{K}
:\pi\pi:\eta\eta$). A flavour singlet glueball would be expected to
show these ratios too if it decays through a flavour singlet
intermediate state.
We can now look into the decays of glueballs by
finding examples of decays where gluons are already believed to play a role.
The data are sparse and do show consistency with the flavour singlet idea;
however, one must exercise caution before applying this too widely. I shall
first illustrate the flavour singlet phenomenon as it manifests itself
for gluonic systems at energies far from the mass scales of light-flavoured
quarkonium. Then I shall investigate what modifications may be expected
for glueballs at mass scales of 1-2GeV where quarkonium states with the same
$J^{PC}$ may contaminate the picture.
\section{PRIMITIVE GLUEBALL DECAYS}
The decays of $c\bar{c}$, in particular $\chi_{0,2}$, provide a
direct window on $G$ dynamics in the $0^{++},2^{++}$ channels
insofar as the hadronic decays are triggered by $c\bar{c}
\rightarrow gg \rightarrow Q\bar{Q}Q\bar{Q}$ (fig.
\ref{3graphs}a). It is necessary to keep in mind that these are in a
different kinematic region to that appropriate to our main
analysis but, nonetheless, they offer some insights into the gluon
dynamics. Mixing between hard gluons
and $0^{++}$, $2^{++}$ $Q\bar{Q}$ states (fig. \ref{3graphs}c)
is improbable at these energies as the latter 1 - 1.5 GeV
states will be far off their mass-shell. Furthermore,the
narrow widths of $\chi_{0,2}$ are consistent with
the hypothesis that the 3.5 GeV region is remote from the
prominent
$0^+,2^+$ glueballs, $G$. Thus we
expect that the dominant decay dynamics is triggered by hard
gluons directly fragmenting into two independent $Q\bar{Q}$
pairs (fig. \ref{3graphs}a) or showering into lower energy gluons
(fig. \ref{3graphs}b). We consider the former case now; mixing
with $Q\bar{Q}$ (fig. 6c) and $G \rightarrow GG$ (fig. 6b) will be discussed in
section 6.
\subsection*{$ G \rightarrow QQ\bar{Q}\bar{Q}$}
This was discussed in ref. \cite{closerev} and
the relative amplitudes for the process shown in fig.
\ref{3graphs}a
read
\begin{eqnarray}
\langle G|V|\pi\pi\rangle & = & 1 \nonumber\\
\langle G|V|K\overline{K}\rangle & = & R \nonumber\\
\langle G|V|\eta\eta\rangle & = & (1+ R^2)/2 \nonumber\\
\langle G|V|\eta\eta^\prime\rangle & = & (1- R^2)/2,
\nonumber\\
\label{b}
\end{eqnarray}
with generalizations for arbitrary pseudoscalar mixing angles
given in ref.\cite{cafe95} and where $R \equiv \langle
g|V|s\bar{s}\rangle/\langle
g|V|d\bar{d}\rangle$. SU(3)$_f$ symmetry corresponds to
$R^2=1$. In this case the relative branching ratios (after weighting
by the number of charge combinations) for the decays
$\chi_{0,2} \rightarrow \pi\pi,\eta\eta,\eta\eta',K\bar{K}$
would be in the relative ratios 3 : 1 : 0 : 4. Data for $\chi_0$ are in
accord with this where the branching ratios are (in parts per mil)
\cite{PDG}:
\begin{eqnarray}
B(\pi^0\pi^0) & = & 3.1 \pm 0.6 \nonumber\\
\frac{1}{2}B(\pi^+\pi^-) & = & 3.7 \pm 1.1 \nonumber\\
\frac{1}{2}B(K^+K^-) & = & 3.5 \pm 1.2 \nonumber\\
B(\eta\eta) & = & 2.5 \pm 1.1. \nonumber\\
\label{chidata}
\end{eqnarray}
No signal has been reported for $\eta\eta'$.
Flavour symmetry is manifested in the decays of $\chi_2$ also:
\begin{eqnarray}
B(\pi^0\pi^0 )& = & 1.1 \pm 0.3 \nonumber\\
\frac{1}{2}B(\pi^+\pi^-) & = & 0.95 \pm 0.50 \nonumber\\
\frac{1}{2}B(K^+K^-) & = & 0.75 \pm 0.55 \nonumber\\
B(\eta\eta) & = & 0.8 \pm 0.5, \nonumber\\
\label{chidata2}
\end{eqnarray}
again in parts per mil. The channel $\eta\eta'$
has not been observed either. These results are natural as they
involve hard gluons away from the kinematic region where $G$
bound states dominate the dynamics. If glueballs occur at lower
energies and mix with nearby $Q\bar{Q}$ states, this will in
general lead to a distortion of the branching ratios from the
``ideal" equal weighting values above. (A pedagogical example
will be igven in the next section). It will
also cause significant mixing
between $n\bar{n}$ and $s\bar{s}$ in the quarkonium
eigenstates. Conversely, ``ideal" nonets, where the
quarkonium eigenstates are $n\bar{n}$ and $s\bar{s}$, are
expected to signal those $J^{PC}$ channels where the masses of the
prominent glueballs are remote from those of the quarkonia.
An example of this is the $2^{++}$ sector where the quarkonium
members are ``ideal" which suggests that $G$ mixing is nugatory in this
channel.
These data collectively suggest
that prominent $2^{++}$ glueballs are not in the $1.2 -
1.6$ GeV region which in turn is consistent with lattice calculations where
the mass of the $2^{++}$ primitive glueball is predicted to be larger than 2
GeV. The sighting of a $2^{++}$ state in the glueball favoured central
production, decaying into $\eta\eta$ with no significant $\pi\pi$\cite{2170}
could be the first evidence for this state. There are also interesting
signals from BES on a narrow state in this mass region seen in $\psi
\rightarrow \gamma MM$ where $MM$ refer to mesons pairs, $\pi\pi,
K\bar{K}$
with branching ratios consistent with flavour symmetry\cite{beijing}.
\section{$Q\bar{Q}$ AND GLUEBALL DECAYS IN STRONG COUPLING QCD}
In the strong coupling ($g\rightarrow\infty$) lattice formulation
of QCD, hadrons consist of quarks and flux links, or flux tubes, on
the lattice. ``Primitive" $Q\bar{Q}$ mesons consist of
a quark and antiquark connected by a tube of coloured
flux whereas primitive glueballs consist of a loop of flux
(fig. \ref{Rivsx}a,b) \cite{paton85}. For finite $g$ these eigenstates remain a
complete basis set for QCD but are perturbed by two types of
interaction \cite{kokoski87}:
\begin{enumerate}
\item
$V_1$ which creates a $Q$ and a $\bar{Q}$ at neighbouring lattice
sites, together with an elementary flux-tube connecting them, as
illustrated in fig. \ref{Rivsx}c,
\item
$V_2$ which creates or destroys a unit of flux around any
plaquette (where a plaquette is an elementary
square with links on its edges), illustrated in fig. \ref{Rivsx}d.
\end{enumerate}
The perturbation $V_1$ in leading order causes decays of
$Q\bar{Q}$ (fig. \ref{Rivsx}e) and also induces mixing
between the ``primitive" glueball $(G_0)$ and $Q\bar{Q}$
(fig. \ref{Rivsx}f). It is perturbation $V_2$ in leading order that
causes glueball decays and leads to a final state consisting of
$G_0G_0$ (fig. \ref{Rivsx}g); decays into $Q\bar{Q}$ pairs occur at
higher order, by application of the perturbation $V_1$ twice. This
latter sequence effectively causes $G_0$ mixing with $Q\bar{Q}$
followed by its decay. Application of $V_1^2$ leads
to a $Q^2\bar{Q}^2$ intermediate state which then turns into
colour singlet mesons by quark rearrangement (fig.
\ref{3graphs}a); application of $V_2$ would lead to direct
coupling to glue in $\eta, \eta'$ or $V_2\times V_1^2$ to their
$Q\overline{Q}$ content (fig. \ref{3graphs}b).
The absolute magnitudes of these various contributions require
commitment to a detailed dynamics and are beyond the scope
of this first survey. We concentrate here on their {\bf relative}
contributions to the various two body pseudoscalar meson final
states available to $0^{++}$ meson decays.For $Q\bar{Q}
\rightarrow Q\bar{q}q\bar{Q}$ decays induced by $V_1$, the
relative branching ratios are given in eqn. \ref{eq:quarkonium0}
where one identifies
\begin{equation}
\rho \equiv \frac{\langle Q\bar{s} s\bar{Q} |V_1| Q\bar{Q}
\rangle }{\langle Q\bar{d} d\bar{Q}|V_1| Q\bar{Q}\rangle}.
\label{rho}
\end{equation}
The magnitude of $\rho$ and its dependence on $J^{PC}$ is a
challenge for the lattice. We turn now to consider the effect of
$V_1$ on the initial ``primitive" glueball $G_0$. Here too we allow
for possible flavour dependence and define
\begin{equation}
R^2 \equiv \frac{\langle s\bar{s} |V_1| G_0 \rangle}
{\langle d\bar{d} |V_1| G_0 \rangle }.
\label{r2}
\end{equation}
The lattice may eventually guide us on this magnitude and also on
the ratio $R^2/\rho$. In the absence of this information we shall
leave $R$ as free parameter and set $\rho=1$.
\subsection{Glueball-$Q\overline{Q}$ mixing at $O(V_1)$}
In this first orientation we shall consider mixing between $G_0$
(the primitive glueball state) and the quarkonia, $n\bar{n}$ and
$s\bar{s}$, at leading order in $V_1$ but will ignore that between
the two different quarkonia which is assumed to be higher order
perturbation.
The mixed glueball state is then
\begin{equation}
G = |G_0\rangle + \frac{|n\bar{n}\rangle \langle
n\bar{n}|V_1|G_0 \rangle}{E_{G_0}-E_{n\bar{n}}}
+ \frac{|s\bar{s}\rangle \langle s\bar{s}|V_1|G_0 \rangle}{E_{G_0}-
E_{s\bar{s}}}
\label{perturb}
\end{equation}
which may be written as
\begin{equation}
G = |G_0\rangle + \frac{\langle
n\bar{n}|V_1|G_0\rangle}{\sqrt{2}(E_{G_0}-
E_{n\bar{n}})}\{\sqrt{2}
|n\bar{n}\rangle +\omega R^2 |s\bar{s}\rangle \}
\end{equation}
where
\begin{equation}
\omega \equiv \frac{E_{G_0}-E_{n\bar{n}}}{ E_{G_0}-E_{s\bar{s}}}
\label{omega}
\end{equation}
is the ratio
of the energy denominators for the $n\bar{n}$ and $s\bar{s}$
intermediate states in old fashioned perturbation theory (fig.
\ref{3graphs}d).
Denoting the dimensionless mixing parameter by
\begin{equation}
\xi \equiv \frac{\langle d\bar{d}|V_1|G_0 \rangle }{E_{G_0}-
E_{n\bar{n}}},
\end{equation}
the eigenstate becomes, to leading order in the perturbation,
\begin{eqnarray}
N_G |G \rangle = |G_0 \rangle + \xi \{\sqrt{2}|n\bar{n}\rangle +
\omega R^2 |s\bar{s} \rangle \}
\equiv |G_0 \rangle + \sqrt{2}\xi |Q\overline{Q}\rangle \nonumber\\
\label{3states}
\end{eqnarray}
with the normalization
\begin{eqnarray}
N_G = \sqrt{1 + \xi^2 (2 + \omega^2 R^4)} ,\nonumber\\
\end{eqnarray}
Recalling our definition of quarkonium mixing
\begin{equation}
|Q\overline{Q} \rangle = {\rm cos} \alpha |n\bar{n} \rangle -
{\rm sin} \alpha |s\bar{s}
\rangle
\end{equation}
we see that $G_0$ has mixed with an effective quarkonium of
mixing angle $\alpha$ where $\sqrt{2}{\rm tan} \alpha = -
\omega R^2$ (eqn . \ref{eq:quarkonium0}). For example, if
$\omega R^2 \equiv 1$, the SU(3)$_f$ flavour symmetry
maps a glueball onto quarkonium where tan$\alpha = -
1/\sqrt{2}$ hence $\theta=-90^{\circ}$, leading to the familiar
flavour singlet
\begin{equation}
|Q\overline{Q} \rangle = |u\bar{u} +d\bar{d} +s\bar{s} \rangle /\sqrt{3}.
\end{equation}
When the glueball is far removed in mass from the $Q\bar{Q}$,
$\omega \rightarrow 1$ and flavour symmetry ensues;
the $\chi_{0,2}$ decay and the $2^{++}$ analysis earlier
are examples of this ``ideal" situation. However, when $\omega
\neq 1$, as will tend to be the case when $G_0$ is in the vicinity
of the primitive $Q\bar{Q}$ nonet (the $0^{++}$ case of interest
here), significant distortion from naive flavour singlet can arise.
In particular lattice QCD
suggests that the ``primitive" scalar glueball $G_0$
lies at or above 1500 MeV, hence above the $I=1$ $Q\bar{Q}$
state $a_0(1450)$ and the (presumed) associated $n\bar{n}$
$f_0(1370)$. Hence $E_{G_0}-E_{n\bar{n}} >0$ in the numerator of
$\omega$. The $\Delta m = m_{s\bar{s}} - m_{n\bar{n}} \approx 200-300$
MeV suggests that the primitive $s\bar{s}$ state is in the region
1600-1700 MeV. Hence it is quite possible that the primtive glueball is
in the vicinity of the quarkonium nonet, maybe in the middle of it. Indeed,
the suppression of $K\bar{K}$ in the $f_0(1500)$ decays suggests
a destructive interference between $n\bar{n}$ and $s\bar{s}$
such that $\omega R^2 < 0$. This
arises naturally if the primitive glueball mass is
between those of $n\bar{n}$ and the primitive $s\bar{s}$. As the
mass of $G_0 \rightarrow m_{n\bar{n}}$ or $m_{s\bar{s}}$, the
$K\bar{K}$ remains suppressed though non-zero; thus eventual
quantification of the $K\bar{K}$ signal will be important.
The decay into pairs of glueballs, or states such as $\eta$ that
appear to couple to gluons, is triggered by the perturbation $V_2$.
This can drive decays into $\eta\eta$ and is discussed in ref.\cite{cafe95}.
This breaks the connection between $\eta\eta$ and $K\bar{K}$ that is
a signature for quarkonium as illustrated earlier. The phenomenology
of the $f_0(1500)$ appears to have these features.
If the $f_0(1550 \pm 50)$ becomes accepted as a
scalar glueball, consistent with the predictions of the lattice, then
searches for the $0^{-+}$ and especially the $2^{++}$ at mass
2.22 $\pm$ 0.13 GeV \cite{teper} may become
seminal for establishing the lattice as a successful calculational laboratory.
There are tantalising indications of a state produced in $\psi \rightarrow
\gamma 0^- 0^-$ at BES whose decays may be consistent with those of
a flavour blind glueball (flavour blind as it is removed from the
prominent quarkonia of the same quantum numbers)\cite{beijing}.
It also adds confidence to the predictions that gluonic degrees of freedom
are excited in the 2GeV mass region when $q\bar{q}$ ``seeds" are already
present. Such states are known as hybrids and these too may be showing up
(see later).
\section{PRODUCTION RATES}
There are two main phenomenological pillars on which glueball
phenomenology now tend to agree. These are their mass spectroscopy
(at least for the lightest few states), and their optimised production
mechanisms. We shall see that the ``interesting" states appear to share
these properties.
Meson spectroscopy has been studied for several decades and the spectrum of
$q\bar{q}$ states has emerged. Why in all this time has it been so hard
to identify glueballs and hybrids if they exist below 2GeV?
Some time ago I suggested \cite{closerev} this
to be due to the experimental concentration on a restricted class of
production mechanisms and on final states
with charged pions and kaons. We will consider each of these in turn.
Experiments historically have tended to use beams of quarks (contained within
hadrons) hitting targets which are also quark favoured. The emergence
of states made from quarks was thereby emphasised. To enhance any
gluonic signal above the quark ``noise" required one to destroy the
quarks. Hence the focussing on three particular production mechanisms,
\cite{closerev}
in each of which the candidate scalar glueball\cite{cafe95} has been seen.
\begin{enumerate}
\item
Radiative $J/\psi$ decay: $J/\psi \rightarrow \gamma+G$
\cite{bugg}
\item
Collisions in the central region away from quark beams and target:
$pp \rightarrow p_f(G)p_s$ \cite{Kirk,Gentral}.
\item
Proton-antiproton annihilation where the destruction of quarks
creates opportunity for gluons to be manifested. This is the Crystal
Barrel \cite{Anis}-\cite{Enhan} and E760
\cite{Hasan1,Hasan2} production mechanism in which detailed
decay systematics of $f_0(1500)$ have been studied.
\item
Tantalising further hints come from the claimed sighting \cite{had95}
of the $f_0(1500)$ in decays of the hybrid meson candidate \cite{cp94}
$\pi(1800) \rightarrow \pi f_0(1500) \rightarrow \pi \eta \eta$.
\end{enumerate}
The signals appear to be prominent in decay channels such as
$\eta\eta$ and $\eta\eta'$ that are traditionally regarded as
glueball signatures. This recent emphasis on
neutral final states (involving $\pi^0$, $\eta$, $\eta'$) was
inspired by the possibiblity that $\eta$ and $\eta'$ are strongly
coupled to glue and reinforced by the earlier concentrations on
charged particles. This dedicated study of neutrals was a new
direction pioneered by the GAMS Collaboration at CERN
announcing new states decaying to $\eta\eta$ and
$\eta\eta'$ \cite{Aldeall}. Note from the decays of quarkonia, fig 4,
the channels $\eta\eta$ and $K\bar{K}$ are strongly correlated for
quarkonia. Thus observation of states that couple strongly to $\eta$
are signatures for non-quarkonia and, to the extent that $\eta$ couples
to glue, may be a glueball signature.
These qualitative remarks are now becoming more quantitative following
work on $\psi$ radiative decays that is currently being extended
\cite{cak,zpli}
By combining the known B.R.
$(\psi\rightarrow\gamma R$) for any resonance $R$ with perturbative QCD
calculation of $\psi\rightarrow\gamma (gg)_R$ where the two gluons are
projected onto the $J^{PC}$ of $R$,
one may estimate the gluon branching
ratio $B(R\rightarrow gg$). One may expect that
\begin{equation}
\begin{array}{lcl}
B(R[Q\bar{Q}] \rightarrow gg)& =& 0(\alpha^2_s) \simeq 0.1\nonumber\\
B(R[G] \rightarrow gg)& =& \frac{1}{2} \; \mbox{to} \; 1\nonumber\\
\end{array}
\end{equation}
Known $Q\bar{Q}$ resonances (such as $f_2$(1270)) satisfy the former; we seek
examples of the latter.
For example, perturbative QCD gives
$$
B(\psi \rightarrow \gamma^3P_J) = \frac{128}{5} \frac{\alpha \alpha_s}{\pi}
\frac{1}{(\pi^2-9)} \frac{|R^\prime_p (0) |^2}{m^3 M^2} x|H_J |^2
$$
where $m, M$ are the resonance and $\psi$ masses respectively, $R^\prime_p
(0) $ is the derivative of the $P-$state wavefunction at the origin and the $J$
dependent quantity $x|H |^2$ is plotted in fig 7. One can manipulate the
above formula into the form, for scalar mesons
$$
10^3 B(\psi \rightarrow \gamma 0^{++}) = (\frac{m}{1.5\; GeV})
(\frac{\Gamma_{R\rightarrow gg}}{96\; MeV}) \frac{x|H|^2}{35}
$$
The analysis of ref. \cite{bugg} suggests
$ B(\psi \rightarrow \gamma f_0 (1500)\simeq 10^{-3})$.
Thus a very broad $Q\bar{Q}$ state (width
$\sim$ 500 MeV) could be present at this level, but for $f_0$(1500) with
$\Gamma_T$ = 100-150 MeV, one infers $B(f_0\rightarrow gg)$ = 0.6 to 0.9
which is far from $Q\bar{Q}$.
Such arguments need more
careful study but do add to the interest in the $f_0$(1500).
Thus
the $f_0$(1500) has the right mass and is produced in the right places to be a
glueball and with a strength (in $\psi \rightarrow \gamma f_0$) consistent
with a glueball.
Its total width is out of line with expectations for a $Q\bar{Q}$\cite{cafe95}.
Its branching
ratios are interesting and may also signify a glueball that is mixed in
with the neighbouring $Q\bar{Q}$ nonet. It is a state for
which data are accumulating and will be worth watching.
\section{THE HYBRID CANDIDATES}
The
origins of the masses of gluonic excitations on the lattice are known only to
the computer. Those in the flux tube have some heuristic underpinning. The
$Q\bar{Q}$
are connected by a colour flux with tension 1 GeV/fm which leads to
a linear potential in accord with the conventional spectroscopy (section 3).
The
simplest glue loop is based on four lattice points that are the corners of a
square.
As lattice spacing tends to zero one has a circle, the diameter is $\simeq$
0.5 fm, the circle of flux length is then $\simeq$ 1.5 fm and, at 1 GeV/fm, the
ballpark
1.5 GeV mass emerges. In the limit of lattice spacing vanishing, its 3-D
realisation is a sphere, and hence it is natural that this is $0^{++}$.
The next simplest configuration is based on an oblong, one link across and two
links long.
The total length of flux is $\simeq \frac{3}{2} $ larger than the square
and the ensuing mass $\simeq \frac{3}{2} \times $ 1.5 GeV $\simeq $ 2.2 GeV.
In the
3-D continuum
limit this rotates into a rugby ball shape rather than a sphere. A
decomposition in spherical harmonics contains $L \geq 0$, in particular
$2^{++}$.
This is by no means rigorous (!) but may help to give a feeling for the
origin of the glueball systematics in this picture, inspired by the lattice.
Finally
one has the prediction that there exist states where the gluonic degrees of
freedom
are excited in the presence of $Q\bar{Q}$. With the 1 GeV/fm setting the
scale,
one finds \cite{bcs,paton85}
that the lightest of these ``hybrid" states have masses of
order 1 GeV above their conventional $q\bar{q}$ counterparts. Thus hybrid
charmonium may exist at around 4 GeV, just above the $D\bar{D}$ pair
production
threshold. More immediately accessible
are light quark hybrids that are expected
in the 1.5 to 2 GeV range after spin dependent mass splittings are allowed for.
There are
tantalising sightings of an emerging spectroscopy as I shall now review.
It is well known that hybrid mesons can have $J^{PC}$ quantum numbers
in combinations such as $0^{--},0^{+-},
1^{-+}, 2^{+-}$ etc. which are unavailable to conventional mesons and as
such provide a potentially sharp signature.
It was noted in ref.\cite{kokoski85} and confirmed in ref.\cite{cp95}
that the best opportunity for isolating exotic hybrids appears
to be in the $1^{-+}$ wave where, for the I=1 state with mass around 2 GeV,
partial widths are typically
\begin{equation}
\label{bnlwidth}
\pi b_1 : \pi f_1 : \pi \rho \;
= \; 170 \; MeV : 60 \; MeV : 10 \; MeV
\end{equation}
The narrow $f_1(1285)$ provides a useful tag for the
$1^{-+} \rightarrow \pi f_1$ and ref.\cite{lee94} has recently reported a
signal
in $\pi^- p \rightarrow (\pi f_1) p$ at around 2.0 GeV
that appears to have a resonant phase.
Note
the prediction that the $\pi \rho$ channel is
not negligible relative to the signal channel
$\pi f_1$
thereby resolving the puzzle of the production
mechanism that was commented upon in ref. \cite{lee94}.
This state may also have been sighted in photoproduction \cite{utk}
with $M=1750$ and may be the $X(1775)$ of the Data Tables, ref.\cite{PDG}.
A recent development
is the realisation that even when hybrid and conventional mesons
have the {\bf same} $J^{PC}$ quantum numbers, they may still be distinguished
\cite{cp95} due to their different internal structures which give rise to
characteristic selection rules\cite{pene,paton85,cp95}. As an example
consider the $\rho(1460)$.
(i) If $q\bar{q}$ in either hybrid or conventional mesons
are in a net spin singlet configuration then the dynamics of the flux-tube
forbids decay into final states consisting only of spin singlet mesons.
For $J^{PC}=1^{--}$ this selection rule
distinguishes between conventional vector mesons
which are $^3S_1$ or $^3D_1$ states and hybrid vector mesons where the
$Q\bar{Q}$ are coupled to a spin singlet. This implies that in the decays of
hybrid $\rho$, the channel
$\pi h_1$ is forbidden whereas $\pi a_1$ is allowed and
that $\pi b_1$ is analogously suppressed for hybrid $\omega$ decays; this
is quite opposite to the case of $^3L_1$ conventional mesons where the
$\pi a_1$ channel is relatively suppressed and $\pi h_1$ or $\pi b_1$
are allowed\cite{busetto,kokoski87}. The extensive analysis of data
in ref.\cite{don2} revealed the clear presence of $\rho(1460)$\cite{PDG}
with a strong $\pi a_1$ mode but no sign of $\pi h_1$,
in accord with the hybrid situation. Furthermore, ref.\cite{don2}
finds evidence for $\omega (1440)$ with no visible
decays into $\pi b_1$ which again contrasts with
conventional $q\bar{q}$ $(^3S_1$ or $^3D_1$) initial states and in accord
with the hybrid configuration.
(ii) The dynamics of the excited flux-tube in the {\bf hybrid} state
suppresses the decay to mesons where the $q\bar{q}$ are $^3S_1$ or $^1S_0$
$``L=0"$ states. The preferred decay channels are to ($L=0$) + ($L=1$)
pairs\cite{paton85,kokoski85}. Thus
in the decays of hybrid $\rho \rightarrow 4\pi$ the $\pi a_1$ content is
predicted to be dominant and the $\rho \rho$ to be absent. The
analysis of ref.\cite{don2} finds such a pattern for $\rho(1460)$.
(iii) The selection rule forbidding ($L=0$) + ($L=0$) final states
no longer operates if the internal structure or size of the two ($L=0$) states
differ\cite{paton85,pene}.
Thus, for example, decays to $\pi + \rho$, $\pi + \omega$ or $K + K^*$
may be significant in some cases\cite{cp95,cp951},
and it is possible that the {\bf production}
strength could be significant where an exchanged
$\pi, \rho$ or $\omega$ is involved,
as the
exchanged off mass-shell state may have different structure to the incident
on-shell beam particle.
This may be particularly pronounced in the case
of {\bf photoproduction} where couplings to $\rho \omega$ or $\rho \pi$
could be considerable when the $\rho$ is effectively replaced
by a photon and the $\omega$ or $\pi$ is exchanged.
This may explain the production of the candidate exotic
$J^{PC}=1^{-+}$ (ref.\cite{lee94}) and a variety of anomalous
signals in photoproduction.
The first calculation of the widths and branching ratios of hybrid
mesons with conventional quantum numbers is in ref.\cite{cp95}:
the
$0^{-+},2^{-+}$ and the
$1^{--}$ are predicted to be potentially accessible.
It is therefore interesting that each of these $J^{PC}$ combinations
shows rather clear
signals with features characteristic of hybrid dynamics and which do not
fit naturally into a tidy $Q\bar{Q}$ conventional classification.
We have already mentioned the $1^{--}$.
Turning to the $0^{-+}$ wave,
the VES Collaboration at Protvino confirm their enigmatic
and clear $0^{-+}$ signal in diffractive production with 37 GeV
incident pions on beryllium \cite{had95}. Its mass and decays typify those
expected for a hybrid: $M \approx 1790$ MeV, $\Gamma \approx 200$ MeV
in the $(L=0)$ + $(L=1)$ $\bar{q}q$
channels $\pi^- + f_0; \; K^- + K^*_0, \; K {( K \pi )}_S $ with no
corresponding strong signal in the kinematically allowed $L=0$ two body
channels $\pi + \rho; \; K + K^*$. This confirms the earlier sighting
by Bellini et al\cite{bellini}, listed in the Particle Data group\cite{PDG}
as $\pi(1770)$.
The resonance also appears to couple as strongly to
the enigmatic $f_0(980)$ as it does to $f_0(1300)$,
which was commented upon with some surprise in ref. \cite{had95}.
This may be natural for a hybrid at this mass due to the
predicted dominant $KK_0^*$ channel which will feed
the $(KK\pi)_S$ (as observed \cite{had95}) and hence the channel
$\pi f_0(980)$ through the strong affinity of $K\bar{K} \rightarrow f_0(980)$.
Thus the overall expectations for hybrid $0^{-+}$ are in line with
the data of ref.\cite{had95}. Important tests are now that there should be
a measureable decay to the $\pi \rho$ channel with only a small
$\pi f_2$ or $KK^*$ branching ratio.
At the Hadron95 conference it was learned
that in the $\pi \eta\eta$ final state the glueball
candidate is seen: $\pi(1.8) \rightarrow \pi f_0(1500)
\rightarrow \pi \eta\eta$.
This leaves us with the $2^{-+}$.
There are clear signals of unexplained activity in the
$2^{-+}$ wave in several experiments for which a hybrid interpretation
may offer advantages. These are discussed in ref.\cite{cp95}.
These various signals in the desired channels provide a potentially consistent
picture. The challenge now is to test it. Dedicated high statistics
experiments with the power of modern detection and analysis should re-
examine
these channels. Ref.\cite{cp951} suggests that the hybrid couplings are
especially
favourable in
{\it low-energy} photoproduction and as such offer a rich opportunity
for the programme at an upgraded CEBAF
or possibly even at HERA. If the results of ref.\cite{atkinson}
are a guide, then photoproduction may be an important gateway at a range of
energies and the channel $\gamma + N \rightarrow (b_1 \pi) + N$ can
discriminate
hybrid $1^{--}$ and $2^{-+}$ from their conventional counterparts.
Thus to summarise, we suggest that data are consistent with the existence of
low lying multiplets of hybrid mesons based on the mass spectroscopic
predictions of ref.\cite{paton85} and the production and decay dynamics of ref.
\cite{cp95}. Specifically the data include
\begin{eqnarray}
0^{-+} & (1790 \; MeV; \Gamma = 200 \; MeV) & \rightarrow
\hspace{0.2cm} \pi f_0 ; K\bar{K}\pi \\
\nonumber
1^{-+} & (\sim 2 \; GeV; \Gamma \sim 300 \; MeV) & \rightarrow
\hspace{0.2cm} \pi f_1 ; \pi b_1 (?) \\
\nonumber
2^{-+} & (\sim 1.8 \; GeV; \Gamma \sim 200 \; MeV) & \rightarrow
\hspace{0.2cm} \pi b_1; \pi f_2 \\
\nonumber
1^{--} & (1460 \; MeV; \Gamma \sim 300 \; MeV) & \rightarrow
\hspace{0.2cm} \pi a_1
\end{eqnarray}
Detailed studies of these and other relevant channels are called for together
with analogous searches for their hybrid charmonium analogues, especially in
photoproduction or $e^+ e^-$ annihilation.
\section{RADIALOGY}
If these states are not glueballs and hybrids, what are they? On masses alone
they could be radial excitations as we have already noted. The decay patterns
have been seen to fit well with gluonic excitations but we need to close
the argument by considering the decays in the radial hypothesis. Only if
the hybrid succeeds and radial fails can one be sure to have a convincing
argument.
As an illustration consider the $0^{-+}$ (1800) which could be either
the hybrid or the radial
$3S (^1S_0)$ quarkonium (denoted $\pi_{RR}$). In fig 8 we see the width in an
S.H.O. calculation as a function of the oscillator strength $\beta$.
The $\pi \rho$ channel is small near the preferred value of $\beta \simeq$ 0.35
- 0.4 GeV and so both radial and hybrid share the property of suppressed $\pi
\rho$ and, to some extent, $KK^*$. It is therefore encouraging that data
show a
clear absence of $\pi \rho$ in the 1800 region in
contrast to $\pi f_0$ which
shows two clear bumps at 1300 and 1800 MeV. Notice that for $\pi_{RR}$ the
$\pi f_0$ is small for all $\beta$ in dramatic contrast to the hybrid, where
this
channel is predicted to dominate, and also apparently in contrast to data.
The
$\pi \rho_R$ (radial $\rho$) is predicted to be large for $\pi_{RR}$ and hence
one would expect a significant branching ratio $\pi_{RR}\rightarrow \pi
\rho_R \rightarrow 5 \pi$ which is not apparent in data though more study is
warranted.
A discriminator between $\pi_{RR}$ and hybrid $\pi_H$ is the
$\rho \omega$ channel. This is a dominant channel for $\pi_{RR}$ whereas
it is predicted to be absent for $\pi_H$ \cite{cp95}.
Another example that distinguishes hybrid and radial is in the $1^{++}$ sector.
There is a clear signal in $\pi f_1$ \cite{lee94}; $a_{1H}$ is forbidden to
decay into
$\pi b_1$ due to the spin selection rule \cite{cp95} whereas $a_{1R}\rightarrow
\pi b_1$ with a branching ratio equal to $a_{1R} \rightarrow \pi f_2$ over the
full $\beta$ range and moreover $\Gamma (a_{1R} \rightarrow \pi b_1)
\gapproxeq 2 \Gamma (a_{1R} \rightarrow \pi f_1)$. (see fig 9) The $\pi f_2$
channel may be easier experimentally. In any event we see that there are
characteristic differences in the branching ratios for radials and hybrids
states that
should enable a clear separation to be made.
After years of searching, at last we have some potential candidates for mesons
where the gluonic degrees of freedom are excited. Furthermore there are
some clear selection rules
and other discriminators in their decay branching ratios that can help to
verify
their existence and thereby complete the strong QCD sector of the
standard model.
\vskip 0.2in
\noindent {\bf Acknowledgements}
We thank the organisers for having arranged such a stimulating school, for
having made me feel at home by arranging British weather
and for introducing us to the delightful beaches of Ubatuba.
\pagebreak
\newpage
| 21,818 |
\section{Introduction}
The inclusive production of heavy-flavor quark-antiquark ($Q\bar Q$) pairs
in high-energy hadron-hadron collisions is one of the standard
applications of perturbative QCD.
Calculations, available in next to leading order
in the strong coupling constant $\alpha_s$,
agree reasonably well with experimental data (for a recent review see
ref.\cite{Fixiea94}).
In the perturbative treatment of heavy-flavor production from
nuclear targets, nuclear modifications of the
total production cross section are usually neglected.
Although this is certainly sufficient on a qualitative level, it is
hard to justify quantitatively within the framework of perturbative
QCD,
especially since it demands the calculation of higher order Feynman
diagrams which involve different nucleons inside the nucleus.
However the light-cone wave function formalism outlined in this paper
is ideal for an investigation of nuclear effects.
In this formulation heavy-quark production at large energies resembles
diffractive processes.
More insight is gained about the propagation of $Q\bar Q$ pairs
through the nuclear medium.
This matter is closely related to the effects of ``color
transparency'' and ``color opacity'', which are discussed
in a large variety of processes, ranging from quasielastic
electron scattering to heavy quarkonium production
(see e.g. \cite{Nikola92,PiMuWe93} and references therein).
Let $s$ be the squared center of mass energy of the collision process.
The Feynman variable $x_F$ is defined as $x_F = 2 P_{Q\bar Q}/\sqrt {s}$,
where $P_{Q\bar Q}$ is the longitudinal momentum of the $Q\bar Q$ pair
in the center of mass of the reaction.
It is common to introduce the variables $x_t$ and $x_b$ via:
\begin{eqnarray}
x_t x_b s &=& M_{Q\bar Q}^2,\\
x_F &=& x_b - x_t\,,
\label{eq:xF}
\end{eqnarray}
with $M_{Q\bar Q}$ being the invariant mass of the produced $Q\bar Q$.
In perturbative QCD $x_b$ and $x_t$ are identical
to the light-cone momentum fractions of the active beam and target
parton, respectively.
Within the leading order perturbative scheme they enter in the heavy-quark
production cross section as follows:
\begin{equation}
\sigma_{Q\bar{Q}}(s)
=\sum_{i,j}\int dx_{t} dx_{b}\,
f_{t}^{i}(x_{t},\mu^{2})\,
f_{b}^{j}(x_{b},\mu^{2})\,
\sigma_{ij}(x_{t}x_{b}s,\mu^{2}) \, .
\label{eq:pQCD}
\end{equation}
Here $f_{t}^{i}(x_{t},\mu^{2})$ and
$f_{b}^{j}(x_{b},\mu^{2})$ are the
densities of partons "i" and "j", carrying fractions $x_{t}$ and
$x_{b}$ of the light-cone momenta of the colliding target and projectile.
The partonic subprocess $i+j\rightarrow Q+\bar{Q}$
is described by the cross section $\sigma_{ij}(x_{t}x_{b} s,\mu^{2})$.
It requires a squared center of mass energy $x_t x_b s = M_{Q\bar Q}^2
\geq 4 m_Q^2$, where $m_Q$ is the invariant mass of the heavy quark.
The factorization in parton densities and hard partonic subprocesses
is carried out at a typical scale $\mu^2 \sim 4\,m_Q^2$.
In this work we will be concerned with nuclear effects in inclusive
hadro\-pro\-duction of $Q\bar{Q}$ pairs which carry a high energy in the
laboratory frame.
In particular we consider processes at small target light-cone
momentum fractions $x_t < 0.1$.
Furthermore we will restrict ourselves to moderate $x_F$.
Thus we concentrate on the kinematic domain where $Q\bar{Q}$ production
is dominated by the gluon fusion subprocess $g+g\rightarrow Q+\bar{Q}$
(see e.g. ref.\cite{PilTho95})
and neither the annihilation of light-quarks nor the
excitation of higher-twist intrinsic heavy-quark components \cite{VoBrHo92}
are of importance.
The perturbative QCD cross section (\ref{eq:pQCD})
is then proportional to the density of gluons in beam and target.
In the case of open-charm production at typical Tevatron energies
($s \sim 1600\,GeV^2$)
this implies that we focus on the region $0<x_F<0.5$.
The paper is organized as follows: In Sec.~2 we discuss the
space-time picture of heavy-quark production at small $x_t$, as seen from the
lab frame. Section~3 introduces the light-cone wave function
formulation of heavy-flavor production from free nucleon targets.
This approach is extended to nuclear targets in Sec.~4. Finally
we summarize the main results in Sec.~5.
\section{Lab frame picture of $Q\bar Q$ production at \protect \\
small $x_t$}
At moderate values of $x_F$
heavy-quark production proceeds via the fusion of a projectile
and target gluon.
In the lab frame, where the target is at rest, the projectile gluon
interacts at small $x_t<0.1$ via quark-antiquark fluctuations present
in its wave function.
In heavy-flavor production only heavy $Q\bar Q$
Fock states of the incident gluon are of relevance.
Their propagation length is typically of the order:
\begin{equation}
l_{Q\bar Q} \sim \frac{2\, \nu_G}{4\,m_Q^2} \approx \frac{1}{M\,x_t},
\end{equation}
where $\nu_G = x_b E_{lab}$ denotes the lab frame energy of
the projectile gluon and $E_{lab}$ stands for the beam energy.
For $Q\bar Q$ production from a nuclear target with mass number $A$
and radius $R_A$ the propagation length $l_{Q\bar Q}$ plays an
important role.
If $l_{Q\bar Q}$ is less then the average nucleon-nucleon
distance in nuclei, $Q\bar Q$ production
takes place incoherently on all nucleons inside the
target.
Consequently then nuclear effects in inclusive heavy-flavor
production will be absent.
However if the energy of the incident gluon is large, or equivalently
$x_t < 0.1\cdot A^{-1/3}$ is small, the coherence length
will exceed the nuclear size, $l_{Q\bar Q} > R_A$.
Then the formation of $Q\bar Q$ pairs takes place coherently
on the whole nucleus.
This may in principle lead to nuclear modifications
of the heavy-flavor total production cross section.
\footnote{Note, that this picture is quite similar to deep-inelastic
scattering at small values of the Bjorken variable $x$, leading to nuclear
shadowing (see e.g. \cite{NikZak91,PiRaWe95}).}
The magnitude of these nuclear effects however is controlled by
the transverse size of the $Q\bar Q$ fluctuations and turns out
to be small as discussed below.
Once produced, a $Q\bar{Q}$ pair evolves
into hadrons after a typical formation length
\begin{equation}
l_{f} \sim \frac{2 \pi }{\Delta M_{Q\bar Q}}{\nu_{G} \over M_{Q\bar Q}}
\gg l_{Q\bar{Q}}.
\end{equation}
Here $\Delta M_{Q\bar Q}$ is the characteristic mass difference of
heavy $Q\bar Q$ states.
Since the formation time $l_f$ is much larger than the coherence
length $l_{Q\bar Q}$, it is justified to neglect the
intrinsic evolution of a produced
$Q\bar{Q}$ pair during its propagation through the nucleus.
At large $x_F \sim 1$ the momentum of the projectile is transfered
collectively to a heavy $Q\bar Q$ state by several partons in the
beam hadron \cite{VoBrHo92}.
In this case the heavy quark pair can evidently not be assigned
to a single projectile gluon. Consequently, the excitation
of heavy-quark components at $x_F \sim 1$ is quite
different
from the $Q\bar Q$ production mechanism at moderate $x_F$.
In this work we focus on moderate $x_F$ only, where the above
mechanism is not important.
\section{$Q\bar Q$ production from free nucleons}
As a first step we consider the production of
heavy quark-antiquark pairs through the interaction of
projectile gluons with free nucleons.
The total $Q\bar Q$ production cross section for free nucleon targets
is then obtained by a convolution with the incident gluon flux.
To leading order in the
QCD coupling constant $\alpha_s$, the incoming projectile gluon
has a simple Fock state decomposition.
It includes bare gluons, two-gluon states and quark-antiquark
components.
At small $x_t < 0.1$
the propagation lengths of these Fock
states exceed the target size
as pointed out in the previous section.
Furthermore, at large projectile energies the transverse
separations and the longitudinal
momenta of all partons in a certain Fock component are
conserved during its interaction with the target \cite{NikZak91}.
Consequently it is most convenient to describe the different
Fock components in terms of light-cone wave functions in a ``mixed''
representation, given by these conserved quantities.
In this representation the incident, dressed gluon $|G(k_b^2)\rangle$
can be decomposed as:~\footnote{
Wherever it does not lead to confusion we suppress the virtuality
of the incident gluon $k_b^2$.}
\begin{equation}
|G(k_{b}^{2})\rangle = \sqrt{1-n_{Q\bar{Q}}-n_{\xi}}|g\rangle +
\sum_{z,\vec{r}\,} \Psi_{G}(z,\vec{r}\,)|Q\bar{Q};z,\vec{r}\,\rangle
+ \sum_{\xi} \Psi(\xi)|\xi\rangle \, .
\label{eq:Gluon}
\end{equation}
Here $\Psi_{G}(z,\vec{r}\,)$ is the projection of the dressed gluon wave
function onto a $Q\bar Q$ state in the $(z,\vec{r}\,)$-representation,
where the light-cone variable $z$ represents the fraction of the
gluon momentum carried by the quark and $\vec{r}$ is the transverse
separation of the $Q$--$\bar{Q}$ in the impact parameter space.
The light quark-antiquark ($q\bar{q}$) and gluon-gluon ($gg$)
Fock components are written as
$\sum_{\xi}\Psi(\xi~)|\xi\rangle$.
Since they are of no importance for our considerations we will omit
them from now on.
The normalization of the $Q\bar Q$ Fock states is given by:
\begin{equation}
n_{Q\bar{Q}}= \int_{0}^{1} dz \int d^{2}\vec{r}\,
|\Psi_{G}(z,\vec{r}\,)|^{2}.
\label{eq:n_QQbar}
\end{equation}
Similarly, $n_{\xi}$ denotes the normalization of the light-flavor
($q\bar q$ and $gg$) components.
The $Q\bar Q$ wave function $\Psi_G$ of the gluon can be obtained
from the $Q\bar Q$ wave function of a photon $\Psi_{\gamma^*}$,
as derived in ref.\cite{NikZak91}.
The only difference is a color factor and the substitution
of the strong coupling constant for the electromagnetic one:
\begin{eqnarray}
|\Psi_{G}(z,\vec{r}\,)|^{2} &=&
{\alpha_{s}(r) \over 6\,\alpha_{em}}|\Psi_{\gamma^{*}}(z,\vec{r}\,)|^{2}
\nonumber\\
&=&{\alpha_{s}(r)
\over (2\pi)^{2}}\left\{
[z^{2}+(1-z)^{2}]\epsilon^{2}K_1^{2}(\epsilon r) +
m_{Q}^{2}K_{0}^{2}(\epsilon r)\right\}\,.
\label{eq:psi_G}
\end{eqnarray}
$K_{0,1}$ are the modified Bessel functions,
$\alpha_s(r)$ is the running coupling constant in
coordinate-space,
$r = |\vec r\,|$, and $\epsilon^{2}=z(1-z)k_b^{2} +m_{Q}^{2}$.
The scattering of an incident gluon carrying color $a$ from a
target nucleon at an impact parameter $\vec b$
is described by the scattering matrix $\hat S(\vec b)$:
\begin{equation} \label{eq:SGN}
|G^a,N ;\vec{b}\, \rangle \Longrightarrow
\hat{S}(\vec b\,)|G^a,N;\vec{b}\,\rangle \,.
\end{equation}
In the impact parameter representation
we use the S-matrix in the eikonal form from ref.\cite{GunSop77}:
\begin{equation}
\hat{S}(\vec b\,)=\exp\left[-i\sum_{i,j}V(\vec{b}+\vec{b}_{i}-\vec{b}_{j})
\hat{T}_{i}^{b}\hat{T}_{j}^{b}\right] \, ,
\label{eq:Smatrix}
\end{equation}
with the one-gluon exchange potential (eikonal function)
\begin{equation}
V(\vec{b}\,)={1 \over \pi}\int d^{2}\vec{k}
\,{\alpha_{s}({\vec k\,}^2)
\,e^{i\vec{k}\cdot\vec{b}} \over \vec{k}^{2} + \mu_{g}^{2}}.
\label{eq:eikonalfunction}
\end{equation}
$\hat{T}_{i,j}^{b}$ are color SU(3) generators
acting on the individual partons of the projectile and target
at transverse coordinates
$\vec{b}_{i}$ and $\vec{b}_{j}$ respectively.
The effective gluon mass $\mu_g$ is introduced as an infrared
regulator.
In heavy-flavor production $m_{Q} \gg \mu_{g}$,
which ensures that our final results will
not depend on the exact choice of $\mu_g$.
Furthermore, as we will show below, $\mu_g$ can be absorbed into the
definition of the target gluon density.
In Eq.(\ref{eq:Smatrix}) and below
an implicit summation over repeated color indices is understood.
In the lowest non-trivial
order the scattering matrix in (\ref{eq:Smatrix})
accounts for the exchange of two t-channel
gluons between the target and projectile.
This has turned out to be successful in describing
high energy hadron-hadron forward scattering processes
\cite{GunSop77}, deep-inelastic scattering at
small values of the Bjorken variable $x$ \cite{NikZak91},
and the diffractive photoproduction of vector mesons \cite{NeNiZa94}.
In Eq.(\ref{eq:SGN}) the leading term is:
\begin{eqnarray}
\left(\hat{S}(\vec b\,)-1\right)&&\hspace{-0.6cm}
|G^a,N;\vec{b}\,\rangle = \nonumber \\
&&\hspace{-1.2cm}
\frac{-i}{\pi}
\int d^{2}\vec{k}
\,{\alpha_{s}({\vec k\,}^2)
\, e^{i\vec{k}\cdot\vec{b}} \over \vec{k}^{2} + \mu_{g}^{2}}
\left[\sum_j e^{-i\vec k\cdot \vec b_j}
\hat T^b_j \left|N \right \rangle \right]
\left[\sum_i e^{ i\vec k\cdot \vec b_i}
\hat T^b_i \left| G^a \right\rangle \right]\!.
\label{eq:SG2}
\end{eqnarray}
The last term in Eq.(\ref{eq:SG2}) represents the coupling of an exchanged
t-channel gluon with color $b$ to the incident gluon, while the
second to last term describes the coupling to the nucleon target.
The interaction of the incident gluon can be expressed in terms
of its Fock components specified in Eq.(\ref{eq:Gluon}).
For this purpose, note that
the quark and antiquark of a certain $Q\bar Q$ Fock state with
transverse distance $\vec r$ and light-cone momentum
fraction $z$ are located at impact parameters
$\vec b_{Q} = \vec r \,(1- z)$ and
$\vec b_{\bar Q} = -\vec r \,z $
with respect to the parent gluon.
Neglecting light quark-antiquark and gluon-gluon components we obtain
(see Fig. 1):
\begin{eqnarray}
\hspace*{-2cm}\sum_i e^{i\vec k\cdot \vec b_i} \hat T_i^b
&&\hspace{-0.6cm}\left|G^a \right\rangle =
i f_{abc} \sqrt{1-n_{Q\bar Q}}\left| g^c\right\rangle
\nonumber \\
+\sum_{\vec r,z} &&\hspace{-0.6cm}\Psi_G(z,\vec r)
\left\{
\frac{1}{2} \left( e^{i\vec k \cdot \vec r (1-z)} +
e^{-i\vec k \cdot \vec r z} \right)
i f_{abc} \left| Q\bar Q_{[8]}^c\right\rangle \right.
\nonumber \\
&&\hspace*{+0.7cm}\left.+\frac{1}{2} \left( e^{i\vec k \cdot \vec r (1-z)} -
e^{-i\vec k \cdot \vec r z} \right)
d_{abc} \left|Q\bar Q_{[8]}^c \right\rangle\right.
\nonumber \\
&&\hspace*{+0.7cm}\left.+\frac{1}{\sqrt{6}}
\left( e^{i\vec k \cdot \vec r (1-z)} -
e^{-i\vec k \cdot \vec r z} \right)
\delta_{ab} \left|Q\bar Q_{[1]}\right\rangle
\right\}.
\label{eq:finalQQ1}
\end{eqnarray}
As usual $f_{abc}$ and $d_{abc}$ are the
antisymmetric and symmetric SU$(3)$ structure constants.
In terms of the color wave functions of its quark and
antiquark constituents the color wave function of an octet $Q\bar Q$ pair
with color $c$ is:
\begin{equation}
\left|Q\bar Q_{[8]}^c\right\rangle = \sqrt{2}
\sum_{k,l} \hat T^c_{k l} \left|Q_k\right \rangle
\left|\bar Q_l\right \rangle.
\end{equation}
The indices $k$ and $l$ specify the color of the
quark and antiquark.
Similarly, a color singlet $Q\bar Q$ state is given by:
\begin{equation}
\left|Q\bar Q_{[1]}\right\rangle = \frac{1}{\sqrt{3}}
\sum_k \left|Q_k\right \rangle \left|\bar Q_k\right \rangle.
\end{equation}
The scattering state
(\ref{eq:finalQQ1}) can be decomposed
into final state, dressed gluons and heavy-quark pairs:
\begin{eqnarray}
\hspace*{-2cm}\sum_i e^{i\vec k\cdot \vec b_i} \hat T_i^b
&&\hspace{-0.6cm}\left|
G^a\right\rangle = i f_{abc} \left| G^c\right\rangle \nonumber \\
+\sum_{\vec r,z}
&&\hspace{-0.6cm}\Psi_G(z,\vec r) \left\{ \frac{1}{2} \left( e^{i\vec
k \cdot \vec r (1-z)} + e^{-i\vec k \cdot \vec r z} - 2 \right) i f_{abc}
\left| Q\bar Q_{[8]}^c\right\rangle \right. \nonumber \\
&&\hspace*{+0.7cm}\left.+\frac{1}{2} \left( e^{i\vec k \cdot \vec r (1-z)} -
e^{-i\vec k \cdot \vec r z} \right) d_{abc} \left|Q\bar Q_{[8]}^c
\right\rangle\right. \nonumber \\ &&\hspace*{+0.7cm}\left.+\frac{1}{\sqrt{6}}
\left( e^{i\vec k \cdot \vec r (1-z)} - e^{-i\vec k \cdot \vec r z} \right)
\delta_{ab} \left|Q\bar Q_{[1]}\right\rangle \right\}.
\label{eq:finalQQ2}
\end{eqnarray}
In this decomposition the last three terms
represent heavy-quark pairs pro\-du\-ced via the interaction of the incident
gluon with the exchanged t-channel gluon.
According to the derivation leading to Eq.(\ref{eq:finalQQ2})
they have to be seen as
diffractive excitations of heavy-flavor $Q\bar{Q}$ Fock components
present in the wave function of the incident dressed gluon.
The contribution of these $Q\bar Q$ Fock states to the
heavy-flavor production amplitude disappears if their transverse
size $\vec r$ vanishes, since then
they cannot be resolved by the exchanged t-channel gluon.
This is due to the fact that
interactions of point-like color octet $Q\bar{Q}$ states are
indistinguishable from interactions of color octet gluons, as
a consequence of color gauge invariance.
Each of the color octet $Q\bar Q$ terms proportional to $f$ and $d$,
as well as the color singlet term in Eq.(\ref{eq:finalQQ2}),
vanishes for $\vec r\rightarrow 0$.
There is a one-to-one correspondence between the
configuration-space derivation of heavy-flavor production,
and the conventional perturbative QCD approach (see Fig.~1).
The excitation of color singlet $Q\bar{Q}$ states receives
contributions only from $t$- and $u$-channel quark
exchange diagrams (Fig.~1a and Fig.~1b). The same is true for
the excitation of color octet $Q\bar{Q}$ pairs proportional
to the color factor $d$.
The excitation of color octet $Q\bar{Q}$ states
proportional to $f$ receives contributions also from the $s$-channel
gluon exchange process in Fig.~1c.
They yield the term ``$-2$'' within the factor
$( e^{i\vec k \cdot \vec r (1-z)} + e^{-i\vec k \cdot \vec r z} - 2)$
in Eq.(\ref{eq:finalQQ2}).
This $s$-channel gluon-exchange contribution is crucial for the
disappearance of the corresponding heavy-quark excitation
amplitude at $\vec{r}\rightarrow 0$.
In the following we focus on the diffractively produced $Q\bar Q$
states and omit the gluon contribution in Eq.(\ref{eq:finalQQ2}).
In the light of our discussion in Sec. 2 this should yield the
main contribution to heavy-flavor production at high energies
and moderate $x_F$.
In the lowest non-trivial order we obtain the
heavy-flavor production cross section in gluon-nucleon collisions
by inserting the $Q\bar{Q}$ component of
Eq.(\ref{eq:finalQQ2}) into Eq.(\ref{eq:SG2}), multiplying
with the complex conjugate,
integrating over the impact parameter space,
summing over the color of the final state and averaging
over the color of the incident gluon:
\begin{eqnarray}
\sigma(G N \rightarrow Q\bar Q\, X)
= \frac{2}{3}\int d^2\vec r \int_0^1 d z
\left|\Psi_G(z,\vec r)\right|^2
\int d^2 \vec k \,
\frac{\alpha_s^2({\vec k\,}^2)\,
{\cal F}(k^2)}{(k^2 + \mu_g)^2}
\nonumber\\
\times \left( 17 - 9 e^{-i \vec k \cdot \vec r z}
- 9 e^{i \vec k \cdot \vec r (1-z)} + e^{i\vec k \cdot \vec r}
\right).
\label{eq:sigmaGN1}
\end{eqnarray}
All relevant information about the target is
in ${\cal F}(k^2)$, which is linked
to the form factor $G_2$
related to the coupling of two gluons to the nucleon:
\begin{equation}
{\cal F}(k^2) = 1 - G_2(k, - k) =
1 - \left\langle N \right| e^{i\vec k\cdot(\vec b_1 - \vec b_2)}
\left| N\right\rangle.
\end{equation}
The vectors $\vec b_1$ and $\vec b_2$ specify the
coordinates of the interacting
quarks in the impact parameter space.
If the size of the target, which carries no
net color, would shrink to zero, the exchanged gluons would decouple
from the target and ${\cal F} = 0$.
In actual calculations we determine $G_2$ using
a constituent quark wave function for the nucleon of Gaussian
shape, fitted to the electromagnetic charge radius of the nucleon.
The result in Eq.(\ref{eq:sigmaGN1}) can be illustrated
as follows:
Consider a color singlet gluon-quark-antiquark
$(gQ\bar Q)$ state with a
$Q$-$g$ and $\bar Q$-$g$ se\-pa\-ration $\vec \rho$ and $\vec R$
respectively.
The $Q$-$\bar Q$ separation is then $\vec r = \vec \rho - \vec R$.
Furthermore let $\sigma(r)$ be the cross section for the interaction of
a $Q\bar{Q}$ color dipole of size $r$ with the nucleon target.
In the Born approximation, i.e. with two gluon exchange, one has
\cite{GunSop77,NikZak91}:
\begin{equation}
\sigma(r) = \frac{16}{3} \int d^2\vec k \,\frac{
\alpha_s^2({\vec k\,}^2)
\,{\cal F}(k^2)}
{(k^2+\mu_g^2)^2} \left( 1- e^{i\vec k\cdot \vec r}\right).
\label{eq:dipole}
\end{equation}
Then the $gQ\bar Q$-nucleon cross section
can be written as \cite{NikZak94JETP}
\begin{equation}
\sigma^N_{g Q\bar Q}(r,R,\rho) = \frac{9}{8}
\left[\sigma(R) + \sigma(\rho) \right] - \frac{1}{8} \sigma(r)\,.
\label{eq:gQQ}
\end{equation}
Comparing with Eq.~(\ref{eq:sigmaGN1}) we find:
\begin{equation}
\sigma(GN\rightarrow Q\bar Q \,X) =
\int d^2 \vec r \int_0^1 d z \,|\Psi_G(z,\vec r)|^2
\sigma^N_{gQ\bar Q} \left( r, - zr, (1-z) r\right).
\label{eq:sigmaGN2}
\end{equation}
Another derivation of this result, based on unitarity, can be found
in ref.\cite{NiPiZa94}.
Let us briefly discuss the properties of $\sigma(GN\rightarrow Q\bar Q \,X)$.
In this respect it is useful to observe that Eq.(\ref{eq:sigmaGN2})
resembles heavy-flavor contributions to real and virtual
photoproduction \cite{NikZak91,NikZak94JETP}.
The wave functions corresponding to these processes
are equal, up to normalization
factors (see Eq.(\ref{eq:psi_G})).
While the three-parton cross section (\ref{eq:gQQ})
enters in hadroproduction,
the dipole cross section (\ref{eq:dipole})
is present in photoproduction.
Both cross sections are closely related through
Eq.(\ref{eq:gQQ}).
In the leading $\log Q^2$ approximation the dipole
cross section (\ref{eq:dipole}) is proportional
to the gluon distribution of the target
\cite{NikZak94JETP,Nikoea93ZP58,Nikoea93InA8}:
\begin{equation}
\sigma(r) \rightarrow \sigma(x_{t},r)={\pi^{2} \over 3}\,r^{2}\,
\alpha_{s}(r)
\left[
x_{t}\,g_t(x_{t},k_{t}^{2} \sim {1 \over r^{2}})\right].
\label{eq:gluon}
\end{equation}
The explicit
dependence on $x_t$ results from higher order Fock
components of the projectile parton, e.g. $Q\bar Q g$ states.
Note that the gluon distribution in Eq.({\ref{eq:gluon}) absorbs
the infrared regularization $\mu_g$. Most important for our
further discussion is the color transparency property of
the dipole cross section, i.e. its proportionality to
$r^2$.
{}From the asymptotic properties of the modified Bessel functions
one finds immediately that the squared $Q\bar Q$ wave function
decreases exponentially for $r > 1/\epsilon$.
We therefore conclude that for
$k_b^{2} \lsim 4m_{Q}^{2}$ small transverse sizes
are relevant for the $Q\bar Q$ production process:
\begin{equation}
r^{2}\lsim {1\over m_{Q}^{2}} \, .
\label{eq:size}
\end{equation}
For highly virtual gluons, $k_b^{2} \gg 4m_{Q}^{2}$,
the $Q\bar Q$ wave function selects quark pairs with a
transverse size $r^2 \ll 1/4 m_Q^2$.
Their contribution to the production cross section
vanishes like $\sim 1/k_b^2$ due to the color transparency
property of the dipole cross section in Eq.(\ref{eq:gluon}).
Combining Eqs.(\ref{eq:sigmaGN2},\ref{eq:gluon}, \ref{eq:size})
we find:
\begin{equation}
\sigma(GN\rightarrow Q\bar{Q}X) \propto x_{t}\,g_t(x_{t},\mu^{2}
\sim 4m_{Q}^{2})\, .
\label{eq:targetgluon}
\end{equation}
Equation (\ref{eq:targetgluon}) will later be important for a comparison
with the parton model description of heavy-flavor
production in Eq.(\ref{eq:pQCD}).
An important property of
$\sigma(GN \rightarrow Q\bar Q\,X)$ is its infrared stability,
i.e. its convergence in the limit $\vec k \rightarrow 0$.
This is due to the fact that soft t-channel gluons with
$\vec k \rightarrow 0$ cannot resolve the $Q\bar Q$ component of the
incident beam gluon, and therefore cannot contribute to heavy-flavor
production.
Although $n_{Q\bar Q}$, the normalization of a dressed gluon
to be found in a $Q\bar Q$ Fock state (\ref{eq:n_QQbar}),
diverges logarithmically at $r \rightarrow 0$,
the cross section $\sigma(GN \rightarrow Q\bar Q\,X)$ is finite.
This is again implied by color transparency,
since small size $Q\bar Q$ pairs
cannot be resolved by interacting t-channel gluons with
wavelengths $\lambda \gsim r$. They are therefore indistinguishable
from bare gluons and cannot be excited into final $Q\bar Q$ states.
We are now in the position to write down the heavy-quark production
cross section $\sigma_{Q\bar Q}(h,N)$ for hadron-nucleon collisions.
For this purpose we have to
multiply $\sigma(GN\rightarrow Q\bar Q\,X)$ from
Eq.~(\ref{eq:sigmaGN2}) by the gluon density of the incoming hadron
projectile
and integrate over the virtuality $k_b^2$ of the projectile gluon:
\begin{equation}
{d\sigma_{Q\bar{Q}}(h,N) \over dx_b}=
\int {dk_b^{2} \over k_b^{2}}
{\partial[g_{b}(x_{b},k_b^{2})]
\over \partial \log k_b^{2}}\,
\sigma(GN \rightarrow Q\bar{Q}\,X;x_t,k_b^{2}).
\end{equation}
As mentioned above, the leading contributions to
$\sigma(GN\rightarrow Q\bar Q\,X)$ result from the region
$k_b^2 \lsim 4 m_Q^2$.
With the approximation
$\sigma(\!GN\!\rightarrow\!Q\bar Q\,X\!;\!x_t,\!k_b^{2}\!) \approx
\sigma(GN\rightarrow Q\bar Q\,X;x_t,k_b^{2}=0)$ we obtain:
\begin{equation}
{d\sigma_{Q\bar{Q}}(h,N) \over dx_{b}}\approx
g_{b}(x_{b},\mu^{2} = 4m_{Q}^{2})\,
\sigma(GN\rightarrow Q\bar{Q}\,X;x_{t},k_b^{2} = 0) \,.
\label{eq:sigGN}
\end{equation}
Equations (\ref{eq:targetgluon},\ref{eq:sigGN}) describe
the dependence of the production cross section on the
beam and target gluon densities, in close
correspondence to the conventional
parton model ansatz of Eq.(\ref{eq:pQCD}).
This demonstrates the similarity between the light-cone wave
function formulation and the conventional parton model approach.
Since the $g+g\rightarrow Q+\bar{Q}$ cross section
decreases rapidly with the invariant mass of the heavy-quark pair, one finds
dominant contributions to heavy-flavor production for
$x_t x_b s \approx 4 m_Q^2$.
This leads to:
\begin{equation}
{d\sigma_{Q\bar{Q}}(h,N) \over dx_{F}}\approx
\left(1 + \frac{4 m_Q^2}{s x_b^2} \right)
g_{b}(x_{b},4m_{Q}^{2})\,
\sigma(GN\rightarrow Q\bar{Q}\,X;x_{b}-x_{F}) \,,
\label{eq:sighN}
\end{equation}
with $x_b \approx x_F/2 + \sqrt{16 m_Q^2 s + s^2 x_F^2}/2s$.
Let us explore the result (\ref{eq:sighN}) for charm production
in nucleon-nucleon
collisions. To be specific we choose a beam energy
$E_{lab} = 800$\,GeV,
as used in the Fermilab E743 experiment \cite{E74388}.
For the dipole cross section which enters in
Eqs.(\ref{eq:gQQ},\ref{eq:sigmaGN2}) we
employ two different parameterizations.
First we use $\sigma(r)$ from Eq.(\ref{eq:dipole})
with $\mu_{g} = 0.140$\,GeV.
This choice reproduces measured hadron-nucleon
cross sections at high energies.
Furthermore it has been
successfully applied to nucleon structure functions at small
$x$ and moderate $Q^2$,
and to nuclear shadowing \cite{NikZak91}.
As an alternative we
use a parameterization from ref.\cite{NikZak94PLB327}
which includes the effects of higher order Fock states
of the projectile parton.
This choice has been shown to yield a good description of the
small-$x$ proton structure function measured at HERA.
For transverse sizes $r^2 \sim 1/m_c^2$ one finds
in the region $x_{t} \lsim x_{0} = 0.03$ \cite{NikZak94PLB327}:
\begin{equation}
\sigma(x_{t},r) \approx \sigma(r)
\left({x_{0}\over x_{t}}\right)^{\Delta}\, ,
\label{eq:dipoleext}
\end{equation}
with $\mu_{g}=0.75$\,GeV and $\Delta = 0.4$.
In both parameterizations the squared coupling constant
$\alpha_s^2({\vec k\,}^2)$
appearing in Eq.(\ref{eq:dipole}) has been replaced by
\linebreak
$\alpha_s(r) \,\alpha_s ({\vec k\,}^2)$.
The invariant mass of the charm quark is fixed at $m_c = 1.5$\,GeV.
The gluon distribution of the nucleon projectile enters at
moderate to large values of $x_{b}$. We therefore use the
parameterization of ref.\cite{Owens91}.
In Fig.2 we compare our results with the
data from the E743 proton-proton
experiment \cite{E74388}.
In the kinematic region $0<x_F<0.5$, where our approach is well founded,
both parameterizations of the dipole cross section lead to a
reasonable agreement with the experimental data.
\section{Hadroproduction of $Q\bar{Q}$ pairs from nuclear
\protect \\ targets }
An important aspect of the light-cone wave function formulation
of heavy-flavor production is the factorization of the
$Q\bar Q$ wave function and its interaction cross section
(see Eq.(\ref{eq:sigmaGN2})).
This feature is crucial for the generalization
to nuclear targets.
The derivation of the $Q\bar Q$ production cross section in
(\ref{eq:sigmaGN2}) has been based upon the observation that
the coherence length $l_{Q\bar Q}$ of a $Q\bar Q$ fluctuation,
belonging to the incident gluon, is larger than the target size
for high energies and small $x_t$.
As a consequence the transverse size of the heavy-quark pair
is frozen during the interaction process.
This has led to a diagonalization
of the scattering $\hat{S}$-matrix
in a mixed $(z,\vec{r}\,)$-representation.
In high energy hadron-nucleus collisions
with $l_{Q\bar{Q}} \gsim R_{A}$,
this diagonalization of the $S$-matrix
is also possible.
Consequently we obtain the $Q\bar Q$
production cross section for nuclear
targets by substituting
$\sigma^A_{gQ\bar Q}$ for $\sigma^N_{gQ\bar Q}$ in Eq.~(\ref{eq:sigmaGN2}).
In the frozen size approximation the cross section
$\sigma^A_{gQ\bar Q}$ for the scattering of a
color singlet $gQ\bar Q$ state from
a nucleus with mass number $A$ is given by the
conventional Glauber formalism:
\begin{eqnarray}
\sigma^A_{gQ\bar Q}&=&
2\int d^{2}\vec{b} \left\{
1-\left[1-{1\over 2A}\
\sigma^N_{gQ\bar Q}T(\vec b\,)\right]^{A}\right\} \nonumber\\
&\approx &
2\int d^{2}\vec{b} \left\{
1-\exp\left[-{1\over 2}\
\sigma^N_{gQ\bar Q}T(\vec b\,)\right]\right\} \, .
\label{eq:siggQQA}
\end{eqnarray}
Here $\vec{b}$ is the impact parameter of the $gQ\bar{Q}$-nucleus
scattering process,
which must not be confused with the impact parameter of the
$gQ\bar{Q}$-nucleon
interaction in Sec.~3. $T(\vec b\,)$ stands for the optical thickness of the
nucleus:
\begin{equation}
T(\vec b\,)=\int_{-\infty}^{+\infty} dz \,n_{A}(\vec b,z) \,,
\end{equation}
with the nuclear density ${n_A}(\vec b,z)$ normalized to
$\int d^{3}\vec{r} \,n_{A}(\vec r\,)=A$.
We then obtain:
\begin{equation}
{d\sigma_{Q\bar{Q}}(h,A) \over dx_{b}} \approx
g_{b}(x_{b},\mu^{2} = 4m_{Q}^{2})\,
\sigma(GA\rightarrow Q\bar{Q}\,X;x_t,k_b^{2}=0) \, ,
\label{eq:dsigQQA/dxf}
\end{equation}
where
\begin{eqnarray}
\sigma(GA\rightarrow && \hspace*{-1cm}Q\bar{Q}\,X;x_t,k_b^{2}) =
2 \int d^{2}\vec r \int_0^1 dz \,|\Psi_{G}(z,r)|^{2}
\nonumber\\
&\times& \int d^{2}\vec{b} \left\{
1-\exp\left[-{1\over 2}\
\sigma^N_{gQ\bar Q}T(\vec b\,)\right]\right\}
\equiv
\left\langle \sigma^A_{gQ\bar Q}\right\rangle
\, .
\label{eq:GA}
\end{eqnarray}
In the multiple scattering series (\ref{eq:siggQQA},
\ref{eq:GA})
nuclear coherence effects are controlled by the parameter
\begin{equation}
\tau_{A}=\sigma^N_{gQ\bar Q}\,T(\vec b\,)
\propto \sigma^N_{gQ\bar Q}\,A^{1/3}.
\label{eq:4.5}
\end{equation}
Expanding the exponential in Eq.(\ref{eq:GA})
in powers of $\tau_{A}$ one can identify terms
proportional to $\tau_{A}^{n}$ which describe contributions to the
total production cross section resulting from the coherent
interaction of the $gQ\bar Q$ state with $n$ nucleons inside the
target nucleus.
In leading order, $n = 1$,
we have the incoherent sum over
the nucleon production cross sections:
\begin{eqnarray}
\sigma(GA\rightarrow Q\bar{Q}\,X) &=&
\int d^{2}\vec{b}\,
T(\vec b\,)\,
\sigma(GN\rightarrow Q\bar{Q}\,X) \nonumber \\
&=&A\,\sigma(GN\rightarrow Q\bar{Q}\,X).
\end{eqnarray}
This is the conventional impulse approximation component
of the nuclear production cross section, proportional to the
nuclear mass number.
Effects of coherent higher order interactions are usually discussed
in terms of the nuclear transparency
\begin{equation}
T_{A}= {\sigma_{Q\bar Q}(h,A) \over {A\, \sigma_{Q\bar Q}(h,N)} } \, .
\label{eq:TA1}
\end{equation}
In the impulse approximation $T_{A} = 1$. The driving contribution to
nuclear attenuation results from the coherent interaction of the
three-parton $gQ\bar{Q}$ state with two target nucleons:
\begin{equation}
T_{A}=1-{1\over 4} \,
{\left\langle \left(\sigma^N_{gQ\bar Q}\right)^{2} \right\rangle
\over
\langle \sigma^N_{gQ\bar Q}\rangle}\int d^{2}\vec{b}\,T^{2}(\vec b\,)
\, .
\label{eq:TA2}
\end{equation}
Using Gaussian nuclear densities fitted to the measured
electromagnetic charge radii, $R_{ch} \approx r_0 A^{1/3}\approx
1.1\,fm \,A^{1/3}$, we obtain:
\begin{equation}
T_{A}=1-
\frac{3}{16 \pi r_0^2}
{\left\langle \left(\sigma^N_{gQ\bar Q}\right)^{2} \right\rangle
\over \langle \sigma^N_{gQ\bar Q}\rangle}\,A^{1/3}
\, .
\label{eq:TA3}
\end{equation}
In Fig.3 we present results for the transparency ratio $T_A$
using the two different parameterizations for the
dipole cross section, introduced in the previous section
(see Eqs.(\ref{eq:dipole},\ref{eq:dipoleext})).
The dipole cross section from Eq.(\ref{eq:dipoleext}) depends
in general on $x_t$, which is however restricted by the kinematic
constraint $x_t x_b s \geq 4 m_c^2$. We choose $x_t = 0.01$
--- a typical value for current Tevatron energies ($s \sim 1600\,GeV^2$).
For both parameterizations of the dipole cross section we
find only small nuclear effects.
In the commonly used parameterization
$T_A = A^{\alpha -1}$ they translate to $\alpha \sim 0.99$.
This is in agreement with recent experiments at CERN and Fermilab:
In the WA82 experiment \cite{WA8292} with a $340\,GeV$ $\pi^-$ beam one
finds $\alpha = 0.92\pm 0.06$, at an average value of $\bar x_F = 0.24$.
{}From the E769 measurement \cite{E76993}, using a
$250\, GeV$ $\pi^{\pm}$ beam,
$\alpha = 1.00\pm 0.05$ was obtained for $0<x_F<0.5$.
\section{Summary}
We have presented a light-cone wave
function formulation of heavy-flavor production
in high energy hadron-hadron collisions.
At moderate values of $x_F$ we have found that heavy-flavor production
can be viewed as the diffractive excitation of heavy $Q\bar Q$ Fock states
present in the wave function of the interacting projectile gluon.
In the region of applicability ($0<x_F<0.5$) the energy dependence of
recent data on open-charm production in proton-proton collisions is
described reasonably well.
The light-cone wave function formulation is most appropriate to
address the production of heavy-flavors from nuclear targets.
In accordance with recent experiments on open-charm production
we have found small nuclear effects.
High precision measurements involving heavy nuclear targets
would be necessary to observe any significant nuclear modification of
heavy-flavor production rates.
\bigskip
\bigskip
{\bf Acknowledgments:} G. P. and B. G. Z.
would like to thank J. Speth and the theoretical physics group at
the Institut f\"ur Kernphysik, KFA J\"ulich, for their hospitality
during several visits.
We also thank W. Melnitchouk and W. Weise for a careful reading
of the manuscript.
This work was partially supported by the INTAS grant 93-239 and
grant N9S000 from the International Science Foundation.
\pagebreak\\
| 12,850 |
\section{Introduction}
\setcounter{equation}{0}
\setcounter{subsection}{0}
Scalar integrable hierarchies can be introduced in terms of
(pseudo)differential operators by means of a formalism first
introduced by Gelfand and Dickey (see \cite{Dickey}). This is the
most `disembodied'
form in which such hierarchies can appear, and it can be taken
as a reference form. One can then consider realizations of
these hierarchies in physical systems.
A comprehensive realization is the one studied by
Drinfeld and Sokolov in terms of linear systems defined on
Lie algebras, \cite{DS}; let us refer
to it as the Drinfeld--Sokolov realization (DSR).
In this letter we present a new general realization of
integrable hierarchies in terms of the Toda lattice hierarchy (TLH).
We call it Toda lattice realization (TLR), and it looks as general
as the DSR. While the DSR is contiguous to (reduced) WZNW models
and Toda field theories in 2D,
the TLR is inspired by matrix models, see \cite{BX1},\cite{BCX}.
The letter is organized as follows. In section 2 we introduce the TLR.
We do not give a general proof of it, but in section 3 we verify it
on a large number of examples among KP, n--KdV and other classes
of hierarchies. Section 5 is devoted to some comments.
\section{The Toda lattice realization of integrable hierarchies.}
\setcounter{equation}{0}
\setcounter{subsection}{0}
In the Gelfand--Dickey (GD) formalism an integrable hierarchy can
be entirely specified in terms of the Lax operator
\begin{eqnarray}
L = {\partial}^N + Na_1 {\partial}^{N-2}+ Na_2 {\partial}^{N-3}+...+Na_{N-1}+ N a_N {\partial}^{-1}+...
\label{Lax}
\end{eqnarray}
where ${\partial} = {{\partial}\over{{\partial} x}}$. The operator $L$ may be purely
differential, in which case $a_k=0$ for $ k\geq N$, and we get
the $N$-KdV hierarchy. The fields $a_k$
may be either elementary or composite of more elementary fields,
as in the case of the $(N,M)$--KdV hierarchies studied in
\cite{BX2},\cite{BX3},\cite{BLX}.
If the hierarchy is integrable, the flows are given by
\begin{eqnarray}
\frac {{\partial} L} {{\partial} t_k} = [ (L^{k/N})_+, L] \label{GDflows}
\end{eqnarray}
where the subscript + denotes the differential part of a
pseudodifferential operator, $t_1$ is identified with $x$ and
$k$ spans a specific subset of the positive integers.
The Toda lattice hierarchy is defined in terms of a semi--infinite
Jacobi matrix $\hat Q$ \footnote{In this paper we limit ourselves
to a simple version
of the TLH, in which only one matrix $\hat Q$ and one
set of parameters intervene,
instead of two or more \cite{UT},\cite{BX1}}. We parametrize it as
follows
\begin{eqnarray}
\hat Q=\sum_{j=0}^\infty \Big( E_{j,j+1} + \sum_{l=0}^{\infty}
\hat a_l(j)E_{j+l,j}\Big),
\qquad (E_{j,m})_{k,l}= \delta_{j,k}\delta_{m,l}\label{jacobi}
\end{eqnarray}
and consider $\hat a_l$ as fields defined on a lattice. The flows
are given by
\begin{eqnarray}
\frac {{\partial} \hat Q} {{\partial} t_k} = [(\hat Q^k)_+ , \hat Q] ,
\qquad k=1,2,...\label{Tflows}
\end{eqnarray}
where the subscript + denotes the upper triangular part of a matrix,
including the main diagonal. (\ref{Tflows}) represents a hierarchy of
differential--difference equations for the fields $a_l$. In particular
the first flows are
\begin{eqnarray}
\hat a_l(j)' = \hat a_{l+1}(j+1)- \hat a_{l+1}(j) + \hat a_l(j)\Big(\hat a_0(j)-
\hat a_0(j-l)\Big)\label{ff}
\end{eqnarray}
where we have adopted the notation $\frac {{\partial}}{{\partial} t_1}f \equiv f'\equiv {\partial} f$,
for any function $f$. The parameter $t_k$ of the TLH will be identified
later on with the
corresponding parameter $t_k$ in (\ref{GDflows}) whenever the latter exists;
therefore, in particular, $t_1$ will be identified with $x$.
Next, integrability permits us to introduce the function $\hat F(n,t)$
(the free energy in matrix models) via
\begin{eqnarray}
\frac{{\partial}^2}{{\partial} t_{k}{\partial} t_{l}} \hat F(n,t)
= {\rm Tr}\Big([\hat Q^k_+, \hat Q^l]\Big)
\label{F}
\end{eqnarray}
where ${\rm Tr} (X)$ denotes the finite trace $\sum_{j=0}^{n-1}X_{j,j}$. In
particular (\ref{F}) leads to
\begin{eqnarray}
\frac{{\partial}^2}{{\partial} t_{1}^2} \hat F(n,t)=\hat a_1(n)\label{a1}
\end{eqnarray}
It is clear that by means of (\ref{Tflows}) we can compute the derivatives
of any order of $\hat F$ in terms of the entries of $\hat Q$. In general we will
denote by $\hat F_{k_1,...,k_s}$ the derivative of $\hat F$ with respect to
$t_{k_1},...,t_{k_s}$.
Next we introduce the operator $D_0$, defined by its action on any discrete
function $f(n)$
\begin{eqnarray}
(D_0 f)(n) = f(n+1)\nonumber
\end{eqnarray}
For later use we remark that, if $f_0=0$, the operation
${\rm Tr}$ is the inverse of the operation $D_0-1$.
We will also use the notation $e^{{\partial}_0}$ instead of $D_0$,
with the following difference: $D_0$ is meant to be applied to the nearest
right neighbour, while $e^{{\partial}_0}$ acts on whatever is on its right.
Now we can equivalently represent the matrix $\hat Q$ by the following operator
\begin{eqnarray}
\hat Q(j) = e^{{\partial}_0} + \sum_{l=0}^\infty \hat a_l(j) e^{-l{\partial}_0}\label{Qj}
\end{eqnarray}
The contact between (\ref{Qj}) and (\ref{jacobi}) is made by acting with the
former on a discrete function $\xi(j)$; then $\hat Q(j)\xi(j)$
is the same as
the $j$--th component of $\hat Q \xi$, where $\xi$ is a column vector with
components
$\xi(0),\xi(1),...$. We will generally drop the dependence on $j$
in (\ref{Qj}) and merge the two symbols.
After this short introduction to the GD formalism and the Toda lattice
hierarchy, let us come to the presentation of the TLR of the integrable
hierarchy defined
by the Lax operator (\ref{Lax}), i.e. to the problem of embedding the latter
into the TLH. The prescription consists of several steps.
{\it Step 1}. In $\hat Q$ we set $\hat a_0=0$ and replace the first flows
(\ref{ff})
with
\begin{eqnarray}
D_0\hat a_1 = \hat a_1,\qquad D_0 \hat a_i= \hat a_i + \hat a_{i-1}',
\qquad i=2,3,...\label{ff'}
\end{eqnarray}
{\it Step 2}. We compute
\begin{eqnarray}
\frac {{\partial} \hat a_1}{{\partial} t_k} = {\partial} {\rm Tr}\Big([\hat Q_+, \hat Q^k]\Big)\equiv{\partial} \hat F_{1,k}
\label{dta1}
\end{eqnarray}
The right hand side will be a polynomial of the fields $\hat a_k$
to which monomials of $D_0$ and $D_0^{-1}$ are applied. Next we substitute
the first flows (\ref{ff'}) to eliminate the presence of $D_0$.
Examples:
\begin{eqnarray}
&&\hat F_{1,1} = \hat a_1, \nonumber\\
&&\hat F_{1,2} = (D_0+1)\hat a_2 = 2\hat a_2 + \hat a_1',\label{A1k}\\
&&\hat F_{1,3} = (D_0^2 + D_0 +1) \hat a_3 + D_0\hat a_1\hat a_1+
\hat a_1\hat a_1 +\hat a_1D_0^{-1}\hat a_1= 3\hat a_3 +3\hat a_2' +\hat a_1'' +3\hat a_1^2\nonumber
\end{eqnarray}
and so on.
Next we recall that
\begin{eqnarray}
\hat F_{1,k} = \frac{{\partial}^2}{{\partial} t_{k}{\partial} t_{1}} \hat F\nonumber
\end{eqnarray}
Using this and (\ref{A1k}), we can recursively write all the derivatives
of $\hat a_l$ with respect to the couplings $t_k$ (and in particular the flows)
in terms of derivatives
of $\hat F$, which, in turn, can be expressed as functions of the entries
of $\hat Q$. Example:
\begin{eqnarray}
\frac {{\partial} }{{\partial} t_k}\hat a_2 = {1\over 2} {\partial} {\rm Tr} \Big([\hat Q^2_+, \hat Q^k]\Big)
-{1\over 2} \frac{{\partial}}{{\partial} t_k} \hat a_1'\nonumber
\end{eqnarray}
In general we will need all $\hat F_{k_1,...,k_n}$. Here are some general
formulas. Let us introduce the symbols $\hat A^{[k]}_j$ as follows
\begin{eqnarray}
\hat Q^k={e^{k\partial}}+ka_1{e^{(k-2)\partial}}+\hat A^{[k]}_2{e^{(k-3)\partial}}+\hat A^{[k]}_3{e^{(k-4)\partial}}+\hat A^{[k]}_4{e^{(k-5)\partial}}+\ldots
\end{eqnarray}
The explicit form of the first few is:
\begin{eqnarray}
\hat A^{[k]}_2&=&\left(\begin{array}{c}k\\2\end{array}\right) a_1'+k\hat a_2\label{Akj}\\
\hat A^{[k]}_3&=&\left(\begin{array}{c}k\\3\end{array}\right) a_1''+\left(\begin{array}{c}k\\2\end{array}\right)\hat a_2'
+k\hat a_3+\left(\begin{array}{c}k\\2\end{array}\right) a^2_1\nonumber\\
\hat A^{[k]}_4&=&\left(\begin{array}{c}k\\4\end{array}\right) a_1'''+\left(\begin{array}{c}k\\3\end{array}\right)\hat a_2''+
\left(\begin{array}{c}k\\2\end{array}\right)\hat a_3'+
k\hat a_4+(3\left(\begin{array}{c}k\\3\end{array}\right)-\left(\begin{array}{c}k\\2\end{array}\right))a_1a_1'+
2\left(\begin{array}{c}k\\2\end{array}\right) a_1\hat a_2\label{Akk}\nonumber\\
\hat A^{[k]}_5&=&\left(\begin{array}{c}k\\5\end{array}\right) a_1^{(4)}+
\left(\begin{array}{c}k\\4\end{array}\right)\hat a_2'''+
\left(\begin{array}{c}k\\3\end{array}\right)\hat a_3''+
\left(\begin{array}{c}k\\2\end{array}\right) \hat a_4'+k\hat a_5+
\left(\begin{array}{c}k\\2\end{array}\right) \hat a_2\hat a_2+
2\left(\begin{array}{c}k\\2\end{array}\right) a_1\hat a_3\nonumber\\
&&+\left(\begin{array}{c}k\\3\end{array}\right) a_1^3+
(3\left(\begin{array}{c}k\\4\end{array}\right)-\left(\begin{array}{c}k\\3\end{array}\right))a_1'a_1'+
(4\left(\begin{array}{c}k\\4\end{array}\right)-2\left(\begin{array}{c}k\\3\end{array}\right)+
\left(\begin{array}{c}k\\2\end{array}\right))a_1a_1''+\nonumber\\
&& (3\left(\begin{array}{c}k\\3\end{array}\right)-\left(\begin{array}{c}k\\2\end{array}\right)) a_1\hat a_2'+
(3\left(\begin{array}{c}k\\3\end{array}\right)-2\left(\begin{array}{c}k\\2\end{array}\right))\hat a_2 a_1'\nonumber
\end{eqnarray}
and so on. In terms of these coefficients we can compute all the derivatives
of $\hat F$. For example
\begin{eqnarray}
\hat F_{1,k} &=& \hat A^{[k]}_k\nonumber\\
\hat F_{2,k}&=&(D_0+1)\hat A^{[k]}_{k+1}\nonumber\\
\hat F_{3,k}&=&(D_0^2+D_0+1)\hat A^{[k]}_{k+2}+3a_1\hat A^{[k]}_k\label{Fkk}\\
\hat F_{4,k}&=&(D_0^3+D_0^2+D_0+1)\hat A^{[k]}_{k+3}+
4a_1(D_0+1)\hat A^{[k]}_{k+1}+a_2^{(4)}\hat A^{[k]}_{k}\nonumber
\end{eqnarray}
This procedure allows us to compute all the derivatives of the fields $\hat a_l$
in terms of the same fields and their derivatives with respect to
$x\equiv t_1$ -- therefore, in particular, the flows.
So far all our moves have been completely general (except for setting
$\hat a_0=0$,
but see the comment at the end of this section). The next step is instead
a `gauge choice', that is we make a particular choice for the matrix $\hat Q$.
The word `gauge' is not merely colorful. In fact gauge transformations play
here a role analogous to gauge transformations in \cite{DS}. The relevant
gauge transformations in the present case are defined by $\hat Q$ $\to$
$G_-\hat Q G_-^{-1}$, where $G_-$ is a strictly lower triangular semi--infinite
matrix.
{\it Step 3}. We fix the gauge by imposing the condition
\begin{eqnarray}
\hat Q^N= e^{N{\partial}_0} + \sum_{l=1}a_l e^{(N-1-l){\partial}_0}\label{gf}
\end{eqnarray}
where the $a_l$ are the same as in eq.(\ref{Lax}).
The matrix $\hat Q$ that satisfies such condition will be referred to as $\bar Q$.
It is clear that $\bar Q^N$ exactly mimics the Lax operator $L$.
The condition (\ref{gf}) recursively determines
$\hat a_k$ in terms of the fields $a_l$ that appear in $L$.
\begin{eqnarray}
\hat a_k = \bar a_k \equiv P_k(a_l)\nonumber
\end{eqnarray}
where $P_k$ are differential polynomials of $a_l$.
In particular we always have $\hat a_1= \bar a_1\equiv a_1$.
{\it Step 4}. Then we evaluate both sides of the flows found in {\it Step 2}
at $ \hat a_k = \bar a_k $. The order here is crucial. The gauge fixing of the
flows must be the last operation.
Now we claim:
{\bf Claim}. {\it The flows obtained in this way coincide with the flows
(\ref{GDflows}) for corresponding couplings.}
We will substantiate this claim with a large number of examples in the next
section.
It is perhaps useful to summarize our method: {\it start from the TLH flows,
use the first flows (\ref{ff'}) and impose the relevant gauge fixing; the
resulting flows are the desired differential integrable flows.}
We would like to end this section with a remark concerning the restriction
$\hat a_0=0$ we imposed at the very beginning.
This can be avoided at the price of
working with very encumbering formulas. One can keep
$\hat a_0\neq 0$ provided one
uses the first flows (\ref{ff}) instead of (\ref{ff'}) in {\it Step 1}.
In this way it is possible, in general, to eliminate $D_0$ in the flows
only when it acts over $\hat a_l$, $l\neq 0$ (see the last section for an
additional comment on this point). We obtain in this way the same equations
as above with the addition of terms involving $\hat a_0$.
We can suppress all these
additional terms at the end ({\it Step 5}) by imposing $\hat a_0=0$ as part of
the gauge choice. The final result is of course the same as before. This
justifies our having imposed $\hat a_0=0$ from the very beginning.
\section{Examples.}
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In this section we present a large number of examples in support of
the claim of the previous section. Of course for obvious reasons of space
we can explicitly exhibit a few cases only, and for each case only a few flows
among those we have checked.
\subsubsection*{The KP hierarchy}
The KP case corresponds to $n=1$ in (\ref{Lax}). Therefore there is no gauge
fixing: $\hat a_l=a_l$. The flows obtained with our method are simply those
in {\it Step 3}. Examples:
\begin{eqnarray}
&&\frac{{\partial} \hat a_1} {{\partial} t_2} = \Big(2\hat a_2+\hat a_1'\Big)', \qquad\qquad
\frac{{\partial} \hat a_1} {{\partial} t_3} = \Big(3\hat a_3 +3\hat a_2' +\hat a_1''
+3\hat a_1^2\Big)'\nonumber\\
&&\frac{{\partial} \hat a_2} {{\partial} t_2}= \Big(2\hat a_3 +\hat a_2' + \hat a_1^2\Big)',\qquad
\frac{{\partial} \hat a_2} {{\partial} t_3}= \Big(3\hat a_4 +3\hat a_3' +\hat a_2'' +6 \hat a_1
\hat a_2\Big)'\nonumber
\end{eqnarray}
and so on. Setting $\hat a_l=a_l$, these are exactly the KP flows.
\subsubsection*{The $N$-KdV hierarchy case}
In \cite{BCX} we have explicitly shown that our claim is true for the 3--KdV
hierarchy. In this section we generalize that result. To start with we
pick a generic $N$. The relevant differential operator is
\begin{eqnarray}
L=D^N+Na_1D^{N-2}+Na_2D^{N-3}+\ldots +Na_{N-1}\label{nkdv}
\end{eqnarray}
We also write
\begin{eqnarray}
L^{k/N}=D^k+ka_1D^{k-2}+b_2^{[k]}D^{k-3}+\ldots +b_{j-1}^{[k]}D^{k-j}
+...\label{Lk}
\end{eqnarray}
The coefficients $b_j^{[k]}$ are differential polynomials in $a_l$,
$l=1,...,a_{N-2}$.
Working out the commutator in relation (\ref{GDflows}), we can
write down the general formula for arbitrary flow $t_m$:
\begin{eqnarray}
{{\partial} a_{j-1}\over {\partial} t_m}&=&
\sum_{k=0}^{m-1}\left(\begin{array}{c}m\\k\end{array}\right)a_{j+k-1}^{(m-k)}-
\sum_{k=0}^{m-2}{1\over N}\left(\begin{array}{c}N\\j+k\end{array}\right)
(b_{m-k-1}^{(m)})^{(j+k)}+\label{nflows}\\
&+&\sum_{k=0}^{m-3}(\left(\begin{array}{c}m-2\\k\end{array}\right)
b_{k-1}^{(m)}a_{j-k-1}^{(m-2-k)}-
\sum_{l=0}^{m-2}\left(\begin{array}{c}N-j-k\\l-k\end{array}\right)
a_{k-1}(b_{m-l-1}^{(m)})^{(l-k)})\nonumber\\
&-&\sum_{k=2}^{j-1}\sum_{l=0}^{m-2}\left(\begin{array}{c}N-k\\j-k+l\end{array}\right) a_{k-1}
(b_{m-l-1}^{(m)})^{(j-k+l)}\nonumber
\end{eqnarray}
Now let us pass to the TLR of this hierarchy. We recall eqs.(\ref{Akj})
and (\ref{Fkk}). We fix the gauge by imposing $\hat A^{[N]}_j=Na_j$. We solve
the equations for $\hat a_j$ in terms of $a_j$ and obtain $\bar a_j$. Next we
insert back the result in the formulas of the coefficients $\hat A^{[k]}_j$ so that
they become functions of $a_j$. We call the result $\bar A^{[k]}_j$.
Examples:
\begin{eqnarray}
\bar A^{[k]}_2/k&=&a_2-{N-k\over 2}a_1'\nonumber\\
\bar A^{[k]}_3/k&=&a_3-{N-k\over 2}a_2'+{(N-2k+3)(N-k)\over 12}a_1''-
{N-k\over 2}a_1^2\\
\bar A^{[k]}_4/k&=&a_4-{N-k\over 2}a_3'+{(N-2k+3)(N-k)\over 12}a_2''-
{(N-k+2)(k-2)(N-k)\over 24}a_1'''\nonumber\\
&&- (N-k)a_1 a_2+{(N-k+2)(N-k)\over 2}a_1 a_1'\nonumber
\end{eqnarray}
Then, using our recipe, we obtain
\begin{eqnarray}
{\partial}^{-1}{{\partial} a_1\over{\partial} t_k}&=&\frac {{\partial}^2 F}{{\partial} t_1{\partial} t_k}|_{\hat a=\bar a}
=\bar A^{[k]}_k\nonumber\\
{\partial}^{-1}{{\partial} a_2\over{\partial} t_k}&=&\Big({1\over 2}
\frac {{\partial}^2 F}{{\partial} t_2{\partial} t_k}\Big)|_{\hat a=\bar a}
+{N-2\over 2}{{\partial} a_1\over {\partial} t_k}=
\bar A^{[k]}_{k+1}+ {N-1\over 2}(\bar A^{[k]}_k)'\nonumber\\
{\partial}^{-1}{{\partial} a_3\over{\partial} t_k}&=&\Big({1\over 3}
\frac {{\partial}^2 F}{{\partial} t_3{\partial} t_k}\Big)|_{\hat a=\bar a}+
{N-3\over 2}{{\partial} a_2\over {\partial} t_k}-{(N-3)^2\over 12}{{\partial} a_1'\over {\partial} t_k}
+(N-3){\partial}^{-1}(a_1{{\partial} a_1\over {\partial} t_k})\nonumber\\
&=& \bar A^{[k]}_{k+2}+ {N-1\over 2}(\bar A^{[k]}_{k+1})'+{(N-1)(N-2)\over 6}(\bar A^{[k]}_k)''+
a_1 \bar A^{[k]}_k+(N-3){\partial}^{-1}(a_1(\bar A^{[k]}_k)')\nonumber\\
{\partial}^{-1}{{\partial} a_4\over{\partial} t_k}&=&\Big({1\over 4}
\frac {{\partial}^2 F}{{\partial} t_4{\partial} t_k}+
{N-4\over 2}{{\partial} a_3\over {\partial} t_k}\Big)|_{\hat a=\bar a}
-{(N-4)(N-5)\over 12}{{\partial} a_2'\over {\partial} t_k}
+(N-4){\partial}^{-1}{{\partial}( a_1a_2)\over{\partial} t_k}\nonumber\\
&-& {(N-4)(N-2)\over 2}({1\over 6}{{\partial} a_1''\over {\partial} t_k}
+a_1{{\partial} a_1\over {\partial} t_k})=
\bar A^{[k]}_{k+3}+ {N-1\over 2}(\bar A^{[k]}_{k+2})'\nonumber\\
&+&{(3N-11)(N-1)(N-2)\over 24}(\bar A^{[k]}_{k})'''-
{(N-2)(N-4)\over 2}a_1(\bar A^{[k]}_k)'+\nonumber\\
&+&{(5N-16)(N-1)\over 12}(\bar A^{[k]}_{k+1})''+ (N-4){\partial}^{-1}(a_1 (\bar A^{[k]}_{k+1})'+
{N-1\over 2}(a_1(\bar A^{[k]}_{k})''+
a_2(\hat A^{[k]}_k)')\nonumber
\label{toda234}
\end{eqnarray}
and so on, where ${\hat a=\bar a}$ denotes gauge fixing.
We give a few concrete examples of the second and third flows:
\begin{eqnarray}
{\partial}^{-1}{{\partial} a_1\over {\partial} t_2}&=&2a_2-(N-2)a_1'\nonumber\\
{\partial}^{-1}{{\partial} a_2\over {\partial} t_2}&=&2a_3+a_2'-{(N-1)(N-2)\over 3}a_1''-
(N-2)a_1^2\nonumber\\
{{\partial} a_3\over {\partial} t_2}&=&2a_4'+a_3''-{(N-1)(N-2)(N-3)\over 12}a_1^{(4)}
-(N-2)(N-3)a_1a_1''-2(N-3)a_2a_1'\nonumber
\end{eqnarray}
\begin{eqnarray}
{\partial}^{-1}{{\partial} a_1\over {\partial} t_3}&=&3a_3-{3\over 2}(N-3)a_2'+
{(N-3)^2\over 4}a_1''-{3\over 2}(N-3)a_1^2\nonumber\\
{\partial}^{-1}{{\partial} a_2\over {\partial} t_3}&=&3a_4+3a_3'-{N(N-3)\over 2}a_2''+
{(N-1)(N-2)(N-3)\over 8}a_1'''-3(N-3)a_1a_2\nonumber\\
{{\partial} a_3\over {\partial} t_3}&=&3a_5'+3a_4''+a_3'''-
{3\over N}\left(\begin{array}{c}N\\4\end{array}\right)a_2^{(4)}+
{3(3N-7)\over 10N}\left(\begin{array}{c}N\\4\end{array}\right)a_1^{(5)}+
3a_1a_3'-3(N-4)a_1'a_3\nonumber\\
&&-3(N-3)a_2a_2'-{3\over 2}(N-2)(N-3)a_1a_2''+
{3\over 2}(N-3)a_2a_1''+
{6\over N}\left(\begin{array}{c}N\\4\end{array}\right)a_1a_1'''\nonumber
\end{eqnarray}
These are flows pertinent to the $N$--KdV hierarchy with $N>3$.
In general the formulas of the $N$--KdV hierarchy and the corresponding
formulas obtained with our method coincide since
\begin{eqnarray}
b_k^{[m]}=\bar A_k^{[m]}\nonumber
\end{eqnarray}
We have checked these identities case by case up to the 5-KdV and
for $m\leq 5$. For the dispersionless case we have verified the
correspondence up to the 8--KdV flows.
\subsubsection*{The DS hierarchies}
Drinfeld and Sokolov, \cite{DS}, introduced a large set of
generalized KdV systems in terms of
the pair $(G,c_m)$, where $G$ is a classical Kac-Moody algebra and
$c_m$ is a vertex of the Dynkin diagram of $G$. From each choice
of the pair $(G,c_m)$ they were able
to construct a pseudo--differential operator $L$ which give rise to a
hierarchy of integrable equations.
We have studied all the examples corresponding to the operator
$L$ of orders 3,4,5 and found a complete agreement with our method.
For simplicity here we present a few examples of order 4 and 5, corresponding
to the cases with a pseudodifferential Lax operator. The
cases with a differential Lax operator are restriction of the 4-- and 5--KdV
hierarchies, and will be omitted.
In each case we give the
explicit form of the (pseudo--)differential operator $L$, the
gauge--fixed matrix $\bar Q$ and the first significant flows:
\noindent{\bf Order 4.}
\vskip 0.2 cm
\noindent {\bf Case} $B_2^{(1)}$:
\begin{eqnarray}
c_0,c_1~~:L&=&D^4+2u_1D^2+u_1'D+2(u_0+u_1'')-D^{-1}(u_0+u_1'')'\\
\hat Q &=&{e^{\partial}}+{v_1\over 4}e^{-\partial}-{v_1'\over 4}e^{-2\partial}+({1\over 4}v_0+{v_1''\over 8}-
{3\over 32}v_1^2)e^{-3\partial}+\nonumber\\
&&+({3\over 8}v_1v_1'-{1\over 2}v_0')e^{-4\partial}+\ldots\nonumber\\
c_2~~:L&=&D^4+2u_1D^2+u_1'D+u_0^2-u_0D^{-1}u_0'\nonumber
\end{eqnarray}
For $c_2$ we have, up to the order $e^{-4\partial}$, the same expression for $\hat Q$
with $v_0=u_0^2$. The first non--trivial flows are:
\begin{eqnarray}
{{\partial} v_1\over {\partial} t_3}&=&-{1\over 2}v_1'''-{3\over 4}v_1v_1'+3v_0'\nonumber\\
{{\partial} v_0\over {\partial} t_3}&=&v_0'''+{3\over 4}v_1v_0'
\end{eqnarray}
where for $c_0,c_1:v_1=2u_1,v_0=2(u_0+u_1'')$ and for
$c_2:~v_1=2u_1,v_0=u_0^2$.
\noindent{\bf Case} $D_3^{(1)}$:
\begin{eqnarray}
c_0,c_1:~~L&=&D^4+2u_2D^2+u_2'D+2u_2''+2u_1-D^{-1}(u_1'+u_2'')'+
(D^{-1}u_0)^2\label{D31}\\
\hat Q&=&{e^{\partial}}+{u_2\over 2}e^{-\partial}-{1\over 2}u_2'e^{-2\partial}+
({3\over 4}u_2''-{3\over 8}{u_2^2}-{1\over 2}u_1)e^{-3\partial}+\nonumber\\
&&+({3\over 2}u_2u_2'-u_1'-u_2''')e^{-4\partial}+\ldots\nonumber\\
c_2,c_3:~~L&=&D^4+2u_2D^2+3u_2'D+(2u_1+3u_2'')+(u_1'+u_2''')D^{-1}+
u_0D^{-1}u_0D^{-1}\nonumber\\
\hat Q&=&{e^{\partial}}+{u_2\over 2}e^{-\partial}+
({1\over 4}u_2''-{3\over 8}u_2^2+{1\over 2}u_1)e^{-3\partial}+\nonumber\\
&&+({3\over 4}u_2u_2'-{1\over 2}u_1'-{1\over 4}u_2''')e^{-4\partial}+\ldots\nonumber
\end{eqnarray}
The first non--trivial equations for $c_0,c_1$ are:
\begin{eqnarray}
{{\partial} u_1\over {\partial} t_3}&=&3u_0u_0'-{3\over 2}u_2^{(5)}-2u_1'''+{3\over 2}u_1'u_2
+3u_2u_2'''+{9 \over 2}u_2'u_2''\label{fD31}\\
{{\partial} u_2\over {\partial} t_3}&=&{5\over 2}u_2'''-{3\over 2}u_2u_2'+3u_1'\nonumber
\end{eqnarray}
and for $c_2,c_3$ are
\begin{eqnarray}
{{\partial} u_1\over {\partial} t_3}&=&3u_0u_0'-{3\over 2}u_2^{(5)}-2u_1'''-
{3\over 2}u_1'u_2+{9\over 2}u_2'u_2''+3u_2u_2'''\label{fD31'}\\
{{\partial} u_2\over {\partial} t_3}&=&{5\over 2}u_2'''-{3\over 2}u_2u_2'+3u_1'\nonumber
\end{eqnarray}
\vskip.2cm
\noindent {\bf Order 5.}
\vskip.2cm
\noindent {\bf Case} $A_5^{(2)}$:
\begin{eqnarray}
c_0,c_1~~:&&L= D^5+2u_2D^3+2u_2'D^2+(2u_1+4u_2'')D+D^{-1}(2u_0+u_1''+u_2''')\\
&&\hat Q={e^{\partial}}+{2\over 5}u_2e^{-\partial}-{2\over 5}u_2'e^{-2\partial}+
{2\over 5}(u_1+2u_2''-{4\over 5}u_2^2)e^{-3\partial}+\ldots\nonumber\\
c_2~~:&&L=D^5+2(v_0+u_1)D^3+(6v_0'+u_1')D^2+(6v_0''+u_0^2+4v_0u_1)D+\nonumber\\
&&+(2v_0'''-u_0u_0'+4u_1v_0'+2v_0u_1')+u_0D^{-1}(u_0''+2u_0v_0)\nonumber\\
&&\hat Q={e^{\partial}}+{2\over 5}(u_1+v_0)e^{-\partial}+{1\over 5}(2v_0'-3u_1')e^{-2\partial}+\nonumber\\
&&+{1\over 5}(2u_1''-2v_0''+{4\over 5}v_0u_1+u_0^2-
{8\over 5}u_1^2-{8\over 5} v_0^2)e^{-3\partial}+\ldots\nonumber\\
c_3~~:&&L=D^5+2u_2D^3+3u_2'D^2+(2u_1+3u_2'')D+u_1'+u_2'''+u_0D^{-1}u_0\nonumber\\
&&\hat Q={e^{\partial}}+{2\over 5}u_2e^{-\partial}-{1\over 5}u_2'e^{-2\partial}+{1\over 5}
(2u_1+u_2''-{8\over 5}u_2^2)e^{-3\partial}+\ldots\nonumber
\end{eqnarray}
The equations are for $c_0,c_1$:
\begin{eqnarray}
{{\partial} u_2\over {\partial} t_3}&=&4u_2'''+3u_1'-{12\over 5}u_2u_2'\\
{{\partial} u_1\over {\partial} t_3}&=&-{7\over 2}u_1'''+6u_2u_2'''+{54\over 5}u_2'u_2''
+{6\over 5}(u_2u_1'-u_1u_2')-{{51}\over {10}}u_2^{(5)}+3 u_0'\nonumber
\end{eqnarray}
for $c_2$:
\begin{eqnarray}
5{{\partial} v_0\over {\partial} t_3}&=&-v_0'''+
{3\over 2}u_1'''-12v_0v_0'+6v_0'u_1+12v_0u_1'\\
5{{\partial} u_1\over {\partial} t_3}&=&6v_0'''-
4u_1'''+15u_0u_0'+6v_0u_1'+12v_0'u_1-12u_1u_1'\nonumber
\end{eqnarray}
and for $c_3$ are:
\begin{eqnarray}
{{\partial} u_2\over {\partial} t_3}&=&u_2'''+3u_1'-{12\over 5}u_2u_2'\\
{{\partial} u_1\over {\partial} t_3}&=&-2u_1'''+{27\over 5}u_2'u_2''+{12\over 5}u_2u_2'''
+{6\over 5}(u_2u_1'-u_1u_2')-{3\over 5}u_2^{(5)}+3u_0u_0'\nonumber
\end{eqnarray}
\subsubsection*{The (N,M)--KdV hierarchies}
The $(N,M)$--KdV hierarchies are defined by the pseudodifferential
operator
\begin{eqnarray}
L={\partial}^{N}+N\sum_{l=1}^{N-1} a_l{\partial}^{N-l-1}
+N\sum_{l=1}^M a_{N+l-1}{1\over{{\partial}-S_l}}{1\over{{\partial}-S_{l-1}}}
\ldots {1\over{{\partial}-S_1}},\quad N\ge1,~M\ge0 \label{pdo}
\end{eqnarray}
The case $(N,0)$ coincides with the $N$--KdV case. These hierarchies
were studied in \cite{BX2},\cite{BX3},\cite{BLX},\cite{D}. In \cite{BLX}
it was shown that they can be embedded in the DS construction. Now we show
that this class of integrable hierarchies can be entirely embedded in the
TLH. Let us see, for example, the $(2,1)$ case. The Lax operator is
\begin{eqnarray}
L={\partial}^2+2a_1+2a_2\frac{1}{{\partial}-S}\nonumber
\end{eqnarray}
The gauge fixing gives
\begin{eqnarray}
\bar a_1= a_1,\qquad\bar a_2 = a_2 -{1\over 2} a_1',\qquad
&&\bar a_3 = - {1\over 2}a_2' +{1\over 4} a_1''-{1\over 2}a_1^2
+a_2S\nonumber
\end{eqnarray}
and so on. It leads, via our recipe, to the following flows
\begin{eqnarray}
&&{\partial}^{-1} \frac{{\partial} a_1} {{\partial} t_2} = 2a_2,\qquad
{\partial}^{-1} \frac{{\partial} a_2} {{\partial} t_2} = a_2' + 2a_2S,\qquad
{\partial}^{-1} \frac{{\partial} S} {{\partial} t_2}= S^2 + 2a_1 -S,'\nonumber\\
&&{\partial}^{-1} \frac{{\partial} a_1} {{\partial} t_3}= {3\over 2} a_2' +{1\over 4}a_1''
+{3\over 2}a_1^2 +3 a_2S,\qquad
{\partial}^{-1} \frac{{\partial} a_2} {{\partial} t_3}= a_2''+3a_1a_2+3a_2'S+3a_2S^2\nonumber
\end{eqnarray}
and so on. These are exactly the flows of the (2,1)--KdV hierarchy.
\section{Comments and conclusion}.
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\setcounter{subsection}{0}
The examples we have considered in the previous section do not
exhaust all possible integrable hierarchies (for an updating on this
subject see \cite{Grimm}). However they are very numerous and they
leave very little doubt that whatever scalar Lax operator (\ref{Lax}),
defining an integrable hierarchy, we may think of, it can be embedded
in the Toda lattice hierarchy in the way we showed above.
Anyhow, thus far we have not found any counterexample.
Therefore our construction looks at
least as general as the DS realization. The fact that we are dealing
with semi--infinite matrices may suggest additional possibilities.
We also remark that the TLH, in its general formulation, may
encompass several $\hat Q$ matrices (not only one, as in this paper).
Therefore there is room for `tensor products of integrable hierarchies
in interaction'.
We end the paper by recalling that in the case of the
${(1,M)}$--KdV hierarchies
there is a variant to the realization of section 2. This was already
pointed out in section 6.2 of \cite{BCX} and, implicitly, in \cite{BX2}.
If one does not set $\hat a_0=0$ and replaces the first flows (\ref{ff})
in the Toda lattice flows, one gets exactly the $(1,M)$ hierarchies if
the gauge fixing simply consists of setting $\hat a_l=0$ for $l>M$.
It was shown in \cite{BX2}
that $(N,M)$--KdV hierarchies can then be extracted from the $(1,M)$
via a cascade Hamiltonian reduction. However
it is not clear whether this method can be generalized to other hierarchies,
and, anyhow, it does not seem to be appropriate to call it a realization
of differential hierarchies, at least in the same sense this terminology
has been used in this paper.
\vskip.5cm
{\bf Acknowledgements}. One of us (C.P.C.) would like to thank CNPq and FAPESP
for financial support.
| 12,231 |
\section*{Abstract}
\else \small
\begin{center}
{\bf ABSTRACT}
\end{center}
\quotation
\fi}
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\begin{document}
\setcounter{page}0
\title{\large\bf Enhanced Gauge Symmetries \\and Calabi--Yau Threefolds\\[10mm]}
\author{
Paul S. Aspinwall\\[0.7cm]
\normalsize F.R.~Newman Lab.~of Nuclear Studies,\\
\normalsize Cornell University,\\
\normalsize Ithaca, NY 14853\\[10mm]
}
{\hfuzz=10cm\maketitle}
\def\large{\large}
\def\large\bf{\large\bf}
\vskip 1.5cm
\vskip 1cm
\begin{abstract}
We consider the general case of a type IIA string compactified on a
Calabi--Yau manifold which has a heterotic dual description. It is
shown that the nonabelian gauge symmetries which can appear
nonperturbatively in the type II string but which are understood
perturbatively in the
heterotic string are purely a result of string-string duality in six
dimensions. We illustrate this with some examples.
\end{abstract}
\vfil\break
\section{Introduction} \label{s:intro}
Our understanding of the dynamics of $N=2$ Yang-Mills theories in four
dimensions has greatly improved recently due to the work of Seiberg
and Witten
\cite{SW:I,SW:II}. The moduli of such theories appear in two kinds of
supermultiplets, i.e., hypermultiplets and vector multiplets. Our
attention will focus on the vector multiplets in this letter. Consider
a theory with $n$ vector multiplets. This will correspond to a
Yang-Mills theory with a gauge group of rank $n$.
Roughly speaking, the vector moduli live in the Cartan subalgebra of
the gauge group and will break this group down to the elements that
commute with the value of the moduli. Thus, the effective gauge group,
$G$, will always be of rank $n$ and will generically be $U(1)^n$. At
particular points and subspaces within the moduli space the gauge group
will become nonabelian. This is the classical picture but quantum
effects modify this behaviour \cite{SW:I} (see also
\cite{KLTY:An,AF:An}). Instead of enhancement of the gauge group at
special points, the gauge group remains $U(1)^n$ but hypermultiplets
which are massive at generic points in the moduli space can become
massless. We want to consider the classical limit in which the gauge
group becomes enhanced but in the context of string theory. Having found
such points in the moduli space we can assume that Seiberg-Witten
theory will take over in the case of nonzero coupling.
Until recent developments in string duality it was generally believed
that a type II superstring compactified on a Calabi-Yau\ manifold would not
exhibit a nonabelian gauge symmetry. When it was realized that the
type IIA string compactified on a K3 surface could naturally be
identified as the dual of the heterotic string on a four-torus
\cite{HT:unity,W:dyn,HS:sol} then it followed that, since the heterotic
string can have nonabelian gauge symmetries, the same must be true for
the type IIA string on a K3 surface. The way in which this was
possible was discussed in \cite{W:dyn} and was further developed in
\cite{me:enhg,W:dyn2,BSV:D-man}.
A similar duality in four dimensions has been conjectured
\cite{KV:N=2,FHSV:N=2} in which the type II string compactified on a
Calabi-Yau\ threefold is considered equivalent by some kind of duality to a
heterotic string compactified on K3$\times T^2$ (or some variant
thereof). Since this heterotic string can again lead to nonabelian
gauge groups the same must be true of type II strings on a Calabi-Yau\
threefold. Based on these conjectured dualities, points in the moduli
space for specific examples of compactification of type II strings on
Calabi-Yau\ manifolds where the gauge group should become enhanced were identified.
In these cases the reduction to Seiberg-Witten theory for nonzero
coupling has been explicitly shown in \cite{KKL:limit}.
Our goal here will be to make general statements about the appearance
of enhanced gauge groups in type IIA compactifications.
The basic idea is actually very simple thanks to the results of
\cite{AL:ubiq}. We will restrict our attention (until some comments at
the end) to gauge groups which are visible perturbatively in the
heterotic string picture. The dual picture to this is a type IIA
string theory compactified on $X$ which must be a K3 fibration and the
gauge group arises from properties of the generic fibre. It was
shown that the dilaton modulus in the heterotic string corresponds to
the size of the base ${\Bbb P}^1$ of the fibration and that the
weak-coupling limit corresponds to the size of this base space being
taken to infinity. Clearly if we look at a generic fibre and the base
space becomes infinitely large then we have effectively decompactified
the 4-dimensional picture to 6 dimensions and we have reduced the
question the that studied in \cite{W:dyn,me:enhg}.
In a recent paper \cite{BSV:D-man} some questions regarding enhanced
gauge groups were studied using $D$-branes, particularly in the type
IIB context. Our results here concern type IIA strings but clearly
have some overlap with the conjectures and examples of
\cite{BSV:D-man}. The $D$-brane approach is probably the most powerful
tool for answering questions regarding type II string
compactifications but the purpose of this letter is to demonstrate how
simply many of the properties can be derived from what we already know
about 6-dimensional duality.
In section \ref{s:K3f} we will review the machinery we need to apply
our picture to examples and in section \ref{s:egs} we give
some examples. Finally we present some concluding remarks in section
\ref{s:conc}.
\section{K3 Fibrations} \label{s:K3f}
We will be concerned with the case that $X$ can be written as a
fibration over ${\Bbb P}^1$ with generic fibre a K3 surface. The importance
of such a class in the context of string duality was first realized in
\cite{KLM:K3f,VW:pairs}.
Let us consider the general case of a dual pair consisting of a type
IIA string compactified on a Calabi-Yau\ manifold, $X$, and a heterotic string such
that the weakly-coupled heterotic string is identified with $X$ at
some kind of large radius limit. It was shown in \cite{AL:ubiq} that
$X$ must be a K3-fibration in this case.
For a generic dual pair, the gauge group is $U(1)^{r+1}$. This comes
from $r$ vector multiplets and the graviphoton. In the type IIA
picture, the vector multiplets come from $H^{1,1}(X)$ and
$r=h^{1,1}(X)$. The group $H^{1,1}(X)\cap H^2(X,{\Bbb Z})$ is known as the
Picard group of $X$. The rank of the Picard group is the Picard
number. In our case we will assume that $h^{2,0}(X)=0$
and so the Picard group of $X$ is simply $H^2(X,{\Bbb Z})$. That is, the
generators of the Picard group of $X$ can be considered to form a basis of
the space of vector multiplets.
The Picard group of $X$ when $X$ is a K3 fibration essentially has
three different contributions. We work in terms of the dual group
$H_4(X)$. The generic K3 fibre itself gives one contribution. Another
source is from elements of the Picard group of the K3 fibre. In this
case a curve in the fibre is transported over the whole base ${\Bbb P}^1$ to
build up a 4-cycle. Note that such curves may have monodromy under
such a transport. One can see that monodromy-invariant elements of the
Picard group of the generic K3 fibre contribute to the Picard group of
$X$. Note that the Picard group of a K3 surface is a more subtle
object than the Picard group of a Calabi-Yau\ threefold. The Picard group of
a K3 surface can depend upon the complex structure of the K3 surface
since $h^{2,0}\neq0$.
Lastly, there will be degenerate fibres over a finite number of points
in the base ${\Bbb P}^1$. In some cases such fibres can also contribute to
the Picard group of $X$.
In \cite{AL:ubiq} these three sources for elements of the Picard group
of $X$ were given different interpretations in terms of the dual
heterotic string. The generic fibre element was identified with the
dilaton, which lies in a vector multiplet. The elements coming from
the Picard group of the fibre were matched with the rest of the gauge
group that was visible perturbatively in the heterotic string. Lastly,
the contributions from the degenerate fibres were expected to have
some nonperturbative origin in the heterotic string.
We are interested in the gauge group that is perturbatively visible in
the heterotic string --- that is the part associated to the Picard
group of the K3 fibres. We would like to know if we can obtain
nonabelian groups by varying the moduli in these vector multiplets in
the weakly-coupled limit. The weakly coupled limit corresponds to the
base ${\Bbb P}^1$ becoming very large. Clearly in this limit, any question
about the generic fibres can be treated purely in terms of the type
IIA string compactified on a K3 surface along the lines of
\cite{W:dyn,me:enhg}.
This means that the analysis is actually very simple --- the question
of enhanced groups for $N=2$ theories in four dimensions is actually
completely reducible to the case of six-dimensions, at least as far as
the perturbative heterotic string is concerned.
To simplify our
discussion we will make an assumption concerning the way in which $X$
is written as a K3 fibration. We will assert that the Picard lattice
of a generic fibre is invariant, i.e., does not undergo any monodromy
transformation, as we move about the base space. One can certainly
find examples which will not obey our assumption but simple examples,
such as the ones we discuss later, do not have monodromy. It should
not be difficult
to extend the analysis here to nontrivial monodromy.
We are thus concerned with the enhanced gauge groups that can appear
on a K3 surface as we vary its K\"ahler form. The fibration of $X$
will restrict the K3 fibre to be of a particular type. Indeed, the generic
fibre will be an algebraic K3 surface which may be considered to be
embedded in some higher-dimensional projective space. An abstract K3
surface has variations in its complex structure and K\"ahler structure
which naturally fill out a space of 80 real dimensions in string
theory \cite{Sei:K3}. In the case of a specific algebraic K3 surface
however only some of these deformations are allowed. In particular,
the K\"ahler form is only allowed to vary within the space spanned by
the Picard group.
For the general case, the stringy moduli space of K3 surfaces is
given locally by the Grassmanian of space-like 4-planes in ${\Bbb R}^{4,20}$. The
space ${\Bbb R}^{4,20}$ may be viewed as the space of total real cohomology
$H^*({\rm K3},{\Bbb R})$ which contains the lattice $H^*({\rm K3},{\Bbb Z})$
which has intersection form $E_8\oplus E_8\oplus H\oplus H\oplus
H\oplus H$ \cite{AM:K3p}. Here $E_8$ denotes {\em minus\/} the
Cartan matrix of the Lie group $E_8$ and $H$ is the hyperbolic
plane. The global form of the moduli space is obtained by dividing
this Grassmanian by the group of isometries of this lattice.
In the case of an algebraic K3 surface, this moduli space is
restricted as follows.\footnote{This analysis rests heavily on work
done in collaboration with D.~Morrison \cite{AM:K3m}. Aspects have
also been discussed in \cite{Mart:CCC,Dol:K3m}.} Divide the
lattice as
\begin{equation}
H^*({\rm K3},{\Bbb Z}) \supseteq \Lambda_K \oplus \Lambda_c.
\label{eq:decomp}
\end{equation}
That is, take $\Lambda_K$ to be a sublattice of the integral
cohomology of the K3 surface and $\Lambda_c$ to be its orthogonal
complement. We demand that the sum of the ranks of these lattices is
equal to the rank of $H^*({\rm K3},{\Bbb Z})$. That is, the lattice
$\Lambda_K \oplus \Lambda_c$ is equal to $H^*({\rm K3},{\Bbb Z})$ or is a
sublattice of finite index.
Let $V_K=\Lambda_K\otimes_{\Bbb Z}{\Bbb R}$ be the real vector spanned by the
generators of $\Lambda_K$ with $V_c$ similarly defined. Now we may
consider the Grassmanian of space-like 2-planes in $V_K$ to be our
restricted moduli space of complexified K\"ahler forms and the
Grassmanian of space-like 2-planes in $V_c$ to be the restricted
moduli space of complex structures. Clearly the product of these two
moduli spaces is a subspace of the total stringy moduli space of K3
surfaces locally. Globally some of the isometries of $H^*({\rm
K3},{\Bbb Z})$ will descend to isometries of $\Lambda_K$ and $\Lambda_c$ to
give identifications within these Grassmanians.
By restricting to a special class of K3 surfaces we have thus managed
to locally factorize the moduli space into deformations of complex
structure and deformations of K\"ahler form. If we want to make contact with
classical geometry then we must insist that the moduli space of
K\"ahler forms contains the large radius limit. In this case
\begin{equation}
\Lambda_K = \operatorname{Pic}{} \oplus H, \label{eq:LK}
\end{equation}
where Pic is the Picard lattice of the algebraic K3 surface. The $H$
factor is then identified with $H^0\oplus H^4$.
Let us denote the Picard number of the fibre by $r$. Since the
signature of the Picard lattice is $(1,r-1)$, we see immediately from
(\ref{eq:LK}) that the moduli space of K\"ahler forms within the fibre
is given by
\begin{equation}
\frac{O(2,r)}{O(2)\times O(r)},
\end{equation}
divided by the group of isometries of $\Lambda_K$. Thus we give a
geometrical interpretation to the result of \cite{FvP:Ka}.
Note that the mirror map exchanges the r\^oles of $\Lambda_K$ and
$\Lambda_c$. Thus the mirror of one algebraic K3 surface is generally a
different algebraic K3 surface and the Picard numbers of these two
surfaces will add up to 20.
Now recall how, using string-string duality in six dimensions, we find
the points in the moduli space of type IIA strings on a K3 surface
where we have enhanced gauge groups \cite{W:dyn,me:enhg}. Let $\Pi$ be
the space-like 4-plane in $H^*({\rm K3},{\Bbb R})$. The set of vectors
\begin{equation}
\{\alpha\in H^*({\rm K3},{\Bbb Z}); \alpha^2=-2 \;{\rm and}\;
\alpha\in\Pi^\perp\},
\end{equation}
give the roots of the semi-simple part of the gauge group $G$.
For the case of interest to us, we are only concerned with the slice
of the moduli space given by deformations of the K\"ahler form on the
K3 surface. Given the decomposition (\ref{eq:decomp}) we see that our
gauge group now has roots
\begin{equation}
\{\alpha\in \Lambda_K; \alpha^2=-2 \;{\rm and}\;
\alpha\in\mho^\perp\}, \label{eq:xG}
\end{equation}
where $\mho$ is the space-like 2-plane\footnote{The reason for this
notation comes from \cite{Cand:mir} as explained in \cite{AM:K3p}.} in
$V_K$.
This essentially gives the full description of how the generic fibres
can enhance the gauge group in the limit that the base space becomes
infinitely large, i.e., when the dual heterotic string becomes
weakly-coupled. First find the Picard group of the generic fibre, then
build $\Lambda_K$ from (\ref{eq:LK}). The gauge group is then given by
(\ref{eq:xG}).
\section{Examples} \label{s:egs}
Let us clarify the discussion of the last section by giving two
examples. The first example we take from \cite{KV:N=2}. Let $X_0$ be the
Calabi-Yau\ hypersurface of degree 24 in the weighted projective space
${\Bbb P}^4_{\{1,1,2,8,12\}}$. The first
two homogeneous coordinates may be used as the homogeneous coordinates
of the base ${\Bbb P}^1$ to form a K3-fibration. The generic fibre may then
be written as a degree 12 hypersurface in ${\Bbb P}^3_{\{1,1,4,6\}}$ which
is indeed a K3 surface which we denote $F_0$.
The space ${\Bbb P}^3_{\{1,1,4,6\}}$ contains a curve of ${\Bbb Z}_2$-quotient
singularities which $F_0$ intersects once at a point. We blow-up the quotient
singularities in $X_0$ to obtain $X$ and this induces a blow-up of
$F_0$ which we call $F$. The latter is locally a blow-up of a ${\Bbb Z}_2$-quotient
singularity and so gives us a rational curve of self-intersection $-2$ within
$F$. Let us call the homology class of this curve $C$. Clearly $C$
lies in the Picard group of $F$. The other contribution to the Picard group
comes from hyperplanes of ${\Bbb P}^3_{\{1,1,4,6\}}$ slicing $F_0$. Let us
denote the resulting curve $A$. Representatives of $A$ pass through
the quotient singularity of $F_0$. When $F_0$ is blown-up such
self-intersections are removed and thus $A$ has self-intersection 0 in
$F$. Since $A$ passed through this singularity in $F_0$, the
intersection number between $A$ and $C$ is equal to 1.
We claim then that $F$ has Picard number 2 with intersection lattice
\begin{equation}
\left(\begin{array}{cc} 0&\phantom{-}1\\1&-2 \end{array}\right).
\end{equation}
Clearly by replacing $C$ with the cycle $B=A+C$ we obtain an
intersection form between $A$ and $B$ equal to $H$, the hyperbolic
plane.
Thus we obtain
\begin{equation}
\Lambda_K \cong H\oplus H
\end{equation}
and the part of the moduli space coming from the fibre is given by
\begin{equation}
\frac{O(2,2)}{O(2)\times O(2)}
\end{equation}
divided by $O(2,2;{\Bbb Z})$ in the usual language. This is, of course, the
Narain moduli space for a string on the 2-torus. If the conjecture in
\cite{KV:N=2} is true then this is no accident as explained in
\cite{VW:pairs}. This part of the moduli space would arise from
deformations of the $T^2$ of the K3$\times T^2$ on which the dual
heterotic string is compactified.
Now we can look for places in the moduli space where the gauge group
is enhanced. The easiest case is when the K3 fibre can be taken to be
at large radius. Thus corresponds to a direction in $\mho$ becoming
almost light-like being close to a null vector in the $H$ factor in
(\ref{eq:LK}) \cite{AM:K3p}. Consistent with such a limit be may take
$\mho$ to be perpendicular to $C$. This means that the K\"ahler form
or $B$-field when integrated over the curve $C$ is zero. That is, we
have blown down $F$ back to $F_0$. The roots corresponding to $\pm C$
give the root lattice of $SU(2)$. Thus we expect an $SU(2)$ gauge
symmetry on the space $X_0$ (for suitable $B$-field).
$X_0$ contains a curve of ${\Bbb Z}_2$-quotient singularities. Note that
this is easy to generalize --- when we can enhance the gauge symmetry
by blowing down a curve in the K3 generic fibre then every fibre will
have a singular point meaning that $X$ will contain a curve of
singularities. The fact that curves of singularities can be associated
with enhanced gauge symmetries in a type IIB context has been
discussed in \cite{BSV:D-man}.
We can also embed the root lattice of $SU(2)\times SU(2)$ or $SU(3)$
into $\Lambda_K$ for the case at hand. In both cases the plane $\mho$
is fixed and so these are isolated points in the moduli space. Note
also that in both cases we cannot be near the large radius limit of
the K3 fibre. These further enhancements of the gauge group thus
correspond to effects of quantum geometry within the K3 fibre when the
volume of the fibre will be of the order of $(\alpha^\prime)^2$.
We can make closer contact with the conjecture of \cite{KV:N=2} by
going to the mirror picture of a type IIB string compactified on $Y$,
the mirror of $X$. In this case, $Y$ is an orbifold of the
hypersurface
\begin{equation}
x_1^{24}+x_2^{24}+x_3^{12}+x_4^3+x_5^2+a_0x_1x_2x_3x_4x_5
+a_1x_1^6x_2^6x_3^6+a_2x_1^{12}x_2^{12}=0
\end{equation}
in ${\Bbb P}^4_{\{1,1,2,8,12\}}$. Application of the monomial-divisor mirror
map in the Calabi-Yau\ phase as described in \cite{AGM:sd} immediately tells
us that, to leading order, the size of the base ${\Bbb P}^1$ is given by
$\log(a_2^2)$; the size of the blown-up curve, $C$, within the K3
fibre is $\log(a_1^2/a_2)$; and the size of the fibre itself is given
by $\log(a_0^6/a_1)$.
Thus, the weak-coupling limit of the heterotic
string is given by $a_2\to\infty$ from \cite{AL:ubiq} in agreement
with the conjecture of \cite{KV:N=2}.
Keeping the K3 fibre big, by keeping $a_0^6/a_1$ big, we can blow down
$C$ by decreasing $a_1^2/a_2$. In the limit that $a_1^2/a_2$ becomes
zero we reach the conformal field theory orbifold. This is not what we
want however. In order to get the enhanced gauge symmetry we need the
component of $B$ along the blow-up to be zero as explained in
\cite{me:enhg}, whereas the conformal field theory orbifold gives a
value of $\ff12$. Fortunately for this blow-up, the $B$-field value for the
mirror map has been explicitly worked out in section 5.5 of
\cite{AGM:sd}. To obtain a zero-sized exceptional divisor and zero
$B$-field we require $a_1^2/a_2=4$. Thus we expect an $SU(2)$ enhanced
gauge group here again in agreement with \cite{KV:N=2}.
Determination of the $SU(2)\times SU(2)$ or $SU(3)$ points of enhanced
gauged symmetry is harder to analyze in this direct manner and we will
not pursue it here. It is clear that we should again reproduce the
results of \cite{KV:N=2} however.
As a second example we turn to one of the conjectured dual pairs of
\cite{AFIQ:chains} which has also been analyzed in
\cite{HM:alg}. Consider the Calabi-Yau\ hypersurface of degree 84 in
${\Bbb P}^4_{\{1,1,12,28,42\}}$. This is a K3 fibration where the generic
fibre is a hypersurface of degree 42 in ${\Bbb P}^3_{\{1,6,14,21\}}$. This K3
surface has Picard number 10 and the Picard lattice has intersection
form $E_8\oplus H$. This is most easily seen following the methods
explained in \cite{Mart:CCC}. Thus we have
\begin{equation}
\Lambda_K \cong E_8\oplus H\oplus H.
\end{equation}
It is easy then to see that there will be dual heterotic strings with
an $E_8$ factor in the gauge group in agreement with
\cite{AFIQ:chains}. We can also say in more detail how to obtain this
gauge group. Let us label the simple roots of $E_8$ as follows:
\begin{equation}
\hbox{\unitlength=1mm\begin{picture}(42,14)(0,0)
\multiput(0,3)(7,0){7}{\circle{1.8}}
\multiput(0.9,3)(7,0){6}{\line(1,0){5.2}}
\put(14,10){\circle{1.8}}
\put(14,3.9){\line(0,1){5.2}}
\put(-1.0,-1.0){\makebox(3.0,1.8)[b]{$e_2$}}
\put(6.0,-1.0){\makebox(3.0,1.8)[b]{$e_3$}}
\put(13.0,-1.0){\makebox(3.0,1.8)[b]{$e_4$}}
\put(20.0,-1.0){\makebox(3.0,1.8)[b]{$e_5$}}
\put(27.0,-1.0){\makebox(3.0,1.8)[b]{$e_6$}}
\put(34.0,-1.0){\makebox(3.0,1.8)[b]{$e_7$}}
\put(41.0,-1.0){\makebox(3.0,1.8)[b]{$e_8$}}
\put(15.5,8.8){\makebox(3.0,1.8){$e_1$}}
\end{picture}} \label{eq:E8}
\end{equation}
Each of these roots is associated with a rational curve in the K3
fibre. All the roots, with the exception of $e_4$, come from
blowing-up the quotient singularities of the ambient
${\Bbb P}^4_{\{1,1,12,28,42\}}$. Thus we can blow these down to get a gauge
group $SU(2)\times SU(3)\times SU(5)$ while keeping the fibre at the
large radius limit. The curve corresponding to $e_4$ comes from the
hyperplane section from the ambient projective space however. To blow
this down we have to shrink down the whole K3 fibre to take us into
the realms of quantum geometry. Thus in order to obtain the full $E_8$
symmetry the fibre must be shrunk down to volume of order
$(\alpha^\prime)^2$.
Actually one can enhance the gauge group beyond
this using the $H$ factors in $\Lambda_K$. For example
$E_8\times SU(3)$ or $SU(10)$ should appear in the moduli space.
One can see that we can reproduce all of
the gauge groups for type IIA strings compactified on hypersurfaces
listed in \cite{AFIQ:chains}. Note also that any enhanced gauge groups
appearing in the dual pairs studied in \cite{FHSV:N=2,me:flower} can also
be recovered by the same methods.
\section{Comments} \label{s:conc}
We have seen how enhanced gauge groups appearing on a type IIA string
compactified on a Calabi-Yau\ manifold that can be seen perturbatively in the
dual heterotic string can be understood purely in terms of
string-string duality in six-dimensions. We used this fact to
reproduce all the currently known results about gauge groups from
conjectured dual pairs.
We should emphasize that we have not completed the problem of
understanding the appearance of nonabelian groups for type II strings
on Calabi-Yau\ manifolds however. Firstly there can be parts of the gauge
group which cannot be understood perturbatively from either the type
II or the heterotic string point of view. Such groups have been
analyzed recently in six dimensions by using nonperturbative methods
for a heterotic string compactified on a K3 surface
\cite{W:small-i}. Similar effects must be expected for the case
considered in this letter.
It is tempting to conjecture how they will appear. We know from
\cite{AL:ubiq} that the vector multiplets that cannot be seen
perturbatively in the heterotic string arise from contributions to the
Picard group of $X$ from the degenerate fibres. We also have seen
above that curves of quotient singularities lead to nonabelian groups
in the type IIA string when this curve can be fibred over the base
${\Bbb P}^1$. If we suppose that the appearance of nonabelian groups is a
result purely of singular curves and not whether they fibre properly
over the base ${\Bbb P}^1$ then we can consider the case that we pick up a
curve of singularities within the degenerate fibre. Thus, if we
have a type IIA string compactified on $X$ and $X$ is a K3 fibration
with curves of quotient singularities within the degenerate fibres,
then the dual heterotic string will have nonperturbatively enhanced
gauge groups as in \cite{W:small-i}. We should add however that one
may have to worry about what one means by the ``weak-coupling limit''
in which one actually sees these nonabelina gauge groups.
This should be investigated further.
Another aspect which we ignored above concerns hypermultiplets. We
have explored questions involving vector multiplets
only. Hypermultiplets can become massless when the gauge group gets
enhanced. This is essential for analysis of phase transitions as
discussed in \cite{GMS:con,FHSV:N=2,me:flower}. On a related point,
gauge groups can also become enhanced as we move about in the moduli
space of hypermultiplets.
The $D$-brane picture should be of help here. We may also need to
worry about discrete R-R degrees of freedom since precisely these
issues concerning hypermultiplets were sensitive to such effects in
the example of \cite{FHSV:N=2}.
Clearly we must address these problems before
the subject of enhanced gauge symmetries on Calabi-Yau\ manifolds is
completely understood.
\section*{Acknowledgements}
It is a pleasure to thank D. Morrison
for useful conversations.
The work of the author is supported by a grant from the National
Science Foundation.
| 10,243 |
\part{This is a Part}
\chapterauthor{Jeremy Kepner$^{1}$, Kenjiro Cho$^{2}$, KC Claffy$^{3}$, Vijay Gadepally$^{1}$, Sarah McGuire$^{1}$, Lauren Milechin$^{4}$, William Arcand$^{1}$, David Bestor$^{1}$, William Bergeron$^{1}$, Chansup Byun$^{1}$, Matthew Hubbell$^{1}$, Michael Houle$^{1}$, Michael Jones$^{1}$, Andrew Prout$^{1}$, Albert Reuther$^{1}$, Antonio Rosa$^{1}$, Siddharth Samsi$^{1}$, Charles Yee$^{1}$, Peter Michaleas$^{1}$}
{$^{1}$MIT Lincoln Laboratory, $^{2}$Research Laboratory, Internet Initiative Japan, Inc., $^{3}$UCSD Center for Applied Internet Data Analysis, $^{4}$MIT Dept. of Earth, Atmospheric and Planetary Sciences}
\chapter[New Phenomena in Large-Scale Internet Traffic]{New Phenomena in Large-Scale Internet Traffic}
\blfootnote{This material is based, in part, upon the work supported by the NSF under grants DMS-1312831, CCF-1533644, and CNS-1513283, DHS cooperative agreement FA8750-18-2-0049, and ASD(R\&E) under contract FA8702-15-D-0001. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF, DHS, or ASD(R\&E).}
\chapterinitial{The}{Internet} is transforming our society, necessitating a quantitative understanding of Internet traffic. Our team collects and curates the largest publicly available Internet traffic data sets. An analysis of 50 billion packets using 10,000 processors in the MIT SuperCloud reveals a new phenomenon: the importance of otherwise unseen leaf nodes and isolated links in Internet traffic. Our analysis further shows that a two-parameter modified Zipf--Mandelbrot distribution accurately describes a wide variety of source/destination statistics on moving sample windows ranging from 100{,}000 to 100{,}000{,}000 packets over collections that span years and continents. The measured model parameters distinguish different network streams, and the model leaf parameter strongly correlates with the fraction of the traffic in different underlying network topologies.
\section{Introduction}\label{intro}
Our civilization is now dependent on the Internet, necessitating a scientific understanding of this virtual universe \cite{hilbert2011world,li2013survey}. The two largest efforts to capture, curate, and share Internet packet traffic data for scientific analysis are led by our team via the Widely Integrated Distributed Environment (WIDE) project \cite{cho2000tr} and the Center for Applied Internet Data Analysis (CAIDA) \cite{claffy1999internet}. These data have been used for a wide variety of research projects, resulting in hundreds of peer-reviewed publications \cite{CAIDApubs}, ranging from characterizing the global state of Internet traffic, to specific studies on the prevalence of peer-to-peer file sharing traffic, to testing prototype software designed to stop the spread of Internet worms.
The stochastic network structure of Internet traffic is a core property of great interest to a wide range of Internet stakeholders \cite{li2013survey} and network scientists \cite{barabasi2016network}. Of particular interest is the probability distribution $p(d)$, where $d$ is the degree (or count) of several network quantities, such as source packets, packets over a unique source-destination pair (or link), and destination packets collected over specified time intervals. Among the earliest and most widely cited results of virtual Internet topology analysis has been the observation that $p(d) \propto 1/d^\alpha$ with a model exponent $1 < \alpha < 3$ for large values of $d$ \cite{barabasi1999emergence,albert1999internet,leskovec2005graphs} fit a range of network characteristics.
Many Internet models are based on the data obtained from crawling the network from a number of starting points \cite{olston2010web}. These webcrawls naturally sample the supernodes of the network \cite{cao2009identifying}, and their resulting $p(d)$ are accurately fit at large values of $d$ by single-parameter power-law models. However, as we will show, for our streaming samples of the Internet there are other topologies that contribute significant traffic. Characterizing a network by a single power-law exponent provides one view of Internet phenomena, but more accurate and complex models are required to understand the diverse topologies seen in streaming samples of the Internet. Improving model accuracy while also increasing model complexity requires overcoming a number of challenges, including acquisition of larger, rigorously collected data sets \cite{soule2004identify,zhang2005estimating}; the enormous computational cost of processing large network traffic graphs \cite{lumsdaine2007challenges,bader2013graph,tune2013internet}; careful filtering, binning, and normalization of the data; and fitting of nonlinear models to the data.
\section{Methodology}\label{method}
This work aims to improve model accuracy through several techniques. First, for over a decade we have scientifically collected and curated the largest publicly available Internet packet traffic data sets and this work analyzes the very largest collections in our corpora containing 49.6 billion packets (Table~\ref{tab:TrafficData}). Second, utilizing recent innovations in interactive supercomputing \cite{Kepner2009,reuther2018interactive}, matrix-based graph theory \cite{kolda2009tensor,kepner2011graph}, and big data mathematics (Figure~\ref{fig:AssociativeArrays}) \cite{kepner2018mathematics}, we have developed a scalable Internet traffic processing pipeline that runs efficiently on more than 10{,}000 processors in the MIT SuperCloud \cite{gadepally2018hyperscaling}. This pipeline allows us, for the first time, to process our largest traffic collections as network traffic graphs. Third, since not all packets have both source and destination Internet Protocol version 4 (IPv4) addresses, the data have been filtered so that for any chosen time window all data sets have the same number of valid IPv4 packets, denoted $N_V$ (Figure~\ref{fig:ValidPackets} and Eq.~\ref{eq:Valid}). All computed probability distributions also use the same binary logarithmic binning to allow for consistent statistical comparison across data sets (Eq.~\ref{eq:Cumulative})\cite{clauset2009power,barabasi2016network}. Fourth, to accurately model the data over the full range of $d$, we employ a modified Zipf--Mandelbrot distribution \cite{mandelbrot1953informational,montemurro2001beyond,saleh2006modeling}
\begin{equation}\label{eq:ZipfMandelbrot}
p(d;\alpha,\delta) \propto 1/(d + \delta)^\alpha
\end{equation}
The inclusion of a second model offset parameter $\delta$ allows the model to accurately fit small values of $d$, in particular $d=1$, which has the highest observed probability in these streaming data. The modified Zipf--Mandelbrot model is a special case of the more general saturation/cutoff models used to model a variety of network phenomena (Eq.~\ref{eq:satcut}) \cite{clauset2009power,barabasi2016network}. Finally, nonlinear fitting techniques are used to achieve quality fits over the entire range of $d$ (Eq.~\ref{eq:NonLinFit}).
Throughout this chapter, we defer to the terminology of network science. Network operators use many similar terms with significant differences in meaning. We use network topology to refer to the graph-theoretic virtual topology of sources and destinations observed communicating and not the underlying physical topology of the Internet (Table~\ref{tab:Terms})
\begin{table}[htp]
\caption{Network Terminology Used by Computer Network Operators and Network Scientists. Throughout this Work, the Network Science Meanings are Employed}
\begin{center}
\begin{tabular}{p{0.75in}p{2.15in}p{2.15in}}
{Term} & {Network Operations Meaning} & {Network Science Meaning} \\
\hline
Network &
The \emph{physical} links, wires, routers, switches, and endpoints used to transmit data. &
Any system that can be represented as a \emph{graph} of connections (links/edges) among entities (nodes/vertices). \\
\hline
Topology &
The \emph{layout} of the physical network. &
The specific \emph{geometries} of a graph and its sub-graphs. \\
\hline
Stream &
The \emph{flow of data} over a specific physical communication link. &
A time-ordered \emph{sequence of pairs} of entities (nodes/vertices) representing distinction in time connections (links/edges) between entities. \\
\hline
\end{tabular}
\end{center}
\label{tab:Terms}
\end{table}%
\subsection{MAWI and CAIDA Internet Traffic Collection}\label{mawicaida}
For the analysis in following sections, the data utilized are summarized in Table~\ref{tab:TrafficData} with data from Tokyo coming from the MAWI Internet Traffic Collection and the data from Chicago from the CAIDA Internet Traffic Collection.
The Tokyo data sets are publicly available packet traces provided by the WIDE project (aka the MAWI traces). The WIDE project is a research consortium in Japan established in 1988 \cite{cho2000tr}. The members of the project include network engineers, researchers, university students, and industrial partners. The focus of WIDE is on the empirical study of the large-scale Internet. WIDE operates an Internet testbed both for commercial traffic and for conducting research experiments. These data have enabled quantitative analysis of Internet traffic spanning years illustrating trends such as the emergence of residential usage, peer-to-peer networks, probe scanning, and botnets \cite{cho2006impact,borgnat2009seven,fontugne2017scaling}. The Tokyo data sets are publicly available packet traces provided by the WIDE project (aka the MAWI traces). The traces are collected from a 1~Gbps academic backbone connection in Japan. The 2015 and 2017 data sets are 48-hour-long traces captured during December 2--3, 2015, and April 12--13, 2017, in JST. The IP addresses appearing in the traces are anonymized using a prefix-preserving method \cite{fan2004prefix}.
The MAWI repository is an ongoing collection of Internet traffic traces, captured within the WIDE backbone network (AS2500) that connects Japanese universities and research institutes to the Internet. Each trace consists of captured packets observed from within WIDE and includes the packet headers of each packet along with the captured timestamp. Anonymized versions of the traces (with anonymized IP addresses and with transport layer payload removed) are made publicly available at http://mawi.wide.ad.jp/.
WIDE carries a variety of traffic including academic and commercial traffic. These data have enabled quantitative analysis of Internet traffic spanning years illustrating trends such as the emergence of residential usage, peer-to-peer networks, probe scanning, and botnets \cite{cho2006impact,cho2008observing,fontugne2017scaling}. WIDE is mostly dominated by HTTP traffic, but is influenced by global anomalies. For example, Code Red, Blaster, and Sasser are worms that disrupted Internet traffic \cite{allman2007brief}. Of these, Sasser (2005) impacted MAWI traffic the most, accounting for two-thirds of packets at its peak. Conversely, the ICMP traffic surge in 2003 and the SYN Flood in 2012 were more local in nature, each revealing attacks on targets within WIDE that lasted several months.
CAIDA collects several different data types at geographically and topologically diverse locations and makes these data available to the research community to the extent possible while preserving the privacy of individuals and organizations who donate data or network access \cite{claffy1999internet,claffy2000measuring}. CAIDA has (and had) monitoring locations in Internet service providers (ISPs) in the USA. CAIDA's passive traces data set contains traces collected from high-speed monitors on a commercial backbone link. The data collection started in April 2008 and is ongoing. These data are useful for research on the characteristics of Internet traffic, including application breakdown (based on TCP/IP ports), security events, geographic and topological distribution, flow volume, and duration. For an overview of all traces, see the trace statistics page \cite{CAIDAstats}.
Collectively, our consortium has enabled the scientific analysis of Internet traffic, resulting in hundreds of peer-reviewed publications with over 30,000 citations \cite{CAIDApubs}. These include early work on Internet threats such as the Code Red worm \cite{moore2002code} and Slammer worm \cite{moore2003inside} and how quarantining might mitigate threats \cite{moore2003internet}. Subsequent work explored various techniques, such as dispersion, for measuring Internet capacity and bandwidth \cite{dovrolis2001packet,dovrolis2004packet,prasad2003bandwidth}. The next major area of research provided significant results on the dispersal of the Internet via the emergence of peer-to-peer networks \cite{karagiannis2004transport,karagiannis2004p2p}, edge devices \cite{kohno2005remote}, and corresponding denial-of-service attacks \cite{moore2006inferring}, which drove the need for new ways to categorize traffic \cite{kim2008internet}. The incorporation of network science and statistical physics concepts into the analysis of the Internet produced new results on the hyperbolic geometry of complex networks \cite{krioukov2009curvature,krioukov2010hyperbolic} and sustaining the Internet with hyperbolic mapping \cite{boguna2009navigating,boguna2009navigability,boguna2010sustaining}. Likewise, a new understanding also emerged on the identification of influential spreaders in complex networks \cite{kitsak2010identification}, the relationship of popularity versus similarity in growing networks \cite{papadopoulos2012popularity}, and overall network cosmology \cite{krioukov2012network}. More recent work has developed new ideas for Internet classification \cite{dainotti2012issues} and future data centric architectures \cite{zhang2014named}.
\begin{table}[htp]
\caption{Network Traffic Packet Data Sets from MAWI (Tokyo data sets) and CAIDA (Chicago data sets) Collected at Different Times and Durations over Two Years}
\begin{center}
\begin{tabular}{lcccc}
{Location} & {Date} & {Duration} & {Bandwidth} & {Packets} \\
\hline
Tokyo & 2015 Dec 02 & 2 days & $10^9$~ bits/second & $17.0{\times}10^9$ \\
Tokyo & 2017 Apr 12 & 2 days & $10^9$~ bits/second & $16.8{\times}10^9$ \\
Chicago A & 2016 Jan 21 & 1 hour & $10^{10}$ bits/second & ~~$2.0{\times}10^9$ \\
Chicago A & 2016 Feb 18 & 1 hour & $10^{10}$ bits/second & ~~$2.0{\times}10^9$ \\
Chicago A & 2016 Mar 17 & 1 hour & $10^{10}$ bits/second & ~~$1.8{\times}10^9$ \\
Chicago A & 2016 Apr 06 & 1 hour & $10^{10}$ bits/second & ~~$1.8{\times}10^9$ \\
Chicago B & 2016 Jan 21 & 1 hour & $10^{10}$ bits/second & ~~$2.3{\times}10^9$ \\
Chicago B & 2016 Feb 18 & 1 hour & $10^{10}$ bits/second & ~~$1.7{\times}10^9$ \\
Chicago B & 2016 Mar 17 & 1 hour & $10^{10}$ bits/second & ~~$2.0{\times}10^9$ \\
Chicago B & 2016 Apr 06 & 1 hour & $10^{10}$ bits/second & ~~$2.1{\times}10^9$ \\
\hline
\end{tabular}
\end{center}
\label{tab:TrafficData}
\end{table}%
\subsection{Network Quantities from Matrices}\label{networkquantities}
In our analysis, the network traffic packet data are reduced to origin--destination traffic matrices. These matrices can be used to compute a wide range of network statistics useful in the analysis, monitoring, and control of the Internet. Such an analysis includes the temporal fluctuations of the supernodes \cite{soule2004identify} and inferring the presence of unobserved traffic \cite{zhang2005estimating,bharti2010inferring}.
To create the matrices, at a given time $t$, $N_V$ consecutive valid packets are aggregated from the traffic into a sparse matrix ${\bf A}_t$, where ${\bf A}_t(i,j)$ is the number of valid packets between the source $i$ and destination $j$ \cite{mucha2010community}. The sum of all the entries in ${\bf A}_t$ is equal to $N_V$
\begin{equation}\label{eq:Valid}
\sum_{i,j} {\bf A}_t(i,j) = N_V
\end{equation}
All the network quantities depicted in Figure~\ref{fig:NetworkDistribution}a can be readily computed from ${\bf A}_t$ as specified in Tables~\ref{tab:Aggregates} and ~\ref{tab:Filters}, including the number of unique sources and destinations, along with many other network statistics \cite{soule2004identify,zhang2005estimating,tune2013internet}.
\begin{table}
\caption{Aggregate Network Properties}
\begin{center}
\begin{tabular}{p{1.5in}p{1.5in}p{1.0in}}
\hline
{Aggregate Property} & {Summation Notation} & {Matrix Notation} \\
\hline
Valid packets $N_V$ & $\sum_i ~ \sum_j ~ {\bf A}_t(i,j)$ & $~{\bf 1}^{\sf T} {\bf A}_t {\bf 1}$ \\
Unique links & $\sum_i ~ \sum_j |{\bf A}_t(i,j)|_0$ & ${\bf 1}^{\sf T}|{\bf A}_t|_0 {\bf 1}$ \\
Unique sources & $\sum_i |\sum_j ~ {\bf A}_t(i,j)|_0$ & ${\bf 1}^{\sf T}|{\bf A}_t {\bf 1}|_0$ \\
Unique destinations & $\sum_j |\sum_i ~ {\bf A}_t(i,j)|_0$ & $|{\bf 1}^{\sf T} {\bf A}_t|_0 {\bf 1}$ \\
\hline
\end{tabular}{Formulas for computing aggregates from a sparse network image ${\bf A}_t$ at time $t$ in both summation and matrix notations. ${\bf 1}$ is a column vector of all 1's, $^{\sf T}$ is the transpose operation, and $|~|_0$ is the zero-norm that sets each nonzero value of its argument to 1 \cite{karvanen2003measuring}.}
\end{center}
\label{tab:Aggregates}
\end{table}%
\begin{table}[h]
\caption{Neural Network Image Convolution Filters}
\begin{center}
\begin{tabular}{p{2.0in}p{1.5in}p{1.0in}}
\hline
{Network Quantity} & {Summation Notation} & {Matrix Notation} \\
\hline
Source packets from $i$ & $\sum_j ~ {\bf A}_t(i,j)$ & ~~~$~{\bf A}_t ~~ {\bf 1}$ \\
Source fan-out from $i$ & $\sum_j |{\bf A}_t(i,j)|_0$ & ~~~$|{\bf A}_t|_0 {\bf 1}$ \\
Link packets from $i$ to $j$ & $~~~~~~{\bf A}_t(i,j)$ & ~~~$~{\bf A}_t$ \\
Destination fan-in to $j$ & $\sum_i |{\bf A}_t(i,j)|_0$ & ${\bf 1}^{\sf T}~{\bf A}_t$ \\
Destination packets to $j$ & $\sum_i ~ {\bf A}_t(i,j)$ & ${\bf 1}^{\sf T}|{\bf A}_t|_0$ \\
\hline
\end{tabular}{Different network quantities are extracted from a sparse traffic image ${\bf A}_t$ at time $t$ via convolution with different filters. Formulas for the filters are given in both summation and matrix notations. ${\bf 1}$ is a column vector of all 1's, $^{\sf T}$ is the transpose operation, and $|~|_0$ is the zero-norm that sets each nonzero value of its argument to 1 \cite{karvanen2003measuring}.}
\end{center}
\label{tab:Filters}
\end{table}%
Figure~\ref{fig:NetworkTopology}a depicts the major topological structures in the network traffic. Isolated links are sources and destinations that each have only one connection (Table~\ref{tab:Isolatedlinks}). The first, second, third, $\ldots$ supernodes are the source or destination with the first, second, third, $\ldots$ most links (Table~\ref{tab:Supernodes}). The core of a network can be defined in a variety of ways \cite{schaeffer2007graph,benson2016higher}. In this work, the network core conveys the concept of a collection of sources and destinations that are not isolated and are multiply connected. The core is defined as the collection of sources and destinations in which every source and destination has more than one connection. The core, as computed here, does not include the first five supernodes although only the first supernode is significant, and whether or not the other supernodes are included has minimal impact on the core in these data. The core leaves are sources and destinations that have only one connection to a core source or destination (Tables~\ref{tab:Core} and \ref{tab:Coreleaves}).
\begin{table}[h]
\caption{Properties of Isolated Links}
\begin{center}
\begin{tabular}{p{2.75in}p{1.25in}}
\hline
{Network Quantity} & {Matrix Notation} \\
\hline
Isolated links & ${\bf A}_t(i_1,j_1)$\\
Number of isolated link sources & ${\bf 1}^{\sf T}|{\bf A}_t(i_1,j_1) {\bf 1}|_0$\\
Number of packets traversing isolated links & ${\bf 1}^{\sf T}{\bf A}_t(i_1,j_1){\bf 1}$\\
Number of unique isolated links & ${\bf 1}^{\sf T}|{\bf A}_t(i_1,j_1)|_0 {\bf 1}$ \\
Number of isolated link destinations & $|{\bf 1}^{\sf T} {\bf A}_t(i_1,j_1)|_0 {\bf 1}$ \\
\hline
\end{tabular}{Different characteristics related to isolated links are extracted from a sparse traffic image ${\bf A}_t$ at time $t$. Formulas are in matrix notation. The set of sources that send to only one destination are $ i_1 = {\rm arg}({\bf d}_{\rm out} = 1)$, and the set of destinations that receive from only one destination are $j_1 = {\rm arg}({\bf d}_{\rm in} = 1)$.}
\end{center}
\label{tab:Isolatedlinks}
\end{table}%
\begin{table}[h]
\caption{Properties of Supernodes}
\begin{center}
\begin{tabular}{p{2.75in}p{2.25in}}
\hline
{\bf Network} & {\bf ~Matrix} \\
{\bf Quantity} & {\bf Notation} \\
\hline
Supernode source leaves & ${\bf A}_t(i_1,k_{\rm max})$\\
Supernode destination leaves & ${\bf A}_t(k_{\rm max},j_1)$\\
Number of supernode leaf sources & ${\bf 1}^{\sf T}|{\bf A}_t(i_1,k_{\rm max}) {\bf 1}|_0$\\
Number of packets traversing supernode leaves & $ {\bf 1}^{\sf T}{\bf A}_t(i_1,k_{\rm max}) + {\bf A}_t(k_{\rm max},j_1){\bf 1}$ \\
Number of unique supernode leaf links & $ {\bf 1}^{\sf T}|{\bf A}_t(i_1,k_{\rm max})|_0 + |{\bf A}_t(k_{\rm max},j_1)|_0 {\bf 1}$ \\
Number of supernode leaf destinations & $|{\bf 1}^{\sf T} {\bf A}_t(k_{\rm max},j_1)|_0 {\bf 1}$\\
\hline
\end{tabular}{Different characteristics related to supernodes are extracted from a sparse traffic image ${\bf A}_t$ at time $t$. Formulas are in matrix notation. The identity of the first supernode is given by $k_{\rm max} = {\rm argmax}({\bf d}_{\rm out} + {\bf d}_{\rm in})$. The leaves of a supernode are those sources and destinations whose only connection is to the supernode.}
\end{center}
\label{tab:Supernodes}
\end{table}%
\begin{table}[h]
\caption{Properties of Network Core}
\begin{center}
\begin{tabular}{p{2.75in}p{2.25in}}
\hline
{\bf Network} & {\bf ~Matrix} \\
{\bf Quantity} & {\bf Notation} \\
\hline
Core links & ${\bf A}_t(i_{\rm core},j_{\rm core})$\\
Number of core sources & ${\bf 1}^{\sf T}|{\bf A}_t(i_{\rm core},j_{\rm core}) {\bf 1}|_0$\\
Number of core packets & $ {\bf 1}^{\sf T}{\bf A}_t(i_{\rm core},j_{\rm core}){\bf 1}$\\
Number of unique core links & ${\bf 1}^{\sf T}|{\bf A}_t(i_{\rm core},j_{\rm core})|_0 {\bf 1}$\\
Number of core destinations & $|{\bf 1}^{\sf T} {\bf A}_t(i_{\rm core},j_{\rm core})|_0 {\bf 1}$\\
\hline
\end{tabular}{Different characteristics related to the core are extracted from a sparse traffic image ${\bf A}_t$ at time $t$. Formulas are in matrix notation. The set of sources that send to more than one destination, excluding the supernode(s), is $i_{\rm core} = {\rm arg}(1 < {\bf d}_{\rm out} < {\bf d}_{\rm out}(k_{\rm max}))$. The set of destinations that receive from more than one source, excluding the supernode(s), is $ j_{\rm core} = {\rm arg}(1 < {\bf d}_{\rm in} < {\bf d}_{\rm in}(k_{\rm max}))$.}
\end{center}
\label{tab:Core}
\end{table}%
\begin{table}[h]
\caption{Properties of Core Leaves}
\begin{center}
\begin{tabular}{p{2.75in}p{2.25in}}
\hline
{\bf Network} & {\bf ~Matrix} \\
{\bf Quantity} & {\bf Notation} \\
\hline
Core source leaves & ${\bf A}_t(i_1,k_{\rm core})$\\
Core destination leaves & ${\bf A}_t(k_{\rm core},j_1)$\\
Number of core leaf sources & ${\bf 1}^{\sf T}|{\bf A}_t(i_1,k_{\rm core}) {\bf 1}|_0$\\
Number of core leaf packets & ${\bf 1}^{\sf T}{\bf A}_t(i_1,k_{\rm core}) + {\bf A}_t(k_{\rm core},j_1){\bf 1}$\\
Number of unique core leaf links & ${\bf 1}^{\sf T}|{\bf A}_t(i_1,k_{\rm core})|_0 + |{\bf A}_t(k_{\rm core},j_1)|_0 {\bf 1}$\\
Number of core leaf destination & $ |{\bf 1}^{\sf T} {\bf A}_t(k_{\rm core},j_1)|_0 {\bf 1}$\\
\hline
\end{tabular}{Different characteristics related to the core leaves are extracted from a sparse traffic image ${\bf A}_t$ at time $t$. Formulas are in matrix notation. The core leaves are sources and destinations that have one connection to a core source or destination.}
\end{center}
\label{tab:Coreleaves}
\end{table}%
An essential step for increasing the accuracy of the statistical measures of Internet traffic is using windows with the same number of valid packets $N_V$. For this analysis, a valid packet is defined as TCP over IPv4, which includes more than 95\% of the data in the collection and eliminates a small amount of data that use other protocols or contain anomalies. Using packet windows with the same number of valid packets produces quantities that are consistent over a wide range from $N_V = 100{,}000$ to $N_V = 100{,}000{,}000$ (Figure~\ref{fig:ValidPackets}).
\begin{figure}
\includegraphics[width=\columnwidth]{chapters/chaptercy/figs/ValidPackets-Tokyo2015.pdf}
\includegraphics[width=\columnwidth]{chapters/chaptercy/figs/ValidPackets-Tokyo2017.pdf}
\caption{{\bf Valid packets.} Analyzing packet windows with the same numbers of valid packets produces consistent fractions of unique links, unique destinations, and unique sources over a wide range of packet sizes for the Tokyo 2015 (a) and Tokyo 2017 (b) data sets. The plots show these fractions for moving packet windows of $N_V$ = 100{,}000 packets (left) and $N_V$ = 100{,}000{,}000 packets (right). The packet windows correspond to time windows of approximately 1.5 seconds and 25 minutes.
}
\label{fig:ValidPackets}
\end{figure}
\subsection{Memory and Computation Requirements}
Processing 50 billion Internet packets with a variety of algorithms presents numerous computational challenges. Dividing the data set into combinable units of approximately 100{,}000 consecutive packets made the analysis amenable to processing on a massively parallel supercomputer. The detailed architecture of the parallel processing system and its corresponding performance are described in \cite{gadepally2018hyperscaling}. The resulting processing pipeline was able to efficiently use over 10{,}000 processors on the MIT SuperCloud and was essential to this first-ever complete analysis of these data.
A key element of our analysis is the use of novel sparse matrix mathematics in concert with the MIT SuperCloud. Construction and analysis of network traffic matrices of the entire Internet address space have been considered impractical for its massive size \cite{tune2013internet}. Internet Protocol version 4 (IPv4) has $2^{32}$ unique addresses, but at any given collection point, only a fraction of these addresses will be observed. Exploiting this property to save memory can be accomplished by extending traditional sparse matrices so that new rows and columns can be added dynamically.
The algebra of associative arrays \cite{kepner2018mathematics} and its corresponding implementation in the Dynamic Distributed Dimensional Data Model (D4M) software library (d4m.mit.edu) allows the row and columns of a sparse matrix to be any sortable value, in this case character string representations of the Internet addresses (Figure~\ref{fig:AssociativeArrays}). Associative arrays extend sparse matrices to have database table properties with dynamically insertable and removable rows and columns that adjust as new data are added or subtracted to the matrix. Using these properties, the memory requirements of forming network traffic matrices can be reduced at the cost of increasing the required computation necessary to resort the rows and columns.
A network matrix ${\bf A}_t$ with $N_V = 100{,}000{,}000$ represented as an associative array typically requires 2 gigabytes of memory. A complete analysis of the statistics and topologies of ${\bf A}_t$ typically takes 10 minutes on a single MIT SuperCloud Intel Knights Landing processor core. Using increments of $100{,}000$ packets means that this analysis is repeated over 500{,}000 times to process all 49.6 billion packets. Using 10{,}000 processors on the MIT SuperCloud shortens the runtime of one of these analyses to approximately 8 hours. The results presented within this chapter are products of a discovery process that required hundreds of such runs that would not have been possible without these computational resources. Fortunately, the utilization of these results by Internet stakeholders can be significantly accelerated by creating optimized embedded implementations that only compute the desired statistics and are not required to support a discovery process \cite{liu2010tcam,liu2016packet}.
\begin{figure}
\includegraphics[width=\columnwidth]{chapters/chaptercy/figs/AssociativeArrays.pdf}
\caption{{\bf Associative arrays.} (a) Tabular representation of raw network traffic and corresponding database query to find all records beginning with source 1.1.1.1. (b) Network graph highlighting nearest neighbors of source node 1.1.1.1. (c) Corresponding associative array representation of the network graph illustrating how the neighbors of source node 1.1.1.1 are computed with matrix vector multiplication.}
\label{fig:AssociativeArrays}
\end{figure}
\section{Internet Traffic Modeling}
Quantitative measurements of the Internet \cite{rabinovich2016measuring} have provided Internet stakeholders information on the Internet since its inception. Early work has explored the early growth of the Internet \cite{claffy1994tracking}, the distribution of packet arrival times \cite{paxson1995wide}, the power-law distribution of network outages \cite{paxson1996end}, the self-similar behavior of traffic \cite{leland1994self,willinger1997self,willinger2002scaling}, formation processes of power-law networks \cite{faloutsos1999power,medina2000origin,broder2000graph,willinger2009mathematics}, and the topologies of Internet service providers \cite{spring2002measuring}. Subsequent work has examined the technological properties of Internet topologies \cite{li2004first}, the diameter of the Internet \cite{leskovec2005graphs}, applying rank index-based Zipf--Mandelbrot modeling to peer-to-peer traffic \cite{saleh2006modeling}, and extending topology measurements to edge hosts \cite{heidemann2008census}. More recent work looks to continued measurement of power-law phenomena \cite{mahanti2013tale,kitsak2015long,lischke2016analyzing}, exploiting emerging topologies for optimizing network traffic \cite{dhamdhere2010internet,labovitz2011internet,chiu2015we}, using network data to locate disruptions \cite{fontugne2017pinpointing}, the impact of inter-domain congestion \cite{dhamdhere2018inferring}, and studying the completeness of passive sources to determine how well they can observe microscopic phenomena \cite{mirkovic2017you}.
The above sample of many years of Internet research has provided significant qualitative insights into Internet phenomenology. Single-parameter power-law fits have extensively been explored and shown to adequately fit higher-degree tails of the observations. However, more complex models are required to fit the entire range of observations. Figure~\ref{fig:PowerLawFits}a adapted from figure 8H \cite{clauset2009power} shows the number of bytes of data received in response to $2.3\times10^5$ HTTP (web) requests from computers at a large research laboratory and shows a strong agreement with a power law at large values, but diverges with the single-parameter model at small values. Figure~\ref{fig:PowerLawFits}b adapted from figure 9W \cite{clauset2009power} shows the distribution of $1.2\times10^5$ hits on web sites from AOL users and shows a strong agreement with a power law at small values, but diverges with the single-parameter model at large values. Figure~\ref{fig:PowerLawFits}c adapted from figure 9X \cite{clauset2009power} shows the distribution of $2.4\times10^8$ web hyperlinks and has a reasonable model agreement across the entire range, except for the smallest values. Figure~\ref{fig:PowerLawFits}d adapted from figure 4B \cite{mahanti2013tale} shows the distribution of visitors arriving at YouTube from referring web sites appears to be best represented by two very different power-law models with significant difference as the smallest values. Figure~\ref{fig:PowerLawFits}e adapted from figure 3A \cite{kitsak2015long} shows the distribution of the number of Border Gateway Protocol updates received by the 4 monitors in 1-minute intervals and shows a strong agreement with a power law at large values, but diverges with the single-parameter model at small values. Figure~\ref{fig:PowerLawFits}f adapted from figure 21 in \cite{lischke2016analyzing} shows the distribution of the Bitcoin network in 2011 and shows a strong agreement with a power law at small values, but diverges with the single-parameter model at large values.
The results shown in Figure~\ref{fig:PowerLawFits} represent some of the best and most carefully executed fits to Internet data and clearly show the difficulty of fitting the entire range with a single-parameter power law. It is also worth mentioning that in each case the cumulative distribution is used, which naturally provides a smoother curve (in contrast to the differential cumulative distribution used in our analysis), but provides less detail on the underlying phenomena. Furthermore, the data in Figure~\ref{fig:PowerLawFits} are typically isolated collections such that the error bars are not readily computable, which limits the ability to assess both the quality of the measurements and the model fits.
\begin{figure}
\includegraphics[width=\columnwidth]{chapters/chaptercy/figs/PowerLawFits.pdf}
\caption{{\bf Single-parameter power-law fits of Internet data.} Single-parameter fits of the cumulative distributions of Internet data have difficulty modeling the entire range. The estimated ratio between the model and the data at the model extremes is shown.
({a}) Figure 8H \cite{clauset2009power}. ({b}) Figure 9W \cite{clauset2009power}. ({c}) Figure 9X \cite{clauset2009power}. ({d}) Figure 4B \cite{mahanti2013tale}. ({e}) Figure 3A \cite{kitsak2015long}. ({f}) Figure 21 \cite{lischke2016analyzing}.}
\label{fig:PowerLawFits}
\end{figure}
Regrettably, the best publicly available data about the global interconnection system that carries most of the world's communications traffic are incomplete and of unknown accuracy. There is no map of physical link locations, capacity, traffic, or interconnection arrangements. This opacity of the Internet infrastructure hinders research and development efforts to model network behavior and topology; design protocols and new architectures; and study real-world properties such as robustness, resilience, and economic sustainability. There are good reasons for the dearth of information: complexity and scale of the infrastructure; information-hiding properties of the routing system; security and commercial sensitivities; costs of storing and processing the data; and lack of incentives to gather or share data in the first place, including cost-effective ways to use it operationally. But understanding the Internet's history and present, much less its future, is impossible without realistic and representative data sets and measurement infrastructure on which to support sustained longitudinal measurements as well as new experiments. The MAWI and CAIDA data collection efforts are the largest efforts to provide the data necessary to begin to answer these questions.
\subsection{Logarithmic Pooling}\label{prob}
In this analysis before model fitting, the differential cumulative probabilities are calculated.
For a network quantity $d$, the histogram of this quantity computed from ${\bf A}_t$ is denoted by $n_t(d)$, with corresponding probability
\begin{equation}\label{eq:Probability}
p_t(d) = n_t(d)/\sum_d n_t(d)
\end{equation}
and cumulative probability
\begin{equation}\label{eq:Cumulative}
P_t(d) = \sum_{i=1,d} p_t(d)
\end{equation}
Because of the relatively large values of $d$ observed due to a single supernode, the measured probability at large $d$ often exhibits large fluctuations. However, the cumulative probability lacks sufficient detail to see variations around specific values of $d$, so it is typical to use the differential cumulative probability with logarithmic bins in $d$
\begin{equation}\label{eq:LogBin}
D_t(d_i) = P_t(d_i) - P_t(d_{i-1})
\end{equation}
where $d_i = 2^i$ \cite{clauset2009power}. The corresponding mean and standard deviation of $D_t(d_i)$ over many different consecutive values of $t$ for a given data set are denoted $D(d_i)$ and $\sigma(d_i)$. These quantities strike a balance between accuracy and detail for subsequent model fitting as demonstrated in the daily structural variations revealed in the Tokyo data (Figures~\ref{fig:DailyVariation} and \ref{fig:DailyLimits}).
Diurnal variations in supernode network traffic are well known \cite{soule2004identify}. The Tokyo packet data were collected over a period spanning two days and allow the daily variations in packet traffic to be observed. The precision and accuracy of our measurements allow these variations to be observed across a wide range of nodes. Figure~\ref{fig:DailyVariation} shows the fraction of source fan-outs in each of various bin ranges. The fluctuations show the network evolving between two envelopes occurring between noon and midnight that are shown in Figure~\ref{fig:DailyLimits}.
\begin{figure}
\centering
\includegraphics[width=0.75\columnwidth]{chapters/chaptercy/figs/DailyVariation.pdf}
\caption{{\bf Daily variation in Internet traffic.} The fraction of source nodes with a given range of fan-out is shown as a function of time for the Tokyo 2015 data. The $p(d = 1)$ value is plotted on a separate linear scale because of the larger magnitude relative to the other points. Each point is the mean of many neighboring points in time, and the error bars are the measured $\pm$1-$\sigma$. The daily variations of the distributions oscillate between extremes corresponding to approximately local noon and midnight.
}
\label{fig:DailyVariation}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.75\columnwidth]{chapters/chaptercy/figs/DailyLimits.pdf}
\caption{{\bf Daily limits in Internet traffic.} The fraction of source nodes versus fan-out is shown for two noons and two midnights for the Tokyo 2015 data. The overlap among the noons and the midnights shows the relative day-to-day consistency in these data and shows the limits of the two extremes in daily variation. During the day, there is more traffic among nodes with intermediate fan-out. At night, the traffic is more dominated by leaf nodes and the supernode.
}
\label{fig:DailyLimits}
\end{figure}
\subsection{Modified Zipf--Mandelbrot Model}\label{zipf}
Measurements of $D(d_i)$ can reveal many properties of network traffic, such as the number of nodes with only one connection $D(d = 1)$ and the size of the supernode $d_{\rm max}={\rm argmax}(D(d) > 0)$. An effective low-parameter model allows these and many other properties to be summarized and computed efficiently. In the standard Zipf--Mandelbrot model typically used in linguistic contexts, the value $d$ in Eq.~\ref{eq:ZipfMandelbrot} is a ranking with $d=1$ corresponding to the most popular value \cite{mandelbrot1953informational,montemurro2001beyond,saleh2006modeling}. In our analysis, the Zipf--Mandelbrot model is modified so that $d$ is a measured network quantity instead of a rank index (Eq. \ref{eq:ZipfMandelbrot}). The model exponent $\alpha$ has a larger impact on the model at large values of $d$, while the model offset $\delta$ has a larger impact on the model at small values of $d$ and in particular at $d=1$.
The general saturation/cutoff models used to model a variety of network phenomena is denoted \cite{clauset2009power,barabasi2016network}
\begin{equation}\label{eq:satcut}
p(d) \propto \frac{1}{(d + \delta)^\alpha \exp[\lambda d]}
\end{equation}
where $\delta$ is the low-$d$ saturation and $1/\lambda$ is the high-$d$ cutoff that bounds the power-law regime of the distribution. The modified Zipf--Mandelbrot is a special case of this distribution that accurately models our observations. The unnormalized modified Zipf--Mandelbrot model is denoted
\begin{equation}\label{eq:rho}
\rho(d;\alpha,\delta) = \frac{1}{(d + \delta)^\alpha}
\end{equation}
with corresponding derivative with respect to $\delta$
\begin{equation}\label{eq:drho}
\partial_\delta \rho(d;\alpha,\delta) = \frac{-\alpha}{(d + \delta)^{\alpha+1}} = -\alpha \rho(d;\alpha+1,\delta)
\end{equation}
The normalized model probability is given by
\begin{equation}\label{eq:ZM}
p(d;\alpha,\delta) = \frac{\rho(d;\alpha,\delta)}{\sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta)}
\end{equation}
where $d_{\rm{max}}$ is the largest value of the network quantity $d$. The cumulative model probability is the sum
\begin{equation}\label{eq:ZMcum}
P(d_i;\alpha,\delta) = \sum_{d=1}^{d_i} p(d;\alpha,\delta)
\end{equation}
The corresponding differential cumulative model probability is
\begin{equation}\label{eq:ZMdiff}
D(d_i;\alpha,\delta) = P(d_i;\alpha,\delta) - P(d_{i-1};\alpha,\delta)
\end{equation}
where $d_i = 2^i$. In terms of $\rho$, the differential cumulative model probability is
\begin{equation}\label{eq:ZMdiffrho}
D(d_i;\alpha,\delta) = \frac{\sum_{d=d_{i-1}+1}^{d=d_i} \rho(d;\alpha,\delta)}{\sum_{d=1}^{d = d_{\rm max}}~~\rho(d;\alpha,\delta)}
\end{equation}
The above function is closely related to the Hurwitz zeta function \cite{NIST:DLMF,clauset2009power,yu2017link} \begin{equation}\label{eq:HZ}
\zeta(\alpha,\delta_1) = \sum_{d=0}^{\infty}~~\rho(d;\alpha,\delta_1)
\end{equation}
where $\delta_1 = \delta+1$. The differential cumulative model probability in terms of the Hurwitz zeta function is
\begin{equation}\label{eq:ZMdiffrho}
D(d_i;\alpha,\delta) = \frac{\zeta(\alpha,\delta+3+d_{i-1}) - \zeta(\alpha,\delta+2+d_i)}{\zeta(\alpha,\delta+) - \zeta(\alpha,\delta+)}
\end{equation}
\subsection{Nonlinear Model Fitting}\label{nonlinear}
The model exponent $\alpha$ has a larger impact on the model at large values of $d$, while the model offset $\delta$ has a larger impact on the model at small values of $d$ and in particular at $d=1$. A nonlinear fitting technique is used to obtain accurate model fits across the entire range of $d$. Initially, a set of candidate exponent values is selected, typically $\alpha = 0.10, 0.11,\ldots,3.99,4.00$. For each value of $\alpha$, a value of $\delta$ is computed that exactly matches the model with the data at $D(1)$. Finding the value of $\delta$ corresponding to a give $D(1)$ is done using Newton's method as follows. Setting the measured value of $D(1)$ equal to the model value $D(1;\alpha,\delta)$ gives
\begin{equation}\label{eq:ZM1}
D(1) = D(1;\alpha,\delta) = \frac{1}{(1 + \delta)^{\alpha} \sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta)}
\end{equation}
Newton's method works on functions of the form $f(\delta) = 0$. Rewriting the above expression produces
\begin{equation}\label{eq:ZMnewton}
f(\delta) = D(1) (1 + \delta)^\alpha \sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta) - 1 = 0
\end{equation}
For given value of $\alpha$, $\delta$ can be computed from the following iterative equation
\begin{equation}\label{eq:NewtonIteration}
\delta \rightarrow \delta - \frac{f(\delta)}{\partial_\delta f(\delta)}
\end{equation}
where the partial derivative $\partial_\delta f(\delta)$ is
\begin{eqnarray}\label{eq:NewtonDerivative}
\partial_\delta f(\delta)
& = & D(1) ~ \partial_\delta [(1 + \delta)^\alpha ~~ \sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta)] \nonumber \\
& = & D(1) [[\alpha (1 + \delta)^{\alpha-1} \sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta)] +
[(1 + \delta)^\alpha \sum_{d=1}^{d_{\rm max}} ~~ \partial_\delta \rho(d;\alpha,\delta)]]
\nonumber \\
& = & D(1) [[\alpha (1 + \delta)^{\alpha-1} \sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta)] +
[(1 + \delta)^\alpha \sum_{d=1}^{d_{\rm max}} -\alpha \rho(d;\alpha+1,\delta)]] \nonumber \\
& = & \alpha D(1) (1 + \delta)^\alpha [(1 + \delta)^{-1} \sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta) -
\sum_{d=1}^{d_{\rm max}} \rho(d;\alpha+1,\delta)]
\end{eqnarray}
Using a starting value of $\delta=1$ and bounds of $0 < \delta < 10$, Newton's method can be iterated until the differences in successive values of $\delta$ fall below a specified error (typically 0.001), which is usually achieved in less than five iterations.
If faster evaluation is required, the sums in the above formulas can be accelerated using the integral approximations
\begin{eqnarray}\label{eq:SumApprox}
\sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta)
&\approx& \sum_{d=1}^{d_{\rm sum}} \rho(d;\alpha,\delta) + \int_{d_{\rm sum} + 0.5}^{d_{\rm max} + 0.5} \rho(x;\alpha,\delta) dx \nonumber \\
&=& \sum_{d=1}^{d_{\rm sum}} \rho(d;\alpha,\delta) + \frac{\rho(d_{\rm sum} + 0.5;\alpha-1,\delta) - \rho(d_{\rm max} + 0.5;\alpha-1,\delta) }{\alpha-1} \nonumber \\
\sum_{d=1}^{d_{\rm max}} \rho(d;\alpha+1,\delta)
&\approx& \sum_{d=1}^{d_{\rm sum}} \rho(d;\alpha+1,\delta) + \int_{d_{\rm sum} + 0.5}^{d_{\rm max} + 0.5} \rho(x;\alpha+1,\delta) dx \nonumber \\
&=& \sum_{d=1}^{d_{\rm sum}} \rho(d;\alpha+1,\delta) + \frac{\rho(d_{\rm sum} + 0.5;\alpha,\delta) - \rho(d_{\rm max} + 0.5;\alpha,\delta) }{\alpha}
\end{eqnarray}
where the parameter $d_{\rm sum}$ can be adjusted to exchange speed for accuracy. For typical values of $\alpha$, $\delta$, and $d_{\rm max}$ used in this work, the accuracy is approximately $1/d_{\rm sum}$.
The best-fit $\alpha$ (and corresponding $\delta$) is chosen by minimizing the $|~|^{1/2}$ metric over logarithmic differences between the candidate models $D(d_i;\alpha,\delta)$ and the data
\begin{equation}\label{eq:NonLinFit}
{\rm argmin}_{\alpha} \sum_{d_i}|\log(D(d_i)) - \log(D(d_i;\alpha,\delta))|^{1/2}
\end{equation}
The $|~|^{1/2}$ metric (or $|~|_p$-norm with $p = 1/2$) favors maximizing error sparsity over minimizing outliers \cite{donoho2006compressed,chartrand2007exact,xu2012,karvanen2003measuring,saito2000sparsity,Brbic2018,Rahimi2018scale}. Several authors have recently shown that it is possible to reconstruct a nearly sparse signal from fewer linear measurements than would be expected from traditional sampling theory. Furthermore, by replacing the $|~|_1$ norm with the $|~|^p$ with $p < 1$, reconstruction is possible with substantially fewer measurements.
Using logarithmic values more evenly weights their contribution to the model fit and more accurately reflects the number of packets used to compute each value of $D(d_i)$. Lower-accuracy data points are avoided by limiting the fitting procedure to data points where the value is greater than the standard deviation: $D(d_i) > \sigma(d_i)$.
\section{Results}\label{Results}
Figure~\ref{fig:NetworkDistribution}b shows five representative model fits out of the 350 performed on 10 data sets, 5 network quantities, and 7 valid packet windows: $N_V = 10^5$, $3{\times}10^5$, $10^6$, $3{\times}10^6$, $10^7$, $3{\times}10^7$, $10^8$. The model fits are valid over the entire range of $d$ and provide parameter estimates with precisions of 0.01. In every case, the high value of $p(d=1)$ is indicative of a large contribution from a combination of supernode leaves, core leaves, and isolated links (Figure~\ref{fig:NetworkTopology}a). The breadth and accuracy of these data allow a detailed comparison of the model parameters. Figure~\ref{fig:NetworkDistribution}c shows the model offset $\delta$ versus the model exponent $\alpha$ for all 350 fits. The different collection locations are clearly distinguishable in this model parameter space. The Tokyo collections have smaller offsets and are more tightly clustered than the Chicago collections. Chicago B has a consistently smaller source and link packet model offset than Chicago A. All the collections have source, link, and destination packet model exponents in the relatively narrow $1.5 < \alpha < 2$ range. The source fan-out and destination fan-in model exponents are in the broader $1.5 < \alpha < 2.5$ range and are consistent with the prior literature \cite{clauset2009power}. These results represent an entirely new approach to characterizing Internet traffic that allows the distributions to be projected into a low-dimensional space and enables accurate comparisons among packet collections with different locations, dates, durations, and sizes. Figure~\ref{fig:NetworkDistribution}c indicates that the distributions of the different collection points occupy different parts of the modified Zipf--Mandelbrot model parameter space. Figures~\ref{fig:ModelFitsA}--\ref{fig:ModelFitsJ} show the measured and modeled differential cumulative distributions for the source fan-out, source packets, destination fan-in, destination packets, and link packets for all the collected data.
\begin{figure}
\vspace*{-0.5cm}
\hspace*{-1cm}
\includegraphics[height = 19cm, width=1.1\columnwidth]{chapters/chaptercy/figs/NetworkDistribution.pdf}
\vspace*{-1cm}
\caption{{\bf Streaming network traffic quantities, distributions, and model fits.} ({a}) Internet traffic streams of $N_V$ valid packets are divided into a variety of quantities for analysis. ({b}) A selection of 5 of the 350 measured differential cumulative probabilities spanning different locations, dates, and packet windows. Blue circles are measured data with $\pm$1-$\sigma$ error bars. Black lines are the best-fit modified Zipf--Mandelbrot models with parameters $\alpha$ and $\delta$. Red dots highlight the large contribution of leaf nodes and isolated links. ({c}) Model fit parameters for all 350 measured probability distributions.}
\label{fig:NetworkDistribution}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitA.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Tokyo 2015 Dec 02.
}
\label{fig:ModelFitsA}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitB.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Tokyo 2017 Apr 12.
}
\label{fig:ModelFitsB}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitC.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago A 2016 Jan 21.
}
\label{fig:ModelFitsC}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitD.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago A 2016 Feb 18.
}
\label{fig:ModelFitsD}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitE.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago A 2016 Mar 17.
}
\label{fig:ModelFitsE}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitF.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago A 2016 Apr 06.
}
\label{fig:ModelFitsF}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitG.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago B 2016 Jan 21.
}
\label{fig:ModelFitsG}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitH.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago B 2016 Feb 18.
}
\label{fig:ModelFitsH}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitI.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago B 2016 Mar 17.
}
\label{fig:ModelFitsI}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitJ.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago B 2016 Apr 06.
}
\label{fig:ModelFitsJ}
\end{figure}
Figure~\ref{fig:NetworkTopology}b shows the average relative fractions of sources, total packets, total links, and the number of destinations in each of the five topologies for the ten data sets, and seven valid packet windows: $N_V = 10^5$, $3{\times}10^5$, $10^6$, $3{\times}10^6$, $10^7$, $3{\times}10^7$, $10^8$. The four projections in Figure~\ref{fig:NetworkTopology}b are chosen from Figures~\ref{fig:NetTopoA}--\ref{fig:NetTopD} to highlight the differences in the collection locations. The distinct regions in the various projections shown in Figure~\ref{fig:NetworkTopology}b indicate that underlying topological differences are present in the data. The Tokyo collections have much larger supernode leaf components than the Chicago collections. The Chicago collections have much larger core and core leaves components than the Tokyo collections. Chicago A consistently has fewer isolated links than Chicago B. Comparing the modified Zipf--Mandelbrot model parameters in Figure~\ref{fig:NetworkDistribution}c and underlying topologies in Figure~\ref{fig:NetworkTopology}b suggests that the model parameters are a more compact way to distinguish the network traffic.
\begin{figure}
\vspace*{-2cm}
\hspace*{-2cm}
\includegraphics[width=1.2\columnwidth]{chapters/chaptercy/figs/NetworkTopology.pdf}
\caption{{\bf Distribution of traffic among network topologies.} ({a}) Internet traffic forms networks consisting of a variety of topologies: isolated links, supernode leaves connected to a supernode, and densely connected core(s) with corresponding core leaves. ({b}) A selection of four projections showing the fraction of data in various underlying topologies. Horizontal and vertical axes are the corresponding fraction of the sources, links, total packets, and destinations that are in various topologies for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$). These data reveal the differences in the network traffic topologies in the data collected in Tokyo (dominated by supernode leaves), Chicago A (dominated by core leaves), and Chicago B (between Tokyo and Chicago A).}
\label{fig:NetworkTopology}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.2cm}
\includegraphics[clip, trim=0.9cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/NetTopoA.pdf}
\caption{{\bf Fraction of data in different network topologies.} Fraction of data in isolated links, core leaves, and supernode leaves versus the fraction of data in the core for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:NetTopoA}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.2cm}
\includegraphics[clip, trim=0.9cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/NetTopoB.pdf}
\caption{{\bf Fraction of data in different network topologies.} Fraction of data in the core, core leaves, and supernode leaves versus the fraction of data in isolated links for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:NetTopoB}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.2cm}
\includegraphics[clip, trim=0.9cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/NetTopoC.pdf}
\caption{{\bf Fraction of data in different network topologies.} Fraction of data in the core, isolated links, and supernode leaves versus the fraction of data in core leaves for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:NetTopC}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.2cm}
\includegraphics[clip, trim=0.9cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/NetTopoD.pdf}
\caption{{\bf Fraction of data in different network topologies.} Fraction of data in the core, isolated links, and core leaves versus the fraction of data in supernode leaves for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:NetTopD}
\end{figure}
Figures~\ref{fig:NetworkDistribution}c and \ref{fig:NetworkTopology}b indicate that different collection points produce different model parameters $\alpha$ and $\delta$ and that these collection points also have different underlying topologies. Figure~\ref{fig:TopoModel} connects the model fits and topology observations by plotting the topology fraction as a function of the model leaf parameter $1/(1+\delta)^\alpha$ which corresponds to the relative strength of the distribution at $p(d=1)$
\begin{equation}\label{eq:ZipfMandelbrot}
1/(1 + \delta)^\alpha \propto p(d=1;\alpha,\delta)
\end{equation}
The correlations revealed in Figure~\ref{fig:TopoModel} suggest that the model leaf parameter strongly correlates with the fraction of the traffic in different underlying network topologies and is a potentially new and beneficial way to characterize networks. Figure~\ref{fig:TopoModel} indicates that the fraction of sources, links, and destinations in the core shrinks as the relative importance of the leaf parameter in the source fan-out and destination fan-in increases. In other words, more source and destination leaves mean a smaller core. Likewise, the fraction of links and total packets in the supernode leaves grows as the leaf parameter in the link packets and source packets increases. Interestingly, the fraction of sources in the core leaves and isolated links decreases as the leaf parameter in the source and destination packets increases indicating a shift of sources away from the core leaves and isolated links into supernode leaves. Thus, the modified Zipf--Mandelbrot model and its leaf parameter provide a direct connection with the network topology, underscoring the value of having accurate model fits across the entire range of values and in particular for $d=1$.
\begin{figure}
\vspace*{-1cm}
\hspace*{-2cm}
\includegraphics[width=1.3\columnwidth]{chapters/chaptercy/figs/TopoModel.pdf}
\caption{{\bf Topology versus model leaf parameter.} Network topology is highly correlated with the modified Zipf--Mandelbrot model leaf parameter $1/(1+\delta)^\alpha$. A selection of eight projections showing the fraction of data in various underlying topologies. Vertical axis is the corresponding fraction of the sources, links, total packets, and destinations that are in various topologies. Horizontal axis is the value of the model parameter taken from either the source packet, source fan-out, link packet, destination fan-in, or destination packet fits. Data points are for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).}
\label{fig:TopoModel}
\end{figure}
Figures~\ref{fig:TopoModelA}--\ref{fig:TopoModelE} show the fraction of the sources, links, total packets, and destinations in each of the measured topologies for all the locations as a function of the modified Zipf--Mandelbrot leaf parameter computed from the model fits of the source packets, source fan-out, link packets, destination fan-in, and destination packets taken from Figures~\ref{fig:ModelFitsA}--\ref{fig:ModelFitsJ}.
\begin{figure}[bh]
\vspace*{-0.5cm}
\includegraphics[clip, trim=1.4cm 0cm 0cm 0cm, angle=-90,origin=c,height=0.75\textheight,width=\columnwidth, ]{chapters/chaptercy/figs/TopoModelA.pdf}
\caption{{\bf Topology versus modified Zipf--Mandelbrot model leaf parameter.} Fraction of data in the core, isolated links, core leaves, and supernode leaves versus the model leaf parameter computed from the source packet modified Zipf--Mandelbrot model fits for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:TopoModelA}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\includegraphics[clip, trim=1.4cm 0cm 0cm 0cm, angle=-90,origin=c,height=0.75\textheight,width=\columnwidth, ]{chapters/chaptercy/figs/TopoModelB.pdf}
\caption{{\bf Topology versus modified Zipf--Mandelbrot model leaf parameter.} Fraction of data in the core, isolated links, core leaves, and supernode leaves versus the model leaf parameter computed from the source fan-out modified Zipf--Mandelbrot model fits for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:TopoModelB}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\includegraphics[clip, trim=1.4cm 0cm 0cm 0cm, angle=-90,origin=c,height=0.75\textheight,width=\columnwidth, ]{chapters/chaptercy/figs/TopoModelC.pdf}
\caption{{\bf Topology versus modified Zipf--Mandelbrot model leaf parameter.} Fraction of data in the core, isolated links, core leaves, and supernode leaves versus the model leaf parameter computed from the link packets modified Zipf--Mandelbrot model fits for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:TopoModelC}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\includegraphics[clip, trim=1.4cm 0cm 0cm 0cm, angle=-90,origin=c,height=0.75\textheight,width=\columnwidth, ]{chapters/chaptercy/figs/TopoModelD.pdf}
\caption{{\bf Topology versus modified Zipf--Mandelbrot model leaf parameter.} Fraction of data in the core, isolated links, core leaves, and supernode leaves versus the model leaf parameter computed from the destination fan-out modified Zipf--Mandelbrot model fits for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:TopoModelD}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\includegraphics[clip, trim=1.4cm 0cm 0cm 0cm, angle=-90,origin=c,height=0.75\textheight,width=\columnwidth, ]{chapters/chaptercy/figs/TopoModelE.pdf}
\caption{{\bf Topology versus modified Zipf--Mandelbrot model leaf parameter.} Fraction of data in the core, isolated links, core leaves, and supernode leaves versus the model leaf parameter computed from the destination packets modified Zipf--Mandelbrot model fits for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:TopoModelE}
\end{figure}
\section{Discussion}\label{Discussion}
Measurements of Internet traffic are useful for informing policy, identifying and preventing outages, defeating attacks, planning for future loads, and protecting the Domain Name System \cite{clark20179th}.
On a given day, millions of IPs are engaged in scanning behavior. Our improved models can aid cybersecurity analysts in determining which of these IPs are nefarious \cite{yu2012predicted}, the distribution of attacks in particular critical sectors \cite{husak2018assessing}, identifying spamming behavior \cite{fonseca2016measuring}, how to vaccinate against computer viruses \cite{balthrop2004technological}, obscuring web sources \cite{javed2015measurement}, identifying significant flow aggregates in traffic \cite{cho2017recursive}, and sources of rumors \cite{paluch2018fast}.
The results presented here have a number of potential practical applications for Internet stakeholders. The methods presented of collecting, filtering, computing, and binning the data to produce accurate measurements of a variety of network quantities are generally applicable to Internet measurements and have the potential to produce more accurate measures of these quantities. The accurate fits of the two-parameter modified Zipf--Mandelbrot distribution offer all the usual benefits of low-parameter models: measuring parameters with far less data, accurate predictions of network quantities based on a few parameters, observing changes in the underlying distribution, and using modeled distributions to detect anomalies in the data.
From a scientific perspective, improved knowledge of how Internet traffic flows can inform our understanding of how economics, topology, and demand shape the Internet over time. As with all scientific disciplines, the ability of theoreticians to develop and test theories of the Internet and network phenomena is bounded by the scale and accuracy of measured phenomena \cite{adamic2000power,bohman2009emergence,stumpf2012critical,virkar2014power}. The connections among dynamic evolution \cite{bianconi2001competition}, network topology \cite{mucha2010community}\cite{boccaletti2014structure,lu2016vital}, network robustness \cite{li2017fundamental}, controllability \cite{liu2016control}, community formation \cite{perc2017statistical}, and spreading phenomena \cite{holme2015modern} have emerged in many contexts \cite{barabasi2009scale,wang2016statistical,koliba2018governance}. Many first-principles theories for Internet and network phenomena have been proposed, such as Poisson models \cite{paxson1995wide}, fractional Brownian motion \cite{willinger1997self}, preferential attachment \cite{barabasi1999emergence,albert1999internet}\cite{newman2001clustering,sheridan2018preferential}, statistical mechanics \cite{albert2002statistical}, percolation \cite{achlioptas2009explosive}, hyperbolic geometries \cite{krioukov2009curvature,krioukov2010hyperbolic}, non-global greedy routing \cite{boguna2009navigating,boguna2009navigability,boguna2010sustaining}, interacting particle systems \cite{antonopoulos2018opinion}, higher-order organization of complex networks from graph motifs \cite{benson2016higher}, and minimum control energy \cite{lindmark2018minimum}. All of these models require data to test them. In contrast to previous network models that have principally been based on data obtained from network crawls from a variety of start points on the network, our network traffic data are collected from observations of network streams. Both viewpoints provide important network observations. Observations of a network stream provide complementary data on network dynamics and highlight the contribution of leaves and isolated edges, which are less sampled in network crawls.
The aggregated data sets our teams have collected provide a unique window into these questions. The nonlinear fitting techniques described are a novel approach to fitting power-law data and have potential applications to power-law networks in diverse domains. The model fit parameters present new opportunities to connect the distributions to underlying theoretical models of networks. That the model fit parameters distinguish the different collection points and are reflective of different network topologies in the data at these points suggests a deeper underlying connection between the models and the network topologies.
\section{Conclusions}\label{Conclusions}
Our society critically depends on the Internet for our professional, personal, and political lives. This dependence has rapidly grown much stronger than our comprehension of its underlying structure, performance limits, dynamics, and evolution. The fundamental characteristics of the Internet are perpetually challenging to research and analyze, and we must admit we know little about what keeps the system stable. As a result, researchers and policymakers deal with a multi-trillion-dollar ecosystem essentially in the dark, and agencies charged with infrastructure protection have little situational awareness regarding global dynamics and operational threats. This paper has presented an analysis of the largest publicly available collection of Internet traffic consisting of 50 billion packets and reveals a new phenomenon: the importance of otherwise unseen leaf nodes and isolated links in Internet traffic. Our analysis further shows that a two-parameter modified Zipf--Mandelbrot distribution accurately describes a wide variety of source/destination statistics on moving sample windows ranging from 100{,}000 to 100{,}000{,}000 packets over collections that span years and continents. The measured model parameters distinguish different network streams, and the model leaf parameter strongly correlates with the fraction of the traffic in different underlying network topologies. These results represent a significant improvement in Internet modeling accuracy, improve our understanding of the Internet, and show the importance of stream sampling for measuring network phenomena.
\let\cleardoublepage\clearpage
\section*{Acknowledgements}
The authors wish to acknowledge the following individuals for their contributions and support: Shohei Araki, William Arcand, David Bestor, William Bergeron, Bob Bond, Paul Burkhardt, Chansup Byun, Cary Conrad, Alan Edelman, Sterling Foster, Bo Hu, Matthew Hubbell, Micheal Houle, Micheal Jones, Anne Klein, Charles Leiserson, Dave Martinez, Mimi McClure, Julie Mullen, Steve Pritchard, Andrew Prout, Albert Reuther, Antonio Rosa, Victor Roytburd, Siddharth Samsi, Koichi Suzuki, Kenji Takahashi, Michael Wright, Charles Yee, and Michitoshi Yoshida.
\part{This is a Part}
\chapter[New Phenomena in Large-Scale Internet Traffic]{New Phenomena in Large-Scale Internet Traffic}
\noindent{\large\bf Jeremy Kepner$^{1}$, Kenjiro Cho$^{2}$, KC Claffy$^{3}$, Vijay Gadepally$^{1}$, Sarah McGuire$^{1}$, Lauren Milechin$^{4}$, William Arcand$^{1}$, David Bestor$^{1}$, William Bergeron$^{1}$, Chansup Byun$^{1}$, Matthew Hubbell$^{1}$, Michael Houle$^{1}$, Michael Jones$^{1}$, Andrew Prout$^{1}$, Albert Reuther$^{1}$, Antonio Rosa$^{1}$, Siddharth Samsi$^{1}$, Charles Yee$^{1}$, Peter Michaleas$^{1}$}
{\it $^{1}$MIT Lincoln Laboratory, $^{2}$Research Laboratory, Internet Initiative Japan, Inc., $^{3}$UCSD Center for Applied Internet Data Analysis, $^{4}$MIT Dept. of Earth, Atmospheric and Planetary Sciences}
\blfootnote{This material is based, in part, upon the work supported by the NSF under grants DMS-1312831, CCF-1533644, and CNS-1513283, DHS cooperative agreement FA8750-18-2-0049, and ASD(R\&E) under contract FA8702-15-D-0001. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF, DHS, or ASD(R\&E).}
\chapterinitial{The}{Internet} is transforming our society, necessitating a quantitative understanding of Internet traffic. Our team collects and curates the largest publicly available Internet traffic data sets. An analysis of 50 billion packets using 10,000 processors in the MIT SuperCloud reveals a new phenomenon: the importance of otherwise unseen leaf nodes and isolated links in Internet traffic. Our analysis further shows that a two-parameter modified Zipf--Mandelbrot distribution accurately describes a wide variety of source/destination statistics on moving sample windows ranging from 100{,}000 to 100{,}000{,}000 packets over collections that span years and continents. The measured model parameters distinguish different network streams, and the model leaf parameter strongly correlates with the fraction of the traffic in different underlying network topologies.
\section{Introduction}\label{intro}
Our civilization is now dependent on the Internet, necessitating a scientific understanding of this virtual universe \cite{hilbert2011world,li2013survey}. The two largest efforts to capture, curate, and share Internet packet traffic data for scientific analysis are led by our team via the Widely Integrated Distributed Environment (WIDE) project \cite{cho2000tr} and the Center for Applied Internet Data Analysis (CAIDA) \cite{claffy1999internet}. These data have been used for a wide variety of research projects, resulting in hundreds of peer-reviewed publications \cite{CAIDApubs}, ranging from characterizing the global state of Internet traffic, to specific studies on the prevalence of peer-to-peer file sharing traffic, to testing prototype software designed to stop the spread of Internet worms.
The stochastic network structure of Internet traffic is a core property of great interest to a wide range of Internet stakeholders \cite{li2013survey} and network scientists \cite{barabasi2016network}. Of particular interest is the probability distribution $p(d)$, where $d$ is the degree (or count) of several network quantities, such as source packets, packets over a unique source-destination pair (or link), and destination packets collected over specified time intervals. Among the earliest and most widely cited results of virtual Internet topology analysis has been the observation that $p(d) \propto 1/d^\alpha$ with a model exponent $1 < \alpha < 3$ for large values of $d$ \cite{barabasi1999emergence,albert1999internet,leskovec2005graphs} fit a range of network characteristics.
Many Internet models are based on the data obtained from crawling the network from a number of starting points \cite{olston2010web}. These webcrawls naturally sample the supernodes of the network \cite{cao2009identifying}, and their resulting $p(d)$ are accurately fit at large values of $d$ by single-parameter power-law models. However, as we will show, for our streaming samples of the Internet there are other topologies that contribute significant traffic. Characterizing a network by a single power-law exponent provides one view of Internet phenomena, but more accurate and complex models are required to understand the diverse topologies seen in streaming samples of the Internet. Improving model accuracy while also increasing model complexity requires overcoming a number of challenges, including acquisition of larger, rigorously collected data sets \cite{soule2004identify,zhang2005estimating}; the enormous computational cost of processing large network traffic graphs \cite{lumsdaine2007challenges,bader2013graph,tune2013internet}; careful filtering, binning, and normalization of the data; and fitting of nonlinear models to the data.
\section{Methodology}\label{method}
This work aims to improve model accuracy through several techniques. First, for over a decade we have scientifically collected and curated the largest publicly available Internet packet traffic data sets and this work analyzes the very largest collections in our corpora containing 49.6 billion packets (Table~\ref{tab:TrafficData}). Second, utilizing recent innovations in interactive supercomputing \cite{Kepner2009,reuther2018interactive}, matrix-based graph theory \cite{kolda2009tensor,kepner2011graph}, and big data mathematics (Figure~\ref{fig:AssociativeArrays}) \cite{kepner2018mathematics}, we have developed a scalable Internet traffic processing pipeline that runs efficiently on more than 10{,}000 processors in the MIT SuperCloud \cite{gadepally2018hyperscaling}. This pipeline allows us, for the first time, to process our largest traffic collections as network traffic graphs. Third, since not all packets have both source and destination Internet Protocol version 4 (IPv4) addresses, the data have been filtered so that for any chosen time window all data sets have the same number of valid IPv4 packets, denoted $N_V$ (Figure~\ref{fig:ValidPackets} and Eq.~\ref{eq:Valid}). All computed probability distributions also use the same binary logarithmic binning to allow for consistent statistical comparison across data sets (Eq.~\ref{eq:Cumulative})\cite{clauset2009power,barabasi2016network}. Fourth, to accurately model the data over the full range of $d$, we employ a modified Zipf--Mandelbrot distribution \cite{mandelbrot1953informational,montemurro2001beyond,saleh2006modeling}
\begin{equation}\label{eq:ZipfMandelbrot}
p(d;\alpha,\delta) \propto 1/(d + \delta)^\alpha
\end{equation}
The inclusion of a second model offset parameter $\delta$ allows the model to accurately fit small values of $d$, in particular $d=1$, which has the highest observed probability in these streaming data. The modified Zipf--Mandelbrot model is a special case of the more general saturation/cutoff models used to model a variety of network phenomena (Eq.~\ref{eq:satcut}) \cite{clauset2009power,barabasi2016network}. Finally, nonlinear fitting techniques are used to achieve quality fits over the entire range of $d$ (Eq.~\ref{eq:NonLinFit}).
Throughout this chapter, we defer to the terminology of network science. Network operators use many similar terms with significant differences in meaning. We use network topology to refer to the graph-theoretic virtual topology of sources and destinations observed communicating and not the underlying physical topology of the Internet (Table~\ref{tab:Terms})
\begin{table}[htp]
\caption{Network Terminology Used by Computer Network Operators and Network Scientists. Throughout this Work, the Network Science Meanings are Employed}
\begin{center}
\begin{tabular}{p{0.75in}p{2.15in}p{2.15in}}
{Term} & {Network Operations Meaning} & {Network Science Meaning} \\
\hline
Network &
The \emph{physical} links, wires, routers, switches, and endpoints used to transmit data. &
Any system that can be represented as a \emph{graph} of connections (links/edges) among entities (nodes/vertices). \\
\hline
Topology &
The \emph{layout} of the physical network. &
The specific \emph{geometries} of a graph and its sub-graphs. \\
\hline
Stream &
The \emph{flow of data} over a specific physical communication link. &
A time-ordered \emph{sequence of pairs} of entities (nodes/vertices) representing distinction in time connections (links/edges) between entities. \\
\hline
\end{tabular}
\end{center}
\label{tab:Terms}
\end{table}%
\subsection{MAWI and CAIDA Internet Traffic Collection}\label{mawicaida}
For the analysis in following sections, the data utilized are summarized in Table~\ref{tab:TrafficData} with data from Tokyo coming from the MAWI Internet Traffic Collection and the data from Chicago from the CAIDA Internet Traffic Collection.
The Tokyo data sets are publicly available packet traces provided by the WIDE project (aka the MAWI traces). The WIDE project is a research consortium in Japan established in 1988 \cite{cho2000tr}. The members of the project include network engineers, researchers, university students, and industrial partners. The focus of WIDE is on the empirical study of the large-scale Internet. WIDE operates an Internet testbed both for commercial traffic and for conducting research experiments. These data have enabled quantitative analysis of Internet traffic spanning years illustrating trends such as the emergence of residential usage, peer-to-peer networks, probe scanning, and botnets \cite{cho2006impact,borgnat2009seven,fontugne2017scaling}. The Tokyo data sets are publicly available packet traces provided by the WIDE project (aka the MAWI traces). The traces are collected from a 1~Gbps academic backbone connection in Japan. The 2015 and 2017 data sets are 48-hour-long traces captured during December 2--3, 2015, and April 12--13, 2017, in JST. The IP addresses appearing in the traces are anonymized using a prefix-preserving method \cite{fan2004prefix}.
The MAWI repository is an ongoing collection of Internet traffic traces, captured within the WIDE backbone network (AS2500) that connects Japanese universities and research institutes to the Internet. Each trace consists of captured packets observed from within WIDE and includes the packet headers of each packet along with the captured timestamp. Anonymized versions of the traces (with anonymized IP addresses and with transport layer payload removed) are made publicly available at http://mawi.wide.ad.jp/.
WIDE carries a variety of traffic including academic and commercial traffic. These data have enabled quantitative analysis of Internet traffic spanning years illustrating trends such as the emergence of residential usage, peer-to-peer networks, probe scanning, and botnets \cite{cho2006impact,cho2008observing,fontugne2017scaling}. WIDE is mostly dominated by HTTP traffic, but is influenced by global anomalies. For example, Code Red, Blaster, and Sasser are worms that disrupted Internet traffic \cite{allman2007brief}. Of these, Sasser (2005) impacted MAWI traffic the most, accounting for two-thirds of packets at its peak. Conversely, the ICMP traffic surge in 2003 and the SYN Flood in 2012 were more local in nature, each revealing attacks on targets within WIDE that lasted several months.
CAIDA collects several different data types at geographically and topologically diverse locations and makes these data available to the research community to the extent possible while preserving the privacy of individuals and organizations who donate data or network access \cite{claffy1999internet,claffy2000measuring}. CAIDA has (and had) monitoring locations in Internet service providers (ISPs) in the USA. CAIDA's passive traces data set contains traces collected from high-speed monitors on a commercial backbone link. The data collection started in April 2008 and is ongoing. These data are useful for research on the characteristics of Internet traffic, including application breakdown (based on TCP/IP ports), security events, geographic and topological distribution, flow volume, and duration. For an overview of all traces, see the trace statistics page \cite{CAIDAstats}.
Collectively, our consortium has enabled the scientific analysis of Internet traffic, resulting in hundreds of peer-reviewed publications with over 30,000 citations \cite{CAIDApubs}. These include early work on Internet threats such as the Code Red worm \cite{moore2002code} and Slammer worm \cite{moore2003inside} and how quarantining might mitigate threats \cite{moore2003internet}. Subsequent work explored various techniques, such as dispersion, for measuring Internet capacity and bandwidth \cite{dovrolis2001packet,dovrolis2004packet,prasad2003bandwidth}. The next major area of research provided significant results on the dispersal of the Internet via the emergence of peer-to-peer networks \cite{karagiannis2004transport,karagiannis2004p2p}, edge devices \cite{kohno2005remote}, and corresponding denial-of-service attacks \cite{moore2006inferring}, which drove the need for new ways to categorize traffic \cite{kim2008internet}. The incorporation of network science and statistical physics concepts into the analysis of the Internet produced new results on the hyperbolic geometry of complex networks \cite{krioukov2009curvature,krioukov2010hyperbolic} and sustaining the Internet with hyperbolic mapping \cite{boguna2009navigating,boguna2009navigability,boguna2010sustaining}. Likewise, a new understanding also emerged on the identification of influential spreaders in complex networks \cite{kitsak2010identification}, the relationship of popularity versus similarity in growing networks \cite{papadopoulos2012popularity}, and overall network cosmology \cite{krioukov2012network}. More recent work has developed new ideas for Internet classification \cite{dainotti2012issues} and future data centric architectures \cite{zhang2014named}.
\begin{table}[htp]
\caption{Network Traffic Packet Data Sets from MAWI (Tokyo data sets) and CAIDA (Chicago data sets) Collected at Different Times and Durations over Two Years}
\begin{center}
\begin{tabular}{lcccc}
{Location} & {Date} & {Duration} & {Bandwidth} & {Packets} \\
\hline
Tokyo & 2015 Dec 02 & 2 days & $10^9$~ bits/second & $17.0{\times}10^9$ \\
Tokyo & 2017 Apr 12 & 2 days & $10^9$~ bits/second & $16.8{\times}10^9$ \\
Chicago A & 2016 Jan 21 & 1 hour & $10^{10}$ bits/second & ~~$2.0{\times}10^9$ \\
Chicago A & 2016 Feb 18 & 1 hour & $10^{10}$ bits/second & ~~$2.0{\times}10^9$ \\
Chicago A & 2016 Mar 17 & 1 hour & $10^{10}$ bits/second & ~~$1.8{\times}10^9$ \\
Chicago A & 2016 Apr 06 & 1 hour & $10^{10}$ bits/second & ~~$1.8{\times}10^9$ \\
Chicago B & 2016 Jan 21 & 1 hour & $10^{10}$ bits/second & ~~$2.3{\times}10^9$ \\
Chicago B & 2016 Feb 18 & 1 hour & $10^{10}$ bits/second & ~~$1.7{\times}10^9$ \\
Chicago B & 2016 Mar 17 & 1 hour & $10^{10}$ bits/second & ~~$2.0{\times}10^9$ \\
Chicago B & 2016 Apr 06 & 1 hour & $10^{10}$ bits/second & ~~$2.1{\times}10^9$ \\
\hline
\end{tabular}
\end{center}
\label{tab:TrafficData}
\end{table}%
\subsection{Network Quantities from Matrices}\label{networkquantities}
In our analysis, the network traffic packet data are reduced to origin--destination traffic matrices. These matrices can be used to compute a wide range of network statistics useful in the analysis, monitoring, and control of the Internet. Such an analysis includes the temporal fluctuations of the supernodes \cite{soule2004identify} and inferring the presence of unobserved traffic \cite{zhang2005estimating,bharti2010inferring}.
To create the matrices, at a given time $t$, $N_V$ consecutive valid packets are aggregated from the traffic into a sparse matrix ${\bf A}_t$, where ${\bf A}_t(i,j)$ is the number of valid packets between the source $i$ and destination $j$ \cite{mucha2010community}. The sum of all the entries in ${\bf A}_t$ is equal to $N_V$
\begin{equation}\label{eq:Valid}
\sum_{i,j} {\bf A}_t(i,j) = N_V
\end{equation}
All the network quantities depicted in Figure~\ref{fig:NetworkDistribution}a can be readily computed from ${\bf A}_t$ as specified in Tables~\ref{tab:Aggregates} and ~\ref{tab:Filters}, including the number of unique sources and destinations, along with many other network statistics \cite{soule2004identify,zhang2005estimating,tune2013internet}.
\begin{table}
\caption{Aggregate Network Properties}
\begin{center}
\begin{tabular}{p{1.5in}p{1.5in}p{1.0in}}
\hline
{Aggregate Property} & {Summation Notation} & {Matrix Notation} \\
\hline
Valid packets $N_V$ & $\sum_i ~ \sum_j ~ {\bf A}_t(i,j)$ & $~{\bf 1}^{\sf T} {\bf A}_t {\bf 1}$ \\
Unique links & $\sum_i ~ \sum_j |{\bf A}_t(i,j)|_0$ & ${\bf 1}^{\sf T}|{\bf A}_t|_0 {\bf 1}$ \\
Unique sources & $\sum_i |\sum_j ~ {\bf A}_t(i,j)|_0$ & ${\bf 1}^{\sf T}|{\bf A}_t {\bf 1}|_0$ \\
Unique destinations & $\sum_j |\sum_i ~ {\bf A}_t(i,j)|_0$ & $|{\bf 1}^{\sf T} {\bf A}_t|_0 {\bf 1}$ \\
\hline
\end{tabular}{Formulas for computing aggregates from a sparse network image ${\bf A}_t$ at time $t$ in both summation and matrix notations. ${\bf 1}$ is a column vector of all 1's, $^{\sf T}$ is the transpose operation, and $|~|_0$ is the zero-norm that sets each nonzero value of its argument to 1 \cite{karvanen2003measuring}.}
\end{center}
\label{tab:Aggregates}
\end{table}%
\begin{table}[h]
\caption{Neural Network Image Convolution Filters}
\begin{center}
\begin{tabular}{p{2.0in}p{1.5in}p{1.0in}}
\hline
{Network Quantity} & {Summation Notation} & {Matrix Notation} \\
\hline
Source packets from $i$ & $\sum_j ~ {\bf A}_t(i,j)$ & ~~~$~{\bf A}_t ~~ {\bf 1}$ \\
Source fan-out from $i$ & $\sum_j |{\bf A}_t(i,j)|_0$ & ~~~$|{\bf A}_t|_0 {\bf 1}$ \\
Link packets from $i$ to $j$ & $~~~~~~{\bf A}_t(i,j)$ & ~~~$~{\bf A}_t$ \\
Destination fan-in to $j$ & $\sum_i |{\bf A}_t(i,j)|_0$ & ${\bf 1}^{\sf T}~{\bf A}_t$ \\
Destination packets to $j$ & $\sum_i ~ {\bf A}_t(i,j)$ & ${\bf 1}^{\sf T}|{\bf A}_t|_0$ \\
\hline
\end{tabular}{Different network quantities are extracted from a sparse traffic image ${\bf A}_t$ at time $t$ via convolution with different filters. Formulas for the filters are given in both summation and matrix notations. ${\bf 1}$ is a column vector of all 1's, $^{\sf T}$ is the transpose operation, and $|~|_0$ is the zero-norm that sets each nonzero value of its argument to 1 \cite{karvanen2003measuring}.}
\end{center}
\label{tab:Filters}
\end{table}%
Figure~\ref{fig:NetworkTopology}a depicts the major topological structures in the network traffic. Isolated links are sources and destinations that each have only one connection (Table~\ref{tab:Isolatedlinks}). The first, second, third, $\ldots$ supernodes are the source or destination with the first, second, third, $\ldots$ most links (Table~\ref{tab:Supernodes}). The core of a network can be defined in a variety of ways \cite{schaeffer2007graph,benson2016higher}. In this work, the network core conveys the concept of a collection of sources and destinations that are not isolated and are multiply connected. The core is defined as the collection of sources and destinations in which every source and destination has more than one connection. The core, as computed here, does not include the first five supernodes although only the first supernode is significant, and whether or not the other supernodes are included has minimal impact on the core in these data. The core leaves are sources and destinations that have only one connection to a core source or destination (Tables~\ref{tab:Core} and \ref{tab:Coreleaves}).
\begin{table}[h]
\caption{Properties of Isolated Links}
\begin{center}
\begin{tabular}{p{2.75in}p{1.25in}}
\hline
{Network Quantity} & {Matrix Notation} \\
\hline
Isolated links & ${\bf A}_t(i_1,j_1)$\\
Number of isolated link sources & ${\bf 1}^{\sf T}|{\bf A}_t(i_1,j_1) {\bf 1}|_0$\\
Number of packets traversing isolated links & ${\bf 1}^{\sf T}{\bf A}_t(i_1,j_1){\bf 1}$\\
Number of unique isolated links & ${\bf 1}^{\sf T}|{\bf A}_t(i_1,j_1)|_0 {\bf 1}$ \\
Number of isolated link destinations & $|{\bf 1}^{\sf T} {\bf A}_t(i_1,j_1)|_0 {\bf 1}$ \\
\hline
\end{tabular}{Different characteristics related to isolated links are extracted from a sparse traffic image ${\bf A}_t$ at time $t$. Formulas are in matrix notation. The set of sources that send to only one destination are $ i_1 = {\rm arg}({\bf d}_{\rm out} = 1)$, and the set of destinations that receive from only one destination are $j_1 = {\rm arg}({\bf d}_{\rm in} = 1)$.}
\end{center}
\label{tab:Isolatedlinks}
\end{table}%
\begin{table}[h]
\caption{Properties of Supernodes}
\begin{center}
\begin{tabular}{p{2.75in}p{2.25in}}
\hline
{\bf Network} & {\bf ~Matrix} \\
{\bf Quantity} & {\bf Notation} \\
\hline
Supernode source leaves & ${\bf A}_t(i_1,k_{\rm max})$\\
Supernode destination leaves & ${\bf A}_t(k_{\rm max},j_1)$\\
Number of supernode leaf sources & ${\bf 1}^{\sf T}|{\bf A}_t(i_1,k_{\rm max}) {\bf 1}|_0$\\
Number of packets traversing supernode leaves & $ {\bf 1}^{\sf T}{\bf A}_t(i_1,k_{\rm max}) + {\bf A}_t(k_{\rm max},j_1){\bf 1}$ \\
Number of unique supernode leaf links & $ {\bf 1}^{\sf T}|{\bf A}_t(i_1,k_{\rm max})|_0 + |{\bf A}_t(k_{\rm max},j_1)|_0 {\bf 1}$ \\
Number of supernode leaf destinations & $|{\bf 1}^{\sf T} {\bf A}_t(k_{\rm max},j_1)|_0 {\bf 1}$\\
\hline
\end{tabular}{Different characteristics related to supernodes are extracted from a sparse traffic image ${\bf A}_t$ at time $t$. Formulas are in matrix notation. The identity of the first supernode is given by $k_{\rm max} = {\rm argmax}({\bf d}_{\rm out} + {\bf d}_{\rm in})$. The leaves of a supernode are those sources and destinations whose only connection is to the supernode.}
\end{center}
\label{tab:Supernodes}
\end{table}%
\begin{table}[h]
\caption{Properties of Network Core}
\begin{center}
\begin{tabular}{p{2.75in}p{2.25in}}
\hline
{\bf Network} & {\bf ~Matrix} \\
{\bf Quantity} & {\bf Notation} \\
\hline
Core links & ${\bf A}_t(i_{\rm core},j_{\rm core})$\\
Number of core sources & ${\bf 1}^{\sf T}|{\bf A}_t(i_{\rm core},j_{\rm core}) {\bf 1}|_0$\\
Number of core packets & $ {\bf 1}^{\sf T}{\bf A}_t(i_{\rm core},j_{\rm core}){\bf 1}$\\
Number of unique core links & ${\bf 1}^{\sf T}|{\bf A}_t(i_{\rm core},j_{\rm core})|_0 {\bf 1}$\\
Number of core destinations & $|{\bf 1}^{\sf T} {\bf A}_t(i_{\rm core},j_{\rm core})|_0 {\bf 1}$\\
\hline
\end{tabular}{Different characteristics related to the core are extracted from a sparse traffic image ${\bf A}_t$ at time $t$. Formulas are in matrix notation. The set of sources that send to more than one destination, excluding the supernode(s), is $i_{\rm core} = {\rm arg}(1 < {\bf d}_{\rm out} < {\bf d}_{\rm out}(k_{\rm max}))$. The set of destinations that receive from more than one source, excluding the supernode(s), is $ j_{\rm core} = {\rm arg}(1 < {\bf d}_{\rm in} < {\bf d}_{\rm in}(k_{\rm max}))$.}
\end{center}
\label{tab:Core}
\end{table}%
\begin{table}[h]
\caption{Properties of Core Leaves}
\begin{center}
\begin{tabular}{p{2.75in}p{2.25in}}
\hline
{\bf Network} & {\bf ~Matrix} \\
{\bf Quantity} & {\bf Notation} \\
\hline
Core source leaves & ${\bf A}_t(i_1,k_{\rm core})$\\
Core destination leaves & ${\bf A}_t(k_{\rm core},j_1)$\\
Number of core leaf sources & ${\bf 1}^{\sf T}|{\bf A}_t(i_1,k_{\rm core}) {\bf 1}|_0$\\
Number of core leaf packets & ${\bf 1}^{\sf T}{\bf A}_t(i_1,k_{\rm core}) + {\bf A}_t(k_{\rm core},j_1){\bf 1}$\\
Number of unique core leaf links & ${\bf 1}^{\sf T}|{\bf A}_t(i_1,k_{\rm core})|_0 + |{\bf A}_t(k_{\rm core},j_1)|_0 {\bf 1}$\\
Number of core leaf destination & $ |{\bf 1}^{\sf T} {\bf A}_t(k_{\rm core},j_1)|_0 {\bf 1}$\\
\hline
\end{tabular}{Different characteristics related to the core leaves are extracted from a sparse traffic image ${\bf A}_t$ at time $t$. Formulas are in matrix notation. The core leaves are sources and destinations that have one connection to a core source or destination.}
\end{center}
\label{tab:Coreleaves}
\end{table}%
An essential step for increasing the accuracy of the statistical measures of Internet traffic is using windows with the same number of valid packets $N_V$. For this analysis, a valid packet is defined as TCP over IPv4, which includes more than 95\% of the data in the collection and eliminates a small amount of data that use other protocols or contain anomalies. Using packet windows with the same number of valid packets produces quantities that are consistent over a wide range from $N_V = 100{,}000$ to $N_V = 100{,}000{,}000$ (Figure~\ref{fig:ValidPackets}).
\begin{figure}
\includegraphics[width=\columnwidth]{chapters/chaptercy/figs/ValidPackets-Tokyo2015.pdf}
\includegraphics[width=\columnwidth]{chapters/chaptercy/figs/ValidPackets-Tokyo2017.pdf}
\caption{{\bf Valid packets.} Analyzing packet windows with the same numbers of valid packets produces consistent fractions of unique links, unique destinations, and unique sources over a wide range of packet sizes for the Tokyo 2015 (a) and Tokyo 2017 (b) data sets. The plots show these fractions for moving packet windows of $N_V$ = 100{,}000 packets (left) and $N_V$ = 100{,}000{,}000 packets (right). The packet windows correspond to time windows of approximately 1.5 seconds and 25 minutes.
}
\label{fig:ValidPackets}
\end{figure}
\subsection{Memory and Computation Requirements}
Processing 50 billion Internet packets with a variety of algorithms presents numerous computational challenges. Dividing the data set into combinable units of approximately 100{,}000 consecutive packets made the analysis amenable to processing on a massively parallel supercomputer. The detailed architecture of the parallel processing system and its corresponding performance are described in \cite{gadepally2018hyperscaling}. The resulting processing pipeline was able to efficiently use over 10{,}000 processors on the MIT SuperCloud and was essential to this first-ever complete analysis of these data.
A key element of our analysis is the use of novel sparse matrix mathematics in concert with the MIT SuperCloud. Construction and analysis of network traffic matrices of the entire Internet address space have been considered impractical for its massive size \cite{tune2013internet}. Internet Protocol version 4 (IPv4) has $2^{32}$ unique addresses, but at any given collection point, only a fraction of these addresses will be observed. Exploiting this property to save memory can be accomplished by extending traditional sparse matrices so that new rows and columns can be added dynamically.
The algebra of associative arrays \cite{kepner2018mathematics} and its corresponding implementation in the Dynamic Distributed Dimensional Data Model (D4M) software library (d4m.mit.edu) allows the row and columns of a sparse matrix to be any sortable value, in this case character string representations of the Internet addresses (Figure~\ref{fig:AssociativeArrays}). Associative arrays extend sparse matrices to have database table properties with dynamically insertable and removable rows and columns that adjust as new data are added or subtracted to the matrix. Using these properties, the memory requirements of forming network traffic matrices can be reduced at the cost of increasing the required computation necessary to resort the rows and columns.
A network matrix ${\bf A}_t$ with $N_V = 100{,}000{,}000$ represented as an associative array typically requires 2 gigabytes of memory. A complete analysis of the statistics and topologies of ${\bf A}_t$ typically takes 10 minutes on a single MIT SuperCloud Intel Knights Landing processor core. Using increments of $100{,}000$ packets means that this analysis is repeated over 500{,}000 times to process all 49.6 billion packets. Using 10{,}000 processors on the MIT SuperCloud shortens the runtime of one of these analyses to approximately 8 hours. The results presented within this chapter are products of a discovery process that required hundreds of such runs that would not have been possible without these computational resources. Fortunately, the utilization of these results by Internet stakeholders can be significantly accelerated by creating optimized embedded implementations that only compute the desired statistics and are not required to support a discovery process \cite{liu2010tcam,liu2016packet}.
\begin{figure}
\includegraphics[width=\columnwidth]{chapters/chaptercy/figs/AssociativeArrays.pdf}
\caption{{\bf Associative arrays.} (a) Tabular representation of raw network traffic and corresponding database query to find all records beginning with source 1.1.1.1. (b) Network graph highlighting nearest neighbors of source node 1.1.1.1. (c) Corresponding associative array representation of the network graph illustrating how the neighbors of source node 1.1.1.1 are computed with matrix vector multiplication.}
\label{fig:AssociativeArrays}
\end{figure}
\section{Internet Traffic Modeling}
Quantitative measurements of the Internet \cite{rabinovich2016measuring} have provided Internet stakeholders information on the Internet since its inception. Early work has explored the early growth of the Internet \cite{claffy1994tracking}, the distribution of packet arrival times \cite{paxson1995wide}, the power-law distribution of network outages \cite{paxson1996end}, the self-similar behavior of traffic \cite{leland1994self,willinger1997self,willinger2002scaling}, formation processes of power-law networks \cite{faloutsos1999power,medina2000origin,broder2000graph,willinger2009mathematics}, and the topologies of Internet service providers \cite{spring2002measuring}. Subsequent work has examined the technological properties of Internet topologies \cite{li2004first}, the diameter of the Internet \cite{leskovec2005graphs}, applying rank index-based Zipf--Mandelbrot modeling to peer-to-peer traffic \cite{saleh2006modeling}, and extending topology measurements to edge hosts \cite{heidemann2008census}. More recent work looks to continued measurement of power-law phenomena \cite{mahanti2013tale,kitsak2015long,lischke2016analyzing}, exploiting emerging topologies for optimizing network traffic \cite{dhamdhere2010internet,labovitz2011internet,chiu2015we}, using network data to locate disruptions \cite{fontugne2017pinpointing}, the impact of inter-domain congestion \cite{dhamdhere2018inferring}, and studying the completeness of passive sources to determine how well they can observe microscopic phenomena \cite{mirkovic2017you}.
The above sample of many years of Internet research has provided significant qualitative insights into Internet phenomenology. Single-parameter power-law fits have extensively been explored and shown to adequately fit higher-degree tails of the observations. However, more complex models are required to fit the entire range of observations. Figure~\ref{fig:PowerLawFits}a adapted from figure 8H \cite{clauset2009power} shows the number of bytes of data received in response to $2.3\times10^5$ HTTP (web) requests from computers at a large research laboratory and shows a strong agreement with a power law at large values, but diverges with the single-parameter model at small values. Figure~\ref{fig:PowerLawFits}b adapted from figure 9W \cite{clauset2009power} shows the distribution of $1.2\times10^5$ hits on web sites from AOL users and shows a strong agreement with a power law at small values, but diverges with the single-parameter model at large values. Figure~\ref{fig:PowerLawFits}c adapted from figure 9X \cite{clauset2009power} shows the distribution of $2.4\times10^8$ web hyperlinks and has a reasonable model agreement across the entire range, except for the smallest values. Figure~\ref{fig:PowerLawFits}d adapted from figure 4B \cite{mahanti2013tale} shows the distribution of visitors arriving at YouTube from referring web sites appears to be best represented by two very different power-law models with significant difference as the smallest values. Figure~\ref{fig:PowerLawFits}e adapted from figure 3A \cite{kitsak2015long} shows the distribution of the number of Border Gateway Protocol updates received by the 4 monitors in 1-minute intervals and shows a strong agreement with a power law at large values, but diverges with the single-parameter model at small values. Figure~\ref{fig:PowerLawFits}f adapted from figure 21 in \cite{lischke2016analyzing} shows the distribution of the Bitcoin network in 2011 and shows a strong agreement with a power law at small values, but diverges with the single-parameter model at large values.
The results shown in Figure~\ref{fig:PowerLawFits} represent some of the best and most carefully executed fits to Internet data and clearly show the difficulty of fitting the entire range with a single-parameter power law. It is also worth mentioning that in each case the cumulative distribution is used, which naturally provides a smoother curve (in contrast to the differential cumulative distribution used in our analysis), but provides less detail on the underlying phenomena. Furthermore, the data in Figure~\ref{fig:PowerLawFits} are typically isolated collections such that the error bars are not readily computable, which limits the ability to assess both the quality of the measurements and the model fits.
\begin{figure}
\includegraphics[width=\columnwidth]{chapters/chaptercy/figs/PowerLawFits.pdf}
\caption{{\bf Single-parameter power-law fits of Internet data.} Single-parameter fits of the cumulative distributions of Internet data have difficulty modeling the entire range. The estimated ratio between the model and the data at the model extremes is shown.
({a}) Figure 8H \cite{clauset2009power}. ({b}) Figure 9W \cite{clauset2009power}. ({c}) Figure 9X \cite{clauset2009power}. ({d}) Figure 4B \cite{mahanti2013tale}. ({e}) Figure 3A \cite{kitsak2015long}. ({f}) Figure 21 \cite{lischke2016analyzing}.}
\label{fig:PowerLawFits}
\end{figure}
Regrettably, the best publicly available data about the global interconnection system that carries most of the world's communications traffic are incomplete and of unknown accuracy. There is no map of physical link locations, capacity, traffic, or interconnection arrangements. This opacity of the Internet infrastructure hinders research and development efforts to model network behavior and topology; design protocols and new architectures; and study real-world properties such as robustness, resilience, and economic sustainability. There are good reasons for the dearth of information: complexity and scale of the infrastructure; information-hiding properties of the routing system; security and commercial sensitivities; costs of storing and processing the data; and lack of incentives to gather or share data in the first place, including cost-effective ways to use it operationally. But understanding the Internet's history and present, much less its future, is impossible without realistic and representative data sets and measurement infrastructure on which to support sustained longitudinal measurements as well as new experiments. The MAWI and CAIDA data collection efforts are the largest efforts to provide the data necessary to begin to answer these questions.
\subsection{Logarithmic Pooling}\label{prob}
In this analysis before model fitting, the differential cumulative probabilities are calculated.
For a network quantity $d$, the histogram of this quantity computed from ${\bf A}_t$ is denoted by $n_t(d)$, with corresponding probability
\begin{equation}\label{eq:Probability}
p_t(d) = n_t(d)/\sum_d n_t(d)
\end{equation}
and cumulative probability
\begin{equation}\label{eq:Cumulative}
P_t(d) = \sum_{i=1,d} p_t(d)
\end{equation}
Because of the relatively large values of $d$ observed due to a single supernode, the measured probability at large $d$ often exhibits large fluctuations. However, the cumulative probability lacks sufficient detail to see variations around specific values of $d$, so it is typical to use the differential cumulative probability with logarithmic bins in $d$
\begin{equation}\label{eq:LogBin}
D_t(d_i) = P_t(d_i) - P_t(d_{i-1})
\end{equation}
where $d_i = 2^i$ \cite{clauset2009power}. The corresponding mean and standard deviation of $D_t(d_i)$ over many different consecutive values of $t$ for a given data set are denoted $D(d_i)$ and $\sigma(d_i)$. These quantities strike a balance between accuracy and detail for subsequent model fitting as demonstrated in the daily structural variations revealed in the Tokyo data (Figures~\ref{fig:DailyVariation} and \ref{fig:DailyLimits}).
Diurnal variations in supernode network traffic are well known \cite{soule2004identify}. The Tokyo packet data were collected over a period spanning two days and allow the daily variations in packet traffic to be observed. The precision and accuracy of our measurements allow these variations to be observed across a wide range of nodes. Figure~\ref{fig:DailyVariation} shows the fraction of source fan-outs in each of various bin ranges. The fluctuations show the network evolving between two envelopes occurring between noon and midnight that are shown in Figure~\ref{fig:DailyLimits}.
\begin{figure}
\centering
\includegraphics[width=0.75\columnwidth]{chapters/chaptercy/figs/DailyVariation.pdf}
\caption{{\bf Daily variation in Internet traffic.} The fraction of source nodes with a given range of fan-out is shown as a function of time for the Tokyo 2015 data. The $p(d = 1)$ value is plotted on a separate linear scale because of the larger magnitude relative to the other points. Each point is the mean of many neighboring points in time, and the error bars are the measured $\pm$1-$\sigma$. The daily variations of the distributions oscillate between extremes corresponding to approximately local noon and midnight.
}
\label{fig:DailyVariation}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.75\columnwidth]{chapters/chaptercy/figs/DailyLimits.pdf}
\caption{{\bf Daily limits in Internet traffic.} The fraction of source nodes versus fan-out is shown for two noons and two midnights for the Tokyo 2015 data. The overlap among the noons and the midnights shows the relative day-to-day consistency in these data and shows the limits of the two extremes in daily variation. During the day, there is more traffic among nodes with intermediate fan-out. At night, the traffic is more dominated by leaf nodes and the supernode.
}
\label{fig:DailyLimits}
\end{figure}
\subsection{Modified Zipf--Mandelbrot Model}\label{zipf}
Measurements of $D(d_i)$ can reveal many properties of network traffic, such as the number of nodes with only one connection $D(d = 1)$ and the size of the supernode $d_{\rm max}={\rm argmax}(D(d) > 0)$. An effective low-parameter model allows these and many other properties to be summarized and computed efficiently. In the standard Zipf--Mandelbrot model typically used in linguistic contexts, the value $d$ in Eq.~\ref{eq:ZipfMandelbrot} is a ranking with $d=1$ corresponding to the most popular value \cite{mandelbrot1953informational,montemurro2001beyond,saleh2006modeling}. In our analysis, the Zipf--Mandelbrot model is modified so that $d$ is a measured network quantity instead of a rank index (Eq. \ref{eq:ZipfMandelbrot}). The model exponent $\alpha$ has a larger impact on the model at large values of $d$, while the model offset $\delta$ has a larger impact on the model at small values of $d$ and in particular at $d=1$.
The general saturation/cutoff models used to model a variety of network phenomena is denoted \cite{clauset2009power,barabasi2016network}
\begin{equation}\label{eq:satcut}
p(d) \propto \frac{1}{(d + \delta)^\alpha \exp[\lambda d]}
\end{equation}
where $\delta$ is the low-$d$ saturation and $1/\lambda$ is the high-$d$ cutoff that bounds the power-law regime of the distribution. The modified Zipf--Mandelbrot is a special case of this distribution that accurately models our observations. The unnormalized modified Zipf--Mandelbrot model is denoted
\begin{equation}\label{eq:rho}
\rho(d;\alpha,\delta) = \frac{1}{(d + \delta)^\alpha}
\end{equation}
with corresponding derivative with respect to $\delta$
\begin{equation}\label{eq:drho}
\partial_\delta \rho(d;\alpha,\delta) = \frac{-\alpha}{(d + \delta)^{\alpha+1}} = -\alpha \rho(d;\alpha+1,\delta)
\end{equation}
The normalized model probability is given by
\begin{equation}\label{eq:ZM}
p(d;\alpha,\delta) = \frac{\rho(d;\alpha,\delta)}{\sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta)}
\end{equation}
where $d_{\rm{max}}$ is the largest value of the network quantity $d$. The cumulative model probability is the sum
\begin{equation}\label{eq:ZMcum}
P(d_i;\alpha,\delta) = \sum_{d=1}^{d_i} p(d;\alpha,\delta)
\end{equation}
The corresponding differential cumulative model probability is
\begin{equation}\label{eq:ZMdiff}
D(d_i;\alpha,\delta) = P(d_i;\alpha,\delta) - P(d_{i-1};\alpha,\delta)
\end{equation}
where $d_i = 2^i$. In terms of $\rho$, the differential cumulative model probability is
\begin{equation}\label{eq:ZMdiffrho}
D(d_i;\alpha,\delta) = \frac{\sum_{d=d_{i-1}+1}^{d=d_i} \rho(d;\alpha,\delta)}{\sum_{d=1}^{d = d_{\rm max}}~~\rho(d;\alpha,\delta)}
\end{equation}
The above function is closely related to the Hurwitz zeta function \cite{NIST:DLMF,clauset2009power,yu2017link} \begin{equation}\label{eq:HZ}
\zeta(\alpha,\delta_1) = \sum_{d=0}^{\infty}~~\rho(d;\alpha,\delta_1)
\end{equation}
where $\delta_1 = \delta+1$. The differential cumulative model probability in terms of the Hurwitz zeta function is
\begin{equation}\label{eq:ZMdiffrho}
D(d_i;\alpha,\delta) = \frac{\zeta(\alpha,\delta+3+d_{i-1}) - \zeta(\alpha,\delta+2+d_i)}{\zeta(\alpha,\delta+) - \zeta(\alpha,\delta+)}
\end{equation}
\subsection{Nonlinear Model Fitting}\label{nonlinear}
The model exponent $\alpha$ has a larger impact on the model at large values of $d$, while the model offset $\delta$ has a larger impact on the model at small values of $d$ and in particular at $d=1$. A nonlinear fitting technique is used to obtain accurate model fits across the entire range of $d$. Initially, a set of candidate exponent values is selected, typically $\alpha = 0.10, 0.11,\ldots,3.99,4.00$. For each value of $\alpha$, a value of $\delta$ is computed that exactly matches the model with the data at $D(1)$. Finding the value of $\delta$ corresponding to a give $D(1)$ is done using Newton's method as follows. Setting the measured value of $D(1)$ equal to the model value $D(1;\alpha,\delta)$ gives
\begin{equation}\label{eq:ZM1}
D(1) = D(1;\alpha,\delta) = \frac{1}{(1 + \delta)^{\alpha} \sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta)}
\end{equation}
Newton's method works on functions of the form $f(\delta) = 0$. Rewriting the above expression produces
\begin{equation}\label{eq:ZMnewton}
f(\delta) = D(1) (1 + \delta)^\alpha \sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta) - 1 = 0
\end{equation}
For given value of $\alpha$, $\delta$ can be computed from the following iterative equation
\begin{equation}\label{eq:NewtonIteration}
\delta \rightarrow \delta - \frac{f(\delta)}{\partial_\delta f(\delta)}
\end{equation}
where the partial derivative $\partial_\delta f(\delta)$ is
\begin{eqnarray}\label{eq:NewtonDerivative}
\partial_\delta f(\delta)
& = & D(1) ~ \partial_\delta [(1 + \delta)^\alpha ~~ \sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta)] \nonumber \\
& = & D(1) [[\alpha (1 + \delta)^{\alpha-1} \sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta)] +
[(1 + \delta)^\alpha \sum_{d=1}^{d_{\rm max}} ~~ \partial_\delta \rho(d;\alpha,\delta)]]
\nonumber \\
& = & D(1) [[\alpha (1 + \delta)^{\alpha-1} \sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta)] +
[(1 + \delta)^\alpha \sum_{d=1}^{d_{\rm max}} -\alpha \rho(d;\alpha+1,\delta)]] \nonumber \\
& = & \alpha D(1) (1 + \delta)^\alpha [(1 + \delta)^{-1} \sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta) -
\sum_{d=1}^{d_{\rm max}} \rho(d;\alpha+1,\delta)]
\end{eqnarray}
Using a starting value of $\delta=1$ and bounds of $0 < \delta < 10$, Newton's method can be iterated until the differences in successive values of $\delta$ fall below a specified error (typically 0.001), which is usually achieved in less than five iterations.
If faster evaluation is required, the sums in the above formulas can be accelerated using the integral approximations
\begin{eqnarray}\label{eq:SumApprox}
\sum_{d=1}^{d_{\rm max}} \rho(d;\alpha,\delta)
&\approx& \sum_{d=1}^{d_{\rm sum}} \rho(d;\alpha,\delta) + \int_{d_{\rm sum} + 0.5}^{d_{\rm max} + 0.5} \rho(x;\alpha,\delta) dx \nonumber \\
&=& \sum_{d=1}^{d_{\rm sum}} \rho(d;\alpha,\delta) + \frac{\rho(d_{\rm sum} + 0.5;\alpha-1,\delta) - \rho(d_{\rm max} + 0.5;\alpha-1,\delta) }{\alpha-1} \nonumber \\
\sum_{d=1}^{d_{\rm max}} \rho(d;\alpha+1,\delta)
&\approx& \sum_{d=1}^{d_{\rm sum}} \rho(d;\alpha+1,\delta) + \int_{d_{\rm sum} + 0.5}^{d_{\rm max} + 0.5} \rho(x;\alpha+1,\delta) dx \nonumber \\
&=& \sum_{d=1}^{d_{\rm sum}} \rho(d;\alpha+1,\delta) + \frac{\rho(d_{\rm sum} + 0.5;\alpha,\delta) - \rho(d_{\rm max} + 0.5;\alpha,\delta) }{\alpha}
\end{eqnarray}
where the parameter $d_{\rm sum}$ can be adjusted to exchange speed for accuracy. For typical values of $\alpha$, $\delta$, and $d_{\rm max}$ used in this work, the accuracy is approximately $1/d_{\rm sum}$.
The best-fit $\alpha$ (and corresponding $\delta$) is chosen by minimizing the $|~|^{1/2}$ metric over logarithmic differences between the candidate models $D(d_i;\alpha,\delta)$ and the data
\begin{equation}\label{eq:NonLinFit}
{\rm argmin}_{\alpha} \sum_{d_i}|\log(D(d_i)) - \log(D(d_i;\alpha,\delta))|^{1/2}
\end{equation}
The $|~|^{1/2}$ metric (or $|~|_p$-norm with $p = 1/2$) favors maximizing error sparsity over minimizing outliers \cite{donoho2006compressed,chartrand2007exact,xu2012,karvanen2003measuring,saito2000sparsity,Brbic2018,Rahimi2018scale}. Several authors have recently shown that it is possible to reconstruct a nearly sparse signal from fewer linear measurements than would be expected from traditional sampling theory. Furthermore, by replacing the $|~|_1$ norm with the $|~|^p$ with $p < 1$, reconstruction is possible with substantially fewer measurements.
Using logarithmic values more evenly weights their contribution to the model fit and more accurately reflects the number of packets used to compute each value of $D(d_i)$. Lower-accuracy data points are avoided by limiting the fitting procedure to data points where the value is greater than the standard deviation: $D(d_i) > \sigma(d_i)$.
\section{Results}\label{Results}
Figure~\ref{fig:NetworkDistribution}b shows five representative model fits out of the 350 performed on 10 data sets, 5 network quantities, and 7 valid packet windows: $N_V = 10^5$, $3{\times}10^5$, $10^6$, $3{\times}10^6$, $10^7$, $3{\times}10^7$, $10^8$. The model fits are valid over the entire range of $d$ and provide parameter estimates with precisions of 0.01. In every case, the high value of $p(d=1)$ is indicative of a large contribution from a combination of supernode leaves, core leaves, and isolated links (Figure~\ref{fig:NetworkTopology}a). The breadth and accuracy of these data allow a detailed comparison of the model parameters. Figure~\ref{fig:NetworkDistribution}c shows the model offset $\delta$ versus the model exponent $\alpha$ for all 350 fits. The different collection locations are clearly distinguishable in this model parameter space. The Tokyo collections have smaller offsets and are more tightly clustered than the Chicago collections. Chicago B has a consistently smaller source and link packet model offset than Chicago A. All the collections have source, link, and destination packet model exponents in the relatively narrow $1.5 < \alpha < 2$ range. The source fan-out and destination fan-in model exponents are in the broader $1.5 < \alpha < 2.5$ range and are consistent with the prior literature \cite{clauset2009power}. These results represent an entirely new approach to characterizing Internet traffic that allows the distributions to be projected into a low-dimensional space and enables accurate comparisons among packet collections with different locations, dates, durations, and sizes. Figure~\ref{fig:NetworkDistribution}c indicates that the distributions of the different collection points occupy different parts of the modified Zipf--Mandelbrot model parameter space. Figures~\ref{fig:ModelFitsA}--\ref{fig:ModelFitsJ} show the measured and modeled differential cumulative distributions for the source fan-out, source packets, destination fan-in, destination packets, and link packets for all the collected data.
\begin{figure}
\vspace*{-0.5cm}
\hspace*{-1cm}
\includegraphics[height = 19cm, width=1.1\columnwidth]{chapters/chaptercy/figs/NetworkDistribution.pdf}
\vspace*{-1cm}
\caption{{\bf Streaming network traffic quantities, distributions, and model fits.} ({a}) Internet traffic streams of $N_V$ valid packets are divided into a variety of quantities for analysis. ({b}) A selection of 5 of the 350 measured differential cumulative probabilities spanning different locations, dates, and packet windows. Blue circles are measured data with $\pm$1-$\sigma$ error bars. Black lines are the best-fit modified Zipf--Mandelbrot models with parameters $\alpha$ and $\delta$. Red dots highlight the large contribution of leaf nodes and isolated links. ({c}) Model fit parameters for all 350 measured probability distributions.}
\label{fig:NetworkDistribution}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitA.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Tokyo 2015 Dec 02.
}
\label{fig:ModelFitsA}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitB.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Tokyo 2017 Apr 12.
}
\label{fig:ModelFitsB}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitC.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago A 2016 Jan 21.
}
\label{fig:ModelFitsC}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitD.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago A 2016 Feb 18.
}
\label{fig:ModelFitsD}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitE.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago A 2016 Mar 17.
}
\label{fig:ModelFitsE}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitF.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago A 2016 Apr 06.
}
\label{fig:ModelFitsF}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitG.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago B 2016 Jan 21.
}
\label{fig:ModelFitsG}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitH.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago B 2016 Feb 18.
}
\label{fig:ModelFitsH}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitI.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago B 2016 Mar 17.
}
\label{fig:ModelFitsI}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.9cm}
\includegraphics[clip, trim=2cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/ModelFitJ.pdf}
\caption{{\bf Measured differential cumulative probabilities.} Blue circles with $\pm$1-$\sigma$ error bars, along with their best-fit modified Zipf--Mandelbrot models (black line) and parameters $\alpha$ and $\delta$ performed for Chicago B 2016 Apr 06.
}
\label{fig:ModelFitsJ}
\end{figure}
Figure~\ref{fig:NetworkTopology}b shows the average relative fractions of sources, total packets, total links, and the number of destinations in each of the five topologies for the ten data sets, and seven valid packet windows: $N_V = 10^5$, $3{\times}10^5$, $10^6$, $3{\times}10^6$, $10^7$, $3{\times}10^7$, $10^8$. The four projections in Figure~\ref{fig:NetworkTopology}b are chosen from Figures~\ref{fig:NetTopoA}--\ref{fig:NetTopD} to highlight the differences in the collection locations. The distinct regions in the various projections shown in Figure~\ref{fig:NetworkTopology}b indicate that underlying topological differences are present in the data. The Tokyo collections have much larger supernode leaf components than the Chicago collections. The Chicago collections have much larger core and core leaves components than the Tokyo collections. Chicago A consistently has fewer isolated links than Chicago B. Comparing the modified Zipf--Mandelbrot model parameters in Figure~\ref{fig:NetworkDistribution}c and underlying topologies in Figure~\ref{fig:NetworkTopology}b suggests that the model parameters are a more compact way to distinguish the network traffic.
\begin{figure}
\vspace*{-2cm}
\hspace*{-2cm}
\includegraphics[width=1.2\columnwidth]{chapters/chaptercy/figs/NetworkTopology.pdf}
\caption{{\bf Distribution of traffic among network topologies.} ({a}) Internet traffic forms networks consisting of a variety of topologies: isolated links, supernode leaves connected to a supernode, and densely connected core(s) with corresponding core leaves. ({b}) A selection of four projections showing the fraction of data in various underlying topologies. Horizontal and vertical axes are the corresponding fraction of the sources, links, total packets, and destinations that are in various topologies for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$). These data reveal the differences in the network traffic topologies in the data collected in Tokyo (dominated by supernode leaves), Chicago A (dominated by core leaves), and Chicago B (between Tokyo and Chicago A).}
\label{fig:NetworkTopology}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.2cm}
\includegraphics[clip, trim=0.9cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/NetTopoA.pdf}
\caption{{\bf Fraction of data in different network topologies.} Fraction of data in isolated links, core leaves, and supernode leaves versus the fraction of data in the core for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:NetTopoA}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.2cm}
\includegraphics[clip, trim=0.9cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/NetTopoB.pdf}
\caption{{\bf Fraction of data in different network topologies.} Fraction of data in the core, core leaves, and supernode leaves versus the fraction of data in isolated links for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:NetTopoB}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.2cm}
\includegraphics[clip, trim=0.9cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/NetTopoC.pdf}
\caption{{\bf Fraction of data in different network topologies.} Fraction of data in the core, isolated links, and supernode leaves versus the fraction of data in core leaves for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:NetTopC}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\hspace*{-1cm}
\vspace*{-.2cm}
\includegraphics[clip, trim=0.9cm 0cm 0cm 0cm, angle=-90,origin=c,width=\columnwidth]{chapters/chaptercy/figs/NetTopoD.pdf}
\caption{{\bf Fraction of data in different network topologies.} Fraction of data in the core, isolated links, and core leaves versus the fraction of data in supernode leaves for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:NetTopD}
\end{figure}
Figures~\ref{fig:NetworkDistribution}c and \ref{fig:NetworkTopology}b indicate that different collection points produce different model parameters $\alpha$ and $\delta$ and that these collection points also have different underlying topologies. Figure~\ref{fig:TopoModel} connects the model fits and topology observations by plotting the topology fraction as a function of the model leaf parameter $1/(1+\delta)^\alpha$ which corresponds to the relative strength of the distribution at $p(d=1)$
\begin{equation}\label{eq:ZipfMandelbrot}
1/(1 + \delta)^\alpha \propto p(d=1;\alpha,\delta)
\end{equation}
The correlations revealed in Figure~\ref{fig:TopoModel} suggest that the model leaf parameter strongly correlates with the fraction of the traffic in different underlying network topologies and is a potentially new and beneficial way to characterize networks. Figure~\ref{fig:TopoModel} indicates that the fraction of sources, links, and destinations in the core shrinks as the relative importance of the leaf parameter in the source fan-out and destination fan-in increases. In other words, more source and destination leaves mean a smaller core. Likewise, the fraction of links and total packets in the supernode leaves grows as the leaf parameter in the link packets and source packets increases. Interestingly, the fraction of sources in the core leaves and isolated links decreases as the leaf parameter in the source and destination packets increases indicating a shift of sources away from the core leaves and isolated links into supernode leaves. Thus, the modified Zipf--Mandelbrot model and its leaf parameter provide a direct connection with the network topology, underscoring the value of having accurate model fits across the entire range of values and in particular for $d=1$.
\begin{figure}
\vspace*{-1cm}
\hspace*{-2cm}
\includegraphics[width=1.3\columnwidth]{chapters/chaptercy/figs/TopoModel.pdf}
\caption{{\bf Topology versus model leaf parameter.} Network topology is highly correlated with the modified Zipf--Mandelbrot model leaf parameter $1/(1+\delta)^\alpha$. A selection of eight projections showing the fraction of data in various underlying topologies. Vertical axis is the corresponding fraction of the sources, links, total packets, and destinations that are in various topologies. Horizontal axis is the value of the model parameter taken from either the source packet, source fan-out, link packet, destination fan-in, or destination packet fits. Data points are for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).}
\label{fig:TopoModel}
\end{figure}
Figures~\ref{fig:TopoModelA}--\ref{fig:TopoModelE} show the fraction of the sources, links, total packets, and destinations in each of the measured topologies for all the locations as a function of the modified Zipf--Mandelbrot leaf parameter computed from the model fits of the source packets, source fan-out, link packets, destination fan-in, and destination packets taken from Figures~\ref{fig:ModelFitsA}--\ref{fig:ModelFitsJ}.
\begin{figure}[bh]
\vspace*{-0.5cm}
\includegraphics[clip, trim=1.4cm 0cm 0cm 0cm, angle=-90,origin=c,height=0.75\textheight,width=\columnwidth, ]{chapters/chaptercy/figs/TopoModelA.pdf}
\caption{{\bf Topology versus modified Zipf--Mandelbrot model leaf parameter.} Fraction of data in the core, isolated links, core leaves, and supernode leaves versus the model leaf parameter computed from the source packet modified Zipf--Mandelbrot model fits for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:TopoModelA}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\includegraphics[clip, trim=1.4cm 0cm 0cm 0cm, angle=-90,origin=c,height=0.75\textheight,width=\columnwidth, ]{chapters/chaptercy/figs/TopoModelB.pdf}
\caption{{\bf Topology versus modified Zipf--Mandelbrot model leaf parameter.} Fraction of data in the core, isolated links, core leaves, and supernode leaves versus the model leaf parameter computed from the source fan-out modified Zipf--Mandelbrot model fits for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:TopoModelB}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\includegraphics[clip, trim=1.4cm 0cm 0cm 0cm, angle=-90,origin=c,height=0.75\textheight,width=\columnwidth, ]{chapters/chaptercy/figs/TopoModelC.pdf}
\caption{{\bf Topology versus modified Zipf--Mandelbrot model leaf parameter.} Fraction of data in the core, isolated links, core leaves, and supernode leaves versus the model leaf parameter computed from the link packets modified Zipf--Mandelbrot model fits for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:TopoModelC}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\includegraphics[clip, trim=1.4cm 0cm 0cm 0cm, angle=-90,origin=c,height=0.75\textheight,width=\columnwidth, ]{chapters/chaptercy/figs/TopoModelD.pdf}
\caption{{\bf Topology versus modified Zipf--Mandelbrot model leaf parameter.} Fraction of data in the core, isolated links, core leaves, and supernode leaves versus the model leaf parameter computed from the destination fan-out modified Zipf--Mandelbrot model fits for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:TopoModelD}
\end{figure}
\begin{figure}[bh]
\vspace*{-0.5cm}
\includegraphics[clip, trim=1.4cm 0cm 0cm 0cm, angle=-90,origin=c,height=0.75\textheight,width=\columnwidth, ]{chapters/chaptercy/figs/TopoModelE.pdf}
\caption{{\bf Topology versus modified Zipf--Mandelbrot model leaf parameter.} Fraction of data in the core, isolated links, core leaves, and supernode leaves versus the model leaf parameter computed from the destination packets modified Zipf--Mandelbrot model fits for each location, time, and seven packet windows ($N_V = 10^5, \ldots, 10^8$).
}
\label{fig:TopoModelE}
\end{figure}
\section{Discussion}\label{Discussion}
Measurements of Internet traffic are useful for informing policy, identifying and preventing outages, defeating attacks, planning for future loads, and protecting the Domain Name System \cite{clark20179th}.
On a given day, millions of IPs are engaged in scanning behavior. Our improved models can aid cybersecurity analysts in determining which of these IPs are nefarious \cite{yu2012predicted}, the distribution of attacks in particular critical sectors \cite{husak2018assessing}, identifying spamming behavior \cite{fonseca2016measuring}, how to vaccinate against computer viruses \cite{balthrop2004technological}, obscuring web sources \cite{javed2015measurement}, identifying significant flow aggregates in traffic \cite{cho2017recursive}, and sources of rumors \cite{paluch2018fast}.
The results presented here have a number of potential practical applications for Internet stakeholders. The methods presented of collecting, filtering, computing, and binning the data to produce accurate measurements of a variety of network quantities are generally applicable to Internet measurements and have the potential to produce more accurate measures of these quantities. The accurate fits of the two-parameter modified Zipf--Mandelbrot distribution offer all the usual benefits of low-parameter models: measuring parameters with far less data, accurate predictions of network quantities based on a few parameters, observing changes in the underlying distribution, and using modeled distributions to detect anomalies in the data.
From a scientific perspective, improved knowledge of how Internet traffic flows can inform our understanding of how economics, topology, and demand shape the Internet over time. As with all scientific disciplines, the ability of theoreticians to develop and test theories of the Internet and network phenomena is bounded by the scale and accuracy of measured phenomena \cite{adamic2000power,bohman2009emergence,stumpf2012critical,virkar2014power}. The connections among dynamic evolution \cite{bianconi2001competition}, network topology \cite{mucha2010community}\cite{boccaletti2014structure,lu2016vital}, network robustness \cite{li2017fundamental}, controllability \cite{liu2016control}, community formation \cite{perc2017statistical}, and spreading phenomena \cite{holme2015modern} have emerged in many contexts \cite{barabasi2009scale,wang2016statistical,koliba2018governance}. Many first-principles theories for Internet and network phenomena have been proposed, such as Poisson models \cite{paxson1995wide}, fractional Brownian motion \cite{willinger1997self}, preferential attachment \cite{barabasi1999emergence,albert1999internet}\cite{newman2001clustering,sheridan2018preferential}, statistical mechanics \cite{albert2002statistical}, percolation \cite{achlioptas2009explosive}, hyperbolic geometries \cite{krioukov2009curvature,krioukov2010hyperbolic}, non-global greedy routing \cite{boguna2009navigating,boguna2009navigability,boguna2010sustaining}, interacting particle systems \cite{antonopoulos2018opinion}, higher-order organization of complex networks from graph motifs \cite{benson2016higher}, and minimum control energy \cite{lindmark2018minimum}. All of these models require data to test them. In contrast to previous network models that have principally been based on data obtained from network crawls from a variety of start points on the network, our network traffic data are collected from observations of network streams. Both viewpoints provide important network observations. Observations of a network stream provide complementary data on network dynamics and highlight the contribution of leaves and isolated edges, which are less sampled in network crawls.
The aggregated data sets our teams have collected provide a unique window into these questions. The nonlinear fitting techniques described are a novel approach to fitting power-law data and have potential applications to power-law networks in diverse domains. The model fit parameters present new opportunities to connect the distributions to underlying theoretical models of networks. That the model fit parameters distinguish the different collection points and are reflective of different network topologies in the data at these points suggests a deeper underlying connection between the models and the network topologies.
\section{Conclusions}\label{Conclusions}
Our society critically depends on the Internet for our professional, personal, and political lives. This dependence has rapidly grown much stronger than our comprehension of its underlying structure, performance limits, dynamics, and evolution. The fundamental characteristics of the Internet are perpetually challenging to research and analyze, and we must admit we know little about what keeps the system stable. As a result, researchers and policymakers deal with a multi-trillion-dollar ecosystem essentially in the dark, and agencies charged with infrastructure protection have little situational awareness regarding global dynamics and operational threats. This paper has presented an analysis of the largest publicly available collection of Internet traffic consisting of 50 billion packets and reveals a new phenomenon: the importance of otherwise unseen leaf nodes and isolated links in Internet traffic. Our analysis further shows that a two-parameter modified Zipf--Mandelbrot distribution accurately describes a wide variety of source/destination statistics on moving sample windows ranging from 100{,}000 to 100{,}000{,}000 packets over collections that span years and continents. The measured model parameters distinguish different network streams, and the model leaf parameter strongly correlates with the fraction of the traffic in different underlying network topologies. These results represent a significant improvement in Internet modeling accuracy, improve our understanding of the Internet, and show the importance of stream sampling for measuring network phenomena.
\let\cleardoublepage\clearpage
\section*{Acknowledgements}
The authors wish to acknowledge the following individuals for their contributions and support: Shohei Araki, William Arcand, David Bestor, William Bergeron, Bob Bond, Paul Burkhardt, Chansup Byun, Cary Conrad, Alan Edelman, Sterling Foster, Bo Hu, Matthew Hubbell, Micheal Houle, Micheal Jones, Anne Klein, Charles Leiserson, Dave Martinez, Mimi McClure, Julie Mullen, Steve Pritchard, Andrew Prout, Albert Reuther, Antonio Rosa, Victor Roytburd, Siddharth Samsi, Koichi Suzuki, Kenji Takahashi, Michael Wright, Charles Yee, and Michitoshi Yoshida.
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\section{Introduction}
The era of gravitational wave (GW) astronomy has begun with the first detection of a binary black hole (BBH) merger in 2015 \citep{2016PhRvL.116f1102A}. There have been $\sim90$ BBH systems discovered with $\sim3$ years observation in the LIGO/Virgo O1/O2/O3 runs (GWTC-1\&2 catalogues: \cite{2019PhRvX...9c1040A, 2020arXiv201014527A}). The number is catching up on that of the discovered Galactic black holes (BH) in X-ray binaries during the last half century \citep{2016A&A...587A..61C}. LIGO and Virgo are being upgraded towards their designed sensitivity, and new GW observatories such as The Kamioka Gravitational Wave Detector (KAGRA, \citealt{2019NatAs...3...35K}) and LIGO-India \citep{indigo} will join the network in the near future. Furthermore, there are plans for the next generation of GW observatories such as the Einstein Telescope (ET, \citealt{2010CQGra..27s4002P}) and the Cosmic Explore (CE, \citep{2019BAAS...51g..35R}), which will push forward the detecting horizon significantly. With the expectation of the exploding size of the BBH population, the community has been actively studying what science can be extracted from GW observations. Applications include e.g. the cosmic merger rate density of BBH \citep[e.g.][]{2019ApJ...886L...1V}, the mass distribution of stellar mass black hole \citep[e.g.][]{2017PhRvL.119m1301K,2017PhRvD..95j3010K}, the rate of stellar nucleosynthesis \citep[e.g.][]{2020arXiv200606678F} and constraints on Cosmological parameters \citep{2019ApJ...883L..42F}. Simulating a catalogue of observations is the crucial step in the above mentioned work. In all the previous literature, the simulation is performed for specific detectors and observation duration and their catalogues are difficult to extend to other usages. Therefore, we are building a set of simulating tools which has the flexibility to different detectors, observation duration, source populations and cosmological models. The \texttt{GW-Universe Toolbox}\xspace is such a tool set that simulates observations of the GW Universe with different types of GW detectors including Earth-based/Space-borne laser interferometers and Pulsar timing arrays \citep{2021arXiv210613662Y}. It is accessible as a website\footnote{\url{gw-universe.org}}, or run locally as a Python package.
Here we exhibit one application of the \texttt{GW-Universe Toolbox}\xspace by showing how the cosmic merger rate and mass function of BH can be constrained by GW observations using advanced LIGO (aLIGO) and ET. The cosmic merger rate density, or the volumetric merger rate as function of redshift $\mathcal{R}(z)$, provides valuable information on the history of the star formation rate (SFR), which is crucial for understanding the evolution of the Universe, e.g, the chemical enrichment history, and the formation and evolution of galaxies \citep{2014ARA&A..52..415M}. The SFR as function of redshift has been studied for decades mainly with UV/optical/infrared observation of galaxies in different redshift \citep[see][]{2012ARA&A..50..531K,2014ARA&A..52..415M}. Such electromagnetic (EM) wave probes are limited by dust extinction and the completeness of samples at high redshifts is difficult to estimate \citep[e.g.][]{2019A&ARv..27....3M}. Moreover, the connection between the direct observable and the SFR history is model dependent on multiple layers \citep[see e.g.][]{2021MNRAS.508.4994C}. GW provide a unique new probe which suffer less from the above mentioned limitations.
Moreover, $\mathcal{R}(z)$ depends also on the distribution of delay times, i.e., the time lag between the formation of the binary massive stars and the merger of the two compact objects \cite[e.g.][]{2010ApJ...715L.138B,2018MNRAS.481.1908K}. Therefore, a combination of the information of the SFR from EM observations and $\mathcal{R}(z)$ from GW can improve our knowledge of the delay time distribution. $P(\tau)$ together with mass function $p(m_1)$ of BBH helps us to better understand the physics in massive binary evolution, e.g., in the common envelope scenario \cite[e.g.][]{2016Natur.534..512B,2018MNRAS.481.1908K}. Furthermore, the mass function of black holes (BH) implied from GW observation can also be used to search for or place limit on primordial BHs \citep{2017PhRvL.119m1301K,2017PhRvD..95j3010K}.
There are several previous works that studied the prospect of using simulated GW observations to study $\mathcal{R}(z)$ and $p(m_1)$ of BBH. \cite{2019ApJ...886L...1V} shows that with three months observation with a third-generation detector network, $\mathcal{R}(z)$ can be constrained to high precision up to $z\sim 10$. In their work the authors did not take into account the signal-to-noise (SNR) limits of detectors and instead made the assumption that all BBH mergers in the Universe are in the detected catalogue. \cite{2017PhRvD..95j3010K} studied how the observation with aLIGO can be used to give a constraint on $p(m_1)$. They assumed a certain parameterization of $p(m_1)$, and use an approximated method to obtain the covariance of the parameters based on a simulated catalogue of aLIGO observations. Recently, \cite{2021arXiv211204058S} simulated observation with ET on compact object binary mergers, taking into account the effect of the rotation of the Earth. They found that $\mathcal{R}(z)$ and mass distribution can be so well constrained from ET's observation, that mergers from different populations can be distinguished.
In this work, we simulate the observed catalogues of BBH mergers in ten years of aLIGO observation and one month ET observation using the \texttt{GW-Universe Toolbox}\xspace. We also simulate
the observation from a detector that is half as sensitive as the ET in design which represents the early phase of ET. The paper is organised as follows: In section \ref{sec:thoery}, we summarize the method used by the \texttt{GW-Universe Toolbox}\xspace to simulate the catalogue of BBH mergers; In section \ref{sec:constraint}, we use the synthetic catalogues to set constraints on the cosmic merger rate and mass function of the population. We employ two ways to constrain $\mathcal{R}(z)$ along with the mass function of the primary BH. The first assumes parametric forms of $\mathcal{R}(z)$ and the mass function and uses Bayesian inference of their parameters. The other method does not assume a parametric formula and uses weighted Kernel density estimators. The results are presented and compared. We conclude and discuss in section \ref{sec:candd}.
\section{Generating synthetic catalogues}\label{sec:thoery}
\begin{comment}
\subsection{$\mathcal{D}(\mathbf{\Theta})$: the detectability determined by the detector and the waveform of sources}
In the following section two Cartesian frames are introduced. The first one $(\vv{\bm{x}},\vv{\bm{y}},\vv{\bm{z}})$ is attached to the source, in which the $\vv{\bm{z}}$ is towards the observer, $\vv{\bm{x}}$ is parallel to the line of notes of the binary orbit. The second one $(\vv{\bm{X}},\vv{\bm{Y}},\vv{\bm{Z}})$ is attached to the detector, and $\vv{\bm{Z}}$ is vertical to the detector plane. $(\theta,\varphi,\psi)$ are the three Euler angles between the two frames. $\theta$ is the polar angle, $\varphi$ is the azimuth angle, and $\psi$ is the polarization angle.
A series of incoming GW will induce differential displacement on an interferometer as:
\begin{equation}
h=F_+h_++F_{\times}h_{\times},
\end{equation}
where $h_+$ and $h_{\times}$ are the GW strain of the plus and cross modes, which are defined in the $(\vv{\bm{x}},\vv{\bm{y}},\vv{\bm{z}})$ frame. In the inspiral phase which carries most of the power, the frequency domain waveform can be written as:
\begin{align}
h_+(f)&\approx\mathcal{C}\frac{1+\cos^2\iota}{2}f^{-7/6}e^{-i\Psi(f)}\nonumber\\
h_{\times}(f)&\approx\mathcal{C}\cos\iota f^{-7/6}e^{-i(\Psi(f)+\pi/2)}.
\end{align}
In the above equations, $\Psi(f)$ is the phase of the waveform, and the constan
\begin{equation}
\mathcal{C}=\frac{1}{2}\sqrt{\frac{5}{6}}\frac{\left(G\mathcal{M}\right)^{5/6}}{c^{3/2}\pi^{2/3}D_{\rm{L}}},
\end{equation}
where $\mathcal{M}$ is the red-shifted chirp mass of the binary, $D_{\rm{L}}$ is the luminosity distance. The detector response is:
\begin{equation}
h(f)=\mathcal{C}\sqrt{\left(\frac{1+\cos^2\iota}{2}\right)^2F_+^2+\cos\iota^2F_\times^2}f^{-7/6}e^{-i(\Psi(f)+\phi_{\rm{p}})},
\end{equation}
where $\phi_{\rm p}$ is the phase determined by the polarization of GW.
The linear coefficients $F_+$ and $F_{\times}$ are the antenna pattern, which are determined by the shape of the detectors. For LIGO-like interferometers with $90^\circ$ arm-separation, the antenna pattern are:
\begin{align}
F_+&=\frac{1}{2}(1+\cos\theta^2)\cos2\varphi\cos2\psi+\cos\theta\sin2\varphi\sin2\psi \nonumber\\
F_\times&=\frac{1}{2}(1+\cos\theta^2)\cos2\varphi\sin2\psi+\cos\theta\sin2\varphi\cos2\psi,
\end{align}
and for ET-like interferometers with $60^\circ$ arm-separation \citep{2012PhRvD..86l2001R}:,
\begin{align}
F_+&=-\frac{\sqrt{3}}{2}\left(\frac{1}{2}(1+\cos\theta^2)\sin2\varphi\cos2\psi+\cos\theta\cos2\varphi\sin2\psi\right) \nonumber\\
F_\times&=\frac{\sqrt{3}}{2}\left(\frac{1}{2}(1+\cos\theta^2)\sin2\varphi\sin2\psi-\cos\theta\cos2\varphi\cos2\psi\right).
\label{eqn:et-at}
\end{align}
The ET will have three nested interferometers, with $60^\circ$ rotates from each other. The antenna pattern for each interferometers are $F_{i,+,\times}(\theta,\varphi,\psi)=F_{0,+,\times}(\theta,\varphi+2/3i\pi,\psi)$, where $i=0,1,2$ is the index of the interferometers, and $F_{0,+,\times}$ are those in equation (\ref{eqn:et-at}).
Denoting the SNR of the merger as $\rho$, which is determined with the match-filtering technique:
\begin{equation}
\rho^2=(h_{\rm{t}}^\star|h),
\end{equation}
where $h_{\rm{t}}$ is the template used in the match-filtering, and $^\star$ denotes its complex conjunction. In the above equation, the inner production $(...|...)$ is defined as:
\begin{equation}
(a|b)=4\Re{\int\frac{a^\star b}{S_{\rm{n}}(f)}df}.
\end{equation}
where $S_{\rm{n}}$ is the noise power spectrum. For the designed $S_{\rm{n}}$ of aLIGO and ET, we obtain them from \footnote{\url{https://dcc.ligo.org/LIGO-T1800044/public} \\ \url{http://www.et-gw.eu/index.php/etsensitivities}}. We multiply $S_{\rm{n}}$ of the ET with a factor 10.89 to obtain a noise curve with is in the half way between the designed aLIGO and ET, which we refer to as the ``half-ET". The half-ET represent our expectation on the performance of the ET in its early phase. In figure \ref{fig:Sn} , we plot the $\sqrt{S_{\rm{n}}}$ as function of frequency for aLIGO, ET at their designed sensitivity, along with that of the half-ET.
In the most optimized case, when the template $h_{\rm{t}}$ perfectly matched with the signal $h$, the SNR is:
\begin{equation}
\rho^2=(h|h),
\end{equation}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Sn.pdf}
\caption{The squared-root of the noise power spectrum density of the aLIGO (blue) and ET (orange) at their designed sensitivity. The data of the plots are obtained from \url{https://dcc.ligo.org/LIGO-T1800044/public} and \url{http://www.et-gw.eu/index.php/etsensitivities}}
\label{fig:Sn}
\end{figure}
Therefore,
\begin{align}
\rho^2&=4\int\frac{|h^2(f)|}{S_{\rm{n}}(f)}df\nonumber\\
&=4\mathcal{C}^2\left(\left(\frac{1+\cos^2\iota}{2}\right)^2F_+^2+\cos\iota^2F_\times^2\right)\int^{f_{\rm cut}}\frac{f^{-7/3}}{S_{\rm{n}}(f)}df,
\end{align}
where in the above equation, $\mathcal{C}$ is a function of the chirp mass and luminosity distance as defined above. The high frequency cut in the integral is where the inspiral phase ends and the merger begins, which is in general a function of the masses and spins. In the Newtonian approximation, $f_{\rm{cut}}$ is just the Keplerian orbital frequency around the inner most circular orbit. Here we employ the transition frequency from insprial to merger given in \cite{2011PhRvL.106x1101A} as $f_{\rm cut}$. Therefore, $\rho$ is a functional of the waveform $\mathcal{W}$, the antenna patterns $F$ and the detector noise spectrum $S_{\rm{n}}$: $\rho[\mathcal{W}, F, S_{\rm{n}}]$. For ET, the overall $\rho^2$ is the sum of $\rho^2_i$ of the three individual interferometers. We can see that $\Psi(f)$ does not enter when calculating the optimized SNR. However, the phase of the waveform is crucial when estimating the uncertainties. We employ the phase of waveform for non-precessing spinning binaries \citep{2011PhRvL.106x1101A}, and the relevant intrinsic parameters are the masses $m_1$, $m_2$ and the effective spin $\chi$.
We denote intrinsic parameters together with the red-shift of the sources as $\mathbf{\Theta}=(m_1,m_2,\chi,z)$.
For a given detector and a predetermined detection SNR threshold $\rho_\star$, the expected distribution of parameters of the compact binary merger to be detected is:
\begin{equation}
N_{\rm{D}}(\mathbf{\Theta},\theta,\varphi,\psi,\iota)=\frac{T}{1+z}\frac{dV_{\rm{c}}}{4\pi dz}\dot{n}(\mathbf{\Theta},\theta,\varphi,\iota,\psi)\mathcal{H}(\rho^2-\rho^2_\star).\label{eqn:eight}
\end{equation}
In the above equation, $T$ is the time span of observation, $dV_{\rm{c}}/dz$ is the differential cosmic comoving volume (volume per redshift), $\dot{n}$ is the differential cosmic merger rate density of compact binary mergers, $\mathcal{H}$ is the Heaviside step function and $\rho_\star$ is the predefined SNR threshold of detection. Suppose that the spatial distribution of the sources, the viewing angles are all isotropic. As a result, $\dot{n}$ is independent of the angles:
\begin{equation}
\dot{n}(\mathbf{\Theta})=4\pi\dot{n}(\mathbf{\Theta},\theta,\varphi,\psi,\iota).
\end{equation}
The integration of both sides of equation (\ref{eqn:eight}) gives:
\begin{equation}
N_{\rm{D}}(\mathbf{\Theta})=\frac{T}{1+z}\frac{dV_{\rm{c}}}{dz}\dot{n}(\mathbf{\Theta})\mathcal{D}(\mathbf{\Theta}),\label{eqn:detectables}
\end{equation}
where
\begin{equation}
\mathcal{D}(\mathbf{\Theta})=\oiint d\Omega d\Omega^\prime\mathcal{H}(\rho^2-\rho^2_\star)/(4\pi)^2.
\end{equation}
In the above equation, $d\Omega=\sin\theta d\theta d\varphi$ and $d\Omega^\prime=\sin\iota d\iota d\psi$. As shown above, $\mathcal{D}$ is determined with the antenna pattern, the noise spectrum of the telescope and the waveform of the GW source. Given a GW observatory and a certain type of source, $\mathcal{D}(\mathbf{\Theta})$ can be thus determined. The distribution of parameters of the sources to be detected can be determined with equation (\ref{eqn:detectables}).
\end{comment}
\subsection{method}
We use the \texttt{GW-Universe Toolbox}\xspace to simulate observations on BBH with three different detecters, namely, aLIGO (designed sensitivity), ET and ``half-ET". The noise power spectrum $S_{\rm{n}}$ for aLIGO and ET are obtained from \footnote{\url{https://dcc.ligo.org/LIGO-T1800044/public} \\ \url{http://www.et-gw.eu/index.php/etsensitivities}}. The ``half-ET" represents our expectation on the performance of the ET in its early phase.
In figure \ref{fig:Sn}, we plot the $\sqrt{S_{\rm{n}}}$ as function of frequency for aLIGO, ET at their designed sensitivity, along with that of the half-ET. For its $S_{\rm{n}}$, we multiply $S_{\rm{n}}$ of the ET with a factor 10.89 to obtain a noise curve with is in the half way between the designed aLIGO and ET.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Sn.pdf}
\caption{The squared-root of the noise power spectrum density of the aLIGO (blue) and ET (orange) at their designed sensitivity. The data of the plots are obtained from \url{https://dcc.ligo.org/LIGO-T1800044/public} and \url{http://www.et-gw.eu/index.php/etsensitivities}}
\label{fig:Sn}
\end{figure}
Here we will give a brief summary of the method. For details, we refer to \cite{2021arXiv210613662Y}. We first calculate a function $\mathcal{D}(z, m_1, m_2)$, which represents the detectability of a BBH with redshift $z$, primary and secondary masses $m_1$, $m_2$. $\mathcal{D}(z, m_1, m_2)$ depends on the antenna pattern and $S_{\rm{n}}$ of the detector, and the user designated signal-to-noise (SNR) threshold. The other ingredient is the merger rate density distribution of BBH in the Universe, $\dot{n}(z, m_1, m_2)$. In the \texttt{GW-Universe Toolbox}\xspace, we integrate two parameterized function forms for $\dot{n}(z, m_1, m_2)$, namely pop-A and pop-B (see \citealt{2021arXiv210613662Y} for the description on the population models). For more sophisticated $\dot{n}(z, m_1, m_2)$ from population synthesis simulations, see \cite{2002ApJ...572..407B,2017MNRAS.472.2422M,2018MNRAS.479.4391M,2021arXiv210912119A,2021arXiv210804250B,2021arXiv210906222M,2021arXiv211001634V,2021arXiv211204058S}.
The distribution of detectable sources, which we denote as $N(z,m_1,m_2)$ can be calculated with:
\begin{equation}
N_{\rm{D}}(\mathbf{\Theta})=\frac{T}{1+z}\frac{dV_{\rm{c}}}{dz}\dot{n}(\mathbf{\Theta})\mathcal{D}(\mathbf{\Theta}),\label{eqn:detectables}
\end{equation}
where $\mathbf{\Theta}=(z,m_1,m_2)$. An integration of $N_{\rm{D}}(\mathbf{\Theta})$ gives the expectation of total detection number:
\begin{equation}
N_{\rm{tot,exp}}=\int d\mathbf{\Theta}N_D(\mathbf{\Theta}).
\end{equation}
The actual total number in a realization of simulated catalogue is a Poisson random around this expectation. The synthetic catalogue of GW events is drawn from $N_D(\mathbf{\Theta})$ with a Markov-Chain Monte-Carlo (MCMC) algorithm.
We can also give a rough estimation of the uncertainties on the intrinsic parameters of each event in the synthetic catalogue using the Fisher Information Matrix (FIM).
Previous studies ({\it e.g.,} \cite{2013PhRvD..88h4013R}) found that the Fisher information matrix will sometimes severely overestimated the uncertainties, especially when the SNR is close to the threshold. We implement a correction that if the relative uncertainty of masses calculated with FIM is larger than $20\%$, $\delta m_{1,2}=0.2m_{1,2}$ is applied instead.
The uncertainty of the redshift is in correlation with those of other external parameters, such as the localization error. The actual localization uncertainties are largely determined by the triangulation error of the detector network, which can not be estimated with the current version of \texttt{GW-Universe Toolbox}\xspace\ yet. As a result, we do not have a reliable way that accurately estimates the errors on redshift. As a {\it ad hoc} method, we assign $\delta z=0.5z$ as a typical representative for the uncertainties of aLIGO, and $\delta z=0.017z+0.012$ as a typical representative for the uncertainties of ET \citep{2017PhRvD..95f4052V}. For half-ET, we assign $\delta z=0.1z+0.01$.
\begin{comment}
\subsection{$\dot{n}(\mathbf{\Theta})$: the differential merger rate density}\label{pop_model}
Knowing the detectability $\mathcal{D}(\mathbf{\Theta})$, the next piece of $N_{\rm D}(\mathbf{\Theta})$ is the differential merger rate density of the BBH population in the Universe $\dot{n}(\mathbf{\Theta})$. There are more than one equivalent choices of the intrinsic parameters, for instance one can use individual masses ($m_1, m_2$) or equivalently the mass of the primary BH and the mass ratio $q\equiv m_1/m_2$. The transformation of the differential merger rate density under different choice of $\Theta$ is:
\begin{equation}
\dot{n}(\mathbf{\Theta})=\dot{n}(\mathbf{\Theta}^\prime)\frac{d\mathbf{\Theta}^\prime}{d\mathbf{\Theta}},
\end{equation}
where $d\mathbf{\Theta}^\prime/d\mathbf{\Theta}$ is the Jaccobian determinant between $\mathbf{\Theta}$ and $\mathbf{\Theta}^\prime$. For $\mathbf{\Theta}^\prime=(m_1,q,\chi,z)$, the Jaccobian determinant is:
\begin{equation}
\frac{d\mathbf{\Theta}^\prime}{d\mathbf{\Theta}}=\begin{vmatrix}
\partial m_1/\partial m_1 & \partial m_1/\partial m_2\\
\partial q/\partial m_1 & \partial q/\partial m_2
\end{vmatrix}=\frac{1}{m_1}.
\end{equation}
Therefore,
\begin{equation}
\dot{n}(\mathbf{\Theta})=\dot{n}(m_1,q,\chi,z)/m_1.
\end{equation}
In this study, we assume that the variables on $\dot{n}((m_1,q,\chi,z)$ are can be separated, i.e.,
\begin{equation}
\dot{n}(m_1,q,\chi,z)=\mathcal{R}(z)p(m_1)\pi(q)P(\chi),
\end{equation}
where $p(m_1)$ is the mass function of the primary black hole, $\pi(q)$ and $P(\chi)$ are the distribution of $q$ and $\chi$ respectively. The \texttt{GWToolbox} can work with more general form of $\dot{n}(\mathbf{\Theta})$.
The overall normalization factor $\mathcal{R}$ as function of $z$ is often refer to as the cosmic merger rate density, which we take the parameterization form as in \citep{2019ApJ...886L...1V}:
\begin{equation}
\mathcal{R}(z_m)=\mathcal{R}_{\rm{n}}\int^\infty_{z_m}\psi(z_f)P(z_m|z_f)dz_f,
\end{equation}
where $\psi(z)$ is the Madau-Dickinson star formation rate:
\begin{equation}
\psi(z)=\frac{(1+z)^\alpha}{(1+\frac{1+z}{C})^\beta},
\end{equation}
with $\alpha=2.7$, $\beta=5.6$, $C=2.9$ \citep{2014ARA&A..52..415M},
and $P(z_m|z_f,\tau)$ is the probability that the BBH merges at $z_m$ if the binary is formed at $z_f$, which we refer to as the distribution of delay time with the form \citep{2019ApJ...886L...1V}:
\begin{equation}
P(z_m|z_f,\tau)=\frac{1}{\tau}\exp{\left[-\frac{t_f(z_f)-t_m(z_m)}{\tau}\right]}.
\end{equation}
In the above equation, $t_f$ and $t_m$ are the look back time as function of $z_f$ and $z_m$ respectively.
We give plots of $R(z)$ with different $\tau$ in figure \ref{fig:ndot}.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{ndot.eps}
\caption{The cosmic merger rate with different delay time. The normalization factor $R_{\rm{n}}$ are different for each curves, in order to make the resulted $\mathcal{R}(z)$ to have similar values at $z=0$. }
\label{fig:ndot}
\end{figure}
We assume the mass function $p(m_1)$ has the function form:
\begin{eqnarray}
p(m_1)&\nonumber \\
\propto &
\begin{dcases}
\exp\left(-\frac{c}{m_1-\mu}\right)(m_1-\mu)^{-\gamma}, & m_1\le m_{\text{cut}} \\
0, & m_1>m_{\text{cut}}\\
\end{dcases}
\end{eqnarray}
The distribution of $p(m_1)$ is defined for $m_1>\mu$, which has a power law tail of index $-\gamma$ and a cut-off above $m_{\rm cut}$. When $\gamma=3/2$, the distribution becomes a shifted L\'evy distribution. The peak of $p(m_1)$ occurs at $m_1=c/\gamma+\mu$. See figure \ref{fig:PM1} for an illustration, when $\mu=3$, $\gamma=2.5$, $c=15$.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{PM1.pdf}
\caption{$p(m_1)$ when $\mu=3$, $\gamma=2.5$, $c=15$.}
\label{fig:PM1}
\end{figure}
The normalization of $p(m_1)$ is:
\begin{equation}
\int^{m_{\text{cut}}}_\mu p(m_1)dm_1=c^{1-\gamma}Q(\gamma-1, m_{\text{cut}}),
\end{equation}
where $Q(a,b)$ is the regularized upper incomplete gamma function. We use a uniform distribution between [$q_{\rm cut}$, 1] for $\pi(q)$ and assume $\chi$ follows a Gaussian distribution centered at zero with standard deviation $\sigma$.
\end{comment}
\subsection{Catalogues}\label{cat}
Using the above-mentioned method, we generate synthetic catalogues corresponding to 10-years aLIGO, 1-month ET observations and 1-month observation with half-ET. In figures \ref{fig:figure2}, \ref{fig:figure3} and \ref{fig:figure4}, we plot events in those catalogues, along with the corresponding marginalized $N_D(\mathbf{\Theta})$. The numbers in the catalogues are 2072, 1830 and 889 for the 10-year aLIGO, 1-month ET and 1-month half-ET respectively. The catalogues agree with the theoretical number density, which proves the validity of the MCMC sampling process.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{ligo-z-m1.pdf}
\caption{\textbf{LIGO 10 years' catalogue: } Black dashed lines are contours of $N_D(\mathbf{\Theta})$; Blue points with error bars are 2072 events in the simulated catalogues.}
\label{fig:figure2}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{et-z-m1.pdf}
\caption{\textbf{ET 1 month's catalogue: } Black dashed lines are contours of $N_D(\mathbf{\Theta})$; Blue points with error bars are 1830 events in the simulated catalogues.}
\label{fig:figure3}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{half-et-z-m1.pdf}
\caption{\textbf{half-ET 1 month's catalogue: }Black dashed lines are contours of $N_D(\mathbf{\Theta})$; Blue points with error bars are 889 events in the simulated catalogues.}
\label{fig:figure4}
\end{figure}
The underlying population model for $\dot{n}(z,m_1,m_2)$ is pop-A, with parameters: $R_{\rm{n}}=13$\,Gpc$^{-3}$yr$^{-1}$, $\tau=3$\,Gyr, $\mu=3$, $c=6$, $\gamma=2.5$, $m_{\rm cut}=95\,M_\odot$, $q_{\rm cut}=0.4$, $\sigma=0.1$. Those parameters are chosen because they result in simulated catalogues which are in compatible with the real observations (see \citealt{2021arXiv210613662Y}).
\section{Constraints on $\mathcal{R}(z)$ and $p(m_1)$ from BBH catalogues}\label{sec:constraint}
In this section, we study how well $\mathcal{R}$ and $p(m_1)$ can be constrained with BBH catalogues from observations with 2nd and 3rd generations GW detectors. Two distinct methods will be used.
\subsection{Bayesian method}
In the first method, we use a parameterised form of $\dot{n}(\mathbf{\Theta}|\mathcal{B})$. $\mathcal{B}$ are the parameters describing the population rate density, which are $\mathcal{B}=(R_0,\tau,\mu,c,\gamma,q_{\rm cut},\sigma)$ in our case. $\mathbf{\Theta}^k$ are the physical parameters of the $k$-th source in the catalogue, and we use $\{\mathbf{\Theta}\}$ to denote the whole catalogue.
The posterior probability of $\mathcal{B}$ given an observed catalogue $\{\mathbf{\Theta}\}_{\rm obs}$:
\begin{equation}
p(\mathcal{B}|\{\mathbf{\Theta}\}_{\rm obs})\propto p(\{\mathbf{\Theta}\}_{\rm obs}|\mathcal{B}) p(\mathcal{B}),
\label{eqn:bayesian}
\end{equation}
where $p(\{\mathbf{\Theta}\}_{\rm obs}|\mathcal{B})$ is the likelihood and $p(\mathcal{B})$ is the prior. The subscript ``obs" is to distinguish the observed parameters from the true parameters of the sources. The likelihood of detecting a catalogue $\{\mathbf{\Theta}\}_{\rm obs}$ given the parameters $\mathcal{B}$ is a known result as an inhomogeneous Poisson Process \citep{2014ApJ...795...64F}:
\begin{equation}
p(\{\mathbf{\Theta}\}_{\rm obs}|\mathcal{B})=N!\prod_{k=1}^{N}p(\mathbf{\Theta}^k_{\rm obs}|\mathcal{B})p(N|\lambda(\mathcal{B})),
\label{eqn:lkhood}
\end{equation}
where $p(N|\lambda(\mathcal{B}))$ is the probability of detecting $N$ sources if the expected total number is $\lambda$, which is a Poisson function. The probability of detecting a source $k$ with parameters $\mathbf{\Theta}^k$ is:
\begin{equation}
p(\mathbf{\Theta}^k_{\rm obs}|\mathcal{B})=\int p(\mathbf{\Theta}^k_{\rm obs}|\mathbf{\Theta}^k_{\rm true})p(\mathbf{\Theta}^k_{\rm true}|\mathcal{B})d\mathbf{\Theta}^k_{\rm true}.
\label{eqn:integration}
\end{equation}
We further assume that observational errors are Gaussian, therefore we have $p(\mathbf{\Theta}^k_{\rm obs}|\mathbf{\Theta}^k_{\rm true})=p(\mathbf{\Theta}^k_{\rm true}|\mathbf{\Theta}^k_{\rm obs})$. Taking this symmetric relation into equation \ref{eqn:integration}, we have:
\begin{equation}
p(\mathbf{\Theta}^k_{\rm obs}|\mathcal{B})=\int p(\mathbf{\Theta}^k_{\rm true}|\mathbf{\Theta}^k_{\rm obs})p(\mathbf{\Theta}^k_{\rm true}|\mathcal{B})d\mathbf{\Theta}^k_{\rm true}.
\label{eqn:integration_2}
\end{equation}
The above integration is equivalent to the average of $p(\mathbf{\Theta}^k_{\rm true}|\mathcal{B})$ over all possible $\Theta^k$. Therefore:
\begin{equation}
p(\mathbf{\Theta}_{\rm obs}|\mathcal{B})\approx\left<p(\mathbf{\Theta}^k_{\rm true}|\mathcal{B})\right>,
\label{eqn:MCintegrate}
\end{equation}
where the $\left<\cdots\right>$ denotes an average among a sample of $\mathbf{\Theta}^k$ which is drawn from a multivariate Gaussian characterized by the observation uncertainties. Taking equation (\ref{eqn:MCintegrate}) into equation (\ref{eqn:lkhood}), and noting that $p(\mathbf{\Theta}^k|\mathcal{B})\equiv N_D(\mathbf{\Theta}^k|\mathcal{B})/\lambda(\mathcal{B})$, and $p(N|\lambda(\mathcal{B}))$ is the Poisson distribution:
\begin{equation}
p(\{\mathbf{\Theta}\}_{\rm obs}|\mathcal{B})=\prod_{k=1}^{N}\left<N_D(\mathbf{\Theta}^k|\mathcal{B})\right>\exp(-\lambda(\mathcal{B})).
\end{equation}
For the prior distribution $p(\mathcal{B})$, we use the log normal distributions for $\tau$, $R_0$, $c$, $\gamma$ and $\mu$, which center at the true values and large enough standard deviations. The posterior probability distributions of the parameters are obtained with MCMC sampling from the posterior in equation (\ref{eqn:bayesian}). We keep the nuisance parameters $q_{\rm{cut}}$ and $\sigma$ fixed to the true value, in order to reduce the necessary length of chain. The posterior of the interesting parameters would not be influenced significantly by leaving these free.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{chain.pdf}
\caption{The MCMC chains sampled from the posterior distribution of the population parameters. \textbf{The blue line} is for LIGO-10 years observation; \textbf{The green line} is for half-ET's 1 month observation, and \textbf{the orange line} is for full ET's 1 month observation. The black dashed lines mark the true values of the corresponding parameters, which were used to generate the observed catalogue.}
\label{fig:chain}
\end{figure}
In Figure \ref{fig:chain}, we plot the MCMC chains sampled from the posterior distribution of the population parameters. In Figure \ref{fig:cor}, we plot the correlation between the parameters of the red-shift dependence ($R_{\rm{n}}$, $\tau$) and among the parameters of the mass function ($\mu$, $c$, $\gamma$). The correlation between parameters of the red-shift dependence and those of the mass function are small and therefore not plotted.
\begin{figure*}[!h]
\centering
\includegraphics[width=\textwidth]{parameters_scatter.pdf}
\caption{The correlation between the parameters of red-shift dependence ($R_{\rm{n}}$, $\tau$, left panel) and among the parameters of mass function ($\mu$, $c$, $\gamma$, middle and right panel). Blue markers correspond to 10 years observation with LIGO; Green markers correspond to 1 month observation with half-ET, and orange markers correspond to 1 month observation with ET. The red cross in each panel marks the location of the true values, which were used to generate the observed catalogue.}
\label{fig:cor}
\end{figure*}
In figure \ref{fig:reconstruct}, we plot the inferred differential merger rate as function of redshift and mass of the primary BH. The bands correspond to the 95\% confidence level of the posterior distribution of the population parameters. We can see from the panels of figure \ref{fig:reconstruct} that, although ten years of aLIGO observation and 1-month of ET observation result in a similar number of detections, ET can still obtain much better constraints on both $\mathcal{R}(z)$ and $p(M_1)$. That results from 1): the much smaller uncertainties on parameters with ET observation compared to LIGO, and 2): The catalogue observed by ET covers larger redshift and mass ranges than that of LIGO. For half-ET and ET, at low red-shift and high mass, both detectors result in a similar number of detections. Therefore, they give comparable constraints on $\mathcal{R}(z)$ at low redshift and $p(m_1)$ at large mass. While at high red-shift and low mass, the full ET observed more events and with lower uncertainties, therefore gives better constrains than that of half-ET.
We see from figure \ref{fig:figure2} that the maximum red-shift in LIGO's catalogue is $z\sim1.5$, while $\mathcal{R}(z)$ can still be constrained for $z>1.5$, and even be more stringent towards larger red-shift. This constraint on $\mathcal{R}(z)$ at large redshift is not from the observed catalogue, but is intrinsically imposed by the assumed model in the Bayesian method: we have fixed our star formation rate as function of redshift so that $\psi(z_f)$ peaks at $z\sim2$, the peak of BBH merger rate can therefore only appear later.
\begin{figure}[!b]
\centering
\includegraphics[width=0.5\textwidth]{reconstruction.pdf}
\caption{$\mathcal{R}(z)$ (upper panel) and $p(m_1)$ (lower panel) reconstructed with the Bayesian method, from the synthetic aLIGO (blue), ET (orange) and half-ET (green) catalogues. The black dashed curves are the model used to generate the catalogues.}
\label{fig:reconstruct}
\end{figure}
\subsection{Non-parametric method}
From equation (\ref{eqn:detectables}), we know that:
\begin{equation}
\dot{n}(\mathbf{\Theta})=\frac{N_{\rm{D}}(\mathbf{\Theta})}{\mathcal{D(\mathbf{\Theta})}}\frac{1+z}{T}/\frac{dV_c}{dz},\label{eqn:inverse}
\end{equation}
and $N_{\rm{D}}(\mathbf{\Theta})$ can be inferred from the observed catalogue.
Marginalising over $m_1$ and $m_2$ in equation (\ref{eqn:inverse}) on both sides, we have:
\begin{equation}
\mathcal{R}(z)=I(z)|_{w_i=1/\mathcal{D}_i}\frac{1+z}{T}/\frac{dV_c}{dz},
\label{eqn:Rkde}
\end{equation}
where $I(z)$ is the 1-D intensity function (number density) over $z$, of the observed catalogue, with weights $w_i$ equal to the corresponding $1/\mathcal{D}(\Theta_i)$. The intensity can be calculated with an empirical density estimator of the (synthetic) catalogue or with the (unnormalised) weighted Kernel Density Estimator (KDE, see appendix \ref{app:KDE}). By simulating a number of different realisations of the Universe population inferred from $I(z)$, the confidence intervals of the estimated $I(z)$ are obtained.
\noindent
Here, in order to include the observational uncertainties and give at the same time an estimate of the Confidence Intervals of the KDE, a specific bootstrap is employed, as follows:
\begin{comment}
\begin{itemize}
\item Resample from the catalogue with replacement. The size of the new sample is a Poisson random with the mean at the size of the original catalogue.
\item Re-assign parameters of each event in the sample with a Gaussian, which center at the data, and with the standard deviation equals to its associated uncertainty. Thus a new catalogue is obtained.
\item Calculate the intensity with the unnormalised KDE from the new sample.
\item Repeat the above process for a large number of times, in our case, 1000 times, and thus obtain 1000 $I(z)$ curves.
\item Calculate quantiles of $I(z)$ piecewisely to obtain the confident intervals.
\end{itemize}
\end{comment}
\begin{enumerate}
\item The \texttt{GW-Universe Toolbox}\xspace returns a catalogue $\{\mathbf{\Theta}\}_{\rm{true}}$, where $\mathbf{\Theta}^k_{\rm{true}}$ denotes the true parameters of the $k$-th events in the catalogue, including its $m_1$, $m_2$, $z$ and their uncertainties $\delta m_1$, $\delta m_2$, $\delta z$. $k$ goes from 1 to $N$, where $N$ is the total number of events.
\item Shift the true values $\mathbf{\Theta}^k_{\rm{true}}$ according to a multivariate Gaussian centered on the true values and with covariance matrix given by the corresponding uncertainties. Therefore, a new catalogue $\{\mathcal{\mathbf{\Theta}}^k\}_{\rm{obs}}$ is obtained.
\item Estimate the detection probabilities $\mathcal{D}( \mathbf{\Theta}^k_{\rm{obs}})$.
\item Calculate the (unnormalized) weighted KDE of $\{\mathcal{\mathbf{\Theta}}^k_{\rm{obs}}\}$, with weights $1/\mathcal{D}(\mathbf{\Theta}^k_{\rm{obs}})$. The intensity is estimated on the log and then transformed back to the original support; this ensures that there isn't any leakage of the intensity on negative support.
\item Take $M\equiv\sum_{i=k}^{N}1/\mathcal{D}( \mathbf{\Theta}^k_{\rm{obs}})$, as an estimate of the total number of mergers in the Universe in the parameters space.
\item Generate a realization $\widetilde{M}$ according to a Poisson with expected value $M$.
\item Draw a sample of size $\widetilde{M}$ from the KDE calculated in step 4. This is the equivalent of generating a new Universe sample, based on the estimate of the distribution in the Universe from the observations.
\item For each of the events in the above sample, calculate its detection probability. Draw $\widetilde{M}$ uniform random values, $U$, in between 0 and 1, and drop the observations for whose detectability is less than $U$. A smaller sample is left, which represents a new realization of the detected catalogue.
\item Estimate the covariance for each observation of the generated data set $\mathbf{\Sigma_{{\Theta}^k}}$ using interpolation of the original catalogue, and shift them according to a multivariate Gaussian centered on each observation and with covariance matrix $\mathbf{\Sigma_{{\Theta}^k}}$.
\item Estimate their detection probability.
\item Calculate the (unnormalized) weighted KDE of the above mentioned sample.
\item Repeat steps from 6 to 11 for a large number of times $B$ (we find $B$=200 is large enough). Calculate the confidence intervals according to the assigned quantiles.
\item Based on the results above, we can correct for the bias introduced by the measurement uncertainties. We estimate the bias between the bootstrap intensities and the intensity estimated on the observed sample, and use it to rescale the observed sample's estimated intensity.
\end{enumerate}
\begin{comment}
\begin{algorithm}
\SetKwInOut{Output}{Output}
\SetAlgoLined
\KwIn{ $Y=[z, m_1, m_2]$, $\delta Y= [\delta z, \delta m_1, \delta m_2]$, $D(\theta)$ }
\Output{$\mathcal{R}(z), \; \mathcal{R}(m_1)$}
$n=dim(Y)$\\
$N= \sum_{i=1}^{n} 1/D(\theta_i)$\\
$p_i=w_i/\sum(w_i)$\\
$w_i=1/D(\theta_i)$ \\
Apply Weighted KDE on $Y$\\
\For{$b \gets 1$ \KwTo $B$}{
$N^{*} \sim Poisson(N)$ \\
Sample with replacement ${N^*}$'s indexes $i^{*}\in\{1,\dots,n\}$ with probability $p_i$\\
$Y^*=Y[i^*,:]$\\
Sample ${N^*}$ times $U \sim Uniform(0,1)$\\
$i^{\dag}=i^*[U<D(\theta_{i^*})]$\\
$n^{\dag}=length(i^\dag)$ \\
$N^{\dag} = \sum_{i^{\dag}=1}^{n^\dag} 1/D(\theta_{i^{\dag}})$\\
$Y^\dag = Y[i^{\dag},:]$\\
$Y_{shift} = Y^\dag + MVN(\boldsymbol{0},\Sigma_{Y^\dag})$\\
Calculate $D(\theta)$ for $Y_{shift}$\\
Apply Weighted KDE on $Y_{shift}$}
Confidence Intervals\\
Mirroring
\caption{Weighted KDE with Bootstrap Uncertainties}
\end{algorithm}
\end{comment}
Similarly, the merger rate in the Universe as function of the mass of primary BH is:
\begin{equation}
\mathcal{R}(m_1)=p(m_1)\times \dot{n}_{\rm{tot}}=I(m_1)|_{w_i=1/\mathcal{D}_i}/T,
\label{eqn:pm1kde}
\end{equation}
where $p(m_1)$ is the normalized mass function of the primary BH, $\dot{n}_{\rm{tot}}$ is the total number of BBH mergers in the Universe in a unit local time. $I(m_1)$ is the 1-D intensity function of $m_1$, calculated from the observed catalogue. The procedures of obtaining $I(m_1)$ is the same as that in the calculation of $I(z)$.
In figures \ref{fig:Rzkde} and \ref{fig:Pm1kde} we plot the $\mathcal{R}(z)$ and $\mathcal{R}(m_1)$ reconstructed with equations \ref{eqn:Rkde} and \ref{eqn:pm1kde} from the synthetic LIGO, ET and half-ET catalogues. The shaded areas are the $95\%$ confidence intervals.
We note in figure \ref{fig:Rzkde} that, for LIGO's observation, the obtained $\mathcal{R}(z)$ in $z>0.3$ is an underestimate of the underlying model. This is not difficult to understand, given that LIGO's detectable range of BBH masses is increasingly incomplete towards higher redshift. An intuitive illustration is in Figure \ref{fig:2Dtheory}. There, we plot the two dimensional merger rate density as function of $m_1$ and $z$. The underlying black region marks the parameters space region where the detectability is essentially zero, i.e., the region beyond LIGO's horizon. When ploting Figure \ref{fig:2Dtheory}, we set the cut at $\mathcal{D}(z,m_1,m_2=m_1)<10^{-2}$. We can see the fraction of the black region in the mass spectrum is increasing towards high redshift. This corresponds to the fact that BBH with small masses cannot be detected with LIGO at higher redshift.
Similarly, in figure \ref{fig:Pm1kde}, we see $\mathcal{R}(m_1)$ is deviating the theory towards low $m_1$. To compare the estimation with the model, we limit both the LIGO catalogue and the theory in the range $z<0.6$. Therefore, the meaning of $\mathcal{R}(m_1)$ is no longer the merger rate density of the complete Universe, but that of a local Universe up to 0.6, which is calcualted with:
\begin{equation}
\mathcal{R}(m_1)|_{z<0.6}=p(m_1)\int^{0.6}_0\mathcal{R}(z)\frac{dV}{dz}/(1+z)dz,
\end{equation}
where $p(m_1)$ is the normalised mass function.
It is unlike the results with the Bayesian inference method, where constraints can still be placed in regimes with no detection. The difference is rooted in the degree of prior knowledge we assumed with the population model in both methods.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Rzkde.pdf}
\caption{The merger rate as function of the redshift, reconstructed with equation (\ref{eqn:pm1kde}) from the synthetic aLIGO, ET and half-ET catalogues.}
\label{fig:Rzkde}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{2Dtheory.pdf}
\caption{Two-dimensional theoretical merger rate as function of redshift and primary masses. The black region marks the region in the parameters space that is not reachable by LIGO.}
\label{fig:2Dtheory}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Pm1kde.pdf}
\caption{The merger rate as function of the primary BH mass, reconstructed with equation (\ref{eqn:pm1kde}) from the synthetic aLIGO, ET and half-ET catalogues.}
\label{fig:Pm1kde}
\end{figure}
\section{Conclusion and Discussion}\label{sec:candd}
In this paper, we describe the details of how the \texttt{GW-Universe Toolbox}\xspace simulates observations of BBH mergers with different ground-based GW detectors. As a demonstration of its scientific application, we use the synthetic catalogues to reconstruct and constrain the cosmic merger rate and mass function of the BBH population. The comparison amongst the results from one month (half) ET observation and ten years aLIGO observations shows the overwhelming advantages of the 3rd generation detectors over the 2nd generation, especially at a redshift higher than $\sim2$, around where the cosmic merger rate is believed to peak.
The reconstruction and constraining on $\mathcal{R}(z)$ and $p(m_1)$ are performed with two methods, namely, 1) a Bayesian method, assuming a parametric formula of $\mathcal{R}(z)$ and $p(m_1)$ and 2) the KDE method with non-parametric $\mathcal{R}(z)$ and $p(m_1)$. For the catalogue of ET observation, the results from both methods are qualitatively the same. However, for aLIGO, the Bayesian method can put constraints on $\mathcal{R}(z)$ beyond the detecting limit, where the non-parametric method lost its ability completely. The $m_1$ dependence is also much more accurately reconstructed by the Bayesian method than the non-parametric method for aLIGO's catalogue. The difference is due to the extra information placed by assuming a specific parameterisation of the population model in the Bayesian method.
In our example, the underlying $\mathcal{R}(z)$ and $p(m_1)$ used to generate the catalogues are the same with those assumed in the Bayesian method; while in the KDE method, no assumption of the general shape of $\mathcal{R}(z)$ and $f(m_1)$ are placed, not even the assumption of its smoothness. These represent the two extreme situations in the differential merger rate reconstruction.
In reality, the underlying shape of $\mathcal{R}(z)$ and $f(m_1)$ are unknown and they could be very different with what we used. The parameterised Bayesian method may give biased inference and underestimated uncertainties. Furthermore, any unexpected structures in $\mathcal{R}(z)$ and $f(m_1)$ cannot be recovered with the parameterised Bayesian method. For instance, there can be additional peaks in the BH spectrum corresponding to pulsational pair-instability supernovae and primordial BHs. On the other hand, although the function form of the merger rate is unknown, the general trends should be more or less known. A better reconstruction method should lie in between these two extreme methods.
An alternative method is to parameterise $\mathcal{R}(z)$ and $p(m_1)$ as piecewise constant functions and use the Bayesian inference to obtain the posterior of the values in each bin, as in \cite{2019ApJ...886L...1V}. It has the advantage that no assumptions on the shapes of $\mathcal{R}(z)$ and $p(m_1)$ is needed, like the Kernel density estimator method. The disadvantage of this method is that MCMC sampling in high dimension needs to be performed. If one assumes that the $p(m_1)$ is independent of $z$, there are more than $N+M$ free parameters, where $N$ and $M$ are the number of bins in $z$ and $m_1$ respectively. If we want the resolutions to be $0.5$ in $z$ and $1\,M_\odot$ in $m_1$, then the dimension of the parameter space is $>80$ for ET catalogue. In the more general case, when one includes the dependence of $p(m_1)$ on $z$, the number of parameters becomes $N\times M$, therefore the dimension of the parameter space goes catastrophically to $>1200$.
Another conclusion we draw about the early phase ET (half-ET) is that, half-ET can also give a good constraint on $\mathcal{R}(z)$ at high redshift Universe. However, both $\mathcal{R}(z)$ and primary mass function are less accurately constrained compared with the full ET, even using a larger number of detected events from a longer observation duration. The reason is that the full ET determines smaller uncertainties on the physical parameters of the GW events.
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